%------------------------------------------------------------------------------
% File     : Leo-III---1.7.15
% Problem  : SWW325+1 : TPTP v8.2.0. Released v5.2.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : run_Leo-III %s %d THM

% Computer : n003.cluster.edu
% Model    : x86_64 x86_64
% CPU      : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory   : 8042.1875MB
% OS       : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit  : 300s
% DateTime : Mon Jun 24 18:22:03 EDT 2024

% Result   : Theorem 129.48s 49.07s
% Output   : Refutation 134.17s
% Verified : 
% SZS Type : Refutation
%            Derivation depth      :   11
%            Number of leaves      : 2408
% Syntax   : Number of formulae    : 4856 (1770 unt;   0 typ;   0 def)
%            Number of atoms       : 12407 (4615 equ;   0 cnn)
%            Maximal formula atoms :   26 (   2 avg)
%            Number of connectives : 149100 (1280   ~; 259   |;1303   &;140283   @)
%                                         ( 424 <=>;5551  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   34 (  11 avg)
%            Number of types       :    2 (   0 usr)
%            Number of type conns  :    0 (   0   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  243 ( 241 usr;  34 con; 0-10 aty)
%            Number of variables   : 18214 (   0   ^17958   !; 256   ?;18214   :)

% Comments : 
%------------------------------------------------------------------------------
thf(hBOOL_type,type,
    hBOOL: $i > $o ).

thf(hAPP_type,type,
    hAPP: $i > $i > $i ).

thf(c_member_type,type,
    c_member: $i > $i ).

thf(tc_Hoare__Mirabelle_Otriple_type,type,
    tc_Hoare__Mirabelle_Otriple: $i > $i ).

thf(t_a_type,type,
    t_a: $i ).

thf(v_G_type,type,
    v_G: $i ).

thf(c_Hoare__Mirabelle_Otriple__valid_type,type,
    c_Hoare__Mirabelle_Otriple__valid: $i > $i > $i > $o ).

thf(v_P_type,type,
    v_P: $i > $i > $o ).

thf(c_Natural_Oevaln_type,type,
    c_Natural_Oevaln: $i > $i > $i > $i > $o ).

thf(c_Com_Ocom_OWhile_type,type,
    c_Com_Ocom_OWhile: $i > $i > $i ).

thf(v_b_type,type,
    v_b: $i ).

thf(v_c_type,type,
    v_c: $i ).

thf(c_Finite__Set_Ofinite_type,type,
    c_Finite__Set_Ofinite: $i > $i ).

thf(c_Set_OCollect_type,type,
    c_Set_OCollect: $i > $i ).

thf(c_COMBB_type,type,
    c_COMBB: $i > $i > $i > $i ).

thf(tc_fun_type,type,
    tc_fun: $i > $i > $i ).

thf(tc_HOL_Obool_type,type,
    tc_HOL_Obool: $i ).

thf(c_HOL_OEx_type,type,
    c_HOL_OEx: $i > $i ).

thf(c_COMBC_type,type,
    c_COMBC: $i > $i > $i > $i ).

thf(c_COMBS_type,type,
    c_COMBS: $i > $i > $i > $i ).

thf(c_fconj_type,type,
    c_fconj: $i ).

thf(c_fequal_type,type,
    c_fequal: $i ).

thf(c_Divides_Odivmod__int__rel_type,type,
    c_Divides_Odivmod__int__rel: $i > $i > $i ).

thf(c_Product__Type_OPair_type,type,
    c_Product__Type_OPair: $i > $i > $i ).

thf(tc_Int_Oint_type,type,
    tc_Int_Oint: $i ).

thf(c_Groups_Ozero__class_Ozero_type,type,
    c_Groups_Ozero__class_Ozero: $i > $i ).

thf(class_Groups_Ocomm__monoid__add_type,type,
    class_Groups_Ocomm__monoid__add: $i > $o ).

thf(c_Big__Operators_Ocomm__monoid__add__class_Osetsum_type,type,
    c_Big__Operators_Ocomm__monoid__add__class_Osetsum: $i > $i > $i ).

thf(tc_prod_type,type,
    tc_prod: $i > $i > $i ).

thf(c_Product__Type_Oprod_Oprod__case_type,type,
    c_Product__Type_Oprod_Oprod__case: $i > $i > $i > $i ).

thf(c_Product__Type_OSigma_type,type,
    c_Product__Type_OSigma: $i > $i > $i ).

thf(c_List_Oconcat_type,type,
    c_List_Oconcat: $i > $i > $i ).

thf(c_List_Olist_ONil_type,type,
    c_List_Olist_ONil: $i > $i ).

thf(tc_List_Olist_type,type,
    tc_List_Olist: $i > $i ).

thf(c_List_Oset_type,type,
    c_List_Oset: $i > $i ).

thf(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR_type,type,
    c_Complete__Lattice_Ocomplete__lattice__class_OSUPR: $i > $i > $i ).

thf(class_Groups_Oab__group__add_type,type,
    class_Groups_Oab__group__add: $i > $o ).

thf(c_Groups_Ominus__class_Ominus_type,type,
    c_Groups_Ominus__class_Ominus: $i > $i ).

thf(tc_Nat_Onat_type,type,
    tc_Nat_Onat: $i ).

thf(c_SetInterval_Oord__class_OatLeastLessThan_type,type,
    c_SetInterval_Oord__class_OatLeastLessThan: $i > $i > $i ).

thf(c_Nat_OSuc_type,type,
    c_Nat_OSuc: $i ).

thf(c_Set_Oinsert_type,type,
    c_Set_Oinsert: $i > $i ).

thf(c_Groups_Oplus__class_Oplus_type,type,
    c_Groups_Oplus__class_Oplus: $i > $i ).

thf(c_Fun_Oinj__on_type,type,
    c_Fun_Oinj__on: $i > $i > $i > $i > $o ).

thf(c_Set_Oimage_type,type,
    c_Set_Oimage: $i > $i > $i > $i ).

thf(c_Groups_Oone__class_Oone_type,type,
    c_Groups_Oone__class_Oone: $i > $i ).

thf(c_Lattices_Osemilattice__sup__class_Osup_type,type,
    c_Lattices_Osemilattice__sup__class_Osup: $i > $i ).

thf(c_Lattices_Osemilattice__inf__class_Oinf_type,type,
    c_Lattices_Osemilattice__inf__class_Oinf: $i > $i ).

thf(c_Fun_Ocomp_type,type,
    c_Fun_Ocomp: $i > $i > $i > $i > $i ).

thf(c_Relation_ORange_type,type,
    c_Relation_ORange: $i > $i > $i ).

thf(c_If_type,type,
    c_If: $i > $i ).

thf(c_Relation_ODomain_type,type,
    c_Relation_ODomain: $i > $i > $i ).

thf(c_List_Olexn_type,type,
    c_List_Olexn: $i > $i > $i ).

thf(c_Nat_Osize__class_Osize_type,type,
    c_Nat_Osize__class_Osize: $i > $i ).

thf(c_Orderings_Obot__class_Obot_type,type,
    c_Orderings_Obot__class_Obot: $i > $i ).

thf(c_Orderings_Oord__class_Oless__eq_type,type,
    c_Orderings_Oord__class_Oless__eq: $i > $i ).

thf(c_SetInterval_Oord__class_OatLeastAtMost_type,type,
    c_SetInterval_Oord__class_OatLeastAtMost: $i > $i > $i > $i ).

thf(c_Divides_Odiv__class_Omod_type,type,
    c_Divides_Odiv__class_Omod: $i > $i > $i > $i ).

thf(c_List_Ofilter_type,type,
    c_List_Ofilter: $i > $i > $i ).

thf(c_List_Omap_type,type,
    c_List_Omap: $i > $i > $i ).

thf(c_List_Oappend_type,type,
    c_List_Oappend: $i > $i ).

thf(c_Wellfounded_Owf_type,type,
    c_Wellfounded_Owf: $i > $i > $o ).

thf(c_Orderings_Oord__class_Oless_type,type,
    c_Orderings_Oord__class_Oless: $i > $i ).

thf(c_Groups_Otimes__class_Otimes_type,type,
    c_Groups_Otimes__class_Otimes: $i > $i ).

thf(c_Divides_Odiv__class_Odiv_type,type,
    c_Divides_Odiv__class_Odiv: $i > $i ).

thf(c_COMBK_type,type,
    c_COMBK: $i > $i > $i ).

thf(c_List_Olist_OCons_type,type,
    c_List_Olist_OCons: $i > $i ).

thf(class_Groups_Ocomm__monoid__mult_type,type,
    class_Groups_Ocomm__monoid__mult: $i > $o ).

thf(c_Finite__Set_Ofold__image_type,type,
    c_Finite__Set_Ofold__image: $i > $i > $i > $i ).

thf(c_List_Opartition_type,type,
    c_List_Opartition: $i > $i > $i > $i ).

thf(c_SetInterval_Oord__class_OatMost_type,type,
    c_SetInterval_Oord__class_OatMost: $i > $i ).

thf(c_List_Otranspose_type,type,
    c_List_Otranspose: $i > $i > $i ).

thf(c_Map_Odom_type,type,
    c_Map_Odom: $i > $i > $i > $i ).

thf(tc_Option_Ooption_type,type,
    tc_Option_Ooption: $i > $i ).

thf(c_fNot_type,type,
    c_fNot: $i ).

thf(c_Relation_Orel__comp_type,type,
    c_Relation_Orel__comp: $i > $i > $i > $i ).

thf(c_List_Onth_type,type,
    c_List_Onth: $i > $i ).

thf(c_List_Oremdups_type,type,
    c_List_Oremdups: $i > $i > $i ).

thf(c_Finite__Set_Ocard_type,type,
    c_Finite__Set_Ocard: $i > $i ).

thf(c_Map_Oran_type,type,
    c_Map_Oran: $i > $i > $i > $i ).

thf(c_Option_Ooption_OSome_type,type,
    c_Option_Ooption_OSome: $i > $i ).

thf(c_SetInterval_Oord__class_OgreaterThanAtMost_type,type,
    c_SetInterval_Oord__class_OgreaterThanAtMost: $i > $i > $i > $i ).

thf(c_List_Ofoldl_type,type,
    c_List_Ofoldl: $i > $i > $i > $i > $i ).

thf(c_SetInterval_Oord__class_OlessThan_type,type,
    c_SetInterval_Oord__class_OlessThan: $i > $i ).

thf(c_Product__Type_Ofst_type,type,
    c_Product__Type_Ofst: $i > $i > $i ).

thf(c_Finite__Set_Ofolding__image__simple__idem_type,type,
    c_Finite__Set_Ofolding__image__simple__idem: $i > $i > $i > $i > $i > $i > $o ).

thf(c_Complete__Lattice_OSup__class_OSup_type,type,
    c_Complete__Lattice_OSup__class_OSup: $i > $i > $i ).

thf(c_Map_Omap__upds_type,type,
    c_Map_Omap__upds: $i > $i > $i > $i > $i > $i ).

thf(c_Option_Oset_type,type,
    c_Option_Oset: $i > $i > $i ).

thf(c_Option_Ooption_ONone_type,type,
    c_Option_Ooption_ONone: $i > $i ).

thf(c_List_Odrop_type,type,
    c_List_Odrop: $i > $i ).

thf(c_List_Olistrel_type,type,
    c_List_Olistrel: $i > $i > $i ).

thf(c_Finite__Set_Ofold_type,type,
    c_Finite__Set_Ofold: $i > $i > $i > $i ).

thf(c_Product__Type_Omap__pair_type,type,
    c_Product__Type_Omap__pair: $i > $i > $i > $i > $i > $i > $i ).

thf(c_Wellfounded_Olex__prod_type,type,
    c_Wellfounded_Olex__prod: $i > $i > $i > $i > $i ).

thf(c_List_Osublist_type,type,
    c_List_Osublist: $i > $i > $i > $i ).

thf(c_Orderings_Otop__class_Otop_type,type,
    c_Orderings_Otop__class_Otop: $i > $i ).

thf(c_Set_Ovimage_type,type,
    c_Set_Ovimage: $i > $i > $i > $i ).

thf(c_SetInterval_Oord_OlessThan_type,type,
    c_SetInterval_Oord_OlessThan: $i > $i > $i > $i ).

thf(c_Rings_Odvd__class_Odvd_type,type,
    c_Rings_Odvd__class_Odvd: $i > $i ).

thf(c_Product__Type_Osnd_type,type,
    c_Product__Type_Osnd: $i > $i > $i ).

thf(c_List_Ozip_type,type,
    c_List_Ozip: $i > $i > $i ).

thf(c_List_Otake_type,type,
    c_List_Otake: $i > $i ).

thf(c_Finite__Set_Ofold1Set_type,type,
    c_Finite__Set_Ofold1Set: $i > $i > $i > $i ).

thf(c_SetInterval_Oord_OatLeastAtMost_type,type,
    c_SetInterval_Oord_OatLeastAtMost: $i > $i > $i > $i > $i ).

thf(c_Wellfounded_OwfP_type,type,
    c_Wellfounded_OwfP: $i > $i > $o ).

thf(c_Relation_Oconverse_type,type,
    c_Relation_Oconverse: $i > $i > $i ).

thf(c_Transitive__Closure_Ortrancl_type,type,
    c_Transitive__Closure_Ortrancl: $i > $i > $i ).

thf(c_Transitive__Closure_Otrancl_type,type,
    c_Transitive__Closure_Otrancl: $i > $i > $i ).

thf(c_Map_Omap__add_type,type,
    c_Map_Omap__add: $i > $i > $i > $i > $i ).

thf(c_List_OtakeWhile_type,type,
    c_List_OtakeWhile: $i > $i > $i > $i ).

thf(c_Relation_OField_type,type,
    c_Relation_OField: $i > $i ).

thf(class_Groups_Oab__semigroup__mult_type,type,
    class_Groups_Oab__semigroup__mult: $i > $o ).

thf(c_Finite__Set_Ofold__graph_type,type,
    c_Finite__Set_Ofold__graph: $i > $i > $i > $i > $i > $i ).

thf(c_Hilbert__Choice_OEps_type,type,
    c_Hilbert__Choice_OEps: $i > $i > $i ).

thf(c_Finite__Set_Ofolding__one__idem_type,type,
    c_Finite__Set_Ofolding__one__idem: $i > $i > $i > $o ).

thf(c_Relation_OImage_type,type,
    c_Relation_OImage: $i > $i > $i > $i ).

thf(c_Nat_Ocompow_type,type,
    c_Nat_Ocompow: $i > $i > $i ).

thf(c_FunDef_Omax__weak_type,type,
    c_FunDef_Omax__weak: $i ).

thf(c_Finite__Set_Ofolding__idem_type,type,
    c_Finite__Set_Ofolding__idem: $i > $i > $i > $i > $o ).

thf(c_Fun_Ofun__upd_type,type,
    c_Fun_Ofun__upd: $i > $i > $i ).

thf(c_Equiv__Relations_Oequiv_type,type,
    c_Equiv__Relations_Oequiv: $i > $i > $i > $o ).

thf(c_Equiv__Relations_Ocongruent_type,type,
    c_Equiv__Relations_Ocongruent: $i > $i > $i > $i > $o ).

thf(c_Equiv__Relations_Oquotient_type,type,
    c_Equiv__Relations_Oquotient: $i > $i ).

thf(c_Finite__Set_Ofolding__one_type,type,
    c_Finite__Set_Ofolding__one: $i > $i > $i > $o ).

thf(c_SetInterval_Oord_OgreaterThan_type,type,
    c_SetInterval_Oord_OgreaterThan: $i > $i > $i > $i ).

thf(class_Rings_Olinordered__semiring__1__strict_type,type,
    class_Rings_Olinordered__semiring__1__strict: $i > $o ).

thf(class_Rings_Olinordered__semiring__1_type,type,
    class_Rings_Olinordered__semiring__1: $i > $o ).

thf(c_SetInterval_Oord_OatLeastLessThan_type,type,
    c_SetInterval_Oord_OatLeastLessThan: $i > $i > $i > $i > $i > $i ).

thf(c_Relation_Orefl__on_type,type,
    c_Relation_Orefl__on: $i > $i > $i > $o ).

thf(class_Orderings_Owellorder_type,type,
    class_Orderings_Owellorder: $i > $o ).

thf(c_Map_Orestrict__map_type,type,
    c_Map_Orestrict__map: $i > $i > $i > $i > $i ).

thf(c_List_Olexord_type,type,
    c_List_Olexord: $i > $i > $i ).

thf(c_List_Olistrel1_type,type,
    c_List_Olistrel1: $i > $i > $i ).

thf(c_Fun_Ooverride__on_type,type,
    c_Fun_Ooverride__on: $i > $i > $i > $i > $i > $i ).

thf(c_List_Odistinct_type,type,
    c_List_Odistinct: $i > $i ).

thf(c_Relation_OId__on_type,type,
    c_Relation_OId__on: $i > $i > $i ).

thf(c_fTrue_type,type,
    c_fTrue: $i ).

thf(c_HOL_OThe_type,type,
    c_HOL_OThe: $i > $i > $i ).

thf(c_Finite__Set_Ofun__left__comm_type,type,
    c_Finite__Set_Ofun__left__comm: $i > $i > $i > $o ).

thf(c_Int_Onumber__class_Onumber__of_type,type,
    c_Int_Onumber__class_Onumber__of: $i > $i ).

thf(c_Predicate_ODomainP_type,type,
    c_Predicate_ODomainP: $i > $i > $i > $i ).

thf(c_Predicate_ORangeP_type,type,
    c_Predicate_ORangeP: $i > $i > $i > $i ).

thf(c_Nat_Onat_Onat__case_type,type,
    c_Nat_Onat_Onat__case: $i > $i > $i > $i > $i ).

thf(c_List_Oupt_type,type,
    c_List_Oupt: $i > $i > $i ).

thf(c_Finite__Set_Ofold1_type,type,
    c_Finite__Set_Ofold1: $i > $i > $i ).

thf(class_Lattices_Oab__semigroup__idem__mult_type,type,
    class_Lattices_Oab__semigroup__idem__mult: $i > $o ).

thf(c_SetInterval_Oord_OgreaterThanAtMost_type,type,
    c_SetInterval_Oord_OgreaterThanAtMost: $i > $i > $i > $i > $i > $i ).

thf(c_Relation_OId_type,type,
    c_Relation_OId: $i > $i ).

thf(c_Wellfounded_Omeasure_type,type,
    c_Wellfounded_Omeasure: $i > $i ).

thf(c_fdisj_type,type,
    c_fdisj: $i ).

thf(c_Fun_Othe__inv__into_type,type,
    c_Fun_Othe__inv__into: $i > $i > $i > $i > $i ).

thf(c_List_Olistrelp_type,type,
    c_List_Olistrelp: $i > $i > $i > $i > $o ).

thf(c_SetInterval_Oord__class_OgreaterThanLessThan_type,type,
    c_SetInterval_Oord__class_OgreaterThanLessThan: $i > $i > $i > $i ).

thf(c_FunDef_Orp__inv__image_type,type,
    c_FunDef_Orp__inv__image: $i > $i > $i ).

thf(c_Relation_Oinv__image_type,type,
    c_Relation_Oinv__image: $i > $i > $i ).

thf(c_Map_Omap__comp_type,type,
    c_Map_Omap__comp: $i > $i > $i > $i > $i > $i > $i ).

thf(c_FunDef_Oreduction__pair_type,type,
    c_FunDef_Oreduction__pair: $i > $i > $o ).

thf(c_FunDef_Omin__strict_type,type,
    c_FunDef_Omin__strict: $i ).

thf(c_FunDef_Omin__weak_type,type,
    c_FunDef_Omin__weak: $i ).

thf(c_Wellfounded_Ofinite__psubset_type,type,
    c_Wellfounded_Ofinite__psubset: $i > $i ).

thf(c_List_Obutlast_type,type,
    c_List_Obutlast: $i > $i > $i ).

thf(c_SetInterval_Oord_OgreaterThanLessThan_type,type,
    c_SetInterval_Oord_OgreaterThanLessThan: $i > $i > $i > $i > $i ).

thf(c_FunDef_Omax__strict_type,type,
    c_FunDef_Omax__strict: $i ).

thf(c_fFalse_type,type,
    c_fFalse: $i ).

thf(c_Map_Omap__of_type,type,
    c_Map_Omap__of: $i > $i > $i > $i ).

thf(c_Typedef_Otype__definition_type,type,
    c_Typedef_Otype__definition: $i > $i > $i > $i > $i > $o ).

thf(c_Finite__Set_Ofolding__image__simple_type,type,
    c_Finite__Set_Ofolding__image__simple: $i > $i > $i > $i > $i > $i > $o ).

thf(c_fimplies_type,type,
    c_fimplies: $i ).

thf(c_Predicate_Opred__comp_type,type,
    c_Predicate_Opred__comp: $i > $i > $i > $i > $i > $i > $i > $o ).

thf(c_Equiv__Relations_Ocongruent2_type,type,
    c_Equiv__Relations_Ocongruent2: $i > $i > $i > $i > $i > $i > $o ).

thf(c_Relation_Osingle__valued_type,type,
    c_Relation_Osingle__valued: $i > $i > $i > $o ).

thf(c_Hoare__Mirabelle_Otriple_Otriple_type,type,
    c_Hoare__Mirabelle_Otriple_Otriple: $i > $i ).

thf(c_Wellfounded_Opred__nat_type,type,
    c_Wellfounded_Opred__nat: $i ).

thf(c_Wellfounded_Oaccp_type,type,
    c_Wellfounded_Oaccp: $i > $i > $i ).

thf(c_Wellfounded_Oacc_type,type,
    c_Wellfounded_Oacc: $i > $i > $i ).

thf(c_Nitpick_Ocard_H_type,type,
    c_Nitpick_Ocard_H: $i > $i > $i ).

thf(c_Divides_Odivmod__nat_type,type,
    c_Divides_Odivmod__nat: $i > $i > $i ).

thf(c_Equiv__Relations_Oequivp_type,type,
    c_Equiv__Relations_Oequivp: $i > $i > $o ).

thf(c_Relation_Ototal__on_type,type,
    c_Relation_Ototal__on: $i > $i > $i > $o ).

thf(c_Sum__Type_Osum_Osum__case_type,type,
    c_Sum__Type_Osum_Osum__case: $i > $i > $i > $i > $i > $i ).

thf(c_List_Oremove1_type,type,
    c_List_Oremove1: $i > $i > $i > $i ).

thf(c_Relation_Oirrefl_type,type,
    c_Relation_Oirrefl: $i > $i > $o ).

thf(c_Wellfounded_Oacyclic_type,type,
    c_Wellfounded_Oacyclic: $i > $i > $o ).

thf(c_List_Olist_Olist__size_type,type,
    c_List_Olist_Olist__size: $i > $i > $i > $i ).

thf(c_Divides_Odivmod__nat__rel_type,type,
    c_Divides_Odivmod__nat__rel: $i > $i > $i ).

thf(c_FunDef_Opair__leq_type,type,
    c_FunDef_Opair__leq: $i ).

thf(c_FunDef_Opair__less_type,type,
    c_FunDef_Opair__less: $i ).

thf(c_Int_ORep__Integ_type,type,
    c_Int_ORep__Integ: $i ).

thf(c_Int_OInteg_type,type,
    c_Int_OInteg: $i ).

thf(c_Predicate_Oreflp_type,type,
    c_Predicate_Oreflp: $i > $i > $o ).

thf(c_Nitpick_Onat__gcd__rel_type,type,
    c_Nitpick_Onat__gcd__rel: $i ).

thf(c_Nitpick_Opair__box_Opair__box__size_type,type,
    c_Nitpick_Opair__box_Opair__box__size: $i > $i > $i > $i > $i > $i ).

thf(c_Nitpick_Opair__box_OPairBox_type,type,
    c_Nitpick_Opair__box_OPairBox: $i > $i > $i > $i > $i ).

thf(c_Wellfounded_Omlex__prod_type,type,
    c_Wellfounded_Omlex__prod: $i > $i > $i > $i ).

thf(c_Predicate_OPowp_type,type,
    c_Predicate_OPowp: $i > $i > $i ).

thf(c_Natural_Oevalc_type,type,
    c_Natural_Oevalc: $i > $i ).

thf(c_Com_Ocom_OAss_type,type,
    c_Com_Ocom_OAss: $i > $i > $i ).

thf(c_Natural_Oupdate_type,type,
    c_Natural_Oupdate: $i ).

thf(c_Com_Ocom_OSemi_type,type,
    c_Com_Ocom_OSemi: $i > $i > $i ).

thf(c_Com_Ocom_OCond_type,type,
    c_Com_Ocom_OCond: $i > $i > $i > $i ).

thf(c_Com_OWT_type,type,
    c_Com_OWT: $i ).

thf(c_Com_Ocom_OSKIP_type,type,
    c_Com_Ocom_OSKIP: $i ).

thf(c_Com_Ocom_Ocom__rec_type,type,
    c_Com_Ocom_Ocom__rec: $i > $i > $i > $i > $i > $i > $i > $i > $i > $i > $i ).

thf(c_Com_Ocom_OCall_type,type,
    c_Com_Ocom_OCall: $i > $i > $i > $i ).

thf(c_Sum__Type_OInl_type,type,
    c_Sum__Type_OInl: $i > $i > $i ).

thf(c_Com_Ocom_OLocal_type,type,
    c_Com_Ocom_OLocal: $i > $i > $i > $i ).

thf(c_Sum__Type_OInr_type,type,
    c_Sum__Type_OInr: $i > $i > $i ).

thf(c_Com_Ocom_OBODY_type,type,
    c_Com_Ocom_OBODY: $i ).

thf(c_FunDef_Oin__rel_type,type,
    c_FunDef_Oin__rel: $i > $i > $i > $i ).

thf(c_Recdef_Osame__fst_type,type,
    c_Recdef_Osame__fst: $i > $i > $i > $i > $i ).

thf(c_Com_Ocom_Ocom__case_type,type,
    c_Com_Ocom_Ocom__case: $i > $i > $i > $i > $i > $i > $i > $i > $i > $i > $i ).

thf(c_New__Random__Sequence_Opos__empty_type,type,
    c_New__Random__Sequence_Opos__empty: $i > $i > $i > $i > $i ).

thf(c_New__DSequence_Opos__empty_type,type,
    c_New__DSequence_Opos__empty: $i > $i ).

thf(c_Big__Operators_Ocomm__monoid__big_type,type,
    c_Big__Operators_Ocomm__monoid__big: $i > $i > $i > $i > $i > $o ).

thf(c_SetInterval_Oord_OatLeast_type,type,
    c_SetInterval_Oord_OatLeast: $i > $i > $i > $i ).

thf(c_Hoare__Mirabelle_Ohoare__valids_type,type,
    c_Hoare__Mirabelle_Ohoare__valids: $i > $i > $i > $o ).

thf(c_SetInterval_Oord_OatMost_type,type,
    c_SetInterval_Oord_OatMost: $i > $i > $i > $i ).

thf(c_New__Random__Sequence_Oneg__single_type,type,
    c_New__Random__Sequence_Oneg__single: $i > $i ).

thf(c_New__DSequence_Oneg__single_type,type,
    c_New__DSequence_Oneg__single: $i > $i > $i ).

thf(c_Sum__Type_OProjr_type,type,
    c_Sum__Type_OProjr: $i > $i > $i > $i ).

thf(c_Nitpick_Opair__box_Opair__box__rec_type,type,
    c_Nitpick_Opair__box_Opair__box__rec: $i > $i > $i > $i > $i > $i ).

thf(c_Lazy__Sequence_Ohb__single_type,type,
    c_Lazy__Sequence_Ohb__single: $i > $i > $i ).

thf(c_Hoare__Mirabelle_Otriple_Otriple__rec_type,type,
    c_Hoare__Mirabelle_Otriple_Otriple__rec: $i > $i > $i > $i > $i ).

thf(c_Nitpick_Opair__box_Opair__box__case_type,type,
    c_Nitpick_Opair__box_Opair__box__case: $i > $i > $i > $i > $i > $i ).

thf(c_Sum__Type_OProjl_type,type,
    c_Sum__Type_OProjl: $i > $i > $i > $i ).

thf(c_Hoare__Mirabelle_Opeek__and_type,type,
    c_Hoare__Mirabelle_Opeek__and: $i > $i > $i > $i ).

thf(c_Product__Type_Oprod_Oprod__rec_type,type,
    c_Product__Type_Oprod_Oprod__rec: $i > $i > $i > $i > $i > $i ).

thf(c_New__Random__Sequence_Opos__single_type,type,
    c_New__Random__Sequence_Opos__single: $i > $i ).

thf(c_New__DSequence_Opos__single_type,type,
    c_New__DSequence_Opos__single: $i > $i > $i ).

thf(c_Lazy__Sequence_Osingle_type,type,
    c_Lazy__Sequence_Osingle: $i > $i > $i ).

thf(c_New__Random__Sequence_Opos__not__random__dseq_type,type,
    c_New__Random__Sequence_Opos__not__random__dseq: $i > $i > $i > $i > $i ).

thf(c_New__DSequence_Opos__not__seq_type,type,
    c_New__DSequence_Opos__not__seq: $i > $i ).

thf(c_Lazy__Sequence_Oempty_type,type,
    c_Lazy__Sequence_Oempty: $i > $i ).

thf(c_Big__Operators_Osemilattice__big_type,type,
    c_Big__Operators_Osemilattice__big: $i > $i > $i > $o ).

thf(c_Hoare__Mirabelle_Otriple_Otriple__case_type,type,
    c_Hoare__Mirabelle_Otriple_Otriple__case: $i > $i > $i > $i > $i ).

thf(c_Nitpick_Orefl_H_type,type,
    c_Nitpick_Orefl_H: $i > $i > $o ).

thf(sk1_type,type,
    sk1: $i ).

thf(sk2_type,type,
    sk2: $i ).

thf(sk3_type,type,
    sk3: $i ).

thf(sk4_type,type,
    sk4: $i ).

thf(2145,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Finite__Set_Ofolding__one__idem @ E @ D @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ E ) @ B ) )
       => ( ( hAPP @ C @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ A ) @ B ) )
          = ( hAPP @ ( hAPP @ ( c_Finite__Set_Ofold @ E @ E @ D ) @ A ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_folding__one__idem_Oeq__fold__idem_H) ).

thf(10822,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Finite__Set_Ofolding__one__idem @ E @ D @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ E ) @ B ) )
       => ( ( hAPP @ C @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ A ) @ B ) )
          = ( hAPP @ ( hAPP @ ( c_Finite__Set_Ofold @ E @ E @ D ) @ A ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2145]) ).

thf(943,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ C ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) @ A ) ) @ ( hAPP @ c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_list_Osize_I4_J) ).

thf(5810,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ C ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) @ A ) ) @ ( hAPP @ c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[943]) ).

thf(757,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) ) )
    <=> ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ B ) )
        & ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_finite__Un) ).

thf(5116,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ B ) )
          & ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ A ) ) ) )
      & ( ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ B ) )
          & ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ A ) ) )
       => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[757]) ).

thf(2047,axiom,
    ! [A: $i,B: $i] :
      ( ( ( hAPP @ c_Nat_OSuc @ B )
        = ( hAPP @ c_Nat_OSuc @ A ) )
     => ( B = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Suc__inject) ).

thf(10452,plain,
    ! [A: $i,B: $i] :
      ( ( ( hAPP @ c_Nat_OSuc @ B )
        = ( hAPP @ c_Nat_OSuc @ A ) )
     => ( B = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2047]) ).

thf(2320,axiom,
    ! [A: $i,B: $i] :
      ( c_Com_Ocom_OSKIP
     != ( c_Com_Ocom_OWhile @ B @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_com_Osimps_I16_J) ).

thf(11504,plain,
    ! [A: $i,B: $i] :
      ( c_Com_Ocom_OSKIP
     != ( c_Com_Ocom_OWhile @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2320]) ).

thf(2020,axiom,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ B @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ B @ B ) @ A ) @ A ) ) @ ( c_Relation_OId @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_IdI) ).

thf(10363,plain,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ B @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ B @ B ) @ A ) @ A ) ) @ ( c_Relation_OId @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2020]) ).

thf(1078,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( c_Finite__Set_Ofun__left__comm @ F @ E @ D )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ F ) @ C ) )
       => ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ F ) @ B ) @ C ) )
         => ( ( hAPP @ ( hAPP @ ( c_Finite__Set_Ofold @ F @ E @ D ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ F ) @ B ) @ C ) )
            = ( hAPP @ ( hAPP @ ( c_Finite__Set_Ofold @ F @ E @ D ) @ ( hAPP @ ( hAPP @ D @ B ) @ A ) ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_fun__left__comm_Ofold__insert2) ).

thf(6374,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( c_Finite__Set_Ofun__left__comm @ F @ E @ D )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ F ) @ C ) )
       => ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ F ) @ B ) @ C ) )
         => ( ( hAPP @ ( hAPP @ ( c_Finite__Set_Ofold @ F @ E @ D ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ F ) @ B ) @ C ) )
            = ( hAPP @ ( hAPP @ ( c_Finite__Set_Ofold @ F @ E @ D ) @ ( hAPP @ ( hAPP @ D @ B ) @ A ) ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1078]) ).

thf(212,axiom,
    ! [A: $i] :
      ( ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ A ) ) @ ( c_List_Olist_ONil @ A ) )
      = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_list_Osize_I3_J) ).

thf(3172,plain,
    ! [A: $i] :
      ( ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ A ) ) @ ( c_List_Olist_ONil @ A ) )
      = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[212]) ).

thf(915,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ B ) )
     => ( ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ B ) @ A )
          = ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) )
      <=> ( ( B
            = ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) )
          & ( A
            = ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_pos__zmult__eq__1__iff) ).

thf(5686,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ B ) )
     => ( ( ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ B ) @ A )
            = ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) )
         => ( ( B
              = ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) )
            & ( A
              = ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ) ) )
        & ( ( ( B
              = ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) )
            & ( A
              = ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ) )
         => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ B ) @ A )
            = ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[915]) ).

thf(1928,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_Com_Ocom_OCond @ D @ C @ B )
     != ( hAPP @ c_Com_Ocom_OBODY @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_com_Osimps_I54_J) ).

thf(9982,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_Com_Ocom_OCond @ D @ C @ B )
     != ( hAPP @ c_Com_Ocom_OBODY @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1928]) ).

thf(116,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( c_Set_OCollect @ C ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ C @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ C ) @ c_fconj ) @ B ) ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Set_OCollect @ C ) @ B ) ) @ ( hAPP @ ( c_Set_OCollect @ C ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Collect__conj__eq) ).

thf(2822,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( c_Set_OCollect @ C ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ C @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ C ) @ c_fconj ) @ B ) ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Set_OCollect @ C ) @ B ) ) @ ( hAPP @ ( c_Set_OCollect @ C ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[116]) ).

thf(1707,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ B ) )
     => ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ A ) @ B ) )
       => ( ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Nat_Onat ) @ A ) @ B )
          = ( hAPP @ c_Nat_OSuc @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ A ) @ B ) ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_div__geq) ).

thf(9131,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ B ) )
     => ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ A ) @ B ) )
       => ( ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Nat_Onat ) @ A ) @ B )
          = ( hAPP @ c_Nat_OSuc @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ A ) @ B ) ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1707]) ).

thf(1587,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Set_Oimage @ C @ C @ A ) @ B ) ) @ B ) )
       => ( ( c_Fun_Oinj__on @ C @ C @ A @ B )
         => ( ( hAPP @ ( c_Set_Oimage @ C @ C @ A ) @ B )
            = B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_endo__inj__surj) ).

thf(8618,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Set_Oimage @ C @ C @ A ) @ B ) ) @ B ) )
       => ( ( c_Fun_Oinj__on @ C @ C @ A @ B )
         => ( ( hAPP @ ( c_Set_Oimage @ C @ C @ A ) @ B )
            = B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1587]) ).

thf(502,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ B ) @ A ) )
       => ( ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ B @ A )
          = B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mod__pos__pos__trivial) ).

thf(4201,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ B ) @ A ) )
       => ( ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ B @ A )
          = B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[502]) ).

thf(1664,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( c_List_Olist_ONil @ C )
        = ( hAPP @ ( hAPP @ ( c_List_Oappend @ C ) @ B ) @ A ) )
    <=> ( ( B
          = ( c_List_Olist_ONil @ C ) )
        & ( A
          = ( c_List_Olist_ONil @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Nil__is__append__conv) ).

thf(8930,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( c_List_Olist_ONil @ C )
          = ( hAPP @ ( hAPP @ ( c_List_Oappend @ C ) @ B ) @ A ) )
       => ( ( B
            = ( c_List_Olist_ONil @ C ) )
          & ( A
            = ( c_List_Olist_ONil @ C ) ) ) )
      & ( ( ( B
            = ( c_List_Olist_ONil @ C ) )
          & ( A
            = ( c_List_Olist_ONil @ C ) ) )
       => ( ( c_List_Olist_ONil @ C )
          = ( hAPP @ ( hAPP @ ( c_List_Oappend @ C ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1664]) ).

thf(599,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( c_Map_Orestrict__map @ D @ C @ B @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) @ A )
      = ( c_Option_Ooption_ONone @ C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_restrict__map__to__empty) ).

thf(4560,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( c_Map_Orestrict__map @ D @ C @ B @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) @ A )
      = ( c_Option_Ooption_ONone @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[599]) ).

thf(834,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ E ) ) @ ( hAPP @ ( c_List_Ofilter @ E @ D ) @ ( hAPP @ ( hAPP @ ( c_List_Omap @ C @ E ) @ B ) @ A ) ) )
      = ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) @ ( hAPP @ ( c_List_Ofilter @ C @ ( hAPP @ ( c_Fun_Ocomp @ E @ tc_HOL_Obool @ C @ D ) @ B ) ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_length__filter__map) ).

thf(5422,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ E ) ) @ ( hAPP @ ( c_List_Ofilter @ E @ D ) @ ( hAPP @ ( hAPP @ ( c_List_Omap @ C @ E ) @ B ) @ A ) ) )
      = ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) @ ( hAPP @ ( c_List_Ofilter @ C @ ( hAPP @ ( c_Fun_Ocomp @ E @ tc_HOL_Obool @ C @ D ) @ B ) ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[834]) ).

thf(475,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ F @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ D ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ F @ ( tc_fun @ C @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ C @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ F ) @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ C @ ( tc_fun @ E @ tc_HOL_Obool ) ) ) @ B ) ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ C @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ F @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ D ) @ B ) ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_UN__extend__simps_I9_J) ).

thf(4094,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ F @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ D ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ F @ ( tc_fun @ C @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ C @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ F ) @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ C @ ( tc_fun @ E @ tc_HOL_Obool ) ) ) @ B ) ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ C @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ F @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ D ) @ B ) ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[475]) ).

thf(1027,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_DiffD1) ).

thf(6167,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1027]) ).

thf(1549,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
      ( ( c_Finite__Set_Ofolding__image__simple @ H @ G @ F @ E @ D @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ G ) @ B ) )
       => ( ( hAPP @ C @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ G ) @ A ) @ B ) )
          = ( hAPP @ ( hAPP @ F @ ( hAPP @ D @ A ) ) @ ( hAPP @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ G @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ G ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ G @ tc_HOL_Obool ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_folding__image__simple_Oinsert__remove) ).

thf(8398,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
      ( ( c_Finite__Set_Ofolding__image__simple @ H @ G @ F @ E @ D @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ G ) @ B ) )
       => ( ( hAPP @ C @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ G ) @ A ) @ B ) )
          = ( hAPP @ ( hAPP @ F @ ( hAPP @ D @ A ) ) @ ( hAPP @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ G @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ G ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ G @ tc_HOL_Obool ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1549]) ).

thf(135,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ~ ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ C ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ B ) @ ( hAPP @ ( c_Set_Oimage @ D @ B @ A ) @ C ) ) )
       => ? [E: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ E ) @ C ) )
            & ~ ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ ( hAPP @ ( c_Set_OCollect @ D ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ D @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ D ) @ c_fconj ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ ( tc_fun @ D @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( c_member @ D ) ) @ C ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ B @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ B @ ( tc_fun @ B @ tc_HOL_Obool ) @ D ) @ c_fequal ) @ A ) ) @ ( hAPP @ A @ E ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_pigeonhole__infinite) ).

thf(2889,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ~ ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ C ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ B ) @ ( hAPP @ ( c_Set_Oimage @ D @ B @ A ) @ C ) ) )
       => ? [E: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ E ) @ C ) )
            & ~ ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ ( hAPP @ ( c_Set_OCollect @ D ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ D @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ D ) @ c_fconj ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ ( tc_fun @ D @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( c_member @ D ) ) @ C ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ B @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ B @ ( tc_fun @ B @ tc_HOL_Obool ) @ D ) @ c_fequal ) @ A ) ) @ ( hAPP @ A @ E ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[135]) ).

thf(1993,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ F @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ F @ E ) @ D ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ F @ E ) @ B ) @ A ) ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ C ) @ ( hAPP @ A @ D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_SigmaD2) ).

thf(10259,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ F @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ F @ E ) @ D ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ F @ E ) @ B ) @ A ) ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ C ) @ ( hAPP @ A @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1993]) ).

thf(2024,axiom,
    ! [A: $i,B: $i,C: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Natural_Oevalc @ ( c_Com_Ocom_OAss @ C @ B ) ) @ A ) @ ( hAPP @ ( hAPP @ ( hAPP @ c_Natural_Oupdate @ A ) @ C ) @ ( hAPP @ B @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_evalc_OAssign) ).

thf(10373,plain,
    ! [A: $i,B: $i,C: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Natural_Oevalc @ ( c_Com_Ocom_OAss @ C @ B ) ) @ A ) @ ( hAPP @ ( hAPP @ ( hAPP @ c_Natural_Oupdate @ A ) @ C ) @ ( hAPP @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2024]) ).

thf(1431,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Fun_Oinj__on @ E @ D @ C @ B )
     => ( c_Fun_Oinj__on @ E @ D @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_inj__on__diff) ).

thf(7890,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Fun_Oinj__on @ E @ D @ C @ B )
     => ( c_Fun_Oinj__on @ E @ D @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1431]) ).

thf(181,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) @ ( hAPP @ ( c_List_Ofilter @ C @ B ) @ A ) ) ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) @ ( hAPP @ ( c_List_Ofilter @ C @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ tc_HOL_Obool @ C ) @ c_fNot ) @ B ) ) @ A ) ) )
      = ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_sum__length__filter__compl) ).

thf(3053,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) @ ( hAPP @ ( c_List_Ofilter @ C @ B ) @ A ) ) ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) @ ( hAPP @ ( c_List_Ofilter @ C @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ tc_HOL_Obool @ C ) @ c_fNot ) @ B ) ) @ A ) ) )
      = ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[181]) ).

thf(572,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ F @ F ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ F @ F ) @ E ) @ D ) ) @ C ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ F ) @ ( tc_List_Olist @ F ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ F ) @ ( tc_List_Olist @ F ) ) @ B ) @ A ) ) @ ( c_List_Olistrel @ F @ C ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ F ) @ ( tc_List_Olist @ F ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ F ) @ ( tc_List_Olist @ F ) ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ F ) @ E ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ F ) @ D ) @ A ) ) ) @ ( c_List_Olistrel @ F @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_listrel_OCons) ).

thf(4464,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ F @ F ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ F @ F ) @ E ) @ D ) ) @ C ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ F ) @ ( tc_List_Olist @ F ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ F ) @ ( tc_List_Olist @ F ) ) @ B ) @ A ) ) @ ( c_List_Olistrel @ F @ C ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ F ) @ ( tc_List_Olist @ F ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ F ) @ ( tc_List_Olist @ F ) ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ F ) @ E ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ F ) @ D ) @ A ) ) ) @ ( c_List_Olistrel @ F @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[572]) ).

thf(674,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ C ) @ ( tc_List_Olist @ C ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ C ) @ ( tc_List_Olist @ C ) ) @ ( c_List_Olist_ONil @ C ) ) @ B ) ) @ ( c_List_Olistrel @ C @ A ) ) )
     => ( B
        = ( c_List_Olist_ONil @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_listrel__Nil1) ).

thf(4824,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ C ) @ ( tc_List_Olist @ C ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ C ) @ ( tc_List_Olist @ C ) ) @ ( c_List_Olist_ONil @ C ) ) @ B ) ) @ ( c_List_Olistrel @ C @ A ) ) )
     => ( B
        = ( c_List_Olist_ONil @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[674]) ).

thf(1421,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ D ) @ C ) )
       => ( ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ E @ ( tc_Option_Ooption @ B ) ) @ ( c_Map_Orestrict__map @ E @ B @ A @ C ) ) @ D ) @ ( c_Option_Ooption_ONone @ B ) )
          = ( c_Map_Orestrict__map @ E @ B @ A @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ D ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ E @ tc_HOL_Obool ) ) ) ) ) ) )
      & ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ D ) @ C ) )
       => ( ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ E @ ( tc_Option_Ooption @ B ) ) @ ( c_Map_Orestrict__map @ E @ B @ A @ C ) ) @ D ) @ ( c_Option_Ooption_ONone @ B ) )
          = ( c_Map_Orestrict__map @ E @ B @ A @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_fun__upd__None__restrict) ).

thf(7854,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ D ) @ C ) )
       => ( ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ E @ ( tc_Option_Ooption @ B ) ) @ ( c_Map_Orestrict__map @ E @ B @ A @ C ) ) @ D ) @ ( c_Option_Ooption_ONone @ B ) )
          = ( c_Map_Orestrict__map @ E @ B @ A @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ D ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ E @ tc_HOL_Obool ) ) ) ) ) ) )
      & ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ D ) @ C ) )
       => ( ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ E @ ( tc_Option_Ooption @ B ) ) @ ( c_Map_Orestrict__map @ E @ B @ A @ C ) ) @ D ) @ ( c_Option_Ooption_ONone @ B ) )
          = ( c_Map_Orestrict__map @ E @ B @ A @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1421]) ).

thf(700,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( c_Relation_OImage @ E @ D @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Relation_OImage @ E @ D @ C ) @ B ) ) @ ( hAPP @ ( c_Relation_OImage @ E @ D @ C ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Image__Un) ).

thf(4909,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( c_Relation_OImage @ E @ D @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Relation_OImage @ E @ D @ C ) @ B ) ) @ ( hAPP @ ( c_Relation_OImage @ E @ D @ C ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[700]) ).

thf(1344,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ C ) )
     => ! [D: $i] :
          ( ! [E: $i] :
              ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Int_Oint ) @ E ) @ ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Int_Oint @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) @ C ) ) )
             => ! [F: $i] :
                  ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Int_Oint ) @ F ) @ B ) )
                 => ( D
                   != ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ F ) @ E ) ) ) )
         => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ D ) @ A ) )
           => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Int_Oint ) @ D ) @ C ) ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_bset_I6_J) ).

thf(7569,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ C ) )
     => ! [D: $i] :
          ( ! [E: $i] :
              ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Int_Oint ) @ E ) @ ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Int_Oint @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) @ C ) ) )
             => ! [F: $i] :
                  ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Int_Oint ) @ F ) @ B ) )
                 => ( D
                   != ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ F ) @ E ) ) ) )
         => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ D ) @ A ) )
           => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Int_Oint ) @ D ) @ C ) ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1344]) ).

thf(1379,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( c_Finite__Set_Ofun__left__comm @ F @ E @ D )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ F ) @ C ) )
       => ( ( hAPP @ ( hAPP @ D @ B ) @ ( hAPP @ ( hAPP @ ( c_Finite__Set_Ofold @ F @ E @ D ) @ A ) @ C ) )
          = ( hAPP @ ( hAPP @ ( c_Finite__Set_Ofold @ F @ E @ D ) @ ( hAPP @ ( hAPP @ D @ B ) @ A ) ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_fun__left__comm_Ofold__fun__comm) ).

thf(7691,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( c_Finite__Set_Ofun__left__comm @ F @ E @ D )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ F ) @ C ) )
       => ( ( hAPP @ ( hAPP @ D @ B ) @ ( hAPP @ ( hAPP @ ( c_Finite__Set_Ofold @ F @ E @ D ) @ A ) @ C ) )
          = ( hAPP @ ( hAPP @ ( c_Finite__Set_Ofold @ F @ E @ D ) @ ( hAPP @ ( hAPP @ D @ B ) @ A ) ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1379]) ).

thf(790,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) @ A )
      = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_empty__Diff) ).

thf(5260,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) @ A )
      = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[790]) ).

thf(1847,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Odrop @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_List_Otake @ D ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_List_Otake @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ B ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_List_Odrop @ D ) @ C ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_drop__take) ).

thf(9648,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Odrop @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_List_Otake @ D ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_List_Otake @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ B ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_List_Odrop @ D ) @ C ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1847]) ).

thf(979,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ D @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ C ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ D @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ ( tc_fun @ E @ tc_HOL_Obool ) @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ E @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ E @ tc_HOL_Obool ) @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ D ) @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ E @ tc_HOL_Obool ) ) ) @ B ) ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_UN__extend__simps_I4_J) ).

thf(5950,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ D @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ C ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ D @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ ( tc_fun @ E @ tc_HOL_Obool ) @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ E @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ E @ tc_HOL_Obool ) @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ D ) @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ E @ tc_HOL_Obool ) ) ) @ B ) ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[979]) ).

thf(211,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ C @ C ) ) @ B ) @ ( c_Transitive__Closure_Ortrancl @ C @ A ) ) )
    <=> ? [D: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ C @ C ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Nat_Ocompow @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) @ D ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_rtrancl__power) ).

thf(3167,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ C @ C ) ) @ B ) @ ( c_Transitive__Closure_Ortrancl @ C @ A ) ) )
       => ? [D: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ C @ C ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Nat_Ocompow @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) @ D ) @ A ) ) ) )
      & ( ? [D: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ C @ C ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Nat_Ocompow @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) @ D ) @ A ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ C @ C ) ) @ B ) @ ( c_Transitive__Closure_Ortrancl @ C @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[211]) ).

thf(1481,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ C ) @ B ) )
     => ( ( hAPP @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Nat_Onat @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Nat_Onat @ C ) @ B ) ) @ ( hAPP @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Nat_Onat @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_atLeastLessThan__add__Un) ).

thf(8114,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ C ) @ B ) )
     => ( ( hAPP @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Nat_Onat @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Nat_Onat @ C ) @ B ) ) @ ( hAPP @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Nat_Onat @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1481]) ).

thf(1,conjecture,
    ! [A: $i] :
      ( ! [B: $i] :
          ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_Hoare__Mirabelle_Otriple @ t_a ) ) @ B ) @ v_G ) )
         => ( c_Hoare__Mirabelle_Otriple__valid @ t_a @ A @ B ) )
     => ! [B: $i,C: $i] :
          ( ( v_P @ B @ C )
         => ! [D: $i] :
              ( ( c_Natural_Oevaln @ ( c_Com_Ocom_OWhile @ v_b @ v_c ) @ C @ A @ D )
             => ( ( v_P @ B @ D )
                & ~ ( hBOOL @ ( hAPP @ v_b @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',conj_1) ).

thf(2,negated_conjecture,
    ~ ! [A: $i] :
        ( ! [B: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_Hoare__Mirabelle_Otriple @ t_a ) ) @ B ) @ v_G ) )
           => ( c_Hoare__Mirabelle_Otriple__valid @ t_a @ A @ B ) )
       => ! [B: $i,C: $i] :
            ( ( v_P @ B @ C )
           => ! [D: $i] :
                ( ( c_Natural_Oevaln @ ( c_Com_Ocom_OWhile @ v_b @ v_c ) @ C @ A @ D )
               => ( ( v_P @ B @ D )
                  & ~ ( hBOOL @ ( hAPP @ v_b @ D ) ) ) ) ) ),
    inference(neg_conjecture,[status(cth)],[1]) ).

thf(2410,plain,
    ~ ! [A: $i] :
        ( ! [B: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_Hoare__Mirabelle_Otriple @ t_a ) ) @ B ) @ v_G ) )
           => ( c_Hoare__Mirabelle_Otriple__valid @ t_a @ A @ B ) )
       => ! [B: $i,C: $i] :
            ( ( v_P @ B @ C )
           => ! [D: $i] :
                ( ( c_Natural_Oevaln @ ( c_Com_Ocom_OWhile @ v_b @ v_c ) @ C @ A @ D )
               => ( ( v_P @ B @ D )
                  & ~ ( hBOOL @ ( hAPP @ v_b @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2]) ).

thf(2412,plain,
    c_Natural_Oevaln @ ( c_Com_Ocom_OWhile @ v_b @ v_c ) @ sk3 @ sk1 @ sk4,
    inference(cnf,[status(esa)],[2410]) ).

thf(1055,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__Suc) ).

thf(6283,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1055]) ).

thf(1238,axiom,
    ! [A: $i,B: $i] :
      ( ( c_SetInterval_Oord__class_OgreaterThanLessThan @ tc_Nat_Onat @ B @ A )
      = ( hAPP @ ( c_List_Oset @ tc_Nat_Onat ) @ ( c_List_Oupt @ ( hAPP @ c_Nat_OSuc @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_greaterThanLessThan__upt) ).

thf(7100,plain,
    ! [A: $i,B: $i] :
      ( ( c_SetInterval_Oord__class_OgreaterThanLessThan @ tc_Nat_Onat @ B @ A )
      = ( hAPP @ ( c_List_Oset @ tc_Nat_Onat ) @ ( c_List_Oupt @ ( hAPP @ c_Nat_OSuc @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1238]) ).

thf(2059,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_Com_Ocom_OCall @ D @ C @ B )
     != ( hAPP @ c_Com_Ocom_OBODY @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_com_Osimps_I63_J) ).

thf(10495,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_Com_Ocom_OCall @ D @ C @ B )
     != ( hAPP @ c_Com_Ocom_OBODY @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2059]) ).

thf(1517,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ A ) ) )
    <=> ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) @ B ) )
        & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_one__le__mult__iff) ).

thf(8283,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ A ) ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) @ B ) )
          & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) @ A ) ) ) )
      & ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) @ B ) )
          & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) @ A ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1517]) ).

thf(330,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ E ) @ D ) ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ E ) @ B ) ) @ A ) ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ C ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ C ) @ E ) )
         => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ A ) @ E ) )
           => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ D ) @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_unique__quotient__lemma) ).

thf(3581,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ E ) @ D ) ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ E ) @ B ) ) @ A ) ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ C ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ C ) @ E ) )
         => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ A ) @ E ) )
           => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ D ) @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[330]) ).

thf(684,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ ( c_Map_Odom @ C @ B @ A ) ) )
     => ( ~ ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) )
       => ? [D: $i] :
            ( ( hAPP @ A @ D )
            = ( c_Option_Ooption_ONone @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_finite__map__freshness) ).

thf(4852,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ ( c_Map_Odom @ C @ B @ A ) ) )
     => ( ~ ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) )
       => ? [D: $i] :
            ( ( hAPP @ A @ D )
            = ( c_Option_Ooption_ONone @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[684]) ).

thf(2129,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C = B )
     => ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) )
          & ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ A ) @ B ) ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ C ) @ A ) )
          & ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ A ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_dvd_Oord__eq__less__trans) ).

thf(10743,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C = B )
     => ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) )
          & ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ A ) @ B ) ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ C ) @ A ) )
          & ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ A ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2129]) ).

thf(55,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Ocomm__monoid__add @ D )
     => ( ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ B ) )
         => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ C @ D ) @ A ) @ B )
            = ( hAPP @ ( hAPP @ ( hAPP @ ( c_Finite__Set_Ofold__image @ D @ C @ ( c_Groups_Oplus__class_Oplus @ D ) ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ D ) ) @ B ) ) )
        & ( ~ ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ B ) )
         => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ C @ D ) @ A ) @ B )
            = ( c_Groups_Ozero__class_Ozero @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_setsum_OF__eq) ).

thf(2595,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Ocomm__monoid__add @ D )
     => ( ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ B ) )
         => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ C @ D ) @ A ) @ B )
            = ( hAPP @ ( hAPP @ ( hAPP @ ( c_Finite__Set_Ofold__image @ D @ C @ ( c_Groups_Oplus__class_Oplus @ D ) ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ D ) ) @ B ) ) )
        & ( ~ ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ B ) )
         => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ C @ D ) @ A ) @ B )
            = ( c_Groups_Ozero__class_Ozero @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[55]) ).

thf(434,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_IntI) ).

thf(3969,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[434]) ).

thf(1638,axiom,
    ! [A: $i] :
      ( ( A
       != ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
     => ? [B: $i] :
          ( A
          = ( hAPP @ c_Nat_OSuc @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_not0__implies__Suc) ).

thf(8843,plain,
    ! [A: $i] :
      ( ( A
       != ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
     => ? [B: $i] :
          ( A
          = ( hAPP @ c_Nat_OSuc @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1638]) ).

thf(779,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( c_Relation_Oconverse @ C @ B ) @ A )
      = ( hAPP @ ( c_Set_OCollect @ ( tc_prod @ B @ C ) ) @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ B @ C @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ B @ ( tc_fun @ ( tc_prod @ C @ B ) @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ ( tc_fun @ ( tc_fun @ ( tc_prod @ C @ B ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ C @ B ) @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ ( c_COMBC @ C @ ( tc_fun @ ( tc_prod @ C @ B ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ ( tc_prod @ C @ B ) ) @ ( tc_fun @ C @ ( tc_fun @ ( tc_fun @ ( tc_prod @ C @ B ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( c_COMBB @ ( tc_prod @ C @ B ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ C @ B ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ C ) @ ( c_member @ ( tc_prod @ C @ B ) ) ) ) @ ( hAPP @ ( c_COMBC @ C @ B @ ( tc_prod @ C @ B ) ) @ ( c_Product__Type_OPair @ C @ B ) ) ) ) ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_converse__def) ).

thf(5205,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( c_Relation_Oconverse @ C @ B ) @ A )
      = ( hAPP @ ( c_Set_OCollect @ ( tc_prod @ B @ C ) ) @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ B @ C @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ B @ ( tc_fun @ ( tc_prod @ C @ B ) @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ ( tc_fun @ ( tc_fun @ ( tc_prod @ C @ B ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ C @ B ) @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ ( c_COMBC @ C @ ( tc_fun @ ( tc_prod @ C @ B ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ ( tc_prod @ C @ B ) ) @ ( tc_fun @ C @ ( tc_fun @ ( tc_fun @ ( tc_prod @ C @ B ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( c_COMBB @ ( tc_prod @ C @ B ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ C @ B ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ C ) @ ( c_member @ ( tc_prod @ C @ B ) ) ) ) @ ( hAPP @ ( c_COMBC @ C @ B @ ( tc_prod @ C @ B ) ) @ ( c_Product__Type_OPair @ C @ B ) ) ) ) ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[779]) ).

thf(1029,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( hAPP @ ( hAPP @ ( c_List_Oappend @ D ) @ C ) @ B )
        = ( hAPP @ ( hAPP @ ( c_List_Oappend @ D ) @ A ) @ B ) )
    <=> ( C = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_append__same__eq) ).

thf(6172,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( ( hAPP @ ( hAPP @ ( c_List_Oappend @ D ) @ C ) @ B )
          = ( hAPP @ ( hAPP @ ( c_List_Oappend @ D ) @ A ) @ B ) )
       => ( C = A ) )
      & ( ( C = A )
       => ( ( hAPP @ ( hAPP @ ( c_List_Oappend @ D ) @ C ) @ B )
          = ( hAPP @ ( hAPP @ ( c_List_Oappend @ D ) @ A ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1029]) ).

thf(449,axiom,
    c_FunDef_Oreduction__pair @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_fun @ ( tc_prod @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) ) @ tc_HOL_Obool ) ) @ c_FunDef_Omin__strict ) @ c_FunDef_Omin__weak ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_min__rpair__set) ).

thf(4015,plain,
    c_FunDef_Oreduction__pair @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_fun @ ( tc_prod @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) ) @ tc_HOL_Obool ) ) @ c_FunDef_Omin__strict ) @ c_FunDef_Omin__weak ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[449]) ).

thf(44,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Groups_Ocomm__monoid__add @ E )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ C ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ B ) @ C ) )
         => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ A ) @ C )
            = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ E ) @ ( hAPP @ A @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ B ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_setsum__diff1_H) ).

thf(2544,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Groups_Ocomm__monoid__add @ E )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ C ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ B ) @ C ) )
         => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ A ) @ C )
            = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ E ) @ ( hAPP @ A @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ B ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[44]) ).

thf(2413,plain,
    v_P @ sk2 @ sk3,
    inference(cnf,[status(esa)],[2410]) ).

thf(2411,plain,
    ( ~ ( v_P @ sk2 @ sk4 )
    | ( hBOOL @ ( hAPP @ v_b @ sk4 ) ) ),
    inference(cnf,[status(esa)],[2410]) ).

thf(11891,plain,
    ( ( hBOOL @ ( hAPP @ v_b @ sk4 ) )
    | ( ( v_P @ sk2 @ sk4 )
     != ( v_P @ sk2 @ sk3 ) ) ),
    inference(paramod_ordered,[status(thm)],[2413,2411]) ).

thf(11892,plain,
    ( ( hBOOL @ ( hAPP @ v_b @ sk4 ) )
    | ( sk2 != sk2 )
    | ( sk4 != sk3 ) ),
    inference(simp,[status(thm)],[11891]) ).

thf(11893,plain,
    ( ( hBOOL @ ( hAPP @ v_b @ sk4 ) )
    | ( sk4 != sk3 ) ),
    inference(simp,[status(thm)],[11892]) ).

thf(419,axiom,
    ~ ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ tc_Int_Oint ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ tc_Int_Oint @ tc_HOL_Obool ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_infinite__UNIV__int) ).

thf(3917,plain,
    ~ ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ tc_Int_Oint ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ tc_Int_Oint @ tc_HOL_Obool ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[419]) ).

thf(3918,plain,
    ~ ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ tc_Int_Oint ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ tc_Int_Oint @ tc_HOL_Obool ) ) ) ),
    inference(polarity_switch,[status(thm)],[3917]) ).

thf(11895,plain,
    ( ( sk4 != sk3 )
    | ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ tc_Int_Oint ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ tc_Int_Oint @ tc_HOL_Obool ) ) ) )
     != ( hBOOL @ ( hAPP @ v_b @ sk4 ) ) ) ),
    inference(paramod_ordered,[status(thm)],[11893,3918]) ).

thf(11898,plain,
    ( ( sk4 != sk3 )
    | ( ( hAPP @ ( c_Finite__Set_Ofinite @ tc_Int_Oint ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ tc_Int_Oint @ tc_HOL_Obool ) ) )
     != ( hAPP @ v_b @ sk4 ) ) ),
    inference(simp,[status(thm)],[11895]) ).

thf(1205,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( c_Relation_Oconverse @ E @ D ) @ ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ E @ C @ D ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ D @ C @ E ) @ ( hAPP @ ( c_Relation_Oconverse @ C @ D ) @ A ) ) @ ( hAPP @ ( c_Relation_Oconverse @ E @ C ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_converse__rel__comp) ).

thf(6954,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( c_Relation_Oconverse @ E @ D ) @ ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ E @ C @ D ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ D @ C @ E ) @ ( hAPP @ ( c_Relation_Oconverse @ C @ D ) @ A ) ) @ ( hAPP @ ( c_Relation_Oconverse @ E @ C ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1205]) ).

thf(1697,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_finite__subset) ).

thf(9091,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1697]) ).

thf(1309,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( c_Set_Ovimage @ E @ D @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Set_Ovimage @ E @ D @ C ) @ B ) ) @ ( hAPP @ ( c_Set_Ovimage @ E @ D @ C ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_vimage__Diff) ).

thf(7396,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( c_Set_Ovimage @ E @ D @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Set_Ovimage @ E @ D @ C ) @ B ) ) @ ( hAPP @ ( c_Set_Ovimage @ E @ D @ C ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1309]) ).

thf(1755,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ F ) @ E ) @ D ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ F @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ F @ C ) @ E ) @ B ) ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ B ) @ ( hAPP @ ( c_Relation_OImage @ F @ C @ A ) @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_rev__ImageI) ).

thf(9305,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ F ) @ E ) @ D ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ F @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ F @ C ) @ E ) @ B ) ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ B ) @ ( hAPP @ ( c_Relation_OImage @ F @ C @ A ) @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1755]) ).

thf(2170,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ c_Com_OWT @ C ) )
     => ( hBOOL @ ( hAPP @ c_Com_OWT @ ( c_Com_Ocom_OLocal @ B @ A @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_WT_OLocal) ).

thf(10901,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ c_Com_OWT @ C ) )
     => ( hBOOL @ ( hAPP @ c_Com_OWT @ ( c_Com_Ocom_OLocal @ B @ A @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2170]) ).

thf(2324,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( c_Typedef_Otype__definition @ G @ F @ E @ D @ C )
     => ( ( ( hAPP @ E @ B )
          = ( hAPP @ E @ A ) )
      <=> ( B = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_type__definition_ORep__inject) ).

thf(11527,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( c_Typedef_Otype__definition @ G @ F @ E @ D @ C )
     => ( ( ( ( hAPP @ E @ B )
            = ( hAPP @ E @ A ) )
         => ( B = A ) )
        & ( ( B = A )
         => ( ( hAPP @ E @ B )
            = ( hAPP @ E @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2324]) ).

thf(492,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( c_Map_Odom @ C @ B @ A )
        = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) )
    <=> ( A
        = ( hAPP @ ( c_COMBK @ ( tc_Option_Ooption @ B ) @ C ) @ ( c_Option_Ooption_ONone @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_dom__eq__empty__conv) ).

thf(4166,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( c_Map_Odom @ C @ B @ A )
          = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) )
       => ( A
          = ( hAPP @ ( c_COMBK @ ( tc_Option_Ooption @ B ) @ C ) @ ( c_Option_Ooption_ONone @ B ) ) ) )
      & ( ( A
          = ( hAPP @ ( c_COMBK @ ( tc_Option_Ooption @ B ) @ C ) @ ( c_Option_Ooption_ONone @ B ) ) )
       => ( ( c_Map_Odom @ C @ B @ A )
          = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[492]) ).

thf(1168,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( c_Finite__Set_Ofun__left__comm @ G @ F @ E )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofold__graph @ G @ F @ E @ D @ C ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofold__graph @ G @ F @ E @ D @ C ) @ A ) )
         => ( A = B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_fun__left__comm_Ofold__graph__determ) ).

thf(6808,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( c_Finite__Set_Ofun__left__comm @ G @ F @ E )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofold__graph @ G @ F @ E @ D @ C ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofold__graph @ G @ F @ E @ D @ C ) @ A ) )
         => ( A = B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1168]) ).

thf(148,axiom,
    ! [A: $i,B: $i] :
      ( ( B = A )
     => ( ( c_SetInterval_Oord_OatLeastAtMost @ tc_Nat_Onat @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B @ A )
        = ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ tc_Nat_Onat ) @ B ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_dvd_OatLeastAtMost__singleton_H) ).

thf(2940,plain,
    ! [A: $i,B: $i] :
      ( ( B = A )
     => ( ( c_SetInterval_Oord_OatLeastAtMost @ tc_Nat_Onat @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B @ A )
        = ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ tc_Nat_Onat ) @ B ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[148]) ).

thf(1185,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ B ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_diff__cancel) ).

thf(6883,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ B ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1185]) ).

thf(76,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Divides_Odivmod__int__rel @ D @ C ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ tc_Int_Oint @ tc_Int_Oint ) @ B ) @ A ) ) )
     => ( ( C
         != ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) )
       => ( ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ D ) @ C )
          = B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_divmod__int__rel__div) ).

thf(2676,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Divides_Odivmod__int__rel @ D @ C ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ tc_Int_Oint @ tc_Int_Oint ) @ B ) @ A ) ) )
     => ( ( C
         != ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) )
       => ( ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ D ) @ C )
          = B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[76]) ).

thf(642,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Fun_Oinj__on @ ( tc_List_Olist @ C ) @ ( tc_List_Olist @ B ) @ ( hAPP @ ( c_List_Omap @ C @ B ) @ A ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ ( tc_List_Olist @ C ) @ tc_HOL_Obool ) ) )
    <=> ( c_Fun_Oinj__on @ C @ B @ A @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_inj__map) ).

thf(4711,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( c_Fun_Oinj__on @ ( tc_List_Olist @ C ) @ ( tc_List_Olist @ B ) @ ( hAPP @ ( c_List_Omap @ C @ B ) @ A ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ ( tc_List_Olist @ C ) @ tc_HOL_Obool ) ) )
       => ( c_Fun_Oinj__on @ C @ B @ A @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) )
      & ( ( c_Fun_Oinj__on @ C @ B @ A @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ C @ tc_HOL_Obool ) ) )
       => ( c_Fun_Oinj__on @ ( tc_List_Olist @ C ) @ ( tc_List_Olist @ B ) @ ( hAPP @ ( c_List_Omap @ C @ B ) @ A ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ ( tc_List_Olist @ C ) @ tc_HOL_Obool ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[642]) ).

thf(1490,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ C ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ B ) @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ) ) @ A ) )
       => ! [D: $i] :
            ( ! [E: $i] :
                ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Int_Oint ) @ E ) @ ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Int_Oint @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) @ C ) ) )
               => ! [F: $i] :
                    ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Int_Oint ) @ F ) @ A ) )
                   => ( D
                     != ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Int_Oint ) @ F ) @ E ) ) ) )
           => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ D ) @ B ) )
             => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ D ) @ C ) ) @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_aset_I6_J) ).

thf(8169,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ C ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ B ) @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ) ) @ A ) )
       => ! [D: $i] :
            ( ! [E: $i] :
                ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Int_Oint ) @ E ) @ ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Int_Oint @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) @ C ) ) )
               => ! [F: $i] :
                    ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Int_Oint ) @ F ) @ A ) )
                   => ( D
                     != ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Int_Oint ) @ F ) @ E ) ) ) )
           => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ D ) @ B ) )
             => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ D ) @ C ) ) @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1490]) ).

thf(2165,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Wellfounded_Owf @ B @ A )
    <=> ~ ? [C: $i] :
          ! [D: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ B @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ B @ B ) @ ( hAPP @ C @ ( hAPP @ c_Nat_OSuc @ D ) ) ) @ ( hAPP @ C @ D ) ) ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_wf__iff__no__infinite__down__chain) ).

thf(10886,plain,
    ! [A: $i,B: $i] :
      ( ( ( c_Wellfounded_Owf @ B @ A )
       => ~ ? [C: $i] :
            ! [D: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ B @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ B @ B ) @ ( hAPP @ C @ ( hAPP @ c_Nat_OSuc @ D ) ) ) @ ( hAPP @ C @ D ) ) ) @ A ) ) )
      & ( ~ ? [C: $i] :
            ! [D: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ B @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ B @ B ) @ ( hAPP @ C @ ( hAPP @ c_Nat_OSuc @ D ) ) ) @ ( hAPP @ C @ D ) ) ) @ A ) )
       => ( c_Wellfounded_Owf @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2165]) ).

thf(651,axiom,
    ! [A: $i] : ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ tc_Int_Oint ) @ ( hAPP @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_finite__atLeastZeroLessThan__int) ).

thf(4747,plain,
    ! [A: $i] : ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ tc_Int_Oint ) @ ( hAPP @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[651]) ).

thf(275,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( c_Equiv__Relations_Oequiv @ F @ E @ D )
     => ( ( c_Equiv__Relations_Ocongruent @ F @ ( tc_fun @ C @ tc_HOL_Obool ) @ D @ B )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ F ) @ A ) @ E ) )
         => ( ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ F @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Relation_OImage @ F @ F @ D ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ F ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ F @ tc_HOL_Obool ) ) ) ) ) @ B )
            = ( hAPP @ B @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_UN__equiv__class) ).

thf(3382,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( c_Equiv__Relations_Oequiv @ F @ E @ D )
     => ( ( c_Equiv__Relations_Ocongruent @ F @ ( tc_fun @ C @ tc_HOL_Obool ) @ D @ B )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ F ) @ A ) @ E ) )
         => ( ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ F @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Relation_OImage @ F @ F @ D ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ F ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ F @ tc_HOL_Obool ) ) ) ) ) @ B )
            = ( hAPP @ B @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[275]) ).

thf(593,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ D ) ) @ ( hAPP @ ( hAPP @ ( c_List_Omap @ C @ D ) @ B ) @ A ) )
      = ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_length__map) ).

thf(4532,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ D ) ) @ ( hAPP @ ( hAPP @ ( c_List_Omap @ C @ D ) @ B ) @ A ) )
      = ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[593]) ).

thf(2200,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ B ) ) @ A ) )
     => ~ ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ C ) @ A ) )
         => ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_add__leE) ).

thf(11008,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ B ) ) @ A ) )
     => ~ ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ C ) @ A ) )
         => ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2200]) ).

thf(1738,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Oappend @ C ) @ ( hAPP @ ( hAPP @ ( c_List_Otake @ C ) @ B ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_List_Odrop @ C ) @ B ) @ A ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_append__take__drop__id) ).

thf(9241,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Oappend @ C ) @ ( hAPP @ ( hAPP @ ( c_List_Otake @ C ) @ B ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_List_Odrop @ C ) @ B ) @ A ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1738]) ).

thf(1766,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ C ) )
     => ( ( c_Fun_Oinj__on @ D @ B @ A @ C )
      <=> ( ( hAPP @ ( c_Finite__Set_Ocard @ B ) @ ( hAPP @ ( c_Set_Oimage @ D @ B @ A ) @ C ) )
          = ( hAPP @ ( c_Finite__Set_Ocard @ D ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_inj__on__iff__eq__card) ).

thf(9342,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ C ) )
     => ( ( ( c_Fun_Oinj__on @ D @ B @ A @ C )
         => ( ( hAPP @ ( c_Finite__Set_Ocard @ B ) @ ( hAPP @ ( c_Set_Oimage @ D @ B @ A ) @ C ) )
            = ( hAPP @ ( c_Finite__Set_Ocard @ D ) @ C ) ) )
        & ( ( ( hAPP @ ( c_Finite__Set_Ocard @ B ) @ ( hAPP @ ( c_Set_Oimage @ D @ B @ A ) @ C ) )
            = ( hAPP @ ( c_Finite__Set_Ocard @ D ) @ C ) )
         => ( c_Fun_Oinj__on @ D @ B @ A @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1766]) ).

thf(844,axiom,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ tc_Int_Oint ) @ ( hAPP @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Int_Oint @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_finite__atLeastLessThan__int) ).

thf(5454,plain,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ tc_Int_Oint ) @ ( hAPP @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Int_Oint @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[844]) ).

thf(1162,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ D ) @ ( hAPP @ ( c_List_Oset @ E ) @ C ) ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ ( hAPP @ A @ D ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ ( c_List_Olist_Olist__size @ E @ A @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_list__size__estimation) ).

thf(6791,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ D ) @ ( hAPP @ ( c_List_Oset @ E ) @ C ) ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ ( hAPP @ A @ D ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ ( c_List_Olist_Olist__size @ E @ A @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1162]) ).

thf(2283,axiom,
    ! [A: $i,B: $i] :
      ( ~ ( hBOOL @ B )
      | ~ ( hBOOL @ A )
      | ( hBOOL @ ( hAPP @ ( hAPP @ c_fconj @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_c__fconj__1) ).

thf(11352,plain,
    ! [A: $i,B: $i] :
      ( ~ ( hBOOL @ B )
      | ~ ( hBOOL @ A )
      | ( hBOOL @ ( hAPP @ ( hAPP @ c_fconj @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2283]) ).

thf(1683,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ A ) @ B ) ) )
    <=> ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ B ) )
        & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__less__cancel2) ).

thf(9020,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ A ) @ B ) ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ B ) )
          & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ C ) @ A ) ) ) )
      & ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ B ) )
          & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ C ) @ A ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ A ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1683]) ).

thf(1003,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Map_Omap__comp @ E @ D @ C @ B @ ( hAPP @ ( c_COMBK @ ( tc_Option_Ooption @ E ) @ C ) @ ( c_Option_Ooption_ONone @ E ) ) @ A )
      = ( c_Option_Ooption_ONone @ D ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_map__comp__empty_I1_J) ).

thf(6063,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Map_Omap__comp @ E @ D @ C @ B @ ( hAPP @ ( c_COMBK @ ( tc_Option_Ooption @ E ) @ C ) @ ( c_Option_Ooption_ONone @ E ) ) @ A )
      = ( c_Option_Ooption_ONone @ D ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1003]) ).

thf(33,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Groups_Ocomm__monoid__add @ E )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ C ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ B ) )
         => ( ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B )
              = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) )
           => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) )
              = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ E ) @ ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ A ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ A ) @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_setsum__Un__disjoint) ).

thf(2512,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Groups_Ocomm__monoid__add @ E )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ C ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ B ) )
         => ( ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B )
              = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) )
           => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) )
              = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ E ) @ ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ A ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ A ) @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[33]) ).

thf(2282,axiom,
    ! [A: $i,B: $i] :
      ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ c_fimplies @ B ) @ A ) )
      | ~ ( hBOOL @ B )
      | ( hBOOL @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_c__fimplies__3) ).

thf(11350,plain,
    ! [A: $i,B: $i] :
      ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ c_fimplies @ B ) @ A ) )
      | ~ ( hBOOL @ B )
      | ( hBOOL @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2282]) ).

thf(1839,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_termination__basic__simps_I4_J) ).

thf(9621,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1839]) ).

thf(188,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_Fun_Oinj__on @ D @ C @ B @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ D @ tc_HOL_Obool ) ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ A ) @ ( hAPP @ ( c_Set_Oimage @ D @ C @ B ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) ) )
      <=> ? [E: $i] :
            ( ( A
              = ( hAPP @ B @ E ) )
            & ! [F: $i] :
                ( ( A
                  = ( hAPP @ B @ F ) )
               => ( F = E ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_range__ex1__eq) ).

thf(3072,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_Fun_Oinj__on @ D @ C @ B @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ D @ tc_HOL_Obool ) ) )
     => ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ A ) @ ( hAPP @ ( c_Set_Oimage @ D @ C @ B ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) ) )
         => ? [E: $i] :
              ( ( A
                = ( hAPP @ B @ E ) )
              & ! [F: $i] :
                  ( ( A
                    = ( hAPP @ B @ F ) )
                 => ( F = E ) ) ) )
        & ( ? [E: $i] :
              ( ( A
                = ( hAPP @ B @ E ) )
              & ! [F: $i] :
                  ( ( A
                    = ( hAPP @ B @ F ) )
                 => ( F = E ) ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ A ) @ ( hAPP @ ( c_Set_Oimage @ D @ C @ B ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[188]) ).

thf(1649,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ C ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ A ) ) )
      <=> ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nat__mult__dvd__cancel1) ).

thf(8883,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ C ) )
     => ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ A ) ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) ) )
        & ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1649]) ).

thf(1562,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
      ( ( hAPP @ ( c_Product__Type_Omap__pair @ H @ G @ F @ E @ D @ C ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ H @ F ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ G @ E ) @ ( hAPP @ D @ B ) ) @ ( hAPP @ C @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_map__pair__simp) ).

thf(8450,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
      ( ( hAPP @ ( c_Product__Type_Omap__pair @ H @ G @ F @ E @ D @ C ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ H @ F ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ G @ E ) @ ( hAPP @ D @ B ) ) @ ( hAPP @ C @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1562]) ).

thf(1351,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Odrop @ B ) @ A ) @ ( c_List_Olist_ONil @ B ) )
      = ( c_List_Olist_ONil @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_drop__Nil) ).

thf(7596,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Odrop @ B ) @ A ) @ ( c_List_Olist_ONil @ B ) )
      = ( c_List_Olist_ONil @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1351]) ).

thf(1470,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ C ) )
     => ! [D: $i] :
          ( ! [E: $i] :
              ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Int_Oint ) @ E ) @ ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Int_Oint @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) @ C ) ) )
             => ! [F: $i] :
                  ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Int_Oint ) @ F ) @ B ) )
                 => ( D
                   != ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Int_Oint ) @ F ) @ E ) ) ) )
         => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ A ) @ D ) )
           => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ D ) @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_aset_I7_J) ).

thf(8044,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ C ) )
     => ! [D: $i] :
          ( ! [E: $i] :
              ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Int_Oint ) @ E ) @ ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Int_Oint @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) @ C ) ) )
             => ! [F: $i] :
                  ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Int_Oint ) @ F ) @ B ) )
                 => ( D
                   != ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Int_Oint ) @ F ) @ E ) ) ) )
         => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ A ) @ D ) )
           => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ D ) @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1470]) ).

thf(1194,axiom,
    ! [A: $i] : ( hBOOL @ ( hAPP @ ( c_List_Odistinct @ A ) @ ( c_List_Olist_ONil @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_distinct_Osimps_I1_J) ).

thf(6918,plain,
    ! [A: $i] : ( hBOOL @ ( hAPP @ ( c_List_Odistinct @ A ) @ ( c_List_Olist_ONil @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1194]) ).

thf(947,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i,F: $i] :
          ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ D ) @ E ) @ F ) ) @ C ) )
          | ( E = F )
          | ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ D ) @ F ) @ E ) ) @ C ) ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) @ B ) @ A ) ) @ ( c_List_Olexord @ D @ C ) ) )
        | ( B = A )
        | ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) @ A ) @ B ) ) @ ( c_List_Olexord @ D @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_lexord__linear) ).

thf(5820,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i,F: $i] :
          ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ D ) @ E ) @ F ) ) @ C ) )
          | ( E = F )
          | ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ D ) @ F ) @ E ) ) @ C ) ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) @ B ) @ A ) ) @ ( c_List_Olexord @ D @ C ) ) )
        | ( B = A )
        | ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) @ A ) @ B ) ) @ ( c_List_Olexord @ D @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[947]) ).

thf(536,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( c_Relation_Oconverse @ E @ D ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ C @ ( tc_fun @ ( tc_prod @ E @ D ) @ tc_HOL_Obool ) ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ C @ ( tc_fun @ ( tc_prod @ D @ E ) @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ ( tc_prod @ E @ D ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ D @ E ) @ tc_HOL_Obool ) @ C ) @ ( c_Relation_Oconverse @ E @ D ) ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_converse__UNION) ).

thf(4339,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( c_Relation_Oconverse @ E @ D ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ C @ ( tc_fun @ ( tc_prod @ E @ D ) @ tc_HOL_Obool ) ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ C @ ( tc_fun @ ( tc_prod @ D @ E ) @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ ( tc_prod @ E @ D ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ D @ E ) @ tc_HOL_Obool ) @ C ) @ ( c_Relation_Oconverse @ E @ D ) ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[536]) ).

thf(468,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ E @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ C ) @ B ) ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ D @ tc_HOL_Obool ) @ E ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ ( tc_prod @ E @ D ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ E @ D ) @ C ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ D @ tc_HOL_Obool ) @ E ) @ A ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ E @ D ) @ B ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ D @ tc_HOL_Obool ) @ E ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Times__Diff__distrib1) ).

thf(4075,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ E @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ C ) @ B ) ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ D @ tc_HOL_Obool ) @ E ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ ( tc_prod @ E @ D ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ E @ D ) @ C ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ D @ tc_HOL_Obool ) @ E ) @ A ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ E @ D ) @ B ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ D @ tc_HOL_Obool ) @ E ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[468]) ).

thf(1718,axiom,
    ! [A: $i] :
      ( ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat )
     != ( hAPP @ c_Nat_OSuc @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Zero__not__Suc) ).

thf(9163,plain,
    ! [A: $i] :
      ( ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat )
     != ( hAPP @ c_Nat_OSuc @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1718]) ).

thf(1775,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ C @ C ) ) @ B ) @ ( c_Relation_OId__on @ C @ A ) ) )
     => ~ ! [D: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ D ) @ A ) )
           => ( B
             != ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ C @ C ) @ D ) @ D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Id__onE) ).

thf(9373,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ C @ C ) ) @ B ) @ ( c_Relation_OId__on @ C @ A ) ) )
     => ~ ! [D: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ D ) @ A ) )
           => ( B
             != ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ C @ C ) @ D ) @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1775]) ).

thf(2405,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i] :
          ( ( hAPP @ B @ C )
          = ( hAPP @ A @ C ) )
     => ( B = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_ext) ).

thf(11880,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i] :
          ( ( hAPP @ B @ C )
          = ( hAPP @ A @ C ) )
     => ( B = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2405]) ).

thf(531,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( c_Equiv__Relations_Oequiv @ G @ F @ E )
     => ( ( c_Equiv__Relations_Ocongruent @ G @ ( tc_fun @ D @ tc_HOL_Obool ) @ E @ C )
       => ( ( ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ G @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ C )
            = ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ G @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ A ) @ C ) )
         => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_fun @ G @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Equiv__Relations_Oquotient @ G ) @ F ) @ E ) ) )
           => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_fun @ G @ tc_HOL_Obool ) ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Equiv__Relations_Oquotient @ G ) @ F ) @ E ) ) )
             => ( ! [H: $i] :
                    ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ G ) @ H ) @ F ) )
                   => ! [I: $i] :
                        ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ G ) @ I ) @ F ) )
                       => ( ( ( hAPP @ C @ H )
                            = ( hAPP @ C @ I ) )
                         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ G @ G ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ G @ G ) @ H ) @ I ) ) @ E ) ) ) ) )
               => ( B = A ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_UN__equiv__class__inject) ).

thf(4311,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( c_Equiv__Relations_Oequiv @ G @ F @ E )
     => ( ( c_Equiv__Relations_Ocongruent @ G @ ( tc_fun @ D @ tc_HOL_Obool ) @ E @ C )
       => ( ( ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ G @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ C )
            = ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ G @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ A ) @ C ) )
         => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_fun @ G @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Equiv__Relations_Oquotient @ G ) @ F ) @ E ) ) )
           => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_fun @ G @ tc_HOL_Obool ) ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Equiv__Relations_Oquotient @ G ) @ F ) @ E ) ) )
             => ( ! [H: $i] :
                    ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ G ) @ H ) @ F ) )
                   => ! [I: $i] :
                        ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ G ) @ I ) @ F ) )
                       => ( ( ( hAPP @ C @ H )
                            = ( hAPP @ C @ I ) )
                         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ G @ G ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ G @ G ) @ H ) @ I ) ) @ E ) ) ) ) )
               => ( B = A ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[531]) ).

thf(415,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( c_Complete__Lattice_OSup__class_OSup @ ( tc_fun @ C @ tc_HOL_Obool ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Int__Union2) ).

thf(3905,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( c_Complete__Lattice_OSup__class_OSup @ ( tc_fun @ C @ tc_HOL_Obool ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[415]) ).

thf(1011,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( c_Set_Oimage @ E @ D @ ( hAPP @ ( hAPP @ ( c_COMBB @ C @ D @ E ) @ B ) @ A ) ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ E @ tc_HOL_Obool ) ) )
      = ( hAPP @ ( c_Set_Oimage @ C @ D @ B ) @ ( hAPP @ ( c_Set_Oimage @ E @ C @ A ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ E @ tc_HOL_Obool ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_range__composition) ).

thf(6106,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( c_Set_Oimage @ E @ D @ ( hAPP @ ( hAPP @ ( c_COMBB @ C @ D @ E ) @ B ) @ A ) ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ E @ tc_HOL_Obool ) ) )
      = ( hAPP @ ( c_Set_Oimage @ C @ D @ B ) @ ( hAPP @ ( c_Set_Oimage @ E @ C @ A ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ E @ tc_HOL_Obool ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1011]) ).

thf(587,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Omap @ tc_Nat_Onat @ B ) @ ( hAPP @ ( c_List_Onth @ B ) @ A ) ) @ ( c_List_Oupt @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ B ) ) @ A ) ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_map__nth) ).

thf(4515,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Omap @ tc_Nat_Onat @ B ) @ ( hAPP @ ( c_List_Onth @ B ) @ A ) ) @ ( c_List_Oupt @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ B ) ) @ A ) ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[587]) ).

thf(746,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ B ) @ A ) )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ B ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ B ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) )
        = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Diff__insert__absorb) ).

thf(5069,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ B ) @ A ) )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ B ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ B ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[746]) ).

thf(2333,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ C ) @ B ) @ A ) )
    <=> ( ( C = A )
        | ( hBOOL @ ( hAPP @ B @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_insert__code) ).

thf(11571,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ C ) @ B ) @ A ) )
       => ( ( C = A )
          | ( hBOOL @ ( hAPP @ B @ A ) ) ) )
      & ( ( ( C = A )
          | ( hBOOL @ ( hAPP @ B @ A ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ C ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2333]) ).

thf(1836,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Com_Ocom_OLocal @ C @ B @ A )
     != c_Com_Ocom_OSKIP ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_com_Osimps_I11_J) ).

thf(9604,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Com_Ocom_OLocal @ C @ B @ A )
     != c_Com_Ocom_OSKIP ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1836]) ).

thf(105,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( c_List_Ofoldl @ ( tc_List_Olist @ C ) @ ( tc_List_Olist @ C ) @ ( c_List_Oappend @ C ) @ B ) @ A )
      = ( hAPP @ ( hAPP @ ( c_List_Oappend @ C ) @ B ) @ ( c_List_Oconcat @ C @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_foldl__conv__concat) ).

thf(2776,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( c_List_Ofoldl @ ( tc_List_Olist @ C ) @ ( tc_List_Olist @ C ) @ ( c_List_Oappend @ C ) @ B ) @ A )
      = ( hAPP @ ( hAPP @ ( c_List_Oappend @ C ) @ B ) @ ( c_List_Oconcat @ C @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[105]) ).

thf(71,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Ocomm__monoid__add @ D )
     => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ tc_Nat_Onat @ D ) @ C ) @ ( hAPP @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Nat_Onat @ ( hAPP @ c_Nat_OSuc @ B ) ) @ ( hAPP @ c_Nat_OSuc @ A ) ) )
        = ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ tc_Nat_Onat @ D ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_Nat_Onat @ D @ tc_Nat_Onat ) @ C ) @ c_Nat_OSuc ) ) @ ( hAPP @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Nat_Onat @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_setsum__shift__bounds__Suc__ivl) ).

thf(2659,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Ocomm__monoid__add @ D )
     => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ tc_Nat_Onat @ D ) @ C ) @ ( hAPP @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Nat_Onat @ ( hAPP @ c_Nat_OSuc @ B ) ) @ ( hAPP @ c_Nat_OSuc @ A ) ) )
        = ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ tc_Nat_Onat @ D ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_Nat_Onat @ D @ tc_Nat_Onat ) @ C ) @ c_Nat_OSuc ) ) @ ( hAPP @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Nat_Onat @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[71]) ).

thf(1998,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ D ) @ C ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ D ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_add__le__mono) ).

thf(10277,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ D ) @ C ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ D ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1998]) ).

thf(1550,axiom,
    ! [A: $i,B: $i] :
      ( ( c_List_OtakeWhile @ B @ A @ ( c_List_Olist_ONil @ B ) )
      = ( c_List_Olist_ONil @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_takeWhile_Osimps_I1_J) ).

thf(8402,plain,
    ! [A: $i,B: $i] :
      ( ( c_List_OtakeWhile @ B @ A @ ( c_List_Olist_ONil @ B ) )
      = ( c_List_Olist_ONil @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1550]) ).

thf(2175,axiom,
    ! [A: $i] : ( c_Equiv__Relations_Oequivp @ A @ c_fequal ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_identity__equivp) ).

thf(10912,plain,
    ! [A: $i] : ( c_Equiv__Relations_Oequivp @ A @ c_fequal ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2175]) ).

thf(1870,axiom,
    ! [A: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_dvd__1__left) ).

thf(9769,plain,
    ! [A: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1870]) ).

thf(207,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( c_Set_OCollect @ D ) @ A ) ) ) )
    <=> ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ B ) )
        & ( hBOOL @ ( hAPP @ A @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Int__Collect) ).

thf(3153,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( c_Set_OCollect @ D ) @ A ) ) ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ B ) )
          & ( hBOOL @ ( hAPP @ A @ C ) ) ) )
      & ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ B ) )
          & ( hBOOL @ ( hAPP @ A @ C ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( c_Set_OCollect @ D ) @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[207]) ).

thf(1292,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ C ) @ A ) ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__imp__diff__less) ).

thf(7322,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ C ) @ A ) ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1292]) ).

thf(1857,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ c_Com_OWT @ ( c_Com_Ocom_OSemi @ B @ A ) ) )
    <=> ( ( hBOOL @ ( hAPP @ c_Com_OWT @ B ) )
        & ( hBOOL @ ( hAPP @ c_Com_OWT @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_WT_Oequations_I4_J) ).

thf(9704,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ c_Com_OWT @ ( c_Com_Ocom_OSemi @ B @ A ) ) )
       => ( ( hBOOL @ ( hAPP @ c_Com_OWT @ B ) )
          & ( hBOOL @ ( hAPP @ c_Com_OWT @ A ) ) ) )
      & ( ( ( hBOOL @ ( hAPP @ c_Com_OWT @ B ) )
          & ( hBOOL @ ( hAPP @ c_Com_OWT @ A ) ) )
       => ( hBOOL @ ( hAPP @ c_Com_OWT @ ( c_Com_Ocom_OSemi @ B @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1857]) ).

thf(54,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Groups_Ocomm__monoid__mult @ E )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ C ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ B ) )
         => ( ! [F: $i] :
                ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ F ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) ) )
               => ( ( hAPP @ A @ F )
                  = ( c_Groups_Oone__class_Oone @ E ) ) )
           => ( ( hAPP @ ( hAPP @ ( hAPP @ ( c_Finite__Set_Ofold__image @ E @ D @ ( c_Groups_Otimes__class_Otimes @ E ) ) @ A ) @ ( c_Groups_Oone__class_Oone @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) )
              = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Finite__Set_Ofold__image @ E @ D @ ( c_Groups_Otimes__class_Otimes @ E ) ) @ A ) @ ( c_Groups_Oone__class_Oone @ E ) ) @ C ) ) @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Finite__Set_Ofold__image @ E @ D @ ( c_Groups_Otimes__class_Otimes @ E ) ) @ A ) @ ( c_Groups_Oone__class_Oone @ E ) ) @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_fold__image__Un__one) ).

thf(2590,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Groups_Ocomm__monoid__mult @ E )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ C ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ B ) )
         => ( ! [F: $i] :
                ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ F ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) ) )
               => ( ( hAPP @ A @ F )
                  = ( c_Groups_Oone__class_Oone @ E ) ) )
           => ( ( hAPP @ ( hAPP @ ( hAPP @ ( c_Finite__Set_Ofold__image @ E @ D @ ( c_Groups_Otimes__class_Otimes @ E ) ) @ A ) @ ( c_Groups_Oone__class_Oone @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) )
              = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Finite__Set_Ofold__image @ E @ D @ ( c_Groups_Otimes__class_Otimes @ E ) ) @ A ) @ ( c_Groups_Oone__class_Oone @ E ) ) @ C ) ) @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Finite__Set_Ofold__image @ E @ D @ ( c_Groups_Otimes__class_Otimes @ E ) ) @ A ) @ ( c_Groups_Oone__class_Oone @ E ) ) @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[54]) ).

thf(800,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( c_Finite__Set_Ofold__image @ D @ C @ B ) @ A )
      = ( c_Finite__Set_Ofold @ C @ D @ ( hAPP @ ( hAPP @ ( c_COMBB @ D @ ( tc_fun @ D @ D ) @ C ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_fold__image__def) ).

thf(5297,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( c_Finite__Set_Ofold__image @ D @ C @ B ) @ A )
      = ( c_Finite__Set_Ofold @ C @ D @ ( hAPP @ ( hAPP @ ( c_COMBB @ D @ ( tc_fun @ D @ D ) @ C ) @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[800]) ).

thf(1603,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ B ) @ ( hAPP @ c_Nat_OSuc @ A ) ) ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_diff__Suc__less) ).

thf(8682,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ B ) @ ( hAPP @ c_Nat_OSuc @ A ) ) ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1603]) ).

thf(637,axiom,
    ! [A: $i,B: $i,C: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) ) @ ( c_Transitive__Closure_Ortrancl @ C @ B ) ) @ ( c_Transitive__Closure_Ortrancl @ C @ A ) ) ) @ ( c_Transitive__Closure_Ortrancl @ C @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) ) @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_rtrancl__Un__subset) ).

thf(4697,plain,
    ! [A: $i,B: $i,C: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) ) @ ( c_Transitive__Closure_Ortrancl @ C @ B ) ) @ ( c_Transitive__Closure_Ortrancl @ C @ A ) ) ) @ ( c_Transitive__Closure_Ortrancl @ C @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[637]) ).

thf(1164,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Transitive__Closure_Ortrancl @ B @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) ) @ A ) @ ( c_Relation_OId @ B ) ) )
      = ( c_Transitive__Closure_Ortrancl @ B @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_rtrancl__r__diff__Id) ).

thf(6796,plain,
    ! [A: $i,B: $i] :
      ( ( c_Transitive__Closure_Ortrancl @ B @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) ) @ A ) @ ( c_Relation_OId @ B ) ) )
      = ( c_Transitive__Closure_Ortrancl @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1164]) ).

thf(2119,axiom,
    ! [A: $i] : ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ tc_Nat_Onat ) @ ( hAPP @ ( c_SetInterval_Oord__class_OlessThan @ tc_Nat_Onat ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_finite__lessThan) ).

thf(10711,plain,
    ! [A: $i] : ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ tc_Nat_Onat ) @ ( hAPP @ ( c_SetInterval_Oord__class_OlessThan @ tc_Nat_Onat ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2119]) ).

thf(961,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Relation_Oinv__image @ D @ C ) @ B ) @ A )
      = ( hAPP @ ( c_Set_OCollect @ ( tc_prod @ C @ C ) ) @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ C @ C @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ C @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ C ) @ ( c_COMBC @ C @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ ( tc_prod @ D @ D ) ) @ ( tc_fun @ C @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( c_COMBB @ ( tc_prod @ D @ D ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ C ) @ ( c_member @ ( tc_prod @ D @ D ) ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ C @ ( tc_fun @ C @ D ) @ ( tc_fun @ C @ ( tc_prod @ D @ D ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ D @ ( tc_prod @ D @ D ) ) @ ( tc_fun @ ( tc_fun @ C @ D ) @ ( tc_fun @ C @ ( tc_prod @ D @ D ) ) ) @ C ) @ ( c_COMBB @ D @ ( tc_prod @ D @ D ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ D @ ( tc_fun @ D @ ( tc_prod @ D @ D ) ) @ C ) @ ( c_Product__Type_OPair @ D @ D ) ) @ A ) ) ) @ A ) ) ) ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_inv__image__def) ).

thf(5878,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Relation_Oinv__image @ D @ C ) @ B ) @ A )
      = ( hAPP @ ( c_Set_OCollect @ ( tc_prod @ C @ C ) ) @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ C @ C @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ C @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ C ) @ ( c_COMBC @ C @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ ( tc_prod @ D @ D ) ) @ ( tc_fun @ C @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( c_COMBB @ ( tc_prod @ D @ D ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ C ) @ ( c_member @ ( tc_prod @ D @ D ) ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ C @ ( tc_fun @ C @ D ) @ ( tc_fun @ C @ ( tc_prod @ D @ D ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ D @ ( tc_prod @ D @ D ) ) @ ( tc_fun @ ( tc_fun @ C @ D ) @ ( tc_fun @ C @ ( tc_prod @ D @ D ) ) ) @ C ) @ ( c_COMBB @ D @ ( tc_prod @ D @ D ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ D @ ( tc_fun @ D @ ( tc_prod @ D @ D ) ) @ C ) @ ( c_Product__Type_OPair @ D @ D ) ) @ A ) ) ) @ A ) ) ) ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[961]) ).

thf(190,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( c_Finite__Set_Ofolding__idem @ F @ E @ D @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ F ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ F ) @ A ) )
         => ( ( hAPP @ C @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ F @ tc_HOL_Obool ) ) @ B ) @ A ) )
            = ( hAPP @ ( c_Fun_Ocomp @ E @ E @ E @ ( hAPP @ C @ B ) ) @ ( hAPP @ C @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_folding__idem_Ounion__idem) ).

thf(3098,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( c_Finite__Set_Ofolding__idem @ F @ E @ D @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ F ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ F ) @ A ) )
         => ( ( hAPP @ C @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ F @ tc_HOL_Obool ) ) @ B ) @ A ) )
            = ( hAPP @ ( c_Fun_Ocomp @ E @ E @ E @ ( hAPP @ C @ B ) ) @ ( hAPP @ C @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[190]) ).

thf(2117,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ ( tc_prod @ C @ B ) ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ ( hAPP @ ( c_Relation_ODomain @ C @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_finite__Domain) ).

thf(10705,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ ( tc_prod @ C @ B ) ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ ( hAPP @ ( c_Relation_ODomain @ C @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2117]) ).

thf(1994,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ A ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ A ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__le__mono2) ).

thf(10262,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ A ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ A ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1994]) ).

thf(1064,axiom,
    ! [A: $i,B: $i,C: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ C @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ C @ C ) @ B ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Nat_Ocompow @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_rel__pow__0__I) ).

thf(6323,plain,
    ! [A: $i,B: $i,C: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ C @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ C @ C ) @ B ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Nat_Ocompow @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1064]) ).

thf(1275,axiom,
    ! [A: $i] :
      ( ( c_List_Obutlast @ A @ ( c_List_Olist_ONil @ A ) )
      = ( c_List_Olist_ONil @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_butlast_Osimps_I1_J) ).

thf(7253,plain,
    ! [A: $i] :
      ( ( c_List_Obutlast @ A @ ( c_List_Olist_ONil @ A ) )
      = ( c_List_Olist_ONil @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1275]) ).

thf(1561,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( c_Finite__Set_Ofun__left__comm @ F @ E @ D )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ F ) @ C ) )
       => ( ( hAPP @ ( hAPP @ ( c_Finite__Set_Ofold @ F @ E @ D ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ F ) @ A ) @ C ) )
          = ( hAPP @ ( hAPP @ D @ A ) @ ( hAPP @ ( hAPP @ ( c_Finite__Set_Ofold @ F @ E @ D ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ F @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ F ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ F @ tc_HOL_Obool ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_fun__left__comm_Ofold__insert__remove) ).

thf(8447,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( c_Finite__Set_Ofun__left__comm @ F @ E @ D )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ F ) @ C ) )
       => ( ( hAPP @ ( hAPP @ ( c_Finite__Set_Ofold @ F @ E @ D ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ F ) @ A ) @ C ) )
          = ( hAPP @ ( hAPP @ D @ A ) @ ( hAPP @ ( hAPP @ ( c_Finite__Set_Ofold @ F @ E @ D ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ F @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ F ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ F @ tc_HOL_Obool ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1561]) ).

thf(194,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ C @ B ) @ A ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ B @ tc_HOL_Obool ) @ C ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) )
      = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ ( tc_prod @ C @ B ) @ tc_HOL_Obool ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Sigma__empty2) ).

thf(3108,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ C @ B ) @ A ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ B @ tc_HOL_Obool ) @ C ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) )
      = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ ( tc_prod @ C @ B ) @ tc_HOL_Obool ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[194]) ).

thf(829,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ E ) @ ( tc_List_Olist @ E ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ E ) @ ( tc_List_Olist @ E ) ) @ D ) @ C ) ) @ ( c_List_Olexord @ E @ B ) ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ E ) @ ( tc_List_Olist @ E ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ E ) @ ( tc_List_Olist @ E ) ) @ ( hAPP @ ( hAPP @ ( c_List_Oappend @ E ) @ A ) @ D ) ) @ ( hAPP @ ( hAPP @ ( c_List_Oappend @ E ) @ A ) @ C ) ) ) @ ( c_List_Olexord @ E @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_lexord__append__leftI) ).

thf(5405,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ E ) @ ( tc_List_Olist @ E ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ E ) @ ( tc_List_Olist @ E ) ) @ D ) @ C ) ) @ ( c_List_Olexord @ E @ B ) ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ E ) @ ( tc_List_Olist @ E ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ E ) @ ( tc_List_Olist @ E ) ) @ ( hAPP @ ( hAPP @ ( c_List_Oappend @ E ) @ A ) @ D ) ) @ ( hAPP @ ( hAPP @ ( c_List_Oappend @ E ) @ A ) @ C ) ) ) @ ( c_List_Olexord @ E @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[829]) ).

thf(120,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) @ C ) @ B ) ) @ ( c_List_Olistrel @ D @ A ) ) )
     => ( ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ D ) ) @ C )
        = ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ D ) ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_listrel__eq__len) ).

thf(2837,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) @ C ) @ B ) ) @ ( c_List_Olistrel @ D @ A ) ) )
     => ( ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ D ) ) @ C )
        = ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ D ) ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[120]) ).

thf(1403,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ F ) @ ( tc_List_Olist @ F ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ F ) @ ( tc_List_Olist @ F ) ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ F ) @ E ) @ D ) ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ F ) @ C ) @ B ) ) ) @ ( c_List_Olistrel1 @ F @ A ) ) )
    <=> ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ F @ F ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ F @ F ) @ E ) @ C ) ) @ A ) )
          & ( D = B ) )
        | ( ( E = C )
          & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ F ) @ ( tc_List_Olist @ F ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ F ) @ ( tc_List_Olist @ F ) ) @ D ) @ B ) ) @ ( c_List_Olistrel1 @ F @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Cons__listrel1__Cons) ).

thf(7781,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ F ) @ ( tc_List_Olist @ F ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ F ) @ ( tc_List_Olist @ F ) ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ F ) @ E ) @ D ) ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ F ) @ C ) @ B ) ) ) @ ( c_List_Olistrel1 @ F @ A ) ) )
       => ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ F @ F ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ F @ F ) @ E ) @ C ) ) @ A ) )
            & ( D = B ) )
          | ( ( E = C )
            & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ F ) @ ( tc_List_Olist @ F ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ F ) @ ( tc_List_Olist @ F ) ) @ D ) @ B ) ) @ ( c_List_Olistrel1 @ F @ A ) ) ) ) ) )
      & ( ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ F @ F ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ F @ F ) @ E ) @ C ) ) @ A ) )
            & ( D = B ) )
          | ( ( E = C )
            & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ F ) @ ( tc_List_Olist @ F ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ F ) @ ( tc_List_Olist @ F ) ) @ D ) @ B ) ) @ ( c_List_Olistrel1 @ F @ A ) ) ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ F ) @ ( tc_List_Olist @ F ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ F ) @ ( tc_List_Olist @ F ) ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ F ) @ E ) @ D ) ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ F ) @ C ) @ B ) ) ) @ ( c_List_Olistrel1 @ F @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1403]) ).

thf(126,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ tc_Nat_Onat ) @ C ) @ B )
        = ( hAPP @ c_Nat_OSuc @ A ) )
     => ? [E: $i] :
          ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ E ) @ B ) )
          & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ ( hAPP @ C @ E ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_setsum__SucD) ).

thf(2854,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ tc_Nat_Onat ) @ C ) @ B )
        = ( hAPP @ c_Nat_OSuc @ A ) )
     => ? [E: $i] :
          ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ E ) @ B ) )
          & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ ( hAPP @ C @ E ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[126]) ).

thf(786,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ! [F: $i] :
          ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ F ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ D @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ C ) @ B ) ) )
         => ( hBOOL @ ( hAPP @ A @ F ) ) )
    <=> ! [F: $i] :
          ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ F ) @ C ) )
         => ! [G: $i] :
              ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ G ) @ ( hAPP @ B @ F ) ) )
             => ( hBOOL @ ( hAPP @ A @ G ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_ball__UN) ).

thf(5237,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ! [F: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ F ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ D @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ C ) @ B ) ) )
           => ( hBOOL @ ( hAPP @ A @ F ) ) )
       => ! [F: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ F ) @ C ) )
           => ! [G: $i] :
                ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ G ) @ ( hAPP @ B @ F ) ) )
               => ( hBOOL @ ( hAPP @ A @ G ) ) ) ) )
      & ( ! [F: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ F ) @ C ) )
           => ! [G: $i] :
                ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ G ) @ ( hAPP @ B @ F ) ) )
               => ( hBOOL @ ( hAPP @ A @ G ) ) ) )
       => ! [F: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ F ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ D @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ C ) @ B ) ) )
           => ( hBOOL @ ( hAPP @ A @ F ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[786]) ).

thf(995,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) )
    <=> ( A
        = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_subset__empty) ).

thf(6031,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) )
       => ( A
          = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) )
      & ( ( A
          = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[995]) ).

thf(871,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Onth @ D ) @ ( hAPP @ ( hAPP @ ( c_List_Oappend @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ D ) @ B ) @ A ) ) ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ D ) ) @ C ) )
      = B ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nth__append__length) ).

thf(5529,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Onth @ D ) @ ( hAPP @ ( hAPP @ ( c_List_Oappend @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ D ) @ B ) @ A ) ) ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ D ) ) @ C ) )
      = B ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[871]) ).

thf(1264,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) )
    <=> ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) )
        | ( B = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_subset__iff__psubset__eq) ).

thf(7208,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) )
          | ( B = A ) ) )
      & ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) )
          | ( B = A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1264]) ).

thf(1715,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ A ) ) @ B ) )
      <=> ( A
          = ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_dvd__mult__cancel1) ).

thf(9152,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ B ) )
     => ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ A ) ) @ B ) )
         => ( A
            = ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) )
        & ( ( A
            = ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ A ) ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1715]) ).

thf(1540,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Otake @ B ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ A )
      = ( c_List_Olist_ONil @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_take__0) ).

thf(8366,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Otake @ B ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ A )
      = ( c_List_Olist_ONil @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1540]) ).

thf(894,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Int__Un__distrib2) ).

thf(5614,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[894]) ).

thf(1886,axiom,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) )
        & ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ A ) @ B ) ) )
     => ~ ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ A ) @ B ) )
          & ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_dvd_Oless__imp__not__less) ).

thf(9818,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) )
        & ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ A ) @ B ) ) )
     => ~ ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ A ) @ B ) )
          & ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1886]) ).

thf(1209,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ C ) )
       => ( ( c_List_Osublist @ B @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ B ) @ A ) @ ( c_List_Olist_ONil @ B ) ) @ C )
          = ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ B ) @ A ) @ ( c_List_Olist_ONil @ B ) ) ) )
      & ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ C ) )
       => ( ( c_List_Osublist @ B @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ B ) @ A ) @ ( c_List_Olist_ONil @ B ) ) @ C )
          = ( c_List_Olist_ONil @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_sublist__singleton) ).

thf(6971,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ C ) )
       => ( ( c_List_Osublist @ B @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ B ) @ A ) @ ( c_List_Olist_ONil @ B ) ) @ C )
          = ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ B ) @ A ) @ ( c_List_Olist_ONil @ B ) ) ) )
      & ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ C ) )
       => ( ( c_List_Osublist @ B @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ B ) @ A ) @ ( c_List_Olist_ONil @ B ) ) @ C )
          = ( c_List_Olist_ONil @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1209]) ).

thf(348,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ B ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) )
     => ( ( ( hAPP @ ( c_Set_Oimage @ B @ B @ A ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ B @ tc_HOL_Obool ) ) )
          = ( c_Orderings_Otop__class_Otop @ ( tc_fun @ B @ tc_HOL_Obool ) ) )
       => ( c_Fun_Oinj__on @ B @ B @ A @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_finite__UNIV__surj__inj) ).

thf(3648,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ B ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) )
     => ( ( ( hAPP @ ( c_Set_Oimage @ B @ B @ A ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ B @ tc_HOL_Obool ) ) )
          = ( c_Orderings_Otop__class_Otop @ ( tc_fun @ B @ tc_HOL_Obool ) ) )
       => ( c_Fun_Oinj__on @ B @ B @ A @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[348]) ).

thf(1499,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_List_Odistinct @ D ) @ ( hAPP @ ( hAPP @ ( c_List_Omap @ ( tc_prod @ D @ C ) @ D ) @ ( c_Product__Type_Ofst @ D @ C ) ) @ B ) ) )
     => ( ( hBOOL @ ( hAPP @ ( c_List_Odistinct @ D ) @ ( hAPP @ ( hAPP @ ( c_List_Omap @ ( tc_prod @ D @ C ) @ D ) @ ( c_Product__Type_Ofst @ D @ C ) ) @ A ) ) )
       => ( ( ( c_Map_Omap__of @ D @ C @ B )
            = ( c_Map_Omap__of @ D @ C @ A ) )
        <=> ( ( hAPP @ ( c_List_Oset @ ( tc_prod @ D @ C ) ) @ B )
            = ( hAPP @ ( c_List_Oset @ ( tc_prod @ D @ C ) ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_map__of__inject__set) ).

thf(8204,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_List_Odistinct @ D ) @ ( hAPP @ ( hAPP @ ( c_List_Omap @ ( tc_prod @ D @ C ) @ D ) @ ( c_Product__Type_Ofst @ D @ C ) ) @ B ) ) )
     => ( ( hBOOL @ ( hAPP @ ( c_List_Odistinct @ D ) @ ( hAPP @ ( hAPP @ ( c_List_Omap @ ( tc_prod @ D @ C ) @ D ) @ ( c_Product__Type_Ofst @ D @ C ) ) @ A ) ) )
       => ( ( ( ( c_Map_Omap__of @ D @ C @ B )
              = ( c_Map_Omap__of @ D @ C @ A ) )
           => ( ( hAPP @ ( c_List_Oset @ ( tc_prod @ D @ C ) ) @ B )
              = ( hAPP @ ( c_List_Oset @ ( tc_prod @ D @ C ) ) @ A ) ) )
          & ( ( ( hAPP @ ( c_List_Oset @ ( tc_prod @ D @ C ) ) @ B )
              = ( hAPP @ ( c_List_Oset @ ( tc_prod @ D @ C ) ) @ A ) )
           => ( ( c_Map_Omap__of @ D @ C @ B )
              = ( c_Map_Omap__of @ D @ C @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1499]) ).

thf(595,axiom,
    ! [A: $i] :
      ( ( c_Orderings_Otop__class_Otop @ ( tc_fun @ A @ tc_HOL_Obool ) )
      = ( hAPP @ ( c_Set_OCollect @ A ) @ ( hAPP @ ( c_COMBK @ tc_HOL_Obool @ A ) @ c_fTrue ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_UNIV__def) ).

thf(4543,plain,
    ! [A: $i] :
      ( ( c_Orderings_Otop__class_Otop @ ( tc_fun @ A @ tc_HOL_Obool ) )
      = ( hAPP @ ( c_Set_OCollect @ A ) @ ( hAPP @ ( c_COMBK @ tc_HOL_Obool @ A ) @ c_fTrue ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[595]) ).

thf(1101,axiom,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ A ) )
     => ( A
        = ( hAPP @ c_Nat_OSuc @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ A ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Suc__pred_H) ).

thf(6522,plain,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ A ) )
     => ( A
        = ( hAPP @ c_Nat_OSuc @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ A ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1101]) ).

thf(835,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( c_Relation_ODomain @ B @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ ( tc_prod @ B @ A ) @ tc_HOL_Obool ) ) )
      = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Domain__empty) ).

thf(5425,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( c_Relation_ODomain @ B @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ ( tc_prod @ B @ A ) @ tc_HOL_Obool ) ) )
      = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[835]) ).

thf(876,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) )
     => ( ( c_SetInterval_Oord_OgreaterThanAtMost @ tc_Nat_Onat @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ tc_Nat_Onat @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ tc_Nat_Onat @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) @ tc_Nat_Onat ) @ ( c_COMBS @ tc_Nat_Onat @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ tc_Nat_Onat @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ tc_Nat_Onat ) @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ tc_Nat_Onat ) @ c_fconj ) ) @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ tc_Nat_Onat ) @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ tc_HOL_Obool @ tc_Nat_Onat ) @ c_fNot ) ) @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) ) ) ) @ A @ B )
        = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_dvd_OgreaterThanAtMost__empty) ).

thf(5550,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) )
     => ( ( c_SetInterval_Oord_OgreaterThanAtMost @ tc_Nat_Onat @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ tc_Nat_Onat @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ tc_Nat_Onat @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) @ tc_Nat_Onat ) @ ( c_COMBS @ tc_Nat_Onat @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ tc_Nat_Onat @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ tc_Nat_Onat ) @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ tc_Nat_Onat ) @ c_fconj ) ) @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ tc_Nat_Onat ) @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ tc_HOL_Obool @ tc_Nat_Onat ) @ c_fNot ) ) @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) ) ) ) @ A @ B )
        = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[876]) ).

thf(511,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ D ) @ ( c_COMBC @ C @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ ( tc_prod @ D @ C ) ) @ ( tc_fun @ C @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ D ) @ ( hAPP @ ( c_COMBB @ ( tc_prod @ D @ C ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ C ) @ ( c_member @ ( tc_prod @ D @ C ) ) ) ) @ ( c_Product__Type_OPair @ D @ C ) ) ) ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ D ) @ ( c_COMBC @ C @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ ( tc_prod @ D @ C ) ) @ ( tc_fun @ C @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ D ) @ ( hAPP @ ( c_COMBB @ ( tc_prod @ D @ C ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ C ) @ ( c_member @ ( tc_prod @ D @ C ) ) ) ) @ ( c_Product__Type_OPair @ D @ C ) ) ) ) @ A ) ) )
    <=> ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_pred__subset__eq2) ).

thf(4234,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ D ) @ ( c_COMBC @ C @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ ( tc_prod @ D @ C ) ) @ ( tc_fun @ C @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ D ) @ ( hAPP @ ( c_COMBB @ ( tc_prod @ D @ C ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ C ) @ ( c_member @ ( tc_prod @ D @ C ) ) ) ) @ ( c_Product__Type_OPair @ D @ C ) ) ) ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ D ) @ ( c_COMBC @ C @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ ( tc_prod @ D @ C ) ) @ ( tc_fun @ C @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ D ) @ ( hAPP @ ( c_COMBB @ ( tc_prod @ D @ C ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ C ) @ ( c_member @ ( tc_prod @ D @ C ) ) ) ) @ ( c_Product__Type_OPair @ D @ C ) ) ) ) @ A ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) ) @ B ) @ A ) ) )
      & ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) ) @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ D ) @ ( c_COMBC @ C @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ ( tc_prod @ D @ C ) ) @ ( tc_fun @ C @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ D ) @ ( hAPP @ ( c_COMBB @ ( tc_prod @ D @ C ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ C ) @ ( c_member @ ( tc_prod @ D @ C ) ) ) ) @ ( c_Product__Type_OPair @ D @ C ) ) ) ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ D ) @ ( c_COMBC @ C @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ ( tc_prod @ D @ C ) ) @ ( tc_fun @ C @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ D ) @ ( hAPP @ ( c_COMBB @ ( tc_prod @ D @ C ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ C ) @ ( c_member @ ( tc_prod @ D @ C ) ) ) ) @ ( c_Product__Type_OPair @ D @ C ) ) ) ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[511]) ).

thf(735,axiom,
    ! [A: $i,B: $i] :
      ( ( c_List_Opartition @ B @ A @ ( c_List_Olist_ONil @ B ) )
      = ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ B ) @ ( tc_List_Olist @ B ) ) @ ( c_List_Olist_ONil @ B ) ) @ ( c_List_Olist_ONil @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_partition_Osimps_I1_J) ).

thf(5037,plain,
    ! [A: $i,B: $i] :
      ( ( c_List_Opartition @ B @ A @ ( c_List_Olist_ONil @ B ) )
      = ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ B ) @ ( tc_List_Olist @ B ) ) @ ( c_List_Olist_ONil @ B ) ) @ ( c_List_Olist_ONil @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[735]) ).

thf(1493,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( c_Map_Omap__of @ C @ B @ ( c_List_Olist_ONil @ ( tc_prod @ C @ B ) ) ) @ A )
      = ( c_Option_Ooption_ONone @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_map__of__Cons__code_I1_J) ).

thf(8182,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( c_Map_Omap__of @ C @ B @ ( c_List_Olist_ONil @ ( tc_prod @ C @ B ) ) ) @ A )
      = ( c_Option_Ooption_ONone @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1493]) ).

thf(1846,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ D ) @ ( c_Map_Oran @ C @ E @ ( c_Map_Orestrict__map @ C @ E @ B @ A ) ) ) )
     => ? [F: $i] :
          ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ F ) @ A ) )
          & ( ( hAPP @ B @ F )
            = ( hAPP @ ( c_Option_Ooption_OSome @ E ) @ D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_ran__restrictD) ).

thf(9644,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ D ) @ ( c_Map_Oran @ C @ E @ ( c_Map_Orestrict__map @ C @ E @ B @ A ) ) ) )
     => ? [F: $i] :
          ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ F ) @ A ) )
          & ( ( hAPP @ B @ F )
            = ( hAPP @ ( c_Option_Ooption_OSome @ E ) @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1846]) ).

thf(556,axiom,
    ! [A: $i,B: $i] :
      ( ( ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ B @ B @ B ) @ A ) @ A )
        = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) ) )
     => ( c_Wellfounded_Owf @ B @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_wf__no__loop) ).

thf(4403,plain,
    ! [A: $i,B: $i] :
      ( ( ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ B @ B @ B ) @ A ) @ A )
        = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) ) )
     => ( c_Wellfounded_Owf @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[556]) ).

thf(291,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ F @ E ) @ D ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ C @ E ) @ D ) ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ B @ ( tc_fun @ A @ tc_HOL_Obool ) ) ) @ C ) @ F ) @ E ) @ D ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_sup2CI) ).

thf(3439,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ F @ E ) @ D ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ C @ E ) @ D ) ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ B @ ( tc_fun @ A @ tc_HOL_Obool ) ) ) @ C ) @ F ) @ E ) @ D ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[291]) ).

thf(198,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Finite__Set_Ofolding__one__idem @ E @ D @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ E ) @ B ) )
       => ( ( A
           != ( c_Orderings_Obot__class_Obot @ ( tc_fun @ E @ tc_HOL_Obool ) ) )
         => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ A ) @ B ) )
           => ( ( hAPP @ ( hAPP @ D @ ( hAPP @ C @ A ) ) @ ( hAPP @ C @ B ) )
              = ( hAPP @ C @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_folding__one__idem_Osubset__idem) ).

thf(3120,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Finite__Set_Ofolding__one__idem @ E @ D @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ E ) @ B ) )
       => ( ( A
           != ( c_Orderings_Obot__class_Obot @ ( tc_fun @ E @ tc_HOL_Obool ) ) )
         => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ A ) @ B ) )
           => ( ( hAPP @ ( hAPP @ D @ ( hAPP @ C @ A ) ) @ ( hAPP @ C @ B ) )
              = ( hAPP @ C @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[198]) ).

thf(289,axiom,
    ! [A: $i] :
      ( ( ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) )
        = ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) )
      & ( ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ A @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) )
        = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_DIVISION__BY__ZERO) ).

thf(3429,plain,
    ! [A: $i] :
      ( ( ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) )
        = ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) )
      & ( ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ A @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[289]) ).

thf(1262,axiom,
    ! [A: $i] :
      ( ( class_Orderings_Owellorder @ A )
     => ( c_Wellfounded_Owf @ A @ ( hAPP @ ( c_Set_OCollect @ ( tc_prod @ A @ A ) ) @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ A @ A @ tc_HOL_Obool ) @ ( c_Orderings_Oord__class_Oless @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_wf) ).

thf(7203,plain,
    ! [A: $i] :
      ( ( class_Orderings_Owellorder @ A )
     => ( c_Wellfounded_Owf @ A @ ( hAPP @ ( c_Set_OCollect @ ( tc_prod @ A @ A ) ) @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ A @ A @ tc_HOL_Obool ) @ ( c_Orderings_Oord__class_Oless @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1262]) ).

thf(2042,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i] :
          ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ B @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ B @ B ) @ C ) @ C ) ) @ ( c_Transitive__Closure_Otrancl @ B @ A ) ) )
     => ( c_Wellfounded_Oacyclic @ B @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_acyclicI) ).

thf(10433,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i] :
          ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ B @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ B @ B ) @ C ) @ C ) ) @ ( c_Transitive__Closure_Otrancl @ B @ A ) ) )
     => ( c_Wellfounded_Oacyclic @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2042]) ).

thf(1705,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Relation_ODomain @ D @ C ) @ B ) ) @ ( hAPP @ ( c_Relation_ODomain @ D @ C ) @ A ) ) ) @ ( hAPP @ ( c_Relation_ODomain @ D @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) ) @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Domain__Diff__subset) ).

thf(9127,plain,
    ! [A: $i,B: $i,C: $i,D: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Relation_ODomain @ D @ C ) @ B ) ) @ ( hAPP @ ( c_Relation_ODomain @ D @ C ) @ A ) ) ) @ ( hAPP @ ( c_Relation_ODomain @ D @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1705]) ).

thf(889,axiom,
    ( ( c_Orderings_Obot__class_Obot @ tc_Nat_Onat )
    = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_bot__nat__def) ).

thf(5589,plain,
    ( ( c_Orderings_Obot__class_Obot @ tc_Nat_Onat )
    = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[889]) ).

thf(80,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Ocomm__monoid__mult @ B )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ B ) @ ( c_Groups_Oone__class_Oone @ B ) ) @ A )
        = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__1) ).

thf(2686,plain,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Ocomm__monoid__mult @ B )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ B ) @ ( c_Groups_Oone__class_Oone @ B ) ) @ A )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[80]) ).

thf(604,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( c_SetInterval_Oord_OatLeastAtMost @ tc_Nat_Onat @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ C @ B )
        = ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ tc_Nat_Onat ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) ) )
    <=> ( ( C = B )
        & ( B = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_dvd_OatLeastAtMost__singleton__iff) ).

thf(4575,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( c_SetInterval_Oord_OatLeastAtMost @ tc_Nat_Onat @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ C @ B )
          = ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ tc_Nat_Onat ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) ) )
       => ( ( C = B )
          & ( B = A ) ) )
      & ( ( ( C = B )
          & ( B = A ) )
       => ( ( c_SetInterval_Oord_OatLeastAtMost @ tc_Nat_Onat @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ C @ B )
          = ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ tc_Nat_Onat ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[604]) ).

thf(2297,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Com_Ocom_OLocal @ E @ D @ C )
     != ( c_Com_Ocom_OWhile @ B @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_com_Osimps_I38_J) ).

thf(11399,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Com_Ocom_OLocal @ E @ D @ C )
     != ( c_Com_Ocom_OWhile @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2297]) ).

thf(476,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( hAPP @ ( c_Product__Type_Osnd @ G @ F ) @ ( hAPP @ ( c_Product__Type_Omap__pair @ E @ G @ D @ F @ C @ B ) @ A ) )
      = ( hAPP @ B @ ( hAPP @ ( c_Product__Type_Osnd @ E @ D ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_snd__prod__fun) ).

thf(4097,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( hAPP @ ( c_Product__Type_Osnd @ G @ F ) @ ( hAPP @ ( c_Product__Type_Omap__pair @ E @ G @ D @ F @ C @ B ) @ A ) )
      = ( hAPP @ B @ ( hAPP @ ( c_Product__Type_Osnd @ E @ D ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[476]) ).

thf(427,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_List_Olistrelp @ D @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ D @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ D ) @ ( c_COMBC @ D @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ D @ ( tc_prod @ D @ D ) ) @ ( tc_fun @ D @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ D ) @ ( hAPP @ ( c_COMBB @ ( tc_prod @ D @ D ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ D ) @ ( c_member @ ( tc_prod @ D @ D ) ) ) ) @ ( c_Product__Type_OPair @ D @ D ) ) ) ) @ C ) @ B @ A )
    <=> ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) @ B ) @ A ) ) @ ( c_List_Olistrel @ D @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_listrelp__listrel__eq) ).

thf(3942,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( c_List_Olistrelp @ D @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ D @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ D ) @ ( c_COMBC @ D @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ D @ ( tc_prod @ D @ D ) ) @ ( tc_fun @ D @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ D ) @ ( hAPP @ ( c_COMBB @ ( tc_prod @ D @ D ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ D ) @ ( c_member @ ( tc_prod @ D @ D ) ) ) ) @ ( c_Product__Type_OPair @ D @ D ) ) ) ) @ C ) @ B @ A )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) @ B ) @ A ) ) @ ( c_List_Olistrel @ D @ C ) ) ) )
      & ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) @ B ) @ A ) ) @ ( c_List_Olistrel @ D @ C ) ) )
       => ( c_List_Olistrelp @ D @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ D @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ D ) @ ( c_COMBC @ D @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ D @ ( tc_prod @ D @ D ) ) @ ( tc_fun @ D @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ D ) @ ( hAPP @ ( c_COMBB @ ( tc_prod @ D @ D ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ D ) @ ( c_member @ ( tc_prod @ D @ D ) ) ) ) @ ( c_Product__Type_OPair @ D @ D ) ) ) ) @ C ) @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[427]) ).

thf(2354,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Predicate_ODomainP @ D @ C @ B ) @ A ) )
    <=> ? [E: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ B @ A ) @ E ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_DomainP_Oequations) ).

thf(11664,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( c_Predicate_ODomainP @ D @ C @ B ) @ A ) )
       => ? [E: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ B @ A ) @ E ) ) )
      & ( ? [E: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ B @ A ) @ E ) )
       => ( hBOOL @ ( hAPP @ ( c_Predicate_ODomainP @ D @ C @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2354]) ).

thf(2157,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ ( hAPP @ ( c_Set_Oimage @ C @ D @ B ) @ A ) ) )
     => ( ( c_Fun_Oinj__on @ C @ D @ B @ A )
       => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_finite__imageD) ).

thf(10869,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ ( hAPP @ ( c_Set_Oimage @ C @ D @ B ) @ A ) ) )
     => ( ( c_Fun_Oinj__on @ C @ D @ B @ A )
       => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2157]) ).

thf(692,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( c_List_Oset @ C ) @ ( c_List_Osublist @ C @ B @ A ) )
      = ( hAPP @ ( c_Set_OCollect @ C ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ tc_HOL_Obool @ C ) @ ( c_HOL_OEx @ tc_Nat_Onat ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ C @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ tc_Nat_Onat @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) @ C ) @ ( c_COMBS @ tc_Nat_Onat @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ tc_Nat_Onat @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ tc_Nat_Onat ) @ c_fconj ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ C @ ( tc_fun @ tc_Nat_Onat @ C ) @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ tc_Nat_Onat @ C ) @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) @ C ) @ ( c_COMBB @ C @ tc_HOL_Obool @ tc_Nat_Onat ) ) @ c_fequal ) ) @ ( hAPP @ ( c_List_Onth @ C ) @ B ) ) ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ tc_Nat_Onat @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ tc_Nat_Onat ) @ c_fconj ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) @ B ) ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( c_member @ tc_Nat_Onat ) ) @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_set__sublist) ).

thf(4884,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( c_List_Oset @ C ) @ ( c_List_Osublist @ C @ B @ A ) )
      = ( hAPP @ ( c_Set_OCollect @ C ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ tc_HOL_Obool @ C ) @ ( c_HOL_OEx @ tc_Nat_Onat ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ C @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ tc_Nat_Onat @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) @ C ) @ ( c_COMBS @ tc_Nat_Onat @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ tc_Nat_Onat @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ tc_Nat_Onat ) @ c_fconj ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ C @ ( tc_fun @ tc_Nat_Onat @ C ) @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ tc_Nat_Onat @ C ) @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) @ C ) @ ( c_COMBB @ C @ tc_HOL_Obool @ tc_Nat_Onat ) ) @ c_fequal ) ) @ ( hAPP @ ( c_List_Onth @ C ) @ B ) ) ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ tc_Nat_Onat @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ tc_Nat_Onat ) @ c_fconj ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) @ B ) ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( c_member @ tc_Nat_Onat ) ) @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[692]) ).

thf(269,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ F ) ) @ E ) ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ D ) ) @ C ) ) )
     => ( ( c_Map_Omap__upds @ F @ D @ B @ ( hAPP @ ( hAPP @ ( c_List_Oappend @ F ) @ E ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ F ) @ A ) @ ( c_List_Olist_ONil @ F ) ) ) @ C )
        = ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ F @ ( tc_Option_Ooption @ D ) ) @ ( c_Map_Omap__upds @ F @ D @ B @ E @ C ) ) @ A ) @ ( hAPP @ ( c_Option_Ooption_OSome @ D ) @ ( hAPP @ ( hAPP @ ( c_List_Onth @ D ) @ C ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ F ) ) @ E ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_map__upds__append1) ).

thf(3365,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ F ) ) @ E ) ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ D ) ) @ C ) ) )
     => ( ( c_Map_Omap__upds @ F @ D @ B @ ( hAPP @ ( hAPP @ ( c_List_Oappend @ F ) @ E ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ F ) @ A ) @ ( c_List_Olist_ONil @ F ) ) ) @ C )
        = ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ F @ ( tc_Option_Ooption @ D ) ) @ ( c_Map_Omap__upds @ F @ D @ B @ E @ C ) ) @ A ) @ ( hAPP @ ( c_Option_Ooption_OSome @ D ) @ ( hAPP @ ( hAPP @ ( c_List_Onth @ D ) @ C ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ F ) ) @ E ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[269]) ).

thf(95,axiom,
    ! [A: $i] :
      ( ( A
       != ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) )
     => ( hBOOL @ ( hAPP @ ( c_Divides_Odivmod__int__rel @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ tc_Int_Oint @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_divmod__int__rel__0) ).

thf(2740,plain,
    ! [A: $i] :
      ( ( A
       != ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) )
     => ( hBOOL @ ( hAPP @ ( c_Divides_Odivmod__int__rel @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ tc_Int_Oint @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[95]) ).

thf(453,axiom,
    ( ( hAPP @ ( c_SetInterval_Oord__class_OlessThan @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
    = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_lessThan__0) ).

thf(4026,plain,
    ( ( hAPP @ ( c_SetInterval_Oord__class_OlessThan @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
    = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[453]) ).

thf(1676,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ B ) @ A ) ) )
    <=> ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ B ) @ C ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ B ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) ) ) @ A ) ) )
        & ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ B ) @ C ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_subset__insert__iff) ).

thf(8981,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ B ) @ A ) ) )
       => ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ B ) @ C ) )
           => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ B ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) ) ) @ A ) ) )
          & ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ B ) @ C ) )
           => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ A ) ) ) ) )
      & ( ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ B ) @ C ) )
           => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ B ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) ) ) @ A ) ) )
          & ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ B ) @ C ) )
           => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ A ) ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ B ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1676]) ).

thf(1413,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) ) @ D ) @ C ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) ) @ B ) @ D ) ) @ c_FunDef_Opair__less ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) ) @ A ) @ C ) ) @ c_FunDef_Omax__strict ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) ) @ B ) @ A ) ) @ C ) ) @ c_FunDef_Omax__strict ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_smax__insertI) ).

thf(7824,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) ) @ D ) @ C ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) ) @ B ) @ D ) ) @ c_FunDef_Opair__less ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) ) @ A ) @ C ) ) @ c_FunDef_Omax__strict ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) ) @ B ) @ A ) ) @ C ) ) @ c_FunDef_Omax__strict ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1413]) ).

thf(1243,axiom,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ ( hAPP @ c_Nat_OSuc @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__add__Suc1) ).

thf(7115,plain,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ ( hAPP @ c_Nat_OSuc @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1243]) ).

thf(479,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( c_List_Opartition @ E @ D @ C )
        = ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ E ) @ ( tc_List_Olist @ E ) ) @ B ) @ A ) )
     => ( ! [F: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ F ) @ ( hAPP @ ( c_List_Oset @ E ) @ B ) ) )
           => ( hBOOL @ ( hAPP @ D @ F ) ) )
        & ! [F: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ F ) @ ( hAPP @ ( c_List_Oset @ E ) @ A ) ) )
           => ~ ( hBOOL @ ( hAPP @ D @ F ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_partition__P) ).

thf(4110,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( c_List_Opartition @ E @ D @ C )
        = ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ E ) @ ( tc_List_Olist @ E ) ) @ B ) @ A ) )
     => ( ! [F: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ F ) @ ( hAPP @ ( c_List_Oset @ E ) @ B ) ) )
           => ( hBOOL @ ( hAPP @ D @ F ) ) )
        & ! [F: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ F ) @ ( hAPP @ ( c_List_Oset @ E ) @ A ) ) )
           => ~ ( hBOOL @ ( hAPP @ D @ F ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[479]) ).

thf(213,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hAPP @ ( c_Fun_Ocomp @ ( tc_prod @ F @ E ) @ E @ ( tc_prod @ D @ C ) @ ( c_Product__Type_Osnd @ F @ E ) ) @ ( c_Product__Type_Omap__pair @ D @ F @ C @ E @ B @ A ) )
      = ( hAPP @ ( c_Fun_Ocomp @ C @ E @ ( tc_prod @ D @ C ) @ A ) @ ( c_Product__Type_Osnd @ D @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_snd__comp__map__pair) ).

thf(3175,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hAPP @ ( c_Fun_Ocomp @ ( tc_prod @ F @ E ) @ E @ ( tc_prod @ D @ C ) @ ( c_Product__Type_Osnd @ F @ E ) ) @ ( c_Product__Type_Omap__pair @ D @ F @ C @ E @ B @ A ) )
      = ( hAPP @ ( c_Fun_Ocomp @ C @ E @ ( tc_prod @ D @ C ) @ A ) @ ( c_Product__Type_Osnd @ D @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[213]) ).

thf(1474,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Equiv__Relations_Oequiv @ C @ B @ A )
     => ( ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ C @ C @ C ) @ ( hAPP @ ( c_Relation_Oconverse @ C @ C ) @ A ) ) @ A )
        = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_equiv__comp__eq) ).

thf(8057,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Equiv__Relations_Oequiv @ C @ B @ A )
     => ( ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ C @ C @ C ) @ ( hAPP @ ( c_Relation_Oconverse @ C @ C ) @ A ) ) @ A )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1474]) ).

thf(1615,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ B ) @ A ) )
     => ( ( hAPP @ ( c_Finite__Set_Ocard @ B ) @ A )
        = ( hAPP @ ( hAPP @ ( hAPP @ ( c_Finite__Set_Ofold__image @ tc_Nat_Onat @ B @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) ) @ ( hAPP @ ( c_COMBK @ tc_Nat_Onat @ B ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_card_Oeq__fold__g) ).

thf(8747,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ B ) @ A ) )
     => ( ( hAPP @ ( c_Finite__Set_Ocard @ B ) @ A )
        = ( hAPP @ ( hAPP @ ( hAPP @ ( c_Finite__Set_Ofold__image @ tc_Nat_Onat @ B @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) ) @ ( hAPP @ ( c_COMBK @ tc_Nat_Onat @ B ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1615]) ).

thf(2014,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ D ) @ C ) @ B ) ) @ ( c_Transitive__Closure_Otrancl @ D @ A ) ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ D ) @ C ) @ B ) ) @ ( c_Transitive__Closure_Ortrancl @ D @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_trancl__into__rtrancl) ).

thf(10346,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ D ) @ C ) @ B ) ) @ ( c_Transitive__Closure_Otrancl @ D @ A ) ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ D ) @ C ) @ B ) ) @ ( c_Transitive__Closure_Ortrancl @ D @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2014]) ).

thf(2271,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Natural_Oevaln @ ( c_Com_Ocom_OSemi @ E @ D ) @ C @ B @ A )
     => ~ ! [F: $i] :
            ( ( c_Natural_Oevaln @ E @ C @ B @ F )
           => ~ ( c_Natural_Oevaln @ D @ F @ B @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_evaln__elim__cases_I4_J) ).

thf(11303,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Natural_Oevaln @ ( c_Com_Ocom_OSemi @ E @ D ) @ C @ B @ A )
     => ~ ! [F: $i] :
            ( ( c_Natural_Oevaln @ E @ C @ B @ F )
           => ~ ( c_Natural_Oevaln @ D @ F @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2271]) ).

thf(1598,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( hAPP @ ( c_Map_Omap__of @ F @ E @ D ) @ C )
        = ( hAPP @ ( c_Option_Ooption_OSome @ E ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ A @ C ) @ B ) )
       => ( ( hAPP @ ( c_Map_Omap__of @ F @ E @ ( hAPP @ ( c_List_Ofilter @ ( tc_prod @ F @ E ) @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ F @ E @ tc_HOL_Obool ) @ A ) ) @ D ) ) @ C )
          = ( hAPP @ ( c_Option_Ooption_OSome @ E ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_map__of__filter__in) ).

thf(8667,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( hAPP @ ( c_Map_Omap__of @ F @ E @ D ) @ C )
        = ( hAPP @ ( c_Option_Ooption_OSome @ E ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ A @ C ) @ B ) )
       => ( ( hAPP @ ( c_Map_Omap__of @ F @ E @ ( hAPP @ ( c_List_Ofilter @ ( tc_prod @ F @ E ) @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ F @ E @ tc_HOL_Obool ) @ A ) ) @ D ) ) @ C )
          = ( hAPP @ ( c_Option_Ooption_OSome @ E ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1598]) ).

thf(2156,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ B ) )
     => ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ A ) @ B ) )
         => ( ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ A ) @ B ) )
            = ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ B ) ) )
        & ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ A ) @ B ) )
         => ( ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ A ) @ B ) )
            = ( hAPP @ c_Nat_OSuc @ ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_card__insert__if) ).

thf(10864,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ B ) )
     => ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ A ) @ B ) )
         => ( ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ A ) @ B ) )
            = ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ B ) ) )
        & ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ A ) @ B ) )
         => ( ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ A ) @ B ) )
            = ( hAPP @ c_Nat_OSuc @ ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2156]) ).

thf(1581,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( D
          = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
       => ( ( hAPP @ ( hAPP @ ( c_List_Odrop @ C ) @ D ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ C ) @ B ) @ A ) )
          = ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ C ) @ B ) @ A ) ) )
      & ( ( D
         != ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
       => ( ( hAPP @ ( hAPP @ ( c_List_Odrop @ C ) @ D ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ C ) @ B ) @ A ) )
          = ( hAPP @ ( hAPP @ ( c_List_Odrop @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ D ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_drop__Cons_H) ).

thf(8519,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( D
          = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
       => ( ( hAPP @ ( hAPP @ ( c_List_Odrop @ C ) @ D ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ C ) @ B ) @ A ) )
          = ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ C ) @ B ) @ A ) ) )
      & ( ( D
         != ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
       => ( ( hAPP @ ( hAPP @ ( c_List_Odrop @ C ) @ D ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ C ) @ B ) @ A ) )
          = ( hAPP @ ( hAPP @ ( c_List_Odrop @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ D ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1581]) ).

thf(1944,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hAPP @ ( c_Option_Ooption_OSome @ C ) @ B )
        = ( hAPP @ ( c_Option_Ooption_OSome @ C ) @ A ) )
    <=> ( B = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_option_Oinject) ).

thf(10049,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( hAPP @ ( c_Option_Ooption_OSome @ C ) @ B )
          = ( hAPP @ ( c_Option_Ooption_OSome @ C ) @ A ) )
       => ( B = A ) )
      & ( ( B = A )
       => ( ( hAPP @ ( c_Option_Ooption_OSome @ C ) @ B )
          = ( hAPP @ ( c_Option_Ooption_OSome @ C ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1944]) ).

thf(1884,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i] :
          ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ C @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ C @ C ) @ D ) @ B ) ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ D ) @ ( c_Wellfounded_Oacc @ C @ A ) ) ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ B ) @ ( c_Wellfounded_Oacc @ C @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_acc_OaccI) ).

thf(9813,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i] :
          ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ C @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ C @ C ) @ D ) @ B ) ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ D ) @ ( c_Wellfounded_Oacc @ C @ A ) ) ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ B ) @ ( c_Wellfounded_Oacc @ C @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1884]) ).

thf(1372,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_termination__basic__simps_I1_J) ).

thf(7669,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1372]) ).

thf(806,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( c_Set_OCollect @ B ) @ ( hAPP @ c_fequal @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ B ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_singleton__conv2) ).

thf(5316,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( c_Set_OCollect @ B ) @ ( hAPP @ c_fequal @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ B ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[806]) ).

thf(1300,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ E @ D ) @ C ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ B @ A @ tc_HOL_Obool ) @ E ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ B @ A ) @ D ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_splitI) ).

thf(7345,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ E @ D ) @ C ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ B @ A @ tc_HOL_Obool ) @ E ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ B @ A ) @ D ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1300]) ).

thf(9,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Groups_Oab__group__add @ E )
     => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ D @ E @ E ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ E @ ( tc_fun @ E @ E ) @ D ) @ ( c_Groups_Ominus__class_Ominus @ E ) ) @ C ) ) @ B ) ) @ A )
        = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ E ) @ ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ C ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_setsum__subtractf) ).

thf(2434,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Groups_Oab__group__add @ E )
     => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ D @ E @ E ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ E @ ( tc_fun @ E @ E ) @ D ) @ ( c_Groups_Ominus__class_Ominus @ E ) ) @ C ) ) @ B ) ) @ A )
        = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ E ) @ ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ C ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[9]) ).

thf(1520,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ C ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Int_Oint ) @ B ) @ A ) )
       => ! [D: $i] :
            ( ! [E: $i] :
                ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Int_Oint ) @ E ) @ ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Int_Oint @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) @ C ) ) )
               => ! [F: $i] :
                    ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Int_Oint ) @ F ) @ A ) )
                   => ( D
                     != ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Int_Oint ) @ F ) @ E ) ) ) )
           => ( ( D != B )
             => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ D ) @ C )
               != B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_aset_I4_J) ).

thf(8298,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ C ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Int_Oint ) @ B ) @ A ) )
       => ! [D: $i] :
            ( ! [E: $i] :
                ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Int_Oint ) @ E ) @ ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Int_Oint @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) @ C ) ) )
               => ! [F: $i] :
                    ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Int_Oint ) @ F ) @ A ) )
                   => ( D
                     != ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Int_Oint ) @ F ) @ E ) ) ) )
           => ( ( D != B )
             => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ D ) @ C )
               != B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1520]) ).

thf(1320,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ ( c_Finite__Set_Ocard @ D ) @ ( hAPP @ ( c_Set_Oimage @ C @ D @ B ) @ A ) ) ) @ ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ A ) ) )
     => ~ ( c_Fun_Oinj__on @ C @ D @ B @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_pigeonhole) ).

thf(7467,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ ( c_Finite__Set_Ocard @ D ) @ ( hAPP @ ( c_Set_Oimage @ C @ D @ B ) @ A ) ) ) @ ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ A ) ) )
     => ~ ( c_Fun_Oinj__on @ C @ D @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1320]) ).

thf(382,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) ) )
    <=> ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ B ) )
        & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Int__iff) ).

thf(3781,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ B ) )
          & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ A ) ) ) )
      & ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ B ) )
          & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ A ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[382]) ).

thf(1733,axiom,
    ! [A: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_minus__nat_Odiff__0) ).

thf(9218,plain,
    ! [A: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1733]) ).

thf(363,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ E @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ C ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ A @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ E @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_UN__insert) ).

thf(3693,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ E @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ C ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ A @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ E @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[363]) ).

thf(2237,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ D ) @ ( c_SetInterval_Oord_OatLeastAtMost @ E @ C @ B @ A ) ) )
    <=> ( ( hBOOL @ ( hAPP @ ( hAPP @ C @ B ) @ D ) )
        & ( hBOOL @ ( hAPP @ ( hAPP @ C @ D ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_ord_OatLeastAtMost__iff) ).

thf(11160,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ D ) @ ( c_SetInterval_Oord_OatLeastAtMost @ E @ C @ B @ A ) ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ C @ B ) @ D ) )
          & ( hBOOL @ ( hAPP @ ( hAPP @ C @ D ) @ A ) ) ) )
      & ( ( ( hBOOL @ ( hAPP @ ( hAPP @ C @ B ) @ D ) )
          & ( hBOOL @ ( hAPP @ ( hAPP @ C @ D ) @ A ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ D ) @ ( c_SetInterval_Oord_OatLeastAtMost @ E @ C @ B @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2237]) ).

thf(462,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Equiv__Relations_Oquotient @ C ) @ B ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ C @ ( tc_fun @ ( tc_fun @ C @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ C @ ( tc_fun @ ( tc_fun @ C @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ C @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ ( tc_fun @ C @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ C @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ C ) @ ( c_Set_Oinsert @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) @ C ) @ ( c_Relation_OImage @ C @ C @ A ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ C @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( c_Set_Oinsert @ C ) ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) ) ) ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ ( tc_fun @ C @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_quotient__def) ).

thf(4058,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Equiv__Relations_Oquotient @ C ) @ B ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ C @ ( tc_fun @ ( tc_fun @ C @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ C @ ( tc_fun @ ( tc_fun @ C @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ C @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ ( tc_fun @ C @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ C @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ C ) @ ( c_Set_Oinsert @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) @ C ) @ ( c_Relation_OImage @ C @ C @ A ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ C @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( c_Set_Oinsert @ C ) ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) ) ) ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ ( tc_fun @ C @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[462]) ).

thf(2258,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_Natural_Oevaln @ D @ C @ B @ A )
     => ( c_Natural_Oevaln @ D @ C @ ( hAPP @ c_Nat_OSuc @ B ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_evaln__Suc) ).

thf(11239,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_Natural_Oevaln @ D @ C @ B @ A )
     => ( c_Natural_Oevaln @ D @ C @ ( hAPP @ c_Nat_OSuc @ B ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2258]) ).

thf(16,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( c_Relation_ORange @ C @ B ) @ ( hAPP @ ( c_Set_OCollect @ ( tc_prod @ C @ B ) ) @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ C @ B @ tc_HOL_Obool ) @ A ) ) )
      = ( hAPP @ ( c_Set_OCollect @ B ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ tc_HOL_Obool ) @ tc_HOL_Obool @ B ) @ ( c_HOL_OEx @ C ) ) @ ( hAPP @ ( c_COMBC @ C @ B @ tc_HOL_Obool ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Range__Collect__split) ).

thf(2459,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( c_Relation_ORange @ C @ B ) @ ( hAPP @ ( c_Set_OCollect @ ( tc_prod @ C @ B ) ) @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ C @ B @ tc_HOL_Obool ) @ A ) ) )
      = ( hAPP @ ( c_Set_OCollect @ B ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ tc_HOL_Obool ) @ tc_HOL_Obool @ B ) @ ( c_HOL_OEx @ C ) ) @ ( hAPP @ ( c_COMBC @ C @ B @ tc_HOL_Obool ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[16]) ).

thf(2147,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) )
    <=> ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) )
          & ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ A ) @ B ) ) )
        | ( B = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_dvd_Ole__less) ).

thf(10827,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) )
       => ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) )
            & ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ A ) @ B ) ) )
          | ( B = A ) ) )
      & ( ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) )
            & ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ A ) @ B ) ) )
          | ( B = A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2147]) ).

thf(1045,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ E ) @ ( tc_List_Olist @ E ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ E ) @ ( tc_List_Olist @ E ) ) @ D ) @ C ) ) @ ( c_List_Olistrel1 @ E @ B ) ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ E ) @ ( tc_List_Olist @ E ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ E ) @ ( tc_List_Olist @ E ) ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ E ) @ A ) @ D ) ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ E ) @ A ) @ C ) ) ) @ ( c_List_Olistrel1 @ E @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_listrel1I2) ).

thf(6241,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ E ) @ ( tc_List_Olist @ E ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ E ) @ ( tc_List_Olist @ E ) ) @ D ) @ C ) ) @ ( c_List_Olistrel1 @ E @ B ) ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ E ) @ ( tc_List_Olist @ E ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ E ) @ ( tc_List_Olist @ E ) ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ E ) @ A ) @ D ) ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ E ) @ A ) @ C ) ) ) @ ( c_List_Olistrel1 @ E @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1045]) ).

thf(724,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( c_Set_Oimage @ D @ C @ B ) @ ( c_Complete__Lattice_OSup__class_OSup @ ( tc_fun @ D @ tc_HOL_Obool ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ ( tc_fun @ D @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ A ) @ ( c_Set_Oimage @ D @ C @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_image__Union) ).

thf(5005,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( c_Set_Oimage @ D @ C @ B ) @ ( c_Complete__Lattice_OSup__class_OSup @ ( tc_fun @ D @ tc_HOL_Obool ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ ( tc_fun @ D @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ A ) @ ( c_Set_Oimage @ D @ C @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[724]) ).

thf(1215,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( class_Rings_Olinordered__semiring__1__strict @ F )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ F ) @ E ) @ D ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ F ) @ C ) @ D ) )
         => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ F ) @ ( c_Groups_Ozero__class_Ozero @ F ) ) @ B ) )
           => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ F ) @ ( c_Groups_Ozero__class_Ozero @ F ) ) @ A ) )
             => ( ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ F ) @ B ) @ A )
                  = ( c_Groups_Oone__class_Oone @ F ) )
               => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ F ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ F ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ F ) @ B ) @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ F ) @ A ) @ C ) ) ) @ D ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_convex__bound__lt) ).

thf(6996,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( class_Rings_Olinordered__semiring__1__strict @ F )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ F ) @ E ) @ D ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ F ) @ C ) @ D ) )
         => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ F ) @ ( c_Groups_Ozero__class_Ozero @ F ) ) @ B ) )
           => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ F ) @ ( c_Groups_Ozero__class_Ozero @ F ) ) @ A ) )
             => ( ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ F ) @ B ) @ A )
                  = ( c_Groups_Oone__class_Oone @ F ) )
               => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ F ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ F ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ F ) @ B ) @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ F ) @ A ) @ C ) ) ) @ D ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1215]) ).

thf(1970,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ F @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ F @ E ) @ D ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ F @ E ) @ B ) @ A ) ) )
    <=> ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ F ) @ D ) @ B ) )
        & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ C ) @ ( hAPP @ A @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mem__Sigma__iff) ).

thf(10170,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ F @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ F @ E ) @ D ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ F @ E ) @ B ) @ A ) ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ F ) @ D ) @ B ) )
          & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ C ) @ ( hAPP @ A @ D ) ) ) ) )
      & ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ F ) @ D ) @ B ) )
          & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ C ) @ ( hAPP @ A @ D ) ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ F @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ F @ E ) @ D ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ F @ E ) @ B ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1970]) ).

thf(1424,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( c_List_Ofilter @ E @ D ) @ ( hAPP @ ( hAPP @ ( c_List_Omap @ C @ E ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_List_Omap @ C @ E ) @ B ) @ ( hAPP @ ( c_List_Ofilter @ C @ ( hAPP @ ( c_Fun_Ocomp @ E @ tc_HOL_Obool @ C @ D ) @ B ) ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_filter__map) ).

thf(7871,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( c_List_Ofilter @ E @ D ) @ ( hAPP @ ( hAPP @ ( c_List_Omap @ C @ E ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_List_Omap @ C @ E ) @ B ) @ ( hAPP @ ( c_List_Ofilter @ C @ ( hAPP @ ( c_Fun_Ocomp @ E @ tc_HOL_Obool @ C @ D ) @ B ) ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1424]) ).

thf(1069,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ B ) @ A ) ) )
    <=> ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) )
         => ( hBOOL @ ( hAPP @ C @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) )
        & ! [D: $i] :
            ( ( B
              = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ D ) )
           => ( hBOOL @ ( hAPP @ C @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nat__diff__split) ).

thf(6336,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ B ) @ A ) ) )
       => ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) )
           => ( hBOOL @ ( hAPP @ C @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) )
          & ! [D: $i] :
              ( ( B
                = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ D ) )
             => ( hBOOL @ ( hAPP @ C @ D ) ) ) ) )
      & ( ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) )
           => ( hBOOL @ ( hAPP @ C @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) )
          & ! [D: $i] :
              ( ( B
                = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ D ) )
             => ( hBOOL @ ( hAPP @ C @ D ) ) ) )
       => ( hBOOL @ ( hAPP @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ B ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1069]) ).

thf(1922,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ D ) @ C ) @ B ) ) @ ( c_Transitive__Closure_Ortrancl @ D @ A ) ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ B ) @ ( c_Wellfounded_Oacc @ D @ A ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( c_Wellfounded_Oacc @ D @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_acc__downwards__aux) ).

thf(9961,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ D ) @ C ) @ B ) ) @ ( c_Transitive__Closure_Ortrancl @ D @ A ) ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ B ) @ ( c_Wellfounded_Oacc @ D @ A ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( c_Wellfounded_Oacc @ D @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1922]) ).

thf(774,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_List_Odistinct @ C ) @ ( hAPP @ ( hAPP @ ( c_List_Oappend @ C ) @ B ) @ A ) ) )
    <=> ( ( hBOOL @ ( hAPP @ ( c_List_Odistinct @ C ) @ B ) )
        & ( hBOOL @ ( hAPP @ ( c_List_Odistinct @ C ) @ A ) )
        & ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_List_Oset @ C ) @ B ) ) @ ( hAPP @ ( c_List_Oset @ C ) @ A ) )
          = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_distinct__append) ).

thf(5178,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( c_List_Odistinct @ C ) @ ( hAPP @ ( hAPP @ ( c_List_Oappend @ C ) @ B ) @ A ) ) )
       => ( ( hBOOL @ ( hAPP @ ( c_List_Odistinct @ C ) @ B ) )
          & ( hBOOL @ ( hAPP @ ( c_List_Odistinct @ C ) @ A ) )
          & ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_List_Oset @ C ) @ B ) ) @ ( hAPP @ ( c_List_Oset @ C ) @ A ) )
            = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) ) )
      & ( ( ( hBOOL @ ( hAPP @ ( c_List_Odistinct @ C ) @ B ) )
          & ( hBOOL @ ( hAPP @ ( c_List_Odistinct @ C ) @ A ) )
          & ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_List_Oset @ C ) @ B ) ) @ ( hAPP @ ( c_List_Oset @ C ) @ A ) )
            = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) )
       => ( hBOOL @ ( hAPP @ ( c_List_Odistinct @ C ) @ ( hAPP @ ( hAPP @ ( c_List_Oappend @ C ) @ B ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[774]) ).

thf(921,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ ( tc_List_Olist @ C ) ) @ ( hAPP @ ( c_Set_OCollect @ ( tc_List_Olist @ C ) ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ ( tc_List_Olist @ C ) @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( tc_List_Olist @ C ) ) @ c_fconj ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ ( tc_List_Olist @ C ) @ ( tc_fun @ C @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ C @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( tc_List_Olist @ C ) ) @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) @ ( c_List_Oset @ C ) ) ) @ B ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ ( tc_List_Olist @ C ) @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_Nat_Onat @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_List_Olist @ C ) ) @ c_fequal ) @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) ) ) @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_finite__lists__length__eq) ).

thf(5708,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ ( tc_List_Olist @ C ) ) @ ( hAPP @ ( c_Set_OCollect @ ( tc_List_Olist @ C ) ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ ( tc_List_Olist @ C ) @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( tc_List_Olist @ C ) ) @ c_fconj ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ ( tc_List_Olist @ C ) @ ( tc_fun @ C @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ C @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( tc_List_Olist @ C ) ) @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) @ ( c_List_Oset @ C ) ) ) @ B ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ ( tc_List_Olist @ C ) @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_Nat_Onat @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_List_Olist @ C ) ) @ c_fequal ) @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) ) ) @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[921]) ).

thf(1594,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ E @ D ) @ C ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ B @ A @ tc_HOL_Obool ) @ E ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ B @ A ) @ D ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_prod__caseI) ).

thf(8650,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ E @ D ) @ C ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ B @ A @ tc_HOL_Obool ) @ E ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ B @ A ) @ D ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1594]) ).

thf(1456,axiom,
    ! [A: $i] :
      ( ( hAPP @ ( c_Finite__Set_Ocard @ tc_Nat_Onat ) @ ( hAPP @ ( c_SetInterval_Oord__class_OlessThan @ tc_Nat_Onat ) @ A ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_card__lessThan) ).

thf(7980,plain,
    ! [A: $i] :
      ( ( hAPP @ ( c_Finite__Set_Ocard @ tc_Nat_Onat ) @ ( hAPP @ ( c_SetInterval_Oord__class_OlessThan @ tc_Nat_Onat ) @ A ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1456]) ).

thf(2329,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Natural_Oevalc @ ( c_Com_Ocom_OWhile @ D @ C ) ) @ B ) @ A ) )
    <=> ( ( ( A = B )
          & ~ ( hBOOL @ ( hAPP @ D @ B ) ) )
        | ( ( hBOOL @ ( hAPP @ D @ B ) )
          & ? [E: $i] :
              ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Natural_Oevalc @ C ) @ B ) @ E ) )
              & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Natural_Oevalc @ ( c_Com_Ocom_OWhile @ D @ C ) ) @ E ) @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_evalc_Oequations_I8_J) ).

thf(11549,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Natural_Oevalc @ ( c_Com_Ocom_OWhile @ D @ C ) ) @ B ) @ A ) )
       => ( ( ( A = B )
            & ~ ( hBOOL @ ( hAPP @ D @ B ) ) )
          | ( ( hBOOL @ ( hAPP @ D @ B ) )
            & ? [E: $i] :
                ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Natural_Oevalc @ C ) @ B ) @ E ) )
                & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Natural_Oevalc @ ( c_Com_Ocom_OWhile @ D @ C ) ) @ E ) @ A ) ) ) ) ) )
      & ( ( ( ( A = B )
            & ~ ( hBOOL @ ( hAPP @ D @ B ) ) )
          | ( ( hBOOL @ ( hAPP @ D @ B ) )
            & ? [E: $i] :
                ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Natural_Oevalc @ C ) @ B ) @ E ) )
                & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Natural_Oevalc @ ( c_Com_Ocom_OWhile @ D @ C ) ) @ E ) @ A ) ) ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Natural_Oevalc @ ( c_Com_Ocom_OWhile @ D @ C ) ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2329]) ).

thf(2376,axiom,
    ! [A: $i,B: $i] :
      ( ? [C: $i] : ( hBOOL @ ( hAPP @ B @ C ) )
     => ( hBOOL @ ( hAPP @ B @ ( c_Hilbert__Choice_OEps @ A @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_someI__ex) ).

thf(11756,plain,
    ! [A: $i,B: $i] :
      ( ? [C: $i] : ( hBOOL @ ( hAPP @ B @ C ) )
     => ( hBOOL @ ( hAPP @ B @ ( c_Hilbert__Choice_OEps @ A @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2376]) ).

thf(1183,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( hAPP @ ( hAPP @ ( c_List_Oappend @ F ) @ E ) @ D )
        = C )
     => ( ( B
          = ( hAPP @ ( hAPP @ ( c_List_Oappend @ F ) @ D ) @ A ) )
       => ( ( hAPP @ ( hAPP @ ( c_List_Oappend @ F ) @ E ) @ B )
          = ( hAPP @ ( hAPP @ ( c_List_Oappend @ F ) @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_append__eq__appendI) ).

thf(6877,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( hAPP @ ( hAPP @ ( c_List_Oappend @ F ) @ E ) @ D )
        = C )
     => ( ( B
          = ( hAPP @ ( hAPP @ ( c_List_Oappend @ F ) @ D ) @ A ) )
       => ( ( hAPP @ ( hAPP @ ( c_List_Oappend @ F ) @ E ) @ B )
          = ( hAPP @ ( hAPP @ ( c_List_Oappend @ F ) @ C ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1183]) ).

thf(777,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ C ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ B ) ) @ A ) ) )
     => ( ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ B ) @ ( hAPP @ ( hAPP @ ( c_List_Onth @ B ) @ A ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_List_Odrop @ B ) @ ( hAPP @ c_Nat_OSuc @ C ) ) @ A ) )
        = ( hAPP @ ( hAPP @ ( c_List_Odrop @ B ) @ C ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nth__drop_H) ).

thf(5192,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ C ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ B ) ) @ A ) ) )
     => ( ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ B ) @ ( hAPP @ ( hAPP @ ( c_List_Onth @ B ) @ A ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_List_Odrop @ B ) @ ( hAPP @ c_Nat_OSuc @ C ) ) @ A ) )
        = ( hAPP @ ( hAPP @ ( c_List_Odrop @ B ) @ C ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[777]) ).

thf(1803,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_List_Odistinct @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( c_List_Odistinct @ C ) @ ( c_List_Osublist @ C @ B @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_distinct__sublistI) ).

thf(9481,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_List_Odistinct @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( c_List_Odistinct @ C ) @ ( c_List_Osublist @ C @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1803]) ).

thf(831,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Equiv__Relations_Oequiv @ B @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ A )
    <=> ( c_Equiv__Relations_Oequivp @ B @ ( hAPP @ ( hAPP @ ( c_COMBC @ B @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ B @ ( tc_fun @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ B ) @ ( c_COMBC @ B @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ B @ ( tc_prod @ B @ B ) ) @ ( tc_fun @ B @ ( tc_fun @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( c_COMBB @ ( tc_prod @ B @ B ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ B ) @ ( c_member @ ( tc_prod @ B @ B ) ) ) ) @ ( c_Product__Type_OPair @ B @ B ) ) ) ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_equivp__equiv) ).

thf(5412,plain,
    ! [A: $i,B: $i] :
      ( ( ( c_Equiv__Relations_Oequiv @ B @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ A )
       => ( c_Equiv__Relations_Oequivp @ B @ ( hAPP @ ( hAPP @ ( c_COMBC @ B @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ B @ ( tc_fun @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ B ) @ ( c_COMBC @ B @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ B @ ( tc_prod @ B @ B ) ) @ ( tc_fun @ B @ ( tc_fun @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( c_COMBB @ ( tc_prod @ B @ B ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ B ) @ ( c_member @ ( tc_prod @ B @ B ) ) ) ) @ ( c_Product__Type_OPair @ B @ B ) ) ) ) @ A ) ) )
      & ( ( c_Equiv__Relations_Oequivp @ B @ ( hAPP @ ( hAPP @ ( c_COMBC @ B @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ B @ ( tc_fun @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ B ) @ ( c_COMBC @ B @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ B @ ( tc_prod @ B @ B ) ) @ ( tc_fun @ B @ ( tc_fun @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( c_COMBB @ ( tc_prod @ B @ B ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ B ) @ ( c_member @ ( tc_prod @ B @ B ) ) ) ) @ ( c_Product__Type_OPair @ B @ B ) ) ) ) @ A ) )
       => ( c_Equiv__Relations_Oequiv @ B @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[831]) ).

thf(2154,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_List_Odistinct @ E ) @ ( hAPP @ ( hAPP @ ( c_List_Omap @ ( tc_prod @ E @ D ) @ E ) @ ( c_Product__Type_Ofst @ E @ D ) ) @ C ) ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ D ) @ B ) @ A ) ) @ ( hAPP @ ( c_List_Oset @ ( tc_prod @ E @ D ) ) @ C ) ) )
       => ( ( hAPP @ ( c_Map_Omap__of @ E @ D @ C ) @ B )
          = ( hAPP @ ( c_Option_Ooption_OSome @ D ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_map__of__is__SomeI) ).

thf(10859,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_List_Odistinct @ E ) @ ( hAPP @ ( hAPP @ ( c_List_Omap @ ( tc_prod @ E @ D ) @ E ) @ ( c_Product__Type_Ofst @ E @ D ) ) @ C ) ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ D ) @ B ) @ A ) ) @ ( hAPP @ ( c_List_Oset @ ( tc_prod @ E @ D ) ) @ C ) ) )
       => ( ( hAPP @ ( c_Map_Omap__of @ E @ D @ C ) @ B )
          = ( hAPP @ ( c_Option_Ooption_OSome @ D ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2154]) ).

thf(2104,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_New__Random__Sequence_Opos__empty @ D @ C @ B @ A )
      = ( c_New__DSequence_Opos__empty @ D ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_pos__empty__def) ).

thf(10662,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_New__Random__Sequence_Opos__empty @ D @ C @ B @ A )
      = ( c_New__DSequence_Opos__empty @ D ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2104]) ).

thf(2278,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ B ) @ ( hAPP @ ( c_Set_OCollect @ C ) @ A ) ) )
     => ( hBOOL @ ( hAPP @ A @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_CollectD) ).

thf(11335,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ B ) @ ( hAPP @ ( c_Set_OCollect @ C ) @ A ) ) )
     => ( hBOOL @ ( hAPP @ A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2278]) ).

thf(968,axiom,
    ! [A: $i,B: $i] :
      ( ( ( c_Orderings_Obot__class_Obot @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) )
        = ( c_SetInterval_Oord_OatLeastAtMost @ tc_Nat_Onat @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B @ A ) )
    <=> ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_dvd_OatLeastatMost__empty__iff2) ).

thf(5903,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( c_Orderings_Obot__class_Obot @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) )
          = ( c_SetInterval_Oord_OatLeastAtMost @ tc_Nat_Onat @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B @ A ) )
       => ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) ) )
      & ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) )
       => ( ( c_Orderings_Obot__class_Obot @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) )
          = ( c_SetInterval_Oord_OatLeastAtMost @ tc_Nat_Onat @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[968]) ).

thf(299,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) @ B ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_bot2E) ).

thf(3466,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) @ B ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[299]) ).

thf(1918,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_Wellfounded_Oacyclic @ D @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ ( tc_prod @ D @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ D ) @ C ) @ B ) ) @ A ) )
    <=> ( ( c_Wellfounded_Oacyclic @ D @ A )
        & ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ D ) @ B ) @ C ) ) @ ( c_Transitive__Closure_Ortrancl @ D @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_acyclic__insert) ).

thf(9946,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( c_Wellfounded_Oacyclic @ D @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ ( tc_prod @ D @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ D ) @ C ) @ B ) ) @ A ) )
       => ( ( c_Wellfounded_Oacyclic @ D @ A )
          & ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ D ) @ B ) @ C ) ) @ ( c_Transitive__Closure_Ortrancl @ D @ A ) ) ) ) )
      & ( ( ( c_Wellfounded_Oacyclic @ D @ A )
          & ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ D ) @ B ) @ C ) ) @ ( c_Transitive__Closure_Ortrancl @ D @ A ) ) ) )
       => ( c_Wellfounded_Oacyclic @ D @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ ( tc_prod @ D @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ D ) @ C ) @ B ) ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1918]) ).

thf(1354,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Predicate_Oreflp @ B @ A )
    <=> ( c_Relation_Orefl__on @ B @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Set_OCollect @ ( tc_prod @ B @ B ) ) @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ B @ B @ tc_HOL_Obool ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_reflp__def) ).

thf(7605,plain,
    ! [A: $i,B: $i] :
      ( ( ( c_Predicate_Oreflp @ B @ A )
       => ( c_Relation_Orefl__on @ B @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Set_OCollect @ ( tc_prod @ B @ B ) ) @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ B @ B @ tc_HOL_Obool ) @ A ) ) ) )
      & ( ( c_Relation_Orefl__on @ B @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Set_OCollect @ ( tc_prod @ B @ B ) ) @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ B @ B @ tc_HOL_Obool ) @ A ) ) )
       => ( c_Predicate_Oreflp @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1354]) ).

thf(1841,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ ( tc_prod @ B @ B ) ) @ A ) )
     => ( ( c_Wellfounded_Owf @ B @ A )
      <=> ( c_Wellfounded_Oacyclic @ B @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_wf__iff__acyclic__if__finite) ).

thf(9625,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ ( tc_prod @ B @ B ) ) @ A ) )
     => ( ( ( c_Wellfounded_Owf @ B @ A )
         => ( c_Wellfounded_Oacyclic @ B @ A ) )
        & ( ( c_Wellfounded_Oacyclic @ B @ A )
         => ( c_Wellfounded_Owf @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1841]) ).

thf(1792,axiom,
    ! [A: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ A ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_le__square) ).

thf(9443,plain,
    ! [A: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ A ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1792]) ).

thf(2012,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ D ) @ C ) ) @ ( c_Transitive__Closure_Ortrancl @ E @ B ) ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ C ) @ A ) ) @ B ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ D ) @ A ) ) @ ( c_Transitive__Closure_Ortrancl @ E @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_rtrancl_Ortrancl__into__rtrancl) ).

thf(10336,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ D ) @ C ) ) @ ( c_Transitive__Closure_Ortrancl @ E @ B ) ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ C ) @ A ) ) @ B ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ D ) @ A ) ) @ ( c_Transitive__Closure_Ortrancl @ E @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2012]) ).

thf(2040,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( c_Map_Omap__add @ F @ E @ D @ ( c_Map_Omap__upds @ F @ E @ C @ B @ A ) )
      = ( c_Map_Omap__upds @ F @ E @ ( c_Map_Omap__add @ F @ E @ D @ C ) @ B @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_map__add__upds) ).

thf(10427,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( c_Map_Omap__add @ F @ E @ D @ ( c_Map_Omap__upds @ F @ E @ C @ B @ A ) )
      = ( c_Map_Omap__upds @ F @ E @ ( c_Map_Omap__add @ F @ E @ D @ C ) @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2040]) ).

thf(1113,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nat__add__assoc) ).

thf(6570,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1113]) ).

thf(233,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ F @ F ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ F @ F ) @ E ) @ D ) ) @ C ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ F @ F ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ F @ F ) @ D ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Nat_Ocompow @ ( tc_prod @ F @ F ) @ tc_HOL_Obool ) @ A ) @ C ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ F @ F ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ F @ F ) @ E ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Nat_Ocompow @ ( tc_prod @ F @ F ) @ tc_HOL_Obool ) @ ( hAPP @ c_Nat_OSuc @ A ) ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_rel__pow__Suc__I2) ).

thf(3236,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ F @ F ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ F @ F ) @ E ) @ D ) ) @ C ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ F @ F ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ F @ F ) @ D ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Nat_Ocompow @ ( tc_prod @ F @ F ) @ tc_HOL_Obool ) @ A ) @ C ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ F @ F ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ F @ F ) @ E ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Nat_Ocompow @ ( tc_prod @ F @ F ) @ tc_HOL_Obool ) @ ( hAPP @ c_Nat_OSuc @ A ) ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[233]) ).

thf(1193,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ C @ C @ C ) @ ( hAPP @ ( c_Relation_Oconverse @ C @ C ) @ B ) ) @ B )
        = B )
     => ( ( ( hAPP @ ( c_Relation_ODomain @ C @ C ) @ B )
          = A )
       => ( c_Equiv__Relations_Oequiv @ C @ A @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_comp__equivI) ).

thf(6914,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ C @ C @ C ) @ ( hAPP @ ( c_Relation_Oconverse @ C @ C ) @ B ) ) @ B )
        = B )
     => ( ( ( hAPP @ ( c_Relation_ODomain @ C @ C ) @ B )
          = A )
       => ( c_Equiv__Relations_Oequiv @ C @ A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1193]) ).

thf(320,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C
       != ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Int_Oint ) @ B ) @ A ) )
      <=> ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zdvd__mono) ).

thf(3545,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C
       != ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) )
     => ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Int_Oint ) @ B ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ A ) ) ) )
        & ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ A ) ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Int_Oint ) @ B ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[320]) ).

thf(984,axiom,
    ! [A: $i,B: $i] :
      ( ( ( hAPP @ ( c_List_Oset @ B ) @ A )
        = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) )
    <=> ( A
        = ( c_List_Olist_ONil @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_set__empty) ).

thf(5977,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( hAPP @ ( c_List_Oset @ B ) @ A )
          = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) )
       => ( A
          = ( c_List_Olist_ONil @ B ) ) )
      & ( ( A
          = ( c_List_Olist_ONil @ B ) )
       => ( ( hAPP @ ( c_List_Oset @ B ) @ A )
          = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[984]) ).

thf(1048,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ C ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ A ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ A ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_diff__less__mono2) ).

thf(6251,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ C ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ A ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ A ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1048]) ).

thf(1230,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) )
     => ( ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Nat_Onat ) @ B ) @ A )
        = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_div__less) ).

thf(7064,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) )
     => ( ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Nat_Onat ) @ B ) @ A )
        = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1230]) ).

thf(26,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Groups_Ocomm__monoid__add @ E )
     => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ tc_Nat_Onat @ E ) @ D ) @ ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ B ) ) )
        = ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ tc_Nat_Onat @ E ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_Nat_Onat @ E @ tc_Nat_Onat ) @ D ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ tc_Nat_Onat @ tc_Nat_Onat ) @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) ) @ B ) ) ) @ ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Nat_Onat @ C @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_setsum__shift__bounds__cl__nat__ivl) ).

thf(2491,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Groups_Ocomm__monoid__add @ E )
     => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ tc_Nat_Onat @ E ) @ D ) @ ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ B ) ) )
        = ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ tc_Nat_Onat @ E ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_Nat_Onat @ E @ tc_Nat_Onat ) @ D ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ tc_Nat_Onat @ tc_Nat_Onat ) @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) ) @ B ) ) ) @ ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Nat_Onat @ C @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[26]) ).

thf(1655,axiom,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) )
       => ( ( c_Divides_Odiv__class_Omod @ tc_Nat_Onat @ B @ A )
          = B ) )
      & ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) )
       => ( ( c_Divides_Odiv__class_Omod @ tc_Nat_Onat @ B @ A )
          = ( c_Divides_Odiv__class_Omod @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ B ) @ A ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mod__if) ).

thf(8898,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) )
       => ( ( c_Divides_Odiv__class_Omod @ tc_Nat_Onat @ B @ A )
          = B ) )
      & ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) )
       => ( ( c_Divides_Odiv__class_Omod @ tc_Nat_Onat @ B @ A )
          = ( c_Divides_Odiv__class_Omod @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ B ) @ A ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1655]) ).

thf(102,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oab__group__add @ C )
     => ( ( B = A )
      <=> ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ C ) @ B ) @ A )
          = ( c_Groups_Ozero__class_Ozero @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_eq__iff__diff__eq__0) ).

thf(2763,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oab__group__add @ C )
     => ( ( ( B = A )
         => ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ C ) @ B ) @ A )
            = ( c_Groups_Ozero__class_Ozero @ C ) ) )
        & ( ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ C ) @ B ) @ A )
            = ( c_Groups_Ozero__class_Ozero @ C ) )
         => ( B = A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[102]) ).

thf(1051,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ C ) )
     => ( ( C
          = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ A ) ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ B ) @ C ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_self__quotient__aux1) ).

thf(6268,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ C ) )
     => ( ( C
          = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ A ) ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ B ) @ C ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1051]) ).

thf(2394,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( c_Fun_Oinj__on @ F @ E @ D @ C )
     => ( ( ( hAPP @ D @ B )
          = ( hAPP @ D @ A ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ F ) @ B ) @ C ) )
         => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ F ) @ A ) @ C ) )
           => ( B = A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_inj__onD) ).

thf(11834,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( c_Fun_Oinj__on @ F @ E @ D @ C )
     => ( ( ( hAPP @ D @ B )
          = ( hAPP @ D @ A ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ F ) @ B ) @ C ) )
         => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ F ) @ A ) @ C ) )
           => ( B = A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2394]) ).

thf(1527,axiom,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) ) @ ( c_Transitive__Closure_Otrancl @ B @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ B @ B ) @ ( hAPP @ ( c_Relation_OField @ B ) @ A ) ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ B @ tc_HOL_Obool ) @ B ) @ ( hAPP @ ( c_Relation_OField @ B ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_trancl__subset__Field2) ).

thf(8325,plain,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) ) @ ( c_Transitive__Closure_Otrancl @ B @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ B @ B ) @ ( hAPP @ ( c_Relation_OField @ B ) @ A ) ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ B @ tc_HOL_Obool ) @ B ) @ ( hAPP @ ( c_Relation_OField @ B ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1527]) ).

thf(306,axiom,
    ( ( hAPP @ ( c_Nat_Osize__class_Osize @ tc_HOL_Obool ) @ c_fTrue )
    = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_bool_Osize_I3_J) ).

thf(3502,plain,
    ( ( hAPP @ ( c_Nat_Osize__class_Osize @ tc_HOL_Obool ) @ c_fTrue )
    = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[306]) ).

thf(160,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Onth @ C ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ C ) @ B ) @ A ) ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
      = B ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nth__Cons__0) ).

thf(2985,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Onth @ C ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ C ) @ B ) @ A ) ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
      = B ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[160]) ).

thf(2084,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Ozip @ F @ E ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ F ) @ D ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ E ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ ( tc_prod @ F @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ F @ E ) @ D ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_List_Ozip @ F @ E ) @ C ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zip__Cons__Cons) ).

thf(10599,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Ozip @ F @ E ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ F ) @ D ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ E ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ ( tc_prod @ F @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ F @ E ) @ D ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_List_Ozip @ F @ E ) @ C ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2084]) ).

thf(2173,axiom,
    ! [A: $i,B: $i,C: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Natural_Oevalc @ ( c_Com_Ocom_OAss @ C @ B ) ) @ A ) @ ( hAPP @ ( hAPP @ ( hAPP @ c_Natural_Oupdate @ A ) @ C ) @ ( hAPP @ B @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_evalc_Oequations_I2_J) ).

thf(10908,plain,
    ! [A: $i,B: $i,C: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Natural_Oevalc @ ( c_Com_Ocom_OAss @ C @ B ) ) @ A ) @ ( hAPP @ ( hAPP @ ( hAPP @ c_Natural_Oupdate @ A ) @ C ) @ ( hAPP @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2173]) ).

thf(866,axiom,
    hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Nat__Transfer_Otransfer__nat__int__function__closures_I6_J) ).

thf(5518,plain,
    hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[866]) ).

thf(2288,axiom,
    hBOOL @ c_fTrue,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_c__fTrue__1) ).

thf(11368,plain,
    hBOOL @ c_fTrue,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2288]) ).

thf(11903,plain,
    ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ tc_Int_Oint ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ tc_Int_Oint @ tc_HOL_Obool ) ) ) )
   != ( hBOOL @ c_fTrue ) ),
    inference(paramod_ordered,[status(thm)],[11368,3918]) ).

thf(11907,plain,
    ( ( hAPP @ ( c_Finite__Set_Ofinite @ tc_Int_Oint ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ tc_Int_Oint @ tc_HOL_Obool ) ) )
   != c_fTrue ),
    inference(simp,[status(thm)],[11903]) ).

thf(1327,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ G ) @ F ) @ ( hAPP @ ( hAPP @ E @ D ) @ C ) ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ G ) @ F ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ B @ A @ ( tc_fun @ G @ tc_HOL_Obool ) ) @ E ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ B @ A ) @ D ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mem__splitI) ).

thf(7499,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ G ) @ F ) @ ( hAPP @ ( hAPP @ E @ D ) @ C ) ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ G ) @ F ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ B @ A @ ( tc_fun @ G @ tc_HOL_Obool ) ) @ E ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ B @ A ) @ D ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1327]) ).

thf(2332,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( hAPP @ E @ D )
        = ( hAPP @ ( c_Option_Ooption_OSome @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ B ) @ ( c_Map_Oran @ A @ C @ E ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_ranI) ).

thf(11568,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( hAPP @ E @ D )
        = ( hAPP @ ( c_Option_Ooption_OSome @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ B ) @ ( c_Map_Oran @ A @ C @ E ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2332]) ).

thf(517,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ B ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ A ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ A @ tc_HOL_Obool ) ) ) )
       => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ ( tc_prod @ B @ A ) ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ ( tc_prod @ B @ A ) @ tc_HOL_Obool ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_finite__Prod__UNIV) ).

thf(4266,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ B ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ A ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ A @ tc_HOL_Obool ) ) ) )
       => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ ( tc_prod @ B @ A ) ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ ( tc_prod @ B @ A ) @ tc_HOL_Obool ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[517]) ).

thf(2361,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Divides_Odivmod__nat__rel @ D @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Divides_Odivmod__nat__rel @ D @ C ) @ A ) )
       => ( B = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_divmod__nat__rel__unique) ).

thf(11699,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Divides_Odivmod__nat__rel @ D @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Divides_Odivmod__nat__rel @ D @ C ) @ A ) )
       => ( B = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2361]) ).

thf(2343,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i] : ( hBOOL @ ( hAPP @ C @ D ) )
    <=> ! [D: $i,E: $i] : ( hBOOL @ ( hAPP @ C @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ B @ A ) @ D ) @ E ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_split__paired__All) ).

thf(11613,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ! [D: $i] : ( hBOOL @ ( hAPP @ C @ D ) )
       => ! [D: $i,E: $i] : ( hBOOL @ ( hAPP @ C @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ B @ A ) @ D ) @ E ) ) ) )
      & ( ! [D: $i,E: $i] : ( hBOOL @ ( hAPP @ C @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ B @ A ) @ D ) @ E ) ) )
       => ! [D: $i] : ( hBOOL @ ( hAPP @ C @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2343]) ).

thf(311,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_HOL_OThe @ ( tc_prod @ D @ C ) @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ D @ C @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ D ) @ ( c_COMBB @ tc_HOL_Obool @ tc_HOL_Obool @ C ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ D ) @ c_fconj ) @ ( hAPP @ c_fequal @ B ) ) ) ) @ ( hAPP @ c_fequal @ A ) ) ) )
      = ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ C ) @ B ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_The__split__eq) ).

thf(3513,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_HOL_OThe @ ( tc_prod @ D @ C ) @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ D @ C @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ D ) @ ( c_COMBB @ tc_HOL_Obool @ tc_HOL_Obool @ C ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ D ) @ c_fconj ) @ ( hAPP @ c_fequal @ B ) ) ) ) @ ( hAPP @ c_fequal @ A ) ) ) )
      = ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ C ) @ B ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[311]) ).

thf(2265,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ F ) @ E ) @ D ) )
     => ( ( hAPP @ ( c_Fun_Ooverride__on @ F @ C @ B @ A @ D ) @ E )
        = ( hAPP @ B @ E ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_override__on__apply__notin) ).

thf(11273,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ F ) @ E ) @ D ) )
     => ( ( hAPP @ ( c_Fun_Ooverride__on @ F @ C @ B @ A @ D ) @ E )
        = ( hAPP @ B @ E ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2265]) ).

thf(1389,axiom,
    ! [A: $i,B: $i] :
      ( ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ A )
        = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
    <=> ( ( B
          = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
        | ( A
          = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__is__0) ).

thf(7734,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ A )
          = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
       => ( ( B
            = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
          | ( A
            = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) )
      & ( ( ( B
            = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
          | ( A
            = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) )
       => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ A )
          = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1389]) ).

thf(778,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) )
        = ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ C @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) )
    <=> ! [E: $i] :
          ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ E ) @ B ) )
         => ( ( hAPP @ A @ E )
            = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_UNION__empty__conv_I1_J) ).

thf(5195,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) )
          = ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ C @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) )
       => ! [E: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ E ) @ B ) )
           => ( ( hAPP @ A @ E )
              = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) ) )
      & ( ! [E: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ E ) @ B ) )
           => ( ( hAPP @ A @ E )
              = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) )
       => ( ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) )
          = ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ C @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[778]) ).

thf(2236,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_Fun_Oinj__on @ D @ C @ B @ A )
    <=> ! [E: $i] :
          ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ E ) @ A ) )
         => ! [F: $i] :
              ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ F ) @ A ) )
             => ( ( ( hAPP @ B @ E )
                  = ( hAPP @ B @ F ) )
               => ( E = F ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_inj__on__def) ).

thf(11146,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( c_Fun_Oinj__on @ D @ C @ B @ A )
       => ! [E: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ E ) @ A ) )
           => ! [F: $i] :
                ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ F ) @ A ) )
               => ( ( ( hAPP @ B @ E )
                    = ( hAPP @ B @ F ) )
                 => ( E = F ) ) ) ) )
      & ( ! [E: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ E ) @ A ) )
           => ! [F: $i] :
                ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ F ) @ A ) )
               => ( ( ( hAPP @ B @ E )
                    = ( hAPP @ B @ F ) )
                 => ( E = F ) ) ) )
       => ( c_Fun_Oinj__on @ D @ C @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2236]) ).

thf(850,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) )
     => ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ A ) @ B )
        = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Un__absorb2) ).

thf(5476,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) )
     => ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ A ) @ B )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[850]) ).

thf(1699,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hAPP @ ( hAPP @ ( c_List_Otake @ C ) @ B ) @ A )
        = ( c_List_Olist_ONil @ C ) )
    <=> ( ( B
          = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
        | ( A
          = ( c_List_Olist_ONil @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_take__eq__Nil) ).

thf(9096,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( hAPP @ ( hAPP @ ( c_List_Otake @ C ) @ B ) @ A )
          = ( c_List_Olist_ONil @ C ) )
       => ( ( B
            = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
          | ( A
            = ( c_List_Olist_ONil @ C ) ) ) )
      & ( ( ( B
            = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
          | ( A
            = ( c_List_Olist_ONil @ C ) ) )
       => ( ( hAPP @ ( hAPP @ ( c_List_Otake @ C ) @ B ) @ A )
          = ( c_List_Olist_ONil @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1699]) ).

thf(1932,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( D = C )
       => ( ( c_List_Oremove1 @ B @ D @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ B ) @ C ) @ A ) )
          = A ) )
      & ( ( D != C )
       => ( ( c_List_Oremove1 @ B @ D @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ B ) @ C ) @ A ) )
          = ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ B ) @ C ) @ ( c_List_Oremove1 @ B @ D @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_remove1_Osimps_I2_J) ).

thf(9994,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( D = C )
       => ( ( c_List_Oremove1 @ B @ D @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ B ) @ C ) @ A ) )
          = A ) )
      & ( ( D != C )
       => ( ( c_List_Oremove1 @ B @ D @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ B ) @ C ) @ A ) )
          = ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ B ) @ C ) @ ( c_List_Oremove1 @ B @ D @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1932]) ).

thf(1271,axiom,
    ! [A: $i] :
      ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ A ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__irrefl__nat) ).

thf(7240,plain,
    ! [A: $i] :
      ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ A ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1271]) ).

thf(428,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ B ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Diff__insert) ).

thf(3947,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ B ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[428]) ).

thf(482,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Diff__Un) ).

thf(4124,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[482]) ).

thf(1294,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ F ) @ E ) @ D )
        = C )
     => ( ( B
          = ( hAPP @ ( hAPP @ ( c_List_Oappend @ F ) @ D ) @ A ) )
       => ( ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ F ) @ E ) @ B )
          = ( hAPP @ ( hAPP @ ( c_List_Oappend @ F ) @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Cons__eq__appendI) ).

thf(7326,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ F ) @ E ) @ D )
        = C )
     => ( ( B
          = ( hAPP @ ( hAPP @ ( c_List_Oappend @ F ) @ D ) @ A ) )
       => ( ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ F ) @ E ) @ B )
          = ( hAPP @ ( hAPP @ ( c_List_Oappend @ F ) @ C ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1294]) ).

thf(742,axiom,
    class_Groups_Oab__semigroup__mult @ tc_Nat_Onat,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat__Onat__Groups_Oab__semigroup__mult) ).

thf(5056,plain,
    class_Groups_Oab__semigroup__mult @ tc_Nat_Onat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[742]) ).

thf(501,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Map_Odom @ C @ B @ A )
      = ( hAPP @ ( c_Set_OCollect @ C ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ tc_HOL_Obool @ C ) @ c_fNot ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ C @ ( tc_Option_Ooption @ B ) @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_Option_Ooption @ B ) @ ( tc_fun @ ( tc_Option_Ooption @ B ) @ tc_HOL_Obool ) @ C ) @ c_fequal ) @ A ) ) @ ( c_Option_Ooption_ONone @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_dom__def) ).

thf(4198,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Map_Odom @ C @ B @ A )
      = ( hAPP @ ( c_Set_OCollect @ C ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ tc_HOL_Obool @ C ) @ c_fNot ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ C @ ( tc_Option_Ooption @ B ) @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_Option_Ooption @ B ) @ ( tc_fun @ ( tc_Option_Ooption @ B ) @ tc_HOL_Obool ) @ C ) @ c_fequal ) @ A ) ) @ ( c_Option_Ooption_ONone @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[501]) ).

thf(2368,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( c_Natural_Oevaln @ F @ E @ D @ C )
     => ( ( c_Natural_Oevaln @ B @ C @ D @ A )
       => ( c_Natural_Oevaln @ ( c_Com_Ocom_OSemi @ F @ B ) @ E @ D @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_evaln_OSemi) ).

thf(11731,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( c_Natural_Oevaln @ F @ E @ D @ C )
     => ( ( c_Natural_Oevaln @ B @ C @ D @ A )
       => ( c_Natural_Oevaln @ ( c_Com_Ocom_OSemi @ F @ B ) @ E @ D @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2368]) ).

thf(1925,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ C ) @ B ) )
     => ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) )
          & ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ A ) @ B ) ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ C ) @ A ) )
          & ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ A ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_dvd_Ole__less__trans) ).

thf(9971,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ C ) @ B ) )
     => ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) )
          & ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ A ) @ B ) ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ C ) @ A ) )
          & ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ A ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1925]) ).

thf(1800,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ F @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ F @ E ) @ D ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ F @ E ) @ B ) @ A ) ) )
     => ~ ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ F ) @ D ) @ B ) )
         => ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ C ) @ ( hAPP @ A @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_SigmaE2) ).

thf(9465,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ F @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ F @ E ) @ D ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ F @ E ) @ B ) @ A ) ) )
     => ~ ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ F ) @ D ) @ B ) )
         => ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ C ) @ ( hAPP @ A @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1800]) ).

thf(2115,axiom,
    ! [A: $i] :
      ( ( hAPP @ c_Com_Ocom_OBODY @ A )
     != c_Com_Ocom_OSKIP ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_com_Osimps_I19_J) ).

thf(10698,plain,
    ! [A: $i] :
      ( ( hAPP @ c_Com_Ocom_OBODY @ A )
     != c_Com_Ocom_OSKIP ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2115]) ).

thf(1574,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C = B )
    <=> ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ A @ tc_HOL_Obool ) ) @ C ) @ B ) )
        & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ A @ tc_HOL_Obool ) ) @ B ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_set__eq__subset) ).

thf(8487,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( C = B )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ A @ tc_HOL_Obool ) ) @ C ) @ B ) )
          & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ A @ tc_HOL_Obool ) ) @ B ) @ C ) ) ) )
      & ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ A @ tc_HOL_Obool ) ) @ C ) @ B ) )
          & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ A @ tc_HOL_Obool ) ) @ B ) @ C ) ) )
       => ( C = B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1574]) ).

thf(1529,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Set_Oimage @ D @ E @ C ) @ B ) ) @ ( hAPP @ ( c_Set_Oimage @ D @ E @ C ) @ A ) ) ) @ ( hAPP @ ( c_Set_Oimage @ D @ E @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_image__diff__subset) ).

thf(8330,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Set_Oimage @ D @ E @ C ) @ B ) ) @ ( hAPP @ ( c_Set_Oimage @ D @ E @ C ) @ A ) ) ) @ ( hAPP @ ( c_Set_Oimage @ D @ E @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1529]) ).

thf(614,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_List_Odistinct @ D ) @ C ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ D ) ) @ C ) ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ A ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ D ) ) @ C ) ) )
         => ( ( ( hAPP @ ( hAPP @ ( c_List_Onth @ D ) @ C ) @ B )
              = ( hAPP @ ( hAPP @ ( c_List_Onth @ D ) @ C ) @ A ) )
          <=> ( B = A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nth__eq__iff__index__eq) ).

thf(4607,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_List_Odistinct @ D ) @ C ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ D ) ) @ C ) ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ A ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ D ) ) @ C ) ) )
         => ( ( ( ( hAPP @ ( hAPP @ ( c_List_Onth @ D ) @ C ) @ B )
                = ( hAPP @ ( hAPP @ ( c_List_Onth @ D ) @ C ) @ A ) )
             => ( B = A ) )
            & ( ( B = A )
             => ( ( hAPP @ ( hAPP @ ( c_List_Onth @ D ) @ C ) @ B )
                = ( hAPP @ ( hAPP @ ( c_List_Onth @ D ) @ C ) @ A ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[614]) ).

thf(1466,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ C ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ B @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ A ) @ C ) ) @ ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ A @ C ) ) )
      <=> ! [D: $i,E: $i] :
            ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ C ) @ E ) )
              & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ E ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) )
              & ( A
                = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ D ) ) @ E ) ) )
           => ( hBOOL @ ( hAPP @ ( hAPP @ B @ D ) @ E ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_split__neg__lemma) ).

thf(8025,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ C ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) )
     => ( ( ( hBOOL @ ( hAPP @ ( hAPP @ B @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ A ) @ C ) ) @ ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ A @ C ) ) )
         => ! [D: $i,E: $i] :
              ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ C ) @ E ) )
                & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ E ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) )
                & ( A
                  = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ D ) ) @ E ) ) )
             => ( hBOOL @ ( hAPP @ ( hAPP @ B @ D ) @ E ) ) ) )
        & ( ! [D: $i,E: $i] :
              ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ C ) @ E ) )
                & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ E ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) )
                & ( A
                  = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ D ) ) @ E ) ) )
             => ( hBOOL @ ( hAPP @ ( hAPP @ B @ D ) @ E ) ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ B @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ A ) @ C ) ) @ ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ A @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1466]) ).

thf(655,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ B ) @ ( hAPP @ ( c_Set_OCollect @ C ) @ A ) )
      = ( hAPP @ ( c_Set_OCollect @ C ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ C @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ C ) @ c_fimplies ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ tc_HOL_Obool @ C ) @ c_fNot ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ C @ C @ tc_HOL_Obool ) @ c_fequal ) @ B ) ) ) ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_insert__Collect) ).

thf(4760,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ B ) @ ( hAPP @ ( c_Set_OCollect @ C ) @ A ) )
      = ( hAPP @ ( c_Set_OCollect @ C ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ C @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ C ) @ c_fimplies ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ tc_HOL_Obool @ C ) @ c_fNot ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ C @ C @ tc_HOL_Obool ) @ c_fequal ) @ B ) ) ) ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[655]) ).

thf(1394,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_trans__less__add2) ).

thf(7761,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1394]) ).

thf(117,axiom,
    ! [A: $i,B: $i] :
      ( ( ( c_Option_Oset @ B @ A )
        = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) )
    <=> ( A
        = ( c_Option_Ooption_ONone @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_set__empty__eq) ).

thf(2825,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( c_Option_Oset @ B @ A )
          = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) )
       => ( A
          = ( c_Option_Ooption_ONone @ B ) ) )
      & ( ( A
          = ( c_Option_Ooption_ONone @ B ) )
       => ( ( c_Option_Oset @ B @ A )
          = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[117]) ).

thf(626,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Relation_OImage @ D @ E @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Relation_OImage @ D @ E @ C ) @ B ) ) @ ( hAPP @ ( c_Relation_OImage @ D @ E @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Image__Int__subset) ).

thf(4665,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Relation_OImage @ D @ E @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Relation_OImage @ D @ E @ C ) @ B ) ) @ ( hAPP @ ( c_Relation_OImage @ D @ E @ C ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[626]) ).

thf(115,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ F ) ) @ E )
        = ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ D ) ) @ C ) )
     => ( ( c_Map_Omap__upds @ F @ D @ B @ ( hAPP @ ( hAPP @ ( c_List_Oappend @ F ) @ E ) @ A ) @ C )
        = ( c_Map_Omap__upds @ F @ D @ B @ E @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_fun__upds__append__drop) ).

thf(2819,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ F ) ) @ E )
        = ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ D ) ) @ C ) )
     => ( ( c_Map_Omap__upds @ F @ D @ B @ ( hAPP @ ( hAPP @ ( c_List_Oappend @ F ) @ E ) @ A ) @ C )
        = ( c_Map_Omap__upds @ F @ D @ B @ E @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[115]) ).

thf(2312,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Wellfounded_Owf @ B @ A )
    <=> ! [C: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ B ) @ C ) @ ( c_Wellfounded_Oacc @ B @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_wf__acc__iff) ).

thf(11481,plain,
    ! [A: $i,B: $i] :
      ( ( ( c_Wellfounded_Owf @ B @ A )
       => ! [C: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ B ) @ C ) @ ( c_Wellfounded_Oacc @ B @ A ) ) ) )
      & ( ! [C: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ B ) @ C ) @ ( c_Wellfounded_Oacc @ B @ A ) ) )
       => ( c_Wellfounded_Owf @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2312]) ).

thf(932,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Nat_Ocompow @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) @ ( hAPP @ c_Nat_OSuc @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ C @ C @ C ) @ ( hAPP @ ( hAPP @ ( c_Nat_Ocompow @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) @ B ) @ A ) ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_relpow_Osimps_I2_J) ).

thf(5750,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Nat_Ocompow @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) @ ( hAPP @ c_Nat_OSuc @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ C @ C @ C ) @ ( hAPP @ ( hAPP @ ( c_Nat_Ocompow @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) @ B ) @ A ) ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[932]) ).

thf(1780,axiom,
    ! [A: $i] :
      ( ( hAPP @ ( c_SetInterval_Oord__class_OlessThan @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ tc_Nat_Onat ) @ A ) @ ( hAPP @ ( c_SetInterval_Oord__class_OlessThan @ tc_Nat_Onat ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_lessThan__Suc) ).

thf(9394,plain,
    ! [A: $i] :
      ( ( hAPP @ ( c_SetInterval_Oord__class_OlessThan @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ tc_Nat_Onat ) @ A ) @ ( hAPP @ ( c_SetInterval_Oord__class_OlessThan @ tc_Nat_Onat ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1780]) ).

thf(39,axiom,
    class_Groups_Oab__group__add @ tc_Int_Oint,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Groups_Oab__group__add) ).

thf(2531,plain,
    class_Groups_Oab__group__add @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[39]) ).

thf(1258,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( c_Set_Oimage @ ( tc_prod @ C @ B ) @ C @ ( c_Product__Type_Ofst @ C @ B ) ) @ A )
      = ( hAPP @ ( c_Relation_ODomain @ C @ B ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_fst__eq__Domain) ).

thf(7178,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( c_Set_Oimage @ ( tc_prod @ C @ B ) @ C @ ( c_Product__Type_Ofst @ C @ B ) ) @ A )
      = ( hAPP @ ( c_Relation_ODomain @ C @ B ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1258]) ).

thf(1290,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( c_Finite__Set_Ofun__left__comm @ F @ E @ D )
     => ( ( hAPP @ ( hAPP @ D @ C ) @ ( hAPP @ ( hAPP @ D @ B ) @ A ) )
        = ( hAPP @ ( hAPP @ D @ B ) @ ( hAPP @ ( hAPP @ D @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_fun__left__comm_Ofun__left__comm) ).

thf(7311,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( c_Finite__Set_Ofun__left__comm @ F @ E @ D )
     => ( ( hAPP @ ( hAPP @ D @ C ) @ ( hAPP @ ( hAPP @ D @ B ) @ A ) )
        = ( hAPP @ ( hAPP @ D @ B ) @ ( hAPP @ ( hAPP @ D @ C ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1290]) ).

thf(1991,axiom,
    ! [A: $i,B: $i] :
      ( ( ( hAPP @ c_Com_Ocom_OBODY @ B )
        = ( hAPP @ c_Com_Ocom_OBODY @ A ) )
    <=> ( B = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_com_Osimps_I6_J) ).

thf(10248,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( hAPP @ c_Com_Ocom_OBODY @ B )
          = ( hAPP @ c_Com_Ocom_OBODY @ A ) )
       => ( B = A ) )
      & ( ( B = A )
       => ( ( hAPP @ c_Com_Ocom_OBODY @ B )
          = ( hAPP @ c_Com_Ocom_OBODY @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1991]) ).

thf(803,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ C ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ B ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ C ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ C ) @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zdiv__mono2) ).

thf(5306,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ C ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ B ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ C ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ C ) @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[803]) ).

thf(635,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ B ) )
     => ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A )
        = ( hAPP @ ( hAPP @ ( c_Finite__Set_Ofold @ C @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( c_Set_Oinsert @ C ) ) @ A ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_union__fold__insert) ).

thf(4692,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ B ) )
     => ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A )
        = ( hAPP @ ( hAPP @ ( c_Finite__Set_Ofold @ C @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( c_Set_Oinsert @ C ) ) @ A ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[635]) ).

thf(1930,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ C ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_diff__le__mono) ).

thf(9988,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ C ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1930]) ).

thf(205,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( c_Equiv__Relations_Oequiv @ G @ F @ E )
     => ( ( c_Equiv__Relations_Ocongruent @ G @ ( tc_fun @ D @ tc_HOL_Obool ) @ E @ C )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_fun @ G @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Equiv__Relations_Oquotient @ G ) @ F ) @ E ) ) )
         => ( ! [H: $i] :
                ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ G ) @ H ) @ F ) )
               => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ C @ H ) ) @ A ) ) )
           => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ G @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ C ) ) @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_UN__equiv__class__type) ).

thf(3148,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( c_Equiv__Relations_Oequiv @ G @ F @ E )
     => ( ( c_Equiv__Relations_Ocongruent @ G @ ( tc_fun @ D @ tc_HOL_Obool ) @ E @ C )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_fun @ G @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Equiv__Relations_Oquotient @ G ) @ F ) @ E ) ) )
         => ( ! [H: $i] :
                ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ G ) @ H ) @ F ) )
               => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ C @ H ) ) @ A ) ) )
           => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ G @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ C ) ) @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[205]) ).

thf(716,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Relation_Osingle__valued @ C @ C @ B )
     => ( c_Relation_Osingle__valued @ C @ C @ ( hAPP @ ( hAPP @ ( c_Nat_Ocompow @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) @ A ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_single__valued__rel__pow) ).

thf(4984,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Relation_Osingle__valued @ C @ C @ B )
     => ( c_Relation_Osingle__valued @ C @ C @ ( hAPP @ ( hAPP @ ( c_Nat_Ocompow @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) @ A ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[716]) ).

thf(394,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) @ B ) @ A ) )
    <=> ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ C ) @ B ) @ A ) ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_bot__empty__eq2) ).

thf(3820,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ C ) @ B ) @ A ) ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) ) ) ) )
      & ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ C ) @ B ) @ A ) ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[394]) ).

thf(877,axiom,
    class_Groups_Oab__semigroup__mult @ tc_Int_Oint,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Groups_Oab__semigroup__mult) ).

thf(5553,plain,
    class_Groups_Oab__semigroup__mult @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[877]) ).

thf(222,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Nat_Ocompow @ C @ C ) @ ( hAPP @ c_Nat_OSuc @ B ) ) @ A )
      = ( hAPP @ ( c_Fun_Ocomp @ C @ C @ C @ A ) @ ( hAPP @ ( hAPP @ ( c_Nat_Ocompow @ C @ C ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_funpow_Osimps_I2_J) ).

thf(3204,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Nat_Ocompow @ C @ C ) @ ( hAPP @ c_Nat_OSuc @ B ) ) @ A )
      = ( hAPP @ ( c_Fun_Ocomp @ C @ C @ C @ A ) @ ( hAPP @ ( hAPP @ ( c_Nat_Ocompow @ C @ C ) @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[222]) ).

thf(980,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ B ) )
     => ( ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ C @ tc_Nat_Onat ) @ A ) @ B )
          = ( hAPP @ c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) )
      <=> ? [D: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ D ) @ B ) )
            & ( ( hAPP @ A @ D )
              = ( hAPP @ c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) )
            & ! [E: $i] :
                ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ E ) @ B ) )
               => ( ( D != E )
                 => ( ( hAPP @ A @ E )
                    = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_setsum__eq__Suc0__iff) ).

thf(5953,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ B ) )
     => ( ( ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ C @ tc_Nat_Onat ) @ A ) @ B )
            = ( hAPP @ c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) )
         => ? [D: $i] :
              ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ D ) @ B ) )
              & ( ( hAPP @ A @ D )
                = ( hAPP @ c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) )
              & ! [E: $i] :
                  ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ E ) @ B ) )
                 => ( ( D != E )
                   => ( ( hAPP @ A @ E )
                      = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) ) ) )
        & ( ? [D: $i] :
              ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ D ) @ B ) )
              & ( ( hAPP @ A @ D )
                = ( hAPP @ c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) )
              & ! [E: $i] :
                  ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ E ) @ B ) )
                 => ( ( D != E )
                   => ( ( hAPP @ A @ E )
                      = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) ) )
         => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ C @ tc_Nat_Onat ) @ A ) @ B )
            = ( hAPP @ c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[980]) ).

thf(292,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) @ C ) @ B ) ) @ ( c_List_Olistrel1 @ D @ A ) ) )
     => ( ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ D ) ) @ C )
        = ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ D ) ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_listrel1__eq__len) ).

thf(3442,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) @ C ) @ B ) ) @ ( c_List_Olistrel1 @ D @ A ) ) )
     => ( ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ D ) ) @ C )
        = ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ D ) ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[292]) ).

thf(935,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( c_Complete__Lattice_OSup__class_OSup @ ( tc_fun @ C @ tc_HOL_Obool ) @ B ) ) @ A )
        = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) )
    <=> ! [D: $i] :
          ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ D ) @ B ) )
         => ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ D ) @ A )
            = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Union__disjoint) ).

thf(5765,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( c_Complete__Lattice_OSup__class_OSup @ ( tc_fun @ C @ tc_HOL_Obool ) @ B ) ) @ A )
          = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) )
       => ! [D: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ D ) @ B ) )
           => ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ D ) @ A )
              = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) ) )
      & ( ! [D: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ D ) @ B ) )
           => ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ D ) @ A )
              = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) )
       => ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( c_Complete__Lattice_OSup__class_OSup @ ( tc_fun @ C @ tc_HOL_Obool ) @ B ) ) @ A )
          = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[935]) ).

thf(2171,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Otake @ D ) @ ( hAPP @ c_Nat_OSuc @ C ) ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ D ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ D ) @ B ) @ ( hAPP @ ( hAPP @ ( c_List_Otake @ D ) @ C ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_take__Suc__Cons) ).

thf(10903,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Otake @ D ) @ ( hAPP @ c_Nat_OSuc @ C ) ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ D ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ D ) @ B ) @ ( hAPP @ ( hAPP @ ( c_List_Otake @ D ) @ C ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2171]) ).

thf(2009,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( c_Fun_Oinj__on @ G @ F @ E @ ( hAPP @ ( c_Set_Oimage @ D @ G @ C ) @ B ) )
     => ( ( c_Fun_Oinj__on @ D @ G @ C @ B )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ F ) @ A ) @ ( hAPP @ ( c_Set_Oimage @ G @ F @ E ) @ ( hAPP @ ( c_Set_Oimage @ D @ G @ C ) @ B ) ) ) )
         => ( ( hAPP @ ( c_Fun_Othe__inv__into @ D @ F @ B @ ( hAPP @ ( c_Fun_Ocomp @ G @ F @ D @ E ) @ C ) ) @ A )
            = ( hAPP @ ( hAPP @ ( c_Fun_Ocomp @ G @ D @ F @ ( c_Fun_Othe__inv__into @ D @ G @ B @ C ) ) @ ( c_Fun_Othe__inv__into @ G @ F @ ( hAPP @ ( c_Set_Oimage @ D @ G @ C ) @ B ) @ E ) ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_the__inv__into__comp) ).

thf(10326,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( c_Fun_Oinj__on @ G @ F @ E @ ( hAPP @ ( c_Set_Oimage @ D @ G @ C ) @ B ) )
     => ( ( c_Fun_Oinj__on @ D @ G @ C @ B )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ F ) @ A ) @ ( hAPP @ ( c_Set_Oimage @ G @ F @ E ) @ ( hAPP @ ( c_Set_Oimage @ D @ G @ C ) @ B ) ) ) )
         => ( ( hAPP @ ( c_Fun_Othe__inv__into @ D @ F @ B @ ( hAPP @ ( c_Fun_Ocomp @ G @ F @ D @ E ) @ C ) ) @ A )
            = ( hAPP @ ( hAPP @ ( c_Fun_Ocomp @ G @ D @ F @ ( c_Fun_Othe__inv__into @ D @ G @ B @ C ) ) @ ( c_Fun_Othe__inv__into @ G @ F @ ( hAPP @ ( c_Set_Oimage @ D @ G @ C ) @ B ) @ E ) ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2009]) ).

thf(229,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) )
    <=> ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A )
        = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_subset__Un__eq) ).

thf(3221,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) )
       => ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A )
          = A ) )
      & ( ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A )
          = A )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[229]) ).

thf(596,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ F ) ) @ E )
        = ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ D ) ) @ C ) )
     => ( ( ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ F ) ) @ B )
          = ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ D ) ) @ A ) )
       => ( ( hAPP @ ( hAPP @ ( c_List_Ozip @ F @ D ) @ ( hAPP @ ( hAPP @ ( c_List_Oappend @ F ) @ E ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_List_Oappend @ D ) @ C ) @ A ) )
          = ( hAPP @ ( hAPP @ ( c_List_Oappend @ ( tc_prod @ F @ D ) ) @ ( hAPP @ ( hAPP @ ( c_List_Ozip @ F @ D ) @ E ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_List_Ozip @ F @ D ) @ B ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zip__append) ).

thf(4546,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ F ) ) @ E )
        = ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ D ) ) @ C ) )
     => ( ( ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ F ) ) @ B )
          = ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ D ) ) @ A ) )
       => ( ( hAPP @ ( hAPP @ ( c_List_Ozip @ F @ D ) @ ( hAPP @ ( hAPP @ ( c_List_Oappend @ F ) @ E ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_List_Oappend @ D ) @ C ) @ A ) )
          = ( hAPP @ ( hAPP @ ( c_List_Oappend @ ( tc_prod @ F @ D ) ) @ ( hAPP @ ( hAPP @ ( c_List_Ozip @ F @ D ) @ E ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_List_Ozip @ F @ D ) @ B ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[596]) ).

thf(1359,axiom,
    ! [A: $i,B: $i] :
      ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ B ) @ A ) )
        = B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_add__diff__inverse) ).

thf(7620,plain,
    ! [A: $i,B: $i] :
      ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ B ) @ A ) )
        = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1359]) ).

thf(2000,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( c_Map_Omap__comp @ F @ E @ D @ C @ B @ A )
        = ( c_Option_Ooption_ONone @ E ) )
    <=> ( ( ( hAPP @ B @ A )
          = ( c_Option_Ooption_ONone @ F ) )
        | ? [G: $i] :
            ( ( ( hAPP @ B @ A )
              = ( hAPP @ ( c_Option_Ooption_OSome @ F ) @ G ) )
            & ( ( hAPP @ C @ G )
              = ( c_Option_Ooption_ONone @ E ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_map__comp__None__iff) ).

thf(10286,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( ( c_Map_Omap__comp @ F @ E @ D @ C @ B @ A )
          = ( c_Option_Ooption_ONone @ E ) )
       => ( ( ( hAPP @ B @ A )
            = ( c_Option_Ooption_ONone @ F ) )
          | ? [G: $i] :
              ( ( ( hAPP @ B @ A )
                = ( hAPP @ ( c_Option_Ooption_OSome @ F ) @ G ) )
              & ( ( hAPP @ C @ G )
                = ( c_Option_Ooption_ONone @ E ) ) ) ) )
      & ( ( ( ( hAPP @ B @ A )
            = ( c_Option_Ooption_ONone @ F ) )
          | ? [G: $i] :
              ( ( ( hAPP @ B @ A )
                = ( hAPP @ ( c_Option_Ooption_OSome @ F ) @ G ) )
              & ( ( hAPP @ C @ G )
                = ( c_Option_Ooption_ONone @ E ) ) ) )
       => ( ( c_Map_Omap__comp @ F @ E @ D @ C @ B @ A )
          = ( c_Option_Ooption_ONone @ E ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2000]) ).

thf(1155,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ C ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Int_Oint ) @ B ) @ A ) )
       => ! [D: $i] :
            ( ! [E: $i] :
                ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Int_Oint ) @ E ) @ ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Int_Oint @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) @ C ) ) )
               => ! [F: $i] :
                    ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Int_Oint ) @ F ) @ A ) )
                   => ( D
                     != ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ F ) @ E ) ) ) )
           => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ B ) @ D ) )
             => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Int_Oint ) @ D ) @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_bset_I7_J) ).

thf(6766,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ C ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Int_Oint ) @ B ) @ A ) )
       => ! [D: $i] :
            ( ! [E: $i] :
                ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Int_Oint ) @ E ) @ ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Int_Oint @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) @ C ) ) )
               => ! [F: $i] :
                    ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Int_Oint ) @ F ) @ A ) )
                   => ( D
                     != ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ F ) @ E ) ) ) )
           => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ B ) @ D ) )
             => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Int_Oint ) @ D ) @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1155]) ).

thf(57,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Groups_Ocomm__monoid__add @ E )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ C ) )
       => ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ B ) @ C ) )
           => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ E @ E ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ D @ E @ ( tc_fun @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ E @ ( tc_fun @ E @ E ) ) @ D ) @ ( c_If @ E ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ D @ tc_HOL_Obool ) @ c_fequal ) @ B ) ) ) @ A ) ) @ ( c_Groups_Ozero__class_Ozero @ E ) ) ) @ C )
              = ( hAPP @ A @ B ) ) )
          & ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ B ) @ C ) )
           => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ E @ E ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ D @ E @ ( tc_fun @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ E @ ( tc_fun @ E @ E ) ) @ D ) @ ( c_If @ E ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ D @ tc_HOL_Obool ) @ c_fequal ) @ B ) ) ) @ A ) ) @ ( c_Groups_Ozero__class_Ozero @ E ) ) ) @ C )
              = ( c_Groups_Ozero__class_Ozero @ E ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_setsum__delta) ).

thf(2603,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Groups_Ocomm__monoid__add @ E )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ C ) )
       => ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ B ) @ C ) )
           => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ E @ E ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ D @ E @ ( tc_fun @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ E @ ( tc_fun @ E @ E ) ) @ D ) @ ( c_If @ E ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ D @ tc_HOL_Obool ) @ c_fequal ) @ B ) ) ) @ A ) ) @ ( c_Groups_Ozero__class_Ozero @ E ) ) ) @ C )
              = ( hAPP @ A @ B ) ) )
          & ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ B ) @ C ) )
           => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ E @ E ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ D @ E @ ( tc_fun @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ E @ ( tc_fun @ E @ E ) ) @ D ) @ ( c_If @ E ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ D @ tc_HOL_Obool ) @ c_fequal ) @ B ) ) ) @ A ) ) @ ( c_Groups_Ozero__class_Ozero @ E ) ) ) @ C )
              = ( c_Groups_Ozero__class_Ozero @ E ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[57]) ).

thf(142,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A )
        = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) )
    <=> ( ( B
          = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) )
        & ( A
          = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Un__empty) ).

thf(2916,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A )
          = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) )
       => ( ( B
            = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) )
          & ( A
            = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) ) )
      & ( ( ( B
            = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) )
          & ( A
            = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) )
       => ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A )
          = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[142]) ).

thf(590,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Wellfounded_Owf @ C @ B )
     => ( ( c_Wellfounded_Owf @ C @ A )
       => ( ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Relation_ODomain @ C @ C ) @ B ) ) @ ( hAPP @ ( c_Relation_ORange @ C @ C ) @ A ) )
            = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) )
         => ( c_Wellfounded_Owf @ C @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) ) @ B ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_wf__Un) ).

thf(4524,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Wellfounded_Owf @ C @ B )
     => ( ( c_Wellfounded_Owf @ C @ A )
       => ( ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Relation_ODomain @ C @ C ) @ B ) ) @ ( hAPP @ ( c_Relation_ORange @ C @ C ) @ A ) )
            = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) )
         => ( c_Wellfounded_Owf @ C @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) ) @ B ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[590]) ).

thf(1721,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) ) @ B ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( c_Wellfounded_Oacc @ C @ A ) ) @ ( c_Wellfounded_Oacc @ C @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_acc__subset) ).

thf(9172,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) ) @ B ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( c_Wellfounded_Oacc @ C @ A ) ) @ ( c_Wellfounded_Oacc @ C @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1721]) ).

thf(1282,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) ) ) ) @ ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_card__Diff1__le) ).

thf(7277,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) ) ) ) @ ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1282]) ).

thf(174,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Nat_Ocompow @ ( tc_fun @ D @ C ) @ ( tc_fun @ D @ C ) ) @ B ) @ ( c_Fun_Ocomp @ C @ C @ D @ A ) )
      = ( c_Fun_Ocomp @ C @ C @ D @ ( hAPP @ ( hAPP @ ( c_Nat_Ocompow @ C @ C ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_comp__funpow) ).

thf(3029,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Nat_Ocompow @ ( tc_fun @ D @ C ) @ ( tc_fun @ D @ C ) ) @ B ) @ ( c_Fun_Ocomp @ C @ C @ D @ A ) )
      = ( c_Fun_Ocomp @ C @ C @ D @ ( hAPP @ ( hAPP @ ( c_Nat_Ocompow @ C @ C ) @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[174]) ).

thf(1752,axiom,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) )
        & ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ A ) @ B ) ) )
    <=> ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) )
        & ( B != A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_dvd_Oless__le) ).

thf(9287,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) )
          & ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ A ) @ B ) ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) )
          & ( B != A ) ) )
      & ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) )
          & ( B != A ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) )
          & ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ A ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1752]) ).

thf(1283,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_dvd__imp__le) ).

thf(7279,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1283]) ).

thf(2003,axiom,
    ! [A: $i] : ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ tc_Nat_Onat ) @ ( hAPP @ ( c_SetInterval_Oord__class_OatMost @ tc_Nat_Onat ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_finite__atMost) ).

thf(10305,plain,
    ! [A: $i] : ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ tc_Nat_Onat ) @ ( hAPP @ ( c_SetInterval_Oord__class_OatMost @ tc_Nat_Onat ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2003]) ).

thf(2396,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ D ) @ C ) )
     => ( ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ B ) @ ( hAPP @ A @ D ) ) @ ( hAPP @ ( c_Set_Oimage @ E @ B @ A ) @ C ) )
        = ( hAPP @ ( c_Set_Oimage @ E @ B @ A ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_insert__image) ).

thf(11844,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ D ) @ C ) )
     => ( ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ B ) @ ( hAPP @ A @ D ) ) @ ( hAPP @ ( c_Set_Oimage @ E @ B @ A ) @ C ) )
        = ( hAPP @ ( c_Set_Oimage @ E @ B @ A ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2396]) ).

thf(1278,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) )
       => ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ A ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nat__dvd__not__less) ).

thf(7262,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) )
       => ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ A ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1278]) ).

thf(600,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( c_Set_Ovimage @ D @ C @ B ) @ ( c_Complete__Lattice_OSup__class_OSup @ ( tc_fun @ C @ tc_HOL_Obool ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ A ) @ ( c_Set_Ovimage @ D @ C @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_vimage__Union) ).

thf(4563,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( c_Set_Ovimage @ D @ C @ B ) @ ( c_Complete__Lattice_OSup__class_OSup @ ( tc_fun @ C @ tc_HOL_Obool ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ A ) @ ( c_Set_Ovimage @ D @ C @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[600]) ).

thf(1381,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_List_Opartition @ C @ B @ A )
      = ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ C ) @ ( tc_List_Olist @ C ) ) @ ( hAPP @ ( c_List_Ofilter @ C @ B ) @ A ) ) @ ( hAPP @ ( c_List_Ofilter @ C @ ( hAPP @ ( c_Fun_Ocomp @ tc_HOL_Obool @ tc_HOL_Obool @ C @ c_fNot ) @ B ) ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_partition__filter__conv) ).

thf(7701,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_List_Opartition @ C @ B @ A )
      = ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ C ) @ ( tc_List_Olist @ C ) ) @ ( hAPP @ ( c_List_Ofilter @ C @ B ) @ A ) ) @ ( hAPP @ ( c_List_Ofilter @ C @ ( hAPP @ ( c_Fun_Ocomp @ tc_HOL_Obool @ tc_HOL_Obool @ C @ c_fNot ) @ B ) ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1381]) ).

thf(1690,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Diff__idemp) ).

thf(9067,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1690]) ).

thf(2093,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ D ) @ C ) @ B ) ) @ ( c_Transitive__Closure_Ortrancl @ D @ ( hAPP @ ( c_Relation_Oconverse @ D @ D ) @ A ) ) ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ D ) @ B ) @ C ) ) @ ( c_Transitive__Closure_Ortrancl @ D @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_rtrancl__converseD) ).

thf(10630,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ D ) @ C ) @ B ) ) @ ( c_Transitive__Closure_Ortrancl @ D @ ( hAPP @ ( c_Relation_Oconverse @ D @ D ) @ A ) ) ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ D ) @ B ) @ C ) ) @ ( c_Transitive__Closure_Ortrancl @ D @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2093]) ).

thf(2303,axiom,
    ~ ( hBOOL @ c_fFalse ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_c__fFalse__1) ).

thf(11421,plain,
    ~ ( hBOOL @ c_fFalse ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2303]) ).

thf(1556,axiom,
    ! [A: $i,B: $i] :
      ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_not__add__less1) ).

thf(8426,plain,
    ! [A: $i,B: $i] :
      ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1556]) ).

thf(1678,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( ( c_Map_Omap__comp @ G @ F @ E @ D @ C @ B )
        = ( hAPP @ ( c_Option_Ooption_OSome @ F ) @ A ) )
    <=> ? [H: $i] :
          ( ( ( hAPP @ C @ B )
            = ( hAPP @ ( c_Option_Ooption_OSome @ G ) @ H ) )
          & ( ( hAPP @ D @ H )
            = ( hAPP @ ( c_Option_Ooption_OSome @ F ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_map__comp__Some__iff) ).

thf(8996,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( ( ( c_Map_Omap__comp @ G @ F @ E @ D @ C @ B )
          = ( hAPP @ ( c_Option_Ooption_OSome @ F ) @ A ) )
       => ? [H: $i] :
            ( ( ( hAPP @ C @ B )
              = ( hAPP @ ( c_Option_Ooption_OSome @ G ) @ H ) )
            & ( ( hAPP @ D @ H )
              = ( hAPP @ ( c_Option_Ooption_OSome @ F ) @ A ) ) ) )
      & ( ? [H: $i] :
            ( ( ( hAPP @ C @ B )
              = ( hAPP @ ( c_Option_Ooption_OSome @ G ) @ H ) )
            & ( ( hAPP @ D @ H )
              = ( hAPP @ ( c_Option_Ooption_OSome @ F ) @ A ) ) )
       => ( ( c_Map_Omap__comp @ G @ F @ E @ D @ C @ B )
          = ( hAPP @ ( c_Option_Ooption_OSome @ F ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1678]) ).

thf(201,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( hAPP @ ( c_Set_Oimage @ D @ C @ B ) @ A )
        = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) )
    <=> ( A
        = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_image__is__empty) ).

thf(3131,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( ( hAPP @ ( c_Set_Oimage @ D @ C @ B ) @ A )
          = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) )
       => ( A
          = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) )
      & ( ( A
          = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) )
       => ( ( hAPP @ ( c_Set_Oimage @ D @ C @ B ) @ A )
          = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[201]) ).

thf(712,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ B ) )
     => ( ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ C @ tc_Nat_Onat ) @ A ) @ B )
          = ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) )
      <=> ? [D: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ D ) @ B ) )
            & ( ( hAPP @ A @ D )
              = ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) )
            & ! [E: $i] :
                ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ E ) @ B ) )
               => ( ( D != E )
                 => ( ( hAPP @ A @ E )
                    = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_setsum__eq__1__iff) ).

thf(4956,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ B ) )
     => ( ( ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ C @ tc_Nat_Onat ) @ A ) @ B )
            = ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) )
         => ? [D: $i] :
              ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ D ) @ B ) )
              & ( ( hAPP @ A @ D )
                = ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) )
              & ! [E: $i] :
                  ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ E ) @ B ) )
                 => ( ( D != E )
                   => ( ( hAPP @ A @ E )
                      = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) ) ) )
        & ( ? [D: $i] :
              ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ D ) @ B ) )
              & ( ( hAPP @ A @ D )
                = ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) )
              & ! [E: $i] :
                  ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ E ) @ B ) )
                 => ( ( D != E )
                   => ( ( hAPP @ A @ E )
                      = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) ) )
         => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ C @ tc_Nat_Onat ) @ A ) @ B )
            = ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[712]) ).

thf(466,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ F ) @ E ) ) @ D )
        = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ B ) ) @ A ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ B ) ) @ A ) ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ D ) @ F ) )
         => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ A ) )
           => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ C ) )
             => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ C ) @ F ) )
               => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ B ) @ E ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zdiv__mono2__neg__lemma) ).

thf(4070,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ F ) @ E ) ) @ D )
        = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ B ) ) @ A ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ B ) ) @ A ) ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ D ) @ F ) )
         => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ A ) )
           => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ C ) )
             => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ C ) @ F ) )
               => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ B ) @ E ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[466]) ).

thf(1311,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i,I: $i,J: $i,K: $i] :
      ( ( ( hAPP @ K @ ( hAPP @ J @ I ) )
        = ( hAPP @ H @ ( hAPP @ G @ F ) ) )
     => ( ( hAPP @ ( hAPP @ ( c_Fun_Ocomp @ E @ D @ C @ K ) @ J ) @ I )
        = ( hAPP @ ( hAPP @ ( c_Fun_Ocomp @ B @ D @ A @ H ) @ G ) @ F ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_comp__cong) ).

thf(7401,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i,I: $i,J: $i,K: $i] :
      ( ( ( hAPP @ K @ ( hAPP @ J @ I ) )
        = ( hAPP @ H @ ( hAPP @ G @ F ) ) )
     => ( ( hAPP @ ( hAPP @ ( c_Fun_Ocomp @ E @ D @ C @ K ) @ J ) @ I )
        = ( hAPP @ ( hAPP @ ( c_Fun_Ocomp @ B @ D @ A @ H ) @ G ) @ F ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1311]) ).

thf(2197,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ F @ E ) ) @ D ) @ ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ F @ C @ E ) @ B ) @ A ) ) )
     => ~ ! [G: $i,H: $i,I: $i] :
            ( ( D
              = ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ F @ E ) @ G ) @ I ) )
           => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ F @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ F @ C ) @ G ) @ H ) ) @ B ) )
             => ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ C @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ C @ E ) @ H ) @ I ) ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_rel__compE) ).

thf(10997,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ F @ E ) ) @ D ) @ ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ F @ C @ E ) @ B ) @ A ) ) )
     => ~ ! [G: $i,H: $i,I: $i] :
            ( ( D
              = ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ F @ E ) @ G ) @ I ) )
           => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ F @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ F @ C ) @ G ) @ H ) ) @ B ) )
             => ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ C @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ C @ E ) @ H ) @ I ) ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2197]) ).

thf(1106,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( c_Map_Omap__upds @ G @ F @ E @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ G ) @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ F ) @ B ) @ A ) )
      = ( c_Map_Omap__upds @ G @ F @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ G @ ( tc_Option_Ooption @ F ) ) @ E ) @ D ) @ ( hAPP @ ( c_Option_Ooption_OSome @ F ) @ B ) ) @ C @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_map__upds__Cons) ).

thf(6547,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( c_Map_Omap__upds @ G @ F @ E @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ G ) @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ F ) @ B ) @ A ) )
      = ( c_Map_Omap__upds @ G @ F @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ G @ ( tc_Option_Ooption @ F ) ) @ E ) @ D ) @ ( hAPP @ ( c_Option_Ooption_OSome @ F ) @ B ) ) @ C @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1106]) ).

thf(1412,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ D @ C ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ D ) @ A ) )
       => ( hBOOL @ ( hAPP @ A @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_rev__predicate1D) ).

thf(7822,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ D @ C ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ D ) @ A ) )
       => ( hBOOL @ ( hAPP @ A @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1412]) ).

thf(969,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hAPP @ ( c_Relation_OImage @ F @ E @ D ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ C @ ( tc_fun @ F @ tc_HOL_Obool ) ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ C @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ F @ tc_HOL_Obool ) @ ( tc_fun @ E @ tc_HOL_Obool ) @ C ) @ ( c_Relation_OImage @ F @ E @ D ) ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Image__UN) ).

thf(5910,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hAPP @ ( c_Relation_OImage @ F @ E @ D ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ C @ ( tc_fun @ F @ tc_HOL_Obool ) ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ C @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ F @ tc_HOL_Obool ) @ ( tc_fun @ E @ tc_HOL_Obool ) @ C ) @ ( c_Relation_OImage @ F @ E @ D ) ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[969]) ).

thf(1141,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( hAPP @ E @ D )
        = ( c_Option_Ooption_ONone @ C ) )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ ( c_Map_Odom @ B @ C @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ B ) @ D ) @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ ( c_Map_Odom @ B @ C @ E ) ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_dom__minus) ).

thf(6723,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( hAPP @ E @ D )
        = ( c_Option_Ooption_ONone @ C ) )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ ( c_Map_Odom @ B @ C @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ B ) @ D ) @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ ( c_Map_Odom @ B @ C @ E ) ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1141]) ).

thf(2270,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( F
        = ( hAPP @ E @ D ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ D ) @ B ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ A ) @ F ) @ ( hAPP @ ( c_Set_Oimage @ C @ A @ E ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_image__eqI) ).

thf(11299,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( F
        = ( hAPP @ E @ D ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ D ) @ B ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ A ) @ F ) @ ( hAPP @ ( c_Set_Oimage @ C @ A @ E ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2270]) ).

thf(184,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ B ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ B ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_insert__is__Un) ).

thf(3061,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ B ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ B ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[184]) ).

thf(2220,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( c_Typedef_Otype__definition @ F @ E @ D @ C @ B )
     => ( ( hAPP @ C @ ( hAPP @ D @ A ) )
        = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_type__definition_ORep__inverse) ).

thf(11087,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( c_Typedef_Otype__definition @ F @ E @ D @ C @ B )
     => ( ( hAPP @ C @ ( hAPP @ D @ A ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2220]) ).

thf(85,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Groups_Ocomm__monoid__mult @ E )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ C ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ B ) )
         => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Finite__Set_Ofold__image @ E @ D @ ( c_Groups_Otimes__class_Otimes @ E ) ) @ A ) @ ( c_Groups_Oone__class_Oone @ E ) ) @ C ) ) @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Finite__Set_Ofold__image @ E @ D @ ( c_Groups_Otimes__class_Otimes @ E ) ) @ A ) @ ( c_Groups_Oone__class_Oone @ E ) ) @ B ) )
            = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Finite__Set_Ofold__image @ E @ D @ ( c_Groups_Otimes__class_Otimes @ E ) ) @ A ) @ ( c_Groups_Oone__class_Oone @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) ) ) @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Finite__Set_Ofold__image @ E @ D @ ( c_Groups_Otimes__class_Otimes @ E ) ) @ A ) @ ( c_Groups_Oone__class_Oone @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_fold__image__Un__Int) ).

thf(2708,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Groups_Ocomm__monoid__mult @ E )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ C ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ B ) )
         => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Finite__Set_Ofold__image @ E @ D @ ( c_Groups_Otimes__class_Otimes @ E ) ) @ A ) @ ( c_Groups_Oone__class_Oone @ E ) ) @ C ) ) @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Finite__Set_Ofold__image @ E @ D @ ( c_Groups_Otimes__class_Otimes @ E ) ) @ A ) @ ( c_Groups_Oone__class_Oone @ E ) ) @ B ) )
            = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Finite__Set_Ofold__image @ E @ D @ ( c_Groups_Otimes__class_Otimes @ E ) ) @ A ) @ ( c_Groups_Oone__class_Oone @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) ) ) @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Finite__Set_Ofold__image @ E @ D @ ( c_Groups_Otimes__class_Otimes @ E ) ) @ A ) @ ( c_Groups_Oone__class_Oone @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[85]) ).

thf(2122,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Divides_Odivmod__nat__rel @ D @ C ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ tc_Nat_Onat @ tc_Nat_Onat ) @ B ) @ A ) ) )
     => ( ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Nat_Onat ) @ D ) @ C )
        = B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_div__eq) ).

thf(10722,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Divides_Odivmod__nat__rel @ D @ C ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ tc_Nat_Onat @ tc_Nat_Onat ) @ B ) @ A ) ) )
     => ( ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Nat_Onat ) @ D ) @ C )
        = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2122]) ).

thf(1150,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_Map_Omap__of @ D @ C @ ( hAPP @ ( hAPP @ ( c_List_Oappend @ ( tc_prod @ D @ C ) ) @ B ) @ A ) )
      = ( c_Map_Omap__add @ D @ C @ ( c_Map_Omap__of @ D @ C @ A ) @ ( c_Map_Omap__of @ D @ C @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_map__of__append) ).

thf(6749,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_Map_Omap__of @ D @ C @ ( hAPP @ ( hAPP @ ( c_List_Oappend @ ( tc_prod @ D @ C ) ) @ B ) @ A ) )
      = ( c_Map_Omap__add @ D @ C @ ( c_Map_Omap__of @ D @ C @ A ) @ ( c_Map_Omap__of @ D @ C @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1150]) ).

thf(1004,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ C ) @ B ) )
       => ( ( hAPP @ ( c_Set_Oimage @ tc_Nat_Onat @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ tc_Nat_Onat @ tc_Nat_Onat ) @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) ) @ C ) ) @ ( hAPP @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Nat_Onat @ A ) @ B ) )
          = ( hAPP @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ A ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ B ) @ C ) ) ) )
      & ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ C ) @ B ) )
       => ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ A ) @ B ) )
           => ( ( hAPP @ ( c_Set_Oimage @ tc_Nat_Onat @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ tc_Nat_Onat @ tc_Nat_Onat ) @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) ) @ C ) ) @ ( hAPP @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Nat_Onat @ A ) @ B ) )
              = ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) ) ) )
          & ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ A ) @ B ) )
           => ( ( hAPP @ ( c_Set_Oimage @ tc_Nat_Onat @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ tc_Nat_Onat @ tc_Nat_Onat ) @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) ) @ C ) ) @ ( hAPP @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Nat_Onat @ A ) @ B ) )
              = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_image__minus__const__atLeastLessThan__nat) ).

thf(6066,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ C ) @ B ) )
       => ( ( hAPP @ ( c_Set_Oimage @ tc_Nat_Onat @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ tc_Nat_Onat @ tc_Nat_Onat ) @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) ) @ C ) ) @ ( hAPP @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Nat_Onat @ A ) @ B ) )
          = ( hAPP @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ A ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ B ) @ C ) ) ) )
      & ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ C ) @ B ) )
       => ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ A ) @ B ) )
           => ( ( hAPP @ ( c_Set_Oimage @ tc_Nat_Onat @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ tc_Nat_Onat @ tc_Nat_Onat ) @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) ) @ C ) ) @ ( hAPP @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Nat_Onat @ A ) @ B ) )
              = ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) ) ) )
          & ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ A ) @ B ) )
           => ( ( hAPP @ ( c_Set_Oimage @ tc_Nat_Onat @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ tc_Nat_Onat @ tc_Nat_Onat ) @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) ) @ C ) ) @ ( hAPP @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Nat_Onat @ A ) @ B ) )
              = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1004]) ).

thf(341,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) @ A ) )
     => ~ ( ( hBOOL @ ( hAPP @ C @ A ) )
         => ~ ( hBOOL @ ( hAPP @ B @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_inf1E) ).

thf(3617,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) @ A ) )
     => ~ ( ( hBOOL @ ( hAPP @ C @ A ) )
         => ~ ( hBOOL @ ( hAPP @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[341]) ).

thf(1982,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ E ) @ ( tc_List_Olist @ E ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ E ) @ ( tc_List_Olist @ E ) ) @ D ) @ C ) ) @ ( c_Transitive__Closure_Ortrancl @ ( tc_List_Olist @ E ) @ ( c_List_Olistrel1 @ E @ B ) ) ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ E ) @ ( tc_List_Olist @ E ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ E ) @ ( tc_List_Olist @ E ) ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ E ) @ A ) @ D ) ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ E ) @ A ) @ C ) ) ) @ ( c_Transitive__Closure_Ortrancl @ ( tc_List_Olist @ E ) @ ( c_List_Olistrel1 @ E @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_rtrancl__listrel1__ConsI1) ).

thf(10221,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ E ) @ ( tc_List_Olist @ E ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ E ) @ ( tc_List_Olist @ E ) ) @ D ) @ C ) ) @ ( c_Transitive__Closure_Ortrancl @ ( tc_List_Olist @ E ) @ ( c_List_Olistrel1 @ E @ B ) ) ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ E ) @ ( tc_List_Olist @ E ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ E ) @ ( tc_List_Olist @ E ) ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ E ) @ A ) @ D ) ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ E ) @ A ) @ C ) ) ) @ ( c_Transitive__Closure_Ortrancl @ ( tc_List_Olist @ E ) @ ( c_List_Olistrel1 @ E @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1982]) ).

thf(1627,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ C ) @ ( c_Transitive__Closure_Otrancl @ D @ B ) ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) ) @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ C ) @ ( c_Transitive__Closure_Otrancl @ D @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_trancl__mono) ).

thf(8785,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ C ) @ ( c_Transitive__Closure_Otrancl @ D @ B ) ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) ) @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ C ) @ ( c_Transitive__Closure_Otrancl @ D @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1627]) ).

thf(1096,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Equiv__Relations_Oequiv @ E @ D @ C )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Relation_OImage @ E @ E @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ B ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ E @ tc_HOL_Obool ) ) ) ) ) @ ( hAPP @ ( c_Relation_OImage @ E @ E @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ E @ tc_HOL_Obool ) ) ) ) ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ B ) @ D ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ A ) @ B ) ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_subset__equiv__class) ).

thf(6506,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Equiv__Relations_Oequiv @ E @ D @ C )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Relation_OImage @ E @ E @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ B ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ E @ tc_HOL_Obool ) ) ) ) ) @ ( hAPP @ ( c_Relation_OImage @ E @ E @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ E @ tc_HOL_Obool ) ) ) ) ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ B ) @ D ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ A ) @ B ) ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1096]) ).

thf(1628,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Nat_Onat @ ( hAPP @ c_Nat_OSuc @ B ) ) @ A )
      = ( c_SetInterval_Oord__class_OgreaterThanLessThan @ tc_Nat_Onat @ B @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_atLeastSucLessThan__greaterThanLessThan) ).

thf(8787,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Nat_Onat @ ( hAPP @ c_Nat_OSuc @ B ) ) @ A )
      = ( c_SetInterval_Oord__class_OgreaterThanLessThan @ tc_Nat_Onat @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1628]) ).

thf(1771,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i,I: $i,J: $i,K: $i,L: $i] :
      ( ( c_Com_Ocom_Ocom__rec @ L @ K @ J @ I @ H @ G @ F @ E @ D @ ( c_Com_Ocom_OCall @ C @ B @ A ) )
      = ( hAPP @ ( hAPP @ ( hAPP @ D @ C ) @ B ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_com_Orecs_I8_J) ).

thf(9360,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i,I: $i,J: $i,K: $i,L: $i] :
      ( ( c_Com_Ocom_Ocom__rec @ L @ K @ J @ I @ H @ G @ F @ E @ D @ ( c_Com_Ocom_OCall @ C @ B @ A ) )
      = ( hAPP @ ( hAPP @ ( hAPP @ D @ C ) @ B ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1771]) ).

thf(1661,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ E @ D ) ) @ ( hAPP @ E @ C ) ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ B @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ B @ B ) @ D ) @ C ) ) @ ( c_Wellfounded_Omlex__prod @ B @ E @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mlex__less) ).

thf(8919,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ E @ D ) ) @ ( hAPP @ E @ C ) ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ B @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ B @ B ) @ D ) @ C ) ) @ ( c_Wellfounded_Omlex__prod @ B @ E @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1661]) ).

thf(2241,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Com_Ocom_OWhile @ E @ D )
     != ( c_Com_Ocom_OCond @ C @ B @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_com_Osimps_I53_J) ).

thf(11178,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Com_Ocom_OWhile @ E @ D )
     != ( c_Com_Ocom_OCond @ C @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2241]) ).

thf(2247,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( c_Predicate_Opred__comp @ G @ F @ E @ D @ C @ B @ A )
    <=> ? [H: $i] :
          ( ( hBOOL @ ( hAPP @ ( hAPP @ D @ B ) @ H ) )
          & ( hBOOL @ ( hAPP @ ( hAPP @ C @ H ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_pred__comp_Oequations) ).

thf(11200,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( ( c_Predicate_Opred__comp @ G @ F @ E @ D @ C @ B @ A )
       => ? [H: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ D @ B ) @ H ) )
            & ( hBOOL @ ( hAPP @ ( hAPP @ C @ H ) @ A ) ) ) )
      & ( ? [H: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ D @ B ) @ H ) )
            & ( hBOOL @ ( hAPP @ ( hAPP @ C @ H ) @ A ) ) )
       => ( c_Predicate_Opred__comp @ G @ F @ E @ D @ C @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2247]) ).

thf(765,axiom,
    ( c_Wellfounded_Opred__nat
    = ( hAPP @ ( c_Set_OCollect @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) ) @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ tc_Nat_Onat @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_Nat_Onat @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ tc_Nat_Onat ) @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ tc_Nat_Onat @ tc_HOL_Obool ) @ c_fequal ) ) @ c_Nat_OSuc ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_pred__nat__def) ).

thf(5150,plain,
    ( c_Wellfounded_Opred__nat
    = ( hAPP @ ( c_Set_OCollect @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) ) @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ tc_Nat_Onat @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_Nat_Onat @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ tc_Nat_Onat ) @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ tc_Nat_Onat @ tc_HOL_Obool ) @ c_fequal ) ) @ c_Nat_OSuc ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[765]) ).

thf(1129,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_List_Olist_Olist__size @ E @ D @ ( hAPP @ ( hAPP @ ( c_List_Omap @ C @ E ) @ B ) @ A ) )
      = ( c_List_Olist_Olist__size @ C @ ( hAPP @ ( c_Fun_Ocomp @ E @ tc_Nat_Onat @ C @ D ) @ B ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_list__size__map) ).

thf(6676,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_List_Olist_Olist__size @ E @ D @ ( hAPP @ ( hAPP @ ( c_List_Omap @ C @ E ) @ B ) @ A ) )
      = ( c_List_Olist_Olist__size @ C @ ( hAPP @ ( c_Fun_Ocomp @ E @ tc_Nat_Onat @ C @ D ) @ B ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1129]) ).

thf(484,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Equiv__Relations_Oquotient @ B ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) @ A )
      = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ ( tc_fun @ B @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_quotient__empty) ).

thf(4130,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Equiv__Relations_Oquotient @ B ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) @ A )
      = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ ( tc_fun @ B @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[484]) ).

thf(1061,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ D ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ C ) ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ D ) @ B ) ) @ C ) ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_left__add__mult__distrib) ).

thf(6315,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ D ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ C ) ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ D ) @ B ) ) @ C ) ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1061]) ).

thf(540,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ C @ B ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) @ A )
      = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ ( tc_prod @ C @ B ) @ tc_HOL_Obool ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Sigma__empty1) ).

thf(4349,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ C @ B ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) @ A )
      = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ ( tc_prod @ C @ B ) @ tc_HOL_Obool ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[540]) ).

thf(760,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ A ) @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ) ) )
    <=> ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ B ) @ A ) )
        | ( B = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zless__add1__eq) ).

thf(5130,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ A ) @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ) ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ B ) @ A ) )
          | ( B = A ) ) )
      & ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ B ) @ A ) )
          | ( B = A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ A ) @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[760]) ).

thf(768,axiom,
    ! [A: $i,B: $i] : ( c_Fun_Oinj__on @ tc_Nat_Onat @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ tc_Nat_Onat @ tc_Nat_Onat ) @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) ) @ B ) @ A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_inj__on__add__nat) ).

thf(5160,plain,
    ! [A: $i,B: $i] : ( c_Fun_Oinj__on @ tc_Nat_Onat @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ tc_Nat_Onat @ tc_Nat_Onat ) @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) ) @ B ) @ A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[768]) ).

thf(245,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Un__insert__right) ).

thf(3280,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[245]) ).

thf(2391,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Natural_Oevaln @ ( c_Com_Ocom_OWhile @ E @ D ) @ C @ B @ A )
    <=> ( ( ( A = C )
          & ~ ( hBOOL @ ( hAPP @ E @ C ) ) )
        | ( ( hBOOL @ ( hAPP @ E @ C ) )
          & ? [F: $i] :
              ( ( c_Natural_Oevaln @ D @ C @ B @ F )
              & ( c_Natural_Oevaln @ ( c_Com_Ocom_OWhile @ E @ D ) @ F @ B @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_evaln_Oequations_I8_J) ).

thf(11812,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( c_Natural_Oevaln @ ( c_Com_Ocom_OWhile @ E @ D ) @ C @ B @ A )
       => ( ( ( A = C )
            & ~ ( hBOOL @ ( hAPP @ E @ C ) ) )
          | ( ( hBOOL @ ( hAPP @ E @ C ) )
            & ? [F: $i] :
                ( ( c_Natural_Oevaln @ D @ C @ B @ F )
                & ( c_Natural_Oevaln @ ( c_Com_Ocom_OWhile @ E @ D ) @ F @ B @ A ) ) ) ) )
      & ( ( ( ( A = C )
            & ~ ( hBOOL @ ( hAPP @ E @ C ) ) )
          | ( ( hBOOL @ ( hAPP @ E @ C ) )
            & ? [F: $i] :
                ( ( c_Natural_Oevaln @ D @ C @ B @ F )
                & ( c_Natural_Oevaln @ ( c_Com_Ocom_OWhile @ E @ D ) @ F @ B @ A ) ) ) )
       => ( c_Natural_Oevaln @ ( c_Com_Ocom_OWhile @ E @ D ) @ C @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2391]) ).

thf(1138,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Nat_Onat ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Nat_Onat ) @ C ) @ B ) ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_div__mult2__eq) ).

thf(6713,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Nat_Onat ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Nat_Onat ) @ C ) @ B ) ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1138]) ).

thf(1936,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( c_List_Oset @ C ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ C ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ B ) @ ( hAPP @ ( c_List_Oset @ C ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_List_Oset_Osimps_I2_J) ).

thf(10010,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( c_List_Oset @ C ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ C ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ B ) @ ( hAPP @ ( c_List_Oset @ C ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1936]) ).

thf(376,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hAPP @ ( hAPP @ ( c_COMBS @ ( tc_prod @ F @ E ) @ E @ D ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ C @ ( tc_fun @ E @ D ) @ ( tc_prod @ F @ E ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ F @ C @ ( tc_prod @ F @ E ) ) @ A ) @ ( c_Product__Type_Ofst @ F @ E ) ) ) ) @ ( c_Product__Type_Osnd @ F @ E ) )
      = ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ F @ E @ D ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ C @ ( tc_fun @ E @ D ) @ F ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_split__comp__eq) ).

thf(3749,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hAPP @ ( hAPP @ ( c_COMBS @ ( tc_prod @ F @ E ) @ E @ D ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ C @ ( tc_fun @ E @ D ) @ ( tc_prod @ F @ E ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ F @ C @ ( tc_prod @ F @ E ) ) @ A ) @ ( c_Product__Type_Ofst @ F @ E ) ) ) ) @ ( c_Product__Type_Osnd @ F @ E ) )
      = ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ F @ E @ D ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ C @ ( tc_fun @ E @ D ) @ F ) @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[376]) ).

thf(452,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ C ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ B ) )
       => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ tc_Nat_Onat ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) )
          = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ tc_Nat_Onat ) @ A ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ tc_Nat_Onat ) @ A ) @ B ) ) ) @ ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ tc_Nat_Onat ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_setsum__Un__nat) ).

thf(4023,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ C ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ B ) )
       => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ tc_Nat_Onat ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) )
          = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ tc_Nat_Onat ) @ A ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ tc_Nat_Onat ) @ A ) @ B ) ) ) @ ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ tc_Nat_Onat ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[452]) ).

thf(901,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hAPP @ ( hAPP @ ( c_List_Odrop @ C ) @ B ) @ A )
        = ( c_List_Olist_ONil @ C ) )
    <=> ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) @ A ) ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_drop__eq__Nil) ).

thf(5640,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( hAPP @ ( hAPP @ ( c_List_Odrop @ C ) @ B ) @ A )
          = ( c_List_Olist_ONil @ C ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) @ A ) ) @ B ) ) )
      & ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) @ A ) ) @ B ) )
       => ( ( hAPP @ ( hAPP @ ( c_List_Odrop @ C ) @ B ) @ A )
          = ( c_List_Olist_ONil @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[901]) ).

thf(25,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Ocomm__monoid__add @ D )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ C ) @ B ) )
       => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ tc_Nat_Onat @ D ) @ A ) @ ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Nat_Onat @ C @ B ) )
          = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ ( hAPP @ A @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ tc_Nat_Onat @ D ) @ A ) @ ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Nat_Onat @ ( hAPP @ c_Nat_OSuc @ C ) @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_setsum__head__Suc) ).

thf(2488,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Ocomm__monoid__add @ D )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ C ) @ B ) )
       => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ tc_Nat_Onat @ D ) @ A ) @ ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Nat_Onat @ C @ B ) )
          = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ ( hAPP @ A @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ tc_Nat_Onat @ D ) @ A ) @ ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Nat_Onat @ ( hAPP @ c_Nat_OSuc @ C ) @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[25]) ).

thf(138,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ F ) @ E ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ C ) )
       => ( ! [G: $i] :
              ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ G ) @ C ) )
             => ( ( hAPP @ ( c_Finite__Set_Ocard @ F ) @ ( hAPP @ ( c_Set_OCollect @ F ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ F @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ F ) @ c_fconj ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ F @ ( tc_fun @ F @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( c_member @ F ) ) @ E ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ F @ D @ tc_HOL_Obool ) @ B ) @ G ) ) ) )
                = ( hAPP @ A @ G ) ) )
         => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ F @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ D @ tc_HOL_Obool ) @ tc_Nat_Onat @ F ) @ ( c_Finite__Set_Ocard @ D ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ D @ tc_HOL_Obool ) @ ( tc_fun @ D @ tc_HOL_Obool ) @ F ) @ ( c_Set_OCollect @ D ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ D @ tc_HOL_Obool ) @ ( tc_fun @ D @ tc_HOL_Obool ) @ F ) @ ( hAPP @ ( c_COMBS @ D @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ D ) @ c_fconj ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ ( tc_fun @ D @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( c_member @ D ) ) @ C ) ) ) ) @ B ) ) ) ) @ E )
            = ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ tc_Nat_Onat ) @ A ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_setsum__multicount__gen) ).

thf(2897,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ F ) @ E ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ C ) )
       => ( ! [G: $i] :
              ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ G ) @ C ) )
             => ( ( hAPP @ ( c_Finite__Set_Ocard @ F ) @ ( hAPP @ ( c_Set_OCollect @ F ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ F @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ F ) @ c_fconj ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ F @ ( tc_fun @ F @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( c_member @ F ) ) @ E ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ F @ D @ tc_HOL_Obool ) @ B ) @ G ) ) ) )
                = ( hAPP @ A @ G ) ) )
         => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ F @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ D @ tc_HOL_Obool ) @ tc_Nat_Onat @ F ) @ ( c_Finite__Set_Ocard @ D ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ D @ tc_HOL_Obool ) @ ( tc_fun @ D @ tc_HOL_Obool ) @ F ) @ ( c_Set_OCollect @ D ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ D @ tc_HOL_Obool ) @ ( tc_fun @ D @ tc_HOL_Obool ) @ F ) @ ( hAPP @ ( c_COMBS @ D @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ D ) @ c_fconj ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ ( tc_fun @ D @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( c_member @ D ) ) @ C ) ) ) ) @ B ) ) ) ) @ E )
            = ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ tc_Nat_Onat ) @ A ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[138]) ).

thf(1850,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) )
    <=> ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ B ) ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__eq__Suc__le) ).

thf(9668,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ B ) ) @ A ) ) )
      & ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ B ) ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1850]) ).

thf(1068,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) )
     => ? [C: $i] :
          ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ C ) )
          & ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ C )
            = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__imp__add__positive) ).

thf(6332,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) )
     => ? [C: $i] :
          ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ C ) )
          & ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ C )
            = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1068]) ).

thf(1916,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ H @ G ) ) @ F ) @ ( hAPP @ ( c_Set_Oimage @ ( tc_prod @ E @ D ) @ ( tc_prod @ H @ G ) @ ( c_Product__Type_Omap__pair @ E @ H @ D @ G @ C @ B ) ) @ A ) ) )
     => ~ ! [I: $i,J: $i] :
            ( ( F
              = ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ H @ G ) @ ( hAPP @ C @ I ) ) @ ( hAPP @ B @ J ) ) )
           => ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ D ) @ I ) @ J ) ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_prod__fun__imageE) ).

thf(9938,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ H @ G ) ) @ F ) @ ( hAPP @ ( c_Set_Oimage @ ( tc_prod @ E @ D ) @ ( tc_prod @ H @ G ) @ ( c_Product__Type_Omap__pair @ E @ H @ D @ G @ C @ B ) ) @ A ) ) )
     => ~ ! [I: $i,J: $i] :
            ( ( F
              = ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ H @ G ) @ ( hAPP @ C @ I ) ) @ ( hAPP @ B @ J ) ) )
           => ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ D ) @ I ) @ J ) ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1916]) ).

thf(2302,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Set_Ovimage @ E @ D @ C ) @ B ) @ A ) )
    <=> ( hBOOL @ ( hAPP @ B @ ( hAPP @ C @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_vimage__code) ).

thf(11416,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Set_Ovimage @ E @ D @ C ) @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ B @ ( hAPP @ C @ A ) ) ) )
      & ( ( hBOOL @ ( hAPP @ B @ ( hAPP @ C @ A ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Set_Ovimage @ E @ D @ C ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2302]) ).

thf(388,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ A ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ B @ tc_HOL_Obool ) ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Int__UNIV__right) ).

thf(3803,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ A ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ B @ tc_HOL_Obool ) ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[388]) ).

thf(565,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ F @ E @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ ( tc_prod @ E @ D ) @ tc_HOL_Obool ) ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ ( tc_prod @ F @ D ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ F @ E @ D ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ F @ E @ D ) @ C ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_rel__comp__distrib) ).

thf(4428,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ F @ E @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ ( tc_prod @ E @ D ) @ tc_HOL_Obool ) ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ ( tc_prod @ F @ D ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ F @ E @ D ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ F @ E @ D ) @ C ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[565]) ).

thf(404,axiom,
    ! [A: $i,B: $i] :
      ( ( ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ B ) @ A )
        = ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) )
    <=> ( ( A
          = ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) )
        | ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ B ) )
          & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ B ) @ A ) ) )
        | ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ B ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) )
          & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ A ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zdiv__eq__0__iff) ).

thf(3855,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ B ) @ A )
          = ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) )
       => ( ( A
            = ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) )
          | ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ B ) )
            & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ B ) @ A ) ) )
          | ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ B ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) )
            & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ A ) @ B ) ) ) ) )
      & ( ( ( A
            = ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) )
          | ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ B ) )
            & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ B ) @ A ) ) )
          | ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ B ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) )
            & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ A ) @ B ) ) ) )
       => ( ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ B ) @ A )
          = ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[404]) ).

thf(1648,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ C ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_diff__diff__left) ).

thf(8880,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ C ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1648]) ).

thf(1617,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ C @ C @ B ) @ ( c_Relation_OId @ C ) ) @ A )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Id__O__R) ).

thf(8753,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ C @ C @ B ) @ ( c_Relation_OId @ C ) ) @ A )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1617]) ).

thf(1219,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) )
     => ? [C: $i] :
          ( A
          = ( hAPP @ c_Nat_OSuc @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__imp__Suc__add) ).

thf(7011,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) )
     => ? [C: $i] :
          ( A
          = ( hAPP @ c_Nat_OSuc @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1219]) ).

thf(2378,axiom,
    ! [A: $i,B: $i] : ( c_Natural_Oevaln @ c_Com_Ocom_OSKIP @ B @ A @ B ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_evaln_Oequations_I1_J) ).

thf(11760,plain,
    ! [A: $i,B: $i] : ( c_Natural_Oevaln @ c_Com_Ocom_OSKIP @ B @ A @ B ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2378]) ).

thf(1546,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( c_Finite__Set_Ofun__left__comm @ G @ F @ E )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofold__graph @ G @ F @ E @ D @ C ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ G ) @ A ) @ C ) )
         => ? [H: $i] :
              ( ( B
                = ( hAPP @ ( hAPP @ E @ A ) @ H ) )
              & ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofold__graph @ G @ F @ E @ D @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ G @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ G ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ G @ tc_HOL_Obool ) ) ) ) ) @ H ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_fun__left__comm_Ofold__graph__insertE__aux) ).

thf(8389,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( c_Finite__Set_Ofun__left__comm @ G @ F @ E )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofold__graph @ G @ F @ E @ D @ C ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ G ) @ A ) @ C ) )
         => ? [H: $i] :
              ( ( B
                = ( hAPP @ ( hAPP @ E @ A ) @ H ) )
              & ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofold__graph @ G @ F @ E @ D @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ G @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ G ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ G @ tc_HOL_Obool ) ) ) ) ) @ H ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1546]) ).

thf(1084,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ A ) ) )
    <=> ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ C ) )
        & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__less__cancel1) ).

thf(6396,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ A ) ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ C ) )
          & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) ) ) )
      & ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ C ) )
          & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1084]) ).

thf(543,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( c_Set_Oimage @ D @ C @ B ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ D @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ A ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ C @ ( tc_fun @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ D ) @ ( c_Set_Oinsert @ C ) ) @ B ) ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_image__eq__UN) ).

thf(4358,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( c_Set_Oimage @ D @ C @ B ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ D @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ A ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ C @ ( tc_fun @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ D ) @ ( c_Set_Oinsert @ C ) ) @ B ) ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[543]) ).

thf(2105,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( hAPP @ ( c_Relation_ODomain @ D @ B ) @ A ) ) )
    <=> ? [E: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ B ) @ C ) @ E ) ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Domain__iff) ).

thf(10665,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( hAPP @ ( c_Relation_ODomain @ D @ B ) @ A ) ) )
       => ? [E: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ B ) @ C ) @ E ) ) @ A ) ) )
      & ( ? [E: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ B ) @ C ) @ E ) ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( hAPP @ ( c_Relation_ODomain @ D @ B ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2105]) ).

thf(2116,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i,I: $i,J: $i] :
      ( ( c_Com_Ocom_Ocom__rec @ J @ I @ H @ G @ F @ E @ D @ C @ B @ ( hAPP @ c_Com_Ocom_OBODY @ A ) )
      = ( hAPP @ C @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_com_Orecs_I7_J) ).

thf(10702,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i,I: $i,J: $i] :
      ( ( c_Com_Ocom_Ocom__rec @ J @ I @ H @ G @ F @ E @ D @ C @ B @ ( hAPP @ c_Com_Ocom_OBODY @ A ) )
      = ( hAPP @ C @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2116]) ).

thf(1621,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) ) @ B )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_diff__add__inverse) ).

thf(8770,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) ) @ B )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1621]) ).

thf(1657,axiom,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ ( tc_List_Olist @ B ) @ ( tc_List_Olist @ B ) ) @ tc_HOL_Obool ) ) @ ( c_List_Olistrel1 @ B @ ( c_Transitive__Closure_Ortrancl @ B @ A ) ) ) @ ( c_Transitive__Closure_Ortrancl @ ( tc_List_Olist @ B ) @ ( c_List_Olistrel1 @ B @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_listrel1__rtrancl__subset__rtrancl__listrel1) ).

thf(8907,plain,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ ( tc_List_Olist @ B ) @ ( tc_List_Olist @ B ) ) @ tc_HOL_Obool ) ) @ ( c_List_Olistrel1 @ B @ ( c_Transitive__Closure_Ortrancl @ B @ A ) ) ) @ ( c_Transitive__Closure_Ortrancl @ ( tc_List_Olist @ B ) @ ( c_List_Olistrel1 @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1657]) ).

thf(5,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Ocomm__monoid__add @ D )
     => ( ~ ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ B ) )
       => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ C @ D ) @ A ) @ B )
          = ( c_Groups_Ozero__class_Ozero @ D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_setsum_Oinfinite) ).

thf(2420,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Ocomm__monoid__add @ D )
     => ( ~ ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ B ) )
       => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ C @ D ) @ A ) @ B )
          = ( c_Groups_Ozero__class_Ozero @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[5]) ).

thf(1001,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i] :
          ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ B ) @ C ) @ A ) )
    <=> ( A
        = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_all__not__in__conv) ).

thf(6049,plain,
    ! [A: $i,B: $i] :
      ( ( ! [C: $i] :
            ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ B ) @ C ) @ A ) )
       => ( A
          = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) )
      & ( ( A
          = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) )
       => ! [C: $i] :
            ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ B ) @ C ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1001]) ).

thf(1049,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ G @ G ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ G @ G ) @ F ) @ E ) ) @ D ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ G ) @ ( tc_List_Olist @ G ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ G ) @ ( tc_List_Olist @ G ) ) @ ( hAPP @ ( hAPP @ ( c_List_Oappend @ G ) @ C ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ G ) @ F ) @ B ) ) ) @ ( hAPP @ ( hAPP @ ( c_List_Oappend @ G ) @ C ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ G ) @ E ) @ A ) ) ) ) @ ( c_List_Olexord @ G @ D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_lexord__append__left__rightI) ).

thf(6253,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ G @ G ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ G @ G ) @ F ) @ E ) ) @ D ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ G ) @ ( tc_List_Olist @ G ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ G ) @ ( tc_List_Olist @ G ) ) @ ( hAPP @ ( hAPP @ ( c_List_Oappend @ G ) @ C ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ G ) @ F ) @ B ) ) ) @ ( hAPP @ ( hAPP @ ( c_List_Oappend @ G ) @ C ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ G ) @ E ) @ A ) ) ) ) @ ( c_List_Olexord @ G @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1049]) ).

thf(2255,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ~ ( hBOOL @ ( hAPP @ D @ C ) )
     => ( c_Natural_Oevaln @ ( c_Com_Ocom_OWhile @ D @ B ) @ C @ A @ C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_evaln_OWhileFalse) ).

thf(11232,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ~ ( hBOOL @ ( hAPP @ D @ C ) )
     => ( c_Natural_Oevaln @ ( c_Com_Ocom_OWhile @ D @ B ) @ C @ A @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2255]) ).

thf(298,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ B @ A ) ) @ A )
      = ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zmod__zdiv__trivial) ).

thf(3463,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ B @ A ) ) @ A )
      = ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[298]) ).

thf(272,axiom,
    class_Orderings_Owellorder @ tc_Nat_Onat,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat__Onat__Orderings_Owellorder) ).

thf(3374,plain,
    class_Orderings_Owellorder @ tc_Nat_Onat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[272]) ).

thf(276,axiom,
    ! [A: $i] : ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ tc_Nat_Onat ) @ ( hAPP @ ( c_Set_OCollect @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_finite__Collect__less__nat) ).

thf(3385,plain,
    ! [A: $i] : ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ tc_Nat_Onat ) @ ( hAPP @ ( c_Set_OCollect @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[276]) ).

thf(1878,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ B ) ) @ ( hAPP @ c_Nat_OSuc @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Suc__mono) ).

thf(9800,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ B ) ) @ ( hAPP @ c_Nat_OSuc @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1878]) ).

thf(958,axiom,
    ( ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ tc_Nat_Onat @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) ) @ ( c_SetInterval_Oord__class_OlessThan @ tc_Nat_Onat ) )
    = ( c_Orderings_Otop__class_Otop @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_UN__lessThan__UNIV) ).

thf(5868,plain,
    ( ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ tc_Nat_Onat @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) ) @ ( c_SetInterval_Oord__class_OlessThan @ tc_Nat_Onat ) )
    = ( c_Orderings_Otop__class_Otop @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[958]) ).

thf(892,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ D ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ C @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ B ) @ A ) ) )
    <=> ? [F: $i] :
          ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ F ) @ B ) )
          & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ D ) @ ( hAPP @ A @ F ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_UN__iff) ).

thf(5605,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ D ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ C @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ B ) @ A ) ) )
       => ? [F: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ F ) @ B ) )
            & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ D ) @ ( hAPP @ A @ F ) ) ) ) )
      & ( ? [F: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ F ) @ B ) )
            & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ D ) @ ( hAPP @ A @ F ) ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ D ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ C @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ B ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[892]) ).

thf(686,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( hAPP @ ( hAPP @ ( c_COMBC @ D @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ D ) @ ( c_COMBC @ C @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ ( tc_prod @ D @ C ) ) @ ( tc_fun @ C @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ D ) @ ( hAPP @ ( c_COMBB @ ( tc_prod @ D @ C ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ C ) @ ( c_member @ ( tc_prod @ D @ C ) ) ) ) @ ( c_Product__Type_OPair @ D @ C ) ) ) ) @ B )
        = ( hAPP @ ( hAPP @ ( c_COMBC @ D @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ D ) @ ( c_COMBC @ C @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ ( tc_prod @ D @ C ) ) @ ( tc_fun @ C @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ D ) @ ( hAPP @ ( c_COMBB @ ( tc_prod @ D @ C ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ C ) @ ( c_member @ ( tc_prod @ D @ C ) ) ) ) @ ( c_Product__Type_OPair @ D @ C ) ) ) ) @ A ) )
    <=> ( B = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_pred__equals__eq2) ).

thf(4862,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( ( hAPP @ ( hAPP @ ( c_COMBC @ D @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ D ) @ ( c_COMBC @ C @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ ( tc_prod @ D @ C ) ) @ ( tc_fun @ C @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ D ) @ ( hAPP @ ( c_COMBB @ ( tc_prod @ D @ C ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ C ) @ ( c_member @ ( tc_prod @ D @ C ) ) ) ) @ ( c_Product__Type_OPair @ D @ C ) ) ) ) @ B )
          = ( hAPP @ ( hAPP @ ( c_COMBC @ D @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ D ) @ ( c_COMBC @ C @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ ( tc_prod @ D @ C ) ) @ ( tc_fun @ C @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ D ) @ ( hAPP @ ( c_COMBB @ ( tc_prod @ D @ C ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ C ) @ ( c_member @ ( tc_prod @ D @ C ) ) ) ) @ ( c_Product__Type_OPair @ D @ C ) ) ) ) @ A ) )
       => ( B = A ) )
      & ( ( B = A )
       => ( ( hAPP @ ( hAPP @ ( c_COMBC @ D @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ D ) @ ( c_COMBC @ C @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ ( tc_prod @ D @ C ) ) @ ( tc_fun @ C @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ D ) @ ( hAPP @ ( c_COMBB @ ( tc_prod @ D @ C ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ C ) @ ( c_member @ ( tc_prod @ D @ C ) ) ) ) @ ( c_Product__Type_OPair @ D @ C ) ) ) ) @ B )
          = ( hAPP @ ( hAPP @ ( c_COMBC @ D @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ D ) @ ( c_COMBC @ C @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ ( tc_prod @ D @ C ) ) @ ( tc_fun @ C @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ D ) @ ( hAPP @ ( c_COMBB @ ( tc_prod @ D @ C ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ C ) @ ( c_member @ ( tc_prod @ D @ C ) ) ) ) @ ( c_Product__Type_OPair @ D @ C ) ) ) ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[686]) ).

thf(940,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ tc_Nat_Onat @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_SetInterval_Oord__class_OatMost @ tc_Nat_Onat ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_Nat_Onat @ ( tc_fun @ D @ tc_HOL_Obool ) @ tc_Nat_Onat ) @ B ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ tc_Nat_Onat @ tc_Nat_Onat ) @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) ) @ A ) ) )
      = ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ tc_Nat_Onat @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Nat_Onat @ A @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ A ) ) ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_UN__le__add__shift) ).

thf(5802,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ tc_Nat_Onat @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_SetInterval_Oord__class_OatMost @ tc_Nat_Onat ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_Nat_Onat @ ( tc_fun @ D @ tc_HOL_Obool ) @ tc_Nat_Onat ) @ B ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ tc_Nat_Onat @ tc_Nat_Onat ) @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) ) @ A ) ) )
      = ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ tc_Nat_Onat @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Nat_Onat @ A @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ A ) ) ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[940]) ).

thf(1633,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ C ) @ B ) @ A )
     != ( c_List_Olist_ONil @ C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_list_Osimps_I3_J) ).

thf(8819,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ C ) @ B ) @ A )
     != ( c_List_Olist_ONil @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1633]) ).

thf(1023,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ C ) )
     => ( ! [E: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ E ) @ C ) )
           => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ B ) @ ( hAPP @ A @ E ) ) ) )
       => ( ( hAPP @ ( c_Finite__Set_Ocard @ ( tc_prod @ D @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ D @ B ) @ C ) @ A ) )
          = ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ B @ tc_HOL_Obool ) @ tc_Nat_Onat @ D ) @ ( c_Finite__Set_Ocard @ B ) ) @ A ) ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_card__SigmaI) ).

thf(6149,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ C ) )
     => ( ! [E: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ E ) @ C ) )
           => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ B ) @ ( hAPP @ A @ E ) ) ) )
       => ( ( hAPP @ ( c_Finite__Set_Ocard @ ( tc_prod @ D @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ D @ B ) @ C ) @ A ) )
          = ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ B @ tc_HOL_Obool ) @ tc_Nat_Onat @ D ) @ ( c_Finite__Set_Ocard @ B ) ) @ A ) ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1023]) ).

thf(2166,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i,I: $i,J: $i,K: $i] :
      ( ( c_Com_Ocom_Ocom__case @ K @ J @ I @ H @ G @ F @ E @ D @ C @ ( c_Com_Ocom_OAss @ B @ A ) )
      = ( hAPP @ ( hAPP @ I @ B ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_com_Osimps_I65_J) ).

thf(10891,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i,I: $i,J: $i,K: $i] :
      ( ( c_Com_Ocom_Ocom__case @ K @ J @ I @ H @ G @ F @ E @ D @ C @ ( c_Com_Ocom_OAss @ B @ A ) )
      = ( hAPP @ ( hAPP @ I @ B ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2166]) ).

thf(1060,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ D ) @ C ) @ B ) ) @ ( hAPP @ ( c_Wellfounded_Omeasure @ D ) @ A ) ) )
    <=> ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ A @ C ) ) @ ( hAPP @ A @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_in__measure) ).

thf(6310,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ D ) @ C ) @ B ) ) @ ( hAPP @ ( c_Wellfounded_Omeasure @ D ) @ A ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ A @ C ) ) @ ( hAPP @ A @ B ) ) ) )
      & ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ A @ C ) ) @ ( hAPP @ A @ B ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ D ) @ C ) @ B ) ) @ ( hAPP @ ( c_Wellfounded_Omeasure @ D ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1060]) ).

thf(1891,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) )
     => ( ( B != A )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) )
          & ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ A ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_dvd_Ole__neq__trans) ).

thf(9843,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) )
     => ( ( B != A )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) )
          & ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ A ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1891]) ).

thf(1117,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i,I: $i,J: $i] :
      ( ( hAPP @ ( c_Fun_Ocomp @ ( tc_prod @ J @ I ) @ ( tc_prod @ H @ G ) @ ( tc_prod @ F @ E ) @ ( c_Product__Type_Omap__pair @ J @ H @ I @ G @ D @ C ) ) @ ( c_Product__Type_Omap__pair @ F @ J @ E @ I @ B @ A ) )
      = ( c_Product__Type_Omap__pair @ F @ H @ E @ G @ ( hAPP @ ( c_Fun_Ocomp @ J @ H @ F @ D ) @ B ) @ ( hAPP @ ( c_Fun_Ocomp @ I @ G @ E @ C ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_map__pair_Ocomp) ).

thf(6579,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i,I: $i,J: $i] :
      ( ( hAPP @ ( c_Fun_Ocomp @ ( tc_prod @ J @ I ) @ ( tc_prod @ H @ G ) @ ( tc_prod @ F @ E ) @ ( c_Product__Type_Omap__pair @ J @ H @ I @ G @ D @ C ) ) @ ( c_Product__Type_Omap__pair @ F @ J @ E @ I @ B @ A ) )
      = ( c_Product__Type_Omap__pair @ F @ H @ E @ G @ ( hAPP @ ( c_Fun_Ocomp @ J @ H @ F @ D ) @ B ) @ ( hAPP @ ( c_Fun_Ocomp @ I @ G @ E @ C ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1117]) ).

thf(580,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ D @ C @ B ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) ) ) @ A )
      = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ ( tc_prod @ D @ B ) @ tc_HOL_Obool ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_rel__comp__empty1) ).

thf(4492,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ D @ C @ B ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) ) ) @ A )
      = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ ( tc_prod @ D @ B ) @ tc_HOL_Obool ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[580]) ).

thf(897,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ F ) @ ( tc_List_Olist @ F ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ F ) @ ( tc_List_Olist @ F ) ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ F ) @ E ) @ D ) ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ F ) @ C ) @ B ) ) ) @ ( c_List_Olexord @ F @ A ) ) )
    <=> ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ F @ F ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ F @ F ) @ E ) @ C ) ) @ A ) )
        | ( ( E = C )
          & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ F ) @ ( tc_List_Olist @ F ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ F ) @ ( tc_List_Olist @ F ) ) @ D ) @ B ) ) @ ( c_List_Olexord @ F @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_lexord__cons__cons) ).

thf(5622,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ F ) @ ( tc_List_Olist @ F ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ F ) @ ( tc_List_Olist @ F ) ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ F ) @ E ) @ D ) ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ F ) @ C ) @ B ) ) ) @ ( c_List_Olexord @ F @ A ) ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ F @ F ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ F @ F ) @ E ) @ C ) ) @ A ) )
          | ( ( E = C )
            & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ F ) @ ( tc_List_Olist @ F ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ F ) @ ( tc_List_Olist @ F ) ) @ D ) @ B ) ) @ ( c_List_Olexord @ F @ A ) ) ) ) ) )
      & ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ F @ F ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ F @ F ) @ E ) @ C ) ) @ A ) )
          | ( ( E = C )
            & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ F ) @ ( tc_List_Olist @ F ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ F ) @ ( tc_List_Olist @ F ) ) @ D ) @ B ) ) @ ( c_List_Olexord @ F @ A ) ) ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ F ) @ ( tc_List_Olist @ F ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ F ) @ ( tc_List_Olist @ F ) ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ F ) @ E ) @ D ) ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ F ) @ C ) @ B ) ) ) @ ( c_List_Olexord @ F @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[897]) ).

thf(2291,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( hAPP @ ( c_List_Oset @ D ) @ ( c_List_OtakeWhile @ D @ B @ A ) ) ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( hAPP @ ( c_List_Oset @ D ) @ A ) ) )
        & ( hBOOL @ ( hAPP @ B @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_set__takeWhileD) ).

thf(11378,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( hAPP @ ( c_List_Oset @ D ) @ ( c_List_OtakeWhile @ D @ B @ A ) ) ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( hAPP @ ( c_List_Oset @ D ) @ A ) ) )
        & ( hBOOL @ ( hAPP @ B @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2291]) ).

thf(420,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ F ) @ E ) @ D ) )
     => ( ( ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ C @ F ) @ B ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ F @ tc_HOL_Obool ) @ C ) @ D ) )
          = ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ C @ F ) @ A ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ F @ tc_HOL_Obool ) @ C ) @ D ) ) )
      <=> ( B = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Times__eq__cancel2) ).

thf(3919,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ F ) @ E ) @ D ) )
     => ( ( ( ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ C @ F ) @ B ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ F @ tc_HOL_Obool ) @ C ) @ D ) )
            = ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ C @ F ) @ A ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ F @ tc_HOL_Obool ) @ C ) @ D ) ) )
         => ( B = A ) )
        & ( ( B = A )
         => ( ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ C @ F ) @ B ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ F @ tc_HOL_Obool ) @ C ) @ D ) )
            = ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ C @ F ) @ A ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ F @ tc_HOL_Obool ) @ C ) @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[420]) ).

thf(228,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( class_Groups_Oab__semigroup__mult @ F )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ E ) @ D ) )
       => ( ! [G: $i] :
              ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ G ) @ D ) )
             => ( ( hAPP @ C @ G )
                = ( hAPP @ B @ G ) ) )
         => ( ( hAPP @ ( hAPP @ ( hAPP @ ( c_Finite__Set_Ofold__image @ F @ E @ ( c_Groups_Otimes__class_Otimes @ F ) ) @ C ) @ A ) @ D )
            = ( hAPP @ ( hAPP @ ( hAPP @ ( c_Finite__Set_Ofold__image @ F @ E @ ( c_Groups_Otimes__class_Otimes @ F ) ) @ B ) @ A ) @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_fold__image__cong) ).

thf(3216,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( class_Groups_Oab__semigroup__mult @ F )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ E ) @ D ) )
       => ( ! [G: $i] :
              ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ G ) @ D ) )
             => ( ( hAPP @ C @ G )
                = ( hAPP @ B @ G ) ) )
         => ( ( hAPP @ ( hAPP @ ( hAPP @ ( c_Finite__Set_Ofold__image @ F @ E @ ( c_Groups_Otimes__class_Otimes @ F ) ) @ C ) @ A ) @ D )
            = ( hAPP @ ( hAPP @ ( hAPP @ ( c_Finite__Set_Ofold__image @ F @ E @ ( c_Groups_Otimes__class_Otimes @ F ) ) @ B ) @ A ) @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[228]) ).

thf(1075,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( hAPP @ ( c_Fun_Ocomp @ G @ F @ E @ D ) @ ( hAPP @ ( c_Fun_Ocomp @ C @ G @ E @ B ) @ A ) )
      = ( hAPP @ ( c_Fun_Ocomp @ C @ F @ E @ ( hAPP @ ( c_Fun_Ocomp @ G @ F @ C @ D ) @ B ) ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_o__assoc) ).

thf(6365,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( hAPP @ ( c_Fun_Ocomp @ G @ F @ E @ D ) @ ( hAPP @ ( c_Fun_Ocomp @ C @ G @ E @ B ) @ A ) )
      = ( hAPP @ ( c_Fun_Ocomp @ C @ F @ E @ ( hAPP @ ( c_Fun_Ocomp @ G @ F @ C @ D ) @ B ) ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1075]) ).

thf(189,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ? [E: $i] :
          ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ E ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) ) )
          & ( hBOOL @ ( hAPP @ A @ E ) ) )
    <=> ( ? [E: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ E ) @ C ) )
            & ( hBOOL @ ( hAPP @ A @ E ) ) )
        | ? [E: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ E ) @ B ) )
            & ( hBOOL @ ( hAPP @ A @ E ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_bex__Un) ).

thf(3084,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ? [E: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ E ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) ) )
            & ( hBOOL @ ( hAPP @ A @ E ) ) )
       => ( ? [E: $i] :
              ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ E ) @ C ) )
              & ( hBOOL @ ( hAPP @ A @ E ) ) )
          | ? [E: $i] :
              ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ E ) @ B ) )
              & ( hBOOL @ ( hAPP @ A @ E ) ) ) ) )
      & ( ( ? [E: $i] :
              ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ E ) @ C ) )
              & ( hBOOL @ ( hAPP @ A @ E ) ) )
          | ? [E: $i] :
              ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ E ) @ B ) )
              & ( hBOOL @ ( hAPP @ A @ E ) ) ) )
       => ? [E: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ E ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) ) )
            & ( hBOOL @ ( hAPP @ A @ E ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[189]) ).

thf(2226,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Wellfounded_Owf @ C @ B )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ A ) @ ( c_Wellfounded_Oacc @ C @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_acc__wfD) ).

thf(11115,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Wellfounded_Owf @ C @ B )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ A ) @ ( c_Wellfounded_Oacc @ C @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2226]) ).

thf(2338,axiom,
    ! [A: $i,B: $i,C: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_insertI1) ).

thf(11600,plain,
    ! [A: $i,B: $i,C: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2338]) ).

thf(740,axiom,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_empty__subsetI) ).

thf(5052,plain,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[740]) ).

thf(1815,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Transitive__Closure_Ortrancl @ B @ ( c_Transitive__Closure_Otrancl @ B @ A ) )
      = ( c_Transitive__Closure_Ortrancl @ B @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_trancl__rtrancl__absorb) ).

thf(9529,plain,
    ! [A: $i,B: $i] :
      ( ( c_Transitive__Closure_Ortrancl @ B @ ( c_Transitive__Closure_Otrancl @ B @ A ) )
      = ( c_Transitive__Closure_Ortrancl @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1815]) ).

thf(2372,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Equiv__Relations_Oequivp @ C @ B )
     => ( hBOOL @ ( hAPP @ ( hAPP @ B @ A ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_equivp__reflp) ).

thf(11741,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Equiv__Relations_Oequivp @ C @ B )
     => ( hBOOL @ ( hAPP @ ( hAPP @ B @ A ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2372]) ).

thf(1196,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__less__mono1) ).

thf(6923,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1196]) ).

thf(1729,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( c_Finite__Set_Ocard @ tc_Nat_Onat ) @ ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Nat_Onat @ B @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ A ) ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_card__atLeastAtMost) ).

thf(9199,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( c_Finite__Set_Ocard @ tc_Nat_Onat ) @ ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Nat_Onat @ B @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ A ) ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1729]) ).

thf(1586,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( c_Nitpick_Opair__box_Opair__box__size @ F @ E @ D @ C @ ( c_Nitpick_Opair__box_OPairBox @ F @ E @ B @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ D @ B ) ) @ ( hAPP @ C @ A ) ) ) @ ( hAPP @ c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_pair__box_Osize_I1_J) ).

thf(8615,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( c_Nitpick_Opair__box_Opair__box__size @ F @ E @ D @ C @ ( c_Nitpick_Opair__box_OPairBox @ F @ E @ B @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ D @ B ) ) @ ( hAPP @ C @ A ) ) ) @ ( hAPP @ c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1586]) ).

thf(2100,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ c_Com_Ocom_OBODY @ C )
     != ( c_Com_Ocom_OSemi @ B @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_com_Osimps_I49_J) ).

thf(10650,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ c_Com_Ocom_OBODY @ C )
     != ( c_Com_Ocom_OSemi @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2100]) ).

thf(257,axiom,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ B ) @ A ) )
       => ( ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Int_Oint @ A @ B )
          = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ tc_Int_Oint @ tc_HOL_Obool ) ) ) )
      & ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ B ) @ A ) )
       => ( ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Int_Oint @ A @ B )
          = ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ tc_Int_Oint ) @ A ) @ ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Int_Oint @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ A ) @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_simp__from__to) ).

thf(3312,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ B ) @ A ) )
       => ( ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Int_Oint @ A @ B )
          = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ tc_Int_Oint @ tc_HOL_Obool ) ) ) )
      & ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ B ) @ A ) )
       => ( ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Int_Oint @ A @ B )
          = ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ tc_Int_Oint ) @ A ) @ ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Int_Oint @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ A ) @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[257]) ).

thf(953,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Relation_OImage @ D @ D @ A ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ C ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) ) ) @ ( hAPP @ ( hAPP @ ( c_Equiv__Relations_Oquotient @ D ) @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_quotientI) ).

thf(5851,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Relation_OImage @ D @ D @ A ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ C ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) ) ) @ ( hAPP @ ( hAPP @ ( c_Equiv__Relations_Oquotient @ D ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[953]) ).

thf(1387,axiom,
    ! [A: $i] :
      ( ( hAPP @ c_Nat_OSuc @ A )
     != ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Suc__neq__Zero) ).

thf(7728,plain,
    ! [A: $i] :
      ( ( hAPP @ c_Nat_OSuc @ A )
     != ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1387]) ).

thf(414,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ B ) )
     => ( ( hAPP @ c_Nat_OSuc @ ( hAPP @ ( c_Finite__Set_Ocard @ tc_Nat_Onat ) @ ( hAPP @ ( c_Set_OCollect @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ tc_Nat_Onat @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ tc_Nat_Onat ) @ c_fconj ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_Nat_Onat @ ( tc_fun @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ tc_Nat_Onat ) @ ( c_member @ tc_Nat_Onat ) ) @ c_Nat_OSuc ) ) @ B ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) ) @ A ) ) ) ) )
        = ( hAPP @ ( c_Finite__Set_Ocard @ tc_Nat_Onat ) @ ( hAPP @ ( c_Set_OCollect @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ tc_Nat_Onat @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ tc_Nat_Onat ) @ c_fconj ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( c_member @ tc_Nat_Onat ) ) @ B ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) ) @ ( hAPP @ c_Nat_OSuc @ A ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_card__less__Suc) ).

thf(3902,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ B ) )
     => ( ( hAPP @ c_Nat_OSuc @ ( hAPP @ ( c_Finite__Set_Ocard @ tc_Nat_Onat ) @ ( hAPP @ ( c_Set_OCollect @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ tc_Nat_Onat @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ tc_Nat_Onat ) @ c_fconj ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_Nat_Onat @ ( tc_fun @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ tc_Nat_Onat ) @ ( c_member @ tc_Nat_Onat ) ) @ c_Nat_OSuc ) ) @ B ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) ) @ A ) ) ) ) )
        = ( hAPP @ ( c_Finite__Set_Ocard @ tc_Nat_Onat ) @ ( hAPP @ ( c_Set_OCollect @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ tc_Nat_Onat @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ tc_Nat_Onat ) @ c_fconj ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( c_member @ tc_Nat_Onat ) ) @ B ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) ) @ ( hAPP @ c_Nat_OSuc @ A ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[414]) ).

thf(351,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ F @ E ) @ D ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ C @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) @ A ) @ F ) @ E ) @ D ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_sup2I2) ).

thf(3657,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ F @ E ) @ D ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ C @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) @ A ) @ F ) @ E ) @ D ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[351]) ).

thf(3503,plain,
    ( ( hAPP @ ( c_Nat_Osize__class_Osize @ tc_HOL_Obool ) @ c_fTrue )
    = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ),
    inference(lifteq,[status(thm)],[3502]) ).

thf(11422,plain,
    ~ ( hBOOL @ c_fFalse ),
    inference(polarity_switch,[status(thm)],[11421]) ).

thf(11894,plain,
    ( ( sk4 != sk3 )
    | ( ( hBOOL @ ( hAPP @ v_b @ sk4 ) )
     != ( hBOOL @ c_fFalse ) ) ),
    inference(paramod_ordered,[status(thm)],[11893,11422]) ).

thf(11897,plain,
    ( ( sk4 != sk3 )
    | ( ( hAPP @ v_b @ sk4 )
     != c_fFalse ) ),
    inference(simp,[status(thm)],[11894]) ).

thf(11943,plain,
    ( ( sk4 != sk3 )
    | ( ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat )
     != c_fFalse )
    | ( ( hAPP @ ( c_Nat_Osize__class_Osize @ tc_HOL_Obool ) @ c_fTrue )
     != ( hAPP @ v_b @ sk4 ) ) ),
    inference(paramod_ordered,[status(thm)],[3503,11897]) ).

thf(11991,plain,
    ( ( sk4 != sk3 )
    | ( ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat )
     != c_fFalse )
    | ( ( c_Nat_Osize__class_Osize @ tc_HOL_Obool )
     != v_b )
    | ( sk4 != c_fTrue ) ),
    inference(simp,[status(thm)],[11943]) ).

thf(1341,axiom,
    ! [A: $i,B: $i] :
      ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) )
     => ( ( c_Divides_Odiv__class_Omod @ tc_Nat_Onat @ B @ A )
        = ( c_Divides_Odiv__class_Omod @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ B ) @ A ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mod__geq) ).

thf(7560,plain,
    ! [A: $i,B: $i] :
      ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) )
     => ( ( c_Divides_Odiv__class_Omod @ tc_Nat_Onat @ B @ A )
        = ( c_Divides_Odiv__class_Omod @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ B ) @ A ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1341]) ).

thf(977,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_HOL_OThe @ ( tc_prod @ C @ B ) @ A )
      = ( c_HOL_OThe @ ( tc_prod @ C @ B ) @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ C @ B @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ B @ ( tc_prod @ C @ B ) ) @ ( tc_fun @ B @ tc_HOL_Obool ) @ C ) @ ( hAPP @ ( c_COMBB @ ( tc_prod @ C @ B ) @ tc_HOL_Obool @ B ) @ A ) ) @ ( c_Product__Type_OPair @ C @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_split__paired__The) ).

thf(5942,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_HOL_OThe @ ( tc_prod @ C @ B ) @ A )
      = ( c_HOL_OThe @ ( tc_prod @ C @ B ) @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ C @ B @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ B @ ( tc_prod @ C @ B ) ) @ ( tc_fun @ B @ tc_HOL_Obool ) @ C ) @ ( hAPP @ ( c_COMBB @ ( tc_prod @ C @ B ) @ tc_HOL_Obool @ B ) @ A ) ) @ ( c_Product__Type_OPair @ C @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[977]) ).

thf(1896,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ~ ( hBOOL @ ( hAPP @ D @ C ) )
     => ( ( c_List_Oremove1 @ B @ C @ ( hAPP @ ( c_List_Ofilter @ B @ D ) @ A ) )
        = ( hAPP @ ( c_List_Ofilter @ B @ D ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_remove1__filter__not) ).

thf(9869,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ~ ( hBOOL @ ( hAPP @ D @ C ) )
     => ( ( c_List_Oremove1 @ B @ C @ ( hAPP @ ( c_List_Ofilter @ B @ D ) @ A ) )
        = ( hAPP @ ( c_List_Ofilter @ B @ D ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1896]) ).

thf(35,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Wellfounded_Owf @ C @ B )
     => ( c_Wellfounded_Owf @ ( tc_List_Olist @ C ) @ ( hAPP @ ( c_List_Olexn @ C @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_wf__lexn) ).

thf(2520,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Wellfounded_Owf @ C @ B )
     => ( c_Wellfounded_Owf @ ( tc_List_Olist @ C ) @ ( hAPP @ ( c_List_Olexn @ C @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[35]) ).

thf(488,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A )
        = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A )
        = B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Diff__triv) ).

thf(4141,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A )
        = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A )
        = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[488]) ).

thf(2071,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Wellfounded_Owf @ B @ A )
    <=> ! [C: $i] :
          ( ? [D: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ B ) @ D ) @ C ) )
         => ? [D: $i] :
              ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ B ) @ D ) @ C ) )
              & ! [E: $i] :
                  ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ B @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ B @ B ) @ E ) @ D ) ) @ A ) )
                 => ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ B ) @ E ) @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_wf__eq__minimal) ).

thf(10555,plain,
    ! [A: $i,B: $i] :
      ( ( ( c_Wellfounded_Owf @ B @ A )
       => ! [C: $i] :
            ( ? [D: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ B ) @ D ) @ C ) )
           => ? [D: $i] :
                ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ B ) @ D ) @ C ) )
                & ! [E: $i] :
                    ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ B @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ B @ B ) @ E ) @ D ) ) @ A ) )
                   => ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ B ) @ E ) @ C ) ) ) ) ) )
      & ( ! [C: $i] :
            ( ? [D: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ B ) @ D ) @ C ) )
           => ? [D: $i] :
                ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ B ) @ D ) @ C ) )
                & ! [E: $i] :
                    ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ B @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ B @ B ) @ E ) @ D ) ) @ A ) )
                   => ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ B ) @ E ) @ C ) ) ) ) )
       => ( c_Wellfounded_Owf @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2071]) ).

thf(927,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ F @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ D @ ( tc_fun @ F @ tc_HOL_Obool ) ) @ C ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ D @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ ( tc_fun @ F @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ F @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ F @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ D ) @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ F @ ( tc_fun @ E @ tc_HOL_Obool ) ) ) @ B ) ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_UN__simps_I9_J) ).

thf(5727,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ F @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ D @ ( tc_fun @ F @ tc_HOL_Obool ) ) @ C ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ D @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ ( tc_fun @ F @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ F @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ F @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ D ) @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ F @ ( tc_fun @ E @ tc_HOL_Obool ) ) ) @ B ) ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[927]) ).

thf(1590,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ A ) ) @ ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ B ) ) ) @ ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ A ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_diff__card__le__card__Diff) ).

thf(8632,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ A ) ) @ ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ B ) ) ) @ ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ A ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1590]) ).

thf(1809,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i,I: $i] :
      ( ( c_Com_Ocom_Ocom__rec @ I @ H @ G @ F @ E @ D @ C @ B @ A @ c_Com_Ocom_OSKIP )
      = H ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_com_Orecs_I1_J) ).

thf(9497,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i,I: $i] :
      ( ( c_Com_Ocom_Ocom__rec @ I @ H @ G @ F @ E @ D @ C @ B @ A @ c_Com_Ocom_OSKIP )
      = H ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1809]) ).

thf(552,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D
        = ( c_Finite__Set_Ofold1 @ C @ B ) )
     => ( ( hAPP @ D @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) )
        = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_fold1__singleton__def) ).

thf(4390,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D
        = ( c_Finite__Set_Ofold1 @ C @ B ) )
     => ( ( hAPP @ D @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[552]) ).

thf(336,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ B ) )
     => ( ( c_Fun_Oinj__on @ C @ ( tc_fun @ ( tc_fun @ C @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ C @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ C @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ C @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ C ) @ ( c_Equiv__Relations_Oquotient @ C ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ C @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( c_Set_Oinsert @ C ) ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) ) ) @ A ) @ B )
       => ( ( hAPP @ ( c_Finite__Set_Ocard @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Equiv__Relations_Oquotient @ C ) @ B ) @ A ) )
          = ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_card__quotient__disjoint) ).

thf(3602,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ B ) )
     => ( ( c_Fun_Oinj__on @ C @ ( tc_fun @ ( tc_fun @ C @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ C @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ C @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ C @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ C ) @ ( c_Equiv__Relations_Oquotient @ C ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ C @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( c_Set_Oinsert @ C ) ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) ) ) @ A ) @ B )
       => ( ( hAPP @ ( c_Finite__Set_Ocard @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Equiv__Relations_Oquotient @ C ) @ B ) @ A ) )
          = ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[336]) ).

thf(555,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Nat_Ocompow @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ A )
      = ( c_Relation_OId @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_relpow_Osimps_I1_J) ).

thf(4400,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Nat_Ocompow @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ A )
      = ( c_Relation_OId @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[555]) ).

thf(1612,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( F != E )
     => ( ( hAPP @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ D @ C ) @ B ) @ E ) @ A ) @ F )
        = ( hAPP @ B @ F ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_fun__upd__other) ).

thf(8738,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( F != E )
     => ( ( hAPP @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ D @ C ) @ B ) @ E ) @ A ) @ F )
        = ( hAPP @ B @ F ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1612]) ).

thf(659,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( hAPP @ ( hAPP @ ( c_List_Omap @ E @ D ) @ C ) @ B )
        = ( hAPP @ ( hAPP @ ( c_List_Omap @ E @ D ) @ C ) @ A ) )
     => ( ( c_Fun_Oinj__on @ E @ D @ C @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_List_Oset @ E ) @ B ) ) @ ( hAPP @ ( c_List_Oset @ E ) @ A ) ) )
       => ( B = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_map__inj__on) ).

thf(4771,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( hAPP @ ( hAPP @ ( c_List_Omap @ E @ D ) @ C ) @ B )
        = ( hAPP @ ( hAPP @ ( c_List_Omap @ E @ D ) @ C ) @ A ) )
     => ( ( c_Fun_Oinj__on @ E @ D @ C @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_List_Oset @ E ) @ B ) ) @ ( hAPP @ ( c_List_Oset @ E ) @ A ) ) )
       => ( B = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[659]) ).

thf(231,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ B ) )
     => ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ C ) @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Int__insert__right__if0) ).

thf(3231,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ B ) )
     => ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ C ) @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[231]) ).

thf(2111,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( c_Set_Oimage @ E @ D @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ ( hAPP @ C @ B ) ) @ ( hAPP @ ( c_Set_Oimage @ E @ D @ C ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_image__insert) ).

thf(10682,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( c_Set_Oimage @ E @ D @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ ( hAPP @ C @ B ) ) @ ( hAPP @ ( c_Set_Oimage @ E @ D @ C ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2111]) ).

thf(528,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ F @ ( tc_fun @ E @ tc_HOL_Obool ) ) ) @ D ) @ C ) @ B ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ C @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_inf2D2) ).

thf(4302,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ F @ ( tc_fun @ E @ tc_HOL_Obool ) ) ) @ D ) @ C ) @ B ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ C @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[528]) ).

thf(1319,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( c_List_Ofilter @ D @ C ) @ ( hAPP @ ( hAPP @ ( c_List_Oappend @ D ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_List_Oappend @ D ) @ ( hAPP @ ( c_List_Ofilter @ D @ C ) @ B ) ) @ ( hAPP @ ( c_List_Ofilter @ D @ C ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_filter__append) ).

thf(7464,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( c_List_Ofilter @ D @ C ) @ ( hAPP @ ( hAPP @ ( c_List_Oappend @ D ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_List_Oappend @ D ) @ ( hAPP @ ( c_List_Ofilter @ D @ C ) @ B ) ) @ ( hAPP @ ( c_List_Ofilter @ D @ C ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1319]) ).

thf(1536,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C = B )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ A @ tc_HOL_Obool ) ) @ C ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_equalityD1) ).

thf(8352,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C = B )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ A @ tc_HOL_Obool ) ) @ C ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1536]) ).

thf(1487,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ D ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ C ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ B ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ A @ D ) ) ) @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zmult2__lemma__aux1) ).

thf(8161,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ D ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ C ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ B ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ A @ D ) ) ) @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1487]) ).

thf(895,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) ) )
     => ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ B ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_UnE) ).

thf(5617,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) ) )
     => ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ B ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[895]) ).

thf(515,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Int_Oint ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zdiff__zmult__distrib2) ).

thf(4253,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Int_Oint ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[515]) ).

thf(1081,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i] :
          ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ C ) @ B ) )
         => ( hBOOL @ ( hAPP @ A @ C ) ) )
    <=> ! [C: $i] :
          ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Nat_Onat ) @ C ) @ ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ B ) ) )
         => ( hBOOL @ ( hAPP @ A @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_all__nat__less) ).

thf(6382,plain,
    ! [A: $i,B: $i] :
      ( ( ! [C: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ C ) @ B ) )
           => ( hBOOL @ ( hAPP @ A @ C ) ) )
       => ! [C: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Nat_Onat ) @ C ) @ ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ B ) ) )
           => ( hBOOL @ ( hAPP @ A @ C ) ) ) )
      & ( ! [C: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Nat_Onat ) @ C ) @ ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ B ) ) )
           => ( hBOOL @ ( hAPP @ A @ C ) ) )
       => ! [C: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ C ) @ B ) )
           => ( hBOOL @ ( hAPP @ A @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1081]) ).

thf(239,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Map_Oran @ B @ A @ ( hAPP @ ( c_COMBK @ ( tc_Option_Ooption @ A ) @ B ) @ ( c_Option_Ooption_ONone @ A ) ) )
      = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ A @ tc_HOL_Obool ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_ran__empty) ).

thf(3258,plain,
    ! [A: $i,B: $i] :
      ( ( c_Map_Oran @ B @ A @ ( hAPP @ ( c_COMBK @ ( tc_Option_Ooption @ A ) @ B ) @ ( c_Option_Ooption_ONone @ A ) ) )
      = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ A @ tc_HOL_Obool ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[239]) ).

thf(1746,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ C ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ C ) @ A ) ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_diff__commute) ).

thf(9266,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ C ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ C ) @ A ) ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1746]) ).

thf(1644,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Relation_OImage @ C @ C @ B ) @ A ) ) @ A ) )
     => ( ( hAPP @ ( c_Relation_OImage @ C @ C @ ( c_Transitive__Closure_Ortrancl @ C @ B ) ) @ A )
        = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Image__closed__trancl) ).

thf(8870,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Relation_OImage @ C @ C @ B ) @ A ) ) @ A ) )
     => ( ( hAPP @ ( c_Relation_OImage @ C @ C @ ( c_Transitive__Closure_Ortrancl @ C @ B ) ) @ A )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1644]) ).

thf(1830,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Com_Ocom_OSemi @ C @ B )
     != ( hAPP @ c_Com_Ocom_OBODY @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_com_Osimps_I48_J) ).

thf(9580,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Com_Ocom_OSemi @ C @ B )
     != ( hAPP @ c_Com_Ocom_OBODY @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1830]) ).

thf(983,axiom,
    ! [A: $i] :
      ( ( hAPP @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Nat_Onat @ A ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
      = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_atLeastLessThan0) ).

thf(5974,plain,
    ! [A: $i] :
      ( ( hAPP @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Nat_Onat @ A ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
      = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[983]) ).

thf(10913,plain,
    ! [A: $i] : ( c_Equiv__Relations_Oequivp @ A @ c_fequal ),
    inference(cnf,[status(esa)],[10912]) ).

thf(2273,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ H @ G ) @ F ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ E @ F ) @ D ) )
       => ( c_Predicate_Opred__comp @ C @ B @ A @ H @ E @ G @ D ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_pred__comp_Ointros) ).

thf(11314,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ H @ G ) @ F ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ E @ F ) @ D ) )
       => ( c_Predicate_Opred__comp @ C @ B @ A @ H @ E @ G @ D ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2273]) ).

thf(1834,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_Fun_Oinj__on @ D @ ( tc_Option_Ooption @ C ) @ ( c_Map_Omap__add @ D @ C @ B @ A ) @ ( c_Map_Odom @ D @ C @ A ) )
    <=> ( c_Fun_Oinj__on @ D @ ( tc_Option_Ooption @ C ) @ A @ ( c_Map_Odom @ D @ C @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_inj__on__map__add__dom) ).

thf(9593,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( c_Fun_Oinj__on @ D @ ( tc_Option_Ooption @ C ) @ ( c_Map_Omap__add @ D @ C @ B @ A ) @ ( c_Map_Odom @ D @ C @ A ) )
       => ( c_Fun_Oinj__on @ D @ ( tc_Option_Ooption @ C ) @ A @ ( c_Map_Odom @ D @ C @ A ) ) )
      & ( ( c_Fun_Oinj__on @ D @ ( tc_Option_Ooption @ C ) @ A @ ( c_Map_Odom @ D @ C @ A ) )
       => ( c_Fun_Oinj__on @ D @ ( tc_Option_Ooption @ C ) @ ( c_Map_Omap__add @ D @ C @ B @ A ) @ ( c_Map_Odom @ D @ C @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1834]) ).

thf(748,axiom,
    ! [A: $i] : ( c_Fun_Oinj__on @ A @ ( tc_fun @ A @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ A @ ( tc_fun @ A @ tc_HOL_Obool ) @ ( tc_fun @ A @ tc_HOL_Obool ) ) @ ( c_Set_Oinsert @ A ) ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ A @ tc_HOL_Obool ) ) ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ A @ tc_HOL_Obool ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_inj__singleton) ).

thf(5084,plain,
    ! [A: $i] : ( c_Fun_Oinj__on @ A @ ( tc_fun @ A @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ A @ ( tc_fun @ A @ tc_HOL_Obool ) @ ( tc_fun @ A @ tc_HOL_Obool ) ) @ ( c_Set_Oinsert @ A ) ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ A @ tc_HOL_Obool ) ) ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ A @ tc_HOL_Obool ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[748]) ).

thf(526,axiom,
    ( ( hAPP @ ( c_Nat_Osize__class_Osize @ tc_HOL_Obool ) @ c_fFalse )
    = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_bool_Osize_I4_J) ).

thf(4297,plain,
    ( ( hAPP @ ( c_Nat_Osize__class_Osize @ tc_HOL_Obool ) @ c_fFalse )
    = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[526]) ).

thf(812,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ D ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Nat_Ocompow @ ( tc_prod @ E @ E ) @ tc_HOL_Obool ) @ ( hAPP @ c_Nat_OSuc @ B ) ) @ A ) ) )
     => ~ ! [F: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ D ) @ F ) ) @ A ) )
           => ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ F ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Nat_Ocompow @ ( tc_prod @ E @ E ) @ tc_HOL_Obool ) @ B ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_rel__pow__Suc__E2) ).

thf(5333,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ D ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Nat_Ocompow @ ( tc_prod @ E @ E ) @ tc_HOL_Obool ) @ ( hAPP @ c_Nat_OSuc @ B ) ) @ A ) ) )
     => ~ ! [F: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ D ) @ F ) ) @ A ) )
           => ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ F ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Nat_Ocompow @ ( tc_prod @ E @ E ) @ tc_HOL_Obool ) @ B ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[812]) ).

thf(1458,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( c_Fun_Oinj__on @ F @ E @ ( hAPP @ ( c_Fun_Ocomp @ D @ E @ F @ C ) @ B ) @ A )
     => ( c_Fun_Oinj__on @ F @ D @ B @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_inj__on__imageI2) ).

thf(7986,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( c_Fun_Oinj__on @ F @ E @ ( hAPP @ ( c_Fun_Ocomp @ D @ E @ F @ C ) @ B ) @ A )
     => ( c_Fun_Oinj__on @ F @ D @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1458]) ).

thf(1375,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
      ( ( ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ H ) ) @ G )
        = ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ F ) ) @ E ) )
     => ( ( ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ H ) ) @ D )
          = ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ F ) ) @ E ) )
       => ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ F ) @ C ) @ ( hAPP @ ( c_List_Oset @ F ) @ E ) ) )
         => ( ( ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ F @ ( tc_Option_Ooption @ H ) ) @ ( c_Map_Omap__of @ F @ H @ ( hAPP @ ( hAPP @ ( c_List_Ozip @ F @ H ) @ E ) @ G ) ) ) @ C ) @ ( hAPP @ ( c_Option_Ooption_OSome @ H ) @ B ) )
              = ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ F @ ( tc_Option_Ooption @ H ) ) @ ( c_Map_Omap__of @ F @ H @ ( hAPP @ ( hAPP @ ( c_List_Ozip @ F @ H ) @ E ) @ D ) ) ) @ C ) @ ( hAPP @ ( c_Option_Ooption_OSome @ H ) @ A ) ) )
           => ( ( c_Map_Omap__of @ F @ H @ ( hAPP @ ( hAPP @ ( c_List_Ozip @ F @ H ) @ E ) @ G ) )
              = ( c_Map_Omap__of @ F @ H @ ( hAPP @ ( hAPP @ ( c_List_Ozip @ F @ H ) @ E ) @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_map__of__zip__upd) ).

thf(7677,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
      ( ( ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ H ) ) @ G )
        = ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ F ) ) @ E ) )
     => ( ( ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ H ) ) @ D )
          = ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ F ) ) @ E ) )
       => ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ F ) @ C ) @ ( hAPP @ ( c_List_Oset @ F ) @ E ) ) )
         => ( ( ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ F @ ( tc_Option_Ooption @ H ) ) @ ( c_Map_Omap__of @ F @ H @ ( hAPP @ ( hAPP @ ( c_List_Ozip @ F @ H ) @ E ) @ G ) ) ) @ C ) @ ( hAPP @ ( c_Option_Ooption_OSome @ H ) @ B ) )
              = ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ F @ ( tc_Option_Ooption @ H ) ) @ ( c_Map_Omap__of @ F @ H @ ( hAPP @ ( hAPP @ ( c_List_Ozip @ F @ H ) @ E ) @ D ) ) ) @ C ) @ ( hAPP @ ( c_Option_Ooption_OSome @ H ) @ A ) ) )
           => ( ( c_Map_Omap__of @ F @ H @ ( hAPP @ ( hAPP @ ( c_List_Ozip @ F @ H ) @ E ) @ G ) )
              = ( c_Map_Omap__of @ F @ H @ ( hAPP @ ( hAPP @ ( c_List_Ozip @ F @ H ) @ E ) @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1375]) ).

thf(1797,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ D ) @ C ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ D ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__le__mono) ).

thf(9457,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ D ) @ C ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ D ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1797]) ).

thf(941,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ C ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ B ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zdiv__mono1) ).

thf(5805,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ C ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ B ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[941]) ).

thf(633,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) @ A ) )
     => ( ~ ( hBOOL @ ( hAPP @ C @ A ) )
       => ( hBOOL @ ( hAPP @ B @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_sup1E) ).

thf(4686,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) @ A ) )
     => ( ~ ( hBOOL @ ( hAPP @ C @ A ) )
       => ( hBOOL @ ( hAPP @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[633]) ).

thf(1757,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ ( hAPP @ c_Nat_OSuc @ A ) ) )
    <=> ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) )
        | ( B = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__Suc__eq) ).

thf(9309,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ ( hAPP @ c_Nat_OSuc @ A ) ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) )
          | ( B = A ) ) )
      & ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) )
          | ( B = A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ ( hAPP @ c_Nat_OSuc @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1757]) ).

thf(15,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( class_Groups_Ocomm__monoid__add @ F )
     => ( ( c_Fun_Oinj__on @ E @ D @ C @ B )
       => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ F ) @ A ) @ ( hAPP @ ( c_Set_Oimage @ E @ D @ C ) @ B ) )
          = ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ E @ F ) @ ( hAPP @ ( c_Fun_Ocomp @ D @ F @ E @ A ) @ C ) ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_setsum__reindex) ).

thf(2456,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( class_Groups_Ocomm__monoid__add @ F )
     => ( ( c_Fun_Oinj__on @ E @ D @ C @ B )
       => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ F ) @ A ) @ ( hAPP @ ( c_Set_Oimage @ E @ D @ C ) @ B ) )
          = ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ E @ F ) @ ( hAPP @ ( c_Fun_Ocomp @ D @ F @ E @ A ) @ C ) ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[15]) ).

thf(2005,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ C ) @ B ) )
        & ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ C ) ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ C ) @ A ) )
          & ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ A ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_dvd_Oless__le__trans) ).

thf(10309,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ C ) @ B ) )
        & ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ C ) ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ C ) @ A ) )
          & ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ A ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2005]) ).

thf(2038,axiom,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ ( c_List_Odistinct @ tc_Nat_Onat ) @ ( c_List_Oupt @ B @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_distinct__upt) ).

thf(10419,plain,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ ( c_List_Odistinct @ tc_Nat_Onat ) @ ( c_List_Oupt @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2038]) ).

thf(2144,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Com_Ocom_OAss @ C @ B )
     != ( hAPP @ c_Com_Ocom_OBODY @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_com_Osimps_I30_J) ).

thf(10818,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Com_Ocom_OAss @ C @ B )
     != ( hAPP @ c_Com_Ocom_OBODY @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2144]) ).

thf(409,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ D @ C ) @ B ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ D @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ D @ ( tc_fun @ C @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ C @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) ) @ D ) @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ C @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) ) ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ ( tc_fun @ C @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) ) ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ ( tc_fun @ C @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) ) ) @ D ) @ ( c_COMBC @ C @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ ( tc_prod @ D @ C ) ) @ ( tc_fun @ C @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) ) ) @ D ) @ ( hAPP @ ( c_COMBB @ ( tc_prod @ D @ C ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) ) @ C ) @ ( c_Set_Oinsert @ ( tc_prod @ D @ C ) ) ) ) @ ( c_Product__Type_OPair @ D @ C ) ) ) ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Sigma__def) ).

thf(3887,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ D @ C ) @ B ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ D @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ D @ ( tc_fun @ C @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ C @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) ) @ D ) @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ C @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) ) ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ ( tc_fun @ C @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) ) ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ ( tc_fun @ C @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) ) ) @ D ) @ ( c_COMBC @ C @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ ( tc_prod @ D @ C ) ) @ ( tc_fun @ C @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) ) ) @ D ) @ ( hAPP @ ( c_COMBB @ ( tc_prod @ D @ C ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) ) @ C ) @ ( c_Set_Oinsert @ ( tc_prod @ D @ C ) ) ) ) @ ( c_Product__Type_OPair @ D @ C ) ) ) ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[409]) ).

thf(433,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Un__empty__right) ).

thf(3966,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[433]) ).

thf(2217,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( c_SetInterval_Oord_OatLeast @ D @ B @ A ) ) )
    <=> ( hBOOL @ ( hAPP @ ( hAPP @ B @ A ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_ord_OatLeast__iff) ).

thf(11064,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( c_SetInterval_Oord_OatLeast @ D @ B @ A ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ B @ A ) @ C ) ) )
      & ( ( hBOOL @ ( hAPP @ ( hAPP @ B @ A ) @ C ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( c_SetInterval_Oord_OatLeast @ D @ B @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2217]) ).

thf(870,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Int_Oint @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ A ) @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ) )
      = ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Int_Oint @ B @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_atLeastLessThanPlusOne__atLeastAtMost__int) ).

thf(5526,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Int_Oint @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ A ) @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ) )
      = ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Int_Oint @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[870]) ).

thf(493,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) ) )
     => ( B = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_singletonE) ).

thf(4173,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) ) )
     => ( B = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[493]) ).

thf(424,axiom,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ A ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_subset__UNIV) ).

thf(3934,plain,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ A ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[424]) ).

thf(451,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ D @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ D ) @ ( c_COMBC @ D @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ D @ ( tc_prod @ D @ D ) ) @ ( tc_fun @ D @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ D ) @ ( hAPP @ ( c_COMBB @ ( tc_prod @ D @ D ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ D ) @ ( c_member @ ( tc_prod @ D @ D ) ) ) ) @ ( c_Product__Type_OPair @ D @ D ) ) ) ) @ C ) ) @ c_fequal ) @ B ) @ A ) )
    <=> ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ D ) @ B ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) ) @ C ) @ ( c_Relation_OId @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_reflcl__set__eq) ).

thf(4018,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ D @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ D ) @ ( c_COMBC @ D @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ D @ ( tc_prod @ D @ D ) ) @ ( tc_fun @ D @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ D ) @ ( hAPP @ ( c_COMBB @ ( tc_prod @ D @ D ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ D ) @ ( c_member @ ( tc_prod @ D @ D ) ) ) ) @ ( c_Product__Type_OPair @ D @ D ) ) ) ) @ C ) ) @ c_fequal ) @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ D ) @ B ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) ) @ C ) @ ( c_Relation_OId @ D ) ) ) ) )
      & ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ D ) @ B ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) ) @ C ) @ ( c_Relation_OId @ D ) ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ D @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ D ) @ ( c_COMBC @ D @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ D @ ( tc_prod @ D @ D ) ) @ ( tc_fun @ D @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ D ) @ ( hAPP @ ( c_COMBB @ ( tc_prod @ D @ D ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ D ) @ ( c_member @ ( tc_prod @ D @ D ) ) ) ) @ ( c_Product__Type_OPair @ D @ D ) ) ) ) @ C ) ) @ c_fequal ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[451]) ).

thf(1762,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( c_Set_Oimage @ tc_Nat_Onat @ tc_Nat_Onat @ c_Nat_OSuc ) @ ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Nat_Onat @ B @ A ) )
      = ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Nat_Onat @ ( hAPP @ c_Nat_OSuc @ B ) @ ( hAPP @ c_Nat_OSuc @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_image__Suc__atLeastAtMost) ).

thf(9330,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( c_Set_Oimage @ tc_Nat_Onat @ tc_Nat_Onat @ c_Nat_OSuc ) @ ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Nat_Onat @ B @ A ) )
      = ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Nat_Onat @ ( hAPP @ c_Nat_OSuc @ B ) @ ( hAPP @ c_Nat_OSuc @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1762]) ).

thf(2158,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_List_Odistinct @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( c_List_Odistinct @ C ) @ ( hAPP @ ( hAPP @ ( c_List_Odrop @ C ) @ A ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_distinct__drop) ).

thf(10871,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_List_Odistinct @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( c_List_Odistinct @ C ) @ ( hAPP @ ( hAPP @ ( c_List_Odrop @ C ) @ A ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2158]) ).

thf(2090,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ E @ D ) ) @ ( hAPP @ E @ C ) ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ B @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ B @ B ) @ D ) @ C ) ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ B @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ B @ B ) @ D ) @ C ) ) @ ( c_Wellfounded_Omlex__prod @ B @ E @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mlex__leq) ).

thf(10623,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ E @ D ) ) @ ( hAPP @ E @ C ) ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ B @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ B @ B ) @ D ) @ C ) ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ B @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ B @ B ) @ D ) @ C ) ) @ ( c_Wellfounded_Omlex__prod @ B @ E @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2090]) ).

thf(164,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Diff__Int__distrib) ).

thf(2996,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[164]) ).

thf(2080,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( c_Fun_Oinj__on @ F @ E @ D @ C )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ B ) @ ( hAPP @ ( c_Set_Oimage @ F @ E @ D ) @ C ) ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ F @ tc_HOL_Obool ) ) @ C ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ F ) @ ( hAPP @ ( c_Fun_Othe__inv__into @ F @ E @ C @ D ) @ B ) ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_the__inv__into__into) ).

thf(10591,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( c_Fun_Oinj__on @ F @ E @ D @ C )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ B ) @ ( hAPP @ ( c_Set_Oimage @ F @ E @ D ) @ C ) ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ F @ tc_HOL_Obool ) ) @ C ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ F ) @ ( hAPP @ ( c_Fun_Othe__inv__into @ F @ E @ C @ D ) @ B ) ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2080]) ).

thf(1453,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( c_Finite__Set_Ocard @ tc_Nat_Onat ) @ ( c_SetInterval_Oord__class_OgreaterThanLessThan @ tc_Nat_Onat @ B @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ A ) @ ( hAPP @ c_Nat_OSuc @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_card__greaterThanLessThan) ).

thf(7969,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( c_Finite__Set_Ocard @ tc_Nat_Onat ) @ ( c_SetInterval_Oord__class_OgreaterThanLessThan @ tc_Nat_Onat @ B @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ A ) @ ( hAPP @ c_Nat_OSuc @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1453]) ).

thf(251,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( c_Relation_OField @ B ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Relation_ODomain @ B @ B ) @ A ) ) @ ( hAPP @ ( c_Relation_ORange @ B @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Field__def) ).

thf(3295,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( c_Relation_OField @ B ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Relation_ODomain @ B @ B ) @ A ) ) @ ( hAPP @ ( c_Relation_ORange @ B @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[251]) ).

thf(107,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_List_Otranspose @ D @ ( hAPP @ ( hAPP @ ( c_List_Omap @ ( tc_List_Olist @ C ) @ ( tc_List_Olist @ D ) ) @ ( hAPP @ ( c_List_Omap @ C @ D ) @ B ) ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_List_Omap @ ( tc_List_Olist @ C ) @ ( tc_List_Olist @ D ) ) @ ( hAPP @ ( c_List_Omap @ C @ D ) @ B ) ) @ ( c_List_Otranspose @ C @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_transpose__map__map) ).

thf(2782,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_List_Otranspose @ D @ ( hAPP @ ( hAPP @ ( c_List_Omap @ ( tc_List_Olist @ C ) @ ( tc_List_Olist @ D ) ) @ ( hAPP @ ( c_List_Omap @ C @ D ) @ B ) ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_List_Omap @ ( tc_List_Olist @ C ) @ ( tc_List_Olist @ D ) ) @ ( hAPP @ ( c_List_Omap @ C @ D ) @ B ) ) @ ( c_List_Otranspose @ C @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[107]) ).

thf(1486,axiom,
    ! [A: $i] :
      ( ( hAPP @ c_Nat_OSuc @ A )
      = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Suc__eq__plus1) ).

thf(8158,plain,
    ! [A: $i] :
      ( ( hAPP @ c_Nat_OSuc @ A )
      = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1486]) ).

thf(1410,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ D ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ C ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ C ) @ B ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ B ) @ ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ A @ D ) ) ) @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zmult2__lemma__aux3) ).

thf(7818,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ D ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ C ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ C ) @ B ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ B ) @ ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ A @ D ) ) ) @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1410]) ).

thf(315,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Equiv__Relations_Oequiv @ E @ D @ C )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ B ) @ D ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ A ) @ D ) )
         => ( ( ( hAPP @ ( hAPP @ ( c_Equiv__Relations_Oquotient @ E ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ B ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ E @ tc_HOL_Obool ) ) ) ) @ C )
              = ( hAPP @ ( hAPP @ ( c_Equiv__Relations_Oquotient @ E ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ E @ tc_HOL_Obool ) ) ) ) @ C ) )
          <=> ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ B ) @ A ) ) @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_eq__equiv__class__iff2) ).

thf(3527,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Equiv__Relations_Oequiv @ E @ D @ C )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ B ) @ D ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ A ) @ D ) )
         => ( ( ( ( hAPP @ ( hAPP @ ( c_Equiv__Relations_Oquotient @ E ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ B ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ E @ tc_HOL_Obool ) ) ) ) @ C )
                = ( hAPP @ ( hAPP @ ( c_Equiv__Relations_Oquotient @ E ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ E @ tc_HOL_Obool ) ) ) ) @ C ) )
             => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ B ) @ A ) ) @ C ) ) )
            & ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ B ) @ A ) ) @ C ) )
             => ( ( hAPP @ ( hAPP @ ( c_Equiv__Relations_Oquotient @ E ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ B ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ E @ tc_HOL_Obool ) ) ) ) @ C )
                = ( hAPP @ ( hAPP @ ( c_Equiv__Relations_Oquotient @ E ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ E @ tc_HOL_Obool ) ) ) ) @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[315]) ).

thf(1825,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( hAPP @ ( c_Relation_ODomain @ D @ B ) @ A ) ) )
     => ~ ! [E: $i] :
            ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ B ) @ C ) @ E ) ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_DomainE) ).

thf(9557,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( hAPP @ ( c_Relation_ODomain @ D @ B ) @ A ) ) )
     => ~ ! [E: $i] :
            ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ B ) @ C ) @ E ) ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1825]) ).

thf(1033,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_List_Odistinct @ C ) @ B ) )
     => ( ( hAPP @ ( c_List_Ofilter @ C @ ( hAPP @ ( hAPP @ ( c_COMBC @ C @ ( tc_fun @ C @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( c_member @ C ) ) @ ( hAPP @ ( c_List_Oset @ C ) @ ( c_List_Osublist @ C @ B @ A ) ) ) ) @ B )
        = ( c_List_Osublist @ C @ B @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_filter__in__sublist) ).

thf(6189,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_List_Odistinct @ C ) @ B ) )
     => ( ( hAPP @ ( c_List_Ofilter @ C @ ( hAPP @ ( hAPP @ ( c_COMBC @ C @ ( tc_fun @ C @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( c_member @ C ) ) @ ( hAPP @ ( c_List_Oset @ C ) @ ( c_List_Osublist @ C @ B @ A ) ) ) ) @ B )
        = ( c_List_Osublist @ C @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1033]) ).

thf(133,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hAPP @ ( c_List_Ofoldl @ F @ E @ D @ C ) @ ( hAPP @ ( hAPP @ ( c_List_Oappend @ E ) @ B ) @ A ) )
      = ( hAPP @ ( c_List_Ofoldl @ F @ E @ D @ ( hAPP @ ( c_List_Ofoldl @ F @ E @ D @ C ) @ B ) ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_foldl__append) ).

thf(2880,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hAPP @ ( c_List_Ofoldl @ F @ E @ D @ C ) @ ( hAPP @ ( hAPP @ ( c_List_Oappend @ E ) @ B ) @ A ) )
      = ( hAPP @ ( c_List_Ofoldl @ F @ E @ D @ ( hAPP @ ( c_List_Ofoldl @ F @ E @ D @ C ) @ B ) ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[133]) ).

thf(1105,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ D ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Nat_Ocompow @ ( tc_prod @ E @ E ) @ tc_HOL_Obool ) @ B ) @ A ) ) )
     => ( ( ( B
            = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
         => ( D != C ) )
       => ~ ! [F: $i,G: $i] :
              ( ( B
                = ( hAPP @ c_Nat_OSuc @ G ) )
             => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ D ) @ F ) ) @ A ) )
               => ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ F ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Nat_Ocompow @ ( tc_prod @ E @ E ) @ tc_HOL_Obool ) @ G ) @ A ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_rel__pow__E2) ).

thf(6534,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ D ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Nat_Ocompow @ ( tc_prod @ E @ E ) @ tc_HOL_Obool ) @ B ) @ A ) ) )
     => ( ( ( B
            = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
         => ( D != C ) )
       => ~ ! [F: $i,G: $i] :
              ( ( B
                = ( hAPP @ c_Nat_OSuc @ G ) )
             => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ D ) @ F ) ) @ A ) )
               => ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ F ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Nat_Ocompow @ ( tc_prod @ E @ E ) @ tc_HOL_Obool ) @ G ) @ A ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1105]) ).

thf(1304,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Nat_Onat @ A ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ ( hAPP @ ( c_Finite__Set_Ocard @ tc_Nat_Onat ) @ B ) ) ) ) )
     => ( B
        = ( hAPP @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Nat_Onat @ A ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ ( hAPP @ ( c_Finite__Set_Ocard @ tc_Nat_Onat ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_subset__card__intvl__is__intvl) ).

thf(7376,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Nat_Onat @ A ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ ( hAPP @ ( c_Finite__Set_Ocard @ tc_Nat_Onat ) @ B ) ) ) ) )
     => ( B
        = ( hAPP @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Nat_Onat @ A ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ ( hAPP @ ( c_Finite__Set_Ocard @ tc_Nat_Onat ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1304]) ).

thf(1883,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ ( tc_prod @ B @ B ) ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ B ) @ ( hAPP @ ( c_Relation_OField @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_finite__Field) ).

thf(9811,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ ( tc_prod @ B @ B ) ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ B ) @ ( hAPP @ ( c_Relation_OField @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1883]) ).

thf(1419,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Divides_Odiv__class_Omod @ tc_Nat_Onat @ ( hAPP @ c_Nat_OSuc @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ B ) ) @ A ) ) @ B )
      = ( c_Divides_Odiv__class_Omod @ tc_Nat_Onat @ ( hAPP @ c_Nat_OSuc @ A ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mod__mult__self4) ).

thf(7845,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Divides_Odiv__class_Omod @ tc_Nat_Onat @ ( hAPP @ c_Nat_OSuc @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ B ) ) @ A ) ) @ B )
      = ( c_Divides_Odiv__class_Omod @ tc_Nat_Onat @ ( hAPP @ c_Nat_OSuc @ A ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1419]) ).

thf(2357,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( hAPP @ F @ E )
        = D )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ D ) @ B ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ A ) @ E ) @ ( hAPP @ ( c_Set_Ovimage @ A @ C @ F ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_vimageI) ).

thf(11676,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( hAPP @ F @ E )
        = D )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ D ) @ B ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ A ) @ E ) @ ( hAPP @ ( c_Set_Ovimage @ A @ C @ F ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2357]) ).

thf(671,axiom,
    ( ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint )
   != ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_int__0__neq__1) ).

thf(4815,plain,
    ( ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint )
   != ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[671]) ).

thf(2250,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_Natural_Oevaln @ D @ C @ B @ A )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Natural_Oevalc @ D ) @ C ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_evaln__evalc) ).

thf(11215,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_Natural_Oevaln @ D @ C @ B @ A )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Natural_Oevalc @ D ) @ C ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2250]) ).

thf(342,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ A ) @ B ) ) )
      <=> ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_pos__imp__zdiv__pos__iff) ).

thf(3621,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ B ) )
     => ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ A ) @ B ) ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ B ) @ A ) ) )
        & ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ B ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ A ) @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[342]) ).

thf(143,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ B ) @ ( hAPP @ ( c_Set_OCollect @ B ) @ A ) ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ B ) @ ( hAPP @ ( c_Set_OCollect @ B ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ tc_HOL_Obool @ B ) @ c_fNot ) @ A ) ) ) )
      <=> ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ B ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_finite__Collect__not) ).

thf(2925,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ B ) @ ( hAPP @ ( c_Set_OCollect @ B ) @ A ) ) )
     => ( ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ B ) @ ( hAPP @ ( c_Set_OCollect @ B ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ tc_HOL_Obool @ B ) @ c_fNot ) @ A ) ) ) )
         => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ B ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) ) )
        & ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ B ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) )
         => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ B ) @ ( hAPP @ ( c_Set_OCollect @ B ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ tc_HOL_Obool @ B ) @ c_fNot ) @ A ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[143]) ).

thf(1177,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Finite__Set_Ofolding__one @ E @ D @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ E ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ A ) @ B ) )
         => ( ( ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ E @ tc_HOL_Obool ) ) ) )
                = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ E @ tc_HOL_Obool ) ) )
             => ( ( hAPP @ C @ B )
                = A ) )
            & ( ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ E @ tc_HOL_Obool ) ) ) )
               != ( c_Orderings_Obot__class_Obot @ ( tc_fun @ E @ tc_HOL_Obool ) ) )
             => ( ( hAPP @ C @ B )
                = ( hAPP @ ( hAPP @ D @ A ) @ ( hAPP @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ E @ tc_HOL_Obool ) ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_folding__one_Oremove) ).

thf(6841,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Finite__Set_Ofolding__one @ E @ D @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ E ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ A ) @ B ) )
         => ( ( ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ E @ tc_HOL_Obool ) ) ) )
                = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ E @ tc_HOL_Obool ) ) )
             => ( ( hAPP @ C @ B )
                = A ) )
            & ( ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ E @ tc_HOL_Obool ) ) ) )
               != ( c_Orderings_Obot__class_Obot @ ( tc_fun @ E @ tc_HOL_Obool ) ) )
             => ( ( hAPP @ C @ B )
                = ( hAPP @ ( hAPP @ D @ A ) @ ( hAPP @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ E @ tc_HOL_Obool ) ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1177]) ).

thf(903,axiom,
    ! [A: $i] : ( c_Wellfounded_OwfP @ A @ ( hAPP @ ( c_COMBK @ ( tc_fun @ A @ tc_HOL_Obool ) @ A ) @ ( hAPP @ ( c_COMBK @ tc_HOL_Obool @ A ) @ c_fFalse ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_wfP__empty) ).

thf(5650,plain,
    ! [A: $i] : ( c_Wellfounded_OwfP @ A @ ( hAPP @ ( c_COMBK @ ( tc_fun @ A @ tc_HOL_Obool ) @ A ) @ ( hAPP @ ( c_COMBK @ tc_HOL_Obool @ A ) @ c_fFalse ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[903]) ).

thf(1507,axiom,
    ! [A: $i,B: $i] :
      ( ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ A )
        = ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) )
    <=> ( ( B
          = ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) )
        & ( A
          = ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nat__mult__eq__1__iff) ).

thf(8246,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ A )
          = ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) )
       => ( ( B
            = ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) )
          & ( A
            = ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) ) )
      & ( ( ( B
            = ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) )
          & ( A
            = ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) )
       => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ A )
          = ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1507]) ).

thf(47,axiom,
    class_Groups_Ocomm__monoid__mult @ tc_Int_Oint,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Groups_Ocomm__monoid__mult) ).

thf(2555,plain,
    class_Groups_Ocomm__monoid__mult @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[47]) ).

thf(249,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( hAPP @ ( hAPP @ ( c_List_Odrop @ E ) @ D ) @ C )
        = ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ E ) @ B ) @ A ) )
     => ( ( hAPP @ ( hAPP @ ( c_List_Onth @ E ) @ C ) @ D )
        = B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nth__via__drop) ).

thf(3290,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( hAPP @ ( hAPP @ ( c_List_Odrop @ E ) @ D ) @ C )
        = ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ E ) @ B ) @ A ) )
     => ( ( hAPP @ ( hAPP @ ( c_List_Onth @ E ) @ C ) @ D )
        = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[249]) ).

thf(1571,axiom,
    ! [A: $i] :
      ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_not__less0) ).

thf(8479,plain,
    ! [A: $i] :
      ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1571]) ).

thf(819,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Wellfounded_Owf @ C @ B )
     => ( c_Wellfounded_Owf @ C @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) ) @ A ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_wf__Int2) ).

thf(5360,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Wellfounded_Owf @ C @ B )
     => ( c_Wellfounded_Owf @ C @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) ) @ A ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[819]) ).

thf(2126,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( c_Com_Ocom_OCall @ F @ E @ D )
     != ( c_Com_Ocom_OLocal @ C @ B @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_com_Osimps_I43_J) ).

thf(10734,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( c_Com_Ocom_OCall @ F @ E @ D )
     != ( c_Com_Ocom_OLocal @ C @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2126]) ).

thf(1496,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ A ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_one__less__mult) ).

thf(8196,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ A ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1496]) ).

thf(1597,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( c_Divides_Odiv__class_Omod @ tc_Nat_Onat @ C @ B )
        = ( c_Divides_Odiv__class_Omod @ tc_Nat_Onat @ A @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ A ) @ C ) )
       => ? [D: $i] :
            ( C
            = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nat__mod__eq__lemma) ).

thf(8664,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( c_Divides_Odiv__class_Omod @ tc_Nat_Onat @ C @ B )
        = ( c_Divides_Odiv__class_Omod @ tc_Nat_Onat @ A @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ A ) @ C ) )
       => ? [D: $i] :
            ( C
            = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1597]) ).

thf(380,axiom,
    ! [A: $i,B: $i] :
      ( ? [C: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ B ) @ C ) @ A ) )
    <=> ( A
       != ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_ex__in__conv) ).

thf(3771,plain,
    ! [A: $i,B: $i] :
      ( ( ? [C: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ B ) @ C ) @ A ) )
       => ( A
         != ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) )
      & ( ( A
         != ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) )
       => ? [C: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ B ) @ C ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[380]) ).

thf(669,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ A ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Int__commute) ).

thf(4810,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ A ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[669]) ).

thf(2085,axiom,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) )
        | ( B = A ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__or__eq__imp__le) ).

thf(10602,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) )
        | ( B = A ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2085]) ).

thf(1433,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i,I: $i,J: $i] :
      ( ( c_Product__Type_Omap__pair @ J @ I @ H @ G @ ( hAPP @ ( c_Fun_Ocomp @ F @ I @ J @ E ) @ D ) @ ( hAPP @ ( c_Fun_Ocomp @ C @ G @ H @ B ) @ A ) )
      = ( hAPP @ ( c_Fun_Ocomp @ ( tc_prod @ F @ C ) @ ( tc_prod @ I @ G ) @ ( tc_prod @ J @ H ) @ ( c_Product__Type_Omap__pair @ F @ I @ C @ G @ E @ B ) ) @ ( c_Product__Type_Omap__pair @ J @ F @ H @ C @ D @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_map__pair__compose) ).

thf(7899,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i,I: $i,J: $i] :
      ( ( c_Product__Type_Omap__pair @ J @ I @ H @ G @ ( hAPP @ ( c_Fun_Ocomp @ F @ I @ J @ E ) @ D ) @ ( hAPP @ ( c_Fun_Ocomp @ C @ G @ H @ B ) @ A ) )
      = ( hAPP @ ( c_Fun_Ocomp @ ( tc_prod @ F @ C ) @ ( tc_prod @ I @ G ) @ ( tc_prod @ J @ H ) @ ( c_Product__Type_Omap__pair @ F @ I @ C @ G @ E @ B ) ) @ ( c_Product__Type_Omap__pair @ J @ F @ H @ C @ D @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1433]) ).

thf(2033,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ ( hAPP @ c_Nat_OSuc @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__SucI) ).

thf(10404,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ ( hAPP @ c_Nat_OSuc @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2033]) ).

thf(644,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Predicate_ODomainP @ D @ C @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ D ) @ ( c_COMBC @ C @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ ( tc_prod @ D @ C ) ) @ ( tc_fun @ C @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ D ) @ ( hAPP @ ( c_COMBB @ ( tc_prod @ D @ C ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ C ) @ ( c_member @ ( tc_prod @ D @ C ) ) ) ) @ ( c_Product__Type_OPair @ D @ C ) ) ) ) @ B ) ) @ A ) )
    <=> ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ A ) @ ( hAPP @ ( c_Relation_ODomain @ D @ C ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_DomainP__Domain__eq) ).

thf(4718,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( c_Predicate_ODomainP @ D @ C @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ D ) @ ( c_COMBC @ C @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ ( tc_prod @ D @ C ) ) @ ( tc_fun @ C @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ D ) @ ( hAPP @ ( c_COMBB @ ( tc_prod @ D @ C ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ C ) @ ( c_member @ ( tc_prod @ D @ C ) ) ) ) @ ( c_Product__Type_OPair @ D @ C ) ) ) ) @ B ) ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ A ) @ ( hAPP @ ( c_Relation_ODomain @ D @ C ) @ B ) ) ) )
      & ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ A ) @ ( hAPP @ ( c_Relation_ODomain @ D @ C ) @ B ) ) )
       => ( hBOOL @ ( hAPP @ ( c_Predicate_ODomainP @ D @ C @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ D ) @ ( c_COMBC @ C @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ ( tc_prod @ D @ C ) ) @ ( tc_fun @ C @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ D ) @ ( hAPP @ ( c_COMBB @ ( tc_prod @ D @ C ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ C ) @ ( c_member @ ( tc_prod @ D @ C ) ) ) ) @ ( c_Product__Type_OPair @ D @ C ) ) ) ) @ B ) ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[644]) ).

thf(1767,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ A ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ A ) @ B ) )
       => ( B = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_le__antisym) ).

thf(9347,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ A ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ A ) @ B ) )
       => ( B = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1767]) ).

thf(1986,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_termination__basic__simps_I3_J) ).

thf(10233,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1986]) ).

thf(1652,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ D ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ C ) @ B ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ ( c_Divides_Odiv__class_Omod @ tc_Nat_Onat @ A @ D ) ) ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mod__lemma) ).

thf(8892,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ D ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ C ) @ B ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ ( c_Divides_Odiv__class_Omod @ tc_Nat_Onat @ A @ D ) ) ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1652]) ).

thf(1665,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Equiv__Relations_Oequiv @ E @ D @ C )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ B ) @ A ) ) @ C ) )
       => ( ( hAPP @ ( c_Relation_OImage @ E @ E @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ B ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ E @ tc_HOL_Obool ) ) ) )
          = ( hAPP @ ( c_Relation_OImage @ E @ E @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ E @ tc_HOL_Obool ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_equiv__class__eq) ).

thf(8939,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Equiv__Relations_Oequiv @ E @ D @ C )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ B ) @ A ) ) @ C ) )
       => ( ( hAPP @ ( c_Relation_OImage @ E @ E @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ B ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ E @ tc_HOL_Obool ) ) ) )
          = ( hAPP @ ( c_Relation_OImage @ E @ E @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ E @ tc_HOL_Obool ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1665]) ).

thf(281,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ A ) )
       => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) ) ) @ ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) ) )
          = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ B ) ) @ ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_card_Ounion__inter) ).

thf(3406,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ A ) )
       => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) ) ) @ ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) ) )
          = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ B ) ) @ ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[281]) ).

thf(278,axiom,
    ! [A: $i,B: $i] :
      ( ( ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) )
        = ( hAPP @ ( c_List_Oset @ B ) @ A ) )
    <=> ( A
        = ( c_List_Olist_ONil @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_set__empty2) ).

thf(3389,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) )
          = ( hAPP @ ( c_List_Oset @ B ) @ A ) )
       => ( A
          = ( c_List_Olist_ONil @ B ) ) )
      & ( ( A
          = ( c_List_Olist_ONil @ B ) )
       => ( ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) )
          = ( hAPP @ ( c_List_Oset @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[278]) ).

thf(155,axiom,
    ! [A: $i] : ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ tc_Nat_Onat ) @ ( hAPP @ ( c_Set_OCollect @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_finite__Collect__le__nat) ).

thf(2964,plain,
    ! [A: $i] : ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ tc_Nat_Onat ) @ ( hAPP @ ( c_Set_OCollect @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[155]) ).

thf(744,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( hAPP @ ( c_List_Ofoldl @ G @ F @ E @ D ) @ ( hAPP @ ( hAPP @ ( c_List_Omap @ C @ F ) @ B ) @ A ) )
      = ( hAPP @ ( c_List_Ofoldl @ G @ C @ ( hAPP @ ( hAPP @ ( c_COMBC @ G @ ( tc_fun @ C @ F ) @ ( tc_fun @ C @ G ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ F @ G ) @ ( tc_fun @ ( tc_fun @ C @ F ) @ ( tc_fun @ C @ G ) ) @ G ) @ ( c_COMBB @ F @ G @ C ) ) @ E ) ) @ B ) @ D ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_foldl__map) ).

thf(5059,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( hAPP @ ( c_List_Ofoldl @ G @ F @ E @ D ) @ ( hAPP @ ( hAPP @ ( c_List_Omap @ C @ F ) @ B ) @ A ) )
      = ( hAPP @ ( c_List_Ofoldl @ G @ C @ ( hAPP @ ( hAPP @ ( c_COMBC @ G @ ( tc_fun @ C @ F ) @ ( tc_fun @ C @ G ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ F @ G ) @ ( tc_fun @ ( tc_fun @ C @ F ) @ ( tc_fun @ C @ G ) ) @ G ) @ ( c_COMBB @ F @ G @ C ) ) @ E ) ) @ B ) @ D ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[744]) ).

thf(1383,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ F ) @ E ) @ D ) )
       => ( ( hAPP @ ( c_Set_Oimage @ F @ C @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ F @ C ) @ B ) @ E ) @ A ) ) @ D )
          = ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ A ) @ ( hAPP @ ( c_Set_Oimage @ F @ C @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ F @ tc_HOL_Obool ) ) @ D ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ F ) @ E ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ F @ tc_HOL_Obool ) ) ) ) ) ) ) )
      & ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ F ) @ E ) @ D ) )
       => ( ( hAPP @ ( c_Set_Oimage @ F @ C @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ F @ C ) @ B ) @ E ) @ A ) ) @ D )
          = ( hAPP @ ( c_Set_Oimage @ F @ C @ B ) @ D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_fun__upd__image) ).

thf(7707,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ F ) @ E ) @ D ) )
       => ( ( hAPP @ ( c_Set_Oimage @ F @ C @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ F @ C ) @ B ) @ E ) @ A ) ) @ D )
          = ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ A ) @ ( hAPP @ ( c_Set_Oimage @ F @ C @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ F @ tc_HOL_Obool ) ) @ D ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ F ) @ E ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ F @ tc_HOL_Obool ) ) ) ) ) ) ) )
      & ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ F ) @ E ) @ D ) )
       => ( ( hAPP @ ( c_Set_Oimage @ F @ C @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ F @ C ) @ B ) @ E ) @ A ) ) @ D )
          = ( hAPP @ ( c_Set_Oimage @ F @ C @ B ) @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1383]) ).

thf(10,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Ocomm__monoid__add @ C )
     => ( ( ( hAPP @ B @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
          = ( c_Groups_Ozero__class_Ozero @ C ) )
       => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ tc_Nat_Onat @ C ) @ B ) @ ( hAPP @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Nat_Onat @ ( hAPP @ c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) @ A ) )
          = ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ tc_Nat_Onat @ C ) @ B ) @ ( hAPP @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_setsum__shift__lb__Suc0__0__upt) ).

thf(2437,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Ocomm__monoid__add @ C )
     => ( ( ( hAPP @ B @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
          = ( c_Groups_Ozero__class_Ozero @ C ) )
       => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ tc_Nat_Onat @ C ) @ B ) @ ( hAPP @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Nat_Onat @ ( hAPP @ c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) @ A ) )
          = ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ tc_Nat_Onat @ C ) @ B ) @ ( hAPP @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[10]) ).

thf(318,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ C ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ B ) ) @ A ) ) )
     => ( A
        = ( hAPP @ ( hAPP @ ( c_List_Oappend @ B ) @ ( hAPP @ ( hAPP @ ( c_List_Otake @ B ) @ C ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ B ) @ ( hAPP @ ( hAPP @ ( c_List_Onth @ B ) @ A ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_List_Odrop @ B ) @ ( hAPP @ c_Nat_OSuc @ C ) ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_id__take__nth__drop) ).

thf(3537,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ C ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ B ) ) @ A ) ) )
     => ( A
        = ( hAPP @ ( hAPP @ ( c_List_Oappend @ B ) @ ( hAPP @ ( hAPP @ ( c_List_Otake @ B ) @ C ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ B ) @ ( hAPP @ ( hAPP @ ( c_List_Onth @ B ) @ A ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_List_Odrop @ B ) @ ( hAPP @ c_Nat_OSuc @ C ) ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[318]) ).

thf(2106,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_List_Odistinct @ D ) @ C ) )
     => ( hBOOL @ ( hAPP @ ( c_List_Odistinct @ ( tc_prod @ D @ B ) ) @ ( hAPP @ ( hAPP @ ( c_List_Ozip @ D @ B ) @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_distinct__zipI1) ).

thf(10670,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_List_Odistinct @ D ) @ C ) )
     => ( hBOOL @ ( hAPP @ ( c_List_Odistinct @ ( tc_prod @ D @ B ) ) @ ( hAPP @ ( hAPP @ ( c_List_Ozip @ D @ B ) @ C ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2106]) ).

thf(1904,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( hAPP @ ( c_Relation_ORange @ B @ D ) @ A ) ) )
    <=> ? [E: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ B @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ B @ D ) @ E ) @ C ) ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Range__iff) ).

thf(9897,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( hAPP @ ( c_Relation_ORange @ B @ D ) @ A ) ) )
       => ? [E: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ B @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ B @ D ) @ E ) @ C ) ) @ A ) ) )
      & ( ? [E: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ B @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ B @ D ) @ E ) @ C ) ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( hAPP @ ( c_Relation_ORange @ B @ D ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1904]) ).

thf(1125,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( hAPP @ ( hAPP @ ( c_List_Oappend @ E ) @ D ) @ C )
        = ( hAPP @ ( hAPP @ ( c_List_Oappend @ E ) @ B ) @ A ) )
    <=> ? [F: $i] :
          ( ( ( D
              = ( hAPP @ ( hAPP @ ( c_List_Oappend @ E ) @ B ) @ F ) )
            & ( ( hAPP @ ( hAPP @ ( c_List_Oappend @ E ) @ F ) @ C )
              = A ) )
          | ( ( ( hAPP @ ( hAPP @ ( c_List_Oappend @ E ) @ D ) @ F )
              = B )
            & ( C
              = ( hAPP @ ( hAPP @ ( c_List_Oappend @ E ) @ F ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_append__eq__append__conv2) ).

thf(6648,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( ( hAPP @ ( hAPP @ ( c_List_Oappend @ E ) @ D ) @ C )
          = ( hAPP @ ( hAPP @ ( c_List_Oappend @ E ) @ B ) @ A ) )
       => ? [F: $i] :
            ( ( ( D
                = ( hAPP @ ( hAPP @ ( c_List_Oappend @ E ) @ B ) @ F ) )
              & ( ( hAPP @ ( hAPP @ ( c_List_Oappend @ E ) @ F ) @ C )
                = A ) )
            | ( ( ( hAPP @ ( hAPP @ ( c_List_Oappend @ E ) @ D ) @ F )
                = B )
              & ( C
                = ( hAPP @ ( hAPP @ ( c_List_Oappend @ E ) @ F ) @ A ) ) ) ) )
      & ( ? [F: $i] :
            ( ( ( D
                = ( hAPP @ ( hAPP @ ( c_List_Oappend @ E ) @ B ) @ F ) )
              & ( ( hAPP @ ( hAPP @ ( c_List_Oappend @ E ) @ F ) @ C )
                = A ) )
            | ( ( ( hAPP @ ( hAPP @ ( c_List_Oappend @ E ) @ D ) @ F )
                = B )
              & ( C
                = ( hAPP @ ( hAPP @ ( c_List_Oappend @ E ) @ F ) @ A ) ) ) )
       => ( ( hAPP @ ( hAPP @ ( c_List_Oappend @ E ) @ D ) @ C )
          = ( hAPP @ ( hAPP @ ( c_List_Oappend @ E ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1125]) ).

thf(2275,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Natural_Oevalc @ C ) @ B ) @ A ) )
     => ? [D: $i] : ( c_Natural_Oevaln @ C @ B @ D @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_evalc__evaln) ).

thf(11325,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Natural_Oevalc @ C ) @ B ) @ A ) )
     => ? [D: $i] : ( c_Natural_Oevaln @ C @ B @ D @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2275]) ).

thf(1462,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( hAPP @ ( c_Fun_Ocomp @ G @ F @ E @ D ) @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ E @ G ) @ C ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ E @ F ) @ ( hAPP @ ( c_Fun_Ocomp @ G @ F @ E @ D ) @ C ) ) @ B ) @ ( hAPP @ D @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_fun__upd__comp) ).

thf(8001,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( hAPP @ ( c_Fun_Ocomp @ G @ F @ E @ D ) @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ E @ G ) @ C ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ E @ F ) @ ( hAPP @ ( c_Fun_Ocomp @ G @ F @ E @ D ) @ C ) ) @ B ) @ ( hAPP @ D @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1462]) ).

thf(461,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) )
     => ( ( c_SetInterval_Oord_OgreaterThanLessThan @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_COMBS @ tc_Nat_Onat @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ tc_Nat_Onat @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) @ tc_Nat_Onat ) @ ( c_COMBS @ tc_Nat_Onat @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ tc_Nat_Onat @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ tc_Nat_Onat ) @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ tc_Nat_Onat ) @ c_fconj ) ) @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ tc_Nat_Onat ) @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ tc_HOL_Obool @ tc_Nat_Onat ) @ c_fNot ) ) @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) ) ) ) @ A @ B )
        = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_dvd_OgreaterThanLessThan__empty) ).

thf(4055,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) )
     => ( ( c_SetInterval_Oord_OgreaterThanLessThan @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_COMBS @ tc_Nat_Onat @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ tc_Nat_Onat @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) @ tc_Nat_Onat ) @ ( c_COMBS @ tc_Nat_Onat @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ tc_Nat_Onat @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ tc_Nat_Onat ) @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ tc_Nat_Onat ) @ c_fconj ) ) @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ tc_Nat_Onat ) @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ tc_HOL_Obool @ tc_Nat_Onat ) @ c_fNot ) ) @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) ) ) ) @ A @ B )
        = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[461]) ).

thf(842,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Ozip @ H @ G ) @ ( hAPP @ ( hAPP @ ( c_List_Omap @ F @ H ) @ E ) @ D ) ) @ ( hAPP @ ( hAPP @ ( c_List_Omap @ C @ G ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_List_Omap @ ( tc_prod @ F @ C ) @ ( tc_prod @ H @ G ) ) @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ F @ C @ ( tc_prod @ H @ G ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ F @ ( tc_fun @ C @ G ) @ ( tc_fun @ C @ ( tc_prod @ H @ G ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ G @ ( tc_prod @ H @ G ) ) @ ( tc_fun @ ( tc_fun @ C @ G ) @ ( tc_fun @ C @ ( tc_prod @ H @ G ) ) ) @ F ) @ ( c_COMBB @ G @ ( tc_prod @ H @ G ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ H @ ( tc_fun @ G @ ( tc_prod @ H @ G ) ) @ F ) @ ( c_Product__Type_OPair @ H @ G ) ) @ E ) ) ) @ B ) ) ) @ ( hAPP @ ( hAPP @ ( c_List_Ozip @ F @ C ) @ D ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zip__map__map) ).

thf(5448,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Ozip @ H @ G ) @ ( hAPP @ ( hAPP @ ( c_List_Omap @ F @ H ) @ E ) @ D ) ) @ ( hAPP @ ( hAPP @ ( c_List_Omap @ C @ G ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_List_Omap @ ( tc_prod @ F @ C ) @ ( tc_prod @ H @ G ) ) @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ F @ C @ ( tc_prod @ H @ G ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ F @ ( tc_fun @ C @ G ) @ ( tc_fun @ C @ ( tc_prod @ H @ G ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ G @ ( tc_prod @ H @ G ) ) @ ( tc_fun @ ( tc_fun @ C @ G ) @ ( tc_fun @ C @ ( tc_prod @ H @ G ) ) ) @ F ) @ ( c_COMBB @ G @ ( tc_prod @ H @ G ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ H @ ( tc_fun @ G @ ( tc_prod @ H @ G ) ) @ F ) @ ( c_Product__Type_OPair @ H @ G ) ) @ E ) ) ) @ B ) ) ) @ ( hAPP @ ( hAPP @ ( c_List_Ozip @ F @ C ) @ D ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[842]) ).

thf(455,axiom,
    ! [A: $i,B: $i] :
      ( ( c_List_Obutlast @ B @ A )
      = ( hAPP @ ( hAPP @ ( c_List_Otake @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ B ) ) @ A ) ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_butlast__conv__take) ).

thf(4033,plain,
    ! [A: $i,B: $i] :
      ( ( c_List_Obutlast @ B @ A )
      = ( hAPP @ ( hAPP @ ( c_List_Otake @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ B ) ) @ A ) ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[455]) ).

thf(1446,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Fun_Oinj__on @ E @ D @ C @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ B ) @ A ) )
    <=> ( ( c_Fun_Oinj__on @ E @ D @ C @ A )
        & ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ ( hAPP @ C @ B ) ) @ ( hAPP @ ( c_Set_Oimage @ E @ D @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ B ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ E @ tc_HOL_Obool ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_inj__on__insert) ).

thf(7931,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( c_Fun_Oinj__on @ E @ D @ C @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ B ) @ A ) )
       => ( ( c_Fun_Oinj__on @ E @ D @ C @ A )
          & ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ ( hAPP @ C @ B ) ) @ ( hAPP @ ( c_Set_Oimage @ E @ D @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ B ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ E @ tc_HOL_Obool ) ) ) ) ) ) ) ) )
      & ( ( ( c_Fun_Oinj__on @ E @ D @ C @ A )
          & ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ ( hAPP @ C @ B ) ) @ ( hAPP @ ( c_Set_Oimage @ E @ D @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ B ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ E @ tc_HOL_Obool ) ) ) ) ) ) ) )
       => ( c_Fun_Oinj__on @ E @ D @ C @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1446]) ).

thf(1356,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ F @ tc_HOL_Obool ) ) @ E ) @ D ) )
     => ( ! [G: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ F ) @ G ) @ E ) )
           => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ B @ G ) ) @ ( hAPP @ A @ G ) ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ F @ C ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ F @ C ) @ E ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ F @ C ) @ D ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Sigma__mono) ).

thf(7613,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ F @ tc_HOL_Obool ) ) @ E ) @ D ) )
     => ( ! [G: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ F ) @ G ) @ E ) )
           => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ B @ G ) ) @ ( hAPP @ A @ G ) ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ F @ C ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ F @ C ) @ E ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ F @ C ) @ D ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1356]) ).

thf(218,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Finite__Set_Ofolding__one__idem @ E @ D @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ E ) @ B ) )
       => ( ( B
           != ( c_Orderings_Obot__class_Obot @ ( tc_fun @ E @ tc_HOL_Obool ) ) )
         => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ E ) @ A ) )
           => ( ( A
               != ( c_Orderings_Obot__class_Obot @ ( tc_fun @ E @ tc_HOL_Obool ) ) )
             => ( ( hAPP @ C @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ B ) @ A ) )
                = ( hAPP @ ( hAPP @ D @ ( hAPP @ C @ B ) ) @ ( hAPP @ C @ A ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_folding__one__idem_Ounion__idem) ).

thf(3189,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Finite__Set_Ofolding__one__idem @ E @ D @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ E ) @ B ) )
       => ( ( B
           != ( c_Orderings_Obot__class_Obot @ ( tc_fun @ E @ tc_HOL_Obool ) ) )
         => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ E ) @ A ) )
           => ( ( A
               != ( c_Orderings_Obot__class_Obot @ ( tc_fun @ E @ tc_HOL_Obool ) ) )
             => ( ( hAPP @ C @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ B ) @ A ) )
                = ( hAPP @ ( hAPP @ D @ ( hAPP @ C @ B ) ) @ ( hAPP @ C @ A ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[218]) ).

thf(1132,axiom,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ A ) )
       => ( ( hAPP @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Nat_Onat @ B ) @ ( hAPP @ c_Nat_OSuc @ A ) )
          = ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ tc_Nat_Onat ) @ A ) @ ( hAPP @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Nat_Onat @ B ) @ A ) ) ) )
      & ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ A ) )
       => ( ( hAPP @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Nat_Onat @ B ) @ ( hAPP @ c_Nat_OSuc @ A ) )
          = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_atLeastLessThanSuc) ).

thf(6685,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ A ) )
       => ( ( hAPP @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Nat_Onat @ B ) @ ( hAPP @ c_Nat_OSuc @ A ) )
          = ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ tc_Nat_Onat ) @ A ) @ ( hAPP @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Nat_Onat @ B ) @ A ) ) ) )
      & ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ A ) )
       => ( ( hAPP @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Nat_Onat @ B ) @ ( hAPP @ c_Nat_OSuc @ A ) )
          = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1132]) ).

thf(2263,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Com_Ocom_OCall @ E @ D @ C )
     != ( c_Com_Ocom_OWhile @ B @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_com_Osimps_I61_J) ).

thf(11265,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Com_Ocom_OCall @ E @ D @ C )
     != ( c_Com_Ocom_OWhile @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2263]) ).

thf(781,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) @ A ) )
     => ( hBOOL @ ( hAPP @ C @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_inf1D1) ).

thf(5210,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) @ A ) )
     => ( hBOOL @ ( hAPP @ C @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[781]) ).

thf(1152,axiom,
    ! [A: $i] :
      ( ( hAPP @ ( c_SetInterval_Oord__class_OlessThan @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ ( hAPP @ ( c_Set_Oimage @ tc_Nat_Onat @ tc_Nat_Onat @ c_Nat_OSuc ) @ ( hAPP @ ( c_SetInterval_Oord__class_OlessThan @ tc_Nat_Onat ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_lessThan__Suc__eq__insert__0) ).

thf(6755,plain,
    ! [A: $i] :
      ( ( hAPP @ ( c_SetInterval_Oord__class_OlessThan @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ ( hAPP @ ( c_Set_Oimage @ tc_Nat_Onat @ tc_Nat_Onat @ c_Nat_OSuc ) @ ( hAPP @ ( c_SetInterval_Oord__class_OlessThan @ tc_Nat_Onat ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1152]) ).

thf(2228,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( c_SetInterval_Oord_OlessThan @ D @ B @ A ) ) )
    <=> ( hBOOL @ ( hAPP @ ( hAPP @ B @ C ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_ord_OlessThan__iff) ).

thf(11119,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( c_SetInterval_Oord_OlessThan @ D @ B @ A ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ B @ C ) @ A ) ) )
      & ( ( hBOOL @ ( hAPP @ ( hAPP @ B @ C ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( c_SetInterval_Oord_OlessThan @ D @ B @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2228]) ).

thf(1147,axiom,
    ! [A: $i] :
      ( ( c_List_Oupt @ A @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
      = ( c_List_Olist_ONil @ tc_Nat_Onat ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_upt__0) ).

thf(6740,plain,
    ! [A: $i] :
      ( ( c_List_Oupt @ A @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
      = ( c_List_Olist_ONil @ tc_Nat_Onat ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1147]) ).

thf(1306,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ F ) @ ( tc_List_Olist @ F ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ F ) @ ( tc_List_Olist @ F ) ) @ E ) @ D ) ) @ ( c_List_Olistrel1 @ F @ C ) ) )
          & ( B = A ) )
        | ( ( E = D )
          & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ F ) @ ( tc_List_Olist @ F ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ F ) @ ( tc_List_Olist @ F ) ) @ B ) @ A ) ) @ ( c_List_Olistrel1 @ F @ C ) ) ) ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ F ) @ ( tc_List_Olist @ F ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ F ) @ ( tc_List_Olist @ F ) ) @ ( hAPP @ ( hAPP @ ( c_List_Oappend @ F ) @ E ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_List_Oappend @ F ) @ D ) @ A ) ) ) @ ( c_List_Olistrel1 @ F @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_append__listrel1I) ).

thf(7381,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ F ) @ ( tc_List_Olist @ F ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ F ) @ ( tc_List_Olist @ F ) ) @ E ) @ D ) ) @ ( c_List_Olistrel1 @ F @ C ) ) )
          & ( B = A ) )
        | ( ( E = D )
          & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ F ) @ ( tc_List_Olist @ F ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ F ) @ ( tc_List_Olist @ F ) ) @ B ) @ A ) ) @ ( c_List_Olistrel1 @ F @ C ) ) ) ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ F ) @ ( tc_List_Olist @ F ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ F ) @ ( tc_List_Olist @ F ) ) @ ( hAPP @ ( hAPP @ ( c_List_Oappend @ F ) @ E ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_List_Oappend @ F ) @ D ) @ A ) ) ) @ ( c_List_Olistrel1 @ F @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1306]) ).

thf(286,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( c_List_Oset @ C ) @ ( hAPP @ ( c_List_Ofilter @ C @ B ) @ A ) )
      = ( hAPP @ ( c_Set_OCollect @ C ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ C @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ C ) @ c_fconj ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ C @ ( tc_fun @ C @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( c_member @ C ) ) @ ( hAPP @ ( c_List_Oset @ C ) @ A ) ) ) ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_set__filter) ).

thf(3421,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( c_List_Oset @ C ) @ ( hAPP @ ( c_List_Ofilter @ C @ B ) @ A ) )
      = ( hAPP @ ( c_Set_OCollect @ C ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ C @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ C ) @ c_fconj ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ C @ ( tc_fun @ C @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( c_member @ C ) ) @ ( hAPP @ ( c_List_Oset @ C ) @ A ) ) ) ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[286]) ).

thf(1743,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ A ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ A ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_insert__mono) ).

thf(9255,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ A ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ A ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1743]) ).

thf(1407,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ A ) )
       => ( ( c_Divides_Odivmod__nat @ A @ B )
          = ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ tc_Nat_Onat @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ A ) @ B ) ) @ B ) ) ) @ ( c_Divides_Odiv__class_Omod @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ A ) @ B ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_divmod__nat__step) ).

thf(7807,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ A ) )
       => ( ( c_Divides_Odivmod__nat @ A @ B )
          = ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ tc_Nat_Onat @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ A ) @ B ) ) @ B ) ) ) @ ( c_Divides_Odiv__class_Omod @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ A ) @ B ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1407]) ).

thf(1719,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_subset__psubset__trans) ).

thf(9167,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1719]) ).

thf(2281,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ F ) @ E ) @ D ) )
     => ( ( C
          = ( hAPP @ B @ E ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ A ) @ C ) @ ( hAPP @ ( c_Set_Oimage @ F @ A @ B ) @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_rev__image__eqI) ).

thf(11346,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ F ) @ E ) @ D ) )
     => ( ( C
          = ( hAPP @ B @ E ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ A ) @ C ) @ ( hAPP @ ( c_Set_Oimage @ F @ A @ B ) @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2281]) ).

thf(754,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Fun_Oinj__on @ E @ D @ C @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ B ) @ A ) )
    <=> ( ( c_Fun_Oinj__on @ E @ D @ C @ B )
        & ( c_Fun_Oinj__on @ E @ D @ C @ A )
        & ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Set_Oimage @ E @ D @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ B ) @ A ) ) ) @ ( hAPP @ ( c_Set_Oimage @ E @ D @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ A ) @ B ) ) )
          = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_inj__on__Un) ).

thf(5101,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( c_Fun_Oinj__on @ E @ D @ C @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ B ) @ A ) )
       => ( ( c_Fun_Oinj__on @ E @ D @ C @ B )
          & ( c_Fun_Oinj__on @ E @ D @ C @ A )
          & ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Set_Oimage @ E @ D @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ B ) @ A ) ) ) @ ( hAPP @ ( c_Set_Oimage @ E @ D @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ A ) @ B ) ) )
            = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) ) )
      & ( ( ( c_Fun_Oinj__on @ E @ D @ C @ B )
          & ( c_Fun_Oinj__on @ E @ D @ C @ A )
          & ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Set_Oimage @ E @ D @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ B ) @ A ) ) ) @ ( hAPP @ ( c_Set_Oimage @ E @ D @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ A ) @ B ) ) )
            = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) )
       => ( c_Fun_Oinj__on @ E @ D @ C @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[754]) ).

thf(360,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_IntD2) ).

thf(3685,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[360]) ).

thf(1634,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ ( hAPP @ ( c_Finite__Set_Ocard @ B ) @ A ) ) )
     => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_card__ge__0__finite) ).

thf(8823,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ ( hAPP @ ( c_Finite__Set_Ocard @ B ) @ A ) ) )
     => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1634]) ).

thf(1865,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( c_Com_Ocom_OCond @ F @ E @ D )
     != ( c_Com_Ocom_OLocal @ C @ B @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_com_Osimps_I37_J) ).

thf(9743,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( c_Com_Ocom_OCond @ F @ E @ D )
     != ( c_Com_Ocom_OLocal @ C @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1865]) ).

thf(1297,axiom,
    ! [A: $i,B: $i] :
      ( ( c_List_Osublist @ B @ ( c_List_Olist_ONil @ B ) @ A )
      = ( c_List_Olist_ONil @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_sublist__nil) ).

thf(7336,plain,
    ! [A: $i,B: $i] :
      ( ( c_List_Osublist @ B @ ( c_List_Olist_ONil @ B ) @ A )
      = ( c_List_Olist_ONil @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1297]) ).

thf(390,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofold__graph @ E @ D @ C @ B @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ E @ tc_HOL_Obool ) ) ) @ A ) )
     => ( A = B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_empty__fold__graphE) ).

thf(3808,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofold__graph @ E @ D @ C @ B @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ E @ tc_HOL_Obool ) ) ) @ A ) )
     => ( A = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[390]) ).

thf(2389,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ D ) @ ( c_Map_Odom @ E @ C @ B ) ) )
     => ( ( hAPP @ ( c_Map_Omap__add @ E @ C @ B @ A ) @ D )
        = ( hAPP @ A @ D ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_map__add__dom__app__simps_I2_J) ).

thf(11806,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ D ) @ ( c_Map_Odom @ E @ C @ B ) ) )
     => ( ( hAPP @ ( c_Map_Omap__add @ E @ C @ B @ A ) @ D )
        = ( hAPP @ A @ D ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2389]) ).

thf(545,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Int__left__absorb) ).

thf(4364,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[545]) ).

thf(1437,axiom,
    ! [A: $i,B: $i,C: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_List_Oset @ C ) @ ( hAPP @ ( hAPP @ ( c_List_Odrop @ C ) @ B ) @ A ) ) ) @ ( hAPP @ ( c_List_Oset @ C ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_set__drop__subset) ).

thf(7909,plain,
    ! [A: $i,B: $i,C: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_List_Oset @ C ) @ ( hAPP @ ( hAPP @ ( c_List_Odrop @ C ) @ B ) @ A ) ) ) @ ( hAPP @ ( c_List_Oset @ C ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1437]) ).

thf(158,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_List_OtakeWhile @ C @ B @ A )
      = ( hAPP @ ( hAPP @ ( c_List_Otake @ C ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) @ ( c_List_OtakeWhile @ C @ B @ A ) ) ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_takeWhile__eq__take) ).

thf(2979,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_List_OtakeWhile @ C @ B @ A )
      = ( hAPP @ ( hAPP @ ( c_List_Otake @ C ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) @ ( c_List_OtakeWhile @ C @ B @ A ) ) ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[158]) ).

thf(721,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ tc_Nat_Onat @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_SetInterval_Oord__class_OatMost @ tc_Nat_Onat ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ tc_Nat_Onat @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Nat_Onat @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) @ B ) ) @ A ) ) @ ( hAPP @ A @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_UN__le__eq__Un0) ).

thf(4996,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ tc_Nat_Onat @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_SetInterval_Oord__class_OatMost @ tc_Nat_Onat ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ tc_Nat_Onat @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Nat_Onat @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) @ B ) ) @ A ) ) @ ( hAPP @ A @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[721]) ).

thf(1366,axiom,
    ! [A: $i] :
      ( ( hAPP @ c_Nat_OSuc @ A )
     != ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Suc__not__Zero) ).

thf(7645,plain,
    ! [A: $i] :
      ( ( hAPP @ c_Nat_OSuc @ A )
     != ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1366]) ).

thf(1199,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( class_Rings_Olinordered__semiring__1 @ F )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ F ) @ E ) @ D ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ F ) @ C ) @ D ) )
         => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ F ) @ ( c_Groups_Ozero__class_Ozero @ F ) ) @ B ) )
           => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ F ) @ ( c_Groups_Ozero__class_Ozero @ F ) ) @ A ) )
             => ( ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ F ) @ B ) @ A )
                  = ( c_Groups_Oone__class_Oone @ F ) )
               => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ F ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ F ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ F ) @ B ) @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ F ) @ A ) @ C ) ) ) @ D ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_convex__bound__le) ).

thf(6932,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( class_Rings_Olinordered__semiring__1 @ F )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ F ) @ E ) @ D ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ F ) @ C ) @ D ) )
         => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ F ) @ ( c_Groups_Ozero__class_Ozero @ F ) ) @ B ) )
           => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ F ) @ ( c_Groups_Ozero__class_Ozero @ F ) ) @ A ) )
             => ( ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ F ) @ B ) @ A )
                  = ( c_Groups_Oone__class_Oone @ F ) )
               => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ F ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ F ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ F ) @ B ) @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ F ) @ A ) @ C ) ) ) @ D ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1199]) ).

thf(104,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ ( hAPP @ ( c_Set_OCollect @ D ) @ C ) ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ B ) @ ( hAPP @ ( c_Set_OCollect @ B ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ D @ tc_HOL_Obool ) @ tc_HOL_Obool @ B ) @ ( c_HOL_OEx @ D ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ D @ tc_HOL_Obool ) @ ( tc_fun @ D @ tc_HOL_Obool ) @ B ) @ ( hAPP @ ( c_COMBS @ D @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ D ) @ c_fconj ) @ C ) ) ) @ A ) ) ) ) )
      <=> ! [E: $i] :
            ( ( hBOOL @ ( hAPP @ C @ E ) )
           => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ B ) @ ( hAPP @ ( c_Set_OCollect @ B ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ B @ D @ tc_HOL_Obool ) @ A ) @ E ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_finite__Collect__bounded__ex) ).

thf(2772,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ ( hAPP @ ( c_Set_OCollect @ D ) @ C ) ) )
     => ( ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ B ) @ ( hAPP @ ( c_Set_OCollect @ B ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ D @ tc_HOL_Obool ) @ tc_HOL_Obool @ B ) @ ( c_HOL_OEx @ D ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ D @ tc_HOL_Obool ) @ ( tc_fun @ D @ tc_HOL_Obool ) @ B ) @ ( hAPP @ ( c_COMBS @ D @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ D ) @ c_fconj ) @ C ) ) ) @ A ) ) ) ) )
         => ! [E: $i] :
              ( ( hBOOL @ ( hAPP @ C @ E ) )
             => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ B ) @ ( hAPP @ ( c_Set_OCollect @ B ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ B @ D @ tc_HOL_Obool ) @ A ) @ E ) ) ) ) ) )
        & ( ! [E: $i] :
              ( ( hBOOL @ ( hAPP @ C @ E ) )
             => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ B ) @ ( hAPP @ ( c_Set_OCollect @ B ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ B @ D @ tc_HOL_Obool ) @ A ) @ E ) ) ) ) )
         => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ B ) @ ( hAPP @ ( c_Set_OCollect @ B ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ D @ tc_HOL_Obool ) @ tc_HOL_Obool @ B ) @ ( c_HOL_OEx @ D ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ D @ tc_HOL_Obool ) @ ( tc_fun @ D @ tc_HOL_Obool ) @ B ) @ ( hAPP @ ( c_COMBS @ D @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ D ) @ c_fconj ) @ C ) ) ) @ A ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[104]) ).

thf(1534,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Lattices_Oab__semigroup__idem__mult @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ B ) )
       => ( ( hAPP @ ( c_Finite__Set_Ofold1 @ C @ ( c_Groups_Otimes__class_Otimes @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ A ) @ B ) )
          = ( hAPP @ ( hAPP @ ( c_Finite__Set_Ofold @ C @ C @ ( c_Groups_Otimes__class_Otimes @ C ) ) @ A ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_fold1__eq__fold__idem) ).

thf(8347,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Lattices_Oab__semigroup__idem__mult @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ B ) )
       => ( ( hAPP @ ( c_Finite__Set_Ofold1 @ C @ ( c_Groups_Otimes__class_Otimes @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ A ) @ B ) )
          = ( hAPP @ ( hAPP @ ( c_Finite__Set_Ofold @ C @ C @ ( c_Groups_Otimes__class_Otimes @ C ) ) @ A ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1534]) ).

thf(357,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Int_Oint ) @ B ) @ A ) )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ A ) @ B ) )
        = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zdvd__mult__div__cancel) ).

thf(3676,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Int_Oint ) @ B ) @ A ) )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ A ) @ B ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[357]) ).

thf(951,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ A ) @ B ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ A ) ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Un__least) ).

thf(5846,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ A ) @ B ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ A ) ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[951]) ).

thf(966,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ B ) )
     => ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ C ) @ A ) ) @ B )
        = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ A ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Int__insert__left__if0) ).

thf(5897,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ B ) )
     => ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ C ) @ A ) ) @ B )
        = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ A ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[966]) ).

thf(2407,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ E @ D ) @ C ) )
     => ( hBOOL @ ( hAPP @ ( c_Predicate_ODomainP @ B @ A @ E ) @ D ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_DomainP_Ointros) ).

thf(11885,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ E @ D ) @ C ) )
     => ( hBOOL @ ( hAPP @ ( c_Predicate_ODomainP @ B @ A @ E ) @ D ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2407]) ).

thf(1726,axiom,
    ! [A: $i,B: $i] :
      ( ( c_List_Olistrel1 @ B @ ( hAPP @ ( c_Relation_Oconverse @ B @ B ) @ A ) )
      = ( hAPP @ ( c_Relation_Oconverse @ ( tc_List_Olist @ B ) @ ( tc_List_Olist @ B ) ) @ ( c_List_Olistrel1 @ B @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_listrel1__converse) ).

thf(9187,plain,
    ! [A: $i,B: $i] :
      ( ( c_List_Olistrel1 @ B @ ( hAPP @ ( c_Relation_Oconverse @ B @ B ) @ A ) )
      = ( hAPP @ ( c_Relation_Oconverse @ ( tc_List_Olist @ B ) @ ( tc_List_Olist @ B ) ) @ ( c_List_Olistrel1 @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1726]) ).

thf(2029,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Com_Ocom_OCond @ E @ D @ C )
     != ( c_Com_Ocom_OAss @ B @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_com_Osimps_I27_J) ).

thf(10390,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Com_Ocom_OCond @ E @ D @ C )
     != ( c_Com_Ocom_OAss @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2029]) ).

thf(992,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( c_Sum__Type_Osum_Osum__case @ E @ D @ C @ ( hAPP @ ( c_COMBK @ D @ E ) @ B ) @ ( hAPP @ ( c_COMBK @ D @ C ) @ B ) ) @ A )
      = B ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_sum__case__KK) ).

thf(6021,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( c_Sum__Type_Osum_Osum__case @ E @ D @ C @ ( hAPP @ ( c_COMBK @ D @ E ) @ B ) @ ( hAPP @ ( c_COMBK @ D @ C ) @ B ) ) @ A )
      = B ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[992]) ).

thf(2300,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ B ) @ ( hAPP @ ( c_List_Oset @ C ) @ ( c_List_Obutlast @ C @ A ) ) ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ B ) @ ( hAPP @ ( c_List_Oset @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_in__set__butlastD) ).

thf(11408,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ B ) @ ( hAPP @ ( c_List_Oset @ C ) @ ( c_List_Obutlast @ C @ A ) ) ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ B ) @ ( hAPP @ ( c_List_Oset @ C ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2300]) ).

thf(112,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
      ( ( c_Finite__Set_Ofolding__image__simple__idem @ H @ G @ F @ E @ D @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ G ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ G ) @ A ) )
         => ( ( hAPP @ C @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ G @ tc_HOL_Obool ) ) @ B ) @ A ) )
            = ( hAPP @ ( hAPP @ F @ ( hAPP @ C @ B ) ) @ ( hAPP @ C @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_folding__image__simple__idem_Ounion__idem) ).

thf(2809,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
      ( ( c_Finite__Set_Ofolding__image__simple__idem @ H @ G @ F @ E @ D @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ G ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ G ) @ A ) )
         => ( ( hAPP @ C @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ G @ tc_HOL_Obool ) ) @ B ) @ A ) )
            = ( hAPP @ ( hAPP @ F @ ( hAPP @ C @ B ) ) @ ( hAPP @ C @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[112]) ).

thf(1957,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( c_Com_Ocom_OCall @ F @ E @ D )
        = ( c_Com_Ocom_OCall @ C @ B @ A ) )
    <=> ( ( F = C )
        & ( E = B )
        & ( D = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_com_Osimps_I7_J) ).

thf(10106,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( ( c_Com_Ocom_OCall @ F @ E @ D )
          = ( c_Com_Ocom_OCall @ C @ B @ A ) )
       => ( ( F = C )
          & ( E = B )
          & ( D = A ) ) )
      & ( ( ( F = C )
          & ( E = B )
          & ( D = A ) )
       => ( ( c_Com_Ocom_OCall @ F @ E @ D )
          = ( c_Com_Ocom_OCall @ C @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1957]) ).

thf(2139,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ D ) @ C ) ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ C ) @ A ) ) @ ( c_Transitive__Closure_Ortrancl @ E @ B ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ D ) @ A ) ) @ ( c_Transitive__Closure_Otrancl @ E @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_rtrancl__into__trancl2) ).

thf(10801,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ D ) @ C ) ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ C ) @ A ) ) @ ( c_Transitive__Closure_Ortrancl @ E @ B ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ D ) @ A ) ) @ ( c_Transitive__Closure_Otrancl @ E @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2139]) ).

thf(2192,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Divides_Odivmod__nat @ B @ A )
      = ( c_HOL_OThe @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ ( c_Divides_Odivmod__nat__rel @ B @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_divmod__nat__def) ).

thf(10977,plain,
    ! [A: $i,B: $i] :
      ( ( c_Divides_Odivmod__nat @ B @ A )
      = ( c_HOL_OThe @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ ( c_Divides_Odivmod__nat__rel @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2192]) ).

thf(1868,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ~ ( hBOOL @ ( hAPP @ E @ D ) )
     => ( ( c_List_OtakeWhile @ C @ E @ ( hAPP @ ( hAPP @ ( c_List_Oappend @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ C ) @ D ) @ A ) ) )
        = ( c_List_OtakeWhile @ C @ E @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_takeWhile__tail) ).

thf(9764,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ~ ( hBOOL @ ( hAPP @ E @ D ) )
     => ( ( c_List_OtakeWhile @ C @ E @ ( hAPP @ ( hAPP @ ( c_List_Oappend @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ C ) @ D ) @ A ) ) )
        = ( c_List_OtakeWhile @ C @ E @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1868]) ).

thf(695,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Option_Oset @ B @ ( hAPP @ ( c_Option_Ooption_OSome @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ B ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Option_Oset_Osimps_I2_J) ).

thf(4892,plain,
    ! [A: $i,B: $i] :
      ( ( c_Option_Oset @ B @ ( hAPP @ ( c_Option_Ooption_OSome @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ B ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[695]) ).

thf(1912,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ C ) @ B ) )
     => ( ( B = A )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_dvd_Oord__le__eq__trans) ).

thf(9923,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ C ) @ B ) )
     => ( ( B = A )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ C ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1912]) ).

thf(1378,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ D ) @ C ) ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ B ) ) @ A ) ) )
     => ( ( hAPP @ ( hAPP @ ( c_List_Otake @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ D ) @ C ) ) @ A )
        = ( hAPP @ ( hAPP @ ( c_List_Oappend @ B ) @ ( hAPP @ ( hAPP @ ( c_List_Otake @ B ) @ D ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_List_Otake @ B ) @ C ) @ ( hAPP @ ( hAPP @ ( c_List_Odrop @ B ) @ D ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_take__add) ).

thf(7688,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ D ) @ C ) ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ B ) ) @ A ) ) )
     => ( ( hAPP @ ( hAPP @ ( c_List_Otake @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ D ) @ C ) ) @ A )
        = ( hAPP @ ( hAPP @ ( c_List_Oappend @ B ) @ ( hAPP @ ( hAPP @ ( c_List_Otake @ B ) @ D ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_List_Otake @ B ) @ C ) @ ( hAPP @ ( hAPP @ ( c_List_Odrop @ B ) @ D ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1378]) ).

thf(1010,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( c_Relation_ORange @ B @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ ( tc_prod @ B @ A ) @ tc_HOL_Obool ) ) )
      = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ A @ tc_HOL_Obool ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Range__empty) ).

thf(6103,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( c_Relation_ORange @ B @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ ( tc_prod @ B @ A ) @ tc_HOL_Obool ) ) )
      = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ A @ tc_HOL_Obool ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1010]) ).

thf(607,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_Finite__Set_Ofolding__one @ D @ C @ B )
     => ( ( hAPP @ B @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) )
        = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_folding__one_Osingleton) ).

thf(4588,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_Finite__Set_Ofolding__one @ D @ C @ B )
     => ( ( hAPP @ B @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[607]) ).

thf(1748,axiom,
    ! [A: $i] :
      ( ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat )
     != ( hAPP @ c_Nat_OSuc @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nat_Osimps_I2_J) ).

thf(9272,plain,
    ! [A: $i] :
      ( ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat )
     != ( hAPP @ c_Nat_OSuc @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1748]) ).

thf(1318,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
      ( ( c_Finite__Set_Ofolding__image__simple @ H @ G @ F @ E @ D @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ G ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ G ) @ A ) @ B ) )
         => ( ( hAPP @ C @ B )
            = ( hAPP @ ( hAPP @ F @ ( hAPP @ D @ A ) ) @ ( hAPP @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ G @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ G ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ G @ tc_HOL_Obool ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_folding__image__simple_Oremove) ).

thf(7460,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
      ( ( c_Finite__Set_Ofolding__image__simple @ H @ G @ F @ E @ D @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ G ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ G ) @ A ) @ B ) )
         => ( ( hAPP @ C @ B )
            = ( hAPP @ ( hAPP @ F @ ( hAPP @ D @ A ) ) @ ( hAPP @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ G @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ G ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ G @ tc_HOL_Obool ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1318]) ).

thf(810,axiom,
    ! [A: $i,B: $i,C: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( c_Complete__Lattice_OSup__class_OSup @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ ( tc_fun @ C @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ B ) @ A ) ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( c_Complete__Lattice_OSup__class_OSup @ ( tc_fun @ C @ tc_HOL_Obool ) @ B ) ) @ ( c_Complete__Lattice_OSup__class_OSup @ ( tc_fun @ C @ tc_HOL_Obool ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Union__Int__subset) ).

thf(5328,plain,
    ! [A: $i,B: $i,C: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( c_Complete__Lattice_OSup__class_OSup @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ ( tc_fun @ C @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ B ) @ A ) ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( c_Complete__Lattice_OSup__class_OSup @ ( tc_fun @ C @ tc_HOL_Obool ) @ B ) ) @ ( c_Complete__Lattice_OSup__class_OSup @ ( tc_fun @ C @ tc_HOL_Obool ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[810]) ).

thf(619,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( c_Relation_OImage @ ( tc_List_Olist @ B ) @ ( tc_List_Olist @ B ) @ ( c_List_Olistrel @ B @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ ( tc_List_Olist @ B ) ) @ ( c_List_Olist_ONil @ B ) ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ ( tc_List_Olist @ B ) @ tc_HOL_Obool ) ) ) )
      = ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ ( tc_List_Olist @ B ) ) @ ( c_List_Olist_ONil @ B ) ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ ( tc_List_Olist @ B ) @ tc_HOL_Obool ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_listrel__Nil) ).

thf(4628,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( c_Relation_OImage @ ( tc_List_Olist @ B ) @ ( tc_List_Olist @ B ) @ ( c_List_Olistrel @ B @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ ( tc_List_Olist @ B ) ) @ ( c_List_Olist_ONil @ B ) ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ ( tc_List_Olist @ B ) @ tc_HOL_Obool ) ) ) )
      = ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ ( tc_List_Olist @ B ) ) @ ( c_List_Olist_ONil @ B ) ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ ( tc_List_Olist @ B ) @ tc_HOL_Obool ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[619]) ).

thf(1833,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( c_Com_Ocom_OLocal @ F @ E @ D )
     != ( c_Com_Ocom_OCall @ C @ B @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_com_Osimps_I42_J) ).

thf(9589,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( c_Com_Ocom_OLocal @ F @ E @ D )
     != ( c_Com_Ocom_OCall @ C @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1833]) ).

thf(1037,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Odrop @ tc_Nat_Onat ) @ C ) @ ( c_List_Oupt @ B @ A ) )
      = ( c_List_Oupt @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ C ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_drop__upt) ).

thf(6210,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Odrop @ tc_Nat_Onat ) @ C ) @ ( c_List_Oupt @ B @ A ) )
      = ( c_List_Oupt @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ C ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1037]) ).

thf(159,axiom,
    ! [A: $i] :
      ( ( hAPP @ ( c_Relation_OField @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ ( tc_prod @ A @ A ) @ tc_HOL_Obool ) ) )
      = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ A @ tc_HOL_Obool ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Field__empty) ).

thf(2982,plain,
    ! [A: $i] :
      ( ( hAPP @ ( c_Relation_OField @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ ( tc_prod @ A @ A ) @ tc_HOL_Obool ) ) )
      = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ A @ tc_HOL_Obool ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[159]) ).

thf(1256,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ C ) )
     => ( ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ B )
          = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ A ) )
      <=> ( B = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nat__mult__eq__cancel1) ).

thf(7170,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ C ) )
     => ( ( ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ B )
            = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ A ) )
         => ( B = A ) )
        & ( ( B = A )
         => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ B )
            = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1256]) ).

thf(471,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A )
      = ( hAPP @ ( c_Set_OCollect @ C ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ C @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ C ) @ c_fdisj ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ C @ ( tc_fun @ C @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( c_member @ C ) ) @ B ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ C @ ( tc_fun @ C @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( c_member @ C ) ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Un__def) ).

thf(4082,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A )
      = ( hAPP @ ( c_Set_OCollect @ C ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ C @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ C ) @ c_fdisj ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ C @ ( tc_fun @ C @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( c_member @ C ) ) @ B ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ C @ ( tc_fun @ C @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( c_member @ C ) ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[471]) ).

thf(1831,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ D ) @ C ) @ B ) ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ E ) @ B ) @ C ) ) @ ( hAPP @ ( c_Relation_Oconverse @ E @ D ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_converseI) ).

thf(9584,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ D ) @ C ) @ B ) ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ E ) @ B ) @ C ) ) @ ( hAPP @ ( c_Relation_Oconverse @ E @ D ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1831]) ).

thf(1518,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ B ) @ ( hAPP @ ( c_List_Oset @ C ) @ A ) ) )
       => ( ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) @ ( c_List_Oremove1 @ C @ B @ A ) )
          = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) @ A ) ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) ) )
      & ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ B ) @ ( hAPP @ ( c_List_Oset @ C ) @ A ) ) )
       => ( ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) @ ( c_List_Oremove1 @ C @ B @ A ) )
          = ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_length__remove1) ).

thf(8289,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ B ) @ ( hAPP @ ( c_List_Oset @ C ) @ A ) ) )
       => ( ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) @ ( c_List_Oremove1 @ C @ B @ A ) )
          = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) @ A ) ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) ) )
      & ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ B ) @ ( hAPP @ ( c_List_Oset @ C ) @ A ) ) )
       => ( ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) @ ( c_List_Oremove1 @ C @ B @ A ) )
          = ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1518]) ).

thf(1863,axiom,
    ! [A: $i,B: $i] :
      ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ ( hAPP @ c_Nat_OSuc @ A ) ) )
       => ( A = B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__antisym) ).

thf(9738,plain,
    ! [A: $i,B: $i] :
      ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ ( hAPP @ c_Nat_OSuc @ A ) ) )
       => ( A = B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1863]) ).

thf(413,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Onth @ D ) @ ( hAPP @ ( hAPP @ ( c_List_Oappend @ D ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ D ) ) @ C ) ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_List_Onth @ D ) @ B ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nth__append__length__plus) ).

thf(3899,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Onth @ D ) @ ( hAPP @ ( hAPP @ ( c_List_Oappend @ D ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ D ) ) @ C ) ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_List_Onth @ D ) @ B ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[413]) ).

thf(1861,axiom,
    ! [A: $i,B: $i] :
      ( ( B
       != ( c_Option_Ooption_ONone @ A ) )
    <=> ? [C: $i] :
          ( B
          = ( hAPP @ ( c_Option_Ooption_OSome @ A ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_not__None__eq) ).

thf(9720,plain,
    ! [A: $i,B: $i] :
      ( ( ( B
         != ( c_Option_Ooption_ONone @ A ) )
       => ? [C: $i] :
            ( B
            = ( hAPP @ ( c_Option_Ooption_OSome @ A ) @ C ) ) )
      & ( ? [C: $i] :
            ( B
            = ( hAPP @ ( c_Option_Ooption_OSome @ A ) @ C ) )
       => ( B
         != ( c_Option_Ooption_ONone @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1861]) ).

thf(613,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Map_Omap__upds @ E @ D @ C @ B @ A )
      = ( hAPP @ ( c_List_Ofoldl @ ( tc_fun @ E @ ( tc_Option_Ooption @ D ) ) @ ( tc_prod @ E @ D ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ E @ ( tc_fun @ D @ ( tc_fun @ E @ ( tc_Option_Ooption @ D ) ) ) ) @ ( tc_fun @ ( tc_prod @ E @ D ) @ ( tc_fun @ E @ ( tc_Option_Ooption @ D ) ) ) @ ( tc_fun @ E @ ( tc_Option_Ooption @ D ) ) ) @ ( c_Product__Type_Oprod_Oprod__case @ E @ D @ ( tc_fun @ E @ ( tc_Option_Ooption @ D ) ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ ( tc_fun @ E @ ( tc_Option_Ooption @ D ) ) @ ( tc_fun @ D @ ( tc_Option_Ooption @ D ) ) @ ( tc_fun @ E @ ( tc_fun @ D @ ( tc_fun @ E @ ( tc_Option_Ooption @ D ) ) ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ E @ ( tc_fun @ ( tc_fun @ D @ ( tc_Option_Ooption @ D ) ) @ ( tc_fun @ D @ ( tc_fun @ E @ ( tc_Option_Ooption @ D ) ) ) ) ) @ ( tc_fun @ ( tc_fun @ D @ ( tc_Option_Ooption @ D ) ) @ ( tc_fun @ E @ ( tc_fun @ D @ ( tc_fun @ E @ ( tc_Option_Ooption @ D ) ) ) ) ) @ ( tc_fun @ E @ ( tc_Option_Ooption @ D ) ) ) @ ( c_COMBC @ E @ ( tc_fun @ D @ ( tc_Option_Ooption @ D ) ) @ ( tc_fun @ D @ ( tc_fun @ E @ ( tc_Option_Ooption @ D ) ) ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ E @ ( tc_fun @ ( tc_Option_Ooption @ D ) @ ( tc_fun @ E @ ( tc_Option_Ooption @ D ) ) ) ) @ ( tc_fun @ E @ ( tc_fun @ ( tc_fun @ D @ ( tc_Option_Ooption @ D ) ) @ ( tc_fun @ D @ ( tc_fun @ E @ ( tc_Option_Ooption @ D ) ) ) ) ) @ ( tc_fun @ E @ ( tc_Option_Ooption @ D ) ) ) @ ( hAPP @ ( c_COMBB @ ( tc_fun @ ( tc_Option_Ooption @ D ) @ ( tc_fun @ E @ ( tc_Option_Ooption @ D ) ) ) @ ( tc_fun @ ( tc_fun @ D @ ( tc_Option_Ooption @ D ) ) @ ( tc_fun @ D @ ( tc_fun @ E @ ( tc_Option_Ooption @ D ) ) ) ) @ E ) @ ( c_COMBB @ ( tc_Option_Ooption @ D ) @ ( tc_fun @ E @ ( tc_Option_Ooption @ D ) ) @ D ) ) ) @ ( c_Fun_Ofun__upd @ E @ ( tc_Option_Ooption @ D ) ) ) ) ) @ ( c_Option_Ooption_OSome @ D ) ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_List_Ozip @ E @ D ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_map__upds__fold__map__upd) ).

thf(4604,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Map_Omap__upds @ E @ D @ C @ B @ A )
      = ( hAPP @ ( c_List_Ofoldl @ ( tc_fun @ E @ ( tc_Option_Ooption @ D ) ) @ ( tc_prod @ E @ D ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ E @ ( tc_fun @ D @ ( tc_fun @ E @ ( tc_Option_Ooption @ D ) ) ) ) @ ( tc_fun @ ( tc_prod @ E @ D ) @ ( tc_fun @ E @ ( tc_Option_Ooption @ D ) ) ) @ ( tc_fun @ E @ ( tc_Option_Ooption @ D ) ) ) @ ( c_Product__Type_Oprod_Oprod__case @ E @ D @ ( tc_fun @ E @ ( tc_Option_Ooption @ D ) ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ ( tc_fun @ E @ ( tc_Option_Ooption @ D ) ) @ ( tc_fun @ D @ ( tc_Option_Ooption @ D ) ) @ ( tc_fun @ E @ ( tc_fun @ D @ ( tc_fun @ E @ ( tc_Option_Ooption @ D ) ) ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ E @ ( tc_fun @ ( tc_fun @ D @ ( tc_Option_Ooption @ D ) ) @ ( tc_fun @ D @ ( tc_fun @ E @ ( tc_Option_Ooption @ D ) ) ) ) ) @ ( tc_fun @ ( tc_fun @ D @ ( tc_Option_Ooption @ D ) ) @ ( tc_fun @ E @ ( tc_fun @ D @ ( tc_fun @ E @ ( tc_Option_Ooption @ D ) ) ) ) ) @ ( tc_fun @ E @ ( tc_Option_Ooption @ D ) ) ) @ ( c_COMBC @ E @ ( tc_fun @ D @ ( tc_Option_Ooption @ D ) ) @ ( tc_fun @ D @ ( tc_fun @ E @ ( tc_Option_Ooption @ D ) ) ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ E @ ( tc_fun @ ( tc_Option_Ooption @ D ) @ ( tc_fun @ E @ ( tc_Option_Ooption @ D ) ) ) ) @ ( tc_fun @ E @ ( tc_fun @ ( tc_fun @ D @ ( tc_Option_Ooption @ D ) ) @ ( tc_fun @ D @ ( tc_fun @ E @ ( tc_Option_Ooption @ D ) ) ) ) ) @ ( tc_fun @ E @ ( tc_Option_Ooption @ D ) ) ) @ ( hAPP @ ( c_COMBB @ ( tc_fun @ ( tc_Option_Ooption @ D ) @ ( tc_fun @ E @ ( tc_Option_Ooption @ D ) ) ) @ ( tc_fun @ ( tc_fun @ D @ ( tc_Option_Ooption @ D ) ) @ ( tc_fun @ D @ ( tc_fun @ E @ ( tc_Option_Ooption @ D ) ) ) ) @ E ) @ ( c_COMBB @ ( tc_Option_Ooption @ D ) @ ( tc_fun @ E @ ( tc_Option_Ooption @ D ) ) @ D ) ) ) @ ( c_Fun_Ofun__upd @ E @ ( tc_Option_Ooption @ D ) ) ) ) ) @ ( c_Option_Ooption_OSome @ D ) ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_List_Ozip @ E @ D ) @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[613]) ).

thf(1544,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ F ) @ E ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ D @ C @ ( tc_fun @ F @ tc_HOL_Obool ) ) @ B ) @ A ) ) )
     => ~ ! [G: $i,H: $i] :
            ( ( A
              = ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ C ) @ G ) @ H ) )
           => ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ F ) @ E ) @ ( hAPP @ ( hAPP @ B @ G ) @ H ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mem__splitE) ).

thf(8382,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ F ) @ E ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ D @ C @ ( tc_fun @ F @ tc_HOL_Obool ) ) @ B ) @ A ) ) )
     => ~ ! [G: $i,H: $i] :
            ( ( A
              = ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ C ) @ G ) @ H ) )
           => ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ F ) @ E ) @ ( hAPP @ ( hAPP @ B @ G ) @ H ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1544]) ).

thf(1553,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Finite__Set_Ofun__left__comm @ E @ D @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ E ) @ B ) )
       => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofold__graph @ E @ D @ C @ A @ B ) @ ( hAPP @ ( hAPP @ ( c_Finite__Set_Ofold @ E @ D @ C ) @ A ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_fun__left__comm_Ofold__graph__fold) ).

thf(8418,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Finite__Set_Ofun__left__comm @ E @ D @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ E ) @ B ) )
       => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofold__graph @ E @ D @ C @ A @ B ) @ ( hAPP @ ( hAPP @ ( c_Finite__Set_Ofold @ E @ D @ C ) @ A ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1553]) ).

thf(442,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
      ( ( c_Fun_Oinj__on @ H @ G @ F @ E )
     => ( ( c_Fun_Oinj__on @ D @ C @ B @ A )
       => ( c_Fun_Oinj__on @ ( tc_prod @ H @ D ) @ ( tc_prod @ G @ C ) @ ( c_Product__Type_Omap__pair @ H @ G @ D @ C @ F @ B ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ H @ D ) @ E ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ D @ tc_HOL_Obool ) @ H ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_map__pair__inj__on) ).

thf(3995,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
      ( ( c_Fun_Oinj__on @ H @ G @ F @ E )
     => ( ( c_Fun_Oinj__on @ D @ C @ B @ A )
       => ( c_Fun_Oinj__on @ ( tc_prod @ H @ D ) @ ( tc_prod @ G @ C ) @ ( c_Product__Type_Omap__pair @ H @ G @ D @ C @ F @ B ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ H @ D ) @ E ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ D @ tc_HOL_Obool ) @ H ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[442]) ).

thf(2218,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Hoare__Mirabelle_Ohoare__valids @ C @ B @ A )
    <=> ! [D: $i] :
          ( ! [E: $i] :
              ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_Hoare__Mirabelle_Otriple @ C ) ) @ E ) @ B ) )
             => ( c_Hoare__Mirabelle_Otriple__valid @ C @ D @ E ) )
         => ! [E: $i] :
              ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_Hoare__Mirabelle_Otriple @ C ) ) @ E ) @ A ) )
             => ( c_Hoare__Mirabelle_Otriple__valid @ C @ D @ E ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_hoare__valids__def) ).

thf(11069,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( c_Hoare__Mirabelle_Ohoare__valids @ C @ B @ A )
       => ! [D: $i] :
            ( ! [E: $i] :
                ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_Hoare__Mirabelle_Otriple @ C ) ) @ E ) @ B ) )
               => ( c_Hoare__Mirabelle_Otriple__valid @ C @ D @ E ) )
           => ! [E: $i] :
                ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_Hoare__Mirabelle_Otriple @ C ) ) @ E ) @ A ) )
               => ( c_Hoare__Mirabelle_Otriple__valid @ C @ D @ E ) ) ) )
      & ( ! [D: $i] :
            ( ! [E: $i] :
                ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_Hoare__Mirabelle_Otriple @ C ) ) @ E ) @ B ) )
               => ( c_Hoare__Mirabelle_Otriple__valid @ C @ D @ E ) )
           => ! [E: $i] :
                ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_Hoare__Mirabelle_Otriple @ C ) ) @ E ) @ A ) )
               => ( c_Hoare__Mirabelle_Otriple__valid @ C @ D @ E ) ) )
       => ( c_Hoare__Mirabelle_Ohoare__valids @ C @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2218]) ).

thf(94,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Groups_Ocomm__monoid__add @ E )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ C ) )
       => ( ( c_Fun_Oinj__on @ D @ D @ B @ C )
         => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ A ) @ ( hAPP @ ( c_Set_Oimage @ D @ D @ B ) @ C ) )
            = ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ ( hAPP @ ( c_Fun_Ocomp @ D @ E @ D @ A ) @ B ) ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_setsum_Oreindex) ).

thf(2737,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Groups_Ocomm__monoid__add @ E )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ C ) )
       => ( ( c_Fun_Oinj__on @ D @ D @ B @ C )
         => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ A ) @ ( hAPP @ ( c_Set_Oimage @ D @ D @ B ) @ C ) )
            = ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ ( hAPP @ ( c_Fun_Ocomp @ D @ E @ D @ A ) @ B ) ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[94]) ).

thf(140,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofold1Set @ D @ C @ B ) @ A ) )
     => ( B
       != ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_fold1Set__nonempty) ).

thf(2905,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofold1Set @ D @ C @ B ) @ A ) )
     => ( B
       != ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[140]) ).

thf(1981,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ c_Com_OWT @ ( c_Com_Ocom_OCall @ C @ B @ A ) ) )
    <=> ( hBOOL @ ( hAPP @ c_Com_OWT @ ( hAPP @ c_Com_Ocom_OBODY @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_WT_Oequations_I8_J) ).

thf(10216,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP @ c_Com_OWT @ ( c_Com_Ocom_OCall @ C @ B @ A ) ) )
       => ( hBOOL @ ( hAPP @ c_Com_OWT @ ( hAPP @ c_Com_Ocom_OBODY @ B ) ) ) )
      & ( ( hBOOL @ ( hAPP @ c_Com_OWT @ ( hAPP @ c_Com_Ocom_OBODY @ B ) ) )
       => ( hBOOL @ ( hAPP @ c_Com_OWT @ ( c_Com_Ocom_OCall @ C @ B @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1981]) ).

thf(1643,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ B ) )
     => ( ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ A ) @ B ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) @ ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_card_Oinsert__remove) ).

thf(8867,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ B ) )
     => ( ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ A ) @ B ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) @ ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1643]) ).

thf(1684,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hAPP @ ( hAPP @ ( c_List_Oappend @ C ) @ B ) @ A )
        = B )
    <=> ( A
        = ( c_List_Olist_ONil @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_append__self__conv) ).

thf(9026,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( hAPP @ ( hAPP @ ( c_List_Oappend @ C ) @ B ) @ A )
          = B )
       => ( A
          = ( c_List_Olist_ONil @ C ) ) )
      & ( ( A
          = ( c_List_Olist_ONil @ C ) )
       => ( ( hAPP @ ( hAPP @ ( c_List_Oappend @ C ) @ B ) @ A )
          = B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1684]) ).

thf(1449,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( c_Fun_Oinj__on @ F @ E @ D @ C )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Set_Oimage @ F @ E @ D ) @ C ) ) @ B ) )
       => ( ( c_Fun_Oinj__on @ E @ F @ A @ B )
         => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ F @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Set_Oimage @ E @ F @ A ) @ B ) ) @ C ) )
           => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ F ) @ C ) )
             => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ E ) @ B ) )
               => ( ( hAPP @ ( c_Finite__Set_Ocard @ F ) @ C )
                  = ( hAPP @ ( c_Finite__Set_Ocard @ E ) @ B ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_card__bij__eq) ).

thf(7952,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( c_Fun_Oinj__on @ F @ E @ D @ C )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Set_Oimage @ F @ E @ D ) @ C ) ) @ B ) )
       => ( ( c_Fun_Oinj__on @ E @ F @ A @ B )
         => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ F @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Set_Oimage @ E @ F @ A ) @ B ) ) @ C ) )
           => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ F ) @ C ) )
             => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ E ) @ B ) )
               => ( ( hAPP @ ( c_Finite__Set_Ocard @ F ) @ C )
                  = ( hAPP @ ( c_Finite__Set_Ocard @ E ) @ B ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1449]) ).

thf(1374,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_add__mult__distrib2) ).

thf(7674,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1374]) ).

thf(1675,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ E ) @ D ) )
     => ( ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ E ) @ C ) ) @ B )
          = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ D ) @ C ) ) @ A ) )
      <=> ( B
          = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ D ) @ E ) ) @ C ) ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nat__eq__add__iff2) ).

thf(8975,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ E ) @ D ) )
     => ( ( ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ E ) @ C ) ) @ B )
            = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ D ) @ C ) ) @ A ) )
         => ( B
            = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ D ) @ E ) ) @ C ) ) @ A ) ) )
        & ( ( B
            = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ D ) @ E ) ) @ C ) ) @ A ) )
         => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ E ) @ C ) ) @ B )
            = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ D ) @ C ) ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1675]) ).

thf(2353,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( c_Big__Operators_Ocomm__monoid__big @ G @ F @ E @ D @ C )
     => ( ~ ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ F ) @ B ) )
       => ( ( hAPP @ ( hAPP @ C @ A ) @ B )
          = D ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_comm__monoid__big_Oinfinite) ).

thf(11660,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( c_Big__Operators_Ocomm__monoid__big @ G @ F @ E @ D @ C )
     => ( ~ ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ F ) @ B ) )
       => ( ( hAPP @ ( hAPP @ C @ A ) @ B )
          = D ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2353]) ).

thf(206,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ B ) ) @ A ) ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ A ) @ C ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ C ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_q__pos__lemma) ).

thf(3151,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ B ) ) @ A ) ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ A ) @ C ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ C ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[206]) ).

thf(2123,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Omap @ tc_Nat_Onat @ tc_Nat_Onat ) @ c_Nat_OSuc ) @ ( c_List_Oupt @ B @ A ) )
      = ( c_List_Oupt @ ( hAPP @ c_Nat_OSuc @ B ) @ ( hAPP @ c_Nat_OSuc @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_map__Suc__upt) ).

thf(10725,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Omap @ tc_Nat_Onat @ tc_Nat_Onat ) @ c_Nat_OSuc ) @ ( c_List_Oupt @ B @ A ) )
      = ( c_List_Oupt @ ( hAPP @ c_Nat_OSuc @ B ) @ ( hAPP @ c_Nat_OSuc @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2123]) ).

thf(767,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Map_Omap__add @ C @ B @ A @ ( hAPP @ ( c_COMBK @ ( tc_Option_Ooption @ B ) @ C ) @ ( c_Option_Ooption_ONone @ B ) ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_map__add__empty) ).

thf(5157,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Map_Omap__add @ C @ B @ A @ ( hAPP @ ( c_COMBK @ ( tc_Option_Ooption @ B ) @ C ) @ ( c_Option_Ooption_ONone @ B ) ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[767]) ).

thf(2143,axiom,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ tc_Nat_Onat ) @ A ) )
    <=> ? [B: $i] :
        ! [C: $i] :
          ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Nat_Onat ) @ C ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ C ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_finite__nat__set__iff__bounded__le) ).

thf(10811,plain,
    ! [A: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ tc_Nat_Onat ) @ A ) )
       => ? [B: $i] :
          ! [C: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Nat_Onat ) @ C ) @ A ) )
           => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ C ) @ B ) ) ) )
      & ( ? [B: $i] :
          ! [C: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Nat_Onat ) @ C ) @ A ) )
           => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ C ) @ B ) ) )
       => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ tc_Nat_Onat ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2143]) ).

thf(1109,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ ( c_Divides_Odiv__class_Omod @ tc_Nat_Onat @ B @ A ) )
      = ( c_Divides_Odiv__class_Omod @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mod__mult__distrib2) ).

thf(6560,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ ( c_Divides_Odiv__class_Omod @ tc_Nat_Onat @ B @ A ) )
      = ( c_Divides_Odiv__class_Omod @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1109]) ).

thf(333,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_List_Osublist @ C @ B @ A )
      = ( hAPP @ ( hAPP @ ( c_List_Omap @ ( tc_prod @ C @ tc_Nat_Onat ) @ C ) @ ( c_Product__Type_Ofst @ C @ tc_Nat_Onat ) ) @ ( hAPP @ ( c_List_Ofilter @ ( tc_prod @ C @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ ( tc_prod @ C @ tc_Nat_Onat ) @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_Nat_Onat @ ( tc_fun @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( tc_prod @ C @ tc_Nat_Onat ) ) @ ( c_member @ tc_Nat_Onat ) ) @ ( c_Product__Type_Osnd @ C @ tc_Nat_Onat ) ) ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_List_Ozip @ C @ tc_Nat_Onat ) @ B ) @ ( c_List_Oupt @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_sublist__def) ).

thf(3591,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_List_Osublist @ C @ B @ A )
      = ( hAPP @ ( hAPP @ ( c_List_Omap @ ( tc_prod @ C @ tc_Nat_Onat ) @ C ) @ ( c_Product__Type_Ofst @ C @ tc_Nat_Onat ) ) @ ( hAPP @ ( c_List_Ofilter @ ( tc_prod @ C @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ ( tc_prod @ C @ tc_Nat_Onat ) @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_Nat_Onat @ ( tc_fun @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( tc_prod @ C @ tc_Nat_Onat ) ) @ ( c_member @ tc_Nat_Onat ) ) @ ( c_Product__Type_Osnd @ C @ tc_Nat_Onat ) ) ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_List_Ozip @ C @ tc_Nat_Onat ) @ B ) @ ( c_List_Oupt @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[333]) ).

thf(253,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( c_Relation_Oconverse @ D @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ ( tc_prod @ C @ D ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Relation_Oconverse @ D @ C ) @ B ) ) @ ( hAPP @ ( c_Relation_Oconverse @ D @ C ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_converse__Un) ).

thf(3301,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( c_Relation_Oconverse @ D @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ ( tc_prod @ C @ D ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Relation_Oconverse @ D @ C ) @ B ) ) @ ( hAPP @ ( c_Relation_Oconverse @ D @ C ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[253]) ).

thf(2081,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_List_Odistinct @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( c_List_Odistinct @ C ) @ ( hAPP @ ( c_List_Ofilter @ C @ A ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_distinct__filter) ).

thf(10593,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_List_Odistinct @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( c_List_Odistinct @ C ) @ ( hAPP @ ( c_List_Ofilter @ C @ A ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2081]) ).

thf(1596,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ A ) ) )
    <=> ( ( C
          = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
        | ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nat__mult__dvd__cancel__disj) ).

thf(8655,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ A ) ) )
       => ( ( C
            = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
          | ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) ) ) )
      & ( ( ( C
            = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
          | ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1596]) ).

thf(1498,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ ( hAPP @ c_Nat_OSuc @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_add__Suc__shift) ).

thf(8201,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ ( hAPP @ c_Nat_OSuc @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1498]) ).

thf(1031,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Oappend @ B ) @ A ) @ ( c_List_Olist_ONil @ B ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_append__Nil2) ).

thf(6183,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Oappend @ B ) @ A ) @ ( c_List_Olist_ONil @ B ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1031]) ).

thf(1087,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( c_Finite__Set_Ofun__left__comm @ F @ E @ D )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ F ) @ C ) )
       => ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ F ) @ B ) @ C ) )
         => ( ( hAPP @ ( hAPP @ ( c_Finite__Set_Ofold @ F @ E @ D ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ F ) @ B ) @ C ) )
            = ( hAPP @ ( hAPP @ D @ B ) @ ( hAPP @ ( hAPP @ ( c_Finite__Set_Ofold @ F @ E @ D ) @ A ) @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_fun__left__comm_Ofold__insert) ).

thf(6406,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( c_Finite__Set_Ofun__left__comm @ F @ E @ D )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ F ) @ C ) )
       => ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ F ) @ B ) @ C ) )
         => ( ( hAPP @ ( hAPP @ ( c_Finite__Set_Ofold @ F @ E @ D ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ F ) @ B ) @ C ) )
            = ( hAPP @ ( hAPP @ D @ B ) @ ( hAPP @ ( hAPP @ ( c_Finite__Set_Ofold @ F @ E @ D ) @ A ) @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1087]) ).

thf(2262,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( c_List_OtakeWhile @ C @ B @ A )
        = A )
    <=> ! [D: $i] :
          ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ D ) @ ( hAPP @ ( c_List_Oset @ C ) @ A ) ) )
         => ( hBOOL @ ( hAPP @ B @ D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_takeWhile__eq__all__conv) ).

thf(11255,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( c_List_OtakeWhile @ C @ B @ A )
          = A )
       => ! [D: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ D ) @ ( hAPP @ ( c_List_Oset @ C ) @ A ) ) )
           => ( hBOOL @ ( hAPP @ B @ D ) ) ) )
      & ( ! [D: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ D ) @ ( hAPP @ ( c_List_Oset @ C ) @ A ) ) )
           => ( hBOOL @ ( hAPP @ B @ D ) ) )
       => ( ( c_List_OtakeWhile @ C @ B @ A )
          = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2262]) ).

thf(20,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ E ) @ ( tc_List_Olist @ E ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ E ) @ ( tc_List_Olist @ E ) ) @ D ) @ C ) ) @ ( hAPP @ ( c_List_Olexn @ E @ B ) @ A ) ) )
     => ( ( ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ E ) ) @ D )
          = A )
        & ( ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ E ) ) @ C )
          = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_lexn__length) ).

thf(2473,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ E ) @ ( tc_List_Olist @ E ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ E ) @ ( tc_List_Olist @ E ) ) @ D ) @ C ) ) @ ( hAPP @ ( c_List_Olexn @ E @ B ) @ A ) ) )
     => ( ( ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ E ) ) @ D )
          = A )
        & ( ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ E ) ) @ C )
          = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[20]) ).

thf(2296,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_Com_Ocom_OAss @ D @ C )
     != ( c_Com_Ocom_OWhile @ B @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_com_Osimps_I28_J) ).

thf(11395,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_Com_Ocom_OAss @ D @ C )
     != ( c_Com_Ocom_OWhile @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2296]) ).

thf(1765,axiom,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ tc_Nat_Onat ) @ ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Nat_Onat @ B @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_finite__atLeastAtMost) ).

thf(9340,plain,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ tc_Nat_Onat ) @ ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Nat_Onat @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1765]) ).

thf(327,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ B ) @ ( hAPP @ ( c_List_Oset @ C ) @ A ) ) )
    <=> ? [D: $i] :
          ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ D ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) @ A ) ) )
          & ( ( hAPP @ ( hAPP @ ( c_List_Onth @ C ) @ A ) @ D )
            = B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_in__set__conv__nth) ).

thf(3567,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ B ) @ ( hAPP @ ( c_List_Oset @ C ) @ A ) ) )
       => ? [D: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ D ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) @ A ) ) )
            & ( ( hAPP @ ( hAPP @ ( c_List_Onth @ C ) @ A ) @ D )
              = B ) ) )
      & ( ? [D: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ D ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) @ A ) ) )
            & ( ( hAPP @ ( hAPP @ ( c_List_Onth @ C ) @ A ) @ D )
              = B ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ B ) @ ( hAPP @ ( c_List_Oset @ C ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[327]) ).

thf(2160,axiom,
    ! [A: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__eq__nat_Osimps_I1_J) ).

thf(10875,plain,
    ! [A: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2160]) ).

thf(1505,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ B )
        = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ A ) )
    <=> ( ( C
          = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
        | ( B = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nat__mult__eq__cancel__disj) ).

thf(8233,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ B )
          = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ A ) )
       => ( ( C
            = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
          | ( B = A ) ) )
      & ( ( ( C
            = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
          | ( B = A ) )
       => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ B )
          = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1505]) ).

thf(367,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Omap @ ( tc_prod @ H @ G ) @ F ) @ E ) @ ( hAPP @ ( hAPP @ ( c_List_Ozip @ H @ G ) @ ( hAPP @ ( hAPP @ ( c_List_Omap @ D @ H ) @ C ) @ B ) ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_List_Omap @ ( tc_prod @ D @ G ) @ F ) @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ D @ G @ F ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ G @ ( tc_prod @ H @ G ) ) @ ( tc_fun @ G @ F ) @ D ) @ ( hAPP @ ( c_COMBB @ ( tc_prod @ H @ G ) @ F @ G ) @ E ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ H @ ( tc_fun @ G @ ( tc_prod @ H @ G ) ) @ D ) @ ( c_Product__Type_OPair @ H @ G ) ) @ C ) ) ) ) @ ( hAPP @ ( hAPP @ ( c_List_Ozip @ D @ G ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_map__zip__map) ).

thf(3720,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Omap @ ( tc_prod @ H @ G ) @ F ) @ E ) @ ( hAPP @ ( hAPP @ ( c_List_Ozip @ H @ G ) @ ( hAPP @ ( hAPP @ ( c_List_Omap @ D @ H ) @ C ) @ B ) ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_List_Omap @ ( tc_prod @ D @ G ) @ F ) @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ D @ G @ F ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ G @ ( tc_prod @ H @ G ) ) @ ( tc_fun @ G @ F ) @ D ) @ ( hAPP @ ( c_COMBB @ ( tc_prod @ H @ G ) @ F @ G ) @ E ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ H @ ( tc_fun @ G @ ( tc_prod @ H @ G ) ) @ D ) @ ( c_Product__Type_OPair @ H @ G ) ) @ C ) ) ) ) @ ( hAPP @ ( hAPP @ ( c_List_Ozip @ D @ G ) @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[367]) ).

thf(1144,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) ) @ B ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Relation_ODomain @ D @ C ) @ B ) ) @ ( hAPP @ ( c_Relation_ODomain @ D @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Domain__mono) ).

thf(6731,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) ) @ B ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Relation_ODomain @ D @ C ) @ B ) ) @ ( hAPP @ ( c_Relation_ODomain @ D @ C ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1144]) ).

thf(1478,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Wellfounded_Owf @ C @ B )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) ) @ A ) @ B ) )
       => ( c_Wellfounded_Owf @ C @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_wf__subset) ).

thf(8078,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Wellfounded_Owf @ C @ B )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) ) @ A ) @ B ) )
       => ( c_Wellfounded_Owf @ C @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1478]) ).

thf(1734,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ A ) )
       => ( ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Nat_Onat ) @ A ) @ B )
          = ( hAPP @ c_Nat_OSuc @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ A ) @ B ) ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_le__div__geq) ).

thf(9221,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ A ) )
       => ( ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Nat_Onat ) @ A ) @ B )
          = ( hAPP @ c_Nat_OSuc @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ A ) @ B ) ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1734]) ).

thf(1506,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( hAPP @ F @ E )
        = ( c_Option_Ooption_ONone @ D ) )
     => ( ( c_Map_Omap__comp @ D @ C @ B @ A @ F @ E )
        = ( c_Option_Ooption_ONone @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_map__comp__simps_I1_J) ).

thf(8243,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( hAPP @ F @ E )
        = ( c_Option_Ooption_ONone @ D ) )
     => ( ( c_Map_Omap__comp @ D @ C @ B @ A @ F @ E )
        = ( c_Option_Ooption_ONone @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1506]) ).

thf(1924,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ c_Com_OWT @ ( c_Com_Ocom_OCall @ C @ B @ A ) ) )
     => ( hBOOL @ ( hAPP @ c_Com_OWT @ ( hAPP @ c_Com_Ocom_OBODY @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_WTs__elim__cases_I8_J) ).

thf(9968,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ c_Com_OWT @ ( c_Com_Ocom_OCall @ C @ B @ A ) ) )
     => ( hBOOL @ ( hAPP @ c_Com_OWT @ ( hAPP @ c_Com_Ocom_OBODY @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1924]) ).

thf(577,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
      ( ( c_Finite__Set_Ofolding__image__simple @ H @ G @ F @ E @ D @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ G ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ G ) @ A ) )
         => ( ! [I: $i] :
                ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ G ) @ I ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ G @ tc_HOL_Obool ) ) @ B ) @ A ) ) )
               => ( ( hAPP @ D @ I )
                  = E ) )
           => ( ( hAPP @ C @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ G @ tc_HOL_Obool ) ) @ B ) @ A ) )
              = ( hAPP @ ( hAPP @ F @ ( hAPP @ C @ B ) ) @ ( hAPP @ C @ A ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_folding__image__simple_Ounion__inter__neutral) ).

thf(4480,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
      ( ( c_Finite__Set_Ofolding__image__simple @ H @ G @ F @ E @ D @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ G ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ G ) @ A ) )
         => ( ! [I: $i] :
                ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ G ) @ I ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ G @ tc_HOL_Obool ) ) @ B ) @ A ) ) )
               => ( ( hAPP @ D @ I )
                  = E ) )
           => ( ( hAPP @ C @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ G @ tc_HOL_Obool ) ) @ B ) @ A ) )
              = ( hAPP @ ( hAPP @ F @ ( hAPP @ C @ B ) ) @ ( hAPP @ C @ A ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[577]) ).

thf(1709,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ A ) )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ A ) @ B ) )
        = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_le__add__diff__inverse) ).

thf(9136,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ A ) )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ A ) @ B ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1709]) ).

thf(1000,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ D @ C ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ D ) @ A ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_sup1I1) ).

thf(6047,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ D @ C ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ D ) @ A ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1000]) ).

thf(1057,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ B ) @ ( hAPP @ c_Nat_OSuc @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ B ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_diff__Suc__eq__diff__pred) ).

thf(6296,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ B ) @ ( hAPP @ c_Nat_OSuc @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ B ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1057]) ).

thf(2219,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ D ) @ ( hAPP @ ( c_Set_Oimage @ C @ E @ B ) @ A ) ) )
    <=> ? [F: $i] :
          ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ F ) @ A ) )
          & ( D
            = ( hAPP @ B @ F ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_image__iff) ).

thf(11079,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ D ) @ ( hAPP @ ( c_Set_Oimage @ C @ E @ B ) @ A ) ) )
       => ? [F: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ F ) @ A ) )
            & ( D
              = ( hAPP @ B @ F ) ) ) )
      & ( ? [F: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ F ) @ A ) )
            & ( D
              = ( hAPP @ B @ F ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ D ) @ ( hAPP @ ( c_Set_Oimage @ C @ E @ B ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2219]) ).

thf(403,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Equiv__Relations_Oequiv @ E @ D @ C )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Equiv__Relations_Oquotient @ E ) @ D ) @ C ) ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Equiv__Relations_Oquotient @ E ) @ D ) @ C ) ) )
         => ( ( B = A )
            | ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ B ) @ A )
              = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ E @ tc_HOL_Obool ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_quotient__disj) ).

thf(3852,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Equiv__Relations_Oequiv @ E @ D @ C )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Equiv__Relations_Oquotient @ E ) @ D ) @ C ) ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Equiv__Relations_Oquotient @ E ) @ D ) @ C ) ) )
         => ( ( B = A )
            | ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ B ) @ A )
              = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ E @ tc_HOL_Obool ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[403]) ).

thf(1223,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ D ) @ C ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Set_Oimage @ E @ B @ A ) @ D ) ) @ ( hAPP @ ( c_Set_Oimage @ E @ B @ A ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_image__mono) ).

thf(7037,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ D ) @ C ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Set_Oimage @ E @ B @ A ) @ D ) ) @ ( hAPP @ ( c_Set_Oimage @ E @ B @ A ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1223]) ).

thf(1265,axiom,
    ! [A: $i,B: $i] :
      ( ( ( ( B
            = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
          | ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ A ) @ B ) ) )
       => ( ( c_Divides_Odivmod__nat @ A @ B )
          = ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ tc_Nat_Onat @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ A ) ) )
      & ( ~ ( ( B
              = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
            | ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ A ) @ B ) ) )
       => ( ( c_Divides_Odivmod__nat @ A @ B )
          = ( hAPP @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ tc_Nat_Onat @ tc_Nat_Onat @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_Nat_Onat @ ( tc_fun @ tc_Nat_Onat @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) ) @ tc_Nat_Onat ) @ ( c_Product__Type_OPair @ tc_Nat_Onat @ tc_Nat_Onat ) ) @ c_Nat_OSuc ) ) @ ( c_Divides_Odivmod__nat @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ A ) @ B ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_divmod__nat__if) ).

thf(7217,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( B
            = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
          | ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ A ) @ B ) ) )
       => ( ( c_Divides_Odivmod__nat @ A @ B )
          = ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ tc_Nat_Onat @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ A ) ) )
      & ( ~ ( ( B
              = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
            | ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ A ) @ B ) ) )
       => ( ( c_Divides_Odivmod__nat @ A @ B )
          = ( hAPP @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ tc_Nat_Onat @ tc_Nat_Onat @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_Nat_Onat @ ( tc_fun @ tc_Nat_Onat @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) ) @ tc_Nat_Onat ) @ ( c_Product__Type_OPair @ tc_Nat_Onat @ tc_Nat_Onat ) ) @ c_Nat_OSuc ) ) @ ( c_Divides_Odivmod__nat @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ A ) @ B ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1265]) ).

thf(917,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Relation_OId__on @ B @ A )
      = ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ B @ B @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ B @ tc_HOL_Obool @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ B @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ ( tc_fun @ tc_HOL_Obool @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ B ) @ ( c_COMBC @ B @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ B @ tc_HOL_Obool ) @ ( tc_fun @ B @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ B ) @ c_fconj ) ) @ c_fequal ) ) ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Id__on__def_H) ).

thf(5697,plain,
    ! [A: $i,B: $i] :
      ( ( c_Relation_OId__on @ B @ A )
      = ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ B @ B @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ B @ tc_HOL_Obool @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ B @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ ( tc_fun @ tc_HOL_Obool @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ B ) @ ( c_COMBC @ B @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ B @ tc_HOL_Obool ) @ ( tc_fun @ B @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ B ) @ c_fconj ) ) @ c_fequal ) ) ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[917]) ).

thf(217,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Finite__Set_Ofolding__one @ E @ D @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ E ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ E ) @ A ) )
         => ( ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ B ) @ A )
             != ( c_Orderings_Obot__class_Obot @ ( tc_fun @ E @ tc_HOL_Obool ) ) )
           => ( ( hAPP @ ( hAPP @ D @ ( hAPP @ C @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ B ) @ A ) ) ) @ ( hAPP @ C @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ B ) @ A ) ) )
              = ( hAPP @ ( hAPP @ D @ ( hAPP @ C @ B ) ) @ ( hAPP @ C @ A ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_folding__one_Ounion__inter) ).

thf(3186,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Finite__Set_Ofolding__one @ E @ D @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ E ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ E ) @ A ) )
         => ( ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ B ) @ A )
             != ( c_Orderings_Obot__class_Obot @ ( tc_fun @ E @ tc_HOL_Obool ) ) )
           => ( ( hAPP @ ( hAPP @ D @ ( hAPP @ C @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ B ) @ A ) ) ) @ ( hAPP @ C @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ B ) @ A ) ) )
              = ( hAPP @ ( hAPP @ D @ ( hAPP @ C @ B ) ) @ ( hAPP @ C @ A ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[217]) ).

thf(1631,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) @ ( c_SetInterval_Oord_OatLeastAtMost @ tc_Nat_Onat @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ D @ C ) ) @ ( c_SetInterval_Oord_OatLeastAtMost @ tc_Nat_Onat @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B @ A ) ) )
    <=> ( ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ D ) @ C ) )
          | ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ D ) )
            & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ C ) @ A ) )
            & ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ D ) )
                & ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ D ) @ B ) ) )
              | ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ C ) @ A ) )
                & ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ A ) @ C ) ) ) ) ) )
        & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_dvd_OatLeastatMost__psubset__iff) ).

thf(8795,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) @ ( c_SetInterval_Oord_OatLeastAtMost @ tc_Nat_Onat @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ D @ C ) ) @ ( c_SetInterval_Oord_OatLeastAtMost @ tc_Nat_Onat @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B @ A ) ) )
       => ( ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ D ) @ C ) )
            | ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ D ) )
              & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ C ) @ A ) )
              & ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ D ) )
                  & ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ D ) @ B ) ) )
                | ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ C ) @ A ) )
                  & ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ A ) @ C ) ) ) ) ) )
          & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) ) ) )
      & ( ( ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ D ) @ C ) )
            | ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ D ) )
              & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ C ) @ A ) )
              & ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ D ) )
                  & ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ D ) @ B ) ) )
                | ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ C ) @ A ) )
                  & ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ A ) @ C ) ) ) ) ) )
          & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) @ ( c_SetInterval_Oord_OatLeastAtMost @ tc_Nat_Onat @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ D @ C ) ) @ ( c_SetInterval_Oord_OatLeastAtMost @ tc_Nat_Onat @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1631]) ).

thf(1119,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ C @ ( c_Divides_Odiv__class_Omod @ tc_Nat_Onat @ B @ A ) ) )
    <=> ( ( ( A
            = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
         => ( hBOOL @ ( hAPP @ C @ B ) ) )
        & ( ( A
           != ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
         => ! [D: $i,E: $i] :
              ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ E ) @ A ) )
             => ( ( B
                  = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ A ) @ D ) ) @ E ) )
               => ( hBOOL @ ( hAPP @ C @ E ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_split__mod) ).

thf(6588,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP @ C @ ( c_Divides_Odiv__class_Omod @ tc_Nat_Onat @ B @ A ) ) )
       => ( ( ( A
              = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
           => ( hBOOL @ ( hAPP @ C @ B ) ) )
          & ( ( A
             != ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
           => ! [D: $i,E: $i] :
                ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ E ) @ A ) )
               => ( ( B
                    = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ A ) @ D ) ) @ E ) )
                 => ( hBOOL @ ( hAPP @ C @ E ) ) ) ) ) ) )
      & ( ( ( ( A
              = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
           => ( hBOOL @ ( hAPP @ C @ B ) ) )
          & ( ( A
             != ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
           => ! [D: $i,E: $i] :
                ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ E ) @ A ) )
               => ( ( B
                    = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ A ) @ D ) ) @ E ) )
                 => ( hBOOL @ ( hAPP @ C @ E ) ) ) ) ) )
       => ( hBOOL @ ( hAPP @ C @ ( c_Divides_Odiv__class_Omod @ tc_Nat_Onat @ B @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1119]) ).

thf(1877,axiom,
    ! [A: $i] :
      ( ( c_Divides_Odivmod__nat @ A @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
      = ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ tc_Nat_Onat @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_divmod__nat__zero) ).

thf(9797,plain,
    ! [A: $i] :
      ( ( c_Divides_Odivmod__nat @ A @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
      = ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ tc_Nat_Onat @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1877]) ).

thf(2356,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( c_Fun_Oinj__on @ F @ E @ D @ C )
     => ( ( ( hAPP @ D @ B )
          = A )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ F ) @ B ) @ C ) )
         => ( ( hAPP @ ( c_Fun_Othe__inv__into @ F @ E @ C @ D ) @ A )
            = B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_the__inv__into__f__eq) ).

thf(11672,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( c_Fun_Oinj__on @ F @ E @ D @ C )
     => ( ( ( hAPP @ D @ B )
          = A )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ F ) @ B ) @ C ) )
         => ( ( hAPP @ ( c_Fun_Othe__inv__into @ F @ E @ C @ D ) @ A )
            = B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2356]) ).

thf(762,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) ) @ ( c_Transitive__Closure_Otrancl @ B @ A ) ) @ ( c_Relation_OId @ B ) )
      = ( c_Transitive__Closure_Ortrancl @ B @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_reflcl__trancl) ).

thf(5142,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) ) @ ( c_Transitive__Closure_Otrancl @ B @ A ) ) @ ( c_Relation_OId @ B ) )
      = ( c_Transitive__Closure_Ortrancl @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[762]) ).

thf(598,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Nat_Ocompow @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) @ A )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_rel__pow__1) ).

thf(4557,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Nat_Ocompow @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) @ A )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[598]) ).

thf(1447,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ D ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Nat_Ocompow @ ( tc_prod @ E @ E ) @ tc_HOL_Obool ) @ B ) @ A ) ) )
     => ( ( ( B
            = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
         => ( D != C ) )
       => ~ ! [F: $i,G: $i] :
              ( ( B
                = ( hAPP @ c_Nat_OSuc @ G ) )
             => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ D ) @ F ) ) @ ( hAPP @ ( hAPP @ ( c_Nat_Ocompow @ ( tc_prod @ E @ E ) @ tc_HOL_Obool ) @ G ) @ A ) ) )
               => ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ F ) @ C ) ) @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_rel__pow__E) ).

thf(7937,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ D ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Nat_Ocompow @ ( tc_prod @ E @ E ) @ tc_HOL_Obool ) @ B ) @ A ) ) )
     => ( ( ( B
            = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
         => ( D != C ) )
       => ~ ! [F: $i,G: $i] :
              ( ( B
                = ( hAPP @ c_Nat_OSuc @ G ) )
             => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ D ) @ F ) ) @ ( hAPP @ ( hAPP @ ( c_Nat_Ocompow @ ( tc_prod @ E @ E ) @ tc_HOL_Obool ) @ G ) @ A ) ) )
               => ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ F ) @ C ) ) @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1447]) ).

thf(798,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( c_Product__Type_Osnd @ ( tc_List_Olist @ C ) @ ( tc_List_Olist @ C ) ) @ ( c_List_Opartition @ C @ B @ A ) )
      = ( hAPP @ ( c_List_Ofilter @ C @ ( hAPP @ ( c_Fun_Ocomp @ tc_HOL_Obool @ tc_HOL_Obool @ C @ c_fNot ) @ B ) ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_partition__filter2) ).

thf(5291,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( c_Product__Type_Osnd @ ( tc_List_Olist @ C ) @ ( tc_List_Olist @ C ) ) @ ( c_List_Opartition @ C @ B @ A ) )
      = ( hAPP @ ( c_List_Ofilter @ C @ ( hAPP @ ( c_Fun_Ocomp @ tc_HOL_Obool @ tc_HOL_Obool @ C @ c_fNot ) @ B ) ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[798]) ).

thf(734,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( c_Int_Onumber__class_Onumber__of @ tc_Int_Oint ) @ B ) ) @ ( hAPP @ ( c_Int_Onumber__class_Onumber__of @ tc_Int_Oint ) @ A ) )
      = ( hAPP @ ( c_Int_Onumber__class_Onumber__of @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_plus__numeral__code_I9_J) ).

thf(5034,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( c_Int_Onumber__class_Onumber__of @ tc_Int_Oint ) @ B ) ) @ ( hAPP @ ( c_Int_Onumber__class_Onumber__of @ tc_Int_Oint ) @ A ) )
      = ( hAPP @ ( c_Int_Onumber__class_Onumber__of @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[734]) ).

thf(2187,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( c_Option_Ooption_OSome @ B ) @ A )
     != ( c_Option_Ooption_ONone @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_option_Osimps_I3_J) ).

thf(10947,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( c_Option_Ooption_OSome @ B ) @ A )
     != ( c_Option_Ooption_ONone @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2187]) ).

thf(1059,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( c_Relation_Oconverse @ D @ D ) @ ( hAPP @ ( hAPP @ ( c_Relation_Oinv__image @ C @ D ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Relation_Oinv__image @ C @ D ) @ ( hAPP @ ( c_Relation_Oconverse @ C @ C ) @ B ) ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_converse__inv__image) ).

thf(6307,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( c_Relation_Oconverse @ D @ D ) @ ( hAPP @ ( hAPP @ ( c_Relation_Oinv__image @ C @ D ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Relation_Oinv__image @ C @ D ) @ ( hAPP @ ( c_Relation_Oconverse @ C @ C ) @ B ) ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1059]) ).

thf(1742,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ E ) @ D ) )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ E ) @ C ) ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ D ) @ C ) ) @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ D ) @ E ) ) @ C ) ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nat__diff__add__eq2) ).

thf(9252,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ E ) @ D ) )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ E ) @ C ) ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ D ) @ C ) ) @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ D ) @ E ) ) @ C ) ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1742]) ).

thf(867,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ A @ B ) ) )
        & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ A @ B ) ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_pos__mod__conj) ).

thf(5519,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ A @ B ) ) )
        & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ A @ B ) ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[867]) ).

thf(2096,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i,I: $i,J: $i,K: $i,L: $i] :
      ( ( c_Com_Ocom_Ocom__case @ L @ K @ J @ I @ H @ G @ F @ E @ D @ ( c_Com_Ocom_OCond @ C @ B @ A ) )
      = ( hAPP @ ( hAPP @ ( hAPP @ G @ C ) @ B ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_com_Osimps_I68_J) ).

thf(10639,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i,I: $i,J: $i,K: $i,L: $i] :
      ( ( c_Com_Ocom_Ocom__case @ L @ K @ J @ I @ H @ G @ F @ E @ D @ ( c_Com_Ocom_OCond @ C @ B @ A ) )
      = ( hAPP @ ( hAPP @ ( hAPP @ G @ C ) @ B ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2096]) ).

thf(1233,axiom,
    ! [A: $i,B: $i] :
      ( ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Nat_Onat @ ( hAPP @ c_Nat_OSuc @ B ) @ A )
      = ( c_SetInterval_Oord__class_OgreaterThanAtMost @ tc_Nat_Onat @ B @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_atLeastSucAtMost__greaterThanAtMost) ).

thf(7078,plain,
    ! [A: $i,B: $i] :
      ( ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Nat_Onat @ ( hAPP @ c_Nat_OSuc @ B ) @ A )
      = ( c_SetInterval_Oord__class_OgreaterThanAtMost @ tc_Nat_Onat @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1233]) ).

thf(477,axiom,
    ! [A: $i,B: $i] :
      ( ( ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) )
        = ( hAPP @ ( c_Set_OCollect @ B ) @ A ) )
    <=> ! [C: $i] :
          ~ ( hBOOL @ ( hAPP @ A @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_empty__Collect__eq) ).

thf(4100,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) )
          = ( hAPP @ ( c_Set_OCollect @ B ) @ A ) )
       => ! [C: $i] :
            ~ ( hBOOL @ ( hAPP @ A @ C ) ) )
      & ( ! [C: $i] :
            ~ ( hBOOL @ ( hAPP @ A @ C ) )
       => ( ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) )
          = ( hAPP @ ( c_Set_OCollect @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[477]) ).

thf(1854,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( c_List_Ofilter @ D @ C ) @ ( c_List_Oremove1 @ D @ B @ A ) )
      = ( c_List_Oremove1 @ D @ B @ ( hAPP @ ( c_List_Ofilter @ D @ C ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_filter__remove1) ).

thf(9688,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( c_List_Ofilter @ D @ C ) @ ( c_List_Oremove1 @ D @ B @ A ) )
      = ( c_List_Oremove1 @ D @ B @ ( hAPP @ ( c_List_Ofilter @ D @ C ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1854]) ).

thf(321,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_Fun_Ooverride__on @ D @ C @ B @ A @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) )
      = B ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_override__on__emptyset) ).

thf(3550,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_Fun_Ooverride__on @ D @ C @ B @ A @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) )
      = B ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[321]) ).

thf(2318,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( c_New__DSequence_Opos__single @ C @ B ) @ A )
      = ( c_Lazy__Sequence_Osingle @ C @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_New__DSequence_Opos__single__def) ).

thf(11499,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( c_New__DSequence_Opos__single @ C @ B ) @ A )
      = ( c_Lazy__Sequence_Osingle @ C @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2318]) ).

thf(722,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( c_Map_Orestrict__map @ D @ C @ ( hAPP @ ( c_COMBK @ ( tc_Option_Ooption @ C ) @ D ) @ ( c_Option_Ooption_ONone @ C ) ) @ B ) @ A )
      = ( c_Option_Ooption_ONone @ C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_restrict__map__empty) ).

thf(4999,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( c_Map_Orestrict__map @ D @ C @ ( hAPP @ ( c_COMBK @ ( tc_Option_Ooption @ C ) @ D ) @ ( c_Option_Ooption_ONone @ C ) ) @ B ) @ A )
      = ( c_Option_Ooption_ONone @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[722]) ).

thf(285,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Int_Oint ) @ C ) @ ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ B @ A ) ) @ A )
      = ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Int_Oint ) @ C ) @ B ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zdiff__zmod__right) ).

thf(3418,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Int_Oint ) @ C ) @ ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ B @ A ) ) @ A )
      = ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Int_Oint ) @ C ) @ B ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[285]) ).

thf(1777,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ c_Com_OWT @ ( c_Com_Ocom_OSemi @ B @ A ) ) )
     => ~ ( ( hBOOL @ ( hAPP @ c_Com_OWT @ B ) )
         => ~ ( hBOOL @ ( hAPP @ c_Com_OWT @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_WTs__elim__cases_I4_J) ).

thf(9384,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ c_Com_OWT @ ( c_Com_Ocom_OSemi @ B @ A ) ) )
     => ~ ( ( hBOOL @ ( hAPP @ c_Com_OWT @ B ) )
         => ~ ( hBOOL @ ( hAPP @ c_Com_OWT @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1777]) ).

thf(1760,axiom,
    ! [A: $i,B: $i,C: $i] : ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ ( c_Map_Odom @ C @ B @ ( c_Map_Omap__of @ C @ B @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_finite__dom__map__of) ).

thf(9323,plain,
    ! [A: $i,B: $i,C: $i] : ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ ( c_Map_Odom @ C @ B @ ( c_Map_Omap__of @ C @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1760]) ).

thf(758,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) ) )
     => ( ( B
          = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) )
        | ( B
          = ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_subset__singletonD) ).

thf(5122,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) ) )
     => ( ( B
          = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) )
        | ( B
          = ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[758]) ).

thf(2301,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ F ) @ E ) @ ( c_SetInterval_Oord_OgreaterThanAtMost @ F @ D @ C @ B @ A ) ) )
    <=> ( ( hBOOL @ ( hAPP @ ( hAPP @ C @ B ) @ E ) )
        & ( hBOOL @ ( hAPP @ ( hAPP @ D @ E ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_ord_OgreaterThanAtMost__iff) ).

thf(11410,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ F ) @ E ) @ ( c_SetInterval_Oord_OgreaterThanAtMost @ F @ D @ C @ B @ A ) ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ C @ B ) @ E ) )
          & ( hBOOL @ ( hAPP @ ( hAPP @ D @ E ) @ A ) ) ) )
      & ( ( ( hBOOL @ ( hAPP @ ( hAPP @ C @ B ) @ E ) )
          & ( hBOOL @ ( hAPP @ ( hAPP @ D @ E ) @ A ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ F ) @ E ) @ ( c_SetInterval_Oord_OgreaterThanAtMost @ F @ D @ C @ B @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2301]) ).

thf(705,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( D
          = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) )
       => ( ( hAPP @ ( c_Set_Oimage @ ( tc_prod @ C @ B ) @ B @ ( c_Product__Type_Osnd @ C @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ C @ B ) @ D ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ B @ tc_HOL_Obool ) @ C ) @ A ) ) )
          = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) )
      & ( ( D
         != ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) )
       => ( ( hAPP @ ( c_Set_Oimage @ ( tc_prod @ C @ B ) @ B @ ( c_Product__Type_Osnd @ C @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ C @ B ) @ D ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ B @ tc_HOL_Obool ) @ C ) @ A ) ) )
          = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_snd__image__times) ).

thf(4925,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( D
          = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) )
       => ( ( hAPP @ ( c_Set_Oimage @ ( tc_prod @ C @ B ) @ B @ ( c_Product__Type_Osnd @ C @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ C @ B ) @ D ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ B @ tc_HOL_Obool ) @ C ) @ A ) ) )
          = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) )
      & ( ( D
         != ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) )
       => ( ( hAPP @ ( c_Set_Oimage @ ( tc_prod @ C @ B ) @ B @ ( c_Product__Type_Osnd @ C @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ C @ B ) @ D ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ B @ tc_HOL_Obool ) @ C ) @ A ) ) )
          = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[705]) ).

thf(1900,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ c_Com_OWT @ ( c_Com_Ocom_OCond @ C @ B @ A ) ) )
    <=> ( ( hBOOL @ ( hAPP @ c_Com_OWT @ B ) )
        & ( hBOOL @ ( hAPP @ c_Com_OWT @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_WT_Oequations_I5_J) ).

thf(9881,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP @ c_Com_OWT @ ( c_Com_Ocom_OCond @ C @ B @ A ) ) )
       => ( ( hBOOL @ ( hAPP @ c_Com_OWT @ B ) )
          & ( hBOOL @ ( hAPP @ c_Com_OWT @ A ) ) ) )
      & ( ( ( hBOOL @ ( hAPP @ c_Com_OWT @ B ) )
          & ( hBOOL @ ( hAPP @ c_Com_OWT @ A ) ) )
       => ( hBOOL @ ( hAPP @ c_Com_OWT @ ( c_Com_Ocom_OCond @ C @ B @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1900]) ).

thf(358,axiom,
    ! [A: $i] :
      ( ( hAPP @ ( c_Nat_Osize__class_Osize @ tc_Nat_Onat ) @ A )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nat__size) ).

thf(3679,plain,
    ! [A: $i] :
      ( ( hAPP @ ( c_Nat_Osize__class_Osize @ tc_Nat_Onat ) @ A )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[358]) ).

thf(1036,axiom,
    ! [A: $i,B: $i] :
      ( ( B != A )
    <=> ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) )
        | ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ A ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nat__neq__iff) ).

thf(6200,plain,
    ! [A: $i,B: $i] :
      ( ( ( B != A )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) )
          | ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ A ) @ B ) ) ) )
      & ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) )
          | ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ A ) @ B ) ) )
       => ( B != A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1036]) ).

thf(373,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ ( tc_prod @ D @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ D @ C ) @ B ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ C @ tc_HOL_Obool ) @ D ) @ A ) ) ) )
     => ( ( A
         != ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) )
       => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_finite__cartesian__productD1) ).

thf(3735,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ ( tc_prod @ D @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ D @ C ) @ B ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ C @ tc_HOL_Obool ) @ D ) @ A ) ) ) )
     => ( ( A
         != ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) )
       => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[373]) ).

thf(2323,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D = C )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ B ) @ A ) @ D ) )
      <=> ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ B ) @ A ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_eqset__imp__iff) ).

thf(11520,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D = C )
     => ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ B ) @ A ) @ D ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ B ) @ A ) @ C ) ) )
        & ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ B ) @ A ) @ C ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ B ) @ A ) @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2323]) ).

thf(240,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Map_Odom @ E @ D @ ( c_Map_Omap__upds @ E @ D @ C @ B @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_List_Oset @ E ) @ ( hAPP @ ( hAPP @ ( c_List_Otake @ E ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ D ) ) @ A ) ) @ B ) ) ) @ ( c_Map_Odom @ E @ D @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_dom__map__upds) ).

thf(3261,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Map_Odom @ E @ D @ ( c_Map_Omap__upds @ E @ D @ C @ B @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_List_Oset @ E ) @ ( hAPP @ ( hAPP @ ( c_List_Otake @ E ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ D ) ) @ A ) ) @ B ) ) ) @ ( c_Map_Odom @ E @ D @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[240]) ).

thf(703,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( c_Relation_OField @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Relation_OField @ C ) @ B ) ) @ ( hAPP @ ( c_Relation_OField @ C ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Field__Un) ).

thf(4920,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( c_Relation_OField @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Relation_OField @ C ) @ B ) ) @ ( hAPP @ ( c_Relation_OField @ C ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[703]) ).

thf(582,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ B ) @ A ) ) )
      <=> ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ A ) @ B ) )
          & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nonneg1__imp__zdiv__pos__iff) ).

thf(4499,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ B ) )
     => ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ B ) @ A ) ) )
         => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ A ) @ B ) )
            & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ A ) ) ) )
        & ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ A ) @ B ) )
            & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ A ) ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ B ) @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[582]) ).

thf(2336,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ D ) @ C ) )
     => ( ( hAPP @ ( c_Map_Orestrict__map @ E @ B @ A @ C ) @ D )
        = ( c_Option_Ooption_ONone @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_restrict__out) ).

thf(11592,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ D ) @ C ) )
     => ( ( hAPP @ ( c_Map_Orestrict__map @ E @ B @ A @ C ) @ D )
        = ( c_Option_Ooption_ONone @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2336]) ).

thf(1686,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofold__graph @ G @ F @ E @ D @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ G @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ G ) @ B ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ G @ tc_HOL_Obool ) ) ) ) ) @ A ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ G ) @ B ) @ C ) )
       => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofold__graph @ G @ F @ E @ D @ C ) @ ( hAPP @ ( hAPP @ E @ B ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Diff1__fold__graph) ).

thf(9036,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofold__graph @ G @ F @ E @ D @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ G @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ G ) @ B ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ G @ tc_HOL_Obool ) ) ) ) ) @ A ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ G ) @ B ) @ C ) )
       => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofold__graph @ G @ F @ E @ D @ C ) @ ( hAPP @ ( hAPP @ E @ B ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1686]) ).

thf(75,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( class_Groups_Ocomm__monoid__add @ F )
     => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ E @ F ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ E @ ( tc_fun @ D @ tc_HOL_Obool ) @ F ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ D @ F ) @ ( tc_fun @ ( tc_fun @ D @ tc_HOL_Obool ) @ F ) @ E ) @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ F ) ) @ C ) ) @ B ) ) @ A )
        = ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ ( tc_prod @ E @ D ) @ F ) @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ E @ D @ F ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ E @ D ) @ A ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ D @ tc_HOL_Obool ) @ E ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_setsum__cartesian__product) ).

thf(2673,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( class_Groups_Ocomm__monoid__add @ F )
     => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ E @ F ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ E @ ( tc_fun @ D @ tc_HOL_Obool ) @ F ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ D @ F ) @ ( tc_fun @ ( tc_fun @ D @ tc_HOL_Obool ) @ F ) @ E ) @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ F ) ) @ C ) ) @ B ) ) @ A )
        = ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ ( tc_prod @ E @ D ) @ F ) @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ E @ D @ F ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ E @ D ) @ A ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ D @ tc_HOL_Obool ) @ E ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[75]) ).

thf(1822,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ H @ G ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ H @ G ) @ F ) @ E ) ) @ D ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ G @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ G @ C ) @ E ) @ B ) ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ H @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ H @ C ) @ F ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ H @ G @ C ) @ D ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_rel__compI) ).

thf(9549,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ H @ G ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ H @ G ) @ F ) @ E ) ) @ D ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ G @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ G @ C ) @ E ) @ B ) ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ H @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ H @ C ) @ F ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ H @ G @ C ) @ D ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1822]) ).

thf(84,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( class_Groups_Ocomm__monoid__add @ F )
     => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ E @ F ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ E @ ( tc_fun @ D @ tc_HOL_Obool ) @ F ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ D @ F ) @ ( tc_fun @ ( tc_fun @ D @ tc_HOL_Obool ) @ F ) @ E ) @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ F ) ) @ C ) ) @ B ) ) @ A )
        = ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ F ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ ( tc_fun @ E @ tc_HOL_Obool ) @ F ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ E @ F ) @ ( tc_fun @ ( tc_fun @ E @ tc_HOL_Obool ) @ F ) @ D ) @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ E @ F ) ) @ ( hAPP @ ( c_COMBC @ E @ D @ F ) @ C ) ) ) @ A ) ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_setsum__commute) ).

thf(2705,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( class_Groups_Ocomm__monoid__add @ F )
     => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ E @ F ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ E @ ( tc_fun @ D @ tc_HOL_Obool ) @ F ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ D @ F ) @ ( tc_fun @ ( tc_fun @ D @ tc_HOL_Obool ) @ F ) @ E ) @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ F ) ) @ C ) ) @ B ) ) @ A )
        = ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ F ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ ( tc_fun @ E @ tc_HOL_Obool ) @ F ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ E @ F ) @ ( tc_fun @ ( tc_fun @ E @ tc_HOL_Obool ) @ F ) @ D ) @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ E @ F ) ) @ ( hAPP @ ( c_COMBC @ E @ D @ F ) @ C ) ) ) @ A ) ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[84]) ).

thf(1533,axiom,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) ) @ A ) @ c_Int_OInteg ) )
     => ~ ! [B: $i] :
            ( A
           != ( hAPP @ c_Int_ORep__Integ @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Rep__Integ__cases) ).

thf(8343,plain,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) ) @ A ) @ c_Int_OInteg ) )
     => ~ ! [B: $i] :
            ( A
           != ( hAPP @ c_Int_ORep__Integ @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1533]) ).

thf(573,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ C ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ C ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ B ) @ A ) ) ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ C @ A ) ) @ ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ B @ A ) ) ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zdiv__zadd1__eq) ).

thf(4466,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ C ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ C ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ B ) @ A ) ) ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ C @ A ) ) @ ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ B @ A ) ) ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[573]) ).

thf(1016,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) @ ( c_List_OtakeWhile @ C @ B @ A ) ) ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) @ A ) ) )
     => ~ ( hBOOL @ ( hAPP @ B @ ( hAPP @ ( hAPP @ ( c_List_Onth @ C ) @ A ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) @ ( c_List_OtakeWhile @ C @ B @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nth__length__takeWhile) ).

thf(6124,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) @ ( c_List_OtakeWhile @ C @ B @ A ) ) ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) @ A ) ) )
     => ~ ( hBOOL @ ( hAPP @ B @ ( hAPP @ ( hAPP @ ( c_List_Onth @ C ) @ A ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) @ ( c_List_OtakeWhile @ C @ B @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1016]) ).

thf(1501,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) )
    <=> ? [C: $i] :
          ( A
          = ( hAPP @ c_Nat_OSuc @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__iff__Suc__add) ).

thf(8211,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) )
       => ? [C: $i] :
            ( A
            = ( hAPP @ c_Nat_OSuc @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ C ) ) ) )
      & ( ? [C: $i] :
            ( A
            = ( hAPP @ c_Nat_OSuc @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ C ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1501]) ).

thf(893,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) )
      = B ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Un__Diff__Int) ).

thf(5611,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) )
      = B ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[893]) ).

thf(1510,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ A ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_n__less__m__mult__n) ).

thf(8265,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ A ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1510]) ).

thf(1624,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) ) @ D ) @ C ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) ) @ D ) @ B ) ) @ c_FunDef_Opair__less ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) ) @ C ) @ A ) ) @ c_FunDef_Omin__strict ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) ) @ B ) @ A ) ) ) @ c_FunDef_Omin__strict ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_smin__insertI) ).

thf(8778,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) ) @ D ) @ C ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) ) @ D ) @ B ) ) @ c_FunDef_Opair__less ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) ) @ C ) @ A ) ) @ c_FunDef_Omin__strict ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) ) @ B ) @ A ) ) ) @ c_FunDef_Omin__strict ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1624]) ).

thf(1015,axiom,
    ! [A: $i] : ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ A @ tc_HOL_Obool ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_finite_Oequations_I1_J) ).

thf(6122,plain,
    ! [A: $i] : ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ A @ tc_HOL_Obool ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1015]) ).

thf(7,axiom,
    ! [A: $i] :
      ( ( c_List_Oconcat @ A @ ( c_List_Olist_ONil @ ( tc_List_Olist @ A ) ) )
      = ( c_List_Olist_ONil @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_concat_Osimps_I1_J) ).

thf(2428,plain,
    ! [A: $i] :
      ( ( c_List_Oconcat @ A @ ( c_List_Olist_ONil @ ( tc_List_Olist @ A ) ) )
      = ( c_List_Olist_ONil @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[7]) ).

thf(450,axiom,
    ! [A: $i] : ( c_Wellfounded_Owf @ ( tc_fun @ A @ tc_HOL_Obool ) @ ( c_Wellfounded_Ofinite__psubset @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_wf__finite__psubset) ).

thf(4016,plain,
    ! [A: $i] : ( c_Wellfounded_Owf @ ( tc_fun @ A @ tc_HOL_Obool ) @ ( c_Wellfounded_Ofinite__psubset @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[450]) ).

thf(1066,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_Wellfounded_Owf @ D @ C )
     => ( ( c_Fun_Oinj__on @ D @ B @ A @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ D @ tc_HOL_Obool ) ) )
       => ( c_Wellfounded_Owf @ B @ ( hAPP @ ( c_Set_Oimage @ ( tc_prod @ D @ D ) @ ( tc_prod @ B @ B ) @ ( c_Product__Type_Omap__pair @ D @ B @ D @ B @ A @ A ) ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_wf__map__pair__image) ).

thf(6328,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_Wellfounded_Owf @ D @ C )
     => ( ( c_Fun_Oinj__on @ D @ B @ A @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ D @ tc_HOL_Obool ) ) )
       => ( c_Wellfounded_Owf @ B @ ( hAPP @ ( c_Set_Oimage @ ( tc_prod @ D @ D ) @ ( tc_prod @ B @ B ) @ ( c_Product__Type_Omap__pair @ D @ B @ D @ B @ A @ A ) ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1066]) ).

thf(1107,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( F = E )
       => ( ( hAPP @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ D @ C ) @ B ) @ E ) @ A ) @ F )
          = A ) )
      & ( ( F != E )
       => ( ( hAPP @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ D @ C ) @ B ) @ E ) @ A ) @ F )
          = ( hAPP @ B @ F ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_fun__upd__apply) ).

thf(6550,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( F = E )
       => ( ( hAPP @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ D @ C ) @ B ) @ E ) @ A ) @ F )
          = A ) )
      & ( ( F != E )
       => ( ( hAPP @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ D @ C ) @ B ) @ E ) @ A ) @ F )
          = ( hAPP @ B @ F ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1107]) ).

thf(1172,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C = B )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ A @ tc_HOL_Obool ) ) @ B ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_equalityD2) ).

thf(6820,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C = B )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ A @ tc_HOL_Obool ) ) @ B ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1172]) ).

thf(43,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Ocomm__monoid__add @ C )
     => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ B @ C ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) )
        = ( c_Groups_Ozero__class_Ozero @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_setsum_Oempty) ).

thf(2541,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Ocomm__monoid__add @ C )
     => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ B @ C ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) )
        = ( c_Groups_Ozero__class_Ozero @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[43]) ).

thf(2381,axiom,
    ! [A: $i,B: $i] :
      ( ~ ( hBOOL @ B )
      | ( hBOOL @ ( hAPP @ ( hAPP @ c_fdisj @ A ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_c__fdisj__2) ).

thf(11768,plain,
    ! [A: $i,B: $i] :
      ( ~ ( hBOOL @ B )
      | ( hBOOL @ ( hAPP @ ( hAPP @ c_fdisj @ A ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2381]) ).

thf(554,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( c_Fun_Oinj__on @ G @ F @ E @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ G @ tc_HOL_Obool ) ) )
     => ( ( ( hAPP @ ( c_Map_Omap__of @ G @ D @ C ) @ B )
          = ( hAPP @ ( c_Option_Ooption_OSome @ D ) @ A ) )
       => ( ( hAPP @ ( c_Map_Omap__of @ F @ D @ ( hAPP @ ( hAPP @ ( c_List_Omap @ ( tc_prod @ G @ D ) @ ( tc_prod @ F @ D ) ) @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ G @ D @ ( tc_prod @ F @ D ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ F @ ( tc_fun @ D @ ( tc_prod @ F @ D ) ) @ G ) @ ( c_Product__Type_OPair @ F @ D ) ) @ E ) ) ) @ C ) ) @ ( hAPP @ E @ B ) )
          = ( hAPP @ ( c_Option_Ooption_OSome @ D ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_map__of__mapk__SomeI) ).

thf(4397,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( c_Fun_Oinj__on @ G @ F @ E @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ G @ tc_HOL_Obool ) ) )
     => ( ( ( hAPP @ ( c_Map_Omap__of @ G @ D @ C ) @ B )
          = ( hAPP @ ( c_Option_Ooption_OSome @ D ) @ A ) )
       => ( ( hAPP @ ( c_Map_Omap__of @ F @ D @ ( hAPP @ ( hAPP @ ( c_List_Omap @ ( tc_prod @ G @ D ) @ ( tc_prod @ F @ D ) ) @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ G @ D @ ( tc_prod @ F @ D ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ F @ ( tc_fun @ D @ ( tc_prod @ F @ D ) ) @ G ) @ ( c_Product__Type_OPair @ F @ D ) ) @ E ) ) ) @ C ) ) @ ( hAPP @ E @ B ) )
          = ( hAPP @ ( c_Option_Ooption_OSome @ D ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[554]) ).

thf(723,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ B ) @ A ) )
       => ( ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ B ) @ A )
          = ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_div__pos__pos__trivial) ).

thf(5002,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ B ) @ A ) )
       => ( ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ B ) @ A )
          = ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[723]) ).

thf(938,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( hAPP @ ( c_Int_Onumber__class_Onumber__of @ tc_Int_Oint ) @ B ) ) @ ( hAPP @ ( c_Int_Onumber__class_Onumber__of @ tc_Int_Oint ) @ A ) ) )
    <=> ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__eq__number__of__int__code) ).

thf(5794,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( hAPP @ ( c_Int_Onumber__class_Onumber__of @ tc_Int_Oint ) @ B ) ) @ ( hAPP @ ( c_Int_Onumber__class_Onumber__of @ tc_Int_Oint ) @ A ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ B ) @ A ) ) )
      & ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( hAPP @ ( c_Int_Onumber__class_Onumber__of @ tc_Int_Oint ) @ B ) ) @ ( hAPP @ ( c_Int_Onumber__class_Onumber__of @ tc_Int_Oint ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[938]) ).

thf(567,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( class_Groups_Ocomm__monoid__add @ F )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ E ) @ D ) )
       => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ E @ F ) @ C ) @ D )
          = ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ B @ F ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ E @ tc_HOL_Obool ) @ F @ B ) @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ E @ F ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ E @ tc_HOL_Obool ) @ ( tc_fun @ E @ tc_HOL_Obool ) @ B ) @ ( c_Set_OCollect @ E ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ E @ tc_HOL_Obool ) @ ( tc_fun @ E @ tc_HOL_Obool ) @ B ) @ ( hAPP @ ( c_COMBS @ E @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ E ) @ c_fconj ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ E @ ( tc_fun @ E @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( c_member @ E ) ) @ D ) ) ) ) @ ( hAPP @ ( c_COMBC @ E @ B @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ B @ ( tc_fun @ B @ tc_HOL_Obool ) @ E ) @ c_fequal ) @ A ) ) ) ) ) ) @ ( hAPP @ ( c_Set_Oimage @ E @ B @ A ) @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_setsum__image__gen) ).

thf(4434,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( class_Groups_Ocomm__monoid__add @ F )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ E ) @ D ) )
       => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ E @ F ) @ C ) @ D )
          = ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ B @ F ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ E @ tc_HOL_Obool ) @ F @ B ) @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ E @ F ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ E @ tc_HOL_Obool ) @ ( tc_fun @ E @ tc_HOL_Obool ) @ B ) @ ( c_Set_OCollect @ E ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ E @ tc_HOL_Obool ) @ ( tc_fun @ E @ tc_HOL_Obool ) @ B ) @ ( hAPP @ ( c_COMBS @ E @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ E ) @ c_fconj ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ E @ ( tc_fun @ E @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( c_member @ E ) ) @ D ) ) ) ) @ ( hAPP @ ( c_COMBC @ E @ B @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ B @ ( tc_fun @ B @ tc_HOL_Obool ) @ E ) @ c_fequal ) @ A ) ) ) ) ) ) @ ( hAPP @ ( c_Set_Oimage @ E @ B @ A ) @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[567]) ).

thf(1618,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Int_Oint ) @ D ) @ C ) )
     => ! [E: $i] :
          ( ! [F: $i] :
              ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Int_Oint ) @ F ) @ ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Int_Oint @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) @ C ) ) )
             => ! [G: $i] :
                  ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Int_Oint ) @ G ) @ B ) )
                 => ( E
                   != ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Int_Oint ) @ G ) @ F ) ) ) )
         => ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Int_Oint ) @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ E ) @ A ) ) )
           => ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Int_Oint ) @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ E ) @ C ) ) @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_aset_I10_J) ).

thf(8756,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Int_Oint ) @ D ) @ C ) )
     => ! [E: $i] :
          ( ! [F: $i] :
              ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Int_Oint ) @ F ) @ ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Int_Oint @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) @ C ) ) )
             => ! [G: $i] :
                  ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Int_Oint ) @ G ) @ B ) )
                 => ( E
                   != ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Int_Oint ) @ G ) @ F ) ) ) )
         => ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Int_Oint ) @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ E ) @ A ) ) )
           => ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Int_Oint ) @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ E ) @ C ) ) @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1618]) ).

thf(2208,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( G = F )
     => ( ( hAPP @ ( c_Sum__Type_Osum_Osum__case @ E @ D @ C @ B @ A ) @ G )
        = ( hAPP @ ( c_Sum__Type_Osum_Osum__case @ E @ D @ C @ B @ A ) @ F ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_sum__case__weak__cong) ).

thf(11037,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( G = F )
     => ( ( hAPP @ ( c_Sum__Type_Osum_Osum__case @ E @ D @ C @ B @ A ) @ G )
        = ( hAPP @ ( c_Sum__Type_Osum_Osum__case @ E @ D @ C @ B @ A ) @ F ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2208]) ).

thf(1077,axiom,
    ! [A: $i] :
      ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ A ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__not__refl) ).

thf(6371,plain,
    ! [A: $i] :
      ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ A ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1077]) ).

thf(2194,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i,I: $i] :
      ( ( c_Com_Ocom_Ocom__case @ I @ H @ G @ F @ E @ D @ C @ B @ A @ c_Com_Ocom_OSKIP )
      = H ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_com_Osimps_I64_J) ).

thf(10990,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i,I: $i] :
      ( ( c_Com_Ocom_Ocom__case @ I @ H @ G @ F @ E @ D @ C @ B @ A @ c_Com_Ocom_OSKIP )
      = H ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2194]) ).

thf(887,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ B ) )
     => ( ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ C @ tc_Nat_Onat ) @ A ) @ B )
          = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
      <=> ! [D: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ D ) @ B ) )
           => ( ( hAPP @ A @ D )
              = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_setsum__eq__0__iff) ).

thf(5579,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ B ) )
     => ( ( ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ C @ tc_Nat_Onat ) @ A ) @ B )
            = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
         => ! [D: $i] :
              ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ D ) @ B ) )
             => ( ( hAPP @ A @ D )
                = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) )
        & ( ! [D: $i] :
              ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ D ) @ B ) )
             => ( ( hAPP @ A @ D )
                = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) )
         => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ C @ tc_Nat_Onat ) @ A ) @ B )
            = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[887]) ).

thf(1103,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ ( tc_fun @ B @ A ) ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ ( tc_fun @ B @ A ) @ tc_HOL_Obool ) ) ) )
     => ( ( ( hAPP @ ( c_Finite__Set_Ocard @ A ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ A @ tc_HOL_Obool ) ) )
         != ( hAPP @ c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) )
       => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ B ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_finite__fun__UNIVD1) ).

thf(6528,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ ( tc_fun @ B @ A ) ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ ( tc_fun @ B @ A ) @ tc_HOL_Obool ) ) ) )
     => ( ( ( hAPP @ ( c_Finite__Set_Ocard @ A ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ A @ tc_HOL_Obool ) ) )
         != ( hAPP @ c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) )
       => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ B ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1103]) ).

thf(864,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ B ) @ C ) ) ) @ ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ B @ C ) ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ B ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zdiv__zmod__equality) ).

thf(5512,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ B ) @ C ) ) ) @ ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ B @ C ) ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ B ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[864]) ).

thf(690,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Onth @ D ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ D ) @ C ) @ B ) ) @ ( hAPP @ c_Nat_OSuc @ A ) )
      = ( hAPP @ ( hAPP @ ( c_List_Onth @ D ) @ B ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nth__Cons__Suc) ).

thf(4878,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Onth @ D ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ D ) @ C ) @ B ) ) @ ( hAPP @ c_Nat_OSuc @ A ) )
      = ( hAPP @ ( hAPP @ ( c_List_Onth @ D ) @ B ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[690]) ).

thf(1157,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) ) @ D ) @ C ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) ) @ D ) @ B ) ) @ c_FunDef_Opair__leq ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) ) @ C ) @ A ) ) @ c_FunDef_Omin__weak ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) ) @ B ) @ A ) ) ) @ c_FunDef_Omin__weak ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_wmin__insertI) ).

thf(6773,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) ) @ D ) @ C ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) ) @ D ) @ B ) ) @ c_FunDef_Opair__leq ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) ) @ C ) @ A ) ) @ c_FunDef_Omin__weak ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) ) @ B ) @ A ) ) ) @ c_FunDef_Omin__weak ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1157]) ).

thf(718,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ B ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ B ) @ ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ A @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_neg__mod__bound) ).

thf(4989,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ B ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ B ) @ ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ A @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[718]) ).

thf(1267,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Finite__Set_Ofolding__one @ E @ D @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ E ) @ B ) )
       => ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ A ) @ B ) )
         => ( ( hAPP @ C @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ A ) @ B ) )
            = ( hAPP @ ( hAPP @ ( c_Finite__Set_Ofold @ E @ E @ D ) @ A ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_folding__one_Oeq__fold_H) ).

thf(7229,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Finite__Set_Ofolding__one @ E @ D @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ E ) @ B ) )
       => ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ A ) @ B ) )
         => ( ( hAPP @ C @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ A ) @ B ) )
            = ( hAPP @ ( hAPP @ ( c_Finite__Set_Ofold @ E @ E @ D ) @ A ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1267]) ).

thf(1428,axiom,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ A ) )
     => ( ( hAPP @ c_Nat_OSuc @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ A ) @ ( hAPP @ c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) )
        = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Suc__pred) ).

thf(7881,plain,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ A ) )
     => ( ( hAPP @ c_Nat_OSuc @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ A ) @ ( hAPP @ c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1428]) ).

thf(815,axiom,
    ( ( hAPP @ ( c_Nat_Osize__class_Osize @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
    = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nat_Osize_I3_J) ).

thf(5349,plain,
    ( ( hAPP @ ( c_Nat_Osize__class_Osize @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
    = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[815]) ).

thf(1461,axiom,
    ! [A: $i] :
      ( ( A
       != ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
    <=> ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_neq0__conv) ).

thf(7994,plain,
    ! [A: $i] :
      ( ( ( A
         != ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ A ) ) )
      & ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ A ) )
       => ( A
         != ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1461]) ).

thf(792,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( c_Finite__Set_Ofolding__idem @ G @ F @ E @ D )
     => ( ( hAPP @ ( c_Fun_Ocomp @ F @ F @ C @ ( hAPP @ E @ B ) ) @ ( hAPP @ ( c_Fun_Ocomp @ F @ F @ C @ ( hAPP @ E @ B ) ) @ A ) )
        = ( hAPP @ ( c_Fun_Ocomp @ F @ F @ C @ ( hAPP @ E @ B ) ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_folding__idem_Oidem__left__comp) ).

thf(5266,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( c_Finite__Set_Ofolding__idem @ G @ F @ E @ D )
     => ( ( hAPP @ ( c_Fun_Ocomp @ F @ F @ C @ ( hAPP @ E @ B ) ) @ ( hAPP @ ( c_Fun_Ocomp @ F @ F @ C @ ( hAPP @ E @ B ) ) @ A ) )
        = ( hAPP @ ( c_Fun_Ocomp @ F @ F @ C @ ( hAPP @ E @ B ) ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[792]) ).

thf(2335,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Natural_Oevalc @ C ) @ B ) @ A ) )
    <=> ? [D: $i] : ( c_Natural_Oevaln @ C @ B @ D @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_eval__eq) ).

thf(11587,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Natural_Oevalc @ C ) @ B ) @ A ) )
       => ? [D: $i] : ( c_Natural_Oevaln @ C @ B @ D @ A ) )
      & ( ? [D: $i] : ( c_Natural_Oevaln @ C @ B @ D @ A )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Natural_Oevalc @ C ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2335]) ).

thf(171,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_Fun_Oinj__on @ D @ C @ B @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ D @ tc_HOL_Obool ) ) )
     => ( c_Fun_Oinj__on @ D @ ( tc_fun @ A @ C ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ C @ ( tc_fun @ A @ C ) @ D ) @ ( c_COMBK @ C @ A ) ) @ B ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_inj__fun) ).

thf(3022,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_Fun_Oinj__on @ D @ C @ B @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ D @ tc_HOL_Obool ) ) )
     => ( c_Fun_Oinj__on @ D @ ( tc_fun @ A @ C ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ C @ ( tc_fun @ A @ C ) @ D ) @ ( c_COMBK @ C @ A ) ) @ B ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[171]) ).

thf(945,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ tc_Nat_Onat @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ E ) ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ tc_Nat_Onat @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ E ) @ B ) ) ) @ A ) ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ tc_Nat_Onat @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ tc_Nat_Onat @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_UN__finite2__subset) ).

thf(5815,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ tc_Nat_Onat @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ E ) ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ tc_Nat_Onat @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ E ) @ B ) ) ) @ A ) ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ tc_Nat_Onat @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ tc_Nat_Onat @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[945]) ).

thf(1417,axiom,
    ! [A: $i,B: $i] :
      ( ? [C: $i] :
          ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ C ) @ B ) )
          & ( hBOOL @ ( hAPP @ A @ C ) ) )
    <=> ? [C: $i] :
          ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Nat_Onat ) @ C ) @ ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ B ) ) )
          & ( hBOOL @ ( hAPP @ A @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_ex__nat__less) ).

thf(7834,plain,
    ! [A: $i,B: $i] :
      ( ( ? [C: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ C ) @ B ) )
            & ( hBOOL @ ( hAPP @ A @ C ) ) )
       => ? [C: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Nat_Onat ) @ C ) @ ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ B ) ) )
            & ( hBOOL @ ( hAPP @ A @ C ) ) ) )
      & ( ? [C: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Nat_Onat ) @ C ) @ ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ B ) ) )
            & ( hBOOL @ ( hAPP @ A @ C ) ) )
       => ? [C: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ C ) @ B ) )
            & ( hBOOL @ ( hAPP @ A @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1417]) ).

thf(2251,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i] :
          ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_Hoare__Mirabelle_Otriple @ C ) ) @ D ) @ B ) )
         => ( c_Hoare__Mirabelle_Otriple__valid @ C @ ( hAPP @ c_Nat_OSuc @ A ) @ D ) )
     => ! [D: $i] :
          ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_Hoare__Mirabelle_Otriple @ C ) ) @ D ) @ B ) )
         => ( c_Hoare__Mirabelle_Otriple__valid @ C @ A @ D ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_triples__valid__Suc) ).

thf(11217,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i] :
          ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_Hoare__Mirabelle_Otriple @ C ) ) @ D ) @ B ) )
         => ( c_Hoare__Mirabelle_Otriple__valid @ C @ ( hAPP @ c_Nat_OSuc @ A ) @ D ) )
     => ! [D: $i] :
          ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_Hoare__Mirabelle_Otriple @ C ) ) @ D ) @ B ) )
         => ( c_Hoare__Mirabelle_Otriple__valid @ C @ A @ D ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2251]) ).

thf(1640,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) @ C ) @ B ) ) @ ( c_List_Olistrel1 @ D @ ( hAPP @ ( c_Relation_Oconverse @ D @ D ) @ A ) ) ) )
    <=> ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) @ C ) @ B ) ) @ ( hAPP @ ( c_Relation_Oconverse @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) @ ( c_List_Olistrel1 @ D @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_in__listrel1__converse) ).

thf(8856,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) @ C ) @ B ) ) @ ( c_List_Olistrel1 @ D @ ( hAPP @ ( c_Relation_Oconverse @ D @ D ) @ A ) ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) @ C ) @ B ) ) @ ( hAPP @ ( c_Relation_Oconverse @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) @ ( c_List_Olistrel1 @ D @ A ) ) ) ) )
      & ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) @ C ) @ B ) ) @ ( hAPP @ ( c_Relation_Oconverse @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) @ ( c_List_Olistrel1 @ D @ A ) ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) @ C ) @ B ) ) @ ( c_List_Olistrel1 @ D @ ( hAPP @ ( c_Relation_Oconverse @ D @ D ) @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1640]) ).

thf(1213,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Fun_Ocomp @ F @ E @ D @ C ) @ B ) @ A )
      = ( hAPP @ C @ ( hAPP @ B @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_o__apply) ).

thf(6990,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Fun_Ocomp @ F @ E @ D @ C ) @ B ) @ A )
      = ( hAPP @ C @ ( hAPP @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1213]) ).

thf(924,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ ( tc_prod @ H @ G ) @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ H @ G ) @ D ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ G @ tc_HOL_Obool ) @ H ) @ C ) ) ) @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ H @ G @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ H @ ( tc_fun @ G @ ( tc_fun @ F @ ( tc_fun @ E @ tc_HOL_Obool ) ) ) @ ( tc_fun @ G @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ ( tc_fun @ F @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ G @ ( tc_fun @ F @ ( tc_fun @ E @ tc_HOL_Obool ) ) ) @ ( tc_fun @ G @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) ) ) @ H ) @ ( c_COMBB @ ( tc_fun @ F @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) @ G ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ F @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ F @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) ) @ H ) @ ( c_Product__Type_OSigma @ F @ E ) ) @ B ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ E @ tc_HOL_Obool ) @ ( tc_fun @ F @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ G ) @ ( c_COMBK @ ( tc_fun @ E @ tc_HOL_Obool ) @ F ) ) @ A ) ) ) )
      = ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ F @ E ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ H @ ( tc_fun @ F @ tc_HOL_Obool ) ) @ D ) @ B ) ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ E @ tc_HOL_Obool ) @ F ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ G @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_UN__Times__distrib) ).

thf(5719,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ ( tc_prod @ H @ G ) @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ H @ G ) @ D ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ G @ tc_HOL_Obool ) @ H ) @ C ) ) ) @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ H @ G @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ H @ ( tc_fun @ G @ ( tc_fun @ F @ ( tc_fun @ E @ tc_HOL_Obool ) ) ) @ ( tc_fun @ G @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ ( tc_fun @ F @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ G @ ( tc_fun @ F @ ( tc_fun @ E @ tc_HOL_Obool ) ) ) @ ( tc_fun @ G @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) ) ) @ H ) @ ( c_COMBB @ ( tc_fun @ F @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) @ G ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ F @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ F @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) ) @ H ) @ ( c_Product__Type_OSigma @ F @ E ) ) @ B ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ E @ tc_HOL_Obool ) @ ( tc_fun @ F @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ G ) @ ( c_COMBK @ ( tc_fun @ E @ tc_HOL_Obool ) @ F ) ) @ A ) ) ) )
      = ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ F @ E ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ H @ ( tc_fun @ F @ tc_HOL_Obool ) ) @ D ) @ B ) ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ E @ tc_HOL_Obool ) @ F ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ G @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ C ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[924]) ).

thf(2102,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ ( hAPP @ c_Nat_OSuc @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_le__imp__less__Suc) ).

thf(10657,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ ( hAPP @ c_Nat_OSuc @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2102]) ).

thf(1585,axiom,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ A ) )
       => ( ( c_List_Oupt @ B @ ( hAPP @ c_Nat_OSuc @ A ) )
          = ( hAPP @ ( hAPP @ ( c_List_Oappend @ tc_Nat_Onat ) @ ( c_List_Oupt @ B @ A ) ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ tc_Nat_Onat ) @ A ) @ ( c_List_Olist_ONil @ tc_Nat_Onat ) ) ) ) )
      & ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ A ) )
       => ( ( c_List_Oupt @ B @ ( hAPP @ c_Nat_OSuc @ A ) )
          = ( c_List_Olist_ONil @ tc_Nat_Onat ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_upt__Suc) ).

thf(8608,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ A ) )
       => ( ( c_List_Oupt @ B @ ( hAPP @ c_Nat_OSuc @ A ) )
          = ( hAPP @ ( hAPP @ ( c_List_Oappend @ tc_Nat_Onat ) @ ( c_List_Oupt @ B @ A ) ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ tc_Nat_Onat ) @ A ) @ ( c_List_Olist_ONil @ tc_Nat_Onat ) ) ) ) )
      & ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ A ) )
       => ( ( c_List_Oupt @ B @ ( hAPP @ c_Nat_OSuc @ A ) )
          = ( c_List_Olist_ONil @ tc_Nat_Onat ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1585]) ).

thf(1111,axiom,
    ! [A: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ A )
      = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__0) ).

thf(6565,plain,
    ! [A: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ A )
      = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1111]) ).

thf(157,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ E ) ) @ D )
        = ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) @ B ) )
     => ( ( ( hAPP @ ( hAPP @ ( c_List_Ozip @ E @ C ) @ D ) @ B )
          = A )
      <=> ( ( ( hAPP @ ( hAPP @ ( c_List_Omap @ ( tc_prod @ E @ C ) @ E ) @ ( c_Product__Type_Ofst @ E @ C ) ) @ A )
            = D )
          & ( ( hAPP @ ( hAPP @ ( c_List_Omap @ ( tc_prod @ E @ C ) @ C ) @ ( c_Product__Type_Osnd @ E @ C ) ) @ A )
            = B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zip__eq__conv) ).

thf(2969,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ E ) ) @ D )
        = ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) @ B ) )
     => ( ( ( ( hAPP @ ( hAPP @ ( c_List_Ozip @ E @ C ) @ D ) @ B )
            = A )
         => ( ( ( hAPP @ ( hAPP @ ( c_List_Omap @ ( tc_prod @ E @ C ) @ E ) @ ( c_Product__Type_Ofst @ E @ C ) ) @ A )
              = D )
            & ( ( hAPP @ ( hAPP @ ( c_List_Omap @ ( tc_prod @ E @ C ) @ C ) @ ( c_Product__Type_Osnd @ E @ C ) ) @ A )
              = B ) ) )
        & ( ( ( ( hAPP @ ( hAPP @ ( c_List_Omap @ ( tc_prod @ E @ C ) @ E ) @ ( c_Product__Type_Ofst @ E @ C ) ) @ A )
              = D )
            & ( ( hAPP @ ( hAPP @ ( c_List_Omap @ ( tc_prod @ E @ C ) @ C ) @ ( c_Product__Type_Osnd @ E @ C ) ) @ A )
              = B ) )
         => ( ( hAPP @ ( hAPP @ ( c_List_Ozip @ E @ C ) @ D ) @ B )
            = A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[157]) ).

thf(474,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) @ ( hAPP @ ( c_List_Ofilter @ C @ B ) @ A ) )
      = ( hAPP @ ( c_Finite__Set_Ocard @ tc_Nat_Onat ) @ ( hAPP @ ( c_Set_OCollect @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ tc_Nat_Onat @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ tc_Nat_Onat ) @ c_fconj ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) @ A ) ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ C @ tc_HOL_Obool @ tc_Nat_Onat ) @ B ) @ ( hAPP @ ( c_List_Onth @ C ) @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_length__filter__conv__card) ).

thf(4091,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) @ ( hAPP @ ( c_List_Ofilter @ C @ B ) @ A ) )
      = ( hAPP @ ( c_Finite__Set_Ocard @ tc_Nat_Onat ) @ ( hAPP @ ( c_Set_OCollect @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ tc_Nat_Onat @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ tc_Nat_Onat ) @ c_fconj ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) @ A ) ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ C @ tc_HOL_Obool @ tc_Nat_Onat ) @ B ) @ ( hAPP @ ( c_List_Onth @ C ) @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[474]) ).

thf(377,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( c_Relation_Oconverse @ D @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ ( tc_prod @ C @ D ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Relation_Oconverse @ D @ C ) @ B ) ) @ ( hAPP @ ( c_Relation_Oconverse @ D @ C ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_converse__Int) ).

thf(3752,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( c_Relation_Oconverse @ D @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ ( tc_prod @ C @ D ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Relation_Oconverse @ D @ C ) @ B ) ) @ ( hAPP @ ( c_Relation_Oconverse @ D @ C ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[377]) ).

thf(2035,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C = B )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_dvd_Oord__eq__le__trans) ).

thf(10409,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C = B )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ C ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2035]) ).

thf(589,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Un__Diff) ).

thf(4521,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[589]) ).

thf(440,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ C @ B ) @ ( hAPP @ ( c_Product__Type_Ofst @ C @ B ) @ A ) ) @ ( hAPP @ ( c_Product__Type_Osnd @ C @ B ) @ A ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_pair__collapse) ).

thf(3989,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ C @ B ) @ ( hAPP @ ( c_Product__Type_Ofst @ C @ B ) @ A ) ) @ ( hAPP @ ( c_Product__Type_Osnd @ C @ B ) @ A ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[440]) ).

thf(271,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) @ A )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Un__empty__left) ).

thf(3371,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) @ A )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[271]) ).

thf(848,axiom,
    ! [A: $i] :
      ( ( c_Relation_OId__on @ A @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ A @ tc_HOL_Obool ) ) )
      = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ ( tc_prod @ A @ A ) @ tc_HOL_Obool ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Id__on__empty) ).

thf(5470,plain,
    ! [A: $i] :
      ( ( c_Relation_OId__on @ A @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ A @ tc_HOL_Obool ) ) )
      = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ ( tc_prod @ A @ A ) @ tc_HOL_Obool ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[848]) ).

thf(147,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ D @ C @ B ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ ( tc_prod @ C @ B ) @ tc_HOL_Obool ) ) )
      = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ ( tc_prod @ D @ B ) @ tc_HOL_Obool ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_rel__comp__empty2) ).

thf(2937,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ D @ C @ B ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ ( tc_prod @ C @ B ) @ tc_HOL_Obool ) ) )
      = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ ( tc_prod @ D @ B ) @ tc_HOL_Obool ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[147]) ).

thf(503,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_Fun_Oinj__on @ D @ C @ B @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ D @ tc_HOL_Obool ) ) )
     => ( ( hAPP @ ( c_Fun_Othe__inv__into @ D @ C @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ B @ A ) )
        = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_the__inv__f__f) ).

thf(4204,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_Fun_Oinj__on @ D @ C @ B @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ D @ tc_HOL_Obool ) ) )
     => ( ( hAPP @ ( c_Fun_Othe__inv__into @ D @ C @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ B @ A ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[503]) ).

thf(408,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ B ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ A ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zadd__commute) ).

thf(3884,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ B ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ A ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[408]) ).

thf(1955,axiom,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ tc_Nat_Onat ) @ ( c_SetInterval_Oord__class_OgreaterThanAtMost @ tc_Nat_Onat @ B @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_finite__greaterThanAtMost) ).

thf(10102,plain,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ tc_Nat_Onat ) @ ( c_SetInterval_Oord__class_OgreaterThanAtMost @ tc_Nat_Onat @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1955]) ).

thf(1364,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( c_Relation_ODomain @ C @ B ) @ ( hAPP @ ( c_Relation_Oconverse @ B @ C ) @ A ) )
      = ( hAPP @ ( c_Relation_ORange @ B @ C ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Domain__converse) ).

thf(7640,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( c_Relation_ODomain @ C @ B ) @ ( hAPP @ ( c_Relation_Oconverse @ B @ C ) @ A ) )
      = ( hAPP @ ( c_Relation_ORange @ B @ C ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1364]) ).

thf(2030,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ D ) @ C ) @ B ) ) @ ( hAPP @ ( c_List_Oset @ ( tc_prod @ E @ D ) ) @ A ) ) )
     => ? [F: $i] :
          ( ( hAPP @ ( c_Map_Omap__of @ E @ D @ A ) @ C )
          = ( hAPP @ ( c_Option_Ooption_OSome @ D ) @ F ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_weak__map__of__SomeI) ).

thf(10394,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ D ) @ C ) @ B ) ) @ ( hAPP @ ( c_List_Oset @ ( tc_prod @ E @ D ) ) @ A ) ) )
     => ? [F: $i] :
          ( ( hAPP @ ( c_Map_Omap__of @ E @ D @ A ) @ C )
          = ( hAPP @ ( c_Option_Ooption_OSome @ D ) @ F ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2030]) ).

thf(1952,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ D ) @ ( hAPP @ ( c_Relation_OImage @ C @ E @ B ) @ A ) ) )
    <=> ? [F: $i] :
          ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ F ) @ A ) )
          & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ C @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ C @ E ) @ F ) @ D ) ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Image__iff) ).

thf(10084,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ D ) @ ( hAPP @ ( c_Relation_OImage @ C @ E @ B ) @ A ) ) )
       => ? [F: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ F ) @ A ) )
            & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ C @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ C @ E ) @ F ) @ D ) ) @ B ) ) ) )
      & ( ? [F: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ F ) @ A ) )
            & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ C @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ C @ E ) @ F ) @ D ) ) @ B ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ D ) @ ( hAPP @ ( c_Relation_OImage @ C @ E @ B ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1952]) ).

thf(2331,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Natural_Oevaln @ ( c_Com_Ocom_OAss @ E @ D ) @ C @ B @ A )
     => ( A
        = ( hAPP @ ( hAPP @ ( hAPP @ c_Natural_Oupdate @ C ) @ E ) @ ( hAPP @ D @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_evaln__elim__cases_I2_J) ).

thf(11565,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Natural_Oevaln @ ( c_Com_Ocom_OAss @ E @ D ) @ C @ B @ A )
     => ( A
        = ( hAPP @ ( hAPP @ ( hAPP @ c_Natural_Oupdate @ C ) @ E ) @ ( hAPP @ D @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2331]) ).

thf(1092,axiom,
    ! [A: $i] :
      ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__zeroE) ).

thf(6496,plain,
    ! [A: $i] :
      ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1092]) ).

thf(673,axiom,
    ! [A: $i,B: $i] :
      ( ( c_HOL_OThe @ B @ ( hAPP @ ( hAPP @ ( c_COMBC @ B @ B @ tc_HOL_Obool ) @ c_fequal ) @ A ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_the__eq__trivial) ).

thf(4821,plain,
    ! [A: $i,B: $i] :
      ( ( c_HOL_OThe @ B @ ( hAPP @ ( hAPP @ ( c_COMBC @ B @ B @ tc_HOL_Obool ) @ c_fequal ) @ A ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[673]) ).

thf(826,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Fun_Oinj__on @ E @ D @ C @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ E @ tc_HOL_Obool ) ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Set_Oimage @ E @ D @ C ) @ B ) ) @ ( hAPP @ ( c_Set_Oimage @ E @ D @ C ) @ A ) ) )
      <=> ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_inj__image__subset__iff) ).

thf(5396,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Fun_Oinj__on @ E @ D @ C @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ E @ tc_HOL_Obool ) ) )
     => ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Set_Oimage @ E @ D @ C ) @ B ) ) @ ( hAPP @ ( c_Set_Oimage @ E @ D @ C ) @ A ) ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ B ) @ A ) ) )
        & ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ B ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Set_Oimage @ E @ D @ C ) @ B ) ) @ ( hAPP @ ( c_Set_Oimage @ E @ D @ C ) @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[826]) ).

thf(1134,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_Map_Omap__of @ D @ C @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ ( tc_prod @ D @ C ) ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ D @ ( tc_Option_Ooption @ C ) ) @ ( c_Map_Omap__of @ D @ C @ A ) ) @ ( hAPP @ ( c_Product__Type_Ofst @ D @ C ) @ B ) ) @ ( hAPP @ ( c_Option_Ooption_OSome @ C ) @ ( hAPP @ ( c_Product__Type_Osnd @ D @ C ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_map__of_Osimps_I2_J) ).

thf(6696,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_Map_Omap__of @ D @ C @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ ( tc_prod @ D @ C ) ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ D @ ( tc_Option_Ooption @ C ) ) @ ( c_Map_Omap__of @ D @ C @ A ) ) @ ( hAPP @ ( c_Product__Type_Ofst @ D @ C ) @ B ) ) @ ( hAPP @ ( c_Option_Ooption_OSome @ C ) @ ( hAPP @ ( c_Product__Type_Osnd @ D @ C ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1134]) ).

thf(2414,plain,
    ! [A: $i] :
      ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_Hoare__Mirabelle_Otriple @ t_a ) ) @ A ) @ v_G ) )
      | ( c_Hoare__Mirabelle_Otriple__valid @ t_a @ sk1 @ A ) ),
    inference(cnf,[status(esa)],[2410]) ).

thf(2045,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ c_Com_OWT @ B ) )
     => ( ( hBOOL @ ( hAPP @ c_Com_OWT @ A ) )
       => ( hBOOL @ ( hAPP @ c_Com_OWT @ ( c_Com_Ocom_OSemi @ B @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_WT_OSemi) ).

thf(10441,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ c_Com_OWT @ B ) )
     => ( ( hBOOL @ ( hAPP @ c_Com_OWT @ A ) )
       => ( hBOOL @ ( hAPP @ c_Com_OWT @ ( c_Com_Ocom_OSemi @ B @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2045]) ).

thf(679,axiom,
    ! [A: $i,B: $i,C: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) @ ( hAPP @ ( c_List_Ofilter @ C @ B ) @ A ) ) ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_length__filter__le) ).

thf(4836,plain,
    ! [A: $i,B: $i,C: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) @ ( hAPP @ ( c_List_Ofilter @ C @ B ) @ A ) ) ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[679]) ).

thf(891,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D = C )
    <=> ( ( ( hAPP @ ( c_Product__Type_Ofst @ B @ A ) @ D )
          = ( hAPP @ ( c_Product__Type_Ofst @ B @ A ) @ C ) )
        & ( ( hAPP @ ( c_Product__Type_Osnd @ B @ A ) @ D )
          = ( hAPP @ ( c_Product__Type_Osnd @ B @ A ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Pair__fst__snd__eq) ).

thf(5594,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( D = C )
       => ( ( ( hAPP @ ( c_Product__Type_Ofst @ B @ A ) @ D )
            = ( hAPP @ ( c_Product__Type_Ofst @ B @ A ) @ C ) )
          & ( ( hAPP @ ( c_Product__Type_Osnd @ B @ A ) @ D )
            = ( hAPP @ ( c_Product__Type_Osnd @ B @ A ) @ C ) ) ) )
      & ( ( ( ( hAPP @ ( c_Product__Type_Ofst @ B @ A ) @ D )
            = ( hAPP @ ( c_Product__Type_Ofst @ B @ A ) @ C ) )
          & ( ( hAPP @ ( c_Product__Type_Osnd @ B @ A ) @ D )
            = ( hAPP @ ( c_Product__Type_Osnd @ B @ A ) @ C ) ) )
       => ( D = C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[891]) ).

thf(490,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( c_Set_Oimage @ ( tc_prod @ D @ C ) @ ( tc_prod @ C @ D ) @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ D @ C @ ( tc_prod @ C @ D ) ) @ ( hAPP @ ( c_COMBC @ C @ D @ ( tc_prod @ C @ D ) ) @ ( c_Product__Type_OPair @ C @ D ) ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ D @ C ) @ B ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ C @ tc_HOL_Obool ) @ D ) @ A ) ) )
      = ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ C @ D ) @ A ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ D @ tc_HOL_Obool ) @ C ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_swap__product) ).

thf(4160,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( c_Set_Oimage @ ( tc_prod @ D @ C ) @ ( tc_prod @ C @ D ) @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ D @ C @ ( tc_prod @ C @ D ) ) @ ( hAPP @ ( c_COMBC @ C @ D @ ( tc_prod @ C @ D ) ) @ ( c_Product__Type_OPair @ C @ D ) ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ D @ C ) @ B ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ C @ tc_HOL_Obool ) @ D ) @ A ) ) )
      = ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ C @ D ) @ A ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ D @ tc_HOL_Obool ) @ C ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[490]) ).

thf(38,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Ocomm__monoid__add @ C )
     => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ B @ C ) @ ( hAPP @ ( c_COMBK @ C @ B ) @ ( c_Groups_Ozero__class_Ozero @ C ) ) ) @ A )
        = ( c_Groups_Ozero__class_Ozero @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_setsum__0) ).

thf(2528,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Ocomm__monoid__add @ C )
     => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ B @ C ) @ ( hAPP @ ( c_COMBK @ C @ B ) @ ( c_Groups_Ozero__class_Ozero @ C ) ) ) @ A )
        = ( c_Groups_Ozero__class_Ozero @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[38]) ).

thf(1041,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Nat_Onat ) @ C ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Nat_Onat ) @ B ) @ A ) ) ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( c_Divides_Odiv__class_Omod @ tc_Nat_Onat @ C @ A ) ) @ ( c_Divides_Odiv__class_Omod @ tc_Nat_Onat @ B @ A ) ) ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_div__add1__eq) ).

thf(6232,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Nat_Onat ) @ C ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Nat_Onat ) @ B ) @ A ) ) ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( c_Divides_Odiv__class_Omod @ tc_Nat_Onat @ C @ A ) ) @ ( c_Divides_Odiv__class_Omod @ tc_Nat_Onat @ B @ A ) ) ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1041]) ).

thf(2245,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( c_Nitpick_Opair__box_Opair__box__rec @ F @ E @ D @ C @ ( c_Nitpick_Opair__box_OPairBox @ F @ E @ B @ A ) )
      = ( hAPP @ ( hAPP @ C @ B ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_pair__box_Orecs) ).

thf(11191,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( c_Nitpick_Opair__box_Opair__box__rec @ F @ E @ D @ C @ ( c_Nitpick_Opair__box_OPairBox @ F @ E @ B @ A ) )
      = ( hAPP @ ( hAPP @ C @ B ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2245]) ).

thf(1647,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ A ) ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ C ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_dvd__mult__cancel) ).

thf(8878,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ A ) ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ C ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1647]) ).

thf(223,axiom,
    class_Rings_Olinordered__semiring__1__strict @ tc_Int_Oint,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Olinordered__semiring__1__strict) ).

thf(3207,plain,
    class_Rings_Olinordered__semiring__1__strict @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[223]) ).

thf(359,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ H @ G @ F ) @ E ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ D @ C @ ( tc_prod @ H @ G ) ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ D @ C @ F ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ ( tc_prod @ H @ G ) ) @ ( tc_fun @ C @ F ) @ D ) @ ( hAPP @ ( c_COMBB @ ( tc_prod @ H @ G ) @ F @ C ) @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ H @ G @ F ) @ E ) ) ) @ B ) ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_split__twice) ).

thf(3682,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ H @ G @ F ) @ E ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ D @ C @ ( tc_prod @ H @ G ) ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ D @ C @ F ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ ( tc_prod @ H @ G ) ) @ ( tc_fun @ C @ F ) @ D ) @ ( hAPP @ ( c_COMBB @ ( tc_prod @ H @ G ) @ F @ C ) @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ H @ G @ F ) @ E ) ) ) @ B ) ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[359]) ).

thf(1931,axiom,
    ! [A: $i] :
      ( c_Com_Ocom_OSKIP
     != ( hAPP @ c_Com_Ocom_OBODY @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_com_Osimps_I18_J) ).

thf(9990,plain,
    ! [A: $i] :
      ( c_Com_Ocom_OSKIP
     != ( hAPP @ c_Com_Ocom_OBODY @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1931]) ).

thf(9991,plain,
    ~ ? [A: $i] :
        ( c_Com_Ocom_OSKIP
        = ( hAPP @ c_Com_Ocom_OBODY @ A ) ),
    inference(miniscope,[status(thm)],[9990]) ).

thf(9992,plain,
    ! [A: $i] :
      ( c_Com_Ocom_OSKIP
     != ( hAPP @ c_Com_Ocom_OBODY @ A ) ),
    inference(cnf,[status(esa)],[9991]) ).

thf(9993,plain,
    ! [A: $i] :
      ( ( hAPP @ c_Com_Ocom_OBODY @ A )
     != c_Com_Ocom_OSKIP ),
    inference(lifteq,[status(thm)],[9992]) ).

thf(247,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ B ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Set_OCollect @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ ( tc_fun @ B @ tc_HOL_Obool ) @ ( tc_fun @ B @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_finite__Collect__subsets) ).

thf(3286,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ B ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Set_OCollect @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ ( tc_fun @ B @ tc_HOL_Obool ) @ ( tc_fun @ B @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[247]) ).

thf(1716,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_fun @ C @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( c_Predicate_OPowp @ C @ B ) ) @ ( c_Predicate_OPowp @ C @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Powp__mono) ).

thf(9158,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_fun @ C @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( c_Predicate_OPowp @ C @ B ) ) @ ( c_Predicate_OPowp @ C @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1716]) ).

thf(950,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) )
     => ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A )
        = B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Int__absorb2) ).

thf(5843,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) )
     => ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A )
        = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[950]) ).

thf(1121,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ A ) ) )
    <=> ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nat__add__left__cancel__less) ).

thf(6620,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ A ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) ) )
      & ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1121]) ).

thf(1118,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ E ) @ D ) )
     => ( ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ D ) @ C ) ) @ B )
          = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ E ) @ C ) ) @ A ) )
      <=> ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ D ) @ E ) ) @ C ) ) @ B )
          = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nat__eq__add__iff1) ).

thf(6582,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ E ) @ D ) )
     => ( ( ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ D ) @ C ) ) @ B )
            = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ E ) @ C ) ) @ A ) )
         => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ D ) @ E ) ) @ C ) ) @ B )
            = A ) )
        & ( ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ D ) @ E ) ) @ C ) ) @ B )
            = A )
         => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ D ) @ C ) ) @ B )
            = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ E ) @ C ) ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1118]) ).

thf(1589,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ C ) )
     => ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ B ) ) @ A ) )
          & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ ( hAPP @ c_Nat_OSuc @ B ) ) ) ) )
      <=> ( B
          = ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Nat_Onat ) @ A ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_split__div__lemma) ).

thf(8623,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ C ) )
     => ( ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ B ) ) @ A ) )
            & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ ( hAPP @ c_Nat_OSuc @ B ) ) ) ) )
         => ( B
            = ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Nat_Onat ) @ A ) @ C ) ) )
        & ( ( B
            = ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Nat_Onat ) @ A ) @ C ) )
         => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ B ) ) @ A ) )
            & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ ( hAPP @ c_Nat_OSuc @ B ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1589]) ).

thf(1796,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ D ) @ C ) ) @ ( c_Transitive__Closure_Ortrancl @ E @ B ) ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ C ) @ A ) ) @ ( c_Transitive__Closure_Ortrancl @ E @ B ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ D ) @ A ) ) @ ( c_Transitive__Closure_Ortrancl @ E @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_rtrancl__trans) ).

thf(9455,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ D ) @ C ) ) @ ( c_Transitive__Closure_Ortrancl @ E @ B ) ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ C ) @ A ) ) @ ( c_Transitive__Closure_Ortrancl @ E @ B ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ D ) @ A ) ) @ ( c_Transitive__Closure_Ortrancl @ E @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1796]) ).

thf(1418,axiom,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ ( hAPP @ ( c_Int_Onumber__class_Onumber__of @ tc_Nat_Onat ) @ A ) ) )
     => ( ( hAPP @ ( c_Int_Onumber__class_Onumber__of @ tc_Nat_Onat ) @ A )
        = ( hAPP @ c_Nat_OSuc @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ ( hAPP @ ( c_Int_Onumber__class_Onumber__of @ tc_Nat_Onat ) @ A ) ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_expand__Suc) ).

thf(7842,plain,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ ( hAPP @ ( c_Int_Onumber__class_Onumber__of @ tc_Nat_Onat ) @ A ) ) )
     => ( ( hAPP @ ( c_Int_Onumber__class_Onumber__of @ tc_Nat_Onat ) @ A )
        = ( hAPP @ c_Nat_OSuc @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ ( hAPP @ ( c_Int_Onumber__class_Onumber__of @ tc_Nat_Onat ) @ A ) ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1418]) ).

thf(1512,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
    ? [E: $i] :
      ( ( c_Fun_Oinj__on @ D @ ( tc_prod @ C @ D ) @ E @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ C @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) )
      & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ C @ D ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Set_Oimage @ D @ ( tc_prod @ C @ D ) @ E ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ C @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ C @ D ) @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Ex__inj__on__UNION__Sigma) ).

thf(8270,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
    ? [E: $i] :
      ( ( c_Fun_Oinj__on @ D @ ( tc_prod @ C @ D ) @ E @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ C @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) )
      & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ C @ D ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Set_Oimage @ D @ ( tc_prod @ C @ D ) @ E ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ C @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ C @ D ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1512]) ).

thf(998,axiom,
    c_FunDef_Oreduction__pair @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_fun @ ( tc_prod @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) ) @ tc_HOL_Obool ) ) @ c_FunDef_Omax__strict ) @ c_FunDef_Omax__weak ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_max__rpair__set) ).

thf(6043,plain,
    c_FunDef_Oreduction__pair @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_fun @ ( tc_prod @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) ) @ tc_HOL_Obool ) ) @ c_FunDef_Omax__strict ) @ c_FunDef_Omax__weak ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[998]) ).

thf(1832,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Otake @ E ) @ D ) @ ( hAPP @ ( hAPP @ ( c_List_Omap @ C @ E ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_List_Omap @ C @ E ) @ B ) @ ( hAPP @ ( hAPP @ ( c_List_Otake @ C ) @ D ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_take__map) ).

thf(9586,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Otake @ E ) @ D ) @ ( hAPP @ ( hAPP @ ( c_List_Omap @ C @ E ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_List_Omap @ C @ E ) @ B ) @ ( hAPP @ ( hAPP @ ( c_List_Otake @ C ) @ D ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1832]) ).

thf(2078,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_List_Odistinct @ C ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ C ) @ B ) @ A ) ) )
    <=> ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ B ) @ ( hAPP @ ( c_List_Oset @ C ) @ A ) ) )
        & ( hBOOL @ ( hAPP @ ( c_List_Odistinct @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_distinct_Osimps_I2_J) ).

thf(10583,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( c_List_Odistinct @ C ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ C ) @ B ) @ A ) ) )
       => ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ B ) @ ( hAPP @ ( c_List_Oset @ C ) @ A ) ) )
          & ( hBOOL @ ( hAPP @ ( c_List_Odistinct @ C ) @ A ) ) ) )
      & ( ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ B ) @ ( hAPP @ ( c_List_Oset @ C ) @ A ) ) )
          & ( hBOOL @ ( hAPP @ ( c_List_Odistinct @ C ) @ A ) ) )
       => ( hBOOL @ ( hAPP @ ( c_List_Odistinct @ C ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ C ) @ B ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2078]) ).

thf(1021,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ E @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ E @ ( tc_fun @ D @ tc_HOL_Obool ) @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ D @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ D @ tc_HOL_Obool ) @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ E ) @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) @ B ) ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ ( tc_prod @ E @ D ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ E @ D ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ E @ D ) @ C ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Sigma__Int__distrib2) ).

thf(6143,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ E @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ E @ ( tc_fun @ D @ tc_HOL_Obool ) @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ D @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ D @ tc_HOL_Obool ) @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ E ) @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) @ B ) ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ ( tc_prod @ E @ D ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ E @ D ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ E @ D ) @ C ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1021]) ).

thf(497,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ B ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Set_OCollect @ C ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ C @ C @ tc_HOL_Obool ) @ c_fequal ) @ B ) ) ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_insert__def) ).

thf(4187,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ B ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Set_OCollect @ C ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ C @ C @ tc_HOL_Obool ) @ c_fequal ) @ B ) ) ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[497]) ).

thf(714,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ B ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Nat__Transfer_Otransfer__nat__int__function__closures_I1_J) ).

thf(4979,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ B ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[714]) ).

thf(882,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ A ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ B @ tc_HOL_Obool ) ) )
      = ( c_Orderings_Otop__class_Otop @ ( tc_fun @ B @ tc_HOL_Obool ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Un__UNIV__right) ).

thf(5566,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ A ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ B @ tc_HOL_Obool ) ) )
      = ( c_Orderings_Otop__class_Otop @ ( tc_fun @ B @ tc_HOL_Obool ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[882]) ).

thf(153,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Ozip @ C @ B ) @ ( hAPP @ ( hAPP @ ( c_List_Omap @ ( tc_prod @ C @ B ) @ C ) @ ( c_Product__Type_Ofst @ C @ B ) ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_List_Omap @ ( tc_prod @ C @ B ) @ B ) @ ( c_Product__Type_Osnd @ C @ B ) ) @ A ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zip__map__fst__snd) ).

thf(2958,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Ozip @ C @ B ) @ ( hAPP @ ( hAPP @ ( c_List_Omap @ ( tc_prod @ C @ B ) @ C ) @ ( c_Product__Type_Ofst @ C @ B ) ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_List_Omap @ ( tc_prod @ C @ B ) @ B ) @ ( c_Product__Type_Osnd @ C @ B ) ) @ A ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[153]) ).

thf(1237,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_add__mult__distrib) ).

thf(7097,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1237]) ).

thf(255,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ B ) @ A ) ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_int__div__less__self) ).

thf(3307,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ B ) @ A ) ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[255]) ).

thf(243,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_IntD1) ).

thf(3271,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[243]) ).

thf(1270,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_List_Odistinct @ C ) @ ( hAPP @ ( hAPP @ ( c_List_Omap @ ( tc_prod @ C @ B ) @ C ) @ ( c_Product__Type_Ofst @ C @ B ) ) @ A ) ) )
     => ( ( hAPP @ ( c_List_Oset @ ( tc_prod @ C @ B ) ) @ A )
        = ( hAPP @ ( c_Set_OCollect @ ( tc_prod @ C @ B ) ) @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ C @ B @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ C @ ( tc_fun @ B @ ( tc_Option_Ooption @ B ) ) @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ ( tc_Option_Ooption @ B ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ B @ ( tc_Option_Ooption @ B ) ) @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ C ) @ ( c_COMBB @ ( tc_Option_Ooption @ B ) @ tc_HOL_Obool @ B ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_Option_Ooption @ B ) @ ( tc_fun @ ( tc_Option_Ooption @ B ) @ tc_HOL_Obool ) @ C ) @ c_fequal ) @ ( c_Map_Omap__of @ C @ B @ A ) ) ) ) @ ( c_Option_Ooption_OSome @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_set__map__of__compr) ).

thf(7237,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_List_Odistinct @ C ) @ ( hAPP @ ( hAPP @ ( c_List_Omap @ ( tc_prod @ C @ B ) @ C ) @ ( c_Product__Type_Ofst @ C @ B ) ) @ A ) ) )
     => ( ( hAPP @ ( c_List_Oset @ ( tc_prod @ C @ B ) ) @ A )
        = ( hAPP @ ( c_Set_OCollect @ ( tc_prod @ C @ B ) ) @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ C @ B @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ C @ ( tc_fun @ B @ ( tc_Option_Ooption @ B ) ) @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ ( tc_Option_Ooption @ B ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ B @ ( tc_Option_Ooption @ B ) ) @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ C ) @ ( c_COMBB @ ( tc_Option_Ooption @ B ) @ tc_HOL_Obool @ B ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_Option_Ooption @ B ) @ ( tc_fun @ ( tc_Option_Ooption @ B ) @ tc_HOL_Obool ) @ C ) @ c_fequal ) @ ( c_Map_Omap__of @ C @ B @ A ) ) ) ) @ ( c_Option_Ooption_OSome @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1270]) ).

thf(1124,axiom,
    ! [A: $i,B: $i,C: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_List_Oset @ C ) @ ( c_List_Osublist @ C @ B @ A ) ) ) @ ( hAPP @ ( c_List_Oset @ C ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_set__sublist__subset) ).

thf(6646,plain,
    ! [A: $i,B: $i,C: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_List_Oset @ C ) @ ( c_List_Osublist @ C @ B @ A ) ) ) @ ( hAPP @ ( c_List_Oset @ C ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1124]) ).

thf(1632,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) )
    <=> ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) )
        & ( B != A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_psubset__eq) ).

thf(8810,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) )
          & ( B != A ) ) )
      & ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) )
          & ( B != A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1632]) ).

thf(824,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( c_Finite__Set_Ocard @ B ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ B @ tc_Nat_Onat ) @ ( hAPP @ ( c_COMBK @ tc_Nat_Onat @ B ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_card__eq__setsum) ).

thf(5391,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( c_Finite__Set_Ocard @ B ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ B @ tc_Nat_Onat ) @ ( hAPP @ ( c_COMBK @ tc_Nat_Onat @ B ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[824]) ).

thf(1508,axiom,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ ( c_Int_Onumber__class_Onumber__of @ tc_Nat_Onat ) @ B ) ) @ ( hAPP @ ( c_Int_Onumber__class_Onumber__of @ tc_Nat_Onat ) @ A ) ) )
       => ( ( c_List_Oupt @ ( hAPP @ ( c_Int_Onumber__class_Onumber__of @ tc_Nat_Onat ) @ B ) @ ( hAPP @ ( c_Int_Onumber__class_Onumber__of @ tc_Nat_Onat ) @ A ) )
          = ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ tc_Nat_Onat ) @ ( hAPP @ ( c_Int_Onumber__class_Onumber__of @ tc_Nat_Onat ) @ B ) ) @ ( c_List_Oupt @ ( hAPP @ c_Nat_OSuc @ ( hAPP @ ( c_Int_Onumber__class_Onumber__of @ tc_Nat_Onat ) @ B ) ) @ ( hAPP @ ( c_Int_Onumber__class_Onumber__of @ tc_Nat_Onat ) @ A ) ) ) ) )
      & ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ ( c_Int_Onumber__class_Onumber__of @ tc_Nat_Onat ) @ B ) ) @ ( hAPP @ ( c_Int_Onumber__class_Onumber__of @ tc_Nat_Onat ) @ A ) ) )
       => ( ( c_List_Oupt @ ( hAPP @ ( c_Int_Onumber__class_Onumber__of @ tc_Nat_Onat ) @ B ) @ ( hAPP @ ( c_Int_Onumber__class_Onumber__of @ tc_Nat_Onat ) @ A ) )
          = ( c_List_Olist_ONil @ tc_Nat_Onat ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_upt__rec__number__of) ).

thf(8255,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ ( c_Int_Onumber__class_Onumber__of @ tc_Nat_Onat ) @ B ) ) @ ( hAPP @ ( c_Int_Onumber__class_Onumber__of @ tc_Nat_Onat ) @ A ) ) )
       => ( ( c_List_Oupt @ ( hAPP @ ( c_Int_Onumber__class_Onumber__of @ tc_Nat_Onat ) @ B ) @ ( hAPP @ ( c_Int_Onumber__class_Onumber__of @ tc_Nat_Onat ) @ A ) )
          = ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ tc_Nat_Onat ) @ ( hAPP @ ( c_Int_Onumber__class_Onumber__of @ tc_Nat_Onat ) @ B ) ) @ ( c_List_Oupt @ ( hAPP @ c_Nat_OSuc @ ( hAPP @ ( c_Int_Onumber__class_Onumber__of @ tc_Nat_Onat ) @ B ) ) @ ( hAPP @ ( c_Int_Onumber__class_Onumber__of @ tc_Nat_Onat ) @ A ) ) ) ) )
      & ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ ( c_Int_Onumber__class_Onumber__of @ tc_Nat_Onat ) @ B ) ) @ ( hAPP @ ( c_Int_Onumber__class_Onumber__of @ tc_Nat_Onat ) @ A ) ) )
       => ( ( c_List_Oupt @ ( hAPP @ ( c_Int_Onumber__class_Onumber__of @ tc_Nat_Onat ) @ B ) @ ( hAPP @ ( c_Int_Onumber__class_Onumber__of @ tc_Nat_Onat ) @ A ) )
          = ( c_List_Olist_ONil @ tc_Nat_Onat ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1508]) ).

thf(919,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_List_Osublist @ D @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ D ) @ C ) @ B ) @ A )
      = ( hAPP @ ( hAPP @ ( c_List_Oappend @ D ) @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_If @ ( tc_List_Olist @ D ) ) @ ( hAPP @ ( hAPP @ ( c_member @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ D ) @ C ) @ ( c_List_Olist_ONil @ D ) ) ) @ ( c_List_Olist_ONil @ D ) ) ) @ ( c_List_Osublist @ D @ B @ ( hAPP @ ( c_Set_OCollect @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_Nat_Onat @ ( tc_fun @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ tc_Nat_Onat ) @ ( c_member @ tc_Nat_Onat ) ) @ c_Nat_OSuc ) ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_sublist__Cons) ).

thf(5702,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_List_Osublist @ D @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ D ) @ C ) @ B ) @ A )
      = ( hAPP @ ( hAPP @ ( c_List_Oappend @ D ) @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_If @ ( tc_List_Olist @ D ) ) @ ( hAPP @ ( hAPP @ ( c_member @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ D ) @ C ) @ ( c_List_Olist_ONil @ D ) ) ) @ ( c_List_Olist_ONil @ D ) ) ) @ ( c_List_Osublist @ D @ B @ ( hAPP @ ( c_Set_OCollect @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_Nat_Onat @ ( tc_fun @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ tc_Nat_Onat ) @ ( c_member @ tc_Nat_Onat ) ) @ c_Nat_OSuc ) ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[919]) ).

thf(438,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hAPP @ ( c_Relation_ORange @ C @ B ) @ A )
        = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) )
    <=> ( A
        = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ ( tc_prod @ C @ B ) @ tc_HOL_Obool ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Range__empty__iff) ).

thf(3980,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( hAPP @ ( c_Relation_ORange @ C @ B ) @ A )
          = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) )
       => ( A
          = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ ( tc_prod @ C @ B ) @ tc_HOL_Obool ) ) ) )
      & ( ( A
          = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ ( tc_prod @ C @ B ) @ tc_HOL_Obool ) ) )
       => ( ( hAPP @ ( c_Relation_ORange @ C @ B ) @ A )
          = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[438]) ).

thf(2095,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ B ) ) @ ( hAPP @ c_Nat_OSuc @ A ) ) )
    <=> ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Suc__less__eq) ).

thf(10634,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ B ) ) @ ( hAPP @ c_Nat_OSuc @ A ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) ) )
      & ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ B ) ) @ ( hAPP @ c_Nat_OSuc @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2095]) ).

thf(976,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) ) @ A ) )
    <=> ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Diff__subset__conv) ).

thf(5937,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) ) ) )
      & ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[976]) ).

thf(2185,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ ( tc_prod @ B @ B ) ) @ A ) )
     => ( ( c_Wellfounded_Oacyclic @ B @ A )
       => ( c_Wellfounded_Owf @ B @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_finite__acyclic__wf) ).

thf(10939,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ ( tc_prod @ B @ B ) ) @ A ) )
     => ( ( c_Wellfounded_Oacyclic @ B @ A )
       => ( c_Wellfounded_Owf @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2185]) ).

thf(1636,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Finite__Set_Ofold @ E @ D @ C ) @ B ) @ A )
      = ( c_HOL_OThe @ D @ ( c_Finite__Set_Ofold__graph @ E @ D @ C @ B @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_fold__def) ).

thf(8832,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Finite__Set_Ofold @ E @ D @ C ) @ B ) @ A )
      = ( c_HOL_OThe @ D @ ( c_Finite__Set_Ofold__graph @ E @ D @ C @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1636]) ).

thf(2132,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( hAPP @ ( c_Map_Omap__add @ F @ E @ D @ C ) @ B )
        = ( hAPP @ ( c_Option_Ooption_OSome @ E ) @ A ) )
    <=> ( ( ( hAPP @ C @ B )
          = ( hAPP @ ( c_Option_Ooption_OSome @ E ) @ A ) )
        | ( ( ( hAPP @ C @ B )
            = ( c_Option_Ooption_ONone @ E ) )
          & ( ( hAPP @ D @ B )
            = ( hAPP @ ( c_Option_Ooption_OSome @ E ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_map__add__Some__iff) ).

thf(10756,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( ( hAPP @ ( c_Map_Omap__add @ F @ E @ D @ C ) @ B )
          = ( hAPP @ ( c_Option_Ooption_OSome @ E ) @ A ) )
       => ( ( ( hAPP @ C @ B )
            = ( hAPP @ ( c_Option_Ooption_OSome @ E ) @ A ) )
          | ( ( ( hAPP @ C @ B )
              = ( c_Option_Ooption_ONone @ E ) )
            & ( ( hAPP @ D @ B )
              = ( hAPP @ ( c_Option_Ooption_OSome @ E ) @ A ) ) ) ) )
      & ( ( ( ( hAPP @ C @ B )
            = ( hAPP @ ( c_Option_Ooption_OSome @ E ) @ A ) )
          | ( ( ( hAPP @ C @ B )
              = ( c_Option_Ooption_ONone @ E ) )
            & ( ( hAPP @ D @ B )
              = ( hAPP @ ( c_Option_Ooption_OSome @ E ) @ A ) ) ) )
       => ( ( hAPP @ ( c_Map_Omap__add @ F @ E @ D @ C ) @ B )
          = ( hAPP @ ( c_Option_Ooption_OSome @ E ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2132]) ).

thf(905,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( c_Finite__Set_Ofolding__idem @ F @ E @ D @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ F ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ F ) @ A ) @ B ) )
         => ( ( hAPP @ ( c_Fun_Ocomp @ E @ E @ E @ ( hAPP @ C @ B ) ) @ ( hAPP @ D @ A ) )
            = ( hAPP @ C @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_folding__idem_Oin__comp__idem) ).

thf(5655,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( c_Finite__Set_Ofolding__idem @ F @ E @ D @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ F ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ F ) @ A ) @ B ) )
         => ( ( hAPP @ ( c_Fun_Ocomp @ E @ E @ E @ ( hAPP @ C @ B ) ) @ ( hAPP @ D @ A ) )
            = ( hAPP @ C @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[905]) ).

thf(2061,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ C ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_insert__commute) ).

thf(10502,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ C ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2061]) ).

thf(2057,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ B ) @ A ) ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ C ) @ A ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ A ) @ B ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ C ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_dvd__diffD) ).

thf(10490,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ B ) @ A ) ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ C ) @ A ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ A ) @ B ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ C ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2057]) ).

thf(1727,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ G ) @ F ) @ ( hAPP @ ( c_List_Oset @ G ) @ ( hAPP @ ( hAPP @ ( c_List_Otake @ G ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ E ) ) @ D ) ) @ C ) ) ) )
       => ( ( c_Map_Omap__upds @ G @ E @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ G @ ( tc_Option_Ooption @ E ) ) @ B ) @ F ) @ ( hAPP @ ( c_Option_Ooption_OSome @ E ) @ A ) ) @ C @ D )
          = ( c_Map_Omap__upds @ G @ E @ B @ C @ D ) ) )
      & ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ G ) @ F ) @ ( hAPP @ ( c_List_Oset @ G ) @ ( hAPP @ ( hAPP @ ( c_List_Otake @ G ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ E ) ) @ D ) ) @ C ) ) ) )
       => ( ( c_Map_Omap__upds @ G @ E @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ G @ ( tc_Option_Ooption @ E ) ) @ B ) @ F ) @ ( hAPP @ ( c_Option_Ooption_OSome @ E ) @ A ) ) @ C @ D )
          = ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ G @ ( tc_Option_Ooption @ E ) ) @ ( c_Map_Omap__upds @ G @ E @ B @ C @ D ) ) @ F ) @ ( hAPP @ ( c_Option_Ooption_OSome @ E ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_map__upd__upds__conv__if) ).

thf(9190,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ G ) @ F ) @ ( hAPP @ ( c_List_Oset @ G ) @ ( hAPP @ ( hAPP @ ( c_List_Otake @ G ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ E ) ) @ D ) ) @ C ) ) ) )
       => ( ( c_Map_Omap__upds @ G @ E @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ G @ ( tc_Option_Ooption @ E ) ) @ B ) @ F ) @ ( hAPP @ ( c_Option_Ooption_OSome @ E ) @ A ) ) @ C @ D )
          = ( c_Map_Omap__upds @ G @ E @ B @ C @ D ) ) )
      & ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ G ) @ F ) @ ( hAPP @ ( c_List_Oset @ G ) @ ( hAPP @ ( hAPP @ ( c_List_Otake @ G ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ E ) ) @ D ) ) @ C ) ) ) )
       => ( ( c_Map_Omap__upds @ G @ E @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ G @ ( tc_Option_Ooption @ E ) ) @ B ) @ F ) @ ( hAPP @ ( c_Option_Ooption_OSome @ E ) @ A ) ) @ C @ D )
          = ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ G @ ( tc_Option_Ooption @ E ) ) @ ( c_Map_Omap__upds @ G @ E @ B @ C @ D ) ) @ F ) @ ( hAPP @ ( c_Option_Ooption_OSome @ E ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1727]) ).

thf(2050,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ C ) )
     => ( ( ( hAPP @ ( c_Finite__Set_Ocard @ B ) @ ( hAPP @ ( c_Set_Oimage @ D @ B @ A ) @ C ) )
          = ( hAPP @ ( c_Finite__Set_Ocard @ D ) @ C ) )
       => ( c_Fun_Oinj__on @ D @ B @ A @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_eq__card__imp__inj__on) ).

thf(10462,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ C ) )
     => ( ( ( hAPP @ ( c_Finite__Set_Ocard @ B ) @ ( hAPP @ ( c_Set_Oimage @ D @ B @ A ) @ C ) )
          = ( hAPP @ ( c_Finite__Set_Ocard @ D ) @ C ) )
       => ( c_Fun_Oinj__on @ D @ B @ A @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2050]) ).

thf(1666,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ C ) @ B ) )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) ) @ C )
        = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ B ) @ C ) ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_diff__add__assoc2) ).

thf(8942,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ C ) @ B ) )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) ) @ C )
        = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ B ) @ C ) ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1666]) ).

thf(220,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ G @ F @ E ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ D @ ( tc_fun @ ( tc_prod @ G @ F ) @ tc_HOL_Obool ) ) @ C ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ D @ ( tc_fun @ ( tc_prod @ G @ E ) @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ G @ E ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ ( tc_prod @ G @ F ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ G @ E ) @ tc_HOL_Obool ) ) @ D ) @ ( c_Relation_Orel__comp @ G @ F @ E ) ) @ B ) ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_rel__comp__UNION__distrib2) ).

thf(3198,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ G @ F @ E ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ D @ ( tc_fun @ ( tc_prod @ G @ F ) @ tc_HOL_Obool ) ) @ C ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ D @ ( tc_fun @ ( tc_prod @ G @ E ) @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ G @ E ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ ( tc_prod @ G @ F ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ G @ E ) @ tc_HOL_Obool ) ) @ D ) @ ( c_Relation_Orel__comp @ G @ F @ E ) ) @ B ) ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[220]) ).

thf(1545,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ F @ E @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ F @ E ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ C @ B ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_prod_Osimps_I2_J) ).

thf(8386,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ F @ E @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ F @ E ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ C @ B ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1545]) ).

thf(2023,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Com_Ocom_OCall @ E @ D @ C )
     != ( c_Com_Ocom_OAss @ B @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_com_Osimps_I33_J) ).

thf(10369,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Com_Ocom_OCall @ E @ D @ C )
     != ( c_Com_Ocom_OAss @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2023]) ).

thf(329,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ tc_Nat_Onat @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ ( tc_fun @ tc_Nat_Onat @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ tc_Nat_Onat @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ tc_Nat_Onat ) @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ tc_Nat_Onat @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ tc_Nat_Onat @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_UN__UN__finite__eq) ).

thf(3578,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ tc_Nat_Onat @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ ( tc_fun @ tc_Nat_Onat @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ tc_Nat_Onat @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ tc_Nat_Onat ) @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ tc_Nat_Onat @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ tc_Nat_Onat @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[329]) ).

thf(1434,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Wellfounded_Owf @ B @ ( hAPP @ ( c_Relation_Oconverse @ B @ B ) @ A ) )
     => ( c_Wellfounded_Owf @ B @ ( hAPP @ ( c_Relation_Oconverse @ B @ B ) @ ( c_Transitive__Closure_Otrancl @ B @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_wf__converse__trancl) ).

thf(7902,plain,
    ! [A: $i,B: $i] :
      ( ( c_Wellfounded_Owf @ B @ ( hAPP @ ( c_Relation_Oconverse @ B @ B ) @ A ) )
     => ( c_Wellfounded_Owf @ B @ ( hAPP @ ( c_Relation_Oconverse @ B @ B ) @ ( c_Transitive__Closure_Otrancl @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1434]) ).

thf(2315,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hBOOL @ ( hAPP @ F @ E ) )
     => ( ( c_Natural_Oevaln @ D @ E @ C @ B )
       => ( c_Natural_Oevaln @ ( c_Com_Ocom_OCond @ F @ D @ A ) @ E @ C @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_evaln_OIfTrue) ).

thf(11492,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hBOOL @ ( hAPP @ F @ E ) )
     => ( ( c_Natural_Oevaln @ D @ E @ C @ B )
       => ( c_Natural_Oevaln @ ( c_Com_Ocom_OCond @ F @ D @ A ) @ E @ C @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2315]) ).

thf(317,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) ) @ B ) @ A ) )
     => ( ( c_Relation_Orefl__on @ C @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ A )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ ( tc_List_Olist @ C ) @ ( tc_List_Olist @ C ) ) @ tc_HOL_Obool ) ) @ ( c_List_Olistrel1 @ C @ B ) ) @ ( c_List_Olistrel @ C @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_listrel1__subset__listrel) ).

thf(3535,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) ) @ B ) @ A ) )
     => ( ( c_Relation_Orefl__on @ C @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ A )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ ( tc_List_Olist @ C ) @ ( tc_List_Olist @ C ) ) @ tc_HOL_Obool ) ) @ ( c_List_Olistrel1 @ C @ B ) ) @ ( c_List_Olistrel @ C @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[317]) ).

thf(756,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ E @ D ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ C ) @ B ) ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ D @ tc_HOL_Obool ) @ E ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ ( tc_prod @ E @ D ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ E @ D ) @ C ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ D @ tc_HOL_Obool ) @ E ) @ A ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ E @ D ) @ B ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ D @ tc_HOL_Obool ) @ E ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Times__Un__distrib1) ).

thf(5113,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ E @ D ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ C ) @ B ) ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ D @ tc_HOL_Obool ) @ E ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ ( tc_prod @ E @ D ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ E @ D ) @ C ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ D @ tc_HOL_Obool ) @ E ) @ A ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ E @ D ) @ B ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ D @ tc_HOL_Obool ) @ E ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[756]) ).

thf(2022,axiom,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_le__add1) ).

thf(10367,plain,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2022]) ).

thf(1328,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( c_List_Ofilter @ B @ A ) @ ( c_List_Olist_ONil @ B ) )
      = ( c_List_Olist_ONil @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_filter_Osimps_I1_J) ).

thf(7501,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( c_List_Ofilter @ B @ A ) @ ( c_List_Olist_ONil @ B ) )
      = ( c_List_Olist_ONil @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1328]) ).

thf(1935,axiom,
    ! [A: $i] :
      ( A
     != ( hAPP @ c_Nat_OSuc @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_n__not__Suc__n) ).

thf(10006,plain,
    ! [A: $i] :
      ( A
     != ( hAPP @ c_Nat_OSuc @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1935]) ).

thf(561,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ D ) @ C ) )
     => ( ! [F: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ F ) @ D ) )
           => ( ( hBOOL @ ( hAPP @ B @ F ) )
             => ( hBOOL @ ( hAPP @ A @ F ) ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ D ) @ ( hAPP @ ( c_Set_OCollect @ E ) @ B ) ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( c_Set_OCollect @ E ) @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Int__Collect__mono) ).

thf(4417,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ D ) @ C ) )
     => ( ! [F: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ F ) @ D ) )
           => ( ( hBOOL @ ( hAPP @ B @ F ) )
             => ( hBOOL @ ( hAPP @ A @ F ) ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ D ) @ ( hAPP @ ( c_Set_OCollect @ E ) @ B ) ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( c_Set_OCollect @ E ) @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[561]) ).

thf(906,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i] :
          ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Nat_Onat ) @ C ) @ B ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ A ) @ C ) ) )
     => ( c_Fun_Oinj__on @ tc_Nat_Onat @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ tc_Nat_Onat @ tc_Nat_Onat ) @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) ) @ A ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_inj__on__diff__nat) ).

thf(5658,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i] :
          ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Nat_Onat ) @ C ) @ B ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ A ) @ C ) ) )
     => ( c_Fun_Oinj__on @ tc_Nat_Onat @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ tc_Nat_Onat @ tc_Nat_Onat ) @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) ) @ A ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[906]) ).

thf(180,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ B ) @ A ) ) ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ B @ A ) ) ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zdiv__zmult1__eq) ).

thf(3050,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ B ) @ A ) ) ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ B @ A ) ) ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[180]) ).

thf(1085,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ F @ E ) @ D ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) @ F ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ A @ E ) @ D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_rev__predicate2D) ).

thf(6402,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ F @ E ) @ D ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) @ F ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ A @ E ) @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1085]) ).

thf(1987,axiom,
    ! [A: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Natural_Oevalc @ c_Com_Ocom_OSKIP ) @ A ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_evalc_OSkip) ).

thf(10235,plain,
    ! [A: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Natural_Oevalc @ c_Com_Ocom_OSKIP ) @ A ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1987]) ).

thf(45,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Groups_Ocomm__monoid__add @ E )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ C ) )
       => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ A ) )
          = ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ E @ E ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ D @ E @ ( tc_fun @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ E @ ( tc_fun @ E @ E ) ) @ D ) @ ( c_If @ E ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ ( tc_fun @ D @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( c_member @ D ) ) @ A ) ) ) @ B ) ) @ ( c_Groups_Ozero__class_Ozero @ E ) ) ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_setsum__restrict__set) ).

thf(2547,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Groups_Ocomm__monoid__add @ E )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ C ) )
       => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ A ) )
          = ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ E @ E ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ D @ E @ ( tc_fun @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ E @ ( tc_fun @ E @ E ) ) @ D ) @ ( c_If @ E ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ ( tc_fun @ D @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( c_member @ D ) ) @ A ) ) ) @ B ) ) @ ( c_Groups_Ozero__class_Ozero @ E ) ) ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[45]) ).

thf(956,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ D @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ C ) @ B ) ) @ A ) )
    <=> ! [F: $i] :
          ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ F ) @ C ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( hAPP @ B @ F ) ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_UN__subset__iff) ).

thf(5858,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ D @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ C ) @ B ) ) @ A ) )
       => ! [F: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ F ) @ C ) )
           => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( hAPP @ B @ F ) ) @ A ) ) ) )
      & ( ! [F: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ F ) @ C ) )
           => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( hAPP @ B @ F ) ) @ A ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ D @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ C ) @ B ) ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[956]) ).

thf(2088,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ tc_Nat_Onat @ tc_Nat_Onat ) @ B ) @ A ) ) @ ( c_Transitive__Closure_Otrancl @ tc_Nat_Onat @ c_Wellfounded_Opred__nat ) ) )
    <=> ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__eq) ).

thf(10615,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ tc_Nat_Onat @ tc_Nat_Onat ) @ B ) @ A ) ) @ ( c_Transitive__Closure_Otrancl @ tc_Nat_Onat @ c_Wellfounded_Opred__nat ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) ) )
      & ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ tc_Nat_Onat @ tc_Nat_Onat ) @ B ) @ A ) ) @ ( c_Transitive__Closure_Otrancl @ tc_Nat_Onat @ c_Wellfounded_Opred__nat ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2088]) ).

thf(1127,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ E @ D ) @ C ) @ B ) @ A )
        = C )
    <=> ( ( hAPP @ C @ B )
        = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_fun__upd__idem__iff) ).

thf(6666,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ E @ D ) @ C ) @ B ) @ A )
          = C )
       => ( ( hAPP @ C @ B )
          = A ) )
      & ( ( ( hAPP @ C @ B )
          = A )
       => ( ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ E @ D ) @ C ) @ B ) @ A )
          = C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1127]) ).

thf(2346,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( hAPP @ ( c_List_Oset @ D ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ D ) @ B ) @ A ) ) ) )
     => ( ( C = B )
        | ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( hAPP @ ( c_List_Oset @ D ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_set__ConsD) ).

thf(11624,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( hAPP @ ( c_List_Oset @ D ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ D ) @ B ) @ A ) ) ) )
     => ( ( C = B )
        | ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( hAPP @ ( c_List_Oset @ D ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2346]) ).

thf(2362,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Natural_Oevalc @ ( c_Com_Ocom_OWhile @ C @ B ) ) @ A ) @ A ) )
    <=> ( ( hBOOL @ ( hAPP @ C @ A ) )
       => ? [D: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Natural_Oevalc @ B ) @ A ) @ D ) )
            & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Natural_Oevalc @ ( c_Com_Ocom_OWhile @ C @ B ) ) @ D ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_evalc_Oequations_I7_J) ).

thf(11702,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Natural_Oevalc @ ( c_Com_Ocom_OWhile @ C @ B ) ) @ A ) @ A ) )
       => ( ( hBOOL @ ( hAPP @ C @ A ) )
         => ? [D: $i] :
              ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Natural_Oevalc @ B ) @ A ) @ D ) )
              & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Natural_Oevalc @ ( c_Com_Ocom_OWhile @ C @ B ) ) @ D ) @ A ) ) ) ) )
      & ( ( ( hBOOL @ ( hAPP @ C @ A ) )
         => ? [D: $i] :
              ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Natural_Oevalc @ B ) @ A ) @ D ) )
              & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Natural_Oevalc @ ( c_Com_Ocom_OWhile @ C @ B ) ) @ D ) @ A ) ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Natural_Oevalc @ ( c_Com_Ocom_OWhile @ C @ B ) ) @ A ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2362]) ).

thf(355,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( c_Map_Odom @ D @ C @ B )
        = ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) )
    <=> ? [E: $i] :
          ( B
          = ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ D @ ( tc_Option_Ooption @ C ) ) @ ( hAPP @ ( c_COMBK @ ( tc_Option_Ooption @ C ) @ D ) @ ( c_Option_Ooption_ONone @ C ) ) ) @ A ) @ ( hAPP @ ( c_Option_Ooption_OSome @ C ) @ E ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_dom__eq__singleton__conv) ).

thf(3667,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( ( c_Map_Odom @ D @ C @ B )
          = ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) )
       => ? [E: $i] :
            ( B
            = ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ D @ ( tc_Option_Ooption @ C ) ) @ ( hAPP @ ( c_COMBK @ ( tc_Option_Ooption @ C ) @ D ) @ ( c_Option_Ooption_ONone @ C ) ) ) @ A ) @ ( hAPP @ ( c_Option_Ooption_OSome @ C ) @ E ) ) ) )
      & ( ? [E: $i] :
            ( B
            = ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ D @ ( tc_Option_Ooption @ C ) ) @ ( hAPP @ ( c_COMBK @ ( tc_Option_Ooption @ C ) @ D ) @ ( c_Option_Ooption_ONone @ C ) ) ) @ A ) @ ( hAPP @ ( c_Option_Ooption_OSome @ C ) @ E ) ) )
       => ( ( c_Map_Odom @ D @ C @ B )
          = ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[355]) ).

thf(2339,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ B ) @ ( hAPP @ ( c_Set_OCollect @ C ) @ A ) ) )
     => ( hBOOL @ ( hAPP @ A @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_CollectE) ).

thf(11602,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ B ) @ ( hAPP @ ( c_Set_OCollect @ C ) @ A ) ) )
     => ( hBOOL @ ( hAPP @ A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2339]) ).

thf(2222,axiom,
    ! [A: $i,B: $i] : ( c_Natural_Oevaln @ c_Com_Ocom_OSKIP @ B @ A @ B ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_evaln_OSkip) ).

thf(11101,plain,
    ! [A: $i,B: $i] : ( c_Natural_Oevaln @ c_Com_Ocom_OSKIP @ B @ A @ B ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2222]) ).

thf(470,axiom,
    ! [A: $i,B: $i] : ( c_Relation_Orefl__on @ B @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ ( c_Transitive__Closure_Ortrancl @ B @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_refl__rtrancl) ).

thf(4080,plain,
    ! [A: $i,B: $i] : ( c_Relation_Orefl__on @ B @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ ( c_Transitive__Closure_Ortrancl @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[470]) ).

thf(423,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ~ ( hBOOL @ ( hAPP @ D @ C ) )
       => ( hBOOL @ ( hAPP @ B @ C ) ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ A @ tc_HOL_Obool ) ) @ B ) @ D ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_sup1CI) ).

thf(3931,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ~ ( hBOOL @ ( hAPP @ D @ C ) )
       => ( hBOOL @ ( hAPP @ B @ C ) ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ A @ tc_HOL_Obool ) ) @ B ) @ D ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[423]) ).

thf(2248,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( hBOOL @ ( hAPP @ G @ F ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ D ) @ C ) ) @ ( hAPP @ B @ F ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_prod @ A @ E ) @ ( tc_prod @ A @ E ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_prod @ A @ E ) @ ( tc_prod @ A @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ A @ E ) @ F ) @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ A @ E ) @ F ) @ C ) ) ) @ ( c_Recdef_Osame__fst @ A @ E @ G @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_same__fstI) ).

thf(11206,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( hBOOL @ ( hAPP @ G @ F ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ D ) @ C ) ) @ ( hAPP @ B @ F ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_prod @ A @ E ) @ ( tc_prod @ A @ E ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_prod @ A @ E ) @ ( tc_prod @ A @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ A @ E ) @ F ) @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ A @ E ) @ F ) @ C ) ) ) @ ( c_Recdef_Osame__fst @ A @ E @ G @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2248]) ).

thf(631,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Diff__Int) ).

thf(4680,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[631]) ).

thf(811,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Diff__Int__distrib2) ).

thf(5330,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[811]) ).

thf(865,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( c_Set_Oimage @ D @ C @ B ) @ ( hAPP @ ( c_Set_Ovimage @ D @ C @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ A ) @ ( hAPP @ ( c_Set_Oimage @ D @ C @ B ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_image__vimage__eq) ).

thf(5515,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( c_Set_Oimage @ D @ C @ B ) @ ( hAPP @ ( c_Set_Ovimage @ D @ C @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ A ) @ ( hAPP @ ( c_Set_Oimage @ D @ C @ B ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[865]) ).

thf(386,axiom,
    ! [A: $i,B: $i,C: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Int__lower1) ).

thf(3799,plain,
    ! [A: $i,B: $i,C: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[386]) ).

thf(103,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Groups_Ocomm__monoid__add @ E )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ C ) )
       => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ B ) @ ( hAPP @ ( c_Set_OCollect @ D ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ D @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ D ) @ c_fconj ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ ( tc_fun @ D @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( c_member @ D ) ) @ C ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ ( tc_fun @ D @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( c_member @ D ) ) @ A ) ) ) )
          = ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ E @ E ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ D @ E @ ( tc_fun @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ E @ ( tc_fun @ E @ E ) ) @ D ) @ ( c_If @ E ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ ( tc_fun @ D @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( c_member @ D ) ) @ A ) ) ) @ B ) ) @ ( c_Groups_Ozero__class_Ozero @ E ) ) ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_setsum__restrict__set_H) ).

thf(2769,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Groups_Ocomm__monoid__add @ E )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ C ) )
       => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ B ) @ ( hAPP @ ( c_Set_OCollect @ D ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ D @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ D ) @ c_fconj ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ ( tc_fun @ D @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( c_member @ D ) ) @ C ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ ( tc_fun @ D @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( c_member @ D ) ) @ A ) ) ) )
          = ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ E @ E ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ D @ E @ ( tc_fun @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ E @ ( tc_fun @ E @ E ) ) @ D ) @ ( c_If @ E ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ ( tc_fun @ D @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( c_member @ D ) ) @ A ) ) ) @ B ) ) @ ( c_Groups_Ozero__class_Ozero @ E ) ) ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[103]) ).

thf(1538,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) @ C ) @ B ) ) @ ( c_List_Olistrel1 @ D @ A ) ) )
     => ~ ! [E: $i,F: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ D ) @ E ) @ F ) ) @ A ) )
           => ! [G: $i,H: $i] :
                ( ( C
                  = ( hAPP @ ( hAPP @ ( c_List_Oappend @ D ) @ G ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ D ) @ E ) @ H ) ) )
               => ( B
                 != ( hAPP @ ( hAPP @ ( c_List_Oappend @ D ) @ G ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ D ) @ F ) @ H ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_listrel1E) ).

thf(8358,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) @ C ) @ B ) ) @ ( c_List_Olistrel1 @ D @ A ) ) )
     => ~ ! [E: $i,F: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ D ) @ E ) @ F ) ) @ A ) )
           => ! [G: $i,H: $i] :
                ( ( C
                  = ( hAPP @ ( hAPP @ ( c_List_Oappend @ D ) @ G ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ D ) @ E ) @ H ) ) )
               => ( B
                 != ( hAPP @ ( hAPP @ ( c_List_Oappend @ D ) @ G ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ D ) @ F ) @ H ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1538]) ).

thf(293,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ A ) @ C ) )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ A ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Un__Int__crazy) ).

thf(3445,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ A ) @ C ) )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ A ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[293]) ).

thf(1485,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nat__mult__assoc) ).

thf(8155,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1485]) ).

thf(1007,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( ( hAPP @ ( c_Int_Onumber__class_Onumber__of @ tc_Nat_Onat ) @ D )
          = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
       => ( ( hAPP @ ( hAPP @ ( c_List_Onth @ C ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ C ) @ B ) @ A ) ) @ ( hAPP @ ( c_Int_Onumber__class_Onumber__of @ tc_Nat_Onat ) @ D ) )
          = B ) )
      & ( ( ( hAPP @ ( c_Int_Onumber__class_Onumber__of @ tc_Nat_Onat ) @ D )
         != ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
       => ( ( hAPP @ ( hAPP @ ( c_List_Onth @ C ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ C ) @ B ) @ A ) ) @ ( hAPP @ ( c_Int_Onumber__class_Onumber__of @ tc_Nat_Onat ) @ D ) )
          = ( hAPP @ ( hAPP @ ( c_List_Onth @ C ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ ( hAPP @ ( c_Int_Onumber__class_Onumber__of @ tc_Nat_Onat ) @ D ) ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nth__Cons__number__of) ).

thf(6090,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( ( hAPP @ ( c_Int_Onumber__class_Onumber__of @ tc_Nat_Onat ) @ D )
          = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
       => ( ( hAPP @ ( hAPP @ ( c_List_Onth @ C ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ C ) @ B ) @ A ) ) @ ( hAPP @ ( c_Int_Onumber__class_Onumber__of @ tc_Nat_Onat ) @ D ) )
          = B ) )
      & ( ( ( hAPP @ ( c_Int_Onumber__class_Onumber__of @ tc_Nat_Onat ) @ D )
         != ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
       => ( ( hAPP @ ( hAPP @ ( c_List_Onth @ C ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ C ) @ B ) @ A ) ) @ ( hAPP @ ( c_Int_Onumber__class_Onumber__of @ tc_Nat_Onat ) @ D ) )
          = ( hAPP @ ( hAPP @ ( c_List_Onth @ C ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ ( hAPP @ ( c_Int_Onumber__class_Onumber__of @ tc_Nat_Onat ) @ D ) ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1007]) ).

thf(2168,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_List_Odistinct @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( c_List_Odistinct @ C ) @ ( c_List_OtakeWhile @ C @ A @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_distinct__takeWhile) ).

thf(10897,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_List_Odistinct @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( c_List_Odistinct @ C ) @ ( c_List_OtakeWhile @ C @ A @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2168]) ).

thf(1401,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) ) @ B ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) ) @ ( c_Transitive__Closure_Ortrancl @ C @ B ) ) @ ( c_Transitive__Closure_Ortrancl @ C @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_rtrancl__mono) ).

thf(7777,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) ) @ B ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) ) @ ( c_Transitive__Closure_Ortrancl @ C @ B ) ) @ ( c_Transitive__Closure_Ortrancl @ C @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1401]) ).

thf(1837,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) )
    <=> ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ A ) )
        & ( B != A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nat__less__le) ).

thf(9608,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ A ) )
          & ( B != A ) ) )
      & ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ A ) )
          & ( B != A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1837]) ).

thf(1996,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ c_Com_Ocom_OBODY @ C )
     != ( c_Com_Ocom_OAss @ B @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_com_Osimps_I31_J) ).

thf(10266,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ c_Com_Ocom_OBODY @ C )
     != ( c_Com_Ocom_OAss @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1996]) ).

thf(1927,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Wellfounded_OwfP @ B @ A )
    <=> ! [C: $i] : ( hBOOL @ ( hAPP @ ( c_Wellfounded_Oaccp @ B @ A ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_wfP__accp__iff) ).

thf(9977,plain,
    ! [A: $i,B: $i] :
      ( ( ( c_Wellfounded_OwfP @ B @ A )
       => ! [C: $i] : ( hBOOL @ ( hAPP @ ( c_Wellfounded_Oaccp @ B @ A ) @ C ) ) )
      & ( ! [C: $i] : ( hBOOL @ ( hAPP @ ( c_Wellfounded_Oaccp @ B @ A ) @ C ) )
       => ( c_Wellfounded_OwfP @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1927]) ).

thf(1073,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ F ) @ E ) @ ( c_Map_Odom @ F @ D @ C ) ) )
     => ( ( c_Map_Omap__add @ F @ D @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ F @ ( tc_Option_Ooption @ D ) ) @ B ) @ E ) @ ( hAPP @ ( c_Option_Ooption_OSome @ D ) @ A ) ) @ C )
        = ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ F @ ( tc_Option_Ooption @ D ) ) @ ( c_Map_Omap__add @ F @ D @ B @ C ) ) @ E ) @ ( hAPP @ ( c_Option_Ooption_OSome @ D ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_map__add__upd__left) ).

thf(6360,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ F ) @ E ) @ ( c_Map_Odom @ F @ D @ C ) ) )
     => ( ( c_Map_Omap__add @ F @ D @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ F @ ( tc_Option_Ooption @ D ) ) @ B ) @ E ) @ ( hAPP @ ( c_Option_Ooption_OSome @ D ) @ A ) ) @ C )
        = ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ F @ ( tc_Option_Ooption @ D ) ) @ ( c_Map_Omap__add @ F @ D @ B @ C ) ) @ E ) @ ( hAPP @ ( c_Option_Ooption_OSome @ D ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1073]) ).

thf(1557,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ C ) @ ( tc_List_Olist @ C ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ C ) @ ( tc_List_Olist @ C ) ) @ ( c_List_Olist_ONil @ C ) ) @ B ) ) @ ( c_List_Olexord @ C @ A ) ) )
    <=> ? [D: $i,E: $i] :
          ( B
          = ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ C ) @ D ) @ E ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_lexord__Nil__left) ).

thf(8429,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ C ) @ ( tc_List_Olist @ C ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ C ) @ ( tc_List_Olist @ C ) ) @ ( c_List_Olist_ONil @ C ) ) @ B ) ) @ ( c_List_Olexord @ C @ A ) ) )
       => ? [D: $i,E: $i] :
            ( B
            = ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ C ) @ D ) @ E ) ) )
      & ( ? [D: $i,E: $i] :
            ( B
            = ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ C ) @ D ) @ E ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ C ) @ ( tc_List_Olist @ C ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ C ) @ ( tc_List_Olist @ C ) ) @ ( c_List_Olist_ONil @ C ) ) @ B ) ) @ ( c_List_Olexord @ C @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1557]) ).

thf(1660,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ B ) @ C ) ) )
      <=> ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ C ) ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_le__diff__conv2) ).

thf(8916,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ C ) @ B ) )
     => ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ B ) @ C ) ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ C ) ) @ B ) ) )
        & ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ C ) ) @ B ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ B ) @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1660]) ).

thf(858,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oab__semigroup__mult @ C )
     => ( ( B
         != ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ B ) )
         => ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ A ) @ B ) )
           => ( ( hAPP @ ( c_Finite__Set_Ofold1 @ C @ ( c_Groups_Otimes__class_Otimes @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ A ) @ B ) )
              = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ A ) @ ( hAPP @ ( c_Finite__Set_Ofold1 @ C @ ( c_Groups_Otimes__class_Otimes @ C ) ) @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_fold1__insert) ).

thf(5497,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oab__semigroup__mult @ C )
     => ( ( B
         != ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ B ) )
         => ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ A ) @ B ) )
           => ( ( hAPP @ ( c_Finite__Set_Ofold1 @ C @ ( c_Groups_Otimes__class_Otimes @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ A ) @ B ) )
              = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ A ) @ ( hAPP @ ( c_Finite__Set_Ofold1 @ C @ ( c_Groups_Otimes__class_Otimes @ C ) ) @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[858]) ).

thf(2256,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( hAPP @ ( c_List_Oset @ D ) @ B ) ) )
     => ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( hAPP @ ( c_List_Oset @ D ) @ ( c_List_Osublist @ D @ B @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_notin__set__sublistI) ).

thf(11234,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( hAPP @ ( c_List_Oset @ D ) @ B ) ) )
     => ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( hAPP @ ( c_List_Oset @ D ) @ ( c_List_Osublist @ D @ B @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2256]) ).

thf(1961,axiom,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) )
        & ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ A ) @ B ) ) )
     => ( B != A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_dvd_Oless__imp__not__eq) ).

thf(10131,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) )
        & ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ A ) @ B ) ) )
     => ( B != A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1961]) ).

thf(1143,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ C ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ A ) ) )
      <=> ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nat__mult__le__cancel1) ).

thf(6728,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ C ) )
     => ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ A ) ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ A ) ) )
        & ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1143]) ).

thf(371,axiom,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_top1I) ).

thf(3731,plain,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[371]) ).

thf(668,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Int_Oint ) @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Int_Oint ) @ C ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Int_Oint ) @ C ) @ ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ B @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zdvd__zmod) ).

thf(4808,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Int_Oint ) @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Int_Oint ) @ C ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Int_Oint ) @ C ) @ ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ B @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[668]) ).

thf(805,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( hAPP @ ( c_Set_Oimage @ D @ C @ B ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ D @ tc_HOL_Obool ) ) )
        = ( c_Orderings_Otop__class_Otop @ ( tc_fun @ C @ tc_HOL_Obool ) ) )
     => ( ( hAPP @ ( c_Set_Oimage @ D @ C @ B ) @ ( hAPP @ ( c_Set_Ovimage @ D @ C @ B ) @ A ) )
        = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_surj__image__vimage__eq) ).

thf(5313,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( hAPP @ ( c_Set_Oimage @ D @ C @ B ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ D @ tc_HOL_Obool ) ) )
        = ( c_Orderings_Otop__class_Otop @ ( tc_fun @ C @ tc_HOL_Obool ) ) )
     => ( ( hAPP @ ( c_Set_Oimage @ D @ C @ B ) @ ( hAPP @ ( c_Set_Ovimage @ D @ C @ B ) @ A ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[805]) ).

thf(1392,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Fun_Ocomp @ F @ E @ D @ C ) @ B ) @ A )
      = ( hAPP @ C @ ( hAPP @ B @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_o__def) ).

thf(7750,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Fun_Ocomp @ F @ E @ D @ C ) @ B ) @ A )
      = ( hAPP @ C @ ( hAPP @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1392]) ).

thf(1566,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Finite__Set_Ofun__left__comm @ E @ D @ C )
     => ( ( hAPP @ ( c_Fun_Ocomp @ D @ D @ D @ ( hAPP @ C @ B ) ) @ ( hAPP @ C @ A ) )
        = ( hAPP @ ( c_Fun_Ocomp @ D @ D @ D @ ( hAPP @ C @ A ) ) @ ( hAPP @ C @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_fun__left__comm_Ofun__comp__comm) ).

thf(8462,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Finite__Set_Ofun__left__comm @ E @ D @ C )
     => ( ( hAPP @ ( c_Fun_Ocomp @ D @ D @ D @ ( hAPP @ C @ B ) ) @ ( hAPP @ C @ A ) )
        = ( hAPP @ ( c_Fun_Ocomp @ D @ D @ D @ ( hAPP @ C @ A ) ) @ ( hAPP @ C @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1566]) ).

thf(1284,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Otake @ ( tc_prod @ E @ D ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_List_Ozip @ E @ D ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_List_Ozip @ E @ D ) @ ( hAPP @ ( hAPP @ ( c_List_Otake @ E ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_List_Otake @ D ) @ C ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_take__zip) ).

thf(7281,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Otake @ ( tc_prod @ E @ D ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_List_Ozip @ E @ D ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_List_Ozip @ E @ D ) @ ( hAPP @ ( hAPP @ ( c_List_Otake @ E ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_List_Otake @ D ) @ C ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1284]) ).

thf(1807,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Nat_Onat ) @ C ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Nat_Onat ) @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_div__le__mono) ).

thf(9493,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Nat_Onat ) @ C ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Nat_Onat ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1807]) ).

thf(2152,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( c_Com_Ocom_OCall @ F @ E @ D )
     != ( c_Com_Ocom_OCond @ C @ B @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_com_Osimps_I57_J) ).

thf(10848,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( c_Com_Ocom_OCall @ F @ E @ D )
     != ( c_Com_Ocom_OCond @ C @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2152]) ).

thf(1523,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( c_Finite__Set_Ofolding__idem @ F @ E @ D @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ F ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ F @ tc_HOL_Obool ) ) @ A ) @ B ) )
         => ( ( hAPP @ ( c_Fun_Ocomp @ E @ E @ E @ ( hAPP @ C @ B ) ) @ ( hAPP @ C @ A ) )
            = ( hAPP @ C @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_folding__idem_Osubset__comp__idem) ).

thf(8314,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( c_Finite__Set_Ofolding__idem @ F @ E @ D @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ F ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ F @ tc_HOL_Obool ) ) @ A ) @ B ) )
         => ( ( hAPP @ ( c_Fun_Ocomp @ E @ E @ E @ ( hAPP @ C @ B ) ) @ ( hAPP @ C @ A ) )
            = ( hAPP @ C @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1523]) ).

thf(1253,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hAPP @ ( c_Set_Oimage @ F @ E @ ( hAPP @ ( c_Fun_Ocomp @ D @ E @ F @ C ) @ B ) ) @ A )
      = ( hAPP @ ( c_Set_Oimage @ D @ E @ C ) @ ( hAPP @ ( c_Set_Oimage @ F @ D @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_image__compose) ).

thf(7142,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hAPP @ ( c_Set_Oimage @ F @ E @ ( hAPP @ ( c_Fun_Ocomp @ D @ E @ F @ C ) @ B ) ) @ A )
      = ( hAPP @ ( c_Set_Oimage @ D @ E @ C ) @ ( hAPP @ ( c_Set_Oimage @ F @ D @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1253]) ).

thf(397,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) )
        = ( hAPP @ ( c_Set_Oimage @ C @ D @ B ) @ A ) )
    <=> ( A
        = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_empty__is__image) ).

thf(3831,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) )
          = ( hAPP @ ( c_Set_Oimage @ C @ D @ B ) @ A ) )
       => ( A
          = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) )
      & ( ( A
          = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) )
       => ( ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) )
          = ( hAPP @ ( c_Set_Oimage @ C @ D @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[397]) ).

thf(121,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ F @ E @ D ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) ) @ C ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ ( tc_prod @ F @ D ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ F @ E @ D ) @ C ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ F @ E @ D ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_rel__comp__distrib2) ).

thf(2840,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ F @ E @ D ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) ) @ C ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ ( tc_prod @ F @ D ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ F @ E @ D ) @ C ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ F @ E @ D ) @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[121]) ).

thf(763,axiom,
    ! [A: $i] :
      ( ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_Option_Ooption @ A ) ) @ ( c_Option_Ooption_ONone @ A ) )
      = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_option_Osize_I3_J) ).

thf(5145,plain,
    ! [A: $i] :
      ( ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_Option_Ooption @ A ) ) @ ( c_Option_Ooption_ONone @ A ) )
      = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[763]) ).

thf(34,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Groups_Oab__group__add @ E )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ C ) )
       => ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ B ) @ C ) )
           => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ B ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) ) )
              = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ E ) @ ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ A ) @ C ) ) @ ( hAPP @ A @ B ) ) ) )
          & ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ B ) @ C ) )
           => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ B ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) ) )
              = ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ A ) @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_setsum__diff1) ).

thf(2515,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Groups_Oab__group__add @ E )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ C ) )
       => ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ B ) @ C ) )
           => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ B ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) ) )
              = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ E ) @ ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ A ) @ C ) ) @ ( hAPP @ A @ B ) ) ) )
          & ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ B ) @ C ) )
           => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ B ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) ) )
              = ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ A ) @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[34]) ).

thf(238,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C
        = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) )
     => ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ B ) @ A ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_equals0D) ).

thf(3254,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C
        = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) )
     => ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ B ) @ A ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[238]) ).

thf(753,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ D @ D @ D ) @ C ) @ B )
        = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) ) )
     => ( ( ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ D @ D @ D ) @ A ) @ B )
          = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) ) )
       => ( ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ D @ D @ D ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) ) @ C ) @ A ) ) @ B )
          = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_union__comp__emptyL) ).

thf(5098,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ D @ D @ D ) @ C ) @ B )
        = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) ) )
     => ( ( ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ D @ D @ D ) @ A ) @ B )
          = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) ) )
       => ( ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ D @ D @ D ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) ) @ C ) @ A ) ) @ B )
          = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[753]) ).

thf(162,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Hilbert__Choice_OEps @ B @ ( hAPP @ ( hAPP @ ( c_COMBC @ B @ B @ tc_HOL_Obool ) @ c_fequal ) @ A ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_some__eq__trivial) ).

thf(2990,plain,
    ! [A: $i,B: $i] :
      ( ( c_Hilbert__Choice_OEps @ B @ ( hAPP @ ( hAPP @ ( c_COMBC @ B @ B @ tc_HOL_Obool ) @ c_fequal ) @ A ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[162]) ).

thf(1202,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ A ) )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ B ) @ A )
        = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_diff__is__0__eq_H) ).

thf(6940,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ A ) )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ B ) @ A )
        = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1202]) ).

thf(1880,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ A ) )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ A ) ) @ B )
        = ( hAPP @ c_Nat_OSuc @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ A ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Suc__diff__le) ).

thf(9804,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ A ) )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ A ) ) @ B )
        = ( hAPP @ c_Nat_OSuc @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ A ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1880]) ).

thf(793,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) ) @ A )
        = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) ) )
    <=> ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ A ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Un__Int__assoc__eq) ).

thf(5270,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) ) @ A )
          = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ A ) @ C ) ) )
      & ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ A ) @ C ) )
       => ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) ) @ A )
          = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[793]) ).

thf(110,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Groups_Ocomm__monoid__add @ E )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ C ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ B ) )
         => ( ! [F: $i] :
                ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ F ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) ) )
               => ( ( hAPP @ A @ F )
                  = ( c_Groups_Ozero__class_Ozero @ E ) ) )
           => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) )
              = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ E ) @ ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ A ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ A ) @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_setsum__Un__zero) ).

thf(2801,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Groups_Ocomm__monoid__add @ E )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ C ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ B ) )
         => ( ! [F: $i] :
                ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ F ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) ) )
               => ( ( hAPP @ A @ F )
                  = ( c_Groups_Ozero__class_Ozero @ E ) ) )
           => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) )
              = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ E ) @ ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ A ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ A ) @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[110]) ).

thf(1263,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( G != F )
     => ( ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ E @ D ) @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ E @ D ) @ C ) @ G ) @ B ) ) @ F ) @ A )
        = ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ E @ D ) @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ E @ D ) @ C ) @ F ) @ A ) ) @ G ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_fun__upd__twist) ).

thf(7205,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( G != F )
     => ( ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ E @ D ) @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ E @ D ) @ C ) @ G ) @ B ) ) @ F ) @ A )
        = ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ E @ D ) @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ E @ D ) @ C ) @ F ) @ A ) ) @ G ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1263]) ).

thf(1451,axiom,
    ! [A: $i,B: $i] :
      ( ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A )
        = B )
     => ( A
        = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_add__eq__self__zero) ).

thf(7962,plain,
    ! [A: $i,B: $i] :
      ( ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A )
        = B )
     => ( A
        = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1451]) ).

thf(586,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ B ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ A ) @ B ) )
       => ( ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ B @ A )
          = B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mod__neg__neg__trivial) ).

thf(4512,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ B ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ A ) @ B ) )
       => ( ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ B @ A )
          = B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[586]) ).

thf(137,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zle__trans) ).

thf(2895,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ C ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[137]) ).

thf(2365,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_Big__Operators_Osemilattice__big @ D @ C @ B )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ A ) )
       => ( ( hAPP @ B @ A )
          = ( hAPP @ ( c_Finite__Set_Ofold1 @ D @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_semilattice__big_OF__eq) ).

thf(11715,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_Big__Operators_Osemilattice__big @ D @ C @ B )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ A ) )
       => ( ( hAPP @ B @ A )
          = ( hAPP @ ( c_Finite__Set_Ofold1 @ D @ C ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2365]) ).

thf(514,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ F ) @ E ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ C ) )
       => ( ! [G: $i] :
              ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ G ) @ C ) )
             => ( ( hAPP @ ( c_Finite__Set_Ocard @ F ) @ ( hAPP @ ( c_Set_OCollect @ F ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ F @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ F ) @ c_fconj ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ F @ ( tc_fun @ F @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( c_member @ F ) ) @ E ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ F @ D @ tc_HOL_Obool ) @ B ) @ G ) ) ) )
                = A ) )
         => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ F @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ D @ tc_HOL_Obool ) @ tc_Nat_Onat @ F ) @ ( c_Finite__Set_Ocard @ D ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ D @ tc_HOL_Obool ) @ ( tc_fun @ D @ tc_HOL_Obool ) @ F ) @ ( c_Set_OCollect @ D ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ D @ tc_HOL_Obool ) @ ( tc_fun @ D @ tc_HOL_Obool ) @ F ) @ ( hAPP @ ( c_COMBS @ D @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ D ) @ c_fconj ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ ( tc_fun @ D @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( c_member @ D ) ) @ C ) ) ) ) @ B ) ) ) ) @ E )
            = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ A ) @ ( hAPP @ ( c_Finite__Set_Ocard @ D ) @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_setsum__multicount) ).

thf(4248,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ F ) @ E ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ C ) )
       => ( ! [G: $i] :
              ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ G ) @ C ) )
             => ( ( hAPP @ ( c_Finite__Set_Ocard @ F ) @ ( hAPP @ ( c_Set_OCollect @ F ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ F @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ F ) @ c_fconj ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ F @ ( tc_fun @ F @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( c_member @ F ) ) @ E ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ F @ D @ tc_HOL_Obool ) @ B ) @ G ) ) ) )
                = A ) )
         => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ F @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ D @ tc_HOL_Obool ) @ tc_Nat_Onat @ F ) @ ( c_Finite__Set_Ocard @ D ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ D @ tc_HOL_Obool ) @ ( tc_fun @ D @ tc_HOL_Obool ) @ F ) @ ( c_Set_OCollect @ D ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ D @ tc_HOL_Obool ) @ ( tc_fun @ D @ tc_HOL_Obool ) @ F ) @ ( hAPP @ ( c_COMBS @ D @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ D ) @ c_fconj ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ ( tc_fun @ D @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( c_member @ D ) ) @ C ) ) ) ) @ B ) ) ) ) @ E )
            = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ A ) @ ( hAPP @ ( c_Finite__Set_Ocard @ D ) @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[514]) ).

thf(2036,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ D ) @ C ) @ B ) ) @ ( hAPP @ ( c_Relation_Oconverse @ D @ D ) @ ( c_Transitive__Closure_Otrancl @ D @ A ) ) ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ D ) @ C ) @ B ) ) @ ( c_Transitive__Closure_Otrancl @ D @ ( hAPP @ ( c_Relation_Oconverse @ D @ D ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_trancl__converseI) ).

thf(10413,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ D ) @ C ) @ B ) ) @ ( hAPP @ ( c_Relation_Oconverse @ D @ D ) @ ( c_Transitive__Closure_Otrancl @ D @ A ) ) ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ D ) @ C ) @ B ) ) @ ( c_Transitive__Closure_Otrancl @ D @ ( hAPP @ ( c_Relation_Oconverse @ D @ D ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2036]) ).

thf(994,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( c_Relation_OImage @ E @ D @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ ( tc_prod @ E @ D ) @ tc_HOL_Obool ) ) @ C ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Relation_OImage @ E @ D @ C ) @ A ) ) @ ( hAPP @ ( c_Relation_OImage @ E @ D @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Un__Image) ).

thf(6028,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( c_Relation_OImage @ E @ D @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ ( tc_prod @ E @ D ) @ tc_HOL_Obool ) ) @ C ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Relation_OImage @ E @ D @ C ) @ A ) ) @ ( hAPP @ ( c_Relation_OImage @ E @ D @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[994]) ).

thf(1360,axiom,
    ! [A: $i,B: $i] :
      ( ? [C: $i] :
          ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ C ) @ B ) )
          & ( hBOOL @ ( hAPP @ A @ C ) ) )
    <=> ? [C: $i] :
          ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Nat_Onat ) @ C ) @ ( hAPP @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ B ) ) )
          & ( hBOOL @ ( hAPP @ A @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_ex__nat__less__eq) ).

thf(7623,plain,
    ! [A: $i,B: $i] :
      ( ( ? [C: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ C ) @ B ) )
            & ( hBOOL @ ( hAPP @ A @ C ) ) )
       => ? [C: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Nat_Onat ) @ C ) @ ( hAPP @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ B ) ) )
            & ( hBOOL @ ( hAPP @ A @ C ) ) ) )
      & ( ? [C: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Nat_Onat ) @ C ) @ ( hAPP @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ B ) ) )
            & ( hBOOL @ ( hAPP @ A @ C ) ) )
       => ? [C: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ C ) @ B ) )
            & ( hBOOL @ ( hAPP @ A @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1360]) ).

thf(2379,axiom,
    ! [A: $i] :
      ( ! [B: $i] :
          ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_Hoare__Mirabelle_Otriple @ t_a ) ) @ B ) @ v_G ) )
         => ( c_Hoare__Mirabelle_Otriple__valid @ t_a @ A @ B ) )
     => ! [B: $i,C: $i] :
          ( ( ( v_P @ B @ C )
            & ( hBOOL @ ( hAPP @ v_b @ C ) ) )
         => ! [D: $i] :
              ( ( c_Natural_Oevaln @ v_c @ C @ A @ D )
             => ( v_P @ B @ D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',conj_0) ).

thf(11762,plain,
    ! [A: $i] :
      ( ! [B: $i] :
          ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_Hoare__Mirabelle_Otriple @ t_a ) ) @ B ) @ v_G ) )
         => ( c_Hoare__Mirabelle_Otriple__valid @ t_a @ A @ B ) )
     => ! [B: $i,C: $i] :
          ( ( ( v_P @ B @ C )
            & ( hBOOL @ ( hAPP @ v_b @ C ) ) )
         => ! [D: $i] :
              ( ( c_Natural_Oevaln @ v_c @ C @ A @ D )
             => ( v_P @ B @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2379]) ).

thf(578,axiom,
    ! [A: $i,B: $i] :
      ( ( c_List_Olistrel @ B @ ( c_Transitive__Closure_Ortrancl @ B @ A ) )
      = ( c_Transitive__Closure_Ortrancl @ ( tc_List_Olist @ B ) @ ( c_List_Olistrel1 @ B @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_listrel__rtrancl__eq__rtrancl__listrel1) ).

thf(4486,plain,
    ! [A: $i,B: $i] :
      ( ( c_List_Olistrel @ B @ ( c_Transitive__Closure_Ortrancl @ B @ A ) )
      = ( c_Transitive__Closure_Ortrancl @ ( tc_List_Olist @ B ) @ ( c_List_Olistrel1 @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[578]) ).

thf(52,axiom,
    ! [A: $i,B: $i] :
      ( ( ( c_List_Oconcat @ B @ A )
        = ( c_List_Olist_ONil @ B ) )
    <=> ! [C: $i] :
          ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_List_Olist @ B ) ) @ C ) @ ( hAPP @ ( c_List_Oset @ ( tc_List_Olist @ B ) ) @ A ) ) )
         => ( C
            = ( c_List_Olist_ONil @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_concat__eq__Nil__conv) ).

thf(2577,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( c_List_Oconcat @ B @ A )
          = ( c_List_Olist_ONil @ B ) )
       => ! [C: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_List_Olist @ B ) ) @ C ) @ ( hAPP @ ( c_List_Oset @ ( tc_List_Olist @ B ) ) @ A ) ) )
           => ( C
              = ( c_List_Olist_ONil @ B ) ) ) )
      & ( ! [C: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_List_Olist @ B ) ) @ C ) @ ( hAPP @ ( c_List_Oset @ ( tc_List_Olist @ B ) ) @ A ) ) )
           => ( C
              = ( c_List_Olist_ONil @ B ) ) )
       => ( ( c_List_Oconcat @ B @ A )
          = ( c_List_Olist_ONil @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[52]) ).

thf(899,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Relation_Orefl__on @ E @ D @ C )
     => ( ( c_Relation_Orefl__on @ E @ B @ A )
       => ( c_Relation_Orefl__on @ E @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ D ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ ( tc_prod @ E @ E ) @ tc_HOL_Obool ) ) @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_refl__on__Int) ).

thf(5635,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Relation_Orefl__on @ E @ D @ C )
     => ( ( c_Relation_Orefl__on @ E @ B @ A )
       => ( c_Relation_Orefl__on @ E @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ D ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ ( tc_prod @ E @ E ) @ tc_HOL_Obool ) ) @ C ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[899]) ).

thf(1872,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ B ) @ A ) ) )
    <=> ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_finite__insert) ).

thf(9773,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ B ) @ A ) ) )
       => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ A ) ) )
      & ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ B ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1872]) ).

thf(664,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ B ) )
       => ( ( hAPP @ ( c_Set_Ovimage @ A @ D @ ( hAPP @ ( c_COMBK @ D @ A ) @ C ) ) @ B )
          = ( c_Orderings_Otop__class_Otop @ ( tc_fun @ A @ tc_HOL_Obool ) ) ) )
      & ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ B ) )
       => ( ( hAPP @ ( c_Set_Ovimage @ A @ D @ ( hAPP @ ( c_COMBK @ D @ A ) @ C ) ) @ B )
          = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ A @ tc_HOL_Obool ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_vimage__const) ).

thf(4789,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ B ) )
       => ( ( hAPP @ ( c_Set_Ovimage @ A @ D @ ( hAPP @ ( c_COMBK @ D @ A ) @ C ) ) @ B )
          = ( c_Orderings_Otop__class_Otop @ ( tc_fun @ A @ tc_HOL_Obool ) ) ) )
      & ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ B ) )
       => ( ( hAPP @ ( c_Set_Ovimage @ A @ D @ ( hAPP @ ( c_COMBK @ D @ A ) @ C ) ) @ B )
          = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ A @ tc_HOL_Obool ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[664]) ).

thf(459,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Finite__Set_Ofolding__idem @ E @ D @ C @ B )
     => ( ( hAPP @ ( c_Fun_Ocomp @ D @ D @ D @ ( hAPP @ C @ A ) ) @ ( hAPP @ C @ A ) )
        = ( hAPP @ C @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_folding__idem_Oidem__comp) ).

thf(4048,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Finite__Set_Ofolding__idem @ E @ D @ C @ B )
     => ( ( hAPP @ ( c_Fun_Ocomp @ D @ D @ D @ ( hAPP @ C @ A ) ) @ ( hAPP @ C @ A ) )
        = ( hAPP @ C @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[459]) ).

thf(1525,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ A ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ B ) ) @ ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ A ) ) )
         => ( A = B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_card__seteq) ).

thf(8319,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ A ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ B ) ) @ ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ A ) ) )
         => ( A = B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1525]) ).

thf(1739,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ A ) @ B ) )
       => ( ( hAPP @ c_Nat_OSuc @ ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) ) ) )
          = ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_card__Suc__Diff1) ).

thf(9244,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ A ) @ B ) )
       => ( ( hAPP @ c_Nat_OSuc @ ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) ) ) )
          = ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1739]) ).

thf(2098,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ A ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_finite_OinsertI) ).

thf(10646,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ A ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2098]) ).

thf(1042,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_trans__less__add1) ).

thf(6235,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1042]) ).

thf(683,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ B ) )
       => ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ C ) @ A ) )
          = ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) ) ) )
      & ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ B ) )
       => ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ C ) @ A ) )
          = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Int__insert__right) ).

thf(4845,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ B ) )
       => ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ C ) @ A ) )
          = ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) ) ) )
      & ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ B ) )
       => ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ C ) @ A ) )
          = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[683]) ).

thf(836,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ E ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ D ) ) @ C ) ) )
     => ( ( hAPP @ ( hAPP @ ( c_List_Onth @ B ) @ ( hAPP @ ( hAPP @ ( c_List_Omap @ D @ B ) @ A ) @ C ) ) @ E )
        = ( hAPP @ A @ ( hAPP @ ( hAPP @ ( c_List_Onth @ D ) @ C ) @ E ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nth__map) ).

thf(5428,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ E ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ D ) ) @ C ) ) )
     => ( ( hAPP @ ( hAPP @ ( c_List_Onth @ B ) @ ( hAPP @ ( hAPP @ ( c_List_Omap @ D @ B ) @ A ) @ C ) ) @ E )
        = ( hAPP @ A @ ( hAPP @ ( hAPP @ ( c_List_Onth @ D ) @ C ) @ E ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[836]) ).

thf(1465,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( c_List_Oupt @ D @ C )
        = ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ tc_Nat_Onat ) @ B ) @ A ) )
    <=> ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ D ) @ C ) )
        & ( D = B )
        & ( ( c_List_Oupt @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ D ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) @ C )
          = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_upt__eq__Cons__conv) ).

thf(8014,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( ( c_List_Oupt @ D @ C )
          = ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ tc_Nat_Onat ) @ B ) @ A ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ D ) @ C ) )
          & ( D = B )
          & ( ( c_List_Oupt @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ D ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) @ C )
            = A ) ) )
      & ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ D ) @ C ) )
          & ( D = B )
          & ( ( c_List_Oupt @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ D ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) @ C )
            = A ) )
       => ( ( c_List_Oupt @ D @ C )
          = ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ tc_Nat_Onat ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1465]) ).

thf(1915,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_List_Odistinct @ D ) @ C ) )
     => ( hBOOL @ ( hAPP @ ( c_List_Odistinct @ ( tc_prod @ B @ D ) ) @ ( hAPP @ ( hAPP @ ( c_List_Ozip @ B @ D ) @ A ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_distinct__zipI2) ).

thf(9936,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_List_Odistinct @ D ) @ C ) )
     => ( hBOOL @ ( hAPP @ ( c_List_Odistinct @ ( tc_prod @ B @ D ) ) @ ( hAPP @ ( hAPP @ ( c_List_Ozip @ B @ D ) @ A ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1915]) ).

thf(1355,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Map_Orestrict__map @ E @ D @ C @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ E @ ( tc_Option_Ooption @ D ) ) @ ( c_Map_Orestrict__map @ E @ D @ C @ A ) ) @ B ) @ ( hAPP @ C @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_restrict__map__insert) ).

thf(7610,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Map_Orestrict__map @ E @ D @ C @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ E @ ( tc_Option_Ooption @ D ) ) @ ( c_Map_Orestrict__map @ E @ D @ C @ A ) ) @ B ) @ ( hAPP @ C @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1355]) ).

thf(2215,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( hAPP @ ( c_List_Oset @ D ) @ ( c_List_Osublist @ D @ B @ A ) ) ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( hAPP @ ( c_List_Oset @ D ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_in__set__sublistD) ).

thf(11057,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( hAPP @ ( c_List_Oset @ D ) @ ( c_List_Osublist @ D @ B @ A ) ) ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( hAPP @ ( c_List_Oset @ D ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2215]) ).

thf(32,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Ocomm__monoid__add @ D )
     => ( ~ ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ B ) )
       => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ C @ D ) @ A ) @ B )
          = ( c_Groups_Ozero__class_Ozero @ D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_setsum__infinite) ).

thf(2509,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Ocomm__monoid__add @ D )
     => ( ~ ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ B ) )
       => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ C @ D ) @ A ) @ B )
          = ( c_Groups_Ozero__class_Ozero @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[32]) ).

thf(820,axiom,
    ( ( hAPP @ ( c_SetInterval_Oord__class_OatMost @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
    = ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_atMost__0) ).

thf(5362,plain,
    ( ( hAPP @ ( c_SetInterval_Oord__class_OatMost @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
    = ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[820]) ).

thf(313,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Finite__Set_Ofun__left__comm @ E @ D @ C )
     => ( c_Finite__Set_Ofun__left__comm @ B @ D @ ( hAPP @ ( hAPP @ ( c_COMBB @ E @ ( tc_fun @ D @ D ) @ B ) @ C ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_fun__left__comm_Ofun__left__comm__apply) ).

thf(3518,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Finite__Set_Ofun__left__comm @ E @ D @ C )
     => ( c_Finite__Set_Ofun__left__comm @ B @ D @ ( hAPP @ ( hAPP @ ( c_COMBB @ E @ ( tc_fun @ D @ D ) @ B ) @ C ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[313]) ).

thf(432,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( c_Product__Type_Osnd @ D @ C ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ C ) @ B ) @ A ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_snd__conv) ).

thf(3963,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( c_Product__Type_Osnd @ D @ C ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ C ) @ B ) @ A ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[432]) ).

thf(1350,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nat__add__commute) ).

thf(7593,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1350]) ).

thf(972,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Fun_Oinj__on @ E @ D @ C @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ E @ tc_HOL_Obool ) ) )
     => ( ( hAPP @ ( c_Set_Oimage @ E @ D @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ B ) @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Set_Oimage @ E @ D @ C ) @ B ) ) @ ( hAPP @ ( c_Set_Oimage @ E @ D @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_image__set__diff) ).

thf(5919,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Fun_Oinj__on @ E @ D @ C @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ E @ tc_HOL_Obool ) ) )
     => ( ( hAPP @ ( c_Set_Oimage @ E @ D @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ B ) @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Set_Oimage @ E @ D @ C ) @ B ) ) @ ( hAPP @ ( c_Set_Oimage @ E @ D @ C ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[972]) ).

thf(2001,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( c_Com_Ocom_OSKIP
     != ( c_Com_Ocom_OCond @ C @ B @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_com_Osimps_I14_J) ).

thf(10298,plain,
    ! [A: $i,B: $i,C: $i] :
      ( c_Com_Ocom_OSKIP
     != ( c_Com_Ocom_OCond @ C @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2001]) ).

thf(345,axiom,
    ! [A: $i,B: $i] :
      ( ( ( c_SetInterval_Oord_OgreaterThanAtMost @ tc_Nat_Onat @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ tc_Nat_Onat @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ tc_Nat_Onat @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) @ tc_Nat_Onat ) @ ( c_COMBS @ tc_Nat_Onat @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ tc_Nat_Onat @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ tc_Nat_Onat ) @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ tc_Nat_Onat ) @ c_fconj ) ) @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ tc_Nat_Onat ) @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ tc_HOL_Obool @ tc_Nat_Onat ) @ c_fNot ) ) @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) ) ) ) @ B @ A )
        = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) )
    <=> ~ ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) )
          & ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ A ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_dvd_OgreaterThanAtMost__empty__iff) ).

thf(3632,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( c_SetInterval_Oord_OgreaterThanAtMost @ tc_Nat_Onat @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ tc_Nat_Onat @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ tc_Nat_Onat @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) @ tc_Nat_Onat ) @ ( c_COMBS @ tc_Nat_Onat @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ tc_Nat_Onat @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ tc_Nat_Onat ) @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ tc_Nat_Onat ) @ c_fconj ) ) @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ tc_Nat_Onat ) @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ tc_HOL_Obool @ tc_Nat_Onat ) @ c_fNot ) ) @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) ) ) ) @ B @ A )
          = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) )
       => ~ ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) )
            & ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ A ) @ B ) ) ) )
      & ( ~ ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) )
            & ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ A ) @ B ) ) )
       => ( ( c_SetInterval_Oord_OgreaterThanAtMost @ tc_Nat_Onat @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ tc_Nat_Onat @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ tc_Nat_Onat @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) @ tc_Nat_Onat ) @ ( c_COMBS @ tc_Nat_Onat @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ tc_Nat_Onat @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ tc_Nat_Onat ) @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ tc_Nat_Onat ) @ c_fconj ) ) @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ tc_Nat_Onat ) @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ tc_HOL_Obool @ tc_Nat_Onat ) @ c_fNot ) ) @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) ) ) ) @ B @ A )
          = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[345]) ).

thf(2041,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i,I: $i,J: $i] :
      ( ( c_Com_Ocom_Ocom__case @ J @ I @ H @ G @ F @ E @ D @ C @ B @ ( hAPP @ c_Com_Ocom_OBODY @ A ) )
      = ( hAPP @ C @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_com_Osimps_I70_J) ).

thf(10430,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i,I: $i,J: $i] :
      ( ( c_Com_Ocom_Ocom__case @ J @ I @ H @ G @ F @ E @ D @ C @ B @ ( hAPP @ c_Com_Ocom_OBODY @ A ) )
      = ( hAPP @ C @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2041]) ).

thf(2244,axiom,
    ! [A: $i,B: $i] :
      ( ? [C: $i] : ( hBOOL @ ( hAPP @ B @ C ) )
     => ( hBOOL @ ( hAPP @ B @ ( c_Hilbert__Choice_OEps @ A @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_tfl__some) ).

thf(11189,plain,
    ! [A: $i,B: $i] :
      ( ? [C: $i] : ( hBOOL @ ( hAPP @ B @ C ) )
     => ( hBOOL @ ( hAPP @ B @ ( c_Hilbert__Choice_OEps @ A @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2244]) ).

thf(448,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Map_Odom @ B @ A @ ( hAPP @ ( c_COMBK @ ( tc_Option_Ooption @ A ) @ B ) @ ( c_Option_Ooption_ONone @ A ) ) )
      = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_dom__empty) ).

thf(4012,plain,
    ! [A: $i,B: $i] :
      ( ( c_Map_Odom @ B @ A @ ( hAPP @ ( c_COMBK @ ( tc_Option_Ooption @ A ) @ B ) @ ( c_Option_Ooption_ONone @ A ) ) )
      = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[448]) ).

thf(1763,axiom,
    ! [A: $i,B: $i] :
      ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ A ) )
    <=> ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ A ) ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_not__less__eq__eq) ).

thf(9333,plain,
    ! [A: $i,B: $i] :
      ( ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ A ) ) @ B ) ) )
      & ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ A ) ) @ B ) )
       => ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1763]) ).

thf(2163,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Relation_Orefl__on @ E @ D @ C )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ B ) @ A ) ) @ C ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ A ) @ D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_refl__onD2) ).

thf(10882,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Relation_Orefl__on @ E @ D @ C )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ B ) @ A ) ) @ C ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ A ) @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2163]) ).

thf(1906,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_Wellfounded_Owf @ D @ C )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ D ) @ B ) @ A ) ) @ C ) )
       => ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ D ) @ A ) @ B ) ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_wf__not__sym) ).

thf(9904,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_Wellfounded_Owf @ D @ C )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ D ) @ B ) @ A ) ) @ C ) )
       => ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ D ) @ A ) @ B ) ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1906]) ).

thf(1425,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) ) )
     => ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_DiffD2) ).

thf(7874,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) ) )
     => ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1425]) ).

thf(1482,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ F ) @ ( tc_List_Olist @ F ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ F ) @ ( tc_List_Olist @ F ) ) @ ( hAPP @ ( hAPP @ ( c_List_Oappend @ F ) @ E ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ F ) @ D ) @ ( c_List_Olist_ONil @ F ) ) ) ) @ ( hAPP @ ( hAPP @ ( c_List_Oappend @ F ) @ C ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ F ) @ B ) @ ( c_List_Olist_ONil @ F ) ) ) ) ) @ ( c_List_Olistrel1 @ F @ A ) ) )
    <=> ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ F ) @ ( tc_List_Olist @ F ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ F ) @ ( tc_List_Olist @ F ) ) @ E ) @ C ) ) @ ( c_List_Olistrel1 @ F @ A ) ) )
          & ( D = B ) )
        | ( ( E = C )
          & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ F @ F ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ F @ F ) @ D ) @ B ) ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_snoc__listrel1__snoc__iff) ).

thf(8117,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ F ) @ ( tc_List_Olist @ F ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ F ) @ ( tc_List_Olist @ F ) ) @ ( hAPP @ ( hAPP @ ( c_List_Oappend @ F ) @ E ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ F ) @ D ) @ ( c_List_Olist_ONil @ F ) ) ) ) @ ( hAPP @ ( hAPP @ ( c_List_Oappend @ F ) @ C ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ F ) @ B ) @ ( c_List_Olist_ONil @ F ) ) ) ) ) @ ( c_List_Olistrel1 @ F @ A ) ) )
       => ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ F ) @ ( tc_List_Olist @ F ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ F ) @ ( tc_List_Olist @ F ) ) @ E ) @ C ) ) @ ( c_List_Olistrel1 @ F @ A ) ) )
            & ( D = B ) )
          | ( ( E = C )
            & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ F @ F ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ F @ F ) @ D ) @ B ) ) @ A ) ) ) ) )
      & ( ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ F ) @ ( tc_List_Olist @ F ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ F ) @ ( tc_List_Olist @ F ) ) @ E ) @ C ) ) @ ( c_List_Olistrel1 @ F @ A ) ) )
            & ( D = B ) )
          | ( ( E = C )
            & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ F @ F ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ F @ F ) @ D ) @ B ) ) @ A ) ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ F ) @ ( tc_List_Olist @ F ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ F ) @ ( tc_List_Olist @ F ) ) @ ( hAPP @ ( hAPP @ ( c_List_Oappend @ F ) @ E ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ F ) @ D ) @ ( c_List_Olist_ONil @ F ) ) ) ) @ ( hAPP @ ( hAPP @ ( c_List_Oappend @ F ) @ C ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ F ) @ B ) @ ( c_List_Olist_ONil @ F ) ) ) ) ) @ ( c_List_Olistrel1 @ F @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1482]) ).

thf(1039,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( ( hAPP @ ( c_Fun_Ocomp @ G @ F @ E @ D ) @ C )
        = B )
     => ( ( hAPP @ D @ ( hAPP @ C @ A ) )
        = ( hAPP @ B @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_o__eq__dest__lhs) ).

thf(6218,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( ( hAPP @ ( c_Fun_Ocomp @ G @ F @ E @ D ) @ C )
        = B )
     => ( ( hAPP @ D @ ( hAPP @ C @ A ) )
        = ( hAPP @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1039]) ).

thf(2404,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( c_Typedef_Otype__definition @ F @ E @ D @ C @ B )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ ( hAPP @ D @ A ) ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_type__definition_ORep) ).

thf(11877,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( c_Typedef_Otype__definition @ F @ E @ D @ C @ B )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ ( hAPP @ D @ A ) ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2404]) ).

thf(2203,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_Com_Ocom_OWhile @ D @ C )
     != ( c_Com_Ocom_OSemi @ B @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_com_Osimps_I47_J) ).

thf(11019,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_Com_Ocom_OWhile @ D @ C )
     != ( c_Com_Ocom_OSemi @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2203]) ).

thf(4816,plain,
    ( ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint )
   != ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ),
    inference(polarity_switch,[status(thm)],[4815]) ).

thf(4817,plain,
    ( ( c_Groups_Oone__class_Oone @ tc_Int_Oint )
   != ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ),
    inference(lifteq,[status(thm)],[4816]) ).

thf(697,axiom,
    ! [A: $i,B: $i,C: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) @ ( c_List_OtakeWhile @ C @ B @ A ) ) ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_length__takeWhile__le) ).

thf(4898,plain,
    ! [A: $i,B: $i,C: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) @ ( c_List_OtakeWhile @ C @ B @ A ) ) ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[697]) ).

thf(2287,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D = C )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ B ) @ C ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ B ) @ D ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_eq__mem__trans) ).

thf(11364,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D = C )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ B ) @ C ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ B ) @ D ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2287]) ).

thf(2254,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Equiv__Relations_Oequivp @ E @ D )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ D @ C ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ D @ B ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ D @ C ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_equivp__transp) ).

thf(11229,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Equiv__Relations_Oequivp @ E @ D )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ D @ C ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ D @ B ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ D @ C ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2254]) ).

thf(1695,axiom,
    ! [A: $i,B: $i] :
      ( ~ ( hBOOL @ ( hAPP @ B @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) )
     => ( ( hBOOL @ ( hAPP @ B @ A ) )
       => ? [C: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ C ) @ A ) )
            & ! [D: $i] :
                ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ D ) @ C ) )
               => ~ ( hBOOL @ ( hAPP @ B @ D ) ) )
            & ( hBOOL @ ( hAPP @ B @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_ex__least__nat__less) ).

thf(9085,plain,
    ! [A: $i,B: $i] :
      ( ~ ( hBOOL @ ( hAPP @ B @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) )
     => ( ( hBOOL @ ( hAPP @ B @ A ) )
       => ? [C: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ C ) @ A ) )
            & ! [D: $i] :
                ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ D ) @ C ) )
               => ~ ( hBOOL @ ( hAPP @ B @ D ) ) )
            & ( hBOOL @ ( hAPP @ B @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1695]) ).

thf(1346,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( ( hAPP @ ( c_Int_Onumber__class_Onumber__of @ tc_Nat_Onat ) @ D )
          = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
       => ( ( hAPP @ ( hAPP @ ( c_List_Otake @ C ) @ ( hAPP @ ( c_Int_Onumber__class_Onumber__of @ tc_Nat_Onat ) @ D ) ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ C ) @ B ) @ A ) )
          = ( c_List_Olist_ONil @ C ) ) )
      & ( ( ( hAPP @ ( c_Int_Onumber__class_Onumber__of @ tc_Nat_Onat ) @ D )
         != ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
       => ( ( hAPP @ ( hAPP @ ( c_List_Otake @ C ) @ ( hAPP @ ( c_Int_Onumber__class_Onumber__of @ tc_Nat_Onat ) @ D ) ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ C ) @ B ) @ A ) )
          = ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_List_Otake @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ ( hAPP @ ( c_Int_Onumber__class_Onumber__of @ tc_Nat_Onat ) @ D ) ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_take__Cons__number__of) ).

thf(7577,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( ( hAPP @ ( c_Int_Onumber__class_Onumber__of @ tc_Nat_Onat ) @ D )
          = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
       => ( ( hAPP @ ( hAPP @ ( c_List_Otake @ C ) @ ( hAPP @ ( c_Int_Onumber__class_Onumber__of @ tc_Nat_Onat ) @ D ) ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ C ) @ B ) @ A ) )
          = ( c_List_Olist_ONil @ C ) ) )
      & ( ( ( hAPP @ ( c_Int_Onumber__class_Onumber__of @ tc_Nat_Onat ) @ D )
         != ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
       => ( ( hAPP @ ( hAPP @ ( c_List_Otake @ C ) @ ( hAPP @ ( c_Int_Onumber__class_Onumber__of @ tc_Nat_Onat ) @ D ) ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ C ) @ B ) @ A ) )
          = ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_List_Otake @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ ( hAPP @ ( c_Int_Onumber__class_Onumber__of @ tc_Nat_Onat ) @ D ) ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1346]) ).

thf(1630,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ G ) @ F ) @ ( hAPP @ ( c_List_Oset @ G ) @ E ) ) )
     => ( ( c_Map_Omap__upds @ G @ D @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ G @ ( tc_Option_Ooption @ D ) ) @ C ) @ F ) @ ( hAPP @ ( c_Option_Ooption_OSome @ D ) @ B ) ) @ E @ A )
        = ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ G @ ( tc_Option_Ooption @ D ) ) @ ( c_Map_Omap__upds @ G @ D @ C @ E @ A ) ) @ F ) @ ( hAPP @ ( c_Option_Ooption_OSome @ D ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_map__upds__twist) ).

thf(8792,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ G ) @ F ) @ ( hAPP @ ( c_List_Oset @ G ) @ E ) ) )
     => ( ( c_Map_Omap__upds @ G @ D @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ G @ ( tc_Option_Ooption @ D ) ) @ C ) @ F ) @ ( hAPP @ ( c_Option_Ooption_OSome @ D ) @ B ) ) @ E @ A )
        = ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ G @ ( tc_Option_Ooption @ D ) ) @ ( c_Map_Omap__upds @ G @ D @ C @ E @ A ) ) @ F ) @ ( hAPP @ ( c_Option_Ooption_OSome @ D ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1630]) ).

thf(1058,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i] :
          ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ C ) @ B ) )
         => ( hBOOL @ ( hAPP @ A @ C ) ) )
    <=> ! [C: $i] :
          ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Nat_Onat ) @ C ) @ ( hAPP @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ B ) ) )
         => ( hBOOL @ ( hAPP @ A @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_all__nat__less__eq) ).

thf(6299,plain,
    ! [A: $i,B: $i] :
      ( ( ! [C: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ C ) @ B ) )
           => ( hBOOL @ ( hAPP @ A @ C ) ) )
       => ! [C: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Nat_Onat ) @ C ) @ ( hAPP @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ B ) ) )
           => ( hBOOL @ ( hAPP @ A @ C ) ) ) )
      & ( ! [C: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Nat_Onat ) @ C ) @ ( hAPP @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ B ) ) )
           => ( hBOOL @ ( hAPP @ A @ C ) ) )
       => ! [C: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ C ) @ B ) )
           => ( hBOOL @ ( hAPP @ A @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1058]) ).

thf(532,axiom,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ ( tc_Option_Ooption @ A ) ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ ( tc_Option_Ooption @ A ) @ tc_HOL_Obool ) ) ) )
    <=> ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ A ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ A @ tc_HOL_Obool ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_finite__option__UNIV) ).

thf(4320,plain,
    ! [A: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ ( tc_Option_Ooption @ A ) ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ ( tc_Option_Ooption @ A ) @ tc_HOL_Obool ) ) ) )
       => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ A ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ A @ tc_HOL_Obool ) ) ) ) )
      & ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ A ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ A @ tc_HOL_Obool ) ) ) )
       => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ ( tc_Option_Ooption @ A ) ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ ( tc_Option_Ooption @ A ) @ tc_HOL_Obool ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[532]) ).

thf(109,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Groups_Oab__group__add @ E )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ C ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ C ) )
         => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) )
            = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ E ) @ ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ A ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ A ) @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_setsum__diff) ).

thf(2798,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Groups_Oab__group__add @ E )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ C ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ C ) )
         => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) )
            = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ E ) @ ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ A ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ A ) @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[109]) ).

thf(534,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ G @ ( hAPP @ ( c_Product__Type_Osnd @ F @ E ) @ D ) ) @ ( hAPP @ ( c_Product__Type_Ofst @ F @ E ) @ D ) ) )
     => ( ! [H: $i,I: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ G @ I ) @ H ) )
           => ( hBOOL @ ( hAPP @ C @ ( hAPP @ ( hAPP @ B @ H ) @ I ) ) ) )
       => ( hBOOL @ ( hAPP @ C @ ( hAPP @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ F @ E @ A ) @ B ) @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_exE__realizer) ).

thf(4333,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ G @ ( hAPP @ ( c_Product__Type_Osnd @ F @ E ) @ D ) ) @ ( hAPP @ ( c_Product__Type_Ofst @ F @ E ) @ D ) ) )
     => ( ! [H: $i,I: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ G @ I ) @ H ) )
           => ( hBOOL @ ( hAPP @ C @ ( hAPP @ ( hAPP @ B @ H ) @ I ) ) ) )
       => ( hBOOL @ ( hAPP @ C @ ( hAPP @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ F @ E @ A ) @ B ) @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[534]) ).

thf(1179,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( hAPP @ ( hAPP @ ( c_List_Omap @ D @ C ) @ B ) @ A )
        = ( c_List_Olist_ONil @ C ) )
    <=> ( A
        = ( c_List_Olist_ONil @ D ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_map__is__Nil__conv) ).

thf(6862,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( ( hAPP @ ( hAPP @ ( c_List_Omap @ D @ C ) @ B ) @ A )
          = ( c_List_Olist_ONil @ C ) )
       => ( A
          = ( c_List_Olist_ONil @ D ) ) )
      & ( ( A
          = ( c_List_Olist_ONil @ D ) )
       => ( ( hAPP @ ( hAPP @ ( c_List_Omap @ D @ C ) @ B ) @ A )
          = ( c_List_Olist_ONil @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1179]) ).

thf(1897,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( c_Wellfounded_Oacc @ D @ B ) ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ D ) @ A ) @ C ) ) @ ( c_Transitive__Closure_Ortrancl @ D @ B ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ A ) @ ( c_Wellfounded_Oacc @ D @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_acc__downwards) ).

thf(9872,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( c_Wellfounded_Oacc @ D @ B ) ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ D ) @ A ) @ C ) ) @ ( c_Transitive__Closure_Ortrancl @ D @ B ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ A ) @ ( c_Wellfounded_Oacc @ D @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1897]) ).

thf(1235,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Nat_Onat @ B ) @ A )
      = ( hAPP @ ( c_List_Oset @ tc_Nat_Onat ) @ ( c_List_Oupt @ B @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_atLeastLessThan__upt) ).

thf(7091,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Nat_Onat @ B ) @ A )
      = ( hAPP @ ( c_List_Oset @ tc_Nat_Onat ) @ ( c_List_Oupt @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1235]) ).

thf(132,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_Fun_Oinj__on @ D @ C @ B @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ D @ tc_HOL_Obool ) ) )
     => ( ( hAPP @ ( c_Set_Ovimage @ D @ C @ B ) @ ( hAPP @ ( c_Set_Oimage @ D @ C @ B ) @ A ) )
        = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_inj__vimage__image__eq) ).

thf(2877,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_Fun_Oinj__on @ D @ C @ B @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ D @ tc_HOL_Obool ) ) )
     => ( ( hAPP @ ( c_Set_Ovimage @ D @ C @ B ) @ ( hAPP @ ( c_Set_Oimage @ D @ C @ B ) @ A ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[132]) ).

thf(588,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_Fun_Oinj__on @ D @ ( tc_fun @ ( tc_fun @ D @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ D @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ D @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ D @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ D ) @ ( c_Equiv__Relations_Oquotient @ D ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ ( tc_fun @ D @ tc_HOL_Obool ) @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( c_Set_Oinsert @ D ) ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) ) ) @ C ) @ B )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ A ) @ B ) )
       => ( ( hAPP @ ( hAPP @ ( c_Equiv__Relations_Oquotient @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) ) ) @ C )
          = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ ( tc_fun @ D @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Equiv__Relations_Oquotient @ D ) @ B ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Equiv__Relations_Oquotient @ D ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_quotient__diff1) ).

thf(4518,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_Fun_Oinj__on @ D @ ( tc_fun @ ( tc_fun @ D @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ D @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ D @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ D @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ D ) @ ( c_Equiv__Relations_Oquotient @ D ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ ( tc_fun @ D @ tc_HOL_Obool ) @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( c_Set_Oinsert @ D ) ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) ) ) @ C ) @ B )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ A ) @ B ) )
       => ( ( hAPP @ ( hAPP @ ( c_Equiv__Relations_Oquotient @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) ) ) @ C )
          = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ ( tc_fun @ D @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Equiv__Relations_Oquotient @ D ) @ B ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Equiv__Relations_Oquotient @ D ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[588]) ).

thf(1637,axiom,
    ! [A: $i,B: $i] :
      ( ( ( B
          = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
       => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ A )
          = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) )
      & ( ( B
         != ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
       => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ A )
          = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ B ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__eq__if) ).

thf(8835,plain,
    ! [A: $i,B: $i] :
      ( ( ( B
          = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
       => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ A )
          = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) )
      & ( ( B
         != ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
       => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ A )
          = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ B ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1637]) ).

thf(733,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ E @ D @ C ) @ B ) @ A )
      = ( hAPP @ ( hAPP @ B @ ( hAPP @ ( c_Product__Type_Ofst @ E @ D ) @ A ) ) @ ( hAPP @ ( c_Product__Type_Osnd @ E @ D ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_split__beta) ).

thf(5031,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ E @ D @ C ) @ B ) @ A )
      = ( hAPP @ ( hAPP @ B @ ( hAPP @ ( c_Product__Type_Ofst @ E @ D ) @ A ) ) @ ( hAPP @ ( c_Product__Type_Osnd @ E @ D ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[733]) ).

thf(2400,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( hAPP @ ( c_List_Oset @ D ) @ B ) ) )
     => ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( hAPP @ ( c_List_Oset @ D ) @ ( c_List_Oremove1 @ D @ A @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_notin__set__remove1) ).

thf(11860,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( hAPP @ ( c_List_Oset @ D ) @ B ) ) )
     => ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( hAPP @ ( c_List_Oset @ D ) @ ( c_List_Oremove1 @ D @ A @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2400]) ).

thf(1400,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ C ) @ B ) )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ B ) @ C ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ B ) ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_add__diff__assoc) ).

thf(7774,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ C ) @ B ) )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ B ) @ C ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ B ) ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1400]) ).

thf(1844,axiom,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ tc_Nat_Onat ) @ ( hAPP @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Nat_Onat @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_finite__atLeastLessThan) ).

thf(9640,plain,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ tc_Nat_Onat ) @ ( hAPP @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Nat_Onat @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1844]) ).

thf(916,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( C
      = ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ B @ A ) @ ( hAPP @ ( c_Product__Type_Ofst @ B @ A ) @ C ) ) @ ( hAPP @ ( c_Product__Type_Osnd @ B @ A ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_surjective__pairing) ).

thf(5694,plain,
    ! [A: $i,B: $i,C: $i] :
      ( C
      = ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ B @ A ) @ ( hAPP @ ( c_Product__Type_Ofst @ B @ A ) @ C ) ) @ ( hAPP @ ( c_Product__Type_Osnd @ B @ A ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[916]) ).

thf(1908,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i,I: $i,J: $i,K: $i,L: $i] :
      ( ( c_Com_Ocom_Ocom__rec @ L @ K @ J @ I @ H @ G @ F @ E @ D @ ( c_Com_Ocom_OLocal @ C @ B @ A ) )
      = ( hAPP @ ( hAPP @ ( hAPP @ ( hAPP @ I @ C ) @ B ) @ A ) @ ( c_Com_Ocom_Ocom__rec @ L @ K @ J @ I @ H @ G @ F @ E @ D @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_com_Orecs_I3_J) ).

thf(9910,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i,I: $i,J: $i,K: $i,L: $i] :
      ( ( c_Com_Ocom_Ocom__rec @ L @ K @ J @ I @ H @ G @ F @ E @ D @ ( c_Com_Ocom_OLocal @ C @ B @ A ) )
      = ( hAPP @ ( hAPP @ ( hAPP @ ( hAPP @ I @ C ) @ B ) @ A ) @ ( c_Com_Ocom_Ocom__rec @ L @ K @ J @ I @ H @ G @ F @ E @ D @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1908]) ).

thf(1361,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ B ) ) @ ( hAPP @ c_Nat_OSuc @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ B ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_diff__Suc__Suc) ).

thf(7631,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ B ) ) @ ( hAPP @ c_Nat_OSuc @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ B ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1361]) ).

thf(1781,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ D ) @ C ) ) @ ( c_Transitive__Closure_Otrancl @ E @ B ) ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ C ) @ A ) ) @ ( c_Transitive__Closure_Otrancl @ E @ B ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ D ) @ A ) ) @ ( c_Transitive__Closure_Otrancl @ E @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_trancl__trans) ).

thf(9397,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ D ) @ C ) ) @ ( c_Transitive__Closure_Otrancl @ E @ B ) ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ C ) @ A ) ) @ ( c_Transitive__Closure_Otrancl @ E @ B ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ D ) @ A ) ) @ ( c_Transitive__Closure_Otrancl @ E @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1781]) ).

thf(401,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Int__left__commute) ).

thf(3846,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[401]) ).

thf(1958,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ E @ D ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Natural_Oevalc @ C ) @ D ) @ B ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Natural_Oevalc @ ( c_Com_Ocom_OCond @ E @ C @ A ) ) @ D ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_evalc_OIfTrue) ).

thf(10120,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ E @ D ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Natural_Oevalc @ C ) @ D ) @ B ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Natural_Oevalc @ ( c_Com_Ocom_OCond @ E @ C @ A ) ) @ D ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1958]) ).

thf(920,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hAPP @ ( c_List_Ofoldl @ F @ E @ D @ C ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ E ) @ B ) @ A ) )
      = ( hAPP @ ( c_List_Ofoldl @ F @ E @ D @ ( hAPP @ ( hAPP @ D @ C ) @ B ) ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_foldl__Cons) ).

thf(5705,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hAPP @ ( c_List_Ofoldl @ F @ E @ D @ C ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ E ) @ B ) @ A ) )
      = ( hAPP @ ( c_List_Ofoldl @ F @ E @ D @ ( hAPP @ ( hAPP @ D @ C ) @ B ) ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[920]) ).

thf(1385,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Fun_Oinj__on @ E @ D @ C @ B )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ A ) @ B ) )
       => ( c_Fun_Oinj__on @ E @ D @ C @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_subset__inj__on) ).

thf(7717,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Fun_Oinj__on @ E @ D @ C @ B )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ A ) @ B ) )
       => ( c_Fun_Oinj__on @ E @ D @ C @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1385]) ).

thf(225,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Fun_Oinj__on @ E @ D @ C @ B )
     => ( ( c_Fun_Oinj__on @ E @ D @ C @ A )
       => ( c_Fun_Oinj__on @ E @ D @ C @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_inj__on__Int) ).

thf(3210,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Fun_Oinj__on @ E @ D @ C @ B )
     => ( ( c_Fun_Oinj__on @ E @ D @ C @ A )
       => ( c_Fun_Oinj__on @ E @ D @ C @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[225]) ).

thf(1242,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C = B )
     => ( C
        = ( hAPP @ ( hAPP @ ( c_List_Oappend @ A ) @ ( c_List_Olist_ONil @ A ) ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_eq__Nil__appendI) ).

thf(7111,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C = B )
     => ( C
        = ( hAPP @ ( hAPP @ ( c_List_Oappend @ A ) @ ( c_List_Olist_ONil @ A ) ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1242]) ).

thf(1778,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( hAPP @ E @ D )
        = ( hAPP @ ( c_Option_Ooption_OSome @ C ) @ B ) )
     => ( ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ A ) @ D ) @ ( c_Map_Odom @ A @ C @ E ) )
        = ( c_Map_Odom @ A @ C @ E ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_insert__dom) ).

thf(9387,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( hAPP @ E @ D )
        = ( hAPP @ ( c_Option_Ooption_OSome @ C ) @ B ) )
     => ( ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ A ) @ D ) @ ( c_Map_Odom @ A @ C @ E ) )
        = ( c_Map_Odom @ A @ C @ E ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1778]) ).

thf(2298,axiom,
    ! [A: $i] :
      ( ~ ~ ( hBOOL @ A )
      | ( hBOOL @ ( hAPP @ c_fNot @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_c__fNot__2) ).

thf(11403,plain,
    ! [A: $i] :
      ( ( hBOOL @ A )
      | ( hBOOL @ ( hAPP @ c_fNot @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2298]) ).

thf(2317,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_New__Random__Sequence_Opos__single @ E ) @ D ) @ C ) @ B ) @ A )
      = ( c_New__DSequence_Opos__single @ E @ D ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_pos__single__def) ).

thf(11496,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_New__Random__Sequence_Opos__single @ E ) @ D ) @ C ) @ B ) @ A )
      = ( c_New__DSequence_Opos__single @ E @ D ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2317]) ).

thf(988,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i] :
          ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ E ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) ) )
         => ( hBOOL @ ( hAPP @ A @ E ) ) )
    <=> ( ! [E: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ E ) @ C ) )
           => ( hBOOL @ ( hAPP @ A @ E ) ) )
        & ! [E: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ E ) @ B ) )
           => ( hBOOL @ ( hAPP @ A @ E ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_ball__Un) ).

thf(5992,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ! [E: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ E ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) ) )
           => ( hBOOL @ ( hAPP @ A @ E ) ) )
       => ( ! [E: $i] :
              ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ E ) @ C ) )
             => ( hBOOL @ ( hAPP @ A @ E ) ) )
          & ! [E: $i] :
              ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ E ) @ B ) )
             => ( hBOOL @ ( hAPP @ A @ E ) ) ) ) )
      & ( ( ! [E: $i] :
              ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ E ) @ C ) )
             => ( hBOOL @ ( hAPP @ A @ E ) ) )
          & ! [E: $i] :
              ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ E ) @ B ) )
             => ( hBOOL @ ( hAPP @ A @ E ) ) ) )
       => ! [E: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ E ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) ) )
           => ( hBOOL @ ( hAPP @ A @ E ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[988]) ).

thf(1074,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Relation_ORange @ C @ D ) @ B ) ) @ ( hAPP @ ( c_Relation_ORange @ C @ D ) @ A ) ) ) @ ( hAPP @ ( c_Relation_ORange @ C @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ ( tc_prod @ C @ D ) @ tc_HOL_Obool ) ) @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Range__Diff__subset) ).

thf(6363,plain,
    ! [A: $i,B: $i,C: $i,D: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Relation_ORange @ C @ D ) @ B ) ) @ ( hAPP @ ( c_Relation_ORange @ C @ D ) @ A ) ) ) @ ( hAPP @ ( c_Relation_ORange @ C @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ ( tc_prod @ C @ D ) @ tc_HOL_Obool ) ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1074]) ).

thf(370,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Fun_Ocomp @ F @ E @ D @ ( hAPP @ ( c_COMBK @ E @ F ) @ C ) ) @ B ) @ A )
      = C ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_K__record__comp) ).

thf(3728,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Fun_Ocomp @ F @ E @ D @ ( hAPP @ ( c_COMBK @ E @ F ) @ C ) ) @ B ) @ A )
      = C ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[370]) ).

thf(130,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ D @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A )
      = ( c_Complete__Lattice_OSup__class_OSup @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( hAPP @ ( c_Set_Oimage @ D @ ( tc_fun @ C @ tc_HOL_Obool ) @ A ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_UNION__eq__Union__image) ).

thf(2868,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ D @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A )
      = ( c_Complete__Lattice_OSup__class_OSup @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( hAPP @ ( c_Set_Oimage @ D @ ( tc_fun @ C @ tc_HOL_Obool ) @ A ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[130]) ).

thf(1787,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Odrop @ C ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ C ) @ B ) @ A ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_drop__1__Cons) ).

thf(9426,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Odrop @ C ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ C ) @ B ) @ A ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1787]) ).

thf(1606,axiom,
    ! [A: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
      = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__0__right) ).

thf(8687,plain,
    ! [A: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
      = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1606]) ).

thf(2176,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ B ) @ A ) ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ C ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ A ) @ B ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ C ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_dvd__diffD1) ).

thf(10914,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ B ) @ A ) ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ C ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ A ) @ B ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ C ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2176]) ).

thf(4,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Divides_Odivmod__int__rel @ C @ C ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ tc_Int_Oint @ tc_Int_Oint ) @ B ) @ A ) ) )
     => ( ( C
         != ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) )
       => ( A
          = ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_self__remainder) ).

thf(2417,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Divides_Odivmod__int__rel @ C @ C ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ tc_Int_Oint @ tc_Int_Oint ) @ B ) @ A ) ) )
     => ( ( C
         != ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) )
       => ( A
          = ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[4]) ).

thf(2418,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( hBOOL @ ( hAPP @ ( c_Divides_Odivmod__int__rel @ C @ C ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ tc_Int_Oint @ tc_Int_Oint ) @ B ) @ A ) ) )
      | ( C
        = ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) )
      | ( A
        = ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) ),
    inference(cnf,[status(esa)],[2417]) ).

thf(2419,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( C
        = ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) )
      | ( A
        = ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) )
      | ~ ( hBOOL @ ( hAPP @ ( c_Divides_Odivmod__int__rel @ C @ C ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ tc_Int_Oint @ tc_Int_Oint ) @ B ) @ A ) ) ) ),
    inference(lifteq,[status(thm)],[2418]) ).

thf(562,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ A ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_UnI2) ).

thf(4421,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ A ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[562]) ).

thf(167,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( c_Relation_ORange @ C @ B ) @ ( c_Complete__Lattice_OSup__class_OSup @ ( tc_fun @ ( tc_prod @ C @ B ) @ tc_HOL_Obool ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ ( tc_fun @ ( tc_prod @ C @ B ) @ tc_HOL_Obool ) @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ A ) @ ( c_Relation_ORange @ C @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Range__Union) ).

thf(3010,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( c_Relation_ORange @ C @ B ) @ ( c_Complete__Lattice_OSup__class_OSup @ ( tc_fun @ ( tc_prod @ C @ B ) @ tc_HOL_Obool ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ ( tc_fun @ ( tc_prod @ C @ B ) @ tc_HOL_Obool ) @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ A ) @ ( c_Relation_ORange @ C @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[167]) ).

thf(789,axiom,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ B ) @ A ) )
       => ( ( c_Nitpick_Ocard_H @ B @ A )
          = ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ B ) ) @ ( c_Hilbert__Choice_OEps @ ( tc_List_Olist @ B ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ ( tc_List_Olist @ B ) @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( tc_List_Olist @ B ) ) @ c_fconj ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ ( tc_List_Olist @ B ) @ ( tc_fun @ B @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ B @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ B @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( tc_List_Olist @ B ) ) @ c_fequal ) @ ( c_List_Oset @ B ) ) ) @ A ) ) ) @ ( c_List_Odistinct @ B ) ) ) ) ) )
      & ( ~ ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ B ) @ A ) )
       => ( ( c_Nitpick_Ocard_H @ B @ A )
          = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Nitpick_Ocard_H__def) ).

thf(5253,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ B ) @ A ) )
       => ( ( c_Nitpick_Ocard_H @ B @ A )
          = ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ B ) ) @ ( c_Hilbert__Choice_OEps @ ( tc_List_Olist @ B ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ ( tc_List_Olist @ B ) @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( tc_List_Olist @ B ) ) @ c_fconj ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ ( tc_List_Olist @ B ) @ ( tc_fun @ B @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ B @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ B @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( tc_List_Olist @ B ) ) @ c_fequal ) @ ( c_List_Oset @ B ) ) ) @ A ) ) ) @ ( c_List_Odistinct @ B ) ) ) ) ) )
      & ( ~ ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ B ) @ A ) )
       => ( ( c_Nitpick_Ocard_H @ B @ A )
          = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[789]) ).

thf(708,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Fun_Oinj__on @ E @ D @ C @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ E @ tc_HOL_Obool ) ) )
     => ( ( c_Fun_Oinj__on @ B @ E @ A @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ B @ tc_HOL_Obool ) ) )
       => ( c_Fun_Oinj__on @ B @ D @ ( hAPP @ ( c_Fun_Ocomp @ E @ D @ B @ C ) @ A ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_inj__comp) ).

thf(4938,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Fun_Oinj__on @ E @ D @ C @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ E @ tc_HOL_Obool ) ) )
     => ( ( c_Fun_Oinj__on @ B @ E @ A @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ B @ tc_HOL_Obool ) ) )
       => ( c_Fun_Oinj__on @ B @ D @ ( hAPP @ ( c_Fun_Ocomp @ E @ D @ B @ C ) @ A ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[708]) ).

thf(2089,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ C ) @ B ) )
        & ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ C ) ) )
     => ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) )
          & ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ A ) @ B ) ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ C ) @ A ) )
          & ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ A ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_dvd_Oless__trans) ).

thf(10620,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ C ) @ B ) )
        & ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ C ) ) )
     => ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) )
          & ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ A ) @ B ) ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ C ) @ A ) )
          & ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ A ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2089]) ).

thf(791,axiom,
    ! [A: $i] :
      ( ( c_Transitive__Closure_Ortrancl @ A @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ ( tc_prod @ A @ A ) @ tc_HOL_Obool ) ) )
      = ( c_Relation_OId @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_rtrancl__empty) ).

thf(5263,plain,
    ! [A: $i] :
      ( ( c_Transitive__Closure_Ortrancl @ A @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ ( tc_prod @ A @ A ) @ tc_HOL_Obool ) ) )
      = ( c_Relation_OId @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[791]) ).

thf(2184,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i,I: $i,J: $i,K: $i] :
      ( ( c_Com_Ocom_Ocom__case @ K @ J @ I @ H @ G @ F @ E @ D @ C @ ( c_Com_Ocom_OSemi @ B @ A ) )
      = ( hAPP @ ( hAPP @ G @ B ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_com_Osimps_I67_J) ).

thf(10936,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i,I: $i,J: $i,K: $i] :
      ( ( c_Com_Ocom_Ocom__case @ K @ J @ I @ H @ G @ F @ E @ D @ C @ ( c_Com_Ocom_OSemi @ B @ A ) )
      = ( hAPP @ ( hAPP @ G @ B ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2184]) ).

thf(557,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ C @ D ) ) @ ( hAPP @ B @ D ) ) )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ tc_Nat_Onat @ tc_Nat_Onat ) @ B ) @ ( hAPP @ ( c_SetInterval_Oord__class_OlessThan @ tc_Nat_Onat ) @ A ) ) ) @ ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ tc_Nat_Onat @ tc_Nat_Onat ) @ C ) @ ( hAPP @ ( c_SetInterval_Oord__class_OlessThan @ tc_Nat_Onat ) @ A ) ) )
        = ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ tc_Nat_Onat @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ tc_Nat_Onat @ tc_Nat_Onat @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_Nat_Onat @ ( tc_fun @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_Nat_Onat ) @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) ) @ B ) ) @ C ) ) @ ( hAPP @ ( c_SetInterval_Oord__class_OlessThan @ tc_Nat_Onat ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_sum__diff__distrib) ).

thf(4406,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ C @ D ) ) @ ( hAPP @ B @ D ) ) )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ tc_Nat_Onat @ tc_Nat_Onat ) @ B ) @ ( hAPP @ ( c_SetInterval_Oord__class_OlessThan @ tc_Nat_Onat ) @ A ) ) ) @ ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ tc_Nat_Onat @ tc_Nat_Onat ) @ C ) @ ( hAPP @ ( c_SetInterval_Oord__class_OlessThan @ tc_Nat_Onat ) @ A ) ) )
        = ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ tc_Nat_Onat @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ tc_Nat_Onat @ tc_Nat_Onat @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_Nat_Onat @ ( tc_fun @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_Nat_Onat ) @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) ) @ B ) ) @ C ) ) @ ( hAPP @ ( c_SetInterval_Oord__class_OlessThan @ tc_Nat_Onat ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[557]) ).

thf(650,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Oab__semigroup__mult @ D )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ B ) ) @ A )
        = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_ab__semigroup__mult__class_Omult__ac_I1_J) ).

thf(4744,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Oab__semigroup__mult @ D )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ B ) ) @ A )
        = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ D ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[650]) ).

thf(1483,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ E ) @ D ) @ C )
        = ( hAPP @ ( hAPP @ ( c_List_Oappend @ E ) @ B ) @ A ) )
    <=> ( ( ( B
            = ( c_List_Olist_ONil @ E ) )
          & ( ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ E ) @ D ) @ C )
            = A ) )
        | ? [F: $i] :
            ( ( ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ E ) @ D ) @ F )
              = B )
            & ( C
              = ( hAPP @ ( hAPP @ ( c_List_Oappend @ E ) @ F ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Cons__eq__append__conv) ).

thf(8132,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ E ) @ D ) @ C )
          = ( hAPP @ ( hAPP @ ( c_List_Oappend @ E ) @ B ) @ A ) )
       => ( ( ( B
              = ( c_List_Olist_ONil @ E ) )
            & ( ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ E ) @ D ) @ C )
              = A ) )
          | ? [F: $i] :
              ( ( ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ E ) @ D ) @ F )
                = B )
              & ( C
                = ( hAPP @ ( hAPP @ ( c_List_Oappend @ E ) @ F ) @ A ) ) ) ) )
      & ( ( ( ( B
              = ( c_List_Olist_ONil @ E ) )
            & ( ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ E ) @ D ) @ C )
              = A ) )
          | ? [F: $i] :
              ( ( ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ E ) @ D ) @ F )
                = B )
              & ( C
                = ( hAPP @ ( hAPP @ ( c_List_Oappend @ E ) @ F ) @ A ) ) ) )
       => ( ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ E ) @ D ) @ C )
          = ( hAPP @ ( hAPP @ ( c_List_Oappend @ E ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1483]) ).

thf(1026,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( c_Finite__Set_Ofold1 @ C @ B ) @ A )
      = ( c_HOL_OThe @ C @ ( c_Finite__Set_Ofold1Set @ C @ B @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_fold1__def) ).

thf(6164,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( c_Finite__Set_Ofold1 @ C @ B ) @ A )
      = ( c_HOL_OThe @ C @ ( c_Finite__Set_Ofold1Set @ C @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1026]) ).

thf(2210,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ c_Com_OWT @ B ) )
     => ( hBOOL @ ( hAPP @ c_Com_OWT @ ( c_Com_Ocom_OWhile @ A @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_While) ).

thf(11046,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ c_Com_OWT @ B ) )
     => ( hBOOL @ ( hAPP @ c_Com_OWT @ ( c_Com_Ocom_OWhile @ A @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2210]) ).

thf(1514,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_termination__basic__simps_I2_J) ).

thf(8275,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1514]) ).

thf(782,axiom,
    ! [A: $i,B: $i] :
      ( ( ( c_Orderings_Obot__class_Obot @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) )
        = ( c_SetInterval_Oord_OgreaterThanAtMost @ tc_Nat_Onat @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ tc_Nat_Onat @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ tc_Nat_Onat @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) @ tc_Nat_Onat ) @ ( c_COMBS @ tc_Nat_Onat @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ tc_Nat_Onat @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ tc_Nat_Onat ) @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ tc_Nat_Onat ) @ c_fconj ) ) @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ tc_Nat_Onat ) @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ tc_HOL_Obool @ tc_Nat_Onat ) @ c_fNot ) ) @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) ) ) ) @ B @ A ) )
    <=> ~ ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) )
          & ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ A ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_dvd_OgreaterThanAtMost__empty__iff2) ).

thf(5213,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( c_Orderings_Obot__class_Obot @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) )
          = ( c_SetInterval_Oord_OgreaterThanAtMost @ tc_Nat_Onat @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ tc_Nat_Onat @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ tc_Nat_Onat @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) @ tc_Nat_Onat ) @ ( c_COMBS @ tc_Nat_Onat @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ tc_Nat_Onat @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ tc_Nat_Onat ) @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ tc_Nat_Onat ) @ c_fconj ) ) @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ tc_Nat_Onat ) @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ tc_HOL_Obool @ tc_Nat_Onat ) @ c_fNot ) ) @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) ) ) ) @ B @ A ) )
       => ~ ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) )
            & ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ A ) @ B ) ) ) )
      & ( ~ ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) )
            & ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ A ) @ B ) ) )
       => ( ( c_Orderings_Obot__class_Obot @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) )
          = ( c_SetInterval_Oord_OgreaterThanAtMost @ tc_Nat_Onat @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ tc_Nat_Onat @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ tc_Nat_Onat @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) @ tc_Nat_Onat ) @ ( c_COMBS @ tc_Nat_Onat @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ tc_Nat_Onat @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ tc_Nat_Onat ) @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ tc_Nat_Onat ) @ c_fconj ) ) @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ tc_Nat_Onat ) @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ tc_HOL_Obool @ tc_Nat_Onat ) @ c_fNot ) ) @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) ) ) ) @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[782]) ).

thf(175,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ B ) @ ( c_Complete__Lattice_OSup__class_OSup @ ( tc_fun @ C @ tc_HOL_Obool ) @ A ) ) )
     => ~ ! [D: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ B ) @ D ) )
           => ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ D ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_UnionE) ).

thf(3032,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ B ) @ ( c_Complete__Lattice_OSup__class_OSup @ ( tc_fun @ C @ tc_HOL_Obool ) @ A ) ) )
     => ~ ! [D: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ B ) @ D ) )
           => ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ D ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[175]) ).

thf(1154,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nat__add__left__commute) ).

thf(6763,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1154]) ).

thf(1114,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ D ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ C ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ B ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ A @ D ) ) ) @ B ) ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zmult2__lemma__aux2) ).

thf(6573,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ D ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ C ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ B ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ A @ D ) ) ) @ B ) ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1114]) ).

thf(1672,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ B ) @ ( hAPP @ ( c_List_Oset @ C ) @ A ) ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_length__pos__if__in__set) ).

thf(8968,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ B ) @ ( hAPP @ ( c_List_Oset @ C ) @ A ) ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1672]) ).

thf(957,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ D @ ( tc_Option_Ooption @ C ) ) @ ( hAPP @ ( c_COMBK @ ( tc_Option_Ooption @ C ) @ D ) @ ( c_Option_Ooption_ONone @ C ) ) ) @ B ) @ ( c_Option_Ooption_ONone @ C ) ) @ A )
      = ( c_Option_Ooption_ONone @ C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_empty__upd__none) ).

thf(5865,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ D @ ( tc_Option_Ooption @ C ) ) @ ( hAPP @ ( c_COMBK @ ( tc_Option_Ooption @ C ) @ D ) @ ( c_Option_Ooption_ONone @ C ) ) ) @ B ) @ ( c_Option_Ooption_ONone @ C ) ) @ A )
      = ( c_Option_Ooption_ONone @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[957]) ).

thf(2113,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ D ) @ ( hAPP @ ( c_Relation_OImage @ C @ E @ B ) @ A ) ) )
     => ~ ! [F: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ C @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ C @ E ) @ F ) @ D ) ) @ B ) )
           => ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ F ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_ImageE) ).

thf(10688,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ D ) @ ( hAPP @ ( c_Relation_OImage @ C @ E @ B ) @ A ) ) )
     => ~ ! [F: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ C @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ C @ E ) @ F ) @ D ) ) @ B ) )
           => ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ F ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2113]) ).

thf(1210,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_Finite__Set_Ofolding__one @ D @ C @ B )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ A ) )
       => ( ( A
           != ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) )
         => ( ! [E: $i,F: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ ( hAPP @ ( hAPP @ C @ E ) @ F ) ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ E ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ F ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) ) ) )
           => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ ( hAPP @ B @ A ) ) @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_folding__one_Oclosed) ).

thf(6978,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_Finite__Set_Ofolding__one @ D @ C @ B )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ A ) )
       => ( ( A
           != ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) )
         => ( ! [E: $i,F: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ ( hAPP @ ( hAPP @ C @ E ) @ F ) ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ E ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ F ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) ) ) )
           => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ ( hAPP @ B @ A ) ) @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1210]) ).

thf(368,axiom,
    ! [A: $i] :
      ~ ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ ( tc_List_Olist @ A ) ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ ( tc_List_Olist @ A ) @ tc_HOL_Obool ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_infinite__UNIV__listI) ).

thf(3723,plain,
    ! [A: $i] :
      ~ ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ ( tc_List_Olist @ A ) ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ ( tc_List_Olist @ A ) @ tc_HOL_Obool ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[368]) ).

thf(1969,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i] :
          ( B
         != ( hAPP @ ( c_Option_Ooption_OSome @ A ) @ C ) )
    <=> ( B
        = ( c_Option_Ooption_ONone @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_not__Some__eq) ).

thf(10163,plain,
    ! [A: $i,B: $i] :
      ( ( ! [C: $i] :
            ( B
           != ( hAPP @ ( c_Option_Ooption_OSome @ A ) @ C ) )
       => ( B
          = ( c_Option_Ooption_ONone @ A ) ) )
      & ( ( B
          = ( c_Option_Ooption_ONone @ A ) )
       => ! [C: $i] :
            ( B
           != ( hAPP @ ( c_Option_Ooption_OSome @ A ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1969]) ).

thf(1312,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( hAPP @ ( hAPP @ ( c_List_Oappend @ E ) @ D ) @ C )
        = ( hAPP @ ( hAPP @ ( c_List_Oappend @ E ) @ B ) @ A ) )
    <=> ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ E ) ) @ D ) ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ E ) ) @ B ) ) )
         => ( ( D
              = ( hAPP @ ( hAPP @ ( c_List_Otake @ E ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ E ) ) @ D ) ) @ B ) )
            & ( C
              = ( hAPP @ ( hAPP @ ( c_List_Oappend @ E ) @ ( hAPP @ ( hAPP @ ( c_List_Odrop @ E ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ E ) ) @ D ) ) @ B ) ) @ A ) ) ) )
        & ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ E ) ) @ D ) ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ E ) ) @ B ) ) )
         => ( ( ( hAPP @ ( hAPP @ ( c_List_Otake @ E ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ E ) ) @ B ) ) @ D )
              = B )
            & ( ( hAPP @ ( hAPP @ ( c_List_Oappend @ E ) @ ( hAPP @ ( hAPP @ ( c_List_Odrop @ E ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ E ) ) @ B ) ) @ D ) ) @ C )
              = A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_append__eq__append__conv__if) ).

thf(7404,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( ( hAPP @ ( hAPP @ ( c_List_Oappend @ E ) @ D ) @ C )
          = ( hAPP @ ( hAPP @ ( c_List_Oappend @ E ) @ B ) @ A ) )
       => ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ E ) ) @ D ) ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ E ) ) @ B ) ) )
           => ( ( D
                = ( hAPP @ ( hAPP @ ( c_List_Otake @ E ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ E ) ) @ D ) ) @ B ) )
              & ( C
                = ( hAPP @ ( hAPP @ ( c_List_Oappend @ E ) @ ( hAPP @ ( hAPP @ ( c_List_Odrop @ E ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ E ) ) @ D ) ) @ B ) ) @ A ) ) ) )
          & ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ E ) ) @ D ) ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ E ) ) @ B ) ) )
           => ( ( ( hAPP @ ( hAPP @ ( c_List_Otake @ E ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ E ) ) @ B ) ) @ D )
                = B )
              & ( ( hAPP @ ( hAPP @ ( c_List_Oappend @ E ) @ ( hAPP @ ( hAPP @ ( c_List_Odrop @ E ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ E ) ) @ B ) ) @ D ) ) @ C )
                = A ) ) ) ) )
      & ( ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ E ) ) @ D ) ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ E ) ) @ B ) ) )
           => ( ( D
                = ( hAPP @ ( hAPP @ ( c_List_Otake @ E ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ E ) ) @ D ) ) @ B ) )
              & ( C
                = ( hAPP @ ( hAPP @ ( c_List_Oappend @ E ) @ ( hAPP @ ( hAPP @ ( c_List_Odrop @ E ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ E ) ) @ D ) ) @ B ) ) @ A ) ) ) )
          & ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ E ) ) @ D ) ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ E ) ) @ B ) ) )
           => ( ( ( hAPP @ ( hAPP @ ( c_List_Otake @ E ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ E ) ) @ B ) ) @ D )
                = B )
              & ( ( hAPP @ ( hAPP @ ( c_List_Oappend @ E ) @ ( hAPP @ ( hAPP @ ( c_List_Odrop @ E ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ E ) ) @ B ) ) @ D ) ) @ C )
                = A ) ) ) )
       => ( ( hAPP @ ( hAPP @ ( c_List_Oappend @ E ) @ D ) @ C )
          = ( hAPP @ ( hAPP @ ( c_List_Oappend @ E ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1312]) ).

thf(1989,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( c_Equiv__Relations_Ocongruent @ F @ E @ D @ C )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ F @ F ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ F @ F ) @ B ) @ A ) ) @ D ) )
       => ( ( hAPP @ C @ B )
          = ( hAPP @ C @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_congruentD) ).

thf(10239,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( c_Equiv__Relations_Ocongruent @ F @ E @ D @ C )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ F @ F ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ F @ F ) @ B ) @ A ) ) @ D ) )
       => ( ( hAPP @ C @ B )
          = ( hAPP @ C @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1989]) ).

thf(569,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
      ( ( c_Finite__Set_Ofolding__image__simple @ H @ G @ F @ E @ D @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ G ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ G ) @ A ) )
         => ( ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ G @ tc_HOL_Obool ) ) @ B ) @ A )
              = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ G @ tc_HOL_Obool ) ) )
           => ( ( hAPP @ C @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ G @ tc_HOL_Obool ) ) @ B ) @ A ) )
              = ( hAPP @ ( hAPP @ F @ ( hAPP @ C @ B ) ) @ ( hAPP @ C @ A ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_folding__image__simple_Ounion__disjoint) ).

thf(4444,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
      ( ( c_Finite__Set_Ofolding__image__simple @ H @ G @ F @ E @ D @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ G ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ G ) @ A ) )
         => ( ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ G @ tc_HOL_Obool ) ) @ B ) @ A )
              = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ G @ tc_HOL_Obool ) ) )
           => ( ( hAPP @ C @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ G @ tc_HOL_Obool ) ) @ B ) @ A ) )
              = ( hAPP @ ( hAPP @ F @ ( hAPP @ C @ B ) ) @ ( hAPP @ C @ A ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[569]) ).

thf(1122,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ B ) @ A ) ) )
    <=> ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ B ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ A ) ) )
        & ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ B ) @ A ) )
         => ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ B ) @ C ) )
             => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ B ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) ) ) @ A ) ) )
            & ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ B ) @ C ) )
             => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ A ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_psubset__insert__iff) ).

thf(6625,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ B ) @ A ) ) )
       => ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ B ) @ A ) )
           => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ A ) ) )
          & ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ B ) @ A ) )
           => ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ B ) @ C ) )
               => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ B ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) ) ) @ A ) ) )
              & ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ B ) @ C ) )
               => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ A ) ) ) ) ) ) )
      & ( ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ B ) @ A ) )
           => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ A ) ) )
          & ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ B ) @ A ) )
           => ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ B ) @ C ) )
               => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ B ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) ) ) @ A ) ) )
              & ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ B ) @ C ) )
               => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ A ) ) ) ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ B ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1122]) ).

thf(323,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_Complete__Lattice_OSup__class_OSup @ ( tc_fun @ D @ tc_HOL_Obool ) @ ( hAPP @ ( c_Set_Oimage @ C @ ( tc_fun @ D @ tc_HOL_Obool ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ C @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ A ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Union__image__eq) ).

thf(3555,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_Complete__Lattice_OSup__class_OSup @ ( tc_fun @ D @ tc_HOL_Obool ) @ ( hAPP @ ( c_Set_Oimage @ C @ ( tc_fun @ D @ tc_HOL_Obool ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ C @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ A ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[323]) ).

thf(2131,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ C ) ) @ B ) @ ( hAPP @ ( c_Relation_Oconverse @ C @ D ) @ A ) ) )
     => ~ ! [E: $i,F: $i] :
            ( ( B
              = ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ C ) @ F ) @ E ) )
           => ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ C @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ C @ D ) @ E ) @ F ) ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_converseE) ).

thf(10752,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ C ) ) @ B ) @ ( hAPP @ ( c_Relation_Oconverse @ C @ D ) @ A ) ) )
     => ~ ! [E: $i,F: $i] :
            ( ( B
              = ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ C ) @ F ) @ E ) )
           => ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ C @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ C @ D ) @ E ) @ F ) ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2131]) ).

thf(1200,axiom,
    ! [A: $i,B: $i,C: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_List_Oset @ C ) @ ( hAPP @ ( hAPP @ ( c_List_Otake @ C ) @ B ) @ A ) ) ) @ ( hAPP @ ( c_List_Oset @ C ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_set__take__subset) ).

thf(6935,plain,
    ! [A: $i,B: $i,C: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_List_Oset @ C ) @ ( hAPP @ ( hAPP @ ( c_List_Otake @ C ) @ B ) @ A ) ) ) @ ( hAPP @ ( c_List_Oset @ C ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1200]) ).

thf(2016,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ D ) @ C ) @ B ) ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ B ) @ ( hAPP @ ( c_Relation_ORange @ E @ D ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_RangeI) ).

thf(10353,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ D ) @ C ) @ B ) ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ B ) @ ( hAPP @ ( c_Relation_ORange @ E @ D ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2016]) ).

thf(1788,axiom,
    ! [A: $i,B: $i] :
      ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) )
    <=> ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ A ) @ ( hAPP @ c_Nat_OSuc @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_not__less__eq) ).

thf(9429,plain,
    ! [A: $i,B: $i] :
      ( ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ A ) @ ( hAPP @ c_Nat_OSuc @ B ) ) ) )
      & ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ A ) @ ( hAPP @ c_Nat_OSuc @ B ) ) )
       => ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1788]) ).

thf(1052,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Wellfounded_Owf @ B @ A )
    <=> ( c_Wellfounded_Owf @ B @ ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ B @ B @ B ) @ A ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_wf__comp__self) ).

thf(6271,plain,
    ! [A: $i,B: $i] :
      ( ( ( c_Wellfounded_Owf @ B @ A )
       => ( c_Wellfounded_Owf @ B @ ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ B @ B @ B ) @ A ) @ A ) ) )
      & ( ( c_Wellfounded_Owf @ B @ ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ B @ B @ B ) @ A ) @ A ) )
       => ( c_Wellfounded_Owf @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1052]) ).

thf(1333,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ A ) )
     => ( ( c_List_Oupt @ A @ B )
        = ( c_List_Olist_ONil @ tc_Nat_Onat ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_upt__conv__Nil) ).

thf(7523,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ A ) )
     => ( ( c_List_Oupt @ A @ B )
        = ( c_List_Olist_ONil @ tc_Nat_Onat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1333]) ).

thf(1954,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( F = E )
       => ( ( hAPP @ ( c_Map_Omap__of @ D @ C @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ ( tc_prod @ D @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ C ) @ F ) @ B ) ) @ A ) ) @ E )
          = ( hAPP @ ( c_Option_Ooption_OSome @ C ) @ B ) ) )
      & ( ( F != E )
       => ( ( hAPP @ ( c_Map_Omap__of @ D @ C @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ ( tc_prod @ D @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ C ) @ F ) @ B ) ) @ A ) ) @ E )
          = ( hAPP @ ( c_Map_Omap__of @ D @ C @ A ) @ E ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_map__of__Cons__code_I2_J) ).

thf(10094,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( F = E )
       => ( ( hAPP @ ( c_Map_Omap__of @ D @ C @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ ( tc_prod @ D @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ C ) @ F ) @ B ) ) @ A ) ) @ E )
          = ( hAPP @ ( c_Option_Ooption_OSome @ C ) @ B ) ) )
      & ( ( F != E )
       => ( ( hAPP @ ( c_Map_Omap__of @ D @ C @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ ( tc_prod @ D @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ C ) @ F ) @ B ) ) @ A ) ) @ E )
          = ( hAPP @ ( c_Map_Omap__of @ D @ C @ A ) @ E ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1954]) ).

thf(1286,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ E @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ C ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ ( tc_prod @ E @ D ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ E @ D ) @ C ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ E @ D ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Sigma__Diff__distrib1) ).

thf(7293,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ E @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ C ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ ( tc_prod @ E @ D ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ E @ D ) @ C ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ E @ D ) @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1286]) ).

thf(1316,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( c_Wellfounded_Oacc @ D @ B ) ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_List_Olist @ D ) ) @ A ) @ ( c_Wellfounded_Oacc @ ( tc_List_Olist @ D ) @ ( c_List_Olistrel1 @ D @ B ) ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_List_Olist @ D ) ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ D ) @ C ) @ A ) ) @ ( c_Wellfounded_Oacc @ ( tc_List_Olist @ D ) @ ( c_List_Olistrel1 @ D @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Cons__acc__listrel1I) ).

thf(7448,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( c_Wellfounded_Oacc @ D @ B ) ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_List_Olist @ D ) ) @ A ) @ ( c_Wellfounded_Oacc @ ( tc_List_Olist @ D ) @ ( c_List_Olistrel1 @ D @ B ) ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_List_Olist @ D ) ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ D ) @ C ) @ A ) ) @ ( c_Wellfounded_Oacc @ ( tc_List_Olist @ D ) @ ( c_List_Olistrel1 @ D @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1316]) ).

thf(2260,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Com_Ocom_OCond @ E @ D @ C )
     != ( c_Com_Ocom_OWhile @ B @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_com_Osimps_I52_J) ).

thf(11248,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Com_Ocom_OCond @ E @ D @ C )
     != ( c_Com_Ocom_OWhile @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2260]) ).

thf(1888,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Natural_Oevalc @ ( c_Com_Ocom_OCond @ E @ D @ C ) ) @ B ) @ A ) )
     => ( ( ( hBOOL @ ( hAPP @ E @ B ) )
         => ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Natural_Oevalc @ D ) @ B ) @ A ) ) )
       => ~ ( ~ ( hBOOL @ ( hAPP @ E @ B ) )
           => ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Natural_Oevalc @ C ) @ B ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_evalc__elim__cases_I5_J) ).

thf(9825,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Natural_Oevalc @ ( c_Com_Ocom_OCond @ E @ D @ C ) ) @ B ) @ A ) )
     => ( ( ( hBOOL @ ( hAPP @ E @ B ) )
         => ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Natural_Oevalc @ D ) @ B ) @ A ) ) )
       => ~ ( ~ ( hBOOL @ ( hAPP @ E @ B ) )
           => ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Natural_Oevalc @ C ) @ B ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1888]) ).

thf(2264,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Com_Ocom_OWhile @ C @ B )
     != ( hAPP @ c_Com_Ocom_OBODY @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_com_Osimps_I58_J) ).

thf(11269,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Com_Ocom_OWhile @ C @ B )
     != ( hAPP @ c_Com_Ocom_OBODY @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2264]) ).

thf(200,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ E ) @ ( tc_List_Olist @ E ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ E ) @ ( tc_List_Olist @ E ) ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ E ) @ D ) @ C ) ) @ B ) ) @ ( c_List_Olistrel @ E @ A ) ) )
     => ~ ! [F: $i,G: $i] :
            ( ( B
              = ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ E ) @ F ) @ G ) )
           => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ D ) @ F ) ) @ A ) )
             => ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ E ) @ ( tc_List_Olist @ E ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ E ) @ ( tc_List_Olist @ E ) ) @ C ) @ G ) ) @ ( c_List_Olistrel @ E @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_listrel__Cons1) ).

thf(3126,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ E ) @ ( tc_List_Olist @ E ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ E ) @ ( tc_List_Olist @ E ) ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ E ) @ D ) @ C ) ) @ B ) ) @ ( c_List_Olistrel @ E @ A ) ) )
     => ~ ! [F: $i,G: $i] :
            ( ( B
              = ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ E ) @ F ) @ G ) )
           => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ D ) @ F ) ) @ A ) )
             => ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ E ) @ ( tc_List_Olist @ E ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ E ) @ ( tc_List_Olist @ E ) ) @ C ) @ G ) ) @ ( c_List_Olistrel @ E @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[200]) ).

thf(1322,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Omap @ F @ E ) @ D ) @ ( hAPP @ ( hAPP @ ( c_List_Omap @ C @ F ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_List_Omap @ C @ E ) @ ( hAPP @ ( c_Fun_Ocomp @ F @ E @ C @ D ) @ B ) ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_List_Omap_Ocompositionality) ).

thf(7485,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Omap @ F @ E ) @ D ) @ ( hAPP @ ( hAPP @ ( c_List_Omap @ C @ F ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_List_Omap @ C @ E ) @ ( hAPP @ ( c_Fun_Ocomp @ F @ E @ C @ D ) @ B ) ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1322]) ).

thf(1607,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ ( tc_Option_Ooption @ D ) ) @ ( hAPP @ ( c_Set_Oimage @ C @ ( tc_Option_Ooption @ D ) @ B ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) ) )
     => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ ( tc_Option_Ooption @ D ) ) @ ( hAPP @ ( c_Set_Oimage @ C @ ( tc_Option_Ooption @ D ) @ ( c_Map_Omap__add @ C @ D @ B @ ( c_Map_Omap__of @ C @ D @ A ) ) ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_finite__range__map__of__map__add) ).

thf(8690,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ ( tc_Option_Ooption @ D ) ) @ ( hAPP @ ( c_Set_Oimage @ C @ ( tc_Option_Ooption @ D ) @ B ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) ) )
     => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ ( tc_Option_Ooption @ D ) ) @ ( hAPP @ ( c_Set_Oimage @ C @ ( tc_Option_Ooption @ D ) @ ( c_Map_Omap__add @ C @ D @ B @ ( c_Map_Omap__of @ C @ D @ A ) ) ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1607]) ).

thf(1439,axiom,
    ! [A: $i,B: $i] :
      ( ( c_List_Oremove1 @ B @ A @ ( c_List_Olist_ONil @ B ) )
      = ( c_List_Olist_ONil @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_remove1_Osimps_I1_J) ).

thf(7913,plain,
    ! [A: $i,B: $i] :
      ( ( c_List_Oremove1 @ B @ A @ ( c_List_Olist_ONil @ B ) )
      = ( c_List_Olist_ONil @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1439]) ).

thf(699,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ B ) )
       => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ tc_Nat_Onat ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ C ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) ) )
          = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ tc_Nat_Onat ) @ A ) @ B ) ) @ ( hAPP @ A @ C ) ) ) )
      & ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ B ) )
       => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ tc_Nat_Onat ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ C ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) ) )
          = ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ tc_Nat_Onat ) @ A ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_setsum__diff1__nat) ).

thf(4902,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ B ) )
       => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ tc_Nat_Onat ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ C ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) ) )
          = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ tc_Nat_Onat ) @ A ) @ B ) ) @ ( hAPP @ A @ C ) ) ) )
      & ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ B ) )
       => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ tc_Nat_Onat ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ C ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) ) )
          = ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ tc_Nat_Onat ) @ A ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[699]) ).

thf(2178,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ D ) @ C ) @ B ) ) @ ( c_Transitive__Closure_Otrancl @ D @ ( hAPP @ ( c_Relation_Oconverse @ D @ D ) @ A ) ) ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ D ) @ C ) @ B ) ) @ ( hAPP @ ( c_Relation_Oconverse @ D @ D ) @ ( c_Transitive__Closure_Otrancl @ D @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_trancl__converseD) ).

thf(10919,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ D ) @ C ) @ B ) ) @ ( c_Transitive__Closure_Otrancl @ D @ ( hAPP @ ( c_Relation_Oconverse @ D @ D ) @ A ) ) ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ D ) @ C ) @ B ) ) @ ( hAPP @ ( c_Relation_Oconverse @ D @ D ) @ ( c_Transitive__Closure_Otrancl @ D @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2178]) ).

thf(2076,axiom,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) )
    <=> ( A
        = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_le__0__eq) ).

thf(10575,plain,
    ! [A: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) )
       => ( A
          = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) )
      & ( ( A
          = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2076]) ).

thf(1151,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ E @ D ) @ C ) @ B ) @ A ) @ B )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_fun__upd__same) ).

thf(6752,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ E @ D ) @ C ) @ B ) @ A ) @ B )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1151]) ).

thf(886,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( c_Fun_Oinj__on @ F @ E @ D @ C )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ F @ tc_HOL_Obool ) ) @ B ) @ C ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ F @ tc_HOL_Obool ) ) @ A ) @ C ) )
         => ( ( hAPP @ ( c_Set_Oimage @ F @ E @ D ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ F @ tc_HOL_Obool ) ) @ B ) @ A ) )
            = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Set_Oimage @ F @ E @ D ) @ B ) ) @ ( hAPP @ ( c_Set_Oimage @ F @ E @ D ) @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_inj__on__image__Int) ).

thf(5576,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( c_Fun_Oinj__on @ F @ E @ D @ C )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ F @ tc_HOL_Obool ) ) @ B ) @ C ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ F @ tc_HOL_Obool ) ) @ A ) @ C ) )
         => ( ( hAPP @ ( c_Set_Oimage @ F @ E @ D ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ F @ tc_HOL_Obool ) ) @ B ) @ A ) )
            = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Set_Oimage @ F @ E @ D ) @ B ) ) @ ( hAPP @ ( c_Set_Oimage @ F @ E @ D ) @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[886]) ).

thf(2230,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( c_Fun_Oinj__on @ F @ E @ D @ C )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ F ) @ B ) @ C ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ F ) @ A ) @ C ) )
         => ( ( ( hAPP @ D @ B )
              = ( hAPP @ D @ A ) )
          <=> ( B = A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_inj__on__iff) ).

thf(11129,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( c_Fun_Oinj__on @ F @ E @ D @ C )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ F ) @ B ) @ C ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ F ) @ A ) @ C ) )
         => ( ( ( ( hAPP @ D @ B )
                = ( hAPP @ D @ A ) )
             => ( B = A ) )
            & ( ( B = A )
             => ( ( hAPP @ D @ B )
                = ( hAPP @ D @ A ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2230]) ).

thf(339,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ C ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zadd__zmult__distrib) ).

thf(3611,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ C ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[339]) ).

thf(1600,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) ) @ A )
      = B ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_diff__add__inverse2) ).

thf(8673,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) ) @ A )
      = B ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1600]) ).

thf(2341,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] : ( c_Natural_Oevaln @ ( c_Com_Ocom_OAss @ D @ C ) @ B @ A @ ( hAPP @ ( hAPP @ ( hAPP @ c_Natural_Oupdate @ B ) @ D ) @ ( hAPP @ C @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_evaln_OAssign) ).

thf(11608,plain,
    ! [A: $i,B: $i,C: $i,D: $i] : ( c_Natural_Oevaln @ ( c_Com_Ocom_OAss @ D @ C ) @ B @ A @ ( hAPP @ ( hAPP @ ( hAPP @ c_Natural_Oupdate @ B ) @ D ) @ ( hAPP @ C @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2341]) ).

thf(2204,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_Finite__Set_Ofolding__one__idem @ D @ C @ B )
     => ( ( hAPP @ ( hAPP @ C @ A ) @ A )
        = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_folding__one__idem_Oidem) ).

thf(11023,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_Finite__Set_Ofolding__one__idem @ D @ C @ B )
     => ( ( hAPP @ ( hAPP @ C @ A ) @ A )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2204]) ).

thf(13,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Divides_Odivmod__int__rel @ C @ C ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ tc_Int_Oint @ tc_Int_Oint ) @ B ) @ A ) ) )
     => ( ( C
         != ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) )
       => ( B
          = ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_self__quotient) ).

thf(2450,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Divides_Odivmod__int__rel @ C @ C ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ tc_Int_Oint @ tc_Int_Oint ) @ B ) @ A ) ) )
     => ( ( C
         != ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) )
       => ( B
          = ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[13]) ).

thf(350,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_HOL_OThe @ ( tc_prod @ C @ B ) @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ C @ B @ tc_HOL_Obool ) @ A ) )
      = ( c_HOL_OThe @ ( tc_prod @ C @ B ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ ( tc_prod @ C @ B ) @ B @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ C @ ( tc_fun @ B @ tc_HOL_Obool ) @ ( tc_prod @ C @ B ) ) @ A ) @ ( c_Product__Type_Ofst @ C @ B ) ) ) @ ( c_Product__Type_Osnd @ C @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_The__split) ).

thf(3654,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_HOL_OThe @ ( tc_prod @ C @ B ) @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ C @ B @ tc_HOL_Obool ) @ A ) )
      = ( c_HOL_OThe @ ( tc_prod @ C @ B ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ ( tc_prod @ C @ B ) @ B @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ C @ ( tc_fun @ B @ tc_HOL_Obool ) @ ( tc_prod @ C @ B ) ) @ A ) @ ( c_Product__Type_Ofst @ C @ B ) ) ) @ ( c_Product__Type_Osnd @ C @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[350]) ).

thf(1444,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( hAPP @ ( c_List_Oset @ D ) @ B ) ) )
     => ( ~ ( hBOOL @ ( hAPP @ A @ C ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ D ) ) @ ( hAPP @ ( c_List_Ofilter @ D @ A ) @ B ) ) ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ D ) ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_length__filter__less) ).

thf(7927,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( hAPP @ ( c_List_Oset @ D ) @ B ) ) )
     => ( ~ ( hBOOL @ ( hAPP @ A @ C ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ D ) ) @ ( hAPP @ ( c_List_Ofilter @ D @ A ) @ B ) ) ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ D ) ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1444]) ).

thf(1770,axiom,
    hBOOL @ ( hAPP @ c_Com_OWT @ c_Com_Ocom_OSKIP ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_WT_Oequations_I1_J) ).

thf(9359,plain,
    hBOOL @ ( hAPP @ c_Com_OWT @ c_Com_Ocom_OSKIP ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1770]) ).

thf(852,axiom,
    ! [A: $i,B: $i,C: $i] : ( c_Fun_Oinj__on @ C @ B @ A @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_inj__on__empty) ).

thf(5481,plain,
    ! [A: $i,B: $i,C: $i] : ( c_Fun_Oinj__on @ C @ B @ A @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[852]) ).

thf(550,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) ) )
     => ~ ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ B ) )
         => ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_IntE) ).

thf(4384,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) ) )
     => ~ ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ B ) )
         => ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[550]) ).

thf(1008,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( c_Set_OCollect @ B ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ B @ ( tc_fun @ B @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( c_member @ B ) ) @ A ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Collect__mem__eq) ).

thf(6097,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( c_Set_OCollect @ B ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ B @ ( tc_fun @ B @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( c_member @ B ) ) @ A ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1008]) ).

thf(383,axiom,
    ! [A: $i] :
      ( ( hAPP @ ( c_Finite__Set_Ocard @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ A @ tc_HOL_Obool ) ) )
      = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_card_Oempty) ).

thf(3787,plain,
    ! [A: $i] :
      ( ( hAPP @ ( c_Finite__Set_Ocard @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ A @ tc_HOL_Obool ) ) )
      = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[383]) ).

thf(1714,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Divides_Odiv__class_Omod @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ B ) ) @ A ) @ B )
      = ( c_Divides_Odiv__class_Omod @ tc_Nat_Onat @ A @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mod__mult__self3) ).

thf(9149,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Divides_Odiv__class_Omod @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ B ) ) @ A ) @ B )
      = ( c_Divides_Odiv__class_Omod @ tc_Nat_Onat @ A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1714]) ).

thf(2213,axiom,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ ( c_Divides_Odivmod__nat__rel @ B @ A ) @ ( c_Divides_Odivmod__nat @ B @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_divmod__nat__rel__divmod__nat) ).

thf(11052,plain,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ ( c_Divides_Odivmod__nat__rel @ B @ A ) @ ( c_Divides_Odivmod__nat @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2213]) ).

thf(2234,axiom,
    ! [A: $i,B: $i] :
      ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ c_fdisj @ B ) @ A ) )
      | ( hBOOL @ B )
      | ( hBOOL @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_c__fdisj__3) ).

thf(11142,plain,
    ! [A: $i,B: $i] :
      ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ c_fdisj @ B ) @ A ) )
      | ( hBOOL @ B )
      | ( hBOOL @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2234]) ).

thf(1220,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( c_Relation_Oconverse @ B @ B ) @ ( c_Relation_OId__on @ B @ A ) )
      = ( c_Relation_OId__on @ B @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_converse__Id__on) ).

thf(7014,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( c_Relation_Oconverse @ B @ B ) @ ( c_Relation_OId__on @ B @ A ) )
      = ( c_Relation_OId__on @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1220]) ).

thf(183,axiom,
    ! [A: $i,B: $i,C: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Int__lower2) ).

thf(3059,plain,
    ! [A: $i,B: $i,C: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[183]) ).

thf(1167,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( c_Relation_ORange @ C @ B ) @ ( hAPP @ ( c_Relation_Oconverse @ B @ C ) @ A ) )
      = ( hAPP @ ( c_Relation_ODomain @ B @ C ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Range__converse) ).

thf(6805,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( c_Relation_ORange @ C @ B ) @ ( hAPP @ ( c_Relation_Oconverse @ B @ C ) @ A ) )
      = ( hAPP @ ( c_Relation_ODomain @ B @ C ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1167]) ).

thf(24,axiom,
    class_Groups_Ocomm__monoid__add @ tc_Int_Oint,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Groups_Ocomm__monoid__add) ).

thf(2487,plain,
    class_Groups_Ocomm__monoid__add @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[24]) ).

thf(1251,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ D ) @ C ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_List_Oset @ B ) @ ( hAPP @ ( hAPP @ ( c_List_Odrop @ B ) @ C ) @ A ) ) ) @ ( hAPP @ ( c_List_Oset @ B ) @ ( hAPP @ ( hAPP @ ( c_List_Odrop @ B ) @ D ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_set__drop__subset__set__drop) ).

thf(7138,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ D ) @ C ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_List_Oset @ B ) @ ( hAPP @ ( hAPP @ ( c_List_Odrop @ B ) @ C ) @ A ) ) ) @ ( hAPP @ ( c_List_Oset @ B ) @ ( hAPP @ ( hAPP @ ( c_List_Odrop @ B ) @ D ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1251]) ).

thf(265,axiom,
    ! [A: $i] : ( c_Relation_Orefl__on @ A @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ A @ tc_HOL_Obool ) ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ ( tc_prod @ A @ A ) @ tc_HOL_Obool ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_refl__on__empty) ).

thf(3354,plain,
    ! [A: $i] : ( c_Relation_Orefl__on @ A @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ A @ tc_HOL_Obool ) ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ ( tc_prod @ A @ A ) @ tc_HOL_Obool ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[265]) ).

thf(1620,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ C ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ A @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) )
         => ( ! [E: $i,F: $i] :
                ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ F ) )
               => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ E ) @ B ) )
                 => ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ E ) @ F ) )
                   => ( ( hBOOL @ ( hAPP @ A @ F ) )
                     => ( hBOOL @ ( hAPP @ A @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ E ) @ F ) ) ) ) ) ) )
           => ( hBOOL @ ( hAPP @ A @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_finite__subset__induct) ).

thf(8764,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ C ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ A @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) )
         => ( ! [E: $i,F: $i] :
                ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ F ) )
               => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ E ) @ B ) )
                 => ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ E ) @ F ) )
                   => ( ( hBOOL @ ( hAPP @ A @ F ) )
                     => ( hBOOL @ ( hAPP @ A @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ E ) @ F ) ) ) ) ) ) )
           => ( hBOOL @ ( hAPP @ A @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1620]) ).

thf(1216,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ C @ B @ B ) @ A ) @ ( c_Relation_OId @ B ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_R__O__Id) ).

thf(6999,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ C @ B @ B ) @ A ) @ ( c_Relation_OId @ B ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1216]) ).

thf(1452,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ ( hAPP @ c_Nat_OSuc @ A ) )
      = ( hAPP @ c_Nat_OSuc @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_add__Suc__right) ).

thf(7966,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ ( hAPP @ c_Nat_OSuc @ A ) )
      = ( hAPP @ c_Nat_OSuc @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1452]) ).

thf(1330,axiom,
    ! [A: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) ) @ ( hAPP @ c_Int_ORep__Integ @ A ) ) @ c_Int_OInteg ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Rep__Integ) ).

thf(7508,plain,
    ! [A: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) ) @ ( hAPP @ c_Int_ORep__Integ @ A ) ) @ c_Int_OInteg ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1330]) ).

thf(1161,axiom,
    ! [A: $i] :
      ( ( hAPP @ ( c_Relation_Oconverse @ A @ A ) @ ( c_Relation_OId @ A ) )
      = ( c_Relation_OId @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_converse__Id) ).

thf(6788,plain,
    ! [A: $i] :
      ( ( hAPP @ ( c_Relation_Oconverse @ A @ A ) @ ( c_Relation_OId @ A ) )
      = ( c_Relation_OId @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1161]) ).

thf(785,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ C @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ A ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ B @ tc_HOL_Obool ) @ C ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) )
      = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_UN__empty2) ).

thf(5234,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ C @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ A ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ B @ tc_HOL_Obool ) @ C ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) )
      = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[785]) ).

thf(967,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ B @ A )
      = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Int_Oint ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ B ) @ A ) ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zmod__zdiv__equality_H) ).

thf(5900,plain,
    ! [A: $i,B: $i] :
      ( ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ B @ A )
      = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Int_Oint ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ B ) @ A ) ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[967]) ).

thf(1018,axiom,
    ! [A: $i] :
      ( ( c_SetInterval_Oord_OatLeastAtMost @ tc_Nat_Onat @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ A @ A )
      = ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ tc_Nat_Onat ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_dvd_OatLeastAtMost__singleton) ).

thf(6129,plain,
    ! [A: $i] :
      ( ( c_SetInterval_Oord_OatLeastAtMost @ tc_Nat_Onat @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ A @ A )
      = ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ tc_Nat_Onat ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1018]) ).

thf(1348,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ C ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ A ) ) )
      <=> ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nat__mult__less__cancel1) ).

thf(7587,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ C ) )
     => ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ A ) ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) ) )
        & ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1348]) ).

thf(125,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_List_Osublist @ D @ ( hAPP @ ( hAPP @ ( c_List_Oappend @ D ) @ C ) @ B ) @ A )
      = ( hAPP @ ( hAPP @ ( c_List_Oappend @ D ) @ ( c_List_Osublist @ D @ C @ A ) ) @ ( c_List_Osublist @ D @ B @ ( hAPP @ ( c_Set_OCollect @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_Nat_Onat @ ( tc_fun @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ tc_Nat_Onat ) @ ( c_member @ tc_Nat_Onat ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ tc_Nat_Onat @ tc_Nat_Onat ) @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ D ) ) @ C ) ) ) ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_sublist__append) ).

thf(2851,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_List_Osublist @ D @ ( hAPP @ ( hAPP @ ( c_List_Oappend @ D ) @ C ) @ B ) @ A )
      = ( hAPP @ ( hAPP @ ( c_List_Oappend @ D ) @ ( c_List_Osublist @ D @ C @ A ) ) @ ( c_List_Osublist @ D @ B @ ( hAPP @ ( c_Set_OCollect @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_Nat_Onat @ ( tc_fun @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ tc_Nat_Onat ) @ ( c_member @ tc_Nat_Onat ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ tc_Nat_Onat @ tc_Nat_Onat ) @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ D ) ) @ C ) ) ) ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[125]) ).

thf(319,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i] : ( c_Wellfounded_OwfP @ C @ ( hAPP @ B @ D ) )
     => ( ! [D: $i,E: $i] :
            ( ( ( hAPP @ B @ D )
             != ( hAPP @ B @ E ) )
           => ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( c_Predicate_ODomainP @ C @ C @ ( hAPP @ B @ D ) ) ) @ ( c_Predicate_ORangeP @ C @ C @ ( hAPP @ B @ E ) ) )
              = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) )
       => ( c_Wellfounded_OwfP @ C @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ A @ ( tc_fun @ C @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ A @ tc_HOL_Obool ) ) ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_wfP__SUP) ).

thf(3540,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i] : ( c_Wellfounded_OwfP @ C @ ( hAPP @ B @ D ) )
     => ( ! [D: $i,E: $i] :
            ( ( ( hAPP @ B @ D )
             != ( hAPP @ B @ E ) )
           => ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( c_Predicate_ODomainP @ C @ C @ ( hAPP @ B @ D ) ) ) @ ( c_Predicate_ORangeP @ C @ C @ ( hAPP @ B @ E ) ) )
              = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) )
       => ( c_Wellfounded_OwfP @ C @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ A @ ( tc_fun @ C @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ A @ tc_HOL_Obool ) ) ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[319]) ).

thf(28,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Groups_Ocomm__monoid__add @ E )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ C ) )
       => ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ B ) @ C ) )
         => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ B ) @ C ) )
            = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ E ) @ ( hAPP @ A @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ A ) @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_setsum_Oinsert) ).

thf(2497,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Groups_Ocomm__monoid__add @ E )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ C ) )
       => ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ B ) @ C ) )
         => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ B ) @ C ) )
            = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ E ) @ ( hAPP @ A @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ A ) @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[28]) ).

thf(1044,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( c_Divides_Odiv__class_Omod @ tc_Nat_Onat @ A @ B ) ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mod__le__divisor) ).

thf(6239,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( c_Divides_Odiv__class_Omod @ tc_Nat_Onat @ A @ B ) ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1044]) ).

thf(2048,axiom,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ B ) @ ( hAPP @ ( c_List_Oset @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_finite__set) ).

thf(10455,plain,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ B ) @ ( hAPP @ ( c_List_Oset @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2048]) ).

thf(49,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Groups_Oab__group__add @ E )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ D ) @ C ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ C ) @ B ) )
         => ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ E ) @ ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ tc_Nat_Onat @ E ) @ A ) @ ( hAPP @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Nat_Onat @ D ) @ B ) ) ) @ ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ tc_Nat_Onat @ E ) @ A ) @ ( hAPP @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Nat_Onat @ D ) @ C ) ) )
            = ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ tc_Nat_Onat @ E ) @ A ) @ ( hAPP @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Nat_Onat @ C ) @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_setsum__diff__nat__ivl) ).

thf(2566,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Groups_Oab__group__add @ E )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ D ) @ C ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ C ) @ B ) )
         => ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ E ) @ ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ tc_Nat_Onat @ E ) @ A ) @ ( hAPP @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Nat_Onat @ D ) @ B ) ) ) @ ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ tc_Nat_Onat @ E ) @ A ) @ ( hAPP @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Nat_Onat @ D ) @ C ) ) )
            = ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ tc_Nat_Onat @ E ) @ A ) @ ( hAPP @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Nat_Onat @ C ) @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[49]) ).

thf(2065,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ B ) @ ( c_Wellfounded_Oacc @ C @ A ) ) )
     => ~ ! [D: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ C @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ C @ C ) @ D ) @ B ) ) @ A ) )
           => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ D ) @ ( c_Wellfounded_Oacc @ C @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_not__acc__down) ).

thf(10518,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ B ) @ ( c_Wellfounded_Oacc @ C @ A ) ) )
     => ~ ! [D: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ C @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ C @ C ) @ D ) @ B ) ) @ A ) )
           => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ D ) @ ( c_Wellfounded_Oacc @ C @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2065]) ).

thf(1978,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ ( hAPP @ c_Nat_OSuc @ A ) ) )
     => ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) )
       => ( B = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__SucE) ).

thf(10207,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ ( hAPP @ c_Nat_OSuc @ A ) ) )
     => ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) )
       => ( B = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1978]) ).

thf(1702,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ B ) )
     => ( ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ A ) @ B ) )
        = ( hAPP @ c_Nat_OSuc @ ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_card__insert) ).

thf(9112,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ B ) )
     => ( ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ A ) @ B ) )
        = ( hAPP @ c_Nat_OSuc @ ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1702]) ).

thf(1559,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Relation_Ototal__on @ C @ B @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) ) @ A ) @ ( c_Relation_OId @ C ) ) )
    <=> ( c_Relation_Ototal__on @ C @ B @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_total__on__diff__Id) ).

thf(8439,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( c_Relation_Ototal__on @ C @ B @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) ) @ A ) @ ( c_Relation_OId @ C ) ) )
       => ( c_Relation_Ototal__on @ C @ B @ A ) )
      & ( ( c_Relation_Ototal__on @ C @ B @ A )
       => ( c_Relation_Ototal__on @ C @ B @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) ) @ A ) @ ( c_Relation_OId @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1559]) ).

thf(2032,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Odrop @ D ) @ ( hAPP @ c_Nat_OSuc @ C ) ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ D ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_List_Odrop @ D ) @ C ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_drop__Suc__Cons) ).

thf(10401,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Odrop @ D ) @ ( hAPP @ c_Nat_OSuc @ C ) ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ D ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_List_Odrop @ D ) @ C ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2032]) ).

thf(2243,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( c_SetInterval_Oord_OatMost @ D @ B @ A ) ) )
    <=> ( hBOOL @ ( hAPP @ ( hAPP @ B @ C ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_ord_OatMost__iff) ).

thf(11184,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( c_SetInterval_Oord_OatMost @ D @ B @ A ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ B @ C ) @ A ) ) )
      & ( ( hBOOL @ ( hAPP @ ( hAPP @ B @ C ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( c_SetInterval_Oord_OatMost @ D @ B @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2243]) ).

thf(1682,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Nat_Onat ) @ A ) @ B ) )
      = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ A ) @ ( c_Divides_Odiv__class_Omod @ tc_Nat_Onat @ A @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__div__cancel) ).

thf(9017,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Nat_Onat ) @ A ) @ B ) )
      = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ A ) @ ( c_Divides_Odiv__class_Omod @ tc_Nat_Onat @ A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1682]) ).

thf(1654,axiom,
    ! [A: $i,B: $i] : ( c_List_Olistrelp @ B @ A @ ( c_List_Olist_ONil @ B ) @ ( c_List_Olist_ONil @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_listrelp_ONil) ).

thf(8896,plain,
    ! [A: $i,B: $i] : ( c_List_Olistrelp @ B @ A @ ( c_List_Olist_ONil @ B ) @ ( c_List_Olist_ONil @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1654]) ).

thf(1959,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ D ) @ C ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ tc_Nat_Onat @ tc_Nat_Onat ) @ D ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ tc_Nat_Onat @ tc_Nat_Onat ) @ C ) @ A ) ) ) @ c_FunDef_Opair__leq ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_pair__leqI2) ).

thf(10122,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ D ) @ C ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ tc_Nat_Onat @ tc_Nat_Onat ) @ D ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ tc_Nat_Onat @ tc_Nat_Onat ) @ C ) @ A ) ) ) @ c_FunDef_Opair__leq ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1959]) ).

thf(1943,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Natural_Oevalc @ E ) @ D ) @ C ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Natural_Oevalc @ B ) @ C ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Natural_Oevalc @ ( c_Com_Ocom_OSemi @ E @ B ) ) @ D ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_evalc_OSemi) ).

thf(10047,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Natural_Oevalc @ E ) @ D ) @ C ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Natural_Oevalc @ B ) @ C ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Natural_Oevalc @ ( c_Com_Ocom_OSemi @ E @ B ) ) @ D ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1943]) ).

thf(2162,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Transitive__Closure_Otrancl @ B @ ( c_Transitive__Closure_Ortrancl @ B @ A ) )
      = ( c_Transitive__Closure_Ortrancl @ B @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_rtrancl__trancl__absorb) ).

thf(10879,plain,
    ! [A: $i,B: $i] :
      ( ( c_Transitive__Closure_Otrancl @ B @ ( c_Transitive__Closure_Ortrancl @ B @ A ) )
      = ( c_Transitive__Closure_Ortrancl @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2162]) ).

thf(510,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ E ) @ ( tc_List_Olist @ E ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ E ) @ ( tc_List_Olist @ E ) ) @ ( hAPP @ ( hAPP @ ( c_List_Oappend @ E ) @ D ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_List_Oappend @ E ) @ D ) @ B ) ) ) @ ( c_List_Olexord @ E @ A ) ) )
     => ( ! [F: $i] :
            ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ F ) @ F ) ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ E ) @ ( tc_List_Olist @ E ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ E ) @ ( tc_List_Olist @ E ) ) @ C ) @ B ) ) @ ( c_List_Olexord @ E @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_lexord__append__leftD) ).

thf(4231,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ E ) @ ( tc_List_Olist @ E ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ E ) @ ( tc_List_Olist @ E ) ) @ ( hAPP @ ( hAPP @ ( c_List_Oappend @ E ) @ D ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_List_Oappend @ E ) @ D ) @ B ) ) ) @ ( c_List_Olexord @ E @ A ) ) )
     => ( ! [F: $i] :
            ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ F ) @ F ) ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ E ) @ ( tc_List_Olist @ E ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ E ) @ ( tc_List_Olist @ E ) ) @ C ) @ B ) ) @ ( c_List_Olexord @ E @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[510]) ).

thf(113,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Complete__Lattice_OSup__class_OSup @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ ( tc_fun @ C @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( c_Complete__Lattice_OSup__class_OSup @ ( tc_fun @ C @ tc_HOL_Obool ) @ B ) ) @ ( c_Complete__Lattice_OSup__class_OSup @ ( tc_fun @ C @ tc_HOL_Obool ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Union__Un__distrib) ).

thf(2813,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Complete__Lattice_OSup__class_OSup @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ ( tc_fun @ C @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( c_Complete__Lattice_OSup__class_OSup @ ( tc_fun @ C @ tc_HOL_Obool ) @ B ) ) @ ( c_Complete__Lattice_OSup__class_OSup @ ( tc_fun @ C @ tc_HOL_Obool ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[113]) ).

thf(539,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ D ) @ C ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ D ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ C ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Int__mono) ).

thf(4347,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ D ) @ C ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ D ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ C ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[539]) ).

thf(1564,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Wellfounded_OwfP @ B @ A )
    <=> ( c_Wellfounded_Owf @ B @ ( hAPP @ ( c_Set_OCollect @ ( tc_prod @ B @ B ) ) @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ B @ B @ tc_HOL_Obool ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_wfP__def) ).

thf(8455,plain,
    ! [A: $i,B: $i] :
      ( ( ( c_Wellfounded_OwfP @ B @ A )
       => ( c_Wellfounded_Owf @ B @ ( hAPP @ ( c_Set_OCollect @ ( tc_prod @ B @ B ) ) @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ B @ B @ tc_HOL_Obool ) @ A ) ) ) )
      & ( ( c_Wellfounded_Owf @ B @ ( hAPP @ ( c_Set_OCollect @ ( tc_prod @ B @ B ) ) @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ B @ B @ tc_HOL_Obool ) @ A ) ) )
       => ( c_Wellfounded_OwfP @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1564]) ).

thf(1623,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ A ) )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ A ) @ B ) ) @ B )
        = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_le__add__diff__inverse2) ).

thf(8775,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ A ) )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ A ) @ B ) ) @ B )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1623]) ).

thf(2002,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ D ) @ C ) @ B ) ) @ ( c_Transitive__Closure_Otrancl @ D @ A ) ) )
     => ? [E: $i] :
          ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ D ) @ C ) @ E ) ) @ A ) )
          & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ D ) @ E ) @ B ) ) @ ( c_Transitive__Closure_Ortrancl @ D @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_tranclD) ).

thf(10302,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ D ) @ C ) @ B ) ) @ ( c_Transitive__Closure_Otrancl @ D @ A ) ) )
     => ? [E: $i] :
          ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ D ) @ C ) @ E ) ) @ A ) )
          & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ D ) @ E ) @ B ) ) @ ( c_Transitive__Closure_Ortrancl @ D @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2002]) ).

thf(237,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ B ) )
       => ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ C ) @ A ) ) @ B )
          = ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ A ) @ B ) ) ) )
      & ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ B ) )
       => ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ C ) @ A ) ) @ B )
          = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ A ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Int__insert__left) ).

thf(3247,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ B ) )
       => ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ C ) @ A ) ) @ B )
          = ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ A ) @ B ) ) ) )
      & ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ B ) )
       => ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ C ) @ A ) ) @ B )
          = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ A ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[237]) ).

thf(282,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ D ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ C @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ C @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ E @ tc_HOL_Obool ) @ ( tc_fun @ E @ tc_HOL_Obool ) @ C ) @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ D ) ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_UN__extend__simps_I5_J) ).

thf(3409,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ D ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ C @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ C @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ E @ tc_HOL_Obool ) @ ( tc_fun @ E @ tc_HOL_Obool ) @ C ) @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ D ) ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[282]) ).

thf(128,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) )
      = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Int__empty__right) ).

thf(2862,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) )
      = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[128]) ).

thf(1835,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( hAPP @ ( c_Relation_ODomain @ D @ D ) @ B ) ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ D ) @ C ) @ A ) ) @ ( c_Transitive__Closure_Ortrancl @ D @ B ) ) )
      <=> ( C = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Not__Domain__rtrancl) ).

thf(9598,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( hAPP @ ( c_Relation_ODomain @ D @ D ) @ B ) ) )
     => ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ D ) @ C ) @ A ) ) @ ( c_Transitive__Closure_Ortrancl @ D @ B ) ) )
         => ( C = A ) )
        & ( ( C = A )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ D ) @ C ) @ A ) ) @ ( c_Transitive__Closure_Ortrancl @ D @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1835]) ).

thf(711,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) @ A ) )
     => ( hBOOL @ ( hAPP @ B @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_inf1D2) ).

thf(4953,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) @ A ) )
     => ( hBOOL @ ( hAPP @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[711]) ).

thf(185,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zadd__zmult__distrib2) ).

thf(3064,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[185]) ).

thf(2342,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Natural_Oevaln @ c_Com_Ocom_OSKIP @ C @ B @ A )
     => ( A = C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_evaln__elim__cases_I1_J) ).

thf(11610,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Natural_Oevaln @ c_Com_Ocom_OSKIP @ C @ B @ A )
     => ( A = C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2342]) ).

thf(1414,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Divides_Odivmod__nat__rel @ E @ D ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ tc_Nat_Onat @ tc_Nat_Onat ) @ C ) @ B ) ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ D ) )
       => ( hBOOL @ ( hAPP @ ( c_Divides_Odivmod__nat__rel @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ A ) @ E ) @ D ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ tc_Nat_Onat @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ A ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ A ) @ B ) ) @ D ) ) ) @ ( c_Divides_Odiv__class_Omod @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ A ) @ B ) @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_divmod__nat__rel__mult1__eq) ).

thf(7826,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Divides_Odivmod__nat__rel @ E @ D ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ tc_Nat_Onat @ tc_Nat_Onat ) @ C ) @ B ) ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ D ) )
       => ( hBOOL @ ( hAPP @ ( c_Divides_Odivmod__nat__rel @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ A ) @ E ) @ D ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ tc_Nat_Onat @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ A ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ A ) @ B ) ) @ D ) ) ) @ ( c_Divides_Odiv__class_Omod @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ A ) @ B ) @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1414]) ).

thf(1949,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ D ) @ C ) ) @ ( c_Transitive__Closure_Otrancl @ E @ B ) ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ C ) @ A ) ) @ B ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ D ) @ A ) ) @ ( c_Transitive__Closure_Otrancl @ E @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Transitive__Closure_Otrancl__into__trancl) ).

thf(10075,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ D ) @ C ) ) @ ( c_Transitive__Closure_Otrancl @ E @ B ) ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ C ) @ A ) ) @ B ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ D ) @ A ) ) @ ( c_Transitive__Closure_Otrancl @ E @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1949]) ).

thf(634,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Equiv__Relations_Oequiv @ C @ B @ A )
     => ( ( c_Complete__Lattice_OSup__class_OSup @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_Equiv__Relations_Oquotient @ C ) @ B ) @ A ) )
        = B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Union__quotient) ).

thf(4689,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Equiv__Relations_Oequiv @ C @ B @ A )
     => ( ( c_Complete__Lattice_OSup__class_OSup @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_Equiv__Relations_Oquotient @ C ) @ B ) @ A ) )
        = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[634]) ).

thf(1997,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( hAPP @ ( c_List_Oset @ D ) @ B ) ) )
       => ( ( c_List_Oremove1 @ D @ C @ ( hAPP @ ( hAPP @ ( c_List_Oappend @ D ) @ B ) @ A ) )
          = ( hAPP @ ( hAPP @ ( c_List_Oappend @ D ) @ ( c_List_Oremove1 @ D @ C @ B ) ) @ A ) ) )
      & ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( hAPP @ ( c_List_Oset @ D ) @ B ) ) )
       => ( ( c_List_Oremove1 @ D @ C @ ( hAPP @ ( hAPP @ ( c_List_Oappend @ D ) @ B ) @ A ) )
          = ( hAPP @ ( hAPP @ ( c_List_Oappend @ D ) @ B ) @ ( c_List_Oremove1 @ D @ C @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_remove1__append) ).

thf(10270,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( hAPP @ ( c_List_Oset @ D ) @ B ) ) )
       => ( ( c_List_Oremove1 @ D @ C @ ( hAPP @ ( hAPP @ ( c_List_Oappend @ D ) @ B ) @ A ) )
          = ( hAPP @ ( hAPP @ ( c_List_Oappend @ D ) @ ( c_List_Oremove1 @ D @ C @ B ) ) @ A ) ) )
      & ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( hAPP @ ( c_List_Oset @ D ) @ B ) ) )
       => ( ( c_List_Oremove1 @ D @ C @ ( hAPP @ ( hAPP @ ( c_List_Oappend @ D ) @ B ) @ A ) )
          = ( hAPP @ ( hAPP @ ( c_List_Oappend @ D ) @ B ) @ ( c_List_Oremove1 @ D @ C @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1997]) ).

thf(1032,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Transitive__Closure_Otrancl @ B @ A )
      = ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ B @ B @ B ) @ ( c_Transitive__Closure_Ortrancl @ B @ A ) ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_trancl__unfold__right) ).

thf(6186,plain,
    ! [A: $i,B: $i] :
      ( ( c_Transitive__Closure_Otrancl @ B @ A )
      = ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ B @ B @ B ) @ ( c_Transitive__Closure_Ortrancl @ B @ A ) ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1032]) ).

thf(1296,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ B ) )
     => ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ A ) @ B ) )
       => ( ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ A ) @ B ) )
          = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) @ ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_card_Oinsert) ).

thf(7333,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ B ) )
     => ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ A ) @ B ) )
       => ( ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ A ) @ B ) )
          = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) @ ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1296]) ).

thf(1917,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Option_Ooption_ONone @ B )
     != ( hAPP @ ( c_Option_Ooption_OSome @ B ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_option_Osimps_I2_J) ).

thf(9942,plain,
    ! [A: $i,B: $i] :
      ( ( c_Option_Ooption_ONone @ B )
     != ( hAPP @ ( c_Option_Ooption_OSome @ B ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1917]) ).

thf(2403,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( c_Typedef_Otype__definition @ F @ E @ D @ C @ B )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ A ) @ B ) )
       => ( ( hAPP @ D @ ( hAPP @ C @ A ) )
          = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_type__definition_OAbs__inverse) ).

thf(11873,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( c_Typedef_Otype__definition @ F @ E @ D @ C @ B )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ A ) @ B ) )
       => ( ( hAPP @ D @ ( hAPP @ C @ A ) )
          = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2403]) ).

thf(221,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ D ) @ C ) ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ B ) ) @ A ) ) )
     => ( ( hAPP @ ( hAPP @ ( c_List_Onth @ B ) @ ( hAPP @ ( hAPP @ ( c_List_Odrop @ B ) @ D ) @ A ) ) @ C )
        = ( hAPP @ ( hAPP @ ( c_List_Onth @ B ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ D ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nth__drop) ).

thf(3201,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ D ) @ C ) ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ B ) ) @ A ) ) )
     => ( ( hAPP @ ( hAPP @ ( c_List_Onth @ B ) @ ( hAPP @ ( hAPP @ ( c_List_Odrop @ B ) @ D ) @ A ) ) @ C )
        = ( hAPP @ ( hAPP @ ( c_List_Onth @ B ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ D ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[221]) ).

thf(508,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ D @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) @ D ) @ ( hAPP @ ( c_COMBC @ C @ ( tc_fun @ C @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( c_member @ C ) ) ) @ B ) ) @ A ) )
    <=> ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ D @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_SUP__UN__eq) ).

thf(4223,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ D @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) @ D ) @ ( hAPP @ ( c_COMBC @ C @ ( tc_fun @ C @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( c_member @ C ) ) ) @ B ) ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ D @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) @ B ) ) ) )
      & ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ D @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) @ B ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ D @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) @ D ) @ ( hAPP @ ( c_COMBC @ C @ ( tc_fun @ C @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( c_member @ C ) ) ) @ B ) ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[508]) ).

thf(2167,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Hilbert__Choice_OEps @ B @ ( hAPP @ c_fequal @ A ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_some__sym__eq__trivial) ).

thf(10894,plain,
    ! [A: $i,B: $i] :
      ( ( c_Hilbert__Choice_OEps @ B @ ( hAPP @ c_fequal @ A ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2167]) ).

thf(227,axiom,
    class_Rings_Olinordered__semiring__1 @ tc_Int_Oint,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Int__Oint__Rings_Olinordered__semiring__1) ).

thf(3215,plain,
    class_Rings_Olinordered__semiring__1 @ tc_Int_Oint,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[227]) ).

thf(544,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ D ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Nat_Ocompow @ ( tc_prod @ E @ E ) @ tc_HOL_Obool ) @ ( hAPP @ c_Nat_OSuc @ B ) ) @ A ) ) )
     => ~ ! [F: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ D ) @ F ) ) @ ( hAPP @ ( hAPP @ ( c_Nat_Ocompow @ ( tc_prod @ E @ E ) @ tc_HOL_Obool ) @ B ) @ A ) ) )
           => ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ F ) @ C ) ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_rel__pow__Suc__E) ).

thf(4361,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ D ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Nat_Ocompow @ ( tc_prod @ E @ E ) @ tc_HOL_Obool ) @ ( hAPP @ c_Nat_OSuc @ B ) ) @ A ) ) )
     => ~ ! [F: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ D ) @ F ) ) @ ( hAPP @ ( hAPP @ ( c_Nat_Ocompow @ ( tc_prod @ E @ E ) @ tc_HOL_Obool ) @ B ) @ A ) ) )
           => ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ F ) @ C ) ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[544]) ).

thf(851,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Fun_Oinj__on @ C @ B @ A @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ C @ tc_HOL_Obool ) ) )
     => ( c_Fun_Oinj__on @ ( tc_List_Olist @ C ) @ ( tc_List_Olist @ B ) @ ( hAPP @ ( c_List_Omap @ C @ B ) @ A ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ ( tc_List_Olist @ C ) @ tc_HOL_Obool ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_inj__mapI) ).

thf(5479,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Fun_Oinj__on @ C @ B @ A @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ C @ tc_HOL_Obool ) ) )
     => ( c_Fun_Oinj__on @ ( tc_List_Olist @ C ) @ ( tc_List_Olist @ B ) @ ( hAPP @ ( c_List_Omap @ C @ B ) @ A ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ ( tc_List_Olist @ C ) @ tc_HOL_Obool ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[851]) ).

thf(1582,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ C @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ B ) @ A ) ) )
    <=> ( ( ( A
            = ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) )
         => ( hBOOL @ ( hAPP @ C @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) ) )
        & ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ A ) )
         => ! [D: $i] :
              ( ? [E: $i] :
                  ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ E ) )
                  & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ E ) @ A ) )
                  & ( B
                    = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ A ) @ D ) ) @ E ) ) )
             => ( hBOOL @ ( hAPP @ C @ D ) ) ) )
        & ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) )
         => ! [D: $i] :
              ( ? [E: $i] :
                  ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ A ) @ E ) )
                  & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ E ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) )
                  & ( B
                    = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ A ) @ D ) ) @ E ) ) )
             => ( hBOOL @ ( hAPP @ C @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_split__zdiv) ).

thf(8527,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP @ C @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ B ) @ A ) ) )
       => ( ( ( A
              = ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) )
           => ( hBOOL @ ( hAPP @ C @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) ) )
          & ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ A ) )
           => ! [D: $i] :
                ( ? [E: $i] :
                    ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ E ) )
                    & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ E ) @ A ) )
                    & ( B
                      = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ A ) @ D ) ) @ E ) ) )
               => ( hBOOL @ ( hAPP @ C @ D ) ) ) )
          & ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) )
           => ! [D: $i] :
                ( ? [E: $i] :
                    ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ A ) @ E ) )
                    & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ E ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) )
                    & ( B
                      = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ A ) @ D ) ) @ E ) ) )
               => ( hBOOL @ ( hAPP @ C @ D ) ) ) ) ) )
      & ( ( ( ( A
              = ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) )
           => ( hBOOL @ ( hAPP @ C @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) ) )
          & ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ A ) )
           => ! [D: $i] :
                ( ? [E: $i] :
                    ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ E ) )
                    & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ E ) @ A ) )
                    & ( B
                      = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ A ) @ D ) ) @ E ) ) )
               => ( hBOOL @ ( hAPP @ C @ D ) ) ) )
          & ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) )
           => ! [D: $i] :
                ( ? [E: $i] :
                    ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ A ) @ E ) )
                    & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ E ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) )
                    & ( B
                      = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ A ) @ D ) ) @ E ) ) )
               => ( hBOOL @ ( hAPP @ C @ D ) ) ) ) )
       => ( hBOOL @ ( hAPP @ C @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ B ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1582]) ).

thf(1579,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ D ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Nat_Ocompow @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ A ) ) )
     => ( C = B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_rel__pow__0__E) ).

thf(8512,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ D ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Nat_Ocompow @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ A ) ) )
     => ( C = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1579]) ).

thf(990,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP @ C @ B ) )
       => ( ( hAPP @ ( c_Set_OCollect @ A ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ A @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ A ) @ c_fconj ) @ ( hAPP @ c_fequal @ B ) ) ) @ C ) )
          = ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ A ) @ B ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ A @ tc_HOL_Obool ) ) ) ) )
      & ( ~ ( hBOOL @ ( hAPP @ C @ B ) )
       => ( ( hAPP @ ( c_Set_OCollect @ A ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ A @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ A ) @ c_fconj ) @ ( hAPP @ c_fequal @ B ) ) ) @ C ) )
          = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ A @ tc_HOL_Obool ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Collect__conv__if2) ).

thf(6011,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP @ C @ B ) )
       => ( ( hAPP @ ( c_Set_OCollect @ A ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ A @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ A ) @ c_fconj ) @ ( hAPP @ c_fequal @ B ) ) ) @ C ) )
          = ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ A ) @ B ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ A @ tc_HOL_Obool ) ) ) ) )
      & ( ~ ( hBOOL @ ( hAPP @ C @ B ) )
       => ( ( hAPP @ ( c_Set_OCollect @ A ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ A @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ A ) @ c_fconj ) @ ( hAPP @ c_fequal @ B ) ) ) @ C ) )
          = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ A @ tc_HOL_Obool ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[990]) ).

thf(603,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ D @ C ) @ ( c_Complete__Lattice_OSup__class_OSup @ ( tc_fun @ D @ tc_HOL_Obool ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ ( tc_fun @ D @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ ( tc_fun @ D @ tc_HOL_Obool ) @ ( tc_fun @ D @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) ) @ ( c_Product__Type_OSigma @ D @ C ) ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Sigma__Union) ).

thf(4572,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ D @ C ) @ ( c_Complete__Lattice_OSup__class_OSup @ ( tc_fun @ D @ tc_HOL_Obool ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ ( tc_fun @ D @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ ( tc_fun @ D @ tc_HOL_Obool ) @ ( tc_fun @ D @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) ) @ ( c_Product__Type_OSigma @ D @ C ) ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[603]) ).

thf(771,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i,I: $i] :
      ( ( c_Equiv__Relations_Oequiv @ I @ H @ G )
     => ( ( c_Equiv__Relations_Oequiv @ F @ E @ D )
       => ( ( c_Equiv__Relations_Ocongruent2 @ I @ F @ ( tc_fun @ C @ tc_HOL_Obool ) @ G @ D @ B )
         => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ F ) @ A ) @ E ) )
           => ( c_Equiv__Relations_Ocongruent @ I @ ( tc_fun @ C @ tc_HOL_Obool ) @ G @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ F @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( tc_fun @ C @ tc_HOL_Obool ) @ I ) @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ F @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Relation_OImage @ F @ F @ D ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ F ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ F @ tc_HOL_Obool ) ) ) ) ) ) @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_congruent2__implies__congruent__UN) ).

thf(5170,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i,I: $i] :
      ( ( c_Equiv__Relations_Oequiv @ I @ H @ G )
     => ( ( c_Equiv__Relations_Oequiv @ F @ E @ D )
       => ( ( c_Equiv__Relations_Ocongruent2 @ I @ F @ ( tc_fun @ C @ tc_HOL_Obool ) @ G @ D @ B )
         => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ F ) @ A ) @ E ) )
           => ( c_Equiv__Relations_Ocongruent @ I @ ( tc_fun @ C @ tc_HOL_Obool ) @ G @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ F @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( tc_fun @ C @ tc_HOL_Obool ) @ I ) @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ F @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Relation_OImage @ F @ F @ D ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ F ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ F @ tc_HOL_Obool ) ) ) ) ) ) @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[771]) ).

thf(491,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Lattices_Oab__semigroup__idem__mult @ B )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ B ) @ A ) @ A )
        = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_times_Oidem) ).

thf(4163,plain,
    ! [A: $i,B: $i] :
      ( ( class_Lattices_Oab__semigroup__idem__mult @ B )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ B ) @ A ) @ A )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[491]) ).

thf(1964,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
      ( ( c_Finite__Set_Ofolding__image__simple @ H @ G @ F @ E @ D @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ G ) @ B ) )
       => ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ G ) @ A ) @ B ) )
         => ( ( hAPP @ C @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ G ) @ A ) @ B ) )
            = ( hAPP @ ( hAPP @ F @ ( hAPP @ D @ A ) ) @ ( hAPP @ C @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_folding__image__simple_Oinsert) ).

thf(10143,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
      ( ( c_Finite__Set_Ofolding__image__simple @ H @ G @ F @ E @ D @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ G ) @ B ) )
       => ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ G ) @ A ) @ B ) )
         => ( ( hAPP @ C @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ G ) @ A ) @ B ) )
            = ( hAPP @ ( hAPP @ F @ ( hAPP @ D @ A ) ) @ ( hAPP @ C @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1964]) ).

thf(296,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Fun_Oinj__on @ E @ D @ C @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ E @ tc_HOL_Obool ) ) )
     => ( ( ( hAPP @ C @ B )
          = ( hAPP @ C @ A ) )
      <=> ( B = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_inj__eq) ).

thf(3454,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Fun_Oinj__on @ E @ D @ C @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ E @ tc_HOL_Obool ) ) )
     => ( ( ( ( hAPP @ C @ B )
            = ( hAPP @ C @ A ) )
         => ( B = A ) )
        & ( ( B = A )
         => ( ( hAPP @ C @ B )
            = ( hAPP @ C @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[296]) ).

thf(1980,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Com_Ocom_OCall @ E @ D @ C )
     != ( c_Com_Ocom_OSemi @ B @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_com_Osimps_I51_J) ).

thf(10212,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Com_Ocom_OCall @ E @ D @ C )
     != ( c_Com_Ocom_OSemi @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1980]) ).

thf(1369,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Oappend @ B ) @ ( c_List_Olist_ONil @ B ) ) @ A )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_append__Nil) ).

thf(7661,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Oappend @ B ) @ ( c_List_Olist_ONil @ B ) ) @ A )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1369]) ).

thf(2051,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ C ) )
     => ( ! [E: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ E ) @ C ) )
           => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ B ) @ ( hAPP @ A @ E ) ) ) )
       => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ ( tc_prod @ D @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ D @ B ) @ C ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_finite__SigmaI) ).

thf(10465,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ C ) )
     => ( ! [E: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ E ) @ C ) )
           => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ B ) @ ( hAPP @ A @ E ) ) ) )
       => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ ( tc_prod @ D @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ D @ B ) @ C ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2051]) ).

thf(2034,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ F ) @ E ) @ ( hAPP @ ( c_List_Oset @ F ) @ D ) ) )
     => ( ( hAPP @ ( hAPP @ ( c_List_Omap @ F @ C ) @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ F @ C ) @ B ) @ E ) @ A ) ) @ D )
        = ( hAPP @ ( hAPP @ ( c_List_Omap @ F @ C ) @ B ) @ D ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_map__fun__upd) ).

thf(10406,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ F ) @ E ) @ ( hAPP @ ( c_List_Oset @ F ) @ D ) ) )
     => ( ( hAPP @ ( hAPP @ ( c_List_Omap @ F @ C ) @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ F @ C ) @ B ) @ E ) @ A ) ) @ D )
        = ( hAPP @ ( hAPP @ ( c_List_Omap @ F @ C ) @ B ) @ D ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2034]) ).

thf(2063,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Com_Ocom_OAss @ E @ D )
     != ( c_Com_Ocom_OCond @ C @ B @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_com_Osimps_I26_J) ).

thf(10510,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Com_Ocom_OAss @ E @ D )
     != ( c_Com_Ocom_OCond @ C @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2063]) ).

thf(1099,axiom,
    ! [A: $i,B: $i] :
      ( ( B != A )
     => ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ A ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_linorder__neqE__nat) ).

thf(6514,plain,
    ! [A: $i,B: $i] :
      ( ( B != A )
     => ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ A ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1099]) ).

thf(378,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ C @ C ) ) @ B ) @ ( c_Transitive__Closure_Ortrancl @ C @ A ) ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ C @ C ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ tc_Nat_Onat @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) ) @ ( c_Nat_Ocompow @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_rtrancl__imp__UN__rel__pow) ).

thf(3755,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ C @ C ) ) @ B ) @ ( c_Transitive__Closure_Ortrancl @ C @ A ) ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ C @ C ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ tc_Nat_Onat @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) ) @ ( c_Nat_Ocompow @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[378]) ).

thf(594,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( E
          = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) )
       => ( ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ D @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ E ) @ A ) )
          = ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ B ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) ) )
      & ( ( E
         != ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) )
       => ( ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ D @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ E ) @ A ) )
          = ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ D @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ E ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) @ D ) @ ( hAPP @ ( c_Set_Oinsert @ C ) @ B ) ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_UN__extend__simps_I1_J) ).

thf(4535,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( E
          = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) )
       => ( ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ D @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ E ) @ A ) )
          = ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ B ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) ) )
      & ( ( E
         != ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) )
       => ( ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ D @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ E ) @ A ) )
          = ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ D @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ E ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) @ D ) @ ( hAPP @ ( c_Set_Oinsert @ C ) @ B ) ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[594]) ).

thf(1491,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Int_Oint ) @ D ) @ C ) )
     => ! [E: $i] :
          ( ! [F: $i] :
              ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Int_Oint ) @ F ) @ ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Int_Oint @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) @ C ) ) )
             => ! [G: $i] :
                  ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Int_Oint ) @ G ) @ B ) )
                 => ( E
                   != ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Int_Oint ) @ G ) @ F ) ) ) )
         => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Int_Oint ) @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ E ) @ A ) ) )
           => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Int_Oint ) @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ E ) @ C ) ) @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_aset_I9_J) ).

thf(8174,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Int_Oint ) @ D ) @ C ) )
     => ! [E: $i] :
          ( ! [F: $i] :
              ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Int_Oint ) @ F ) @ ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Int_Oint @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) @ C ) ) )
             => ! [G: $i] :
                  ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Int_Oint ) @ G ) @ B ) )
                 => ( E
                   != ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Int_Oint ) @ G ) @ F ) ) ) )
         => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Int_Oint ) @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ E ) @ A ) ) )
           => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Int_Oint ) @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ E ) @ C ) ) @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1491]) ).

thf(1817,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Com_Ocom_OAss @ B @ A )
     != c_Com_Ocom_OSKIP ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_com_Osimps_I9_J) ).

thf(9534,plain,
    ! [A: $i,B: $i] :
      ( ( c_Com_Ocom_OAss @ B @ A )
     != c_Com_Ocom_OSKIP ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1817]) ).

thf(1639,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ C ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Int_Oint ) @ B ) @ A ) )
       => ! [D: $i] :
            ( ! [E: $i] :
                ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Int_Oint ) @ E ) @ ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Int_Oint @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) @ C ) ) )
               => ! [F: $i] :
                    ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Int_Oint ) @ F ) @ A ) )
                   => ( D
                     != ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ F ) @ E ) ) ) )
           => ( ( D != B )
             => ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Int_Oint ) @ D ) @ C )
               != B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_bset_I4_J) ).

thf(8846,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ C ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Int_Oint ) @ B ) @ A ) )
       => ! [D: $i] :
            ( ! [E: $i] :
                ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Int_Oint ) @ E ) @ ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Int_Oint @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) @ C ) ) )
               => ! [F: $i] :
                    ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Int_Oint ) @ F ) @ A ) )
                   => ( D
                     != ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ F ) @ E ) ) ) )
           => ( ( D != B )
             => ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Int_Oint ) @ D ) @ C )
               != B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1639]) ).

thf(98,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Ocomm__monoid__mult @ D )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ B ) )
       => ( ! [E: $i] :
              ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ E ) @ B ) )
             => ( ( hAPP @ A @ E )
                = ( c_Groups_Oone__class_Oone @ D ) ) )
         => ( ( hAPP @ ( hAPP @ ( hAPP @ ( c_Finite__Set_Ofold__image @ D @ C @ ( c_Groups_Otimes__class_Otimes @ D ) ) @ A ) @ ( c_Groups_Oone__class_Oone @ D ) ) @ B )
            = ( c_Groups_Oone__class_Oone @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_fold__image__1) ).

thf(2749,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Ocomm__monoid__mult @ D )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ B ) )
       => ( ! [E: $i] :
              ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ E ) @ B ) )
             => ( ( hAPP @ A @ E )
                = ( c_Groups_Oone__class_Oone @ D ) ) )
         => ( ( hAPP @ ( hAPP @ ( hAPP @ ( c_Finite__Set_Ofold__image @ D @ C @ ( c_Groups_Otimes__class_Otimes @ D ) ) @ A ) @ ( c_Groups_Oone__class_Oone @ D ) ) @ B )
            = ( c_Groups_Oone__class_Oone @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[98]) ).

thf(1088,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_List_OtakeWhile @ E @ D @ ( hAPP @ ( hAPP @ ( c_List_Omap @ C @ E ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_List_Omap @ C @ E ) @ B ) @ ( c_List_OtakeWhile @ C @ ( hAPP @ ( c_Fun_Ocomp @ E @ tc_HOL_Obool @ C @ D ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_takeWhile__map) ).

thf(6409,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_List_OtakeWhile @ E @ D @ ( hAPP @ ( hAPP @ ( c_List_Omap @ C @ E ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_List_Omap @ C @ E ) @ B ) @ ( c_List_OtakeWhile @ C @ ( hAPP @ ( c_Fun_Ocomp @ E @ tc_HOL_Obool @ C @ D ) @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1088]) ).

thf(879,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ B ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ A @ B ) ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) )
        & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ B ) @ ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ A @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_neg__mod__conj) ).

thf(5557,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ B ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ A @ B ) ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) )
        & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ B ) @ ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ A @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[879]) ).

thf(496,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Transitive__Closure_Otrancl @ B @ A )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ B @ B @ B ) @ ( c_Transitive__Closure_Otrancl @ B @ A ) ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_trancl__unfold) ).

thf(4184,plain,
    ! [A: $i,B: $i] :
      ( ( c_Transitive__Closure_Otrancl @ B @ A )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ B @ B @ B ) @ ( c_Transitive__Closure_Otrancl @ B @ A ) ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[496]) ).

thf(541,axiom,
    ! [A: $i] :
      ( ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) @ A )
      = ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zmod__zero) ).

thf(4352,plain,
    ! [A: $i] :
      ( ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) @ A )
      = ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[541]) ).

thf(2083,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_add__le__mono1) ).

thf(10597,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2083]) ).

thf(1398,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ A ) @ B ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) ) ) ) @ ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_card__Diff1__less) ).

thf(7770,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ A ) @ B ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) ) ) ) @ ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1398]) ).

thf(1801,axiom,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_le__add2) ).

thf(9468,plain,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1801]) ).

thf(1909,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ A ) @ B ) )
       => ( B = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_dvd_Oantisym) ).

thf(9913,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ A ) @ B ) )
       => ( B = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1909]) ).

thf(1208,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) @ ( c_SetInterval_Oord_OatLeastAtMost @ tc_Nat_Onat @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ D @ C ) ) @ ( c_SetInterval_Oord_OatLeastAtMost @ tc_Nat_Onat @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B @ A ) ) )
    <=> ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ D ) @ C ) )
        | ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ D ) )
          & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ C ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_dvd_OatLeastatMost__subset__iff) ).

thf(6963,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) @ ( c_SetInterval_Oord_OatLeastAtMost @ tc_Nat_Onat @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ D @ C ) ) @ ( c_SetInterval_Oord_OatLeastAtMost @ tc_Nat_Onat @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B @ A ) ) )
       => ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ D ) @ C ) )
          | ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ D ) )
            & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ C ) @ A ) ) ) ) )
      & ( ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ D ) @ C ) )
          | ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ D ) )
            & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ C ) @ A ) ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) @ ( c_SetInterval_Oord_OatLeastAtMost @ tc_Nat_Onat @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ D @ C ) ) @ ( c_SetInterval_Oord_OatLeastAtMost @ tc_Nat_Onat @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1208]) ).

thf(2399,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ? [D: $i] : ( hBOOL @ ( hAPP @ C @ D ) )
     => ( ( B
          = ( c_Hilbert__Choice_OEps @ A @ C ) )
       => ( hBOOL @ ( hAPP @ C @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_exE__some) ).

thf(11856,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ? [D: $i] : ( hBOOL @ ( hAPP @ C @ D ) )
     => ( ( B
          = ( c_Hilbert__Choice_OEps @ A @ C ) )
       => ( hBOOL @ ( hAPP @ C @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2399]) ).

thf(83,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_List_Oconcat @ C @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ ( tc_List_Olist @ C ) ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_List_Oappend @ C ) @ B ) @ ( c_List_Oconcat @ C @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_concat_Osimps_I2_J) ).

thf(2702,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_List_Oconcat @ C @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ ( tc_List_Olist @ C ) ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_List_Oappend @ C ) @ B ) @ ( c_List_Oconcat @ C @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[83]) ).

thf(2013,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ D ) @ C ) @ B ) ) @ ( hAPP @ ( c_List_Oset @ ( tc_prod @ D @ D ) ) @ ( hAPP @ ( hAPP @ ( c_List_Ozip @ D @ D ) @ A ) @ A ) ) ) )
    <=> ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( hAPP @ ( c_List_Oset @ D ) @ A ) ) )
        & ( C = B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zip__same) ).

thf(10338,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ D ) @ C ) @ B ) ) @ ( hAPP @ ( c_List_Oset @ ( tc_prod @ D @ D ) ) @ ( hAPP @ ( hAPP @ ( c_List_Ozip @ D @ D ) @ A ) @ A ) ) ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( hAPP @ ( c_List_Oset @ D ) @ A ) ) )
          & ( C = B ) ) )
      & ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( hAPP @ ( c_List_Oset @ D ) @ A ) ) )
          & ( C = B ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ D ) @ C ) @ B ) ) @ ( hAPP @ ( c_List_Oset @ ( tc_prod @ D @ D ) ) @ ( hAPP @ ( hAPP @ ( c_List_Ozip @ D @ D ) @ A ) @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2013]) ).

thf(1572,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_Fun_Oinj__on @ D @ C @ B @ A )
     => ( ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ ( hAPP @ ( c_Set_Oimage @ D @ C @ B ) @ A ) )
        = ( hAPP @ ( c_Finite__Set_Ocard @ D ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_card__image) ).

thf(8482,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_Fun_Oinj__on @ D @ C @ B @ A )
     => ( ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ ( hAPP @ ( c_Set_Oimage @ D @ C @ B ) @ A ) )
        = ( hAPP @ ( c_Finite__Set_Ocard @ D ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1572]) ).

thf(343,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ A ) @ A )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Un__absorb) ).

thf(3624,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ A ) @ A )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[343]) ).

thf(828,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Un__Int__distrib) ).

thf(5402,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[828]) ).

thf(694,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Map_Odom @ C @ B @ ( hAPP @ ( hAPP @ ( c_COMBB @ B @ ( tc_Option_Ooption @ B ) @ C ) @ ( c_Option_Ooption_OSome @ B ) ) @ A ) )
      = ( c_Orderings_Otop__class_Otop @ ( tc_fun @ C @ tc_HOL_Obool ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_dom__const) ).

thf(4889,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Map_Odom @ C @ B @ ( hAPP @ ( hAPP @ ( c_COMBB @ B @ ( tc_Option_Ooption @ B ) @ C ) @ ( c_Option_Ooption_OSome @ B ) ) @ A ) )
      = ( c_Orderings_Otop__class_Otop @ ( tc_fun @ C @ tc_HOL_Obool ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[694]) ).

thf(216,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ A ) )
       => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ B ) ) @ ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ A ) )
          = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) ) ) @ ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_card__Un__Int) ).

thf(3183,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ A ) )
       => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ B ) ) @ ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ A ) )
          = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) ) ) @ ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[216]) ).

thf(2128,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ D ) @ C ) @ B ) ) @ ( c_Transitive__Closure_Otrancl @ D @ A ) ) )
     => ? [E: $i] :
          ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ D ) @ C ) @ E ) ) @ ( c_Transitive__Closure_Ortrancl @ D @ A ) ) )
          & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ D ) @ E ) @ B ) ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_tranclD2) ).

thf(10740,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ D ) @ C ) @ B ) ) @ ( c_Transitive__Closure_Otrancl @ D @ A ) ) )
     => ? [E: $i] :
          ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ D ) @ C ) @ E ) ) @ ( c_Transitive__Closure_Ortrancl @ D @ A ) ) )
          & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ D ) @ E ) @ B ) ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2128]) ).

thf(1723,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ A ) @ B ) )
       => ( B = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_equalityI) ).

thf(9176,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ A ) @ B ) )
       => ( B = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1723]) ).

thf(1479,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ H @ H ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ H @ H ) @ G ) @ F ) ) @ E ) )
     => ( ( D
          = ( hAPP @ ( hAPP @ ( c_List_Oappend @ H ) @ C ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ H ) @ G ) @ B ) ) )
       => ( ( A
            = ( hAPP @ ( hAPP @ ( c_List_Oappend @ H ) @ C ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ H ) @ F ) @ B ) ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ H ) @ ( tc_List_Olist @ H ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ H ) @ ( tc_List_Olist @ H ) ) @ D ) @ A ) ) @ ( c_List_Olistrel1 @ H @ E ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_listrel1I) ).

thf(8080,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ H @ H ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ H @ H ) @ G ) @ F ) ) @ E ) )
     => ( ( D
          = ( hAPP @ ( hAPP @ ( c_List_Oappend @ H ) @ C ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ H ) @ G ) @ B ) ) )
       => ( ( A
            = ( hAPP @ ( hAPP @ ( c_List_Oappend @ H ) @ C ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ H ) @ F ) @ B ) ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ H ) @ ( tc_List_Olist @ H ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ H ) @ ( tc_List_Olist @ H ) ) @ D ) @ A ) ) @ ( c_List_Olistrel1 @ H @ E ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1479]) ).

thf(657,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_COMBK @ D @ C ) @ B ) @ A )
      = B ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_c__COMBK__1) ).

thf(4765,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_COMBK @ D @ C ) @ B ) @ A )
      = B ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[657]) ).

thf(134,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Nat_Onat ) @ B ) @ ( c_SetInterval_Oord_OlessThan @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_COMBS @ tc_Nat_Onat @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ tc_Nat_Onat @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) @ tc_Nat_Onat ) @ ( c_COMBS @ tc_Nat_Onat @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ tc_Nat_Onat @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ tc_Nat_Onat ) @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ tc_Nat_Onat ) @ c_fconj ) ) @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ tc_Nat_Onat ) @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ tc_HOL_Obool @ tc_Nat_Onat ) @ c_fNot ) ) @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) ) ) ) @ A ) ) )
    <=> ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) )
        & ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ A ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_dvd_OlessThan__iff) ).

thf(2883,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Nat_Onat ) @ B ) @ ( c_SetInterval_Oord_OlessThan @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_COMBS @ tc_Nat_Onat @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ tc_Nat_Onat @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) @ tc_Nat_Onat ) @ ( c_COMBS @ tc_Nat_Onat @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ tc_Nat_Onat @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ tc_Nat_Onat ) @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ tc_Nat_Onat ) @ c_fconj ) ) @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ tc_Nat_Onat ) @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ tc_HOL_Obool @ tc_Nat_Onat ) @ c_fNot ) ) @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) ) ) ) @ A ) ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) )
          & ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ A ) @ B ) ) ) )
      & ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) )
          & ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ A ) @ B ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Nat_Onat ) @ B ) @ ( c_SetInterval_Oord_OlessThan @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_COMBS @ tc_Nat_Onat @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ tc_Nat_Onat @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) @ tc_Nat_Onat ) @ ( c_COMBS @ tc_Nat_Onat @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ tc_Nat_Onat @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ tc_Nat_Onat ) @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ tc_Nat_Onat ) @ c_fconj ) ) @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ tc_Nat_Onat ) @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ tc_HOL_Obool @ tc_Nat_Onat ) @ c_fNot ) ) @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) ) ) ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[134]) ).

thf(1747,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ C ) @ B ) )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ B ) @ C ) ) ) @ A )
        = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_diff__Suc__diff__eq2) ).

thf(9269,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ C ) @ B ) )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ B ) @ C ) ) ) @ A )
        = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1747]) ).

thf(2384,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( c_Map_Odom @ D @ B @ A ) ) )
    <=> ( ( hAPP @ A @ C )
       != ( c_Option_Ooption_ONone @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_domIff) ).

thf(11787,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( c_Map_Odom @ D @ B @ A ) ) )
       => ( ( hAPP @ A @ C )
         != ( c_Option_Ooption_ONone @ B ) ) )
      & ( ( ( hAPP @ A @ C )
         != ( c_Option_Ooption_ONone @ B ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( c_Map_Odom @ D @ B @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2384]) ).

thf(2026,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ F @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ F @ E ) @ D ) @ C ) ) @ ( hAPP @ ( c_List_Oset @ ( tc_prod @ F @ E ) ) @ ( hAPP @ ( hAPP @ ( c_List_Ozip @ F @ E ) @ B ) @ A ) ) ) )
     => ~ ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ F ) @ D ) @ ( hAPP @ ( c_List_Oset @ F ) @ B ) ) )
         => ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ C ) @ ( hAPP @ ( c_List_Oset @ E ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_in__set__zipE) ).

thf(10377,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ F @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ F @ E ) @ D ) @ C ) ) @ ( hAPP @ ( c_List_Oset @ ( tc_prod @ F @ E ) ) @ ( hAPP @ ( hAPP @ ( c_List_Ozip @ F @ E ) @ B ) @ A ) ) ) )
     => ~ ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ F ) @ D ) @ ( hAPP @ ( c_List_Oset @ F ) @ B ) ) )
         => ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ C ) @ ( hAPP @ ( c_List_Oset @ E ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2026]) ).

thf(1680,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ B )
        = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ A ) )
    <=> ( B = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nat__add__left__cancel) ).

thf(9008,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ B )
          = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ A ) )
       => ( B = A ) )
      & ( ( B = A )
       => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ B )
          = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1680]) ).

thf(1663,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D
        = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ B ) ) @ A ) )
     => ( ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ C ) )
           => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ A ) )
              & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ A ) @ C ) ) ) )
          & ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ C ) )
           => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ C ) @ A ) )
              & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) ) ) ) )
       => ( ( C
           != ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) )
         => ( ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ D @ C )
            = A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_divmod__int__rel__mod__eq) ).

thf(8923,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D
        = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ B ) ) @ A ) )
     => ( ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ C ) )
           => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ A ) )
              & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ A ) @ C ) ) ) )
          & ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ C ) )
           => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ C ) @ A ) )
              & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) ) ) ) )
       => ( ( C
           != ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) )
         => ( ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ D @ C )
            = A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1663]) ).

thf(1139,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hAPP @ ( c_Fun_Ocomp @ ( tc_prod @ F @ E ) @ F @ ( tc_prod @ D @ C ) @ ( c_Product__Type_Ofst @ F @ E ) ) @ ( c_Product__Type_Omap__pair @ D @ F @ C @ E @ B @ A ) )
      = ( hAPP @ ( c_Fun_Ocomp @ D @ F @ ( tc_prod @ D @ C ) @ B ) @ ( c_Product__Type_Ofst @ D @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_fst__comp__map__pair) ).

thf(6716,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hAPP @ ( c_Fun_Ocomp @ ( tc_prod @ F @ E ) @ F @ ( tc_prod @ D @ C ) @ ( c_Product__Type_Ofst @ F @ E ) ) @ ( c_Product__Type_Omap__pair @ D @ F @ C @ E @ B @ A ) )
      = ( hAPP @ ( c_Fun_Ocomp @ D @ F @ ( tc_prod @ D @ C ) @ B ) @ ( c_Product__Type_Ofst @ D @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1139]) ).

thf(2053,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ C ) )
     => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ B ) @ ( hAPP @ ( c_Set_Oimage @ D @ B @ A ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_finite__imageI) ).

thf(10470,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ C ) )
     => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ B ) @ ( hAPP @ ( c_Set_Oimage @ D @ B @ A ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2053]) ).

thf(1471,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( c_Product__Type_Ofst @ D @ C ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ C ) @ B ) @ A ) )
      = B ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_fst__conv) ).

thf(8049,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( c_Product__Type_Ofst @ D @ C ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ C ) @ B ) @ A ) )
      = B ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1471]) ).

thf(884,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( c_Relation_OField @ D ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ ( tc_prod @ D @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ D ) @ C ) @ B ) ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ B ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) ) ) @ ( hAPP @ ( c_Relation_OField @ D ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Field__insert) ).

thf(5571,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( c_Relation_OField @ D ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ ( tc_prod @ D @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ D ) @ C ) @ B ) ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ B ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) ) ) @ ( hAPP @ ( c_Relation_OField @ D ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[884]) ).

thf(2328,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Relation_Oirrefl @ B @ A )
    <=> ! [C: $i] :
          ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ B @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ B @ B ) @ C ) @ C ) ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_irrefl__def) ).

thf(11544,plain,
    ! [A: $i,B: $i] :
      ( ( ( c_Relation_Oirrefl @ B @ A )
       => ! [C: $i] :
            ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ B @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ B @ B ) @ C ) @ C ) ) @ A ) ) )
      & ( ! [C: $i] :
            ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ B @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ B @ B ) @ C ) @ C ) ) @ A ) )
       => ( c_Relation_Oirrefl @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2328]) ).

thf(838,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Int_Oint ) @ A ) @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ) ) )
    <=> ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zle__diff1__eq) ).

thf(5434,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Int_Oint ) @ A ) @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ B ) @ A ) ) )
      & ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Int_Oint ) @ A ) @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[838]) ).

thf(1547,axiom,
    ! [A: $i,B: $i] :
      ( ( c_SetInterval_Oord__class_OgreaterThanAtMost @ tc_Nat_Onat @ B @ A )
      = ( hAPP @ ( c_List_Oset @ tc_Nat_Onat ) @ ( c_List_Oupt @ ( hAPP @ c_Nat_OSuc @ B ) @ ( hAPP @ c_Nat_OSuc @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_greaterThanAtMost__upt) ).

thf(8393,plain,
    ! [A: $i,B: $i] :
      ( ( c_SetInterval_Oord__class_OgreaterThanAtMost @ tc_Nat_Onat @ B @ A )
      = ( hAPP @ ( c_List_Oset @ tc_Nat_Onat ) @ ( c_List_Oupt @ ( hAPP @ c_Nat_OSuc @ B ) @ ( hAPP @ c_Nat_OSuc @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1547]) ).

thf(1054,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( F = E )
     => ( ( hAPP @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ D @ C @ B ) @ A ) @ F )
        = ( hAPP @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ D @ C @ B ) @ A ) @ E ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_split__weak__cong) ).

thf(6279,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( F = E )
     => ( ( hAPP @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ D @ C @ B ) @ A ) @ F )
        = ( hAPP @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ D @ C @ B ) @ A ) @ E ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1054]) ).

thf(259,axiom,
    ! [A: $i] : ( c_Wellfounded_Owf @ A @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ ( tc_prod @ A @ A ) @ tc_HOL_Obool ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_wf__empty) ).

thf(3321,plain,
    ! [A: $i] : ( c_Wellfounded_Owf @ A @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ ( tc_prod @ A @ A ) @ tc_HOL_Obool ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[259]) ).

thf(929,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_UnI1) ).

thf(5737,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[929]) ).

thf(1668,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( c_Relation_OImage @ B @ B @ ( c_Relation_OId @ B ) ) @ A )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Image__Id) ).

thf(8955,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( c_Relation_OImage @ B @ B @ ( c_Relation_OId @ B ) ) @ A )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1668]) ).

thf(1720,axiom,
    ! [A: $i] :
      ( ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ A )
      = ( hAPP @ ( c_SetInterval_Oord__class_OatMost @ tc_Nat_Onat ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_atLeast0AtMost) ).

thf(9169,plain,
    ! [A: $i] :
      ( ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ A )
      = ( hAPP @ ( c_SetInterval_Oord__class_OatMost @ tc_Nat_Onat ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1720]) ).

thf(1463,axiom,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) )
       => ( ( c_List_Oupt @ B @ A )
          = ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ tc_Nat_Onat ) @ B ) @ ( c_List_Oupt @ ( hAPP @ c_Nat_OSuc @ B ) @ A ) ) ) )
      & ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) )
       => ( ( c_List_Oupt @ B @ A )
          = ( c_List_Olist_ONil @ tc_Nat_Onat ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_upt__rec) ).

thf(8004,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) )
       => ( ( c_List_Oupt @ B @ A )
          = ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ tc_Nat_Onat ) @ B ) @ ( c_List_Oupt @ ( hAPP @ c_Nat_OSuc @ B ) @ A ) ) ) )
      & ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) )
       => ( ( c_List_Oupt @ B @ A )
          = ( c_List_Olist_ONil @ tc_Nat_Onat ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1463]) ).

thf(11,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Groups_Ocomm__monoid__add @ E )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ C ) )
       => ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ B ) @ C ) )
         => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ B ) @ C ) )
            = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ E ) @ ( hAPP @ A @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ A ) @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_setsum__insert) ).

thf(2440,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Groups_Ocomm__monoid__add @ E )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ C ) )
       => ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ B ) @ C ) )
         => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ B ) @ C ) )
            = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ E ) @ ( hAPP @ A @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ A ) @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[11]) ).

thf(570,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( hAPP @ ( hAPP @ ( c_List_Oappend @ D ) @ C ) @ B )
        = A )
    <=> ( ( C
          = ( hAPP @ ( hAPP @ ( c_List_Otake @ D ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ D ) ) @ C ) ) @ A ) )
        & ( B
          = ( hAPP @ ( hAPP @ ( c_List_Odrop @ D ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ D ) ) @ C ) ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_append__eq__conv__conj) ).

thf(4448,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( ( hAPP @ ( hAPP @ ( c_List_Oappend @ D ) @ C ) @ B )
          = A )
       => ( ( C
            = ( hAPP @ ( hAPP @ ( c_List_Otake @ D ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ D ) ) @ C ) ) @ A ) )
          & ( B
            = ( hAPP @ ( hAPP @ ( c_List_Odrop @ D ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ D ) ) @ C ) ) @ A ) ) ) )
      & ( ( ( C
            = ( hAPP @ ( hAPP @ ( c_List_Otake @ D ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ D ) ) @ C ) ) @ A ) )
          & ( B
            = ( hAPP @ ( hAPP @ ( c_List_Odrop @ D ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ D ) ) @ C ) ) @ A ) ) )
       => ( ( hAPP @ ( hAPP @ ( c_List_Oappend @ D ) @ C ) @ B )
          = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[570]) ).

thf(1509,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_Map_Omap__upds @ D @ C @ B @ A @ ( c_List_Olist_ONil @ C ) )
      = B ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_map__upds__Nil2) ).

thf(8262,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_Map_Omap__upds @ D @ C @ B @ A @ ( c_List_Olist_ONil @ C ) )
      = B ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1509]) ).

thf(2188,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ D ) @ C ) @ B ) ) @ ( hAPP @ ( c_Relation_Oconverse @ D @ E ) @ A ) ) )
    <=> ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ E ) @ B ) @ C ) ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_converse__iff) ).

thf(10951,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ D ) @ C ) @ B ) ) @ ( hAPP @ ( c_Relation_Oconverse @ D @ E ) @ A ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ E ) @ B ) @ C ) ) @ A ) ) )
      & ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ E ) @ B ) @ C ) ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ D ) @ C ) @ B ) ) @ ( hAPP @ ( c_Relation_Oconverse @ D @ E ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2188]) ).

thf(1751,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ ( hAPP @ c_Nat_OSuc @ A ) ) )
     => ( ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Nat_Onat @ B @ ( hAPP @ c_Nat_OSuc @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ A ) ) @ ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Nat_Onat @ B @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_atLeastAtMostSuc__conv) ).

thf(9284,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ ( hAPP @ c_Nat_OSuc @ A ) ) )
     => ( ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Nat_Onat @ B @ ( hAPP @ c_Nat_OSuc @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ A ) ) @ ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Nat_Onat @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1751]) ).

thf(1667,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ B )
        = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ A ) @ B ) )
    <=> ( ( C = A )
        | ( B
          = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__cancel2) ).

thf(8945,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ B )
          = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ A ) @ B ) )
       => ( ( C = A )
          | ( B
            = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) )
      & ( ( ( C = A )
          | ( B
            = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) )
       => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ B )
          = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ A ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1667]) ).

thf(832,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Onth @ D ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ D ) @ C ) @ B ) ) @ A )
      = ( c_Nat_Onat_Onat__case @ D @ C @ ( hAPP @ ( c_List_Onth @ D ) @ B ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nth_Osimps) ).

thf(5417,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Onth @ D ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ D ) @ C ) @ B ) ) @ A )
      = ( c_Nat_Onat_Onat__case @ D @ C @ ( hAPP @ ( c_List_Onth @ D ) @ B ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[832]) ).

thf(456,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Fun_Oinj__on @ E @ D @ C @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ E @ tc_HOL_Obool ) ) )
     => ( ( ( hAPP @ ( hAPP @ ( c_List_Omap @ E @ D ) @ C ) @ B )
          = ( hAPP @ ( hAPP @ ( c_List_Omap @ E @ D ) @ C ) @ A ) )
      <=> ( B = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_inj__map__eq__map) ).

thf(4036,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Fun_Oinj__on @ E @ D @ C @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ E @ tc_HOL_Obool ) ) )
     => ( ( ( ( hAPP @ ( hAPP @ ( c_List_Omap @ E @ D ) @ C ) @ B )
            = ( hAPP @ ( hAPP @ ( c_List_Omap @ E @ D ) @ C ) @ A ) )
         => ( B = A ) )
        & ( ( B = A )
         => ( ( hAPP @ ( hAPP @ ( c_List_Omap @ E @ D ) @ C ) @ B )
            = ( hAPP @ ( hAPP @ ( c_List_Omap @ E @ D ) @ C ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[456]) ).

thf(1953,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ c_Com_Ocom_OBODY @ D )
     != ( c_Com_Ocom_OCond @ C @ B @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_com_Osimps_I55_J) ).

thf(10090,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ c_Com_Ocom_OBODY @ D )
     != ( c_Com_Ocom_OCond @ C @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1953]) ).

thf(309,axiom,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) )
        & ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ A ) @ B ) ) )
     => ( ( c_SetInterval_Oord_OatLeastAtMost @ tc_Nat_Onat @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ A @ B )
        = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_dvd_OatLeastatMost__empty) ).

thf(3508,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) )
        & ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ A ) @ B ) ) )
     => ( ( c_SetInterval_Oord_OatLeastAtMost @ tc_Nat_Onat @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ A @ B )
        = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[309]) ).

thf(69,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Groups_Ocomm__monoid__add @ E )
     => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ tc_Nat_Onat @ E ) @ D ) @ ( hAPP @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ B ) ) )
        = ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ tc_Nat_Onat @ E ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_Nat_Onat @ E @ tc_Nat_Onat ) @ D ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ tc_Nat_Onat @ tc_Nat_Onat ) @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) ) @ B ) ) ) @ ( hAPP @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Nat_Onat @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_setsum__shift__bounds__nat__ivl) ).

thf(2645,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Groups_Ocomm__monoid__add @ E )
     => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ tc_Nat_Onat @ E ) @ D ) @ ( hAPP @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ B ) ) )
        = ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ tc_Nat_Onat @ E ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_Nat_Onat @ E @ tc_Nat_Onat ) @ D ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ tc_Nat_Onat @ tc_Nat_Onat ) @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) ) @ B ) ) ) @ ( hAPP @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Nat_Onat @ C ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[69]) ).

thf(996,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( c_Finite__Set_Ocard @ ( tc_prod @ D @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ D @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ B ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ C @ tc_HOL_Obool ) @ D ) @ A ) ) )
      = ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_card__cartesian__product__singleton) ).

thf(6038,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( c_Finite__Set_Ocard @ ( tc_prod @ D @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ D @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ B ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ C @ tc_HOL_Obool ) @ D ) @ A ) ) )
      = ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[996]) ).

thf(2327,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( c_Natural_Oevaln @ ( c_Com_Ocom_OCond @ F @ E @ D ) @ C @ B @ A )
     => ( ( ( hBOOL @ ( hAPP @ F @ C ) )
         => ~ ( c_Natural_Oevaln @ E @ C @ B @ A ) )
       => ~ ( ~ ( hBOOL @ ( hAPP @ F @ C ) )
           => ~ ( c_Natural_Oevaln @ D @ C @ B @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_evaln__elim__cases_I5_J) ).

thf(11539,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( c_Natural_Oevaln @ ( c_Com_Ocom_OCond @ F @ E @ D ) @ C @ B @ A )
     => ( ( ( hBOOL @ ( hAPP @ F @ C ) )
         => ~ ( c_Natural_Oevaln @ E @ C @ B @ A ) )
       => ~ ( ~ ( hBOOL @ ( hAPP @ F @ C ) )
           => ~ ( c_Natural_Oevaln @ D @ C @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2327]) ).

thf(399,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ B ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ A @ B ) ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_neg__mod__sign) ).

thf(3841,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ B ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ A @ B ) ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[399]) ).

thf(1454,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ D ) @ C ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_List_Oset @ B ) @ ( hAPP @ ( hAPP @ ( c_List_Otake @ B ) @ D ) @ A ) ) ) @ ( hAPP @ ( c_List_Oset @ B ) @ ( hAPP @ ( hAPP @ ( c_List_Otake @ B ) @ C ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_set__take__subset__set__take) ).

thf(7972,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ D ) @ C ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_List_Oset @ B ) @ ( hAPP @ ( hAPP @ ( c_List_Otake @ B ) @ D ) @ A ) ) ) @ ( hAPP @ ( c_List_Oset @ B ) @ ( hAPP @ ( hAPP @ ( c_List_Otake @ B ) @ C ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1454]) ).

thf(61,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Ocomm__monoid__add @ C )
     => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ tc_Nat_Onat @ C ) @ B ) @ ( hAPP @ ( c_SetInterval_Oord__class_OatMost @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ A ) ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ C ) @ ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ tc_Nat_Onat @ C ) @ B ) @ ( hAPP @ ( c_SetInterval_Oord__class_OatMost @ tc_Nat_Onat ) @ A ) ) ) @ ( hAPP @ B @ ( hAPP @ c_Nat_OSuc @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_setsum__atMost__Suc) ).

thf(2617,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Ocomm__monoid__add @ C )
     => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ tc_Nat_Onat @ C ) @ B ) @ ( hAPP @ ( c_SetInterval_Oord__class_OatMost @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ A ) ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ C ) @ ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ tc_Nat_Onat @ C ) @ B ) @ ( hAPP @ ( c_SetInterval_Oord__class_OatMost @ tc_Nat_Onat ) @ A ) ) ) @ ( hAPP @ B @ ( hAPP @ c_Nat_OSuc @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[61]) ).

thf(1180,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) ) @ B ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ ( tc_List_Olist @ C ) @ ( tc_List_Olist @ C ) ) @ tc_HOL_Obool ) ) @ ( c_List_Olistrel1 @ C @ B ) ) @ ( c_List_Olistrel1 @ C @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_listrel1__mono) ).

thf(6869,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) ) @ B ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ ( tc_List_Olist @ C ) @ ( tc_List_Olist @ C ) ) @ tc_HOL_Obool ) ) @ ( c_List_Olistrel1 @ C @ B ) ) @ ( c_List_Olistrel1 @ C @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1180]) ).

thf(72,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Groups_Oab__group__add @ E )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ C ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ B ) )
         => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) )
            = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ E ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ E ) @ ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ A ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ A ) @ B ) ) ) @ ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_setsum__Un) ).

thf(2662,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Groups_Oab__group__add @ E )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ C ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ B ) )
         => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) )
            = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ E ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ E ) @ ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ A ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ A ) @ B ) ) ) @ ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[72]) ).

thf(1240,axiom,
    ! [A: $i,B: $i] : ( c_Fun_Oinj__on @ B @ ( tc_Option_Ooption @ B ) @ ( c_Option_Ooption_OSome @ B ) @ A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_inj__Some) ).

thf(7106,plain,
    ! [A: $i,B: $i] : ( c_Fun_Oinj__on @ B @ ( tc_Option_Ooption @ B ) @ ( c_Option_Ooption_OSome @ B ) @ A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1240]) ).

thf(1710,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ B ) )
     => ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_DiffI) ).

thf(9139,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ B ) )
     => ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1710]) ).

thf(530,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Finite__Set_Ofolding__one @ E @ D @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ E ) @ B ) )
       => ( ( B
           != ( c_Orderings_Obot__class_Obot @ ( tc_fun @ E @ tc_HOL_Obool ) ) )
         => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ E ) @ A ) )
           => ( ( A
               != ( c_Orderings_Obot__class_Obot @ ( tc_fun @ E @ tc_HOL_Obool ) ) )
             => ( ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ B ) @ A )
                  = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ E @ tc_HOL_Obool ) ) )
               => ( ( hAPP @ C @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ B ) @ A ) )
                  = ( hAPP @ ( hAPP @ D @ ( hAPP @ C @ B ) ) @ ( hAPP @ C @ A ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_folding__one_Ounion__disjoint) ).

thf(4308,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Finite__Set_Ofolding__one @ E @ D @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ E ) @ B ) )
       => ( ( B
           != ( c_Orderings_Obot__class_Obot @ ( tc_fun @ E @ tc_HOL_Obool ) ) )
         => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ E ) @ A ) )
           => ( ( A
               != ( c_Orderings_Obot__class_Obot @ ( tc_fun @ E @ tc_HOL_Obool ) ) )
             => ( ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ B ) @ A )
                  = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ E @ tc_HOL_Obool ) ) )
               => ( ( hAPP @ C @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ B ) @ A ) )
                  = ( hAPP @ ( hAPP @ D @ ( hAPP @ C @ B ) ) @ ( hAPP @ C @ A ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[530]) ).

thf(6,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( class_Groups_Ocomm__monoid__add @ F )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ E ) @ D ) )
       => ( ! [G: $i] :
              ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ G ) @ D ) )
             => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ ( hAPP @ B @ G ) ) ) )
         => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ E @ F ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ E @ ( tc_fun @ C @ tc_HOL_Obool ) @ F ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ F ) @ ( tc_fun @ ( tc_fun @ C @ tc_HOL_Obool ) @ F ) @ E ) @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ C @ F ) ) @ A ) ) @ B ) ) @ D )
            = ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ ( tc_prod @ E @ C ) @ F ) @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ E @ C @ F ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ E @ C ) @ D ) @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_setsum__Sigma) ).

thf(2423,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( class_Groups_Ocomm__monoid__add @ F )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ E ) @ D ) )
       => ( ! [G: $i] :
              ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ G ) @ D ) )
             => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ ( hAPP @ B @ G ) ) ) )
         => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ E @ F ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ E @ ( tc_fun @ C @ tc_HOL_Obool ) @ F ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ F ) @ ( tc_fun @ ( tc_fun @ C @ tc_HOL_Obool ) @ F ) @ E ) @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ C @ F ) ) @ A ) ) @ B ) ) @ D )
            = ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ ( tc_prod @ E @ C ) @ F ) @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ E @ C @ F ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ E @ C ) @ D ) @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[6]) ).

thf(1249,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_subset__trans) ).

thf(7133,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1249]) ).

thf(1314,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ ( tc_Option_Ooption @ E ) ) @ ( hAPP @ ( c_Set_Oimage @ D @ ( tc_Option_Ooption @ E ) @ C ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) ) )
     => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ ( tc_Option_Ooption @ E ) ) @ ( hAPP @ ( c_Set_Oimage @ D @ ( tc_Option_Ooption @ E ) @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ D @ ( tc_Option_Ooption @ E ) ) @ C ) @ B ) @ ( hAPP @ ( c_Option_Ooption_OSome @ E ) @ A ) ) ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_finite__range__updI) ).

thf(7444,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ ( tc_Option_Ooption @ E ) ) @ ( hAPP @ ( c_Set_Oimage @ D @ ( tc_Option_Ooption @ E ) @ C ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) ) )
     => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ ( tc_Option_Ooption @ E ) ) @ ( hAPP @ ( c_Set_Oimage @ D @ ( tc_Option_Ooption @ E ) @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ D @ ( tc_Option_Ooption @ E ) ) @ C ) @ B ) @ ( hAPP @ ( c_Option_Ooption_OSome @ E ) @ A ) ) ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1314]) ).

thf(141,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ C @ B )
        = ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ A @ B ) )
    <=> ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Int_Oint ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Int_Oint ) @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zmod__eq__dvd__iff) ).

thf(2909,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ C @ B )
          = ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ A @ B ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Int_Oint ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Int_Oint ) @ C ) @ A ) ) ) )
      & ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Int_Oint ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Int_Oint ) @ C ) @ A ) ) )
       => ( ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ C @ B )
          = ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ A @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[141]) ).

thf(1034,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Relation_Orefl__on @ C @ B @ ( hAPP @ ( c_Relation_Oconverse @ C @ C ) @ A ) )
    <=> ( c_Relation_Orefl__on @ C @ B @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_refl__on__converse) ).

thf(6192,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( c_Relation_Orefl__on @ C @ B @ ( hAPP @ ( c_Relation_Oconverse @ C @ C ) @ A ) )
       => ( c_Relation_Orefl__on @ C @ B @ A ) )
      & ( ( c_Relation_Orefl__on @ C @ B @ A )
       => ( c_Relation_Orefl__on @ C @ B @ ( hAPP @ ( c_Relation_Oconverse @ C @ C ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1034]) ).

thf(1076,axiom,
    ! [A: $i,B: $i] :
      ( ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Nat_Onat @ B @ A )
      = ( hAPP @ ( c_List_Oset @ tc_Nat_Onat ) @ ( c_List_Oupt @ B @ ( hAPP @ c_Nat_OSuc @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_atLeastAtMost__upt) ).

thf(6368,plain,
    ! [A: $i,B: $i] :
      ( ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Nat_Onat @ B @ A )
      = ( hAPP @ ( c_List_Oset @ tc_Nat_Onat ) @ ( c_List_Oupt @ B @ ( hAPP @ c_Nat_OSuc @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1076]) ).

thf(688,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( hAPP @ ( c_Set_Oimage @ E @ D @ C ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ E @ tc_HOL_Obool ) ) )
        = ( c_Orderings_Otop__class_Otop @ ( tc_fun @ D @ tc_HOL_Obool ) ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Set_Ovimage @ E @ D @ C ) @ B ) ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( c_Set_Oimage @ E @ D @ C ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_vimage__subsetD) ).

thf(4872,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( hAPP @ ( c_Set_Oimage @ E @ D @ C ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ E @ tc_HOL_Obool ) ) )
        = ( c_Orderings_Otop__class_Otop @ ( tc_fun @ D @ tc_HOL_Obool ) ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Set_Ovimage @ E @ D @ C ) @ B ) ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( c_Set_Oimage @ E @ D @ C ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[688]) ).

thf(177,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ F ) @ E ) @ D ) )
     => ( ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ F @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ D ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) @ F ) @ ( hAPP @ ( c_Set_Oinsert @ C ) @ B ) ) @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ F @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ D ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_UN__insert__distrib) ).

thf(3042,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ F ) @ E ) @ D ) )
     => ( ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ F @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ D ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) @ F ) @ ( hAPP @ ( c_Set_Oinsert @ C ) @ B ) ) @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ F @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ D ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[177]) ).

thf(955,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ B ) @ A )
      = ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ B @ A ) ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zmod__zmult1__eq) ).

thf(5855,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ B ) @ A )
      = ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ B @ A ) ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[955]) ).

thf(2202,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( hAPP @ ( c_Relation_ORange @ B @ D ) @ A ) ) )
     => ~ ! [E: $i] :
            ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ B @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ B @ D ) @ E ) @ C ) ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_RangeE) ).

thf(11016,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( hAPP @ ( c_Relation_ORange @ B @ D ) @ A ) ) )
     => ~ ! [E: $i] :
            ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ B @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ B @ D ) @ E ) @ C ) ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2202]) ).

thf(737,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) )
     => ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ A ) @ B ) )
        = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Diff__partition) ).

thf(5043,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) )
     => ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ A ) @ B ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[737]) ).

thf(262,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ ( tc_fun @ D @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( c_member @ D ) ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ ( tc_fun @ D @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( c_member @ D ) ) @ B ) ) @ A ) )
    <=> ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_sup__Un__eq) ).

thf(3343,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ ( tc_fun @ D @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( c_member @ D ) ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ ( tc_fun @ D @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( c_member @ D ) ) @ B ) ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) ) ) )
      & ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ ( tc_fun @ D @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( c_member @ D ) ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ ( tc_fun @ D @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( c_member @ D ) ) @ B ) ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[262]) ).

thf(1820,axiom,
    ! [A: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_le0) ).

thf(9545,plain,
    ! [A: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1820]) ).

thf(2155,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_trans__le__add2) ).

thf(10862,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2155]) ).

thf(365,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Finite__Set_Ofolding__one__idem @ E @ D @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ E ) @ B ) )
       => ( ( B
           != ( c_Orderings_Obot__class_Obot @ ( tc_fun @ E @ tc_HOL_Obool ) ) )
         => ( ( hAPP @ C @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ A ) @ B ) )
            = ( hAPP @ ( hAPP @ D @ A ) @ ( hAPP @ C @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_folding__one__idem_Oinsert__idem) ).

thf(3707,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Finite__Set_Ofolding__one__idem @ E @ D @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ E ) @ B ) )
       => ( ( B
           != ( c_Orderings_Obot__class_Obot @ ( tc_fun @ E @ tc_HOL_Obool ) ) )
         => ( ( hAPP @ C @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ A ) @ B ) )
            = ( hAPP @ ( hAPP @ D @ A ) @ ( hAPP @ C @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[365]) ).

thf(1526,axiom,
    ! [A: $i] :
      ( ( hAPP @ ( c_SetInterval_Oord__class_OlessThan @ tc_Nat_Onat ) @ A )
      = ( hAPP @ ( c_List_Oset @ tc_Nat_Onat ) @ ( c_List_Oupt @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_atLeast__upt) ).

thf(8322,plain,
    ! [A: $i] :
      ( ( hAPP @ ( c_SetInterval_Oord__class_OlessThan @ tc_Nat_Onat ) @ A )
      = ( hAPP @ ( c_List_Oset @ tc_Nat_Onat ) @ ( c_List_Oupt @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1526]) ).

thf(853,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ C @ C ) @ B ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ C @ tc_HOL_Obool ) @ C ) @ B ) ) ) )
       => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Equiv__Relations_Oquotient @ C ) @ B ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_finite__quotient) ).

thf(5483,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ C @ C ) @ B ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ C @ tc_HOL_Obool ) @ C ) @ B ) ) ) )
       => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Equiv__Relations_Oquotient @ C ) @ B ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[853]) ).

thf(2149,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ D ) @ C ) @ B ) ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ C ) @ ( hAPP @ ( c_Relation_ODomain @ E @ D ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_DomainI) ).

thf(10840,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ D ) @ C ) @ B ) ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ C ) @ ( hAPP @ ( c_Relation_ODomain @ E @ D ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2149]) ).

thf(925,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( c_Set_Ovimage @ D @ C @ B ) @ A )
      = ( hAPP @ ( c_Set_OCollect @ D ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ ( tc_fun @ C @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ C @ ( tc_fun @ ( tc_fun @ C @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ D ) @ ( c_member @ C ) ) @ B ) ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_vimage__def) ).

thf(5722,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( c_Set_Ovimage @ D @ C @ B ) @ A )
      = ( hAPP @ ( c_Set_OCollect @ D ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ ( tc_fun @ C @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ C @ ( tc_fun @ ( tc_fun @ C @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ D ) @ ( c_member @ C ) ) @ B ) ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[925]) ).

thf(869,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ D ) @ C ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ D ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ C ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zadd__zless__mono) ).

thf(5524,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ D ) @ C ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ D ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ C ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[869]) ).

thf(2395,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ F ) @ E ) @ D ) )
       => ( ( hAPP @ ( c_Fun_Ooverride__on @ F @ C @ B @ A @ D ) @ E )
          = ( hAPP @ A @ E ) ) )
      & ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ F ) @ E ) @ D ) )
       => ( ( hAPP @ ( c_Fun_Ooverride__on @ F @ C @ B @ A @ D ) @ E )
          = ( hAPP @ B @ E ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_override__on__def) ).

thf(11837,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ F ) @ E ) @ D ) )
       => ( ( hAPP @ ( c_Fun_Ooverride__on @ F @ C @ B @ A @ D ) @ E )
          = ( hAPP @ A @ E ) ) )
      & ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ F ) @ E ) @ D ) )
       => ( ( hAPP @ ( c_Fun_Ooverride__on @ F @ C @ B @ A @ D ) @ E )
          = ( hAPP @ B @ E ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2395]) ).

thf(2374,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ A ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_insertI2) ).

thf(11747,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ A ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2374]) ).

thf(1786,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_prod @ H @ G ) @ ( tc_prod @ H @ G ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_prod @ H @ G ) @ ( tc_prod @ H @ G ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ H @ G ) @ F ) @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ H @ G ) @ D ) @ C ) ) ) @ ( c_Wellfounded_Olex__prod @ H @ G @ B @ A ) ) )
    <=> ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ H @ H ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ H @ H ) @ F ) @ D ) ) @ B ) )
        | ( ( F = D )
          & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ G @ G ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ G @ G ) @ E ) @ C ) ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_in__lex__prod) ).

thf(9416,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_prod @ H @ G ) @ ( tc_prod @ H @ G ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_prod @ H @ G ) @ ( tc_prod @ H @ G ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ H @ G ) @ F ) @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ H @ G ) @ D ) @ C ) ) ) @ ( c_Wellfounded_Olex__prod @ H @ G @ B @ A ) ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ H @ H ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ H @ H ) @ F ) @ D ) ) @ B ) )
          | ( ( F = D )
            & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ G @ G ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ G @ G ) @ E ) @ C ) ) @ A ) ) ) ) )
      & ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ H @ H ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ H @ H ) @ F ) @ D ) ) @ B ) )
          | ( ( F = D )
            & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ G @ G ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ G @ G ) @ E ) @ C ) ) @ A ) ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_prod @ H @ G ) @ ( tc_prod @ H @ G ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_prod @ H @ G ) @ ( tc_prod @ H @ G ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ H @ G ) @ F ) @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ H @ G ) @ D ) @ C ) ) ) @ ( c_Wellfounded_Olex__prod @ H @ G @ B @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1786]) ).

thf(641,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Nat_Onat_Onat__case @ C @ B @ A @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
      = B ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nat__case__0) ).

thf(4708,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Nat_Onat_Onat__case @ C @ B @ A @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
      = B ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[641]) ).

thf(2074,axiom,
    ! [A: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ A ) @ ( hAPP @ c_Nat_OSuc @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_lessI) ).

thf(10571,plain,
    ! [A: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ A ) @ ( hAPP @ c_Nat_OSuc @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2074]) ).

thf(2207,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Com_Ocom_OWhile @ E @ D )
     != ( c_Com_Ocom_OLocal @ C @ B @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_com_Osimps_I39_J) ).

thf(11033,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Com_Ocom_OWhile @ E @ D )
     != ( c_Com_Ocom_OLocal @ C @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2207]) ).

thf(1592,axiom,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ A ) )
    <=> ? [B: $i] :
          ( A
          = ( hAPP @ c_Nat_OSuc @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_gr0__conv__Suc) ).

thf(8641,plain,
    ! [A: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ A ) )
       => ? [B: $i] :
            ( A
            = ( hAPP @ c_Nat_OSuc @ B ) ) )
      & ( ? [B: $i] :
            ( A
            = ( hAPP @ c_Nat_OSuc @ B ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1592]) ).

thf(2086,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Com_Ocom_OCond @ C @ B @ A )
     != c_Com_Ocom_OSKIP ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_com_Osimps_I15_J) ).

thf(10607,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Com_Ocom_OCond @ C @ B @ A )
     != c_Com_Ocom_OSKIP ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2086]) ).

thf(2363,axiom,
    ! [A: $i,B: $i] :
      ( ~ ( hBOOL @ B )
      | ( hBOOL @ ( hAPP @ ( hAPP @ c_fimplies @ A ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_c__fimplies__2) ).

thf(11710,plain,
    ! [A: $i,B: $i] :
      ( ~ ( hBOOL @ B )
      | ( hBOOL @ ( hAPP @ ( hAPP @ c_fimplies @ A ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2363]) ).

thf(2350,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ~ ( hBOOL @ ( hAPP @ C @ B ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Natural_Oevalc @ ( c_Com_Ocom_OWhile @ C @ A ) ) @ B ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_evalc_OWhileFalse) ).

thf(11638,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ~ ( hBOOL @ ( hAPP @ C @ B ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Natural_Oevalc @ ( c_Com_Ocom_OWhile @ C @ A ) ) @ B ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2350]) ).

thf(48,axiom,
    ! [A: $i,B: $i] :
      ( ( ( c_List_Olist_ONil @ B )
        = ( c_List_Oconcat @ B @ A ) )
    <=> ! [C: $i] :
          ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_List_Olist @ B ) ) @ C ) @ ( hAPP @ ( c_List_Oset @ ( tc_List_Olist @ B ) ) @ A ) ) )
         => ( C
            = ( c_List_Olist_ONil @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Nil__eq__concat__conv) ).

thf(2556,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( c_List_Olist_ONil @ B )
          = ( c_List_Oconcat @ B @ A ) )
       => ! [C: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_List_Olist @ B ) ) @ C ) @ ( hAPP @ ( c_List_Oset @ ( tc_List_Olist @ B ) ) @ A ) ) )
           => ( C
              = ( c_List_Olist_ONil @ B ) ) ) )
      & ( ! [C: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_List_Olist @ B ) ) @ C ) @ ( hAPP @ ( c_List_Oset @ ( tc_List_Olist @ B ) ) @ A ) ) )
           => ( C
              = ( c_List_Olist_ONil @ B ) ) )
       => ( ( c_List_Olist_ONil @ B )
          = ( c_List_Oconcat @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[48]) ).

thf(480,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ B ) @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ) ) @ A ) )
    <=> ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_add1__zle__eq) ).

thf(4116,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ B ) @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ) ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ B ) @ A ) ) )
      & ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ B ) @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ) ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[480]) ).

thf(615,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Un__assoc) ).

thf(4613,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[615]) ).

thf(1173,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Relation_Orefl__on @ C @ B @ A )
    <=> ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ C @ C ) @ B ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ C @ tc_HOL_Obool ) @ C ) @ B ) ) ) )
        & ! [D: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ D ) @ B ) )
           => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ C @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ C @ C ) @ D ) @ D ) ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_refl__on__def) ).

thf(6824,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( c_Relation_Orefl__on @ C @ B @ A )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ C @ C ) @ B ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ C @ tc_HOL_Obool ) @ C ) @ B ) ) ) )
          & ! [D: $i] :
              ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ D ) @ B ) )
             => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ C @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ C @ C ) @ D ) @ D ) ) @ A ) ) ) ) )
      & ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ C @ C ) @ B ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ C @ tc_HOL_Obool ) @ C ) @ B ) ) ) )
          & ! [D: $i] :
              ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ D ) @ B ) )
             => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ C @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ C @ C ) @ D ) @ D ) ) @ A ) ) ) )
       => ( c_Relation_Orefl__on @ C @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1173]) ).

thf(609,axiom,
    ! [A: $i,B: $i] : ( c_Relation_Orefl__on @ B @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) ) @ A ) @ ( c_Relation_OId @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_refl__reflcl) ).

thf(4594,plain,
    ! [A: $i,B: $i] : ( c_Relation_Orefl__on @ B @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) ) @ A ) @ ( c_Relation_OId @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[609]) ).

thf(981,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ C ) )
     => ( ! [E: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ E ) @ C ) )
           => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ B ) @ ( hAPP @ A @ E ) ) ) )
       => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ B ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ D @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ C ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_finite__UN__I) ).

thf(5969,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ C ) )
     => ( ! [E: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ E ) @ C ) )
           => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ B ) @ ( hAPP @ A @ E ) ) ) )
       => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ B ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ D @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ C ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[981]) ).

thf(843,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ B ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) ) )
      = ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ B ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_insert__Diff__single) ).

thf(5451,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ B ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) ) )
      = ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ B ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[843]) ).

thf(1135,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ A ) @ B ) )
       => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_rev__finite__subset) ).

thf(6699,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ A ) @ B ) )
       => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1135]) ).

thf(410,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( c_Fun_Othe__inv__into @ E @ D @ C @ B ) @ A )
      = ( c_HOL_OThe @ E @ ( hAPP @ ( hAPP @ ( c_COMBS @ E @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ E ) @ c_fconj ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ E @ ( tc_fun @ E @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( c_member @ E ) ) @ C ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ E @ D @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ D @ ( tc_fun @ D @ tc_HOL_Obool ) @ E ) @ c_fequal ) @ B ) ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_the__inv__into__def) ).

thf(3890,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( c_Fun_Othe__inv__into @ E @ D @ C @ B ) @ A )
      = ( c_HOL_OThe @ E @ ( hAPP @ ( hAPP @ ( c_COMBS @ E @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ E ) @ c_fconj ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ E @ ( tc_fun @ E @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( c_member @ E ) ) @ C ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ E @ D @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ D @ ( tc_fun @ D @ tc_HOL_Obool ) @ E ) @ c_fequal ) @ B ) ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[410]) ).

thf(862,axiom,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ B ) @ A ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_UNIV__I) ).

thf(5508,plain,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ B ) @ A ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[862]) ).

thf(1829,axiom,
    ! [A: $i] :
      ( ( hAPP @ ( c_SetInterval_Oord__class_OatMost @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ A ) ) @ ( hAPP @ ( c_SetInterval_Oord__class_OatMost @ tc_Nat_Onat ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_atMost__Suc) ).

thf(9577,plain,
    ! [A: $i] :
      ( ( hAPP @ ( c_SetInterval_Oord__class_OatMost @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ A ) ) @ ( hAPP @ ( c_SetInterval_Oord__class_OatMost @ tc_Nat_Onat ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1829]) ).

thf(1416,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ C ) @ B ) )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ B ) ) @ C )
        = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ B ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_diff__add__assoc) ).

thf(7831,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ C ) @ B ) )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ B ) ) @ C )
        = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ B ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1416]) ).

thf(559,axiom,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ tc_Nat_Onat ) @ ( hAPP @ ( c_Set_OCollect @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ tc_Nat_Onat @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ tc_Nat_Onat ) @ c_fconj ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_finite__M__bounded__by__nat) ).

thf(4412,plain,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ tc_Nat_Onat ) @ ( hAPP @ ( c_Set_OCollect @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ tc_Nat_Onat @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ tc_Nat_Onat ) @ c_fconj ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[559]) ).

thf(516,axiom,
    ! [A: $i,B: $i] :
      ( ( ( c_SetInterval_Oord_OatLeastLessThan @ tc_Nat_Onat @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ tc_Nat_Onat @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ tc_Nat_Onat @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) @ tc_Nat_Onat ) @ ( c_COMBS @ tc_Nat_Onat @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ tc_Nat_Onat @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ tc_Nat_Onat ) @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ tc_Nat_Onat ) @ c_fconj ) ) @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ tc_Nat_Onat ) @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ tc_HOL_Obool @ tc_Nat_Onat ) @ c_fNot ) ) @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) ) ) ) @ B @ A )
        = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) )
    <=> ~ ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) )
          & ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ A ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_dvd_OatLeastLessThan__empty__iff) ).

thf(4256,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( c_SetInterval_Oord_OatLeastLessThan @ tc_Nat_Onat @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ tc_Nat_Onat @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ tc_Nat_Onat @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) @ tc_Nat_Onat ) @ ( c_COMBS @ tc_Nat_Onat @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ tc_Nat_Onat @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ tc_Nat_Onat ) @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ tc_Nat_Onat ) @ c_fconj ) ) @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ tc_Nat_Onat ) @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ tc_HOL_Obool @ tc_Nat_Onat ) @ c_fNot ) ) @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) ) ) ) @ B @ A )
          = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) )
       => ~ ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) )
            & ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ A ) @ B ) ) ) )
      & ( ~ ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) )
            & ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ A ) @ B ) ) )
       => ( ( c_SetInterval_Oord_OatLeastLessThan @ tc_Nat_Onat @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ tc_Nat_Onat @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ tc_Nat_Onat @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) @ tc_Nat_Onat ) @ ( c_COMBS @ tc_Nat_Onat @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ tc_Nat_Onat @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ tc_Nat_Onat ) @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ tc_Nat_Onat ) @ c_fconj ) ) @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ tc_Nat_Onat ) @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ tc_HOL_Obool @ tc_Nat_Onat ) @ c_fNot ) ) @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) ) ) ) @ B @ A )
          = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[516]) ).

thf(1609,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Wellfounded_Owf @ C @ B )
     => ( ( c_Wellfounded_Owf @ C @ A )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ C @ C @ C ) @ B ) @ A ) ) @ B ) )
         => ( c_Wellfounded_Owf @ C @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) ) @ B ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_wf__union__compatible) ).

thf(8728,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Wellfounded_Owf @ C @ B )
     => ( ( c_Wellfounded_Owf @ C @ A )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ C @ C @ C ) @ B ) @ A ) ) @ B ) )
         => ( c_Wellfounded_Owf @ C @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) ) @ B ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1609]) ).

thf(1393,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( F = E )
       => ( ( hAPP @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ D @ C ) @ B ) @ E ) @ A ) @ F )
          = A ) )
      & ( ( F != E )
       => ( ( hAPP @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ D @ C ) @ B ) @ E ) @ A ) @ F )
          = ( hAPP @ B @ F ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_fun__upd__def) ).

thf(7753,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( F = E )
       => ( ( hAPP @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ D @ C ) @ B ) @ E ) @ A ) @ F )
          = A ) )
      & ( ( F != E )
       => ( ( hAPP @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ D @ C ) @ B ) @ E ) @ A ) @ F )
          = ( hAPP @ B @ F ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1393]) ).

thf(739,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E
        = ( hAPP @ D @ C ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ B ) @ E ) @ ( hAPP @ ( c_Set_Oimage @ A @ B @ D ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ A @ tc_HOL_Obool ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_range__eqI) ).

thf(5048,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( E
        = ( hAPP @ D @ C ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ B ) @ E ) @ ( hAPP @ ( c_Set_Oimage @ A @ B @ D ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ A @ tc_HOL_Obool ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[739]) ).

thf(2305,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( c_Natural_Oevaln @ ( c_Com_Ocom_OCond @ F @ E @ D ) @ C @ B @ A )
    <=> ( ( ( hBOOL @ ( hAPP @ F @ C ) )
          & ( c_Natural_Oevaln @ E @ C @ B @ A ) )
        | ( ~ ( hBOOL @ ( hAPP @ F @ C ) )
          & ( c_Natural_Oevaln @ D @ C @ B @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_evaln_Oequations_I6_J) ).

thf(11427,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( c_Natural_Oevaln @ ( c_Com_Ocom_OCond @ F @ E @ D ) @ C @ B @ A )
       => ( ( ( hBOOL @ ( hAPP @ F @ C ) )
            & ( c_Natural_Oevaln @ E @ C @ B @ A ) )
          | ( ~ ( hBOOL @ ( hAPP @ F @ C ) )
            & ( c_Natural_Oevaln @ D @ C @ B @ A ) ) ) )
      & ( ( ( ( hBOOL @ ( hAPP @ F @ C ) )
            & ( c_Natural_Oevaln @ E @ C @ B @ A ) )
          | ( ~ ( hBOOL @ ( hAPP @ F @ C ) )
            & ( c_Natural_Oevaln @ D @ C @ B @ A ) ) )
       => ( c_Natural_Oevaln @ ( c_Com_Ocom_OCond @ F @ E @ D ) @ C @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2305]) ).

thf(2195,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_Fun_Oinj__on @ D @ C @ B @ A )
     => ( c_Fun_Oinj__on @ C @ D @ ( c_Fun_Othe__inv__into @ D @ C @ A @ B ) @ ( hAPP @ ( c_Set_Oimage @ D @ C @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_inj__on__the__inv__into) ).

thf(10993,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_Fun_Oinj__on @ D @ C @ B @ A )
     => ( c_Fun_Oinj__on @ C @ D @ ( c_Fun_Othe__inv__into @ D @ C @ A @ B ) @ ( hAPP @ ( c_Set_Oimage @ D @ C @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2195]) ).

thf(1046,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ C ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Int_Oint ) @ B ) @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ) ) @ A ) )
       => ! [D: $i] :
            ( ! [E: $i] :
                ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Int_Oint ) @ E ) @ ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Int_Oint @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) @ C ) ) )
               => ! [F: $i] :
                    ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Int_Oint ) @ F ) @ A ) )
                   => ( D
                     != ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ F ) @ E ) ) ) )
           => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ B ) @ D ) )
             => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Int_Oint ) @ D ) @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_bset_I8_J) ).

thf(6243,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ C ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Int_Oint ) @ B ) @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ) ) @ A ) )
       => ! [D: $i] :
            ( ! [E: $i] :
                ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Int_Oint ) @ E ) @ ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Int_Oint @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) @ C ) ) )
               => ! [F: $i] :
                    ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Int_Oint ) @ F ) @ A ) )
                   => ( D
                     != ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ F ) @ E ) ) ) )
           => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ B ) @ D ) )
             => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Int_Oint ) @ D ) @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1046]) ).

thf(1352,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_List_Odistinct @ C ) @ ( hAPP @ ( hAPP @ ( c_List_Omap @ ( tc_prod @ C @ B ) @ C ) @ ( c_Product__Type_Ofst @ C @ B ) ) @ A ) ) )
     => ( ( c_Map_Oran @ C @ B @ ( c_Map_Omap__of @ C @ B @ A ) )
        = ( hAPP @ ( c_Set_Oimage @ ( tc_prod @ C @ B ) @ B @ ( c_Product__Type_Osnd @ C @ B ) ) @ ( hAPP @ ( c_List_Oset @ ( tc_prod @ C @ B ) ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_ran__distinct) ).

thf(7599,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_List_Odistinct @ C ) @ ( hAPP @ ( hAPP @ ( c_List_Omap @ ( tc_prod @ C @ B ) @ C ) @ ( c_Product__Type_Ofst @ C @ B ) ) @ A ) ) )
     => ( ( c_Map_Oran @ C @ B @ ( c_Map_Omap__of @ C @ B @ A ) )
        = ( hAPP @ ( c_Set_Oimage @ ( tc_prod @ C @ B ) @ B @ ( c_Product__Type_Osnd @ C @ B ) ) @ ( hAPP @ ( c_List_Oset @ ( tc_prod @ C @ B ) ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1352]) ).

thf(1324,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Nat_Onat ) @ B ) @ A )
      = ( hAPP @ ( c_Product__Type_Ofst @ tc_Nat_Onat @ tc_Nat_Onat ) @ ( c_Divides_Odivmod__nat @ B @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_div__nat__def) ).

thf(7491,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Nat_Onat ) @ B ) @ A )
      = ( hAPP @ ( c_Product__Type_Ofst @ tc_Nat_Onat @ tc_Nat_Onat ) @ ( c_Divides_Odivmod__nat @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1324]) ).

thf(374,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ B ) @ A ) )
    <=> ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ B ) @ A ) )
        & ( B != A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zless__le) ).

thf(3738,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ B ) @ A ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ B ) @ A ) )
          & ( B != A ) ) )
      & ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ B ) @ A ) )
          & ( B != A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[374]) ).

thf(1363,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) )
     => ( A != B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__not__refl2) ).

thf(7636,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) )
     => ( A != B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1363]) ).

thf(1784,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Natural_Oevalc @ ( c_Com_Ocom_OCond @ E @ D @ C ) ) @ B ) @ A ) )
    <=> ( ( ( hBOOL @ ( hAPP @ E @ B ) )
          & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Natural_Oevalc @ D ) @ B ) @ A ) ) )
        | ( ~ ( hBOOL @ ( hAPP @ E @ B ) )
          & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Natural_Oevalc @ C ) @ B ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_evalc_Oequations_I6_J) ).

thf(9404,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Natural_Oevalc @ ( c_Com_Ocom_OCond @ E @ D @ C ) ) @ B ) @ A ) )
       => ( ( ( hBOOL @ ( hAPP @ E @ B ) )
            & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Natural_Oevalc @ D ) @ B ) @ A ) ) )
          | ( ~ ( hBOOL @ ( hAPP @ E @ B ) )
            & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Natural_Oevalc @ C ) @ B ) @ A ) ) ) ) )
      & ( ( ( ( hBOOL @ ( hAPP @ E @ B ) )
            & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Natural_Oevalc @ D ) @ B ) @ A ) ) )
          | ( ~ ( hBOOL @ ( hAPP @ E @ B ) )
            & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Natural_Oevalc @ C ) @ B ) @ A ) ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Natural_Oevalc @ ( c_Com_Ocom_OCond @ E @ D @ C ) ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1784]) ).

thf(273,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Fun_Oinj__on @ E @ D @ C @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ E @ tc_HOL_Obool ) ) )
     => ( ( ( hAPP @ C @ B )
          = ( hAPP @ C @ A ) )
       => ( B = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_injD) ).

thf(3375,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Fun_Oinj__on @ E @ D @ C @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ E @ tc_HOL_Obool ) ) )
     => ( ( ( hAPP @ C @ B )
          = ( hAPP @ C @ A ) )
       => ( B = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[273]) ).

thf(93,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Ocomm__monoid__add @ D )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ B ) )
       => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ C @ D ) @ A ) @ B )
          = ( hAPP @ ( hAPP @ ( hAPP @ ( c_Finite__Set_Ofold__image @ D @ C @ ( c_Groups_Oplus__class_Oplus @ D ) ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ D ) ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_setsum_Oeq__fold) ).

thf(2734,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Ocomm__monoid__add @ D )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ B ) )
       => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ C @ D ) @ A ) @ B )
          = ( hAPP @ ( hAPP @ ( hAPP @ ( c_Finite__Set_Ofold__image @ D @ C @ ( c_Groups_Oplus__class_Oplus @ D ) ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ D ) ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[93]) ).

thf(728,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_Hoare__Mirabelle_Otriple @ D ) ) @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Hoare__Mirabelle_Otriple_Otriple @ D ) @ C ) @ B ) @ A ) )
      = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_triple_Osize_I2_J) ).

thf(5016,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_Hoare__Mirabelle_Otriple @ D ) ) @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Hoare__Mirabelle_Otriple_Otriple @ D ) @ C ) @ B ) @ A ) )
      = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[728]) ).

thf(583,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Wellfounded_Owf @ C @ B )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ A ) @ ( hAPP @ ( c_Relation_OImage @ C @ C @ B ) @ A ) ) )
       => ( A
          = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_wfE__pf) ).

thf(4503,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Wellfounded_Owf @ C @ B )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ A ) @ ( hAPP @ ( c_Relation_OImage @ C @ C @ B ) @ A ) ) )
       => ( A
          = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[583]) ).

thf(752,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Omap @ ( tc_prod @ D @ tc_Nat_Onat ) @ D ) @ ( c_Product__Type_Ofst @ D @ tc_Nat_Onat ) ) @ ( hAPP @ ( c_List_Ofilter @ ( tc_prod @ D @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ ( tc_prod @ D @ tc_Nat_Onat ) @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_Nat_Onat @ ( tc_fun @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( tc_prod @ D @ tc_Nat_Onat ) ) @ ( c_member @ tc_Nat_Onat ) ) @ ( c_Product__Type_Osnd @ D @ tc_Nat_Onat ) ) ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_List_Ozip @ D @ tc_Nat_Onat ) @ B ) @ ( c_List_Oupt @ A @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ D ) ) @ B ) ) ) ) ) )
      = ( hAPP @ ( hAPP @ ( c_List_Omap @ ( tc_prod @ D @ tc_Nat_Onat ) @ D ) @ ( c_Product__Type_Ofst @ D @ tc_Nat_Onat ) ) @ ( hAPP @ ( c_List_Ofilter @ ( tc_prod @ D @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ ( tc_prod @ D @ tc_Nat_Onat ) @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_Nat_Onat @ ( tc_fun @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( tc_prod @ D @ tc_Nat_Onat ) ) @ ( c_member @ tc_Nat_Onat ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ ( tc_prod @ D @ tc_Nat_Onat ) @ tc_Nat_Onat @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_Nat_Onat @ ( tc_fun @ tc_Nat_Onat @ tc_Nat_Onat ) @ ( tc_prod @ D @ tc_Nat_Onat ) ) @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) ) @ ( c_Product__Type_Osnd @ D @ tc_Nat_Onat ) ) ) @ A ) ) ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_List_Ozip @ D @ tc_Nat_Onat ) @ B ) @ ( c_List_Oupt @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ D ) ) @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_sublist__shift__lemma) ).

thf(5095,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Omap @ ( tc_prod @ D @ tc_Nat_Onat ) @ D ) @ ( c_Product__Type_Ofst @ D @ tc_Nat_Onat ) ) @ ( hAPP @ ( c_List_Ofilter @ ( tc_prod @ D @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ ( tc_prod @ D @ tc_Nat_Onat ) @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_Nat_Onat @ ( tc_fun @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( tc_prod @ D @ tc_Nat_Onat ) ) @ ( c_member @ tc_Nat_Onat ) ) @ ( c_Product__Type_Osnd @ D @ tc_Nat_Onat ) ) ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_List_Ozip @ D @ tc_Nat_Onat ) @ B ) @ ( c_List_Oupt @ A @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ D ) ) @ B ) ) ) ) ) )
      = ( hAPP @ ( hAPP @ ( c_List_Omap @ ( tc_prod @ D @ tc_Nat_Onat ) @ D ) @ ( c_Product__Type_Ofst @ D @ tc_Nat_Onat ) ) @ ( hAPP @ ( c_List_Ofilter @ ( tc_prod @ D @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ ( tc_prod @ D @ tc_Nat_Onat ) @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_Nat_Onat @ ( tc_fun @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( tc_prod @ D @ tc_Nat_Onat ) ) @ ( c_member @ tc_Nat_Onat ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ ( tc_prod @ D @ tc_Nat_Onat ) @ tc_Nat_Onat @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_Nat_Onat @ ( tc_fun @ tc_Nat_Onat @ tc_Nat_Onat ) @ ( tc_prod @ D @ tc_Nat_Onat ) ) @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) ) @ ( c_Product__Type_Osnd @ D @ tc_Nat_Onat ) ) ) @ A ) ) ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_List_Ozip @ D @ tc_Nat_Onat ) @ B ) @ ( c_List_Oupt @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ D ) ) @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[752]) ).

thf(195,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ F @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ D @ ( tc_fun @ F @ tc_HOL_Obool ) ) @ C ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ D @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ ( tc_fun @ F @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ F @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ F @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ D ) @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ F @ ( tc_fun @ E @ tc_HOL_Obool ) ) ) @ B ) ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_UN__UN__flatten) ).

thf(3111,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ F @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ D @ ( tc_fun @ F @ tc_HOL_Obool ) ) @ C ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ D @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ ( tc_fun @ F @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ F @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ F @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ D ) @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ F @ ( tc_fun @ E @ tc_HOL_Obool ) ) ) @ B ) ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[195]) ).

thf(902,axiom,
    ! [A: $i] :
      ( ( hAPP @ ( c_Nat_Osize__class_Osize @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ tc_Nat_Onat ) @ A ) ) @ ( hAPP @ c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nat_Osize_I4_J) ).

thf(5647,plain,
    ! [A: $i] :
      ( ( hAPP @ ( c_Nat_Osize__class_Osize @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ tc_Nat_Onat ) @ A ) ) @ ( hAPP @ c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[902]) ).

thf(1522,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Ozip @ E @ D ) @ ( c_List_OtakeWhile @ E @ C @ B ) ) @ A )
      = ( c_List_OtakeWhile @ ( tc_prod @ E @ D ) @ ( hAPP @ ( c_Fun_Ocomp @ E @ tc_HOL_Obool @ ( tc_prod @ E @ D ) @ C ) @ ( c_Product__Type_Ofst @ E @ D ) ) @ ( hAPP @ ( hAPP @ ( c_List_Ozip @ E @ D ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zip__takeWhile__fst) ).

thf(8311,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Ozip @ E @ D ) @ ( c_List_OtakeWhile @ E @ C @ B ) ) @ A )
      = ( c_List_OtakeWhile @ ( tc_prod @ E @ D ) @ ( hAPP @ ( c_Fun_Ocomp @ E @ tc_HOL_Obool @ ( tc_prod @ E @ D ) @ C ) @ ( c_Product__Type_Ofst @ E @ D ) ) @ ( hAPP @ ( hAPP @ ( c_List_Ozip @ E @ D ) @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1522]) ).

thf(1795,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Com_Ocom_OCall @ C @ B @ A )
     != c_Com_Ocom_OSKIP ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_com_Osimps_I21_J) ).

thf(9451,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Com_Ocom_OCall @ C @ B @ A )
     != c_Com_Ocom_OSKIP ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1795]) ).

thf(364,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i] :
          ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ C ) @ ( tc_List_Olist @ C ) ) ) @ D ) @ ( hAPP @ ( c_List_Oset @ ( tc_prod @ ( tc_List_Olist @ C ) @ ( tc_List_Olist @ C ) ) ) @ ( hAPP @ ( hAPP @ ( c_List_Ozip @ ( tc_List_Olist @ C ) @ ( tc_List_Olist @ C ) ) @ B ) @ A ) ) ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ ( tc_List_Olist @ C ) @ ( tc_List_Olist @ C ) @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ ( tc_List_Olist @ C ) @ ( tc_fun @ ( tc_List_Olist @ C ) @ tc_Nat_Onat ) @ ( tc_fun @ ( tc_List_Olist @ C ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ ( tc_List_Olist @ C ) @ tc_Nat_Onat ) @ ( tc_fun @ ( tc_List_Olist @ C ) @ tc_HOL_Obool ) ) @ ( tc_List_Olist @ C ) ) @ ( c_COMBB @ tc_Nat_Onat @ tc_HOL_Obool @ ( tc_List_Olist @ C ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_Nat_Onat @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_List_Olist @ C ) ) @ c_fequal ) @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) ) ) ) @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) ) ) @ D ) ) )
     => ( ( ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ ( tc_List_Olist @ C ) ) ) @ B )
          = ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ ( tc_List_Olist @ C ) ) ) @ A ) )
       => ( ( ( c_List_Oconcat @ C @ B )
            = ( c_List_Oconcat @ C @ A ) )
        <=> ( B = A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_concat__eq__concat__iff) ).

thf(3696,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i] :
          ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ C ) @ ( tc_List_Olist @ C ) ) ) @ D ) @ ( hAPP @ ( c_List_Oset @ ( tc_prod @ ( tc_List_Olist @ C ) @ ( tc_List_Olist @ C ) ) ) @ ( hAPP @ ( hAPP @ ( c_List_Ozip @ ( tc_List_Olist @ C ) @ ( tc_List_Olist @ C ) ) @ B ) @ A ) ) ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ ( tc_List_Olist @ C ) @ ( tc_List_Olist @ C ) @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ ( tc_List_Olist @ C ) @ ( tc_fun @ ( tc_List_Olist @ C ) @ tc_Nat_Onat ) @ ( tc_fun @ ( tc_List_Olist @ C ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ ( tc_List_Olist @ C ) @ tc_Nat_Onat ) @ ( tc_fun @ ( tc_List_Olist @ C ) @ tc_HOL_Obool ) ) @ ( tc_List_Olist @ C ) ) @ ( c_COMBB @ tc_Nat_Onat @ tc_HOL_Obool @ ( tc_List_Olist @ C ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_Nat_Onat @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_List_Olist @ C ) ) @ c_fequal ) @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) ) ) ) @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) ) ) @ D ) ) )
     => ( ( ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ ( tc_List_Olist @ C ) ) ) @ B )
          = ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ ( tc_List_Olist @ C ) ) ) @ A ) )
       => ( ( ( ( c_List_Oconcat @ C @ B )
              = ( c_List_Oconcat @ C @ A ) )
           => ( B = A ) )
          & ( ( B = A )
           => ( ( c_List_Oconcat @ C @ B )
              = ( c_List_Oconcat @ C @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[364]) ).

thf(1091,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ G @ ( tc_Option_Ooption @ F ) ) @ E ) @ D ) @ ( hAPP @ ( c_Option_Ooption_OSome @ F ) @ C ) )
        = ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ G @ ( tc_Option_Ooption @ F ) ) @ B ) @ D ) @ ( hAPP @ ( c_Option_Ooption_OSome @ F ) @ A ) ) )
     => ( C = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_map__upd__eqD1) ).

thf(6492,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ G @ ( tc_Option_Ooption @ F ) ) @ E ) @ D ) @ ( hAPP @ ( c_Option_Ooption_OSome @ F ) @ C ) )
        = ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ G @ ( tc_Option_Ooption @ F ) ) @ B ) @ D ) @ ( hAPP @ ( c_Option_Ooption_OSome @ F ) @ A ) ) )
     => ( C = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1091]) ).

thf(2091,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ F @ F ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ F @ F ) @ E ) @ D ) ) @ ( c_Transitive__Closure_Ortrancl @ F @ C ) ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ F ) @ ( tc_List_Olist @ F ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ F ) @ ( tc_List_Olist @ F ) ) @ B ) @ A ) ) @ ( c_Transitive__Closure_Ortrancl @ ( tc_List_Olist @ F ) @ ( c_List_Olistrel1 @ F @ C ) ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ F ) @ ( tc_List_Olist @ F ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ F ) @ ( tc_List_Olist @ F ) ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ F ) @ E ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ F ) @ D ) @ A ) ) ) @ ( c_Transitive__Closure_Ortrancl @ ( tc_List_Olist @ F ) @ ( c_List_Olistrel1 @ F @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_rtrancl__listrel1__ConsI2) ).

thf(10625,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ F @ F ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ F @ F ) @ E ) @ D ) ) @ ( c_Transitive__Closure_Ortrancl @ F @ C ) ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ F ) @ ( tc_List_Olist @ F ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ F ) @ ( tc_List_Olist @ F ) ) @ B ) @ A ) ) @ ( c_Transitive__Closure_Ortrancl @ ( tc_List_Olist @ F ) @ ( c_List_Olistrel1 @ F @ C ) ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ F ) @ ( tc_List_Olist @ F ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ F ) @ ( tc_List_Olist @ F ) ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ F ) @ E ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ F ) @ D ) @ A ) ) ) @ ( c_Transitive__Closure_Ortrancl @ ( tc_List_Olist @ F ) @ ( c_List_Olistrel1 @ F @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2091]) ).

thf(579,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Hilbert__Choice_OEps @ ( tc_prod @ C @ B ) @ A )
      = ( c_Hilbert__Choice_OEps @ ( tc_prod @ C @ B ) @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ C @ B @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ B @ ( tc_prod @ C @ B ) ) @ ( tc_fun @ B @ tc_HOL_Obool ) @ C ) @ ( hAPP @ ( c_COMBB @ ( tc_prod @ C @ B ) @ tc_HOL_Obool @ B ) @ A ) ) @ ( c_Product__Type_OPair @ C @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_split__paired__Eps) ).

thf(4489,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Hilbert__Choice_OEps @ ( tc_prod @ C @ B ) @ A )
      = ( c_Hilbert__Choice_OEps @ ( tc_prod @ C @ B ) @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ C @ B @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ B @ ( tc_prod @ C @ B ) ) @ ( tc_fun @ B @ tc_HOL_Obool ) @ C ) @ ( hAPP @ ( c_COMBB @ ( tc_prod @ C @ B ) @ tc_HOL_Obool @ B ) @ A ) ) @ ( c_Product__Type_OPair @ C @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[579]) ).

thf(934,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP @ C @ B ) )
       => ( ( hAPP @ ( c_Set_OCollect @ A ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ A @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ A ) @ c_fconj ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ A @ A @ tc_HOL_Obool ) @ c_fequal ) @ B ) ) ) @ C ) )
          = ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ A ) @ B ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ A @ tc_HOL_Obool ) ) ) ) )
      & ( ~ ( hBOOL @ ( hAPP @ C @ B ) )
       => ( ( hAPP @ ( c_Set_OCollect @ A ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ A @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ A ) @ c_fconj ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ A @ A @ tc_HOL_Obool ) @ c_fequal ) @ B ) ) ) @ C ) )
          = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ A @ tc_HOL_Obool ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Collect__conv__if) ).

thf(5758,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP @ C @ B ) )
       => ( ( hAPP @ ( c_Set_OCollect @ A ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ A @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ A ) @ c_fconj ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ A @ A @ tc_HOL_Obool ) @ c_fequal ) @ B ) ) ) @ C ) )
          = ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ A ) @ B ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ A @ tc_HOL_Obool ) ) ) ) )
      & ( ~ ( hBOOL @ ( hAPP @ C @ B ) )
       => ( ( hAPP @ ( c_Set_OCollect @ A ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ A @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ A ) @ c_fconj ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ A @ A @ tc_HOL_Obool ) @ c_fequal ) @ B ) ) ) @ C ) )
          = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ A @ tc_HOL_Obool ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[934]) ).

thf(1926,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ A ) )
     => ( ( B != A )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_le__neq__implies__less) ).

thf(9974,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ A ) )
     => ( ( B != A )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1926]) ).

thf(1692,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ C ) ) @ B )
        = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ C ) ) @ A ) )
    <=> ( B = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Suc__mult__cancel1) ).

thf(9072,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ C ) ) @ B )
          = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ C ) ) @ A ) )
       => ( B = A ) )
      & ( ( B = A )
       => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ C ) ) @ B )
          = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ C ) ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1692]) ).

thf(2201,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ B ) ) )
    <=> ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_dvd__reduce) ).

thf(11011,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ B ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) ) )
      & ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2201]) ).

thf(279,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ A ) @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ) ) )
    <=> ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zle__add1__eq__le) ).

thf(3396,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ A ) @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ B ) @ A ) ) )
      & ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ A ) @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[279]) ).

thf(1020,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( c_List_Oset @ C ) @ ( hAPP @ ( hAPP @ ( c_List_Oappend @ C ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_List_Oset @ C ) @ B ) ) @ ( hAPP @ ( c_List_Oset @ C ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_set__append) ).

thf(6140,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( c_List_Oset @ C ) @ ( hAPP @ ( hAPP @ ( c_List_Oappend @ C ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_List_Oset @ C ) @ B ) ) @ ( hAPP @ ( c_List_Oset @ C ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1020]) ).

thf(845,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ D ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Nat_Ocompow @ ( tc_prod @ E @ E ) @ tc_HOL_Obool ) @ ( hAPP @ c_Nat_OSuc @ B ) ) @ A ) ) )
     => ? [F: $i] :
          ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ D ) @ F ) ) @ A ) )
          & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ F ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Nat_Ocompow @ ( tc_prod @ E @ E ) @ tc_HOL_Obool ) @ B ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_rel__pow__Suc__D2) ).

thf(5456,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ D ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Nat_Ocompow @ ( tc_prod @ E @ E ) @ tc_HOL_Obool ) @ ( hAPP @ c_Nat_OSuc @ B ) ) @ A ) ) )
     => ? [F: $i] :
          ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ D ) @ F ) ) @ A ) )
          & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ F ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Nat_Ocompow @ ( tc_prod @ E @ E ) @ tc_HOL_Obool ) @ B ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[845]) ).

thf(2190,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( ( hAPP @ ( hAPP @ ( hAPP @ ( c_Hoare__Mirabelle_Otriple_Otriple @ G ) @ F ) @ E ) @ D )
        = ( hAPP @ ( hAPP @ ( hAPP @ ( c_Hoare__Mirabelle_Otriple_Otriple @ G ) @ C ) @ B ) @ A ) )
    <=> ( ( F = C )
        & ( E = B )
        & ( D = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_triple_Oinject) ).

thf(10963,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( ( ( hAPP @ ( hAPP @ ( hAPP @ ( c_Hoare__Mirabelle_Otriple_Otriple @ G ) @ F ) @ E ) @ D )
          = ( hAPP @ ( hAPP @ ( hAPP @ ( c_Hoare__Mirabelle_Otriple_Otriple @ G ) @ C ) @ B ) @ A ) )
       => ( ( F = C )
          & ( E = B )
          & ( D = A ) ) )
      & ( ( ( F = C )
          & ( E = B )
          & ( D = A ) )
       => ( ( hAPP @ ( hAPP @ ( hAPP @ ( c_Hoare__Mirabelle_Otriple_Otriple @ G ) @ F ) @ E ) @ D )
          = ( hAPP @ ( hAPP @ ( hAPP @ ( c_Hoare__Mirabelle_Otriple_Otriple @ G ) @ C ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2190]) ).

thf(2388,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( hAPP @ ( c_List_Oset @ D ) @ ( hAPP @ ( hAPP @ ( c_List_Odrop @ D ) @ B ) @ A ) ) ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( hAPP @ ( c_List_Oset @ D ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_in__set__dropD) ).

thf(11804,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( hAPP @ ( c_List_Oset @ D ) @ ( hAPP @ ( hAPP @ ( c_List_Odrop @ D ) @ B ) @ A ) ) ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( hAPP @ ( c_List_Oset @ D ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2388]) ).

thf(2229,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Nat_Onat ) @ B ) @ ( c_SetInterval_Oord_OatMost @ tc_Nat_Onat @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ A ) ) )
    <=> ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_dvd_OatMost__iff) ).

thf(11124,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Nat_Onat ) @ B ) @ ( c_SetInterval_Oord_OatMost @ tc_Nat_Onat @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ A ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) ) )
      & ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Nat_Onat ) @ B ) @ ( c_SetInterval_Oord_OatMost @ tc_Nat_Onat @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2229]) ).

thf(873,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i] :
          ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ C @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ C @ C ) @ D ) @ D ) ) @ B ) )
     => ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ C ) @ ( tc_List_Olist @ C ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ C ) @ ( tc_List_Olist @ C ) ) @ A ) @ A ) ) @ ( c_List_Olexord @ C @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_lexord__irreflexive) ).

thf(5535,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i] :
          ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ C @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ C @ C ) @ D ) @ D ) ) @ B ) )
     => ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ C ) @ ( tc_List_Olist @ C ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ C ) @ ( tc_List_Olist @ C ) ) @ A ) @ A ) ) @ ( c_List_Olexord @ C @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[873]) ).

thf(1838,axiom,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) )
        & ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ A ) @ B ) ) )
     => ( B != A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_dvd_Oless__imp__neq) ).

thf(9617,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) )
        & ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ A ) @ B ) ) )
     => ( B != A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1838]) ).

thf(489,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ D ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ C ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ E @ tc_HOL_Obool ) ) ) )
        = ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ E @ tc_HOL_Obool ) ) ) ) )
    <=> ( ( ( D = B )
          & ( C = A ) )
        | ( ( D = A )
          & ( C = B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_doubleton__eq__iff) ).

thf(4144,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ D ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ C ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ E @ tc_HOL_Obool ) ) ) )
          = ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ E @ tc_HOL_Obool ) ) ) ) )
       => ( ( ( D = B )
            & ( C = A ) )
          | ( ( D = A )
            & ( C = B ) ) ) )
      & ( ( ( ( D = B )
            & ( C = A ) )
          | ( ( D = A )
            & ( C = B ) ) )
       => ( ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ D ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ C ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ E @ tc_HOL_Obool ) ) ) )
          = ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ E @ tc_HOL_Obool ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[489]) ).

thf(1948,axiom,
    ! [A: $i,B: $i] :
      ( ( ( hAPP @ c_Nat_OSuc @ B )
        = ( hAPP @ c_Nat_OSuc @ A ) )
    <=> ( B = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nat_Oinject) ).

thf(10068,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( hAPP @ c_Nat_OSuc @ B )
          = ( hAPP @ c_Nat_OSuc @ A ) )
       => ( B = A ) )
      & ( ( B = A )
       => ( ( hAPP @ c_Nat_OSuc @ B )
          = ( hAPP @ c_Nat_OSuc @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1948]) ).

thf(444,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( c_Relation_OImage @ D @ C @ B ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ D @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ A ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ D @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) @ D ) @ ( c_Relation_OImage @ D @ C @ B ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ ( tc_fun @ D @ tc_HOL_Obool ) @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( c_Set_Oinsert @ D ) ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Image__eq__UN) ).

thf(4000,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( c_Relation_OImage @ D @ C @ B ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ D @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ A ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ D @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) @ D ) @ ( c_Relation_OImage @ D @ C @ B ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ ( tc_fun @ D @ tc_HOL_Obool ) @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( c_Set_Oinsert @ D ) ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[444]) ).

thf(277,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) ) @ B ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ ( tc_List_Olist @ C ) @ ( tc_List_Olist @ C ) ) @ tc_HOL_Obool ) ) @ ( c_List_Olistrel @ C @ B ) ) @ ( c_List_Olistrel @ C @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_listrel__mono) ).

thf(3387,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) ) @ B ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ ( tc_List_Olist @ C ) @ ( tc_List_Olist @ C ) ) @ tc_HOL_Obool ) ) @ ( c_List_Olistrel @ C @ B ) ) @ ( c_List_Olistrel @ C @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[277]) ).

thf(2077,axiom,
    c_Wellfounded_Owf @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ c_FunDef_Opair__less,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_wf__pair__less) ).

thf(10582,plain,
    c_Wellfounded_Owf @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ c_FunDef_Opair__less,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2077]) ).

thf(1384,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ B ) ) @ A ) )
     => ( ( hAPP @ ( hAPP @ ( c_List_Otake @ tc_Nat_Onat ) @ B ) @ ( c_List_Oupt @ C @ A ) )
        = ( c_List_Oupt @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_take__upt) ).

thf(7714,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ B ) ) @ A ) )
     => ( ( hAPP @ ( hAPP @ ( c_List_Otake @ tc_Nat_Onat ) @ B ) @ ( c_List_Oupt @ C @ A ) )
        = ( c_List_Oupt @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1384]) ).

thf(1349,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ F @ E ) @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ F @ E ) @ D ) @ C ) @ B ) ) @ C ) @ A )
      = ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ F @ E ) @ D ) @ C ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_fun__upd__upd) ).

thf(7590,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ F @ E ) @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ F @ E ) @ D ) @ C ) @ B ) ) @ C ) @ A )
      = ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ F @ E ) @ D ) @ C ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1349]) ).

thf(518,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Un__Int__distrib2) ).

thf(4268,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[518]) ).

thf(187,axiom,
    ~ ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ tc_Nat_Onat ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_infinite__UNIV__nat) ).

thf(3070,plain,
    ~ ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ tc_Nat_Onat ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[187]) ).

thf(1975,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ F @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ F @ E ) @ D ) @ C ) ) @ ( hAPP @ ( c_List_Oset @ ( tc_prod @ F @ E ) ) @ ( hAPP @ ( hAPP @ ( c_List_Ozip @ F @ E ) @ B ) @ A ) ) ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ F ) @ D ) @ ( hAPP @ ( c_List_Oset @ F ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_set__zip__leftD) ).

thf(10194,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ F @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ F @ E ) @ D ) @ C ) ) @ ( hAPP @ ( c_List_Oset @ ( tc_prod @ F @ E ) ) @ ( hAPP @ ( hAPP @ ( c_List_Ozip @ F @ E ) @ B ) @ A ) ) ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ F ) @ D ) @ ( hAPP @ ( c_List_Oset @ F ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1975]) ).

thf(1658,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ A @ B ) )
       => ( ! [D: $i,E: $i] :
              ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ E ) )
             => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ D ) @ E ) )
               => ( ( hBOOL @ ( hAPP @ A @ E ) )
                 => ( hBOOL @ ( hAPP @ A @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ E ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ D ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) ) ) ) ) ) )
         => ( hBOOL @ ( hAPP @ A @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_finite__empty__induct) ).

thf(8909,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ A @ B ) )
       => ( ! [D: $i,E: $i] :
              ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ E ) )
             => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ D ) @ E ) )
               => ( ( hBOOL @ ( hAPP @ A @ E ) )
                 => ( hBOOL @ ( hAPP @ A @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ E ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ D ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) ) ) ) ) ) )
         => ( hBOOL @ ( hAPP @ A @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1658]) ).

thf(2349,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( c_SetInterval_Oord_OgreaterThan @ D @ B @ A ) ) )
    <=> ( hBOOL @ ( hAPP @ ( hAPP @ B @ A ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_ord_OgreaterThan__iff) ).

thf(11633,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( c_SetInterval_Oord_OgreaterThan @ D @ B @ A ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ B @ A ) @ C ) ) )
      & ( ( hBOOL @ ( hAPP @ ( hAPP @ B @ A ) @ C ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( c_SetInterval_Oord_OgreaterThan @ D @ B @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2349]) ).

thf(258,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ B ) ) @ A ) ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ A ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ C ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ B ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_q__neg__lemma) ).

thf(3319,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ B ) ) @ A ) ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ A ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ C ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ B ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[258]) ).

thf(232,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ B @ A ) ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zmod__le__nonneg__dividend) ).

thf(3234,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ B @ A ) ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[232]) ).

thf(421,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Nat_Ocompow @ D @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ B ) ) @ A )
      = ( hAPP @ ( c_Fun_Ocomp @ D @ D @ D @ ( hAPP @ ( hAPP @ ( c_Nat_Ocompow @ D @ D ) @ C ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Nat_Ocompow @ D @ D ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_funpow__add) ).

thf(3925,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Nat_Ocompow @ D @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ B ) ) @ A )
      = ( hAPP @ ( c_Fun_Ocomp @ D @ D @ D @ ( hAPP @ ( hAPP @ ( c_Nat_Ocompow @ D @ D ) @ C ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Nat_Ocompow @ D @ D ) @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[421]) ).

thf(127,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( c_List_Ofoldl @ D @ C @ B @ A ) @ ( c_List_Olist_ONil @ C ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_foldl__Nil) ).

thf(2859,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( c_List_Ofoldl @ D @ C @ B @ A ) @ ( c_List_Olist_ONil @ C ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[127]) ).

thf(2224,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Com_Ocom_OWhile @ B @ A )
     != c_Com_Ocom_OSKIP ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_com_Osimps_I17_J) ).

thf(11105,plain,
    ! [A: $i,B: $i] :
      ( ( c_Com_Ocom_OWhile @ B @ A )
     != c_Com_Ocom_OSKIP ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2224]) ).

thf(307,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Relation_Orefl__on @ E @ D @ C )
     => ( ( c_Relation_Orefl__on @ E @ B @ A )
       => ( c_Relation_Orefl__on @ E @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ D ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ ( tc_prod @ E @ E ) @ tc_HOL_Obool ) ) @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_refl__on__Un) ).

thf(3504,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Relation_Orefl__on @ E @ D @ C )
     => ( ( c_Relation_Orefl__on @ E @ B @ A )
       => ( c_Relation_Orefl__on @ E @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ D ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ ( tc_prod @ E @ E ) @ tc_HOL_Obool ) ) @ C ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[307]) ).

thf(2172,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ ( tc_prod @ C @ B ) ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ B ) @ ( hAPP @ ( c_Relation_ORange @ C @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_finite__Range) ).

thf(10906,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ ( tc_prod @ C @ B ) ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ B ) @ ( hAPP @ ( c_Relation_ORange @ C @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2172]) ).

thf(522,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) )
     => ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A )
        = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Un__absorb1) ).

thf(4283,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) )
     => ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[522]) ).

thf(855,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Diff__empty) ).

thf(5488,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[855]) ).

thf(2267,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ E @ D ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Natural_Oevalc @ C ) @ D ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Natural_Oevalc @ ( c_Com_Ocom_OWhile @ E @ C ) ) @ B ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Natural_Oevalc @ ( c_Com_Ocom_OWhile @ E @ C ) ) @ D ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_evalc_OWhileTrue) ).

thf(11279,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ E @ D ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Natural_Oevalc @ C ) @ D ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Natural_Oevalc @ ( c_Com_Ocom_OWhile @ E @ C ) ) @ B ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Natural_Oevalc @ ( c_Com_Ocom_OWhile @ E @ C ) ) @ D ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2267]) ).

thf(1326,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ A ) @ C ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ A ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_in__mono) ).

thf(7497,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ A ) @ C ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ A ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1326]) ).

thf(1184,axiom,
    ( c_FunDef_Opair__leq
    = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ ( tc_prod @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) ) @ tc_HOL_Obool ) ) @ c_FunDef_Opair__less ) @ ( c_Relation_OId @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_pair__leq__def) ).

thf(6881,plain,
    ( c_FunDef_Opair__leq
    = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ ( tc_prod @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) ) @ tc_HOL_Obool ) ) @ c_FunDef_Opair__less ) @ ( c_Relation_OId @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1184]) ).

thf(37,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Groups_Ocomm__monoid__add @ E )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ C ) )
       => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ A ) @ C ) )
          = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ E ) @ ( hAPP @ B @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_setsum_Oinsert__remove) ).

thf(2525,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Groups_Ocomm__monoid__add @ E )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ C ) )
       => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ A ) @ C ) )
          = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ E ) @ ( hAPP @ B @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[37]) ).

thf(124,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ A ) @ C ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ A ) @ ( c_Complete__Lattice_OSup__class_OSup @ ( tc_fun @ D @ tc_HOL_Obool ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_UnionI) ).

thf(2849,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ A ) @ C ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ A ) @ ( c_Complete__Lattice_OSup__class_OSup @ ( tc_fun @ D @ tc_HOL_Obool ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[124]) ).

thf(2011,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ B ) ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ C ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_add__leD1) ).

thf(10334,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ B ) ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ C ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2011]) ).

thf(301,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( c_Set_Ovimage @ E @ D @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Set_Ovimage @ E @ D @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ B ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) ) ) @ ( hAPP @ ( c_Set_Ovimage @ E @ D @ C ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_vimage__insert) ).

thf(3472,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( c_Set_Ovimage @ E @ D @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Set_Ovimage @ E @ D @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ B ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) ) ) @ ( hAPP @ ( c_Set_Ovimage @ E @ D @ C ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[301]) ).

thf(2390,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Predicate_Oreflp @ C @ B )
     => ( hBOOL @ ( hAPP @ ( hAPP @ B @ A ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_reflpE) ).

thf(11809,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Predicate_Oreflp @ C @ B )
     => ( hBOOL @ ( hAPP @ ( hAPP @ B @ A ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2390]) ).

thf(246,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ B ) @ A ) )
     => ( ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ B ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) ) )
        = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_insert__Diff) ).

thf(3283,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ B ) @ A ) )
     => ( ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ B ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[246]) ).

thf(630,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( hAPP @ ( c_Product__Type_Ofst @ D @ C ) @ B )
        = ( hAPP @ ( c_Product__Type_Ofst @ D @ C ) @ A ) )
     => ( ( ( hAPP @ ( c_Product__Type_Osnd @ D @ C ) @ B )
          = ( hAPP @ ( c_Product__Type_Osnd @ D @ C ) @ A ) )
       => ( B = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_prod__eqI) ).

thf(4677,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( hAPP @ ( c_Product__Type_Ofst @ D @ C ) @ B )
        = ( hAPP @ ( c_Product__Type_Ofst @ D @ C ) @ A ) )
     => ( ( ( hAPP @ ( c_Product__Type_Osnd @ D @ C ) @ B )
          = ( hAPP @ ( c_Product__Type_Osnd @ D @ C ) @ A ) )
       => ( B = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[630]) ).

thf(242,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( c_List_Oconcat @ C @ B )
        = ( c_List_Oconcat @ C @ A ) )
     => ( ( ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ ( tc_List_Olist @ C ) ) ) @ B )
          = ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ ( tc_List_Olist @ C ) ) ) @ A ) )
       => ( ! [D: $i] :
              ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ C ) @ ( tc_List_Olist @ C ) ) ) @ D ) @ ( hAPP @ ( c_List_Oset @ ( tc_prod @ ( tc_List_Olist @ C ) @ ( tc_List_Olist @ C ) ) ) @ ( hAPP @ ( hAPP @ ( c_List_Ozip @ ( tc_List_Olist @ C ) @ ( tc_List_Olist @ C ) ) @ B ) @ A ) ) ) )
             => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ ( tc_List_Olist @ C ) @ ( tc_List_Olist @ C ) @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ ( tc_List_Olist @ C ) @ ( tc_fun @ ( tc_List_Olist @ C ) @ tc_Nat_Onat ) @ ( tc_fun @ ( tc_List_Olist @ C ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ ( tc_List_Olist @ C ) @ tc_Nat_Onat ) @ ( tc_fun @ ( tc_List_Olist @ C ) @ tc_HOL_Obool ) ) @ ( tc_List_Olist @ C ) ) @ ( c_COMBB @ tc_Nat_Onat @ tc_HOL_Obool @ ( tc_List_Olist @ C ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_Nat_Onat @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_List_Olist @ C ) ) @ c_fequal ) @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) ) ) ) @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) ) ) @ D ) ) )
         => ( B = A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_concat__injective) ).

thf(3266,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( c_List_Oconcat @ C @ B )
        = ( c_List_Oconcat @ C @ A ) )
     => ( ( ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ ( tc_List_Olist @ C ) ) ) @ B )
          = ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ ( tc_List_Olist @ C ) ) ) @ A ) )
       => ( ! [D: $i] :
              ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ C ) @ ( tc_List_Olist @ C ) ) ) @ D ) @ ( hAPP @ ( c_List_Oset @ ( tc_prod @ ( tc_List_Olist @ C ) @ ( tc_List_Olist @ C ) ) ) @ ( hAPP @ ( hAPP @ ( c_List_Ozip @ ( tc_List_Olist @ C ) @ ( tc_List_Olist @ C ) ) @ B ) @ A ) ) ) )
             => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ ( tc_List_Olist @ C ) @ ( tc_List_Olist @ C ) @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ ( tc_List_Olist @ C ) @ ( tc_fun @ ( tc_List_Olist @ C ) @ tc_Nat_Onat ) @ ( tc_fun @ ( tc_List_Olist @ C ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ ( tc_List_Olist @ C ) @ tc_Nat_Onat ) @ ( tc_fun @ ( tc_List_Olist @ C ) @ tc_HOL_Obool ) ) @ ( tc_List_Olist @ C ) ) @ ( c_COMBB @ tc_Nat_Onat @ tc_HOL_Obool @ ( tc_List_Olist @ C ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_Nat_Onat @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_List_Olist @ C ) ) @ c_fequal ) @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) ) ) ) @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) ) ) @ D ) ) )
         => ( B = A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[242]) ).

thf(2010,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_List_Odistinct @ E ) @ ( hAPP @ ( hAPP @ ( c_List_Omap @ ( tc_prod @ E @ D ) @ E ) @ ( c_Product__Type_Ofst @ E @ D ) ) @ C ) ) )
     => ( ( ( hAPP @ ( c_Map_Omap__of @ E @ D @ C ) @ B )
          = ( hAPP @ ( c_Option_Ooption_OSome @ D ) @ A ) )
      <=> ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ D ) @ B ) @ A ) ) @ ( hAPP @ ( c_List_Oset @ ( tc_prod @ E @ D ) ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_map__of__eq__Some__iff) ).

thf(10329,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_List_Odistinct @ E ) @ ( hAPP @ ( hAPP @ ( c_List_Omap @ ( tc_prod @ E @ D ) @ E ) @ ( c_Product__Type_Ofst @ E @ D ) ) @ C ) ) )
     => ( ( ( ( hAPP @ ( c_Map_Omap__of @ E @ D @ C ) @ B )
            = ( hAPP @ ( c_Option_Ooption_OSome @ D ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ D ) @ B ) @ A ) ) @ ( hAPP @ ( c_List_Oset @ ( tc_prod @ E @ D ) ) @ C ) ) ) )
        & ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ D ) @ B ) @ A ) ) @ ( hAPP @ ( c_List_Oset @ ( tc_prod @ E @ D ) ) @ C ) ) )
         => ( ( hAPP @ ( c_Map_Omap__of @ E @ D @ C ) @ B )
            = ( hAPP @ ( c_Option_Ooption_OSome @ D ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2010]) ).

thf(1653,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ A ) @ B ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ A ) ) @ ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_card__mono) ).

thf(8894,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ A ) @ B ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ A ) ) @ ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1653]) ).

thf(1604,axiom,
    c_Wellfounded_Owf @ tc_Nat_Onat @ ( hAPP @ ( c_Set_OCollect @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) ) @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ tc_Nat_Onat @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_wf__less) ).

thf(8684,plain,
    c_Wellfounded_Owf @ tc_Nat_Onat @ ( hAPP @ ( c_Set_OCollect @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) ) @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ tc_Nat_Onat @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1604]) ).

thf(1990,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ A ) @ B ) )
      <=> ( A = B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_dvd_Oantisym__conv) ).

thf(10242,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) )
     => ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ A ) @ B ) )
         => ( A = B ) )
        & ( ( A = B )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ A ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1990]) ).

thf(1225,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Map_Odom @ C @ B @ ( c_Map_Omap__of @ C @ B @ A ) )
      = ( hAPP @ ( c_Set_Oimage @ ( tc_prod @ C @ B ) @ C @ ( c_Product__Type_Ofst @ C @ B ) ) @ ( hAPP @ ( c_List_Oset @ ( tc_prod @ C @ B ) ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_dom__map__of__conv__image__fst) ).

thf(7041,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Map_Odom @ C @ B @ ( c_Map_Omap__of @ C @ B @ A ) )
      = ( hAPP @ ( c_Set_Oimage @ ( tc_prod @ C @ B ) @ C @ ( c_Product__Type_Ofst @ C @ B ) ) @ ( hAPP @ ( c_List_Oset @ ( tc_prod @ C @ B ) ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1225]) ).

thf(1382,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ C ) )
     => ( ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ B @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ A ) @ C ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ A ) @ ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ B ) @ A ) @ C ) ) ) @ ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ B @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zmod__zmult2__eq) ).

thf(7704,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ C ) )
     => ( ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ B @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ A ) @ C ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ A ) @ ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ B ) @ A ) @ C ) ) ) @ ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1382]) ).

thf(111,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( c_Product__Type_Ofst @ C @ B ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ C @ B @ C ) @ ( c_COMBK @ C @ B ) ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_fst__def) ).

thf(2806,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( c_Product__Type_Ofst @ C @ B ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ C @ B @ C ) @ ( c_COMBK @ C @ B ) ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[111]) ).

thf(985,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Omap @ ( tc_prod @ H @ G ) @ F ) @ E ) @ ( hAPP @ ( hAPP @ ( c_List_Ozip @ H @ G ) @ D ) @ ( hAPP @ ( hAPP @ ( c_List_Omap @ C @ G ) @ B ) @ A ) ) )
      = ( hAPP @ ( hAPP @ ( c_List_Omap @ ( tc_prod @ H @ C ) @ F ) @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ H @ C @ F ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ ( tc_prod @ H @ G ) ) @ ( tc_fun @ C @ F ) @ H ) @ ( hAPP @ ( c_COMBB @ ( tc_prod @ H @ G ) @ F @ C ) @ E ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ H @ ( tc_fun @ C @ G ) @ ( tc_fun @ C @ ( tc_prod @ H @ G ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ G @ ( tc_prod @ H @ G ) ) @ ( tc_fun @ ( tc_fun @ C @ G ) @ ( tc_fun @ C @ ( tc_prod @ H @ G ) ) ) @ H ) @ ( c_COMBB @ G @ ( tc_prod @ H @ G ) @ C ) ) @ ( c_Product__Type_OPair @ H @ G ) ) ) @ B ) ) ) ) @ ( hAPP @ ( hAPP @ ( c_List_Ozip @ H @ C ) @ D ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_map__zip__map2) ).

thf(5984,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Omap @ ( tc_prod @ H @ G ) @ F ) @ E ) @ ( hAPP @ ( hAPP @ ( c_List_Ozip @ H @ G ) @ D ) @ ( hAPP @ ( hAPP @ ( c_List_Omap @ C @ G ) @ B ) @ A ) ) )
      = ( hAPP @ ( hAPP @ ( c_List_Omap @ ( tc_prod @ H @ C ) @ F ) @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ H @ C @ F ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ ( tc_prod @ H @ G ) ) @ ( tc_fun @ C @ F ) @ H ) @ ( hAPP @ ( c_COMBB @ ( tc_prod @ H @ G ) @ F @ C ) @ E ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ H @ ( tc_fun @ C @ G ) @ ( tc_fun @ C @ ( tc_prod @ H @ G ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ G @ ( tc_prod @ H @ G ) ) @ ( tc_fun @ ( tc_fun @ C @ G ) @ ( tc_fun @ C @ ( tc_prod @ H @ G ) ) ) @ H ) @ ( c_COMBB @ G @ ( tc_prod @ H @ G ) @ C ) ) @ ( c_Product__Type_OPair @ H @ G ) ) ) @ B ) ) ) ) @ ( hAPP @ ( hAPP @ ( c_List_Ozip @ H @ C ) @ D ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[985]) ).

thf(1530,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ C ) @ B ) ) @ A ) )
    <=> ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_le__diff__conv) ).

thf(8332,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ C ) @ B ) ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ B ) ) ) )
      & ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ B ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ C ) @ B ) ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1530]) ).

thf(1677,axiom,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ A ) )
     => ? [B: $i] :
          ( A
          = ( hAPP @ c_Nat_OSuc @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_gr0__implies__Suc) ).

thf(8993,plain,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ A ) )
     => ? [B: $i] :
          ( A
          = ( hAPP @ c_Nat_OSuc @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1677]) ).

thf(549,axiom,
    ! [A: $i] :
      ( ( hAPP @ ( c_Relation_ODomain @ A @ A ) @ ( c_Relation_OId @ A ) )
      = ( c_Orderings_Otop__class_Otop @ ( tc_fun @ A @ tc_HOL_Obool ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Domain__Id) ).

thf(4381,plain,
    ! [A: $i] :
      ( ( hAPP @ ( c_Relation_ODomain @ A @ A ) @ ( c_Relation_OId @ A ) )
      = ( c_Orderings_Otop__class_Otop @ ( tc_fun @ A @ tc_HOL_Obool ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[549]) ).

thf(1395,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_psubset__imp__subset) ).

thf(7763,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1395]) ).

thf(1415,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( c_Fun_Ocomp @ ( tc_List_Olist @ E ) @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ C ) @ ( hAPP @ ( c_List_Omap @ E @ D ) @ B ) ) @ ( hAPP @ ( c_List_Omap @ C @ E ) @ A ) )
      = ( hAPP @ ( c_List_Omap @ C @ D ) @ ( hAPP @ ( c_Fun_Ocomp @ E @ D @ C @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_map__comp__map) ).

thf(7828,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( c_Fun_Ocomp @ ( tc_List_Olist @ E ) @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ C ) @ ( hAPP @ ( c_List_Omap @ E @ D ) @ B ) ) @ ( hAPP @ ( c_List_Omap @ C @ E ) @ A ) )
      = ( hAPP @ ( c_List_Omap @ C @ D ) @ ( hAPP @ ( c_Fun_Ocomp @ E @ D @ C @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1415]) ).

thf(939,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ F @ ( tc_fun @ E @ tc_HOL_Obool ) ) ) @ D ) @ C ) @ B ) @ A ) )
     => ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ D @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ C @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_sup2E) ).

thf(5799,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ F @ ( tc_fun @ E @ tc_HOL_Obool ) ) ) @ D ) @ C ) @ B ) @ A ) )
     => ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ D @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ C @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[939]) ).

thf(1887,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ B ) ) @ ( hAPP @ c_Nat_OSuc @ A ) ) )
    <=> ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Suc__le__mono) ).

thf(9820,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ B ) ) @ ( hAPP @ c_Nat_OSuc @ A ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ A ) ) )
      & ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ B ) ) @ ( hAPP @ c_Nat_OSuc @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1887]) ).

thf(860,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Int__greatest) ).

thf(5503,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[860]) ).

thf(1919,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ C ) @ A ) )
       => ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ B ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ A ) @ C ) )
          = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Nat_Odiff__diff__eq) ).

thf(9952,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ C ) @ A ) )
       => ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ B ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ A ) @ C ) )
          = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1919]) ).

thf(730,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ E ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ D ) ) @ C ) ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ E ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ B ) ) @ A ) ) )
       => ( ( hAPP @ ( hAPP @ ( c_List_Onth @ ( tc_prod @ D @ B ) ) @ ( hAPP @ ( hAPP @ ( c_List_Ozip @ D @ B ) @ C ) @ A ) ) @ E )
          = ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ B ) @ ( hAPP @ ( hAPP @ ( c_List_Onth @ D ) @ C ) @ E ) ) @ ( hAPP @ ( hAPP @ ( c_List_Onth @ B ) @ A ) @ E ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nth__zip) ).

thf(5022,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ E ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ D ) ) @ C ) ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ E ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ B ) ) @ A ) ) )
       => ( ( hAPP @ ( hAPP @ ( c_List_Onth @ ( tc_prod @ D @ B ) ) @ ( hAPP @ ( hAPP @ ( c_List_Ozip @ D @ B ) @ C ) @ A ) ) @ E )
          = ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ B ) @ ( hAPP @ ( hAPP @ ( c_List_Onth @ D ) @ C ) @ E ) ) @ ( hAPP @ ( hAPP @ ( c_List_Onth @ B ) @ A ) @ E ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[730]) ).

thf(1409,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( c_Map_Omap__of @ D @ C @ B )
        = ( c_Map_Omap__of @ D @ C @ A ) )
     => ( ( hAPP @ ( c_Set_Oimage @ ( tc_prod @ D @ C ) @ D @ ( c_Product__Type_Ofst @ D @ C ) ) @ ( hAPP @ ( c_List_Oset @ ( tc_prod @ D @ C ) ) @ B ) )
        = ( hAPP @ ( c_Set_Oimage @ ( tc_prod @ D @ C ) @ D @ ( c_Product__Type_Ofst @ D @ C ) ) @ ( hAPP @ ( c_List_Oset @ ( tc_prod @ D @ C ) ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_map__of__eq__dom) ).

thf(7815,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( c_Map_Omap__of @ D @ C @ B )
        = ( c_Map_Omap__of @ D @ C @ A ) )
     => ( ( hAPP @ ( c_Set_Oimage @ ( tc_prod @ D @ C ) @ D @ ( c_Product__Type_Ofst @ D @ C ) ) @ ( hAPP @ ( c_List_Oset @ ( tc_prod @ D @ C ) ) @ B ) )
        = ( hAPP @ ( c_Set_Oimage @ ( tc_prod @ D @ C ) @ D @ ( c_Product__Type_Ofst @ D @ C ) ) @ ( hAPP @ ( c_List_Oset @ ( tc_prod @ D @ C ) ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1409]) ).

thf(335,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ ( hAPP @ c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) ) @ ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ tc_Nat_Onat @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_Nat_Onat @ tc_Nat_Onat @ tc_Nat_Onat ) @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ tc_Nat_Onat @ tc_Nat_Onat ) @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) ) @ B ) ) ) @ ( hAPP @ ( c_SetInterval_Oord__class_OlessThan @ tc_Nat_Onat ) @ A ) ) )
      = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ A ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_arith__series__nat) ).

thf(3599,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ ( hAPP @ c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) ) @ ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ tc_Nat_Onat @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_Nat_Onat @ tc_Nat_Onat @ tc_Nat_Onat ) @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ tc_Nat_Onat @ tc_Nat_Onat ) @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) ) @ B ) ) ) @ ( hAPP @ ( c_SetInterval_Oord__class_OlessThan @ tc_Nat_Onat ) @ A ) ) )
      = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ A ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[335]) ).

thf(2087,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Com_Ocom_OLocal @ E @ D @ C )
     != ( c_Com_Ocom_OAss @ B @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_com_Osimps_I23_J) ).

thf(10611,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Com_Ocom_OLocal @ E @ D @ C )
     != ( c_Com_Ocom_OAss @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2087]) ).

thf(483,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ A ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Int_Oint ) @ B ) @ A ) )
         => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Int_Oint ) @ A ) @ B ) )
           => ( B = A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zdvd__antisym__nonneg) ).

thf(4127,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ A ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Int_Oint ) @ B ) @ A ) )
         => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Int_Oint ) @ A ) @ B ) )
           => ( B = A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[483]) ).

thf(1201,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ C @ C ) @ A ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ C @ tc_HOL_Obool ) @ C ) @ A ) ) ) )
     => ( ! [D: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ D ) @ A ) )
           => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ C @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ C @ C ) @ D ) @ D ) ) @ B ) ) )
       => ( c_Relation_Orefl__on @ C @ A @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_refl__onI) ).

thf(6937,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ C @ C ) @ A ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ C @ tc_HOL_Obool ) @ C ) @ A ) ) ) )
     => ( ! [D: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ D ) @ A ) )
           => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ C @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ C @ C ) @ D ) @ D ) ) @ B ) ) )
       => ( c_Relation_Orefl__on @ C @ A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1201]) ).

thf(1812,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofold1Set @ E @ D @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ C ) @ B ) ) @ A ) )
    <=> ? [F: $i,G: $i] :
          ( ( ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ C ) @ B )
            = ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ F ) @ G ) )
          & ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofold__graph @ E @ E @ D @ F @ G ) @ A ) )
          & ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ F ) @ G ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_fold1Set_Oequations) ).

thf(9506,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofold1Set @ E @ D @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ C ) @ B ) ) @ A ) )
       => ? [F: $i,G: $i] :
            ( ( ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ C ) @ B )
              = ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ F ) @ G ) )
            & ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofold__graph @ E @ E @ D @ F @ G ) @ A ) )
            & ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ F ) @ G ) ) ) )
      & ( ? [F: $i,G: $i] :
            ( ( ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ C ) @ B )
              = ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ F ) @ G ) )
            & ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofold__graph @ E @ E @ D @ F @ G ) @ A ) )
            & ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ F ) @ G ) ) )
       => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofold1Set @ E @ D @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ C ) @ B ) ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1812]) ).

thf(1301,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( D
          = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
       => ( ( hAPP @ ( hAPP @ ( c_List_Otake @ C ) @ D ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ C ) @ B ) @ A ) )
          = ( c_List_Olist_ONil @ C ) ) )
      & ( ( D
         != ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
       => ( ( hAPP @ ( hAPP @ ( c_List_Otake @ C ) @ D ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ C ) @ B ) @ A ) )
          = ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_List_Otake @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ D ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_take__Cons_H) ).

thf(7347,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( D
          = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
       => ( ( hAPP @ ( hAPP @ ( c_List_Otake @ C ) @ D ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ C ) @ B ) @ A ) )
          = ( c_List_Olist_ONil @ C ) ) )
      & ( ( D
         != ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
       => ( ( hAPP @ ( hAPP @ ( c_List_Otake @ C ) @ D ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ C ) @ B ) @ A ) )
          = ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_List_Otake @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ D ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1301]) ).

thf(1373,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( c_Relation_OField @ B ) @ ( hAPP @ ( c_Relation_Oconverse @ B @ B ) @ A ) )
      = ( hAPP @ ( c_Relation_OField @ B ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Field__converse) ).

thf(7671,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( c_Relation_OField @ B ) @ ( hAPP @ ( c_Relation_Oconverse @ B @ B ) @ A ) )
      = ( hAPP @ ( c_Relation_OField @ B ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1373]) ).

thf(1368,axiom,
    ! [A: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Nat_Onat ) @ A ) @ ( hAPP @ c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_div__1) ).

thf(7658,plain,
    ! [A: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Nat_Onat ) @ A ) @ ( hAPP @ c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1368]) ).

thf(1939,axiom,
    ! [A: $i,B: $i] :
      ( c_Com_Ocom_OSKIP
     != ( c_Com_Ocom_OSemi @ B @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_com_Osimps_I12_J) ).

thf(10024,plain,
    ! [A: $i,B: $i] :
      ( c_Com_Ocom_OSKIP
     != ( c_Com_Ocom_OSemi @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1939]) ).

thf(1047,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ B ) )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ C ) @ A ) ) @ B )
        = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ A ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_insert__Diff1) ).

thf(6248,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ B ) )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ C ) @ A ) ) @ B )
        = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ A ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1047]) ).

thf(2344,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( c_Fun_Oinj__on @ F @ E @ D @ C )
     => ( ( B != A )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ F ) @ B ) @ C ) )
         => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ F ) @ A ) @ C ) )
           => ( ( hAPP @ D @ B )
             != ( hAPP @ D @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_inj__on__contraD) ).

thf(11618,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( c_Fun_Oinj__on @ F @ E @ D @ C )
     => ( ( B != A )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ F ) @ B ) @ C ) )
         => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ F ) @ A ) @ C ) )
           => ( ( hAPP @ D @ B )
             != ( hAPP @ D @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2344]) ).

thf(1804,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Odrop @ E ) @ D ) @ ( hAPP @ ( hAPP @ ( c_List_Omap @ C @ E ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_List_Omap @ C @ E ) @ B ) @ ( hAPP @ ( hAPP @ ( c_List_Odrop @ C ) @ D ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_drop__map) ).

thf(9483,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Odrop @ E ) @ D ) @ ( hAPP @ ( hAPP @ ( c_List_Omap @ C @ E ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_List_Omap @ C @ E ) @ B ) @ ( hAPP @ ( hAPP @ ( c_List_Odrop @ C ) @ D ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1804]) ).

thf(1261,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ ( hAPP @ c_Nat_OSuc @ A ) ) )
    <=> ( ( B
          = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
        | ? [C: $i] :
            ( ( B
              = ( hAPP @ c_Nat_OSuc @ C ) )
            & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ C ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__Suc__eq__0__disj) ).

thf(7191,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ ( hAPP @ c_Nat_OSuc @ A ) ) )
       => ( ( B
            = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
          | ? [C: $i] :
              ( ( B
                = ( hAPP @ c_Nat_OSuc @ C ) )
              & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ C ) @ A ) ) ) ) )
      & ( ( ( B
            = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
          | ? [C: $i] :
              ( ( B
                = ( hAPP @ c_Nat_OSuc @ C ) )
              & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ C ) @ A ) ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ ( hAPP @ c_Nat_OSuc @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1261]) ).

thf(1941,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ C ) @ A ) )
       => ( ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ B ) @ C )
            = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ A ) @ C ) )
        <=> ( B = A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_eq__diff__iff) ).

thf(10034,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ C ) @ A ) )
       => ( ( ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ B ) @ C )
              = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ A ) @ C ) )
           => ( B = A ) )
          & ( ( B = A )
           => ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ B ) @ C )
              = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ A ) @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1941]) ).

thf(1281,axiom,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ B ) @ A ) )
       => ( ( hAPP @ ( c_Finite__Set_Ocard @ B ) @ A )
          = ( hAPP @ ( hAPP @ ( hAPP @ ( c_Finite__Set_Ofold__image @ tc_Nat_Onat @ B @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) ) @ ( hAPP @ ( c_COMBK @ tc_Nat_Onat @ B ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ A ) ) )
      & ( ~ ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ B ) @ A ) )
       => ( ( hAPP @ ( c_Finite__Set_Ocard @ B ) @ A )
          = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_card__def) ).

thf(7270,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ B ) @ A ) )
       => ( ( hAPP @ ( c_Finite__Set_Ocard @ B ) @ A )
          = ( hAPP @ ( hAPP @ ( hAPP @ ( c_Finite__Set_Ofold__image @ tc_Nat_Onat @ B @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) ) @ ( hAPP @ ( c_COMBK @ tc_Nat_Onat @ B ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ A ) ) )
      & ( ~ ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ B ) @ A ) )
       => ( ( hAPP @ ( c_Finite__Set_Ocard @ B ) @ A )
          = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1281]) ).

thf(665,axiom,
    ! [A: $i] :
      ( ( A
       != ( c_Orderings_Obot__class_Obot @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) ) ) ) @ c_FunDef_Omin__strict ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_smin__emptyI) ).

thf(4796,plain,
    ! [A: $i] :
      ( ( A
       != ( c_Orderings_Obot__class_Obot @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) ) ) ) @ c_FunDef_Omin__strict ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[665]) ).

thf(1072,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ A ) @ B ) ) )
      <=> ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_finite__Diff2) ).

thf(6357,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ B ) )
     => ( ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ A ) @ B ) ) )
         => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ A ) ) )
        & ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ A ) @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1072]) ).

thf(726,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ ( tc_fun @ B @ A ) ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ ( tc_fun @ B @ A ) @ tc_HOL_Obool ) ) ) )
     => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ A ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ A @ tc_HOL_Obool ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_finite__fun__UNIVD2) ).

thf(5011,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ ( tc_fun @ B @ A ) ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ ( tc_fun @ B @ A ) @ tc_HOL_Obool ) ) ) )
     => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ A ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ A @ tc_HOL_Obool ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[726]) ).

thf(369,axiom,
    ( ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ tc_Nat_Onat @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) ) @ ( c_SetInterval_Oord__class_OatMost @ tc_Nat_Onat ) )
    = ( c_Orderings_Otop__class_Otop @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_UN__atMost__UNIV) ).

thf(3726,plain,
    ( ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ tc_Nat_Onat @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) ) @ ( c_SetInterval_Oord__class_OatMost @ tc_Nat_Onat ) )
    = ( c_Orderings_Otop__class_Otop @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[369]) ).

thf(1457,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Nat_Onat @ B ) @ ( hAPP @ c_Nat_OSuc @ A ) )
      = ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Nat_Onat @ B @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_atLeastLessThanSuc__atLeastAtMost) ).

thf(7983,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Nat_Onat @ B ) @ ( hAPP @ c_Nat_OSuc @ A ) )
      = ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Nat_Onat @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1457]) ).

thf(1128,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ ( hAPP @ c_Nat_OSuc @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__Suc__right) ).

thf(6673,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ ( hAPP @ c_Nat_OSuc @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1128]) ).

thf(349,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hAPP @ ( c_Set_Oimage @ F @ E @ D ) @ ( hAPP @ ( c_Set_Oimage @ C @ F @ B ) @ A ) )
      = ( hAPP @ ( c_Set_Oimage @ C @ E @ ( hAPP @ ( hAPP @ ( c_COMBB @ F @ E @ C ) @ D ) @ B ) ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_image__image) ).

thf(3651,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hAPP @ ( c_Set_Oimage @ F @ E @ D ) @ ( hAPP @ ( c_Set_Oimage @ C @ F @ B ) @ A ) )
      = ( hAPP @ ( c_Set_Oimage @ C @ E @ ( hAPP @ ( hAPP @ ( c_COMBB @ F @ E @ C ) @ D ) @ B ) ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[349]) ).

thf(2310,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( c_Com_Ocom_OWhile @ D @ C )
        = ( c_Com_Ocom_OWhile @ B @ A ) )
    <=> ( ( D = B )
        & ( C = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_com_Osimps_I5_J) ).

thf(11466,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( ( c_Com_Ocom_OWhile @ D @ C )
          = ( c_Com_Ocom_OWhile @ B @ A ) )
       => ( ( D = B )
          & ( C = A ) ) )
      & ( ( ( D = B )
          & ( C = A ) )
       => ( ( c_Com_Ocom_OWhile @ D @ C )
          = ( c_Com_Ocom_OWhile @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2310]) ).

thf(1386,axiom,
    ! [A: $i,B: $i] :
      ( ( ( c_Groups_Oone__class_Oone @ tc_Nat_Onat )
        = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ A ) )
    <=> ( ( B
          = ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) )
        & ( A
          = ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nat__1__eq__mult__iff) ).

thf(7719,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( c_Groups_Oone__class_Oone @ tc_Nat_Onat )
          = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ A ) )
       => ( ( B
            = ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) )
          & ( A
            = ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) ) )
      & ( ( ( B
            = ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) )
          & ( A
            = ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) )
       => ( ( c_Groups_Oone__class_Oone @ tc_Nat_Onat )
          = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1386]) ).

thf(2319,axiom,
    ! [A: $i,B: $i] :
      ( ~ ( hBOOL @ B )
      | ( hBOOL @ ( hAPP @ ( hAPP @ c_fdisj @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_c__fdisj__1) ).

thf(11502,plain,
    ! [A: $i,B: $i] :
      ( ~ ( hBOOL @ B )
      | ( hBOOL @ ( hAPP @ ( hAPP @ c_fdisj @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2319]) ).

thf(224,axiom,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ B ) @ ( tc_List_Olist @ B ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ B ) @ ( tc_List_Olist @ B ) ) @ ( c_List_Olist_ONil @ B ) ) @ ( c_List_Olist_ONil @ B ) ) ) @ ( c_List_Olistrel @ B @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_listrel_ONil) ).

thf(3208,plain,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ B ) @ ( tc_List_Olist @ B ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ B ) @ ( tc_List_Olist @ B ) ) @ ( c_List_Olist_ONil @ B ) ) @ ( c_List_Olist_ONil @ B ) ) ) @ ( c_List_Olistrel @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[224]) ).

thf(353,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ B @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Divides_Otransfer__nat__int__function__closures_I2_J) ).

thf(3663,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ B @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[353]) ).

thf(506,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ D ) @ C ) )
     => ( ( hAPP @ ( hAPP @ ( c_List_Onth @ B ) @ ( hAPP @ ( hAPP @ ( c_List_Otake @ B ) @ C ) @ A ) ) @ D )
        = ( hAPP @ ( hAPP @ ( c_List_Onth @ B ) @ A ) @ D ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nth__take) ).

thf(4218,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ D ) @ C ) )
     => ( ( hAPP @ ( hAPP @ ( c_List_Onth @ B ) @ ( hAPP @ ( hAPP @ ( c_List_Otake @ B ) @ C ) @ A ) ) @ D )
        = ( hAPP @ ( hAPP @ ( c_List_Onth @ B ) @ A ) @ D ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[506]) ).

thf(2015,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ C ) ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ C ) ) @ A ) ) )
    <=> ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Suc__mult__le__cancel1) ).

thf(10348,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ C ) ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ C ) ) @ A ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ A ) ) )
      & ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ C ) ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ C ) ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2015]) ).

thf(2304,axiom,
    ! [A: $i,B: $i] :
      ( ( B != A )
      | ( hBOOL @ ( hAPP @ ( hAPP @ c_fequal @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_c__fequal__2) ).

thf(11423,plain,
    ! [A: $i,B: $i] :
      ( ( B != A )
      | ( hBOOL @ ( hAPP @ ( hAPP @ c_fequal @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2304]) ).

thf(334,axiom,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ) @ A ) ) @ A ) ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) )
    <=> ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_odd__less__0) ).

thf(3594,plain,
    ! [A: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ) @ A ) ) @ A ) ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) ) )
      & ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ) @ A ) ) @ A ) ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[334]) ).

thf(749,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hAPP @ ( hAPP @ ( hAPP @ ( c_COMBB @ F @ E @ D ) @ C ) @ B ) @ A )
      = ( hAPP @ C @ ( hAPP @ B @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_c__COMBB__1) ).

thf(5086,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hAPP @ ( hAPP @ ( hAPP @ ( c_COMBB @ F @ E @ D ) @ C ) @ B ) @ A )
      = ( hAPP @ C @ ( hAPP @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[749]) ).

thf(764,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Nat_Ocompow @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) @ B ) @ A ) ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ C ) @ ( c_Transitive__Closure_Ortrancl @ D @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_rel__pow__imp__rtrancl) ).

thf(5148,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Nat_Ocompow @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) @ B ) @ A ) ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ C ) @ ( c_Transitive__Closure_Ortrancl @ D @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[764]) ).

thf(1818,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( hAPP @ ( c_Sum__Type_OInr @ D @ C ) @ B )
        = ( hAPP @ ( c_Sum__Type_OInr @ D @ C ) @ A ) )
     => ( B = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Inr__inject) ).

thf(9538,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( hAPP @ ( c_Sum__Type_OInr @ D @ C ) @ B )
        = ( hAPP @ ( c_Sum__Type_OInr @ D @ C ) @ A ) )
     => ( B = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1818]) ).

thf(1236,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) )
       => ( ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ A ) @ B ) )
          = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ A ) ) @ ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_card__Diff__subset) ).

thf(7094,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) )
       => ( ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ A ) @ B ) )
          = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ A ) ) @ ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1236]) ).

thf(199,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ E @ D @ C ) @ B ) @ A )
      = ( hAPP @ ( hAPP @ B @ ( hAPP @ ( c_Product__Type_Ofst @ E @ D ) @ A ) ) @ ( hAPP @ ( c_Product__Type_Osnd @ E @ D ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_split__def) ).

thf(3123,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ E @ D @ C ) @ B ) @ A )
      = ( hAPP @ ( hAPP @ B @ ( hAPP @ ( c_Product__Type_Ofst @ E @ D ) @ A ) ) @ ( hAPP @ ( c_Product__Type_Osnd @ E @ D ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[199]) ).

thf(1443,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ F ) @ E ) @ D ) )
     => ( ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ F @ ( tc_Option_Ooption @ C ) ) @ ( c_Map_Orestrict__map @ F @ C @ B @ D ) ) @ E ) @ A )
        = ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ F @ ( tc_Option_Ooption @ C ) ) @ ( c_Map_Orestrict__map @ F @ C @ B @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ F @ tc_HOL_Obool ) ) @ D ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ F ) @ E ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ F @ tc_HOL_Obool ) ) ) ) ) ) @ E ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_fun__upd__restrict__conv) ).

thf(7924,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ F ) @ E ) @ D ) )
     => ( ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ F @ ( tc_Option_Ooption @ C ) ) @ ( c_Map_Orestrict__map @ F @ C @ B @ D ) ) @ E ) @ A )
        = ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ F @ ( tc_Option_Ooption @ C ) ) @ ( c_Map_Orestrict__map @ F @ C @ B @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ F @ tc_HOL_Obool ) ) @ D ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ F ) @ E ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ F @ tc_HOL_Obool ) ) ) ) ) ) @ E ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1443]) ).

thf(776,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ C ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ ( tc_prod @ D @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ D @ B ) @ C ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ B @ tc_HOL_Obool ) @ D ) @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_finite__cartesian__product) ).

thf(5190,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ C ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ ( tc_prod @ D @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ D @ B ) @ C ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ B @ tc_HOL_Obool ) @ D ) @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[776]) ).

thf(2124,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ B @ B ) ) @ A ) @ ( c_Relation_OId @ B ) ) )
     => ~ ! [C: $i] :
            ( A
           != ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ B @ B ) @ C ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_IdE) ).

thf(10728,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ B @ B ) ) @ A ) @ ( c_Relation_OId @ B ) ) )
     => ~ ! [C: $i] :
            ( A
           != ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ B @ B ) @ C ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2124]) ).

thf(290,axiom,
    ! [A: $i,B: $i,C: $i] :
      ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ C ) @ ( tc_List_Olist @ C ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ C ) @ ( tc_List_Olist @ C ) ) @ B ) @ ( c_List_Olist_ONil @ C ) ) ) @ ( c_List_Olexord @ C @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_lexord__Nil__right) ).

thf(3436,plain,
    ! [A: $i,B: $i,C: $i] :
      ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ C ) @ ( tc_List_Olist @ C ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ C ) @ ( tc_List_Olist @ C ) ) @ B ) @ ( c_List_Olist_ONil @ C ) ) ) @ ( c_List_Olexord @ C @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[290]) ).

thf(1254,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( hAPP @ ( hAPP @ ( c_List_Oappend @ E ) @ D ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ E ) @ C ) @ ( c_List_Olist_ONil @ E ) ) )
        = ( hAPP @ ( hAPP @ ( c_List_Oappend @ E ) @ B ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ E ) @ A ) @ ( c_List_Olist_ONil @ E ) ) ) )
    <=> ( ( D = B )
        & ( C = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_append1__eq__conv) ).

thf(7145,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( ( hAPP @ ( hAPP @ ( c_List_Oappend @ E ) @ D ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ E ) @ C ) @ ( c_List_Olist_ONil @ E ) ) )
          = ( hAPP @ ( hAPP @ ( c_List_Oappend @ E ) @ B ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ E ) @ A ) @ ( c_List_Olist_ONil @ E ) ) ) )
       => ( ( D = B )
          & ( C = A ) ) )
      & ( ( ( D = B )
          & ( C = A ) )
       => ( ( hAPP @ ( hAPP @ ( c_List_Oappend @ E ) @ D ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ E ) @ C ) @ ( c_List_Olist_ONil @ E ) ) )
          = ( hAPP @ ( hAPP @ ( c_List_Oappend @ E ) @ B ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ E ) @ A ) @ ( c_List_Olist_ONil @ E ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1254]) ).

thf(1336,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Set_Oimage @ C @ D @ B ) @ ( hAPP @ ( c_Set_Ovimage @ C @ D @ B ) @ A ) ) ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_image__vimage__subset) ).

thf(7536,plain,
    ! [A: $i,B: $i,C: $i,D: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Set_Oimage @ C @ D @ B ) @ ( hAPP @ ( c_Set_Ovimage @ C @ D @ B ) @ A ) ) ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1336]) ).

thf(1229,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_add__less__mono1) ).

thf(7062,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1229]) ).

thf(2191,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( hAPP @ ( c_Map_Omap__of @ E @ D @ C ) @ B )
        = ( hAPP @ ( c_Option_Ooption_OSome @ D ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ D ) @ B ) @ A ) ) @ ( hAPP @ ( c_List_Oset @ ( tc_prod @ E @ D ) ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_map__of__is__SomeD) ).

thf(10974,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( hAPP @ ( c_Map_Omap__of @ E @ D @ C ) @ B )
        = ( hAPP @ ( c_Option_Ooption_OSome @ D ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ D ) @ B ) @ A ) ) @ ( hAPP @ ( c_List_Oset @ ( tc_prod @ E @ D ) ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2191]) ).

thf(254,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_Hilbert__Choice_OEps @ ( tc_prod @ D @ C ) @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ D @ C @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ D ) @ ( c_COMBB @ tc_HOL_Obool @ tc_HOL_Obool @ C ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ D ) @ c_fconj ) @ ( hAPP @ c_fequal @ B ) ) ) ) @ ( hAPP @ c_fequal @ A ) ) ) )
      = ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ C ) @ B ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Eps__split__eq) ).

thf(3304,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_Hilbert__Choice_OEps @ ( tc_prod @ D @ C ) @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ D @ C @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ D ) @ ( c_COMBB @ tc_HOL_Obool @ tc_HOL_Obool @ C ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ D ) @ c_fconj ) @ ( hAPP @ c_fequal @ B ) ) ) ) @ ( hAPP @ c_fequal @ A ) ) ) )
      = ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ C ) @ B ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[254]) ).

thf(1769,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ ( hAPP @ c_Nat_OSuc @ A ) ) )
    <=> ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__Suc__eq__le) ).

thf(9354,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ ( hAPP @ c_Nat_OSuc @ A ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ A ) ) )
      & ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ ( hAPP @ c_Nat_OSuc @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1769]) ).

thf(2380,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ F ) @ E ) @ ( hAPP @ ( c_List_Oset @ F ) @ D ) ) )
     => ( ( hAPP @ ( c_Map_Omap__upds @ F @ C @ B @ D @ A ) @ E )
        = ( hAPP @ B @ E ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_map__upds__apply__nontin) ).

thf(11765,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ F ) @ E ) @ ( hAPP @ ( c_List_Oset @ F ) @ D ) ) )
     => ( ( hAPP @ ( c_Map_Omap__upds @ F @ C @ B @ D @ A ) @ E )
        = ( hAPP @ B @ E ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2380]) ).

thf(2019,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ B ) ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Suc__lessD) ).

thf(10361,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ B ) ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2019]) ).

thf(693,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ E ) @ D ) ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ E ) @ B ) ) @ A ) ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ E ) @ A ) )
         => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ E ) @ C ) )
           => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ B ) @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_unique__quotient__lemma__neg) ).

thf(4887,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ E ) @ D ) ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ E ) @ B ) ) @ A ) ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ E ) @ A ) )
         => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ E ) @ C ) )
           => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ B ) @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[693]) ).

thf(1291,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ A ) @ B ) ) )
    <=> ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ B ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__le__cancel2) ).

thf(7315,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ A ) @ B ) ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ B ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ C ) @ A ) ) ) )
      & ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ B ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ C ) @ A ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ A ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1291]) ).

thf(1971,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ D ) @ ( hAPP @ ( c_List_Oset @ E ) @ C ) ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ ( hAPP @ A @ D ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ ( c_List_Olist_Olist__size @ E @ A @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_list__size__estimation_H) ).

thf(10176,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ D ) @ ( hAPP @ ( c_List_Oset @ E ) @ C ) ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ ( hAPP @ A @ D ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ ( c_List_Olist_Olist__size @ E @ A @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1971]) ).

thf(513,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_Transitive__Closure_Otrancl @ D @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ ( tc_prod @ D @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ D ) @ C ) @ B ) ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) ) @ ( c_Transitive__Closure_Otrancl @ D @ A ) ) @ ( hAPP @ ( c_Set_OCollect @ ( tc_prod @ D @ D ) ) @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ D @ D @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ ( tc_fun @ D @ tc_HOL_Obool ) @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ D @ tc_HOL_Obool ) @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ D ) @ ( c_COMBB @ tc_HOL_Obool @ tc_HOL_Obool @ D ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ D ) @ c_fconj ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_prod @ D @ D ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ D ) @ ( c_member @ ( tc_prod @ D @ D ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ D @ ( tc_prod @ D @ D ) ) @ ( c_Product__Type_OPair @ D @ D ) ) @ C ) ) ) @ ( c_Transitive__Closure_Ortrancl @ D @ A ) ) ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_prod @ D @ D ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ D ) @ ( c_member @ ( tc_prod @ D @ D ) ) ) @ ( hAPP @ ( c_Product__Type_OPair @ D @ D ) @ B ) ) ) @ ( c_Transitive__Closure_Ortrancl @ D @ A ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_trancl__insert) ).

thf(4245,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_Transitive__Closure_Otrancl @ D @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ ( tc_prod @ D @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ D ) @ C ) @ B ) ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) ) @ ( c_Transitive__Closure_Otrancl @ D @ A ) ) @ ( hAPP @ ( c_Set_OCollect @ ( tc_prod @ D @ D ) ) @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ D @ D @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ ( tc_fun @ D @ tc_HOL_Obool ) @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ D @ tc_HOL_Obool ) @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ D ) @ ( c_COMBB @ tc_HOL_Obool @ tc_HOL_Obool @ D ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ D ) @ c_fconj ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_prod @ D @ D ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ D ) @ ( c_member @ ( tc_prod @ D @ D ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ D @ ( tc_prod @ D @ D ) ) @ ( c_Product__Type_OPair @ D @ D ) ) @ C ) ) ) @ ( c_Transitive__Closure_Ortrancl @ D @ A ) ) ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_prod @ D @ D ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ D ) @ ( c_member @ ( tc_prod @ D @ D ) ) ) @ ( hAPP @ ( c_Product__Type_OPair @ D @ D ) @ B ) ) ) @ ( c_Transitive__Closure_Ortrancl @ D @ A ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[513]) ).

thf(1696,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) ) @ ( c_Relation_OId @ C ) ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ C @ C @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) ) @ ( c_Transitive__Closure_Ortrancl @ C @ A ) ) @ B ) ) @ A ) ) @ B ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) ) @ ( c_Transitive__Closure_Ortrancl @ C @ A ) ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_rtrancl__Int__subset) ).

thf(9089,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) ) @ ( c_Relation_OId @ C ) ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ C @ C @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) ) @ ( c_Transitive__Closure_Ortrancl @ C @ A ) ) @ B ) ) @ A ) ) @ B ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) ) @ ( c_Transitive__Closure_Ortrancl @ C @ A ) ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1696]) ).

thf(1543,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ C @ C ) ) @ B ) @ ( c_Transitive__Closure_Otrancl @ C @ A ) ) )
    <=> ? [D: $i] :
          ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ D ) )
          & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ C @ C ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Nat_Ocompow @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) @ D ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_trancl__power) ).

thf(8376,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ C @ C ) ) @ B ) @ ( c_Transitive__Closure_Otrancl @ C @ A ) ) )
       => ? [D: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ D ) )
            & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ C @ C ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Nat_Ocompow @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) @ D ) @ A ) ) ) ) )
      & ( ? [D: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ D ) )
            & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ C @ C ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Nat_Ocompow @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) @ D ) @ A ) ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ C @ C ) ) @ B ) @ ( c_Transitive__Closure_Otrancl @ C @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1543]) ).

thf(2062,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ F @ F ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ F @ F ) @ E ) @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Relation_Oinv__image @ C @ F ) @ B ) @ A ) ) )
    <=> ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ C @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ C @ C ) @ ( hAPP @ A @ E ) ) @ ( hAPP @ A @ D ) ) ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_in__inv__image) ).

thf(10505,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ F @ F ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ F @ F ) @ E ) @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Relation_Oinv__image @ C @ F ) @ B ) @ A ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ C @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ C @ C ) @ ( hAPP @ A @ E ) ) @ ( hAPP @ A @ D ) ) ) @ B ) ) )
      & ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ C @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ C @ C ) @ ( hAPP @ A @ E ) ) @ ( hAPP @ A @ D ) ) ) @ B ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ F @ F ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ F @ F ) @ E ) @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Relation_Oinv__image @ C @ F ) @ B ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2062]) ).

thf(1756,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ D ) @ C ) @ B ) ) @ ( hAPP @ ( c_Relation_Oconverse @ D @ E ) @ A ) ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ E ) @ B ) @ C ) ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_converseD) ).

thf(9307,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ D ) @ C ) @ B ) ) @ ( hAPP @ ( c_Relation_Oconverse @ D @ E ) @ A ) ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ E ) @ B ) @ C ) ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1756]) ).

thf(1963,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_List_Osublist @ C @ B @ ( hAPP @ ( c_SetInterval_Oord__class_OlessThan @ tc_Nat_Onat ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_List_Otake @ C ) @ A ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_sublist__upt__eq__take) ).

thf(10140,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_List_Osublist @ C @ B @ ( hAPP @ ( c_SetInterval_Oord__class_OlessThan @ tc_Nat_Onat ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_List_Otake @ C ) @ A ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1963]) ).

thf(1945,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i] :
          ( ( hBOOL @ ( hAPP @ D @ E ) )
         => ( c_Wellfounded_Owf @ C @ ( hAPP @ B @ E ) ) )
     => ( c_Wellfounded_Owf @ ( tc_prod @ A @ C ) @ ( c_Recdef_Osame__fst @ A @ C @ D @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_wf__same__fst) ).

thf(10056,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i] :
          ( ( hBOOL @ ( hAPP @ D @ E ) )
         => ( c_Wellfounded_Owf @ C @ ( hAPP @ B @ E ) ) )
     => ( c_Wellfounded_Owf @ ( tc_prod @ A @ C ) @ ( c_Recdef_Osame__fst @ A @ C @ D @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1945]) ).

thf(316,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( c_Set_Ovimage @ C @ B @ A ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ B @ tc_HOL_Obool ) ) )
      = ( c_Orderings_Otop__class_Otop @ ( tc_fun @ C @ tc_HOL_Obool ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_vimage__UNIV) ).

thf(3532,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( c_Set_Ovimage @ C @ B @ A ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ B @ tc_HOL_Obool ) ) )
      = ( c_Orderings_Otop__class_Otop @ ( tc_fun @ C @ tc_HOL_Obool ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[316]) ).

thf(1735,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( C
          = ( c_List_Olist_ONil @ B ) )
       => ( ( c_List_Obutlast @ B @ ( hAPP @ ( hAPP @ ( c_List_Oappend @ B ) @ A ) @ C ) )
          = ( c_List_Obutlast @ B @ A ) ) )
      & ( ( C
         != ( c_List_Olist_ONil @ B ) )
       => ( ( c_List_Obutlast @ B @ ( hAPP @ ( hAPP @ ( c_List_Oappend @ B ) @ A ) @ C ) )
          = ( hAPP @ ( hAPP @ ( c_List_Oappend @ B ) @ A ) @ ( c_List_Obutlast @ B @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_butlast__append) ).

thf(9224,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( C
          = ( c_List_Olist_ONil @ B ) )
       => ( ( c_List_Obutlast @ B @ ( hAPP @ ( hAPP @ ( c_List_Oappend @ B ) @ A ) @ C ) )
          = ( c_List_Obutlast @ B @ A ) ) )
      & ( ( C
         != ( c_List_Olist_ONil @ B ) )
       => ( ( c_List_Obutlast @ B @ ( hAPP @ ( hAPP @ ( c_List_Oappend @ B ) @ A ) @ C ) )
          = ( hAPP @ ( hAPP @ ( c_List_Oappend @ B ) @ A ) @ ( c_List_Obutlast @ B @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1735]) ).

thf(1102,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ D @ C ) @ B ) @ A ) @ ( hAPP @ B @ A ) )
      = B ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_fun__upd__triv) ).

thf(6525,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ D @ C ) @ B ) @ A ) @ ( hAPP @ B @ A ) )
      = B ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1102]) ).

thf(1779,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
      ( ( c_Finite__Set_Ofolding__image__simple__idem @ H @ G @ F @ E @ D @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ G ) @ B ) )
       => ( ( hAPP @ C @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ G ) @ A ) @ B ) )
          = ( hAPP @ ( hAPP @ F @ ( hAPP @ D @ A ) ) @ ( hAPP @ C @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_folding__image__simple__idem_Oinsert__idem) ).

thf(9390,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
      ( ( c_Finite__Set_Ofolding__image__simple__idem @ H @ G @ F @ E @ D @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ G ) @ B ) )
       => ( ( hAPP @ C @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ G ) @ A ) @ B ) )
          = ( hAPP @ ( hAPP @ F @ ( hAPP @ D @ A ) ) @ ( hAPP @ C @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1779]) ).

thf(77,axiom,
    class_Groups_Ocomm__monoid__add @ tc_Nat_Onat,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat__Onat__Groups_Ocomm__monoid__add) ).

thf(2679,plain,
    class_Groups_Ocomm__monoid__add @ tc_Nat_Onat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[77]) ).

thf(3,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ ( hAPP @ ( c_Set_OCollect @ D ) @ C ) ) )
     => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ B ) @ ( hAPP @ ( c_Set_OCollect @ B ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ D @ tc_HOL_Obool ) @ tc_HOL_Obool @ B ) @ ( c_HOL_OEx @ D ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ B @ ( tc_fun @ D @ tc_HOL_Obool ) @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ D @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ D @ tc_HOL_Obool ) @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ ( c_COMBS @ D @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ D @ tc_HOL_Obool ) @ ( tc_fun @ D @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ D ) @ c_fconj ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ B @ ( tc_fun @ D @ B ) @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ B @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ D @ B ) @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ ( c_COMBB @ B @ tc_HOL_Obool @ D ) ) @ c_fequal ) ) @ A ) ) ) ) @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_finite__image__set) ).

thf(2415,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ ( hAPP @ ( c_Set_OCollect @ D ) @ C ) ) )
     => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ B ) @ ( hAPP @ ( c_Set_OCollect @ B ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ D @ tc_HOL_Obool ) @ tc_HOL_Obool @ B ) @ ( c_HOL_OEx @ D ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ B @ ( tc_fun @ D @ tc_HOL_Obool ) @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ D @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ D @ tc_HOL_Obool ) @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ ( c_COMBS @ D @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ D @ tc_HOL_Obool ) @ ( tc_fun @ D @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ D ) @ c_fconj ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ B @ ( tc_fun @ D @ B ) @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ B @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ D @ B ) @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ ( c_COMBB @ B @ tc_HOL_Obool @ D ) ) @ c_fequal ) ) @ A ) ) ) ) @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3]) ).

thf(2049,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ c_Com_OWT @ ( c_Com_Ocom_OLocal @ C @ B @ A ) ) )
    <=> ( hBOOL @ ( hAPP @ c_Com_OWT @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_WT_Oequations_I3_J) ).

thf(10457,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP @ c_Com_OWT @ ( c_Com_Ocom_OLocal @ C @ B @ A ) ) )
       => ( hBOOL @ ( hAPP @ c_Com_OWT @ A ) ) )
      & ( ( hBOOL @ ( hAPP @ c_Com_OWT @ A ) )
       => ( hBOOL @ ( hAPP @ c_Com_OWT @ ( c_Com_Ocom_OLocal @ C @ B @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2049]) ).

thf(2072,axiom,
    ! [A: $i,B: $i] :
      ( ( B = A )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_dvd_Oeq__refl) ).

thf(10565,plain,
    ! [A: $i,B: $i] :
      ( ( B = A )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2072]) ).

thf(1558,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ D ) ) @ C )
        = ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ B ) ) @ A ) )
     => ( ( hBOOL @ ( hAPP @ ( c_List_Odistinct @ D ) @ C ) )
       => ( ( c_Map_Odom @ D @ B @ ( c_Map_Omap__of @ D @ B @ ( hAPP @ ( hAPP @ ( c_List_Ozip @ D @ B ) @ C ) @ A ) ) )
          = ( hAPP @ ( c_List_Oset @ D ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_dom__map__of__zip) ).

thf(8436,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ D ) ) @ C )
        = ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ B ) ) @ A ) )
     => ( ( hBOOL @ ( hAPP @ ( c_List_Odistinct @ D ) @ C ) )
       => ( ( c_Map_Odom @ D @ B @ ( c_Map_Omap__of @ D @ B @ ( hAPP @ ( hAPP @ ( c_List_Ozip @ D @ B ) @ C ) @ A ) ) )
          = ( hAPP @ ( c_List_Oset @ D ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1558]) ).

thf(643,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ C ) )
     => ( ( c_Fun_Oinj__on @ B @ D @ A @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ B @ tc_HOL_Obool ) ) )
       => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ B ) @ ( hAPP @ ( c_Set_Ovimage @ B @ D @ A ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_finite__vimageI) ).

thf(4716,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ C ) )
     => ( ( c_Fun_Oinj__on @ B @ D @ A @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ B @ tc_HOL_Obool ) ) )
       => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ B ) @ ( hAPP @ ( c_Set_Ovimage @ B @ D @ A ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[643]) ).

thf(2193,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hAPP @ ( c_List_Ofilter @ C @ B ) @ A )
        = ( c_List_Olist_ONil @ C ) )
    <=> ! [D: $i] :
          ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ D ) @ ( hAPP @ ( c_List_Oset @ C ) @ A ) ) )
         => ~ ( hBOOL @ ( hAPP @ B @ D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_filter__empty__conv) ).

thf(10980,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( hAPP @ ( c_List_Ofilter @ C @ B ) @ A )
          = ( c_List_Olist_ONil @ C ) )
       => ! [D: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ D ) @ ( hAPP @ ( c_List_Oset @ C ) @ A ) ) )
           => ~ ( hBOOL @ ( hAPP @ B @ D ) ) ) )
      & ( ! [D: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ D ) @ ( hAPP @ ( c_List_Oset @ C ) @ A ) ) )
           => ~ ( hBOOL @ ( hAPP @ B @ D ) ) )
       => ( ( hAPP @ ( c_List_Ofilter @ C @ B ) @ A )
          = ( c_List_Olist_ONil @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2193]) ).

thf(993,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C
        = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) ) @ C ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__by__empty) ).

thf(6024,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C
        = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) ) @ C ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[993]) ).

thf(1469,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) ) @ B ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Relation_OField @ C ) @ B ) ) @ ( hAPP @ ( c_Relation_OField @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mono__Field) ).

thf(8042,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) ) @ B ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Relation_OField @ C ) @ B ) ) @ ( hAPP @ ( c_Relation_OField @ C ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1469]) ).

thf(2253,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Wellfounded_Oaccp @ C @ B ) @ A ) )
    <=> ! [D: $i] :
          ( ( hBOOL @ ( hAPP @ ( hAPP @ B @ D ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( c_Wellfounded_Oaccp @ C @ B ) @ D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_accp_Osimps) ).

thf(11222,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( c_Wellfounded_Oaccp @ C @ B ) @ A ) )
       => ! [D: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ B @ D ) @ A ) )
           => ( hBOOL @ ( hAPP @ ( c_Wellfounded_Oaccp @ C @ B ) @ D ) ) ) )
      & ( ! [D: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ B @ D ) @ A ) )
           => ( hBOOL @ ( hAPP @ ( c_Wellfounded_Oaccp @ C @ B ) @ D ) ) )
       => ( hBOOL @ ( hAPP @ ( c_Wellfounded_Oaccp @ C @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2253]) ).

thf(1826,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Com_Ocom_OSemi @ E @ D )
     != ( c_Com_Ocom_OLocal @ C @ B @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_com_Osimps_I35_J) ).

thf(9560,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Com_Ocom_OSemi @ E @ D )
     != ( c_Com_Ocom_OLocal @ C @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1826]) ).

thf(191,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Un__left__absorb) ).

thf(3101,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[191]) ).

thf(2150,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Fun_Oinj__on @ E @ D @ C @ B )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ A ) @ ( hAPP @ ( c_Set_Oimage @ E @ D @ C ) @ B ) ) )
       => ( ( hAPP @ C @ ( hAPP @ ( c_Fun_Othe__inv__into @ E @ D @ B @ C ) @ A ) )
          = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_f__the__inv__into__f) ).

thf(10842,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Fun_Oinj__on @ E @ D @ C @ B )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ A ) @ ( hAPP @ ( c_Set_Oimage @ E @ D @ C ) @ B ) ) )
       => ( ( hAPP @ C @ ( hAPP @ ( c_Fun_Othe__inv__into @ E @ D @ B @ C ) @ A ) )
          = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2150]) ).

thf(959,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ C ) @ B ) ) @ B ) ) @ ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ C @ B ) ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ C ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zdiv__zmod__equality2) ).

thf(5870,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ C ) @ B ) ) @ B ) ) @ ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ C @ B ) ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ C ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[959]) ).

thf(1840,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Wellfounded_Owf @ C @ B )
     => ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ C @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ C @ C ) @ A ) @ A ) ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_wf__irrefl) ).

thf(9623,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Wellfounded_Owf @ C @ B )
     => ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ C @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ C @ C ) @ A ) @ A ) ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1840]) ).

thf(2179,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i,I: $i,J: $i,K: $i] :
      ( ( c_Com_Ocom_Ocom__rec @ K @ J @ I @ H @ G @ F @ E @ D @ C @ ( c_Com_Ocom_OSemi @ B @ A ) )
      = ( hAPP @ ( hAPP @ ( hAPP @ ( hAPP @ G @ B ) @ A ) @ ( c_Com_Ocom_Ocom__rec @ K @ J @ I @ H @ G @ F @ E @ D @ C @ B ) ) @ ( c_Com_Ocom_Ocom__rec @ K @ J @ I @ H @ G @ F @ E @ D @ C @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_com_Orecs_I4_J) ).

thf(10921,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i,I: $i,J: $i,K: $i] :
      ( ( c_Com_Ocom_Ocom__rec @ K @ J @ I @ H @ G @ F @ E @ D @ C @ ( c_Com_Ocom_OSemi @ B @ A ) )
      = ( hAPP @ ( hAPP @ ( hAPP @ ( hAPP @ G @ B ) @ A ) @ ( c_Com_Ocom_Ocom__rec @ K @ J @ I @ H @ G @ F @ E @ D @ C @ B ) ) @ ( c_Com_Ocom_Ocom__rec @ K @ J @ I @ H @ G @ F @ E @ D @ C @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2179]) ).

thf(2067,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( c_Set_Oimage @ tc_Nat_Onat @ tc_Nat_Onat @ c_Nat_OSuc ) @ ( hAPP @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Nat_Onat @ B ) @ A ) )
      = ( hAPP @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Nat_Onat @ ( hAPP @ c_Nat_OSuc @ B ) ) @ ( hAPP @ c_Nat_OSuc @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_image__Suc__atLeastLessThan) ).

thf(10532,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( c_Set_Oimage @ tc_Nat_Onat @ tc_Nat_Onat @ c_Nat_OSuc ) @ ( hAPP @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Nat_Onat @ B ) @ A ) )
      = ( hAPP @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Nat_Onat @ ( hAPP @ c_Nat_OSuc @ B ) ) @ ( hAPP @ c_Nat_OSuc @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2067]) ).

thf(509,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Map_Omap__add @ C @ B @ ( hAPP @ ( c_COMBK @ ( tc_Option_Ooption @ B ) @ C ) @ ( c_Option_Ooption_ONone @ B ) ) @ A )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_empty__map__add) ).

thf(4228,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Map_Omap__add @ C @ B @ ( hAPP @ ( c_COMBK @ ( tc_Option_Ooption @ B ) @ C ) @ ( c_Option_Ooption_ONone @ B ) ) @ A )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[509]) ).

thf(42,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_List_Oconcat @ D @ ( hAPP @ ( hAPP @ ( c_List_Omap @ C @ ( tc_List_Olist @ D ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ C @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ D @ ( tc_fun @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) @ C ) @ ( c_List_Olist_OCons @ D ) ) @ B ) ) @ ( c_List_Olist_ONil @ D ) ) ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_List_Omap @ C @ D ) @ B ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_concat__map__singleton) ).

thf(2538,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_List_Oconcat @ D @ ( hAPP @ ( hAPP @ ( c_List_Omap @ C @ ( tc_List_Olist @ D ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ C @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ D @ ( tc_fun @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) @ C ) @ ( c_List_Olist_OCons @ D ) ) @ B ) ) @ ( c_List_Olist_ONil @ D ) ) ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_List_Omap @ C @ D ) @ B ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[42]) ).

thf(2198,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ c_Com_OWT @ ( c_Com_Ocom_OLocal @ C @ B @ A ) ) )
     => ( hBOOL @ ( hAPP @ c_Com_OWT @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_WTs__elim__cases_I3_J) ).

thf(11002,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ c_Com_OWT @ ( c_Com_Ocom_OLocal @ C @ B @ A ) ) )
     => ( hBOOL @ ( hAPP @ c_Com_OWT @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2198]) ).

thf(2174,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Wellfounded_Owf @ C @ B )
     => ~ ! [D: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ C @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ C @ C ) @ ( hAPP @ A @ ( hAPP @ c_Nat_OSuc @ D ) ) ) @ ( hAPP @ A @ D ) ) ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_wf__no__infinite__down__chainE) ).

thf(10910,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Wellfounded_Owf @ C @ B )
     => ~ ! [D: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ C @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ C @ C ) @ ( hAPP @ A @ ( hAPP @ c_Nat_OSuc @ D ) ) ) @ ( hAPP @ A @ D ) ) ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2174]) ).

thf(21,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Groups_Ocomm__monoid__add @ E )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ C ) )
       => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ D @ E @ E ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ E @ ( tc_fun @ E @ E ) @ D ) @ ( c_Groups_Oplus__class_Oplus @ E ) ) @ B ) ) @ A ) ) @ C )
          = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ E ) @ ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ B ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ A ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_setsum_Odistrib) ).

thf(2478,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Groups_Ocomm__monoid__add @ E )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ C ) )
       => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ D @ E @ E ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ E @ ( tc_fun @ E @ E ) @ D ) @ ( c_Groups_Oplus__class_Oplus @ E ) ) @ B ) ) @ A ) ) @ C )
          = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ E ) @ ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ B ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ A ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[21]) ).

thf(1248,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_diff__mult__distrib2) ).

thf(7130,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1248]) ).

thf(68,axiom,
    ! [A: $i,B: $i] :
      ( ( B
       != ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) )
     => ( hBOOL @ ( hAPP @ ( c_Divides_Odivmod__int__rel @ A @ B ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ tc_Int_Oint @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ A ) @ B ) ) @ ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ A @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_divmod__int__rel__div__mod) ).

thf(2642,plain,
    ! [A: $i,B: $i] :
      ( ( B
       != ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) )
     => ( hBOOL @ ( hAPP @ ( c_Divides_Odivmod__int__rel @ A @ B ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ tc_Int_Oint @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ A ) @ B ) ) @ ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ A @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[68]) ).

thf(1365,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ D ) @ C ) ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ E ) @ ( tc_List_Olist @ E ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ E ) @ ( tc_List_Olist @ E ) ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ E ) @ D ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ E ) @ C ) @ A ) ) ) @ ( c_List_Olistrel1 @ E @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_listrel1I1) ).

thf(7643,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ D ) @ C ) ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ E ) @ ( tc_List_Olist @ E ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ E ) @ ( tc_List_Olist @ E ) ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ E ) @ D ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ E ) @ C ) @ A ) ) ) @ ( c_List_Olistrel1 @ E @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1365]) ).

thf(2227,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Wellfounded_Oaccp @ D @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ C @ A ) @ B ) )
       => ( hBOOL @ ( hAPP @ ( c_Wellfounded_Oaccp @ D @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_accp__downward) ).

thf(11117,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Wellfounded_Oaccp @ D @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ C @ A ) @ B ) )
       => ( hBOOL @ ( hAPP @ ( c_Wellfounded_Oaccp @ D @ C ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2227]) ).

thf(392,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ B ) @ A )
      = ( hAPP @ ( c_Set_OCollect @ C ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ C @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ C ) @ c_fdisj ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ C @ C @ tc_HOL_Obool ) @ c_fequal ) @ B ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ C @ ( tc_fun @ C @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( c_member @ C ) ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_insert__compr__raw) ).

thf(3815,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ B ) @ A )
      = ( hAPP @ ( c_Set_OCollect @ C ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ C @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ C ) @ c_fdisj ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ C @ C @ tc_HOL_Obool ) @ c_fequal ) @ B ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ C @ ( tc_fun @ C @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( c_member @ C ) ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[392]) ).

thf(1195,axiom,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ A ) )
     => ( ( hAPP @ c_Nat_OSuc @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ A ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) )
        = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Suc__diff__1) ).

thf(6920,plain,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ A ) )
     => ( ( hAPP @ c_Nat_OSuc @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ A ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1195]) ).

thf(234,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ A ) @ A )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Int__absorb) ).

thf(3238,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ A ) @ A )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[234]) ).

thf(1030,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ! [G: $i,H: $i] :
          ( ( F
            = ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ D ) @ G ) @ H ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ B ) @ ( hAPP @ ( hAPP @ A @ G ) @ H ) ) ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ E @ D @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ A ) @ F ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mem__splitI2) ).

thf(6179,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ! [G: $i,H: $i] :
          ( ( F
            = ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ D ) @ G ) @ H ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ B ) @ ( hAPP @ ( hAPP @ A @ G ) @ H ) ) ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ E @ D @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ A ) @ F ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1030]) ).

thf(2321,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hAPP @ ( c_List_Ofilter @ C @ B ) @ A )
        = A )
    <=> ! [D: $i] :
          ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ D ) @ ( hAPP @ ( c_List_Oset @ C ) @ A ) ) )
         => ( hBOOL @ ( hAPP @ B @ D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_filter__id__conv) ).

thf(11508,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( hAPP @ ( c_List_Ofilter @ C @ B ) @ A )
          = A )
       => ! [D: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ D ) @ ( hAPP @ ( c_List_Oset @ C ) @ A ) ) )
           => ( hBOOL @ ( hAPP @ B @ D ) ) ) )
      & ( ! [D: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ D ) @ ( hAPP @ ( c_List_Oset @ C ) @ A ) ) )
           => ( hBOOL @ ( hAPP @ B @ D ) ) )
       => ( ( hAPP @ ( c_List_Ofilter @ C @ B ) @ A )
          = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2321]) ).

thf(1895,axiom,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) )
        & ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ A ) @ B ) ) )
     => ~ ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ A ) @ B ) )
          & ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_dvd_Oless__not__sym) ).

thf(9867,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) )
        & ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ A ) @ B ) ) )
     => ~ ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ A ) @ B ) )
          & ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1895]) ).

thf(67,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Ocomm__monoid__add @ D )
     => ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ C ) ) @ B ) )
         => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ tc_Nat_Onat @ D ) @ A ) @ ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Nat_Onat @ B @ ( hAPP @ c_Nat_OSuc @ C ) ) )
            = ( c_Groups_Ozero__class_Ozero @ D ) ) )
        & ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ C ) ) @ B ) )
         => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ tc_Nat_Onat @ D ) @ A ) @ ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Nat_Onat @ B @ ( hAPP @ c_Nat_OSuc @ C ) ) )
            = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ tc_Nat_Onat @ D ) @ A ) @ ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Nat_Onat @ B @ C ) ) ) @ ( hAPP @ A @ ( hAPP @ c_Nat_OSuc @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_setsum__cl__ivl__Suc) ).

thf(2637,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Ocomm__monoid__add @ D )
     => ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ C ) ) @ B ) )
         => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ tc_Nat_Onat @ D ) @ A ) @ ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Nat_Onat @ B @ ( hAPP @ c_Nat_OSuc @ C ) ) )
            = ( c_Groups_Ozero__class_Ozero @ D ) ) )
        & ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ C ) ) @ B ) )
         => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ tc_Nat_Onat @ D ) @ A ) @ ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Nat_Onat @ B @ ( hAPP @ c_Nat_OSuc @ C ) ) )
            = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ tc_Nat_Onat @ D ) @ A ) @ ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Nat_Onat @ B @ C ) ) ) @ ( hAPP @ A @ ( hAPP @ c_Nat_OSuc @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[67]) ).

thf(809,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Un__Diff__cancel2) ).

thf(5325,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[809]) ).

thf(1776,axiom,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ A ) @ ( hAPP @ c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) )
    <=> ( A
        = ( hAPP @ c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_dvd__1__iff__1) ).

thf(9377,plain,
    ! [A: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ A ) @ ( hAPP @ c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) )
       => ( A
          = ( hAPP @ c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) )
      & ( ( A
          = ( hAPP @ c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ A ) @ ( hAPP @ c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1776]) ).

thf(529,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ E @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ D @ tc_HOL_Obool ) @ ( tc_fun @ D @ tc_HOL_Obool ) @ E ) @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ E @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_UN__simps_I5_J) ).

thf(4305,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ E @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ D @ tc_HOL_Obool ) @ ( tc_fun @ D @ tc_HOL_Obool ) @ E ) @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ E @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[529]) ).

thf(1408,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ B ) @ A ) ) )
    <=> ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ A ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zero__less__diff) ).

thf(7810,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ B ) @ A ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ A ) @ B ) ) )
      & ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ A ) @ B ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ B ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1408]) ).

thf(1436,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ A ) @ B ) ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_diff__less) ).

thf(7907,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ A ) @ B ) ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1436]) ).

thf(952,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ A ) @ B ) ) )
      <=> ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_pos__imp__zdiv__nonneg__iff) ).

thf(5848,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ B ) )
     => ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ A ) @ B ) ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ A ) ) )
        & ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ A ) @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[952]) ).

thf(1673,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( c_Relation_ODomain @ B @ B ) @ ( c_Transitive__Closure_Otrancl @ B @ A ) )
      = ( hAPP @ ( c_Relation_ODomain @ B @ B ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_trancl__domain) ).

thf(8970,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( c_Relation_ODomain @ B @ B ) @ ( c_Transitive__Closure_Otrancl @ B @ A ) )
      = ( hAPP @ ( c_Relation_ODomain @ B @ B ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1673]) ).

thf(1999,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( hAPP @ ( c_Sum__Type_OInl @ D @ C ) @ B )
        = ( hAPP @ ( c_Sum__Type_OInl @ D @ C ) @ A ) )
    <=> ( B = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_sum_Osimps_I1_J) ).

thf(10279,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( ( hAPP @ ( c_Sum__Type_OInl @ D @ C ) @ B )
          = ( hAPP @ ( c_Sum__Type_OInl @ D @ C ) @ A ) )
       => ( B = A ) )
      & ( ( B = A )
       => ( ( hAPP @ ( c_Sum__Type_OInl @ D @ C ) @ B )
          = ( hAPP @ ( c_Sum__Type_OInl @ D @ C ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1999]) ).

thf(2109,axiom,
    ! [A: $i] :
      ( ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ A )
      = ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__eq__Suc__le__raw) ).

thf(10677,plain,
    ! [A: $i] :
      ( ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ A )
      = ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2109]) ).

thf(2108,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ C ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ B ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_dvd__diff__nat) ).

thf(10675,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ C ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ B ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2108]) ).

thf(1094,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) )
     => ( ( c_List_Oupt @ B @ A )
        = ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ tc_Nat_Onat ) @ B ) @ ( c_List_Oupt @ ( hAPP @ c_Nat_OSuc @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_upt__conv__Cons) ).

thf(6500,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) )
     => ( ( c_List_Oupt @ B @ A )
        = ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ tc_Nat_Onat ) @ B ) @ ( c_List_Oupt @ ( hAPP @ c_Nat_OSuc @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1094]) ).

thf(676,axiom,
    ! [A: $i] :
      ( ( c_Wellfounded_Ofinite__psubset @ A )
      = ( hAPP @ ( c_Set_OCollect @ ( tc_prod @ ( tc_fun @ A @ tc_HOL_Obool ) @ ( tc_fun @ A @ tc_HOL_Obool ) ) ) @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ ( tc_fun @ A @ tc_HOL_Obool ) @ ( tc_fun @ A @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ ( tc_fun @ A @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ A @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ A @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ ( tc_fun @ A @ tc_HOL_Obool ) @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ ( tc_fun @ A @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ A @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( tc_fun @ A @ tc_HOL_Obool ) ) @ ( c_COMBS @ ( tc_fun @ A @ tc_HOL_Obool ) @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ ( tc_fun @ A @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ A @ tc_HOL_Obool ) @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ ( tc_fun @ A @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( tc_fun @ A @ tc_HOL_Obool ) ) @ c_fconj ) ) @ ( c_Orderings_Oord__class_Oless @ ( tc_fun @ A @ tc_HOL_Obool ) ) ) ) ) @ ( c_Finite__Set_Ofinite @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_finite__psubset__def) ).

thf(4828,plain,
    ! [A: $i] :
      ( ( c_Wellfounded_Ofinite__psubset @ A )
      = ( hAPP @ ( c_Set_OCollect @ ( tc_prod @ ( tc_fun @ A @ tc_HOL_Obool ) @ ( tc_fun @ A @ tc_HOL_Obool ) ) ) @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ ( tc_fun @ A @ tc_HOL_Obool ) @ ( tc_fun @ A @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ ( tc_fun @ A @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ A @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ A @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ ( tc_fun @ A @ tc_HOL_Obool ) @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ ( tc_fun @ A @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ A @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( tc_fun @ A @ tc_HOL_Obool ) ) @ ( c_COMBS @ ( tc_fun @ A @ tc_HOL_Obool ) @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ ( tc_fun @ A @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ A @ tc_HOL_Obool ) @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ ( tc_fun @ A @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( tc_fun @ A @ tc_HOL_Obool ) ) @ c_fconj ) ) @ ( c_Orderings_Oord__class_Oless @ ( tc_fun @ A @ tc_HOL_Obool ) ) ) ) ) @ ( c_Finite__Set_Ofinite @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[676]) ).

thf(381,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ B ) @ ( c_Complete__Lattice_OSup__class_OSup @ ( tc_fun @ B @ tc_HOL_Obool ) @ A ) ) )
     => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_finite__UnionD) ).

thf(3779,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ B ) @ ( c_Complete__Lattice_OSup__class_OSup @ ( tc_fun @ B @ tc_HOL_Obool ) @ A ) ) )
     => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[381]) ).

thf(1305,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_Wellfounded_Owf @ D @ C )
     => ( ( c_Wellfounded_Owf @ B @ A )
       => ( c_Wellfounded_Owf @ ( tc_prod @ D @ B ) @ ( c_Wellfounded_Olex__prod @ D @ B @ C @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_wf__lex__prod) ).

thf(7379,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_Wellfounded_Owf @ D @ C )
     => ( ( c_Wellfounded_Owf @ B @ A )
       => ( c_Wellfounded_Owf @ ( tc_prod @ D @ B ) @ ( c_Wellfounded_Olex__prod @ D @ B @ C @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1305]) ).

thf(1871,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ c_Com_OWT @ C ) )
     => ( ( hBOOL @ ( hAPP @ c_Com_OWT @ B ) )
       => ( hBOOL @ ( hAPP @ c_Com_OWT @ ( c_Com_Ocom_OCond @ A @ C @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_WT_OIf) ).

thf(9771,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ c_Com_OWT @ C ) )
     => ( ( hBOOL @ ( hAPP @ c_Com_OWT @ B ) )
       => ( hBOOL @ ( hAPP @ c_Com_OWT @ ( c_Com_Ocom_OCond @ A @ C @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1871]) ).

thf(1035,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) )
       => ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ A ) @ C ) )
          = C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_double__diff) ).

thf(6197,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) )
       => ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ A ) @ C ) )
          = C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1035]) ).

thf(1013,axiom,
    ! [A: $i,B: $i] :
      ( ( ( c_SetInterval_Oord_OatLeastAtMost @ tc_Nat_Onat @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B @ A )
        = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) )
    <=> ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_dvd_OatLeastatMost__empty__iff) ).

thf(6112,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( c_SetInterval_Oord_OatLeastAtMost @ tc_Nat_Onat @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B @ A )
          = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) )
       => ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) ) )
      & ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) )
       => ( ( c_SetInterval_Oord_OatLeastAtMost @ tc_Nat_Onat @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B @ A )
          = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1013]) ).

thf(638,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ C ) )
     => ( ! [D: $i] :
            ( ( hBOOL @ ( hAPP @ B @ D ) )
           => ( hBOOL @ ( hAPP @ B @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ D ) @ C ) ) ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ A ) )
         => ! [D: $i] :
              ( ( hBOOL @ ( hAPP @ B @ D ) )
             => ( hBOOL @ ( hAPP @ B @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ A ) @ C ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_incr__mult__lemma) ).

thf(4699,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ C ) )
     => ( ! [D: $i] :
            ( ( hBOOL @ ( hAPP @ B @ D ) )
           => ( hBOOL @ ( hAPP @ B @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ D ) @ C ) ) ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ A ) )
         => ! [D: $i] :
              ( ( hBOOL @ ( hAPP @ B @ D ) )
             => ( hBOOL @ ( hAPP @ B @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ A ) @ C ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[638]) ).

thf(802,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_Option_Ooption @ B ) ) @ ( hAPP @ ( c_Option_Ooption_OSome @ B ) @ A ) )
      = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_option_Osize_I4_J) ).

thf(5303,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_Option_Ooption @ B ) ) @ ( hAPP @ ( c_Option_Ooption_OSome @ B ) @ A ) )
      = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[802]) ).

thf(948,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A )
        = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) )
    <=> ! [D: $i] :
          ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ D ) @ B ) )
         => ! [E: $i] :
              ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ E ) @ A ) )
             => ( D != E ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_disjoint__iff__not__equal) ).

thf(5827,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A )
          = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) )
       => ! [D: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ D ) @ B ) )
           => ! [E: $i] :
                ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ E ) @ A ) )
               => ( D != E ) ) ) )
      & ( ! [D: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ D ) @ B ) )
           => ! [E: $i] :
                ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ E ) @ A ) )
               => ( D != E ) ) )
       => ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A )
          = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[948]) ).

thf(954,axiom,
    ! [A: $i] :
      ( ( class_Groups_Oab__semigroup__mult @ A )
     => ( c_Finite__Set_Ofun__left__comm @ A @ A @ ( c_Groups_Otimes__class_Otimes @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_fun__left__comm) ).

thf(5853,plain,
    ! [A: $i] :
      ( ( class_Groups_Oab__semigroup__mult @ A )
     => ( c_Finite__Set_Ofun__left__comm @ A @ A @ ( c_Groups_Otimes__class_Otimes @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[954]) ).

thf(152,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( c_Map_Odom @ D @ C @ B ) ) @ ( c_Map_Odom @ D @ C @ A ) )
        = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) )
     => ( ( c_Map_Omap__add @ D @ C @ B @ A )
        = ( c_Map_Omap__add @ D @ C @ A @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_map__add__comm) ).

thf(2955,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( c_Map_Odom @ D @ C @ B ) ) @ ( c_Map_Odom @ D @ C @ A ) )
        = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) )
     => ( ( c_Map_Omap__add @ D @ C @ B @ A )
        = ( c_Map_Omap__add @ D @ C @ A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[152]) ).

thf(2370,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ D ) @ ( c_Map_Odom @ E @ C @ B ) ) )
     => ( ( hAPP @ ( c_Map_Omap__add @ E @ C @ A @ B ) @ D )
        = ( hAPP @ B @ D ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_map__add__dom__app__simps_I1_J) ).

thf(11735,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ D ) @ ( c_Map_Odom @ E @ C @ B ) ) )
     => ( ( hAPP @ ( c_Map_Omap__add @ E @ C @ A @ B ) @ D )
        = ( hAPP @ B @ D ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2370]) ).

thf(856,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ ( hAPP @ ( c_Set_Ovimage @ D @ C @ B ) @ A ) ) )
     => ( ( ( hAPP @ ( c_Set_Oimage @ D @ C @ B ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ D @ tc_HOL_Obool ) ) )
          = ( c_Orderings_Otop__class_Otop @ ( tc_fun @ C @ tc_HOL_Obool ) ) )
       => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_finite__vimageD) ).

thf(5491,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ ( hAPP @ ( c_Set_Ovimage @ D @ C @ B ) @ A ) ) )
     => ( ( ( hAPP @ ( c_Set_Oimage @ D @ C @ B ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ D @ tc_HOL_Obool ) ) )
          = ( c_Orderings_Otop__class_Otop @ ( tc_fun @ C @ tc_HOL_Obool ) ) )
       => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[856]) ).

thf(612,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) @ B ) ) @ A ) )
     => ( ( hAPP @ ( hAPP @ ( c_List_Otake @ C ) @ A ) @ B )
        = B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_take__all) ).

thf(4601,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) @ B ) ) @ A ) )
     => ( ( hAPP @ ( hAPP @ ( c_List_Otake @ C ) @ A ) @ B )
        = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[612]) ).

thf(193,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ B ) @ A ) )
       => ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Int_Oint ) @ A ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zdvd__not__zless) ).

thf(3106,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ B ) @ A ) )
       => ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Int_Oint ) @ A ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[193]) ).

thf(2369,axiom,
    ! [A: $i,B: $i] :
      ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ c_fconj @ B ) @ A ) )
      | ( hBOOL @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_c__fconj__2) ).

thf(11733,plain,
    ! [A: $i,B: $i] :
      ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ c_fconj @ B ) @ A ) )
      | ( hBOOL @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2369]) ).

thf(1704,axiom,
    ! [A: $i,B: $i] :
      ( ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ A )
        = ( hAPP @ c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) )
    <=> ( ( B
          = ( hAPP @ c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) )
        & ( A
          = ( hAPP @ c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__eq__1__iff) ).

thf(9118,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ A )
          = ( hAPP @ c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) )
       => ( ( B
            = ( hAPP @ c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) )
          & ( A
            = ( hAPP @ c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) ) )
      & ( ( ( B
            = ( hAPP @ c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) )
          & ( A
            = ( hAPP @ c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) )
       => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ A )
          = ( hAPP @ c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1704]) ).

thf(1911,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Transitive__Closure_Ortrancl @ B @ ( c_Transitive__Closure_Ortrancl @ B @ A ) )
      = ( c_Transitive__Closure_Ortrancl @ B @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_rtrancl__idemp) ).

thf(9920,plain,
    ! [A: $i,B: $i] :
      ( ( c_Transitive__Closure_Ortrancl @ B @ ( c_Transitive__Closure_Ortrancl @ B @ A ) )
      = ( c_Transitive__Closure_Ortrancl @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1911]) ).

thf(2393,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ D ) @ ( hAPP @ ( c_Set_Ovimage @ E @ C @ B ) @ A ) ) )
    <=> ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ ( hAPP @ B @ D ) ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_vimage__eq) ).

thf(11829,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ D ) @ ( hAPP @ ( c_Set_Ovimage @ E @ C @ B ) @ A ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ ( hAPP @ B @ D ) ) @ A ) ) )
      & ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ ( hAPP @ B @ D ) ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ D ) @ ( hAPP @ ( c_Set_Ovimage @ E @ C @ B ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2393]) ).

thf(2007,axiom,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ tc_Nat_Onat ) @ A ) )
    <=> ? [B: $i] :
        ! [C: $i] :
          ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Nat_Onat ) @ C ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ C ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_finite__nat__set__iff__bounded) ).

thf(10316,plain,
    ! [A: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ tc_Nat_Onat ) @ A ) )
       => ? [B: $i] :
          ! [C: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Nat_Onat ) @ C ) @ A ) )
           => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ C ) @ B ) ) ) )
      & ( ? [B: $i] :
          ! [C: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Nat_Onat ) @ C ) @ A ) )
           => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ C ) @ B ) ) )
       => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ tc_Nat_Onat ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2007]) ).

thf(1174,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( c_Fun_Oinj__on @ F @ E @ D @ C )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ F @ tc_HOL_Obool ) ) @ B ) @ C ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ F @ tc_HOL_Obool ) ) @ A ) @ C ) )
         => ( ( hAPP @ ( c_Set_Oimage @ F @ E @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ F @ tc_HOL_Obool ) ) @ B ) @ A ) )
            = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Set_Oimage @ F @ E @ D ) @ B ) ) @ ( hAPP @ ( c_Set_Oimage @ F @ E @ D ) @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_inj__on__image__set__diff) ).

thf(6832,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( c_Fun_Oinj__on @ F @ E @ D @ C )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ F @ tc_HOL_Obool ) ) @ B ) @ C ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ F @ tc_HOL_Obool ) ) @ A ) @ C ) )
         => ( ( hAPP @ ( c_Set_Oimage @ F @ E @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ F @ tc_HOL_Obool ) ) @ B ) @ A ) )
            = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Set_Oimage @ F @ E @ D ) @ B ) ) @ ( hAPP @ ( c_Set_Oimage @ F @ E @ D ) @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1174]) ).

thf(741,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Wellfounded_Owf @ C @ B )
     => ( c_Wellfounded_Owf @ C @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_wf__Int1) ).

thf(5054,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Wellfounded_Owf @ C @ B )
     => ( c_Wellfounded_Owf @ C @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) ) @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[741]) ).

thf(1295,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Ozip @ C @ B ) @ ( c_List_Olist_ONil @ C ) ) @ A )
      = ( c_List_Olist_ONil @ ( tc_prod @ C @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zip__Nil) ).

thf(7330,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Ozip @ C @ B ) @ ( c_List_Olist_ONil @ C ) ) @ A )
      = ( c_List_Olist_ONil @ ( tc_prod @ C @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1295]) ).

thf(667,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ B ) @ A ) )
      | ( B = A )
      | ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ A ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zless__linear) ).

thf(4805,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ B ) @ A ) )
      | ( B = A )
      | ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ A ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[667]) ).

thf(606,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ C ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ B ) ) @ A ) ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ B ) @ ( hAPP @ ( hAPP @ ( c_List_Onth @ B ) @ A ) @ C ) ) @ ( hAPP @ ( c_List_Oset @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nth__mem) ).

thf(4586,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ C ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ B ) ) @ A ) ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ B ) @ ( hAPP @ ( hAPP @ ( c_List_Onth @ B ) @ A ) @ C ) ) @ ( hAPP @ ( c_List_Oset @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[606]) ).

thf(1828,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ B ) @ ( c_Wellfounded_Oacc @ C @ A ) ) )
    <=> ! [D: $i] :
          ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ C @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ C @ C ) @ D ) @ B ) ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ D ) @ ( c_Wellfounded_Oacc @ C @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_acc_Osimps) ).

thf(9570,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ B ) @ ( c_Wellfounded_Oacc @ C @ A ) ) )
       => ! [D: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ C @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ C @ C ) @ D ) @ B ) ) @ A ) )
           => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ D ) @ ( c_Wellfounded_Oacc @ C @ A ) ) ) ) )
      & ( ! [D: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ C @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ C @ C ) @ D ) @ B ) ) @ A ) )
           => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ D ) @ ( c_Wellfounded_Oacc @ C @ A ) ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ B ) @ ( c_Wellfounded_Oacc @ C @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1828]) ).

thf(2279,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( c_Hoare__Mirabelle_Otriple_Otriple__rec @ F @ E @ D @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Hoare__Mirabelle_Otriple_Otriple @ F ) @ C ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( hAPP @ D @ C ) @ B ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_triple_Orecs) ).

thf(11338,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( c_Hoare__Mirabelle_Otriple_Otriple__rec @ F @ E @ D @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Hoare__Mirabelle_Otriple_Otriple @ F ) @ C ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( hAPP @ D @ C ) @ B ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2279]) ).

thf(366,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ D @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A )
        = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) )
    <=> ! [E: $i] :
          ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ E ) @ B ) )
         => ( ( hAPP @ A @ E )
            = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_UNION__empty__conv_I2_J) ).

thf(3710,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ D @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A )
          = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) )
       => ! [E: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ E ) @ B ) )
           => ( ( hAPP @ A @ E )
              = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) ) )
      & ( ! [E: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ E ) @ B ) )
           => ( ( hAPP @ A @ E )
              = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) )
       => ( ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ D @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A )
          = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[366]) ).

thf(830,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ E ) @ ( tc_List_Olist @ E ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ E ) @ ( tc_List_Olist @ E ) ) @ D ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ E ) @ C ) @ B ) ) ) @ ( c_List_Olistrel @ E @ A ) ) )
     => ~ ! [F: $i,G: $i] :
            ( ( D
              = ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ E ) @ F ) @ G ) )
           => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ F ) @ C ) ) @ A ) )
             => ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ E ) @ ( tc_List_Olist @ E ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ E ) @ ( tc_List_Olist @ E ) ) @ G ) @ B ) ) @ ( c_List_Olistrel @ E @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_listrel__Cons2) ).

thf(5407,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ E ) @ ( tc_List_Olist @ E ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ E ) @ ( tc_List_Olist @ E ) ) @ D ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ E ) @ C ) @ B ) ) ) @ ( c_List_Olistrel @ E @ A ) ) )
     => ~ ! [F: $i,G: $i] :
            ( ( D
              = ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ E ) @ F ) @ G ) )
           => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ F ) @ C ) ) @ A ) )
             => ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ E ) @ ( tc_List_Olist @ E ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ E ) @ ( tc_List_Olist @ E ) ) @ G ) @ B ) ) @ ( c_List_Olistrel @ E @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[830]) ).

thf(616,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( c_Set_OCollect @ ( tc_prod @ D @ C ) ) @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ D @ C @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ D ) @ ( c_COMBB @ tc_HOL_Obool @ tc_HOL_Obool @ C ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ D ) @ c_fconj ) @ B ) ) ) @ A ) ) )
      = ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ D @ C ) @ ( hAPP @ ( c_Set_OCollect @ D ) @ B ) ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ C @ tc_HOL_Obool ) @ D ) @ ( hAPP @ ( c_Set_OCollect @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Collect__split) ).

thf(4616,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( c_Set_OCollect @ ( tc_prod @ D @ C ) ) @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ D @ C @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ D ) @ ( c_COMBB @ tc_HOL_Obool @ tc_HOL_Obool @ C ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ D ) @ c_fconj ) @ B ) ) ) @ A ) ) )
      = ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ D @ C ) @ ( hAPP @ ( c_Set_OCollect @ D ) @ B ) ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ C @ tc_HOL_Obool ) @ D ) @ ( hAPP @ ( c_Set_OCollect @ C ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[616]) ).

thf(1713,axiom,
    ! [A: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ A )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_plus__nat_Oadd__0) ).

thf(9146,plain,
    ! [A: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ A )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1713]) ).

thf(1681,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Wellfounded_Owf @ C @ B )
     => ( c_Wellfounded_Owf @ C @ ( c_Wellfounded_Omlex__prod @ C @ A @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_wf__mlex) ).

thf(9015,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Wellfounded_Owf @ C @ B )
     => ( c_Wellfounded_Owf @ C @ ( c_Wellfounded_Omlex__prod @ C @ A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1681]) ).

thf(814,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ F @ E ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ F ) @ D ) @ C ) ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ E @ tc_HOL_Obool ) @ F ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ B ) @ A ) ) )
      = ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ ( tc_prod @ F @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ F @ E ) @ D ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ F @ E ) @ C ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ E @ tc_HOL_Obool ) @ F ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ B ) @ A ) ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ F @ E ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ F ) @ D ) @ C ) ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ E @ tc_HOL_Obool ) @ F ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_insert__times__insert) ).

thf(5346,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ F @ E ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ F ) @ D ) @ C ) ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ E @ tc_HOL_Obool ) @ F ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ B ) @ A ) ) )
      = ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ ( tc_prod @ F @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ F @ E ) @ D ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ F @ E ) @ C ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ E @ tc_HOL_Obool ) @ F ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ B ) @ A ) ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ F @ E ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ F ) @ D ) @ C ) ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ E @ tc_HOL_Obool ) @ F ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[814]) ).

thf(467,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ B ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Nat__Transfer_Otransfer__nat__int__function__closures_I2_J) ).

thf(4073,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ B ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[467]) ).

thf(1573,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Wellfounded_OwfP @ B @ A )
     => ( c_Wellfounded_Oacyclic @ B @ ( hAPP @ ( c_Set_OCollect @ ( tc_prod @ B @ B ) ) @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ B @ B @ tc_HOL_Obool ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_wfP__acyclicP) ).

thf(8485,plain,
    ! [A: $i,B: $i] :
      ( ( c_Wellfounded_OwfP @ B @ A )
     => ( c_Wellfounded_Oacyclic @ B @ ( hAPP @ ( c_Set_OCollect @ ( tc_prod @ B @ B ) ) @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ B @ B @ tc_HOL_Obool ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1573]) ).

thf(627,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( c_Set_Oimage @ tc_Nat_Onat @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ tc_Nat_Onat @ tc_Nat_Onat ) @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) ) @ C ) ) @ ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Nat_Onat @ B @ A ) )
      = ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_image__add__atLeastAtMost) ).

thf(4667,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( c_Set_Oimage @ tc_Nat_Onat @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ tc_Nat_Onat @ tc_Nat_Onat ) @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) ) @ C ) ) @ ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Nat_Onat @ B @ A ) )
      = ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[627]) ).

thf(1024,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Equiv__Relations_Oequiv @ E @ D @ C )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ B ) @ D ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ A ) @ D ) )
         => ( ( ( hAPP @ ( c_Relation_OImage @ E @ E @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ B ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ E @ tc_HOL_Obool ) ) ) )
              = ( hAPP @ ( c_Relation_OImage @ E @ E @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ E @ tc_HOL_Obool ) ) ) ) )
          <=> ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ B ) @ A ) ) @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_eq__equiv__class__iff) ).

thf(6154,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Equiv__Relations_Oequiv @ E @ D @ C )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ B ) @ D ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ A ) @ D ) )
         => ( ( ( ( hAPP @ ( c_Relation_OImage @ E @ E @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ B ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ E @ tc_HOL_Obool ) ) ) )
                = ( hAPP @ ( c_Relation_OImage @ E @ E @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ E @ tc_HOL_Obool ) ) ) ) )
             => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ B ) @ A ) ) @ C ) ) )
            & ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ B ) @ A ) ) @ C ) )
             => ( ( hAPP @ ( c_Relation_OImage @ E @ E @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ B ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ E @ tc_HOL_Obool ) ) ) )
                = ( hAPP @ ( c_Relation_OImage @ E @ E @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ E @ tc_HOL_Obool ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1024]) ).

thf(305,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ C ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zadd__strict__right__mono) ).

thf(3500,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ C ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[305]) ).

thf(202,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ E @ ( tc_Option_Ooption @ D ) ) @ C ) @ B ) @ ( hAPP @ ( c_Option_Ooption_OSome @ D ) @ A ) )
     != ( hAPP @ ( c_COMBK @ ( tc_Option_Ooption @ D ) @ E ) @ ( c_Option_Ooption_ONone @ D ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_map__upd__nonempty) ).

thf(3138,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ E @ ( tc_Option_Ooption @ D ) ) @ C ) @ B ) @ ( hAPP @ ( c_Option_Ooption_OSome @ D ) @ A ) )
     != ( hAPP @ ( c_COMBK @ ( tc_Option_Ooption @ D ) @ E ) @ ( c_Option_Ooption_ONone @ D ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[202]) ).

thf(547,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Transitive__Closure_Ortrancl @ B @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) ) @ A ) @ ( c_Relation_OId @ B ) ) )
      = ( c_Transitive__Closure_Ortrancl @ B @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_rtrancl__reflcl) ).

thf(4374,plain,
    ! [A: $i,B: $i] :
      ( ( c_Transitive__Closure_Ortrancl @ B @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) ) @ A ) @ ( c_Relation_OId @ B ) ) )
      = ( c_Transitive__Closure_Ortrancl @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[547]) ).

thf(1303,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C = B )
    <=> ( ( ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ A ) ) @ C )
          = ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ A ) ) @ B ) )
        & ! [D: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ A @ A ) ) @ D ) @ ( hAPP @ ( c_List_Oset @ ( tc_prod @ A @ A ) ) @ ( hAPP @ ( hAPP @ ( c_List_Ozip @ A @ A ) @ C ) @ B ) ) ) )
           => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ A @ A @ tc_HOL_Obool ) @ c_fequal ) @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_list__eq__iff__zip__eq) ).

thf(7362,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( C = B )
       => ( ( ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ A ) ) @ C )
            = ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ A ) ) @ B ) )
          & ! [D: $i] :
              ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ A @ A ) ) @ D ) @ ( hAPP @ ( c_List_Oset @ ( tc_prod @ A @ A ) ) @ ( hAPP @ ( hAPP @ ( c_List_Ozip @ A @ A ) @ C ) @ B ) ) ) )
             => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ A @ A @ tc_HOL_Obool ) @ c_fequal ) @ D ) ) ) ) )
      & ( ( ( ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ A ) ) @ C )
            = ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ A ) ) @ B ) )
          & ! [D: $i] :
              ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ A @ A ) ) @ D ) @ ( hAPP @ ( c_List_Oset @ ( tc_prod @ A @ A ) ) @ ( hAPP @ ( hAPP @ ( c_List_Ozip @ A @ A ) @ C ) @ B ) ) ) )
             => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ A @ A @ tc_HOL_Obool ) @ c_fequal ) @ D ) ) ) )
       => ( C = B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1303]) ).

thf(1146,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( hAPP @ ( c_Relation_OImage @ E @ E @ D ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ C ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ E @ tc_HOL_Obool ) ) ) )
        = ( hAPP @ ( c_Relation_OImage @ E @ E @ D ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ B ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ E @ tc_HOL_Obool ) ) ) ) )
     => ( ( c_Equiv__Relations_Oequiv @ E @ A @ D )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ B ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ C ) @ B ) ) @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_eq__equiv__class) ).

thf(6737,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( hAPP @ ( c_Relation_OImage @ E @ E @ D ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ C ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ E @ tc_HOL_Obool ) ) ) )
        = ( hAPP @ ( c_Relation_OImage @ E @ E @ D ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ B ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ E @ tc_HOL_Obool ) ) ) ) )
     => ( ( c_Equiv__Relations_Oequiv @ E @ A @ D )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ B ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ C ) @ B ) ) @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1146]) ).

thf(982,axiom,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( c_Finite__Set_Ocard @ B ) @ ( hAPP @ ( c_List_Oset @ B ) @ A ) ) ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ B ) ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_card__length) ).

thf(5972,plain,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( c_Finite__Set_Ocard @ B ) @ ( hAPP @ ( c_List_Oset @ B ) @ A ) ) ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ B ) ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[982]) ).

thf(2366,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Hoare__Mirabelle_Otriple__valid @ E @ D @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Hoare__Mirabelle_Otriple_Otriple @ E ) @ C ) @ B ) @ A ) )
    <=> ! [F: $i,G: $i] :
          ( ( hBOOL @ ( hAPP @ ( hAPP @ C @ F ) @ G ) )
         => ! [H: $i] :
              ( ( c_Natural_Oevaln @ B @ G @ D @ H )
             => ( hBOOL @ ( hAPP @ ( hAPP @ A @ F ) @ H ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_triple__valid__def2) ).

thf(11718,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( c_Hoare__Mirabelle_Otriple__valid @ E @ D @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Hoare__Mirabelle_Otriple_Otriple @ E ) @ C ) @ B ) @ A ) )
       => ! [F: $i,G: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ C @ F ) @ G ) )
           => ! [H: $i] :
                ( ( c_Natural_Oevaln @ B @ G @ D @ H )
               => ( hBOOL @ ( hAPP @ ( hAPP @ A @ F ) @ H ) ) ) ) )
      & ( ! [F: $i,G: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ C @ F ) @ G ) )
           => ! [H: $i] :
                ( ( c_Natural_Oevaln @ B @ G @ D @ H )
               => ( hBOOL @ ( hAPP @ ( hAPP @ A @ F ) @ H ) ) ) )
       => ( c_Hoare__Mirabelle_Otriple__valid @ E @ D @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Hoare__Mirabelle_Otriple_Otriple @ E ) @ C ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2366]) ).

thf(1806,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ B ) ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Suc__leI) ).

thf(9491,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ B ) ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1806]) ).

thf(2406,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ C @ C ) ) @ B ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ C @ C ) ) @ B ) @ ( c_Transitive__Closure_Otrancl @ C @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_r__into__trancl_H) ).

thf(11883,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ C @ C ) ) @ B ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ C @ C ) ) @ B ) @ ( c_Transitive__Closure_Otrancl @ C @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2406]) ).

thf(131,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Fun_Oinj__on @ E @ D @ C @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ B ) @ A ) )
     => ( ( ( hAPP @ ( c_Set_Oimage @ E @ D @ C ) @ B )
          = ( hAPP @ ( c_Set_Oimage @ E @ D @ C ) @ A ) )
      <=> ( B = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_inj__on__Un__image__eq__iff) ).

thf(2871,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Fun_Oinj__on @ E @ D @ C @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ B ) @ A ) )
     => ( ( ( ( hAPP @ ( c_Set_Oimage @ E @ D @ C ) @ B )
            = ( hAPP @ ( c_Set_Oimage @ E @ D @ C ) @ A ) )
         => ( B = A ) )
        & ( ( B = A )
         => ( ( hAPP @ ( c_Set_Oimage @ E @ D @ C ) @ B )
            = ( hAPP @ ( c_Set_Oimage @ E @ D @ C ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[131]) ).

thf(585,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Divides_Odivmod__int__rel @ B @ A )
      = ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ tc_Int_Oint @ tc_Int_Oint @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ tc_Int_Oint @ ( tc_fun @ tc_Int_Oint @ tc_HOL_Obool ) @ ( tc_fun @ tc_Int_Oint @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ tc_Int_Oint @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ tc_Int_Oint @ tc_HOL_Obool ) @ ( tc_fun @ tc_Int_Oint @ tc_HOL_Obool ) ) @ tc_Int_Oint ) @ ( c_COMBS @ tc_Int_Oint @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ tc_Int_Oint @ tc_HOL_Obool ) @ ( tc_fun @ tc_Int_Oint @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ tc_Int_Oint ) @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ tc_Int_Oint ) @ c_fconj ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ tc_Int_Oint @ tc_Int_Oint ) @ ( tc_fun @ tc_Int_Oint @ tc_HOL_Obool ) @ tc_Int_Oint ) @ ( hAPP @ ( c_COMBB @ tc_Int_Oint @ tc_HOL_Obool @ tc_Int_Oint ) @ ( hAPP @ c_fequal @ B ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_Int_Oint @ ( tc_fun @ tc_Int_Oint @ tc_Int_Oint ) @ tc_Int_Oint ) @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) ) @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ A ) ) ) ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ tc_Int_Oint @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ tc_Int_Oint ) @ c_fconj ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ tc_HOL_Obool @ tc_Int_Oint ) @ ( hAPP @ c_fimplies @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ A ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ tc_Int_Oint @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ tc_Int_Oint ) @ c_fconj ) @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ tc_Int_Oint @ tc_Int_Oint @ tc_HOL_Obool ) @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) ) @ A ) ) ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ tc_HOL_Obool @ tc_Int_Oint ) @ ( hAPP @ c_fimplies @ ( hAPP @ c_fNot @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ A ) ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ tc_Int_Oint @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ tc_Int_Oint ) @ c_fconj ) @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ A ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ tc_Int_Oint @ tc_Int_Oint @ tc_HOL_Obool ) @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_divmod__int__rel__def) ).

thf(4509,plain,
    ! [A: $i,B: $i] :
      ( ( c_Divides_Odivmod__int__rel @ B @ A )
      = ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ tc_Int_Oint @ tc_Int_Oint @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ tc_Int_Oint @ ( tc_fun @ tc_Int_Oint @ tc_HOL_Obool ) @ ( tc_fun @ tc_Int_Oint @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ tc_Int_Oint @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ tc_Int_Oint @ tc_HOL_Obool ) @ ( tc_fun @ tc_Int_Oint @ tc_HOL_Obool ) ) @ tc_Int_Oint ) @ ( c_COMBS @ tc_Int_Oint @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ tc_Int_Oint @ tc_HOL_Obool ) @ ( tc_fun @ tc_Int_Oint @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ tc_Int_Oint ) @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ tc_Int_Oint ) @ c_fconj ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ tc_Int_Oint @ tc_Int_Oint ) @ ( tc_fun @ tc_Int_Oint @ tc_HOL_Obool ) @ tc_Int_Oint ) @ ( hAPP @ ( c_COMBB @ tc_Int_Oint @ tc_HOL_Obool @ tc_Int_Oint ) @ ( hAPP @ c_fequal @ B ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_Int_Oint @ ( tc_fun @ tc_Int_Oint @ tc_Int_Oint ) @ tc_Int_Oint ) @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) ) @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ A ) ) ) ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ tc_Int_Oint @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ tc_Int_Oint ) @ c_fconj ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ tc_HOL_Obool @ tc_Int_Oint ) @ ( hAPP @ c_fimplies @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ A ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ tc_Int_Oint @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ tc_Int_Oint ) @ c_fconj ) @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ tc_Int_Oint @ tc_Int_Oint @ tc_HOL_Obool ) @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) ) @ A ) ) ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ tc_HOL_Obool @ tc_Int_Oint ) @ ( hAPP @ c_fimplies @ ( hAPP @ c_fNot @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ A ) ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ tc_Int_Oint @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ tc_Int_Oint ) @ c_fconj ) @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ A ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ tc_Int_Oint @ tc_Int_Oint @ tc_HOL_Obool ) @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[585]) ).

thf(1902,axiom,
    ! [A: $i,B: $i] :
      ( ( B = A )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_eq__imp__le) ).

thf(9890,plain,
    ! [A: $i,B: $i] :
      ( ( B = A )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1902]) ).

thf(2272,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_Natural_Oevaln @ ( c_Com_Ocom_OWhile @ D @ C ) @ B @ A @ B )
    <=> ( ( hBOOL @ ( hAPP @ D @ B ) )
       => ? [E: $i] :
            ( ( c_Natural_Oevaln @ C @ B @ A @ E )
            & ( c_Natural_Oevaln @ ( c_Com_Ocom_OWhile @ D @ C ) @ E @ A @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_evaln_Oequations_I7_J) ).

thf(11306,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( c_Natural_Oevaln @ ( c_Com_Ocom_OWhile @ D @ C ) @ B @ A @ B )
       => ( ( hBOOL @ ( hAPP @ D @ B ) )
         => ? [E: $i] :
              ( ( c_Natural_Oevaln @ C @ B @ A @ E )
              & ( c_Natural_Oevaln @ ( c_Com_Ocom_OWhile @ D @ C ) @ E @ A @ B ) ) ) )
      & ( ( ( hBOOL @ ( hAPP @ D @ B ) )
         => ? [E: $i] :
              ( ( c_Natural_Oevaln @ C @ B @ A @ E )
              & ( c_Natural_Oevaln @ ( c_Com_Ocom_OWhile @ D @ C ) @ E @ A @ B ) ) )
       => ( c_Natural_Oevaln @ ( c_Com_Ocom_OWhile @ D @ C ) @ B @ A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2272]) ).

thf(1289,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i] :
          ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ D ) @ C ) )
         => ( ( hBOOL @ ( hAPP @ B @ D ) )
           => ( hBOOL @ ( hAPP @ B @ ( c_Divides_Odiv__class_Omod @ tc_Nat_Onat @ ( hAPP @ c_Nat_OSuc @ D ) @ C ) ) ) ) )
     => ( ( hBOOL @ ( hAPP @ B @ A ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ A ) @ C ) )
         => ( hBOOL @ ( hAPP @ B @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mod__induct__0) ).

thf(7307,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i] :
          ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ D ) @ C ) )
         => ( ( hBOOL @ ( hAPP @ B @ D ) )
           => ( hBOOL @ ( hAPP @ B @ ( c_Divides_Odiv__class_Omod @ tc_Nat_Onat @ ( hAPP @ c_Nat_OSuc @ D ) @ C ) ) ) ) )
     => ( ( hBOOL @ ( hAPP @ B @ A ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ A ) @ C ) )
         => ( hBOOL @ ( hAPP @ B @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1289]) ).

thf(478,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Int_Oint ) @ C ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zdiff__zmult__distrib) ).

thf(4107,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Int_Oint ) @ C ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[478]) ).

thf(1722,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_Equiv__Relations_Oequiv @ D @ C @ B )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ A ) @ C ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ A ) @ ( hAPP @ ( c_Relation_OImage @ D @ D @ B ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_equiv__class__self) ).

thf(9174,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_Equiv__Relations_Oequiv @ D @ C @ B )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ A ) @ C ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ A ) @ ( hAPP @ ( c_Relation_OImage @ D @ D @ B ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1722]) ).

thf(454,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ F @ ( tc_fun @ E @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ F @ tc_HOL_Obool ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ ( tc_prod @ E @ D ) @ tc_HOL_Obool ) @ ( tc_fun @ E @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ F ) @ ( hAPP @ ( c_COMBC @ E @ ( tc_fun @ ( tc_prod @ E @ D ) @ tc_HOL_Obool ) @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ D @ ( tc_fun @ ( tc_fun @ ( tc_prod @ E @ D ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ E @ D ) @ tc_HOL_Obool ) @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ E ) @ ( c_COMBC @ D @ ( tc_fun @ ( tc_prod @ E @ D ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ D @ ( tc_prod @ E @ D ) ) @ ( tc_fun @ D @ ( tc_fun @ ( tc_fun @ ( tc_prod @ E @ D ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ E ) @ ( hAPP @ ( c_COMBB @ ( tc_prod @ E @ D ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ E @ D ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ D ) @ ( c_member @ ( tc_prod @ E @ D ) ) ) ) @ ( c_Product__Type_OPair @ E @ D ) ) ) ) ) @ C ) ) @ B ) @ A ) )
    <=> ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ D ) @ B ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ F @ ( tc_fun @ ( tc_prod @ E @ D ) @ tc_HOL_Obool ) ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ F @ tc_HOL_Obool ) ) ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_SUP__UN__eq2) ).

thf(4028,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ F @ ( tc_fun @ E @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ F @ tc_HOL_Obool ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ ( tc_prod @ E @ D ) @ tc_HOL_Obool ) @ ( tc_fun @ E @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ F ) @ ( hAPP @ ( c_COMBC @ E @ ( tc_fun @ ( tc_prod @ E @ D ) @ tc_HOL_Obool ) @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ D @ ( tc_fun @ ( tc_fun @ ( tc_prod @ E @ D ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ E @ D ) @ tc_HOL_Obool ) @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ E ) @ ( c_COMBC @ D @ ( tc_fun @ ( tc_prod @ E @ D ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ D @ ( tc_prod @ E @ D ) ) @ ( tc_fun @ D @ ( tc_fun @ ( tc_fun @ ( tc_prod @ E @ D ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ E ) @ ( hAPP @ ( c_COMBB @ ( tc_prod @ E @ D ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ E @ D ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ D ) @ ( c_member @ ( tc_prod @ E @ D ) ) ) ) @ ( c_Product__Type_OPair @ E @ D ) ) ) ) ) @ C ) ) @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ D ) @ B ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ F @ ( tc_fun @ ( tc_prod @ E @ D ) @ tc_HOL_Obool ) ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ F @ tc_HOL_Obool ) ) ) @ C ) ) ) )
      & ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ D ) @ B ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ F @ ( tc_fun @ ( tc_prod @ E @ D ) @ tc_HOL_Obool ) ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ F @ tc_HOL_Obool ) ) ) @ C ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ F @ ( tc_fun @ E @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ F @ tc_HOL_Obool ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ ( tc_prod @ E @ D ) @ tc_HOL_Obool ) @ ( tc_fun @ E @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ F ) @ ( hAPP @ ( c_COMBC @ E @ ( tc_fun @ ( tc_prod @ E @ D ) @ tc_HOL_Obool ) @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ D @ ( tc_fun @ ( tc_fun @ ( tc_prod @ E @ D ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ E @ D ) @ tc_HOL_Obool ) @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ E ) @ ( c_COMBC @ D @ ( tc_fun @ ( tc_prod @ E @ D ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ D @ ( tc_prod @ E @ D ) ) @ ( tc_fun @ D @ ( tc_fun @ ( tc_fun @ ( tc_prod @ E @ D ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ E ) @ ( hAPP @ ( c_COMBB @ ( tc_prod @ E @ D ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ E @ D ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ D ) @ ( c_member @ ( tc_prod @ E @ D ) ) ) ) @ ( c_Product__Type_OPair @ E @ D ) ) ) ) ) @ C ) ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[454]) ).

thf(1532,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) ) @ B ) @ A ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) ) @ A ) @ ( c_Transitive__Closure_Ortrancl @ C @ B ) ) )
       => ( ( c_Transitive__Closure_Ortrancl @ C @ A )
          = ( c_Transitive__Closure_Ortrancl @ C @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_rtrancl__subset) ).

thf(8340,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) ) @ B ) @ A ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) ) @ A ) @ ( c_Transitive__Closure_Ortrancl @ C @ B ) ) )
       => ( ( c_Transitive__Closure_Ortrancl @ C @ A )
          = ( c_Transitive__Closure_Ortrancl @ C @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1532]) ).

thf(411,axiom,
    ! [A: $i,B: $i] :
      ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ B ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_emptyE) ).

thf(3893,plain,
    ! [A: $i,B: $i] :
      ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ B ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[411]) ).

thf(881,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Map_Orestrict__map @ E @ D @ ( c_Map_Orestrict__map @ E @ D @ C @ B ) @ A )
      = ( c_Map_Orestrict__map @ E @ D @ C @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_restrict__restrict) ).

thf(5563,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Map_Orestrict__map @ E @ D @ ( c_Map_Orestrict__map @ E @ D @ C @ B ) @ A )
      = ( c_Map_Orestrict__map @ E @ D @ C @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[881]) ).

thf(1754,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Natural_Oevalc @ ( c_Com_Ocom_OAss @ D @ C ) ) @ B ) @ A ) )
     => ( A
        = ( hAPP @ ( hAPP @ ( hAPP @ c_Natural_Oupdate @ B ) @ D ) @ ( hAPP @ C @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_evalc__elim__cases_I2_J) ).

thf(9302,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Natural_Oevalc @ ( c_Com_Ocom_OAss @ D @ C ) ) @ B ) @ A ) )
     => ( A
        = ( hAPP @ ( hAPP @ ( hAPP @ c_Natural_Oupdate @ B ) @ D ) @ ( hAPP @ C @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1754]) ).

thf(1329,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ C ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ B ) @ A ) )
       => ( ? [E: $i] :
              ( ( c_Fun_Oinj__on @ D @ B @ E @ C )
              & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Set_Oimage @ D @ B @ E ) @ C ) ) @ A ) ) )
        <=> ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( c_Finite__Set_Ocard @ D ) @ C ) ) @ ( hAPP @ ( c_Finite__Set_Ocard @ B ) @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_inj__on__iff__card__le) ).

thf(7504,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ C ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ B ) @ A ) )
       => ( ( ? [E: $i] :
                ( ( c_Fun_Oinj__on @ D @ B @ E @ C )
                & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Set_Oimage @ D @ B @ E ) @ C ) ) @ A ) ) )
           => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( c_Finite__Set_Ocard @ D ) @ C ) ) @ ( hAPP @ ( c_Finite__Set_Ocard @ B ) @ A ) ) ) )
          & ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( c_Finite__Set_Ocard @ D ) @ C ) ) @ ( hAPP @ ( c_Finite__Set_Ocard @ B ) @ A ) ) )
           => ? [E: $i] :
                ( ( c_Fun_Oinj__on @ D @ B @ E @ C )
                & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Set_Oimage @ D @ B @ E ) @ C ) ) @ A ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1329]) ).

thf(822,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) @ C ) @ B ) ) @ ( c_List_Olistrel @ D @ A ) ) )
    <=> ( ( ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ D ) ) @ C )
          = ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ D ) ) @ B ) )
        & ! [E: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ E ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ D ) ) @ C ) ) )
           => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ D ) @ ( hAPP @ ( hAPP @ ( c_List_Onth @ D ) @ C ) @ E ) ) @ ( hAPP @ ( hAPP @ ( c_List_Onth @ D ) @ B ) @ E ) ) ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_listrel__iff__nth) ).

thf(5367,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) @ C ) @ B ) ) @ ( c_List_Olistrel @ D @ A ) ) )
       => ( ( ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ D ) ) @ C )
            = ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ D ) ) @ B ) )
          & ! [E: $i] :
              ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ E ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ D ) ) @ C ) ) )
             => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ D ) @ ( hAPP @ ( hAPP @ ( c_List_Onth @ D ) @ C ) @ E ) ) @ ( hAPP @ ( hAPP @ ( c_List_Onth @ D ) @ B ) @ E ) ) ) @ A ) ) ) ) )
      & ( ( ( ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ D ) ) @ C )
            = ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ D ) ) @ B ) )
          & ! [E: $i] :
              ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ E ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ D ) ) @ C ) ) )
             => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ D ) @ ( hAPP @ ( hAPP @ ( c_List_Onth @ D ) @ C ) @ E ) ) @ ( hAPP @ ( hAPP @ ( c_List_Onth @ D ) @ B ) @ E ) ) ) @ A ) ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) @ C ) @ B ) ) @ ( c_List_Olistrel @ D @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[822]) ).

thf(1611,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ C ) @ B ) )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ B ) @ C ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ C ) ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_diff__diff__right) ).

thf(8735,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ C ) @ B ) )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ B ) @ C ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ C ) ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1611]) ).

thf(1006,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hAPP @ ( c_Set_Oimage @ F @ ( tc_prod @ E @ D ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ F @ D @ ( tc_prod @ E @ D ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ E @ ( tc_fun @ D @ ( tc_prod @ E @ D ) ) @ F ) @ ( c_Product__Type_OPair @ E @ D ) ) @ C ) ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ E @ D ) @ ( hAPP @ ( c_Set_Oimage @ F @ E @ C ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ F @ tc_HOL_Obool ) @ ( tc_fun @ D @ tc_HOL_Obool ) @ E ) @ ( c_Set_Oimage @ F @ D @ B ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ E @ ( tc_fun @ F @ tc_HOL_Obool ) @ ( tc_fun @ F @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ F @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ F @ tc_HOL_Obool ) @ ( tc_fun @ F @ tc_HOL_Obool ) ) @ E ) @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ F @ tc_HOL_Obool ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ E @ tc_HOL_Obool ) @ ( tc_fun @ F @ tc_HOL_Obool ) @ E ) @ ( c_Set_Ovimage @ F @ E @ C ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ E @ ( tc_fun @ E @ tc_HOL_Obool ) @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( c_Set_Oinsert @ E ) ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ E @ tc_HOL_Obool ) ) ) ) ) ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_image__split__eq__Sigma) ).

thf(6087,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hAPP @ ( c_Set_Oimage @ F @ ( tc_prod @ E @ D ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ F @ D @ ( tc_prod @ E @ D ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ E @ ( tc_fun @ D @ ( tc_prod @ E @ D ) ) @ F ) @ ( c_Product__Type_OPair @ E @ D ) ) @ C ) ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ E @ D ) @ ( hAPP @ ( c_Set_Oimage @ F @ E @ C ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ F @ tc_HOL_Obool ) @ ( tc_fun @ D @ tc_HOL_Obool ) @ E ) @ ( c_Set_Oimage @ F @ D @ B ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ E @ ( tc_fun @ F @ tc_HOL_Obool ) @ ( tc_fun @ F @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ F @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ F @ tc_HOL_Obool ) @ ( tc_fun @ F @ tc_HOL_Obool ) ) @ E ) @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ F @ tc_HOL_Obool ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ E @ tc_HOL_Obool ) @ ( tc_fun @ F @ tc_HOL_Obool ) @ E ) @ ( c_Set_Ovimage @ F @ E @ C ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ E @ ( tc_fun @ E @ tc_HOL_Obool ) @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( c_Set_Oinsert @ E ) ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ E @ tc_HOL_Obool ) ) ) ) ) ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1006]) ).

thf(1584,axiom,
    ! [A: $i] :
      ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__nat__zero__code) ).

thf(8605,plain,
    ! [A: $i] :
      ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1584]) ).

thf(1706,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Nat_Onat ) @ B ) @ ( hAPP @ ( c_List_Oset @ tc_Nat_Onat ) @ A ) ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ ( hAPP @ ( c_List_Ofoldl @ tc_Nat_Onat @ tc_Nat_Onat @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_elem__le__sum) ).

thf(9129,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Nat_Onat ) @ B ) @ ( hAPP @ ( c_List_Oset @ tc_Nat_Onat ) @ A ) ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ ( hAPP @ ( c_List_Ofoldl @ tc_Nat_Onat @ tc_Nat_Onat @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1706]) ).

thf(2402,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ B @ ( c_Hilbert__Choice_OEps @ A @ B ) ) )
    <=> ? [C: $i] : ( hBOOL @ ( hAPP @ B @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_some__eq__ex) ).

thf(11868,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ B @ ( c_Hilbert__Choice_OEps @ A @ B ) ) )
       => ? [C: $i] : ( hBOOL @ ( hAPP @ B @ C ) ) )
      & ( ? [C: $i] : ( hBOOL @ ( hAPP @ B @ C ) )
       => ( hBOOL @ ( hAPP @ B @ ( c_Hilbert__Choice_OEps @ A @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2402]) ).

thf(1397,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( c_Fun_Oinj__on @ F @ E @ D @ C )
     => ( ( c_Fun_Oinj__on @ E @ B @ A @ ( hAPP @ ( c_Set_Oimage @ F @ E @ D ) @ C ) )
       => ( c_Fun_Oinj__on @ F @ B @ ( hAPP @ ( c_Fun_Ocomp @ E @ B @ F @ A ) @ D ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_comp__inj__on) ).

thf(7768,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( c_Fun_Oinj__on @ F @ E @ D @ C )
     => ( ( c_Fun_Oinj__on @ E @ B @ A @ ( hAPP @ ( c_Set_Oimage @ F @ E @ D ) @ C ) )
       => ( c_Fun_Oinj__on @ F @ B @ ( hAPP @ ( c_Fun_Ocomp @ E @ B @ F @ A ) @ D ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1397]) ).

thf(375,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ tc_Nat_Onat @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ D ) ) @ B ) ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ tc_Nat_Onat @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) ) @ B ) ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_UN__finite__subset) ).

thf(3747,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ tc_Nat_Onat @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ D ) ) @ B ) ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ tc_Nat_Onat @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) ) @ B ) ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[375]) ).

thf(1669,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ F ) ) @ E )
        = ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ D ) ) @ C ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ F @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_List_Oset @ F ) @ E ) ) @ B ) )
       => ( ( c_Map_Orestrict__map @ F @ D @ ( c_Map_Omap__upds @ F @ D @ A @ E @ C ) @ B )
          = ( c_Map_Omap__upds @ F @ D @ ( c_Map_Orestrict__map @ F @ D @ A @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ F @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( c_List_Oset @ F ) @ E ) ) ) @ E @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_restrict__map__upds) ).

thf(8958,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ F ) ) @ E )
        = ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ D ) ) @ C ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ F @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_List_Oset @ F ) @ E ) ) @ B ) )
       => ( ( c_Map_Orestrict__map @ F @ D @ ( c_Map_Omap__upds @ F @ D @ A @ E @ C ) @ B )
          = ( c_Map_Omap__upds @ F @ D @ ( c_Map_Orestrict__map @ F @ D @ A @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ F @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( c_List_Oset @ F ) @ E ) ) ) @ E @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1669]) ).

thf(2355,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( c_New__DSequence_Opos__empty @ B ) @ A )
      = ( c_Lazy__Sequence_Oempty @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_New__DSequence_Opos__empty__def) ).

thf(11669,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( c_New__DSequence_Opos__empty @ B ) @ A )
      = ( c_Lazy__Sequence_Oempty @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2355]) ).

thf(66,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Groups_Ocomm__monoid__add @ E )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ C ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ C ) )
         => ( ! [F: $i] :
                ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ F ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) ) )
               => ( ( hAPP @ A @ F )
                  = ( c_Groups_Ozero__class_Ozero @ E ) ) )
           => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ A ) @ B )
              = ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ A ) @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_setsum__mono__zero__left) ).

thf(2632,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Groups_Ocomm__monoid__add @ E )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ C ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ C ) )
         => ( ! [F: $i] :
                ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ F ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) ) )
               => ( ( hAPP @ A @ F )
                  = ( c_Groups_Ozero__class_Ozero @ E ) ) )
           => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ A ) @ B )
              = ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ A ) @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[66]) ).

thf(2031,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ C ) @ B ) @ A )
     != A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_not__Cons__self2) ).

thf(10397,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ C ) @ B ) @ A )
     != A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2031]) ).

thf(2025,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ D ) @ C ) @ B ) ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ D ) @ C ) @ B ) ) @ ( c_Transitive__Closure_Otrancl @ D @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_trancl_Or__into__trancl) ).

thf(10375,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ D ) @ C ) @ B ) ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ D ) @ C ) @ B ) ) @ ( c_Transitive__Closure_Otrancl @ D @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2025]) ).

thf(447,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hAPP @ ( hAPP @ ( hAPP @ ( c_COMBS @ F @ E @ D ) @ C ) @ B ) @ A )
      = ( hAPP @ ( hAPP @ C @ A ) @ ( hAPP @ B @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_c__COMBS__1) ).

thf(4009,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hAPP @ ( hAPP @ ( hAPP @ ( c_COMBS @ F @ E @ D ) @ C ) @ B ) @ A )
      = ( hAPP @ ( hAPP @ C @ A ) @ ( hAPP @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[447]) ).

thf(928,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofold1Set @ D @ C @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ B ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) ) @ A ) )
    <=> ( B = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_fold1Set__sing) ).

thf(5730,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofold1Set @ D @ C @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ B ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) ) @ A ) )
       => ( B = A ) )
      & ( ( B = A )
       => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofold1Set @ D @ C @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ B ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[928]) ).

thf(678,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ B ) )
     => ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ C ) @ A ) ) @ B )
        = ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ A ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Int__insert__left__if1) ).

thf(4833,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ B ) )
     => ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ C ) @ A ) ) @ B )
        = ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ A ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[678]) ).

thf(965,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Lattices_Oab__semigroup__idem__mult @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ B ) )
       => ( ( B
           != ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) )
         => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ A ) )
           => ( ( A
               != ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) )
             => ( ( hAPP @ ( c_Finite__Set_Ofold1 @ C @ ( c_Groups_Otimes__class_Otimes @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) )
                = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ ( hAPP @ ( c_Finite__Set_Ofold1 @ C @ ( c_Groups_Otimes__class_Otimes @ C ) ) @ B ) ) @ ( hAPP @ ( c_Finite__Set_Ofold1 @ C @ ( c_Groups_Otimes__class_Otimes @ C ) ) @ A ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_fold1__Un2) ).

thf(5894,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Lattices_Oab__semigroup__idem__mult @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ B ) )
       => ( ( B
           != ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) )
         => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ A ) )
           => ( ( A
               != ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) )
             => ( ( hAPP @ ( c_Finite__Set_Ofold1 @ C @ ( c_Groups_Otimes__class_Otimes @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) )
                = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ ( hAPP @ ( c_Finite__Set_Ofold1 @ C @ ( c_Groups_Otimes__class_Otimes @ C ) ) @ B ) ) @ ( hAPP @ ( c_Finite__Set_Ofold1 @ C @ ( c_Groups_Otimes__class_Otimes @ C ) ) @ A ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[965]) ).

thf(2386,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( c_Hoare__Mirabelle_Otriple_Otriple__case @ F @ E @ D @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Hoare__Mirabelle_Otriple_Otriple @ F ) @ C ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( hAPP @ D @ C ) @ B ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_triple_Osimps_I2_J) ).

thf(11796,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( c_Hoare__Mirabelle_Otriple_Otriple__case @ F @ E @ D @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Hoare__Mirabelle_Otriple_Otriple @ F ) @ C ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( hAPP @ D @ C ) @ B ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2386]) ).

thf(2196,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ D ) @ C ) ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ C ) @ A ) ) @ B ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ D ) @ A ) ) @ ( c_Transitive__Closure_Otrancl @ E @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_r__r__into__trancl) ).

thf(10995,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ D ) @ C ) ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ C ) @ A ) ) @ B ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ D ) @ A ) ) @ ( c_Transitive__Closure_Otrancl @ E @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2196]) ).

thf(602,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Fun_Oinj__on @ E @ D @ C @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ E @ tc_HOL_Obool ) ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ ( hAPP @ C @ B ) ) @ ( hAPP @ ( c_Set_Oimage @ E @ D @ C ) @ A ) ) )
      <=> ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_inj__image__mem__iff) ).

thf(4569,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Fun_Oinj__on @ E @ D @ C @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ E @ tc_HOL_Obool ) ) )
     => ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ ( hAPP @ C @ B ) ) @ ( hAPP @ ( c_Set_Oimage @ E @ D @ C ) @ A ) ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ B ) @ A ) ) )
        & ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ B ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ ( hAPP @ C @ B ) ) @ ( hAPP @ ( c_Set_Oimage @ E @ D @ C ) @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[602]) ).

thf(1693,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Otake @ C ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ C ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ C ) @ B ) @ ( c_List_Olist_ONil @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_take__1__Cons) ).

thf(9079,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Otake @ C ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ C ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ C ) @ B ) @ ( c_List_Olist_ONil @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1693]) ).

thf(464,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( c_Relation_ORange @ D @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Relation_ORange @ D @ C ) @ B ) ) @ ( hAPP @ ( c_Relation_ORange @ D @ C ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Range__Un__eq) ).

thf(4064,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( c_Relation_ORange @ D @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Relation_ORange @ D @ C ) @ B ) ) @ ( hAPP @ ( c_Relation_ORange @ D @ C ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[464]) ).

thf(1889,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( c_Com_Ocom_OSemi @ D @ C )
        = ( c_Com_Ocom_OSemi @ B @ A ) )
    <=> ( ( D = B )
        & ( C = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_com_Osimps_I3_J) ).

thf(9830,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( ( c_Com_Ocom_OSemi @ D @ C )
          = ( c_Com_Ocom_OSemi @ B @ A ) )
       => ( ( D = B )
          & ( C = A ) ) )
      & ( ( ( D = B )
          & ( C = A ) )
       => ( ( c_Com_Ocom_OSemi @ D @ C )
          = ( c_Com_Ocom_OSemi @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1889]) ).

thf(1343,axiom,
    ! [A: $i] :
      ( ( hAPP @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ A )
      = ( hAPP @ ( c_SetInterval_Oord__class_OlessThan @ tc_Nat_Onat ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_atLeast0LessThan) ).

thf(7566,plain,
    ! [A: $i] :
      ( ( hAPP @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ A )
      = ( hAPP @ ( c_SetInterval_Oord__class_OlessThan @ tc_Nat_Onat ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1343]) ).

thf(672,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Hilbert__Choice_OEps @ ( tc_prod @ C @ B ) @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ C @ B @ tc_HOL_Obool ) @ A ) )
      = ( c_Hilbert__Choice_OEps @ ( tc_prod @ C @ B ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ ( tc_prod @ C @ B ) @ B @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ C @ ( tc_fun @ B @ tc_HOL_Obool ) @ ( tc_prod @ C @ B ) ) @ A ) @ ( c_Product__Type_Ofst @ C @ B ) ) ) @ ( c_Product__Type_Osnd @ C @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Eps__split) ).

thf(4818,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Hilbert__Choice_OEps @ ( tc_prod @ C @ B ) @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ C @ B @ tc_HOL_Obool ) @ A ) )
      = ( c_Hilbert__Choice_OEps @ ( tc_prod @ C @ B ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ ( tc_prod @ C @ B ) @ B @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ C @ ( tc_fun @ B @ tc_HOL_Obool ) @ ( tc_prod @ C @ B ) ) @ A ) @ ( c_Product__Type_Ofst @ C @ B ) ) ) @ ( c_Product__Type_Osnd @ C @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[672]) ).

thf(2295,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Divides_Odivmod__nat__rel @ C @ B ) @ A ) )
     => ( ( c_Divides_Odivmod__nat @ C @ B )
        = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_divmod__nat__eq) ).

thf(11392,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Divides_Odivmod__nat__rel @ C @ B ) @ A ) )
     => ( ( c_Divides_Odivmod__nat @ C @ B )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2295]) ).

thf(1845,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ A ) )
      | ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ A ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nat__le__linear) ).

thf(9642,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ A ) )
      | ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ A ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1845]) ).

thf(1855,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ A ) )
    <=> ? [C: $i] :
          ( A
          = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_le__iff__add) ).

thf(9691,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ A ) )
       => ? [C: $i] :
            ( A
            = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ C ) ) )
      & ( ? [C: $i] :
            ( A
            = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ C ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1855]) ).

thf(839,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_List_Oset @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) )
      = ( hAPP @ ( c_List_Oset @ C ) @ ( hAPP @ ( c_List_Ofilter @ C @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ tc_HOL_Obool @ C ) @ c_fNot ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ C @ C @ tc_HOL_Obool ) @ c_fequal ) @ A ) ) ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_set__minus__filter__out) ).

thf(5439,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_List_Oset @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) )
      = ( hAPP @ ( c_List_Oset @ C ) @ ( hAPP @ ( c_List_Ofilter @ C @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ tc_HOL_Obool @ C ) @ c_fNot ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ C @ C @ tc_HOL_Obool ) @ c_fequal ) @ A ) ) ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[839]) ).

thf(1411,axiom,
    ! [A: $i,B: $i] : ( c_List_Olistrelp @ B @ A @ ( c_List_Olist_ONil @ B ) @ ( c_List_Olist_ONil @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_listrelp_Oequations_I1_J) ).

thf(7820,plain,
    ! [A: $i,B: $i] : ( c_List_Olistrelp @ B @ A @ ( c_List_Olist_ONil @ B ) @ ( c_List_Olist_ONil @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1411]) ).

thf(1377,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Oappend @ D ) @ ( hAPP @ ( hAPP @ ( c_List_Oappend @ D ) @ C ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_List_Oappend @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_List_Oappend @ D ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_append__assoc) ).

thf(7685,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Oappend @ D ) @ ( hAPP @ ( hAPP @ ( c_List_Oappend @ D ) @ C ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_List_Oappend @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_List_Oappend @ D ) @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1377]) ).

thf(1432,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ B )
        = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ B ) )
    <=> ( C = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nat__add__right__cancel) ).

thf(7892,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ B )
          = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ B ) )
       => ( C = A ) )
      & ( ( C = A )
       => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ B )
          = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1432]) ).

thf(797,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ ( tc_fun @ D @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( c_member @ D ) ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ ( tc_fun @ D @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( c_member @ D ) ) @ B ) ) @ A ) )
    <=> ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_inf__Int__eq) ).

thf(5286,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ ( tc_fun @ D @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( c_member @ D ) ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ ( tc_fun @ D @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( c_member @ D ) ) @ B ) ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) ) ) )
      & ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ ( tc_fun @ D @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( c_member @ D ) ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ ( tc_fun @ D @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( c_member @ D ) ) @ B ) ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[797]) ).

thf(1616,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) @ B ) )
     => ( ( c_Divides_Odiv__class_Omod @ tc_Nat_Onat @ ( hAPP @ c_Nat_OSuc @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ A ) ) @ B )
        = ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Suc__times__mod__eq) ).

thf(8750,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) @ B ) )
     => ( ( c_Divides_Odiv__class_Omod @ tc_Nat_Onat @ ( hAPP @ c_Nat_OSuc @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ A ) ) @ B )
        = ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1616]) ).

thf(521,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ G @ ( tc_fun @ F @ ( tc_fun @ E @ tc_HOL_Obool ) ) ) @ D ) @ C ) @ B ) @ A ) )
    <=> ? [H: $i] :
          ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ G ) @ H ) @ D ) )
          & ( hBOOL @ ( hAPP @ ( hAPP @ ( hAPP @ C @ H ) @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_SUP2__iff) ).

thf(4277,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ G @ ( tc_fun @ F @ ( tc_fun @ E @ tc_HOL_Obool ) ) ) @ D ) @ C ) @ B ) @ A ) )
       => ? [H: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ G ) @ H ) @ D ) )
            & ( hBOOL @ ( hAPP @ ( hAPP @ ( hAPP @ C @ H ) @ B ) @ A ) ) ) )
      & ( ? [H: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ G ) @ H ) @ D ) )
            & ( hBOOL @ ( hAPP @ ( hAPP @ ( hAPP @ C @ H ) @ B ) @ A ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ G @ ( tc_fun @ F @ ( tc_fun @ E @ tc_HOL_Obool ) ) ) @ D ) @ C ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[521]) ).

thf(100,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Ocomm__monoid__add @ C )
     => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ tc_Nat_Onat @ C ) @ B ) @ ( hAPP @ ( c_SetInterval_Oord__class_OlessThan @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ A ) ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ C ) @ ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ tc_Nat_Onat @ C ) @ B ) @ ( hAPP @ ( c_SetInterval_Oord__class_OlessThan @ tc_Nat_Onat ) @ A ) ) ) @ ( hAPP @ B @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_setsum__lessThan__Suc) ).

thf(2757,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Ocomm__monoid__add @ C )
     => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ tc_Nat_Onat @ C ) @ B ) @ ( hAPP @ ( c_SetInterval_Oord__class_OlessThan @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ A ) ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ C ) @ ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ tc_Nat_Onat @ C ) @ B ) @ ( hAPP @ ( c_SetInterval_Oord__class_OlessThan @ tc_Nat_Onat ) @ A ) ) ) @ ( hAPP @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[100]) ).

thf(1370,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ A ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nat__mult__commute) ).

thf(7664,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ A ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1370]) ).

thf(210,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Relation_ORange @ C @ D ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ ( tc_prod @ C @ D ) @ tc_HOL_Obool ) ) @ B ) @ A ) ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Relation_ORange @ C @ D ) @ B ) ) @ ( hAPP @ ( c_Relation_ORange @ C @ D ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Range__Int__subset) ).

thf(3165,plain,
    ! [A: $i,B: $i,C: $i,D: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Relation_ORange @ C @ D ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ ( tc_prod @ C @ D ) @ tc_HOL_Obool ) ) @ B ) @ A ) ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Relation_ORange @ C @ D ) @ B ) ) @ ( hAPP @ ( c_Relation_ORange @ C @ D ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[210]) ).

thf(495,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ G @ F @ E ) @ D ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ C @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ C @ ( tc_fun @ ( tc_prod @ G @ E ) @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ G @ E ) @ tc_HOL_Obool ) @ C ) @ ( hAPP @ ( c_Relation_Orel__comp @ G @ F @ E ) @ D ) ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_rel__comp__UNION__distrib) ).

thf(4181,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ G @ F @ E ) @ D ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ C @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ C @ ( tc_fun @ ( tc_prod @ G @ E ) @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ G @ E ) @ tc_HOL_Obool ) @ C ) @ ( hAPP @ ( c_Relation_Orel__comp @ G @ F @ E ) @ D ) ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[495]) ).

thf(1339,axiom,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ A ) @ ( hAPP @ c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) )
    <=> ( A
        = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__Suc0) ).

thf(7551,plain,
    ! [A: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ A ) @ ( hAPP @ c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) )
       => ( A
          = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) )
      & ( ( A
          = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ A ) @ ( hAPP @ c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1339]) ).

thf(2125,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofold__graph @ E @ E @ D @ C @ B ) @ A ) )
     => ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ C ) @ B ) )
       => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofold1Set @ E @ D @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ C ) @ B ) ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_fold1Set_Ointros) ).

thf(10732,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofold__graph @ E @ E @ D @ C @ B ) @ A ) )
     => ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ C ) @ B ) )
       => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofold1Set @ E @ D @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ C ) @ B ) ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2125]) ).

thf(219,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Nat_Onat ) @ B ) @ ( c_SetInterval_Oord_OgreaterThan @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_COMBS @ tc_Nat_Onat @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ tc_Nat_Onat @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) @ tc_Nat_Onat ) @ ( c_COMBS @ tc_Nat_Onat @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ tc_Nat_Onat @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ tc_Nat_Onat ) @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ tc_Nat_Onat ) @ c_fconj ) ) @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ tc_Nat_Onat ) @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ tc_HOL_Obool @ tc_Nat_Onat ) @ c_fNot ) ) @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) ) ) ) @ A ) ) )
    <=> ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ A ) @ B ) )
        & ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_dvd_OgreaterThan__iff) ).

thf(3192,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Nat_Onat ) @ B ) @ ( c_SetInterval_Oord_OgreaterThan @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_COMBS @ tc_Nat_Onat @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ tc_Nat_Onat @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) @ tc_Nat_Onat ) @ ( c_COMBS @ tc_Nat_Onat @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ tc_Nat_Onat @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ tc_Nat_Onat ) @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ tc_Nat_Onat ) @ c_fconj ) ) @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ tc_Nat_Onat ) @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ tc_HOL_Obool @ tc_Nat_Onat ) @ c_fNot ) ) @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) ) ) ) @ A ) ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ A ) @ B ) )
          & ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) ) ) )
      & ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ A ) @ B ) )
          & ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Nat_Onat ) @ B ) @ ( c_SetInterval_Oord_OgreaterThan @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_COMBS @ tc_Nat_Onat @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ tc_Nat_Onat @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) @ tc_Nat_Onat ) @ ( c_COMBS @ tc_Nat_Onat @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ tc_Nat_Onat @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ tc_Nat_Onat ) @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ tc_Nat_Onat ) @ c_fconj ) ) @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ tc_Nat_Onat ) @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ tc_HOL_Obool @ tc_Nat_Onat ) @ c_fNot ) ) @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) ) ) ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[219]) ).

thf(1824,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( c_Com_Ocom_OSKIP
     != ( c_Com_Ocom_OCall @ C @ B @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_com_Osimps_I20_J) ).

thf(9553,plain,
    ! [A: $i,B: $i,C: $i] :
      ( c_Com_Ocom_OSKIP
     != ( c_Com_Ocom_OCall @ C @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1824]) ).

thf(437,axiom,
    ! [A: $i,B: $i] :
      ( ( c_FunDef_Orp__inv__image @ B @ A )
      = ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ A @ B ) @ ( tc_prod @ ( tc_fun @ ( tc_prod @ A @ A ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ A @ A ) @ tc_HOL_Obool ) ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ A @ B ) @ ( tc_fun @ ( tc_prod @ A @ A ) @ tc_HOL_Obool ) ) ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ A @ B ) @ ( tc_prod @ ( tc_fun @ ( tc_prod @ A @ A ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ A @ A ) @ tc_HOL_Obool ) ) ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ ( tc_fun @ ( tc_fun @ A @ B ) @ ( tc_fun @ ( tc_prod @ A @ A ) @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ A @ B ) @ ( tc_prod @ ( tc_fun @ ( tc_prod @ A @ A ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ A @ A ) @ tc_HOL_Obool ) ) ) ) @ ( tc_fun @ ( tc_fun @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ A @ B ) @ ( tc_fun @ ( tc_prod @ A @ A ) @ tc_HOL_Obool ) ) ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ A @ B ) @ ( tc_prod @ ( tc_fun @ ( tc_prod @ A @ A ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ A @ A ) @ tc_HOL_Obool ) ) ) ) ) @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) ) @ ( c_COMBB @ ( tc_fun @ ( tc_fun @ A @ B ) @ ( tc_fun @ ( tc_prod @ A @ A ) @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ A @ B ) @ ( tc_prod @ ( tc_fun @ ( tc_prod @ A @ A ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ A @ A ) @ tc_HOL_Obool ) ) ) @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ ( tc_fun @ A @ B ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ A @ A ) @ tc_HOL_Obool ) @ ( tc_prod @ ( tc_fun @ ( tc_prod @ A @ A ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ A @ A ) @ tc_HOL_Obool ) ) ) ) @ ( tc_fun @ ( tc_fun @ ( tc_fun @ A @ B ) @ ( tc_fun @ ( tc_prod @ A @ A ) @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ A @ B ) @ ( tc_prod @ ( tc_fun @ ( tc_prod @ A @ A ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ A @ A ) @ tc_HOL_Obool ) ) ) ) @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) ) @ ( c_COMBS @ ( tc_fun @ A @ B ) @ ( tc_fun @ ( tc_prod @ A @ A ) @ tc_HOL_Obool ) @ ( tc_prod @ ( tc_fun @ ( tc_prod @ A @ A ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ A @ A ) @ tc_HOL_Obool ) ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ ( tc_fun @ A @ B ) @ ( tc_fun @ ( tc_prod @ A @ A ) @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ A @ B ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ A @ A ) @ tc_HOL_Obool ) @ ( tc_prod @ ( tc_fun @ ( tc_prod @ A @ A ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ A @ A ) @ tc_HOL_Obool ) ) ) ) @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_COMBB @ ( tc_fun @ ( tc_prod @ A @ A ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ A @ A ) @ tc_HOL_Obool ) @ ( tc_prod @ ( tc_fun @ ( tc_prod @ A @ A ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ A @ A ) @ tc_HOL_Obool ) ) ) @ ( tc_fun @ A @ B ) ) @ ( c_Product__Type_OPair @ ( tc_fun @ ( tc_prod @ A @ A ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ A @ A ) @ tc_HOL_Obool ) ) ) ) @ ( c_Relation_Oinv__image @ B @ A ) ) ) ) ) @ ( c_Relation_Oinv__image @ B @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_rp__inv__image__def) ).

thf(3977,plain,
    ! [A: $i,B: $i] :
      ( ( c_FunDef_Orp__inv__image @ B @ A )
      = ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ A @ B ) @ ( tc_prod @ ( tc_fun @ ( tc_prod @ A @ A ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ A @ A ) @ tc_HOL_Obool ) ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ A @ B ) @ ( tc_fun @ ( tc_prod @ A @ A ) @ tc_HOL_Obool ) ) ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ A @ B ) @ ( tc_prod @ ( tc_fun @ ( tc_prod @ A @ A ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ A @ A ) @ tc_HOL_Obool ) ) ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ ( tc_fun @ ( tc_fun @ A @ B ) @ ( tc_fun @ ( tc_prod @ A @ A ) @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ A @ B ) @ ( tc_prod @ ( tc_fun @ ( tc_prod @ A @ A ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ A @ A ) @ tc_HOL_Obool ) ) ) ) @ ( tc_fun @ ( tc_fun @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ A @ B ) @ ( tc_fun @ ( tc_prod @ A @ A ) @ tc_HOL_Obool ) ) ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ A @ B ) @ ( tc_prod @ ( tc_fun @ ( tc_prod @ A @ A ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ A @ A ) @ tc_HOL_Obool ) ) ) ) ) @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) ) @ ( c_COMBB @ ( tc_fun @ ( tc_fun @ A @ B ) @ ( tc_fun @ ( tc_prod @ A @ A ) @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ A @ B ) @ ( tc_prod @ ( tc_fun @ ( tc_prod @ A @ A ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ A @ A ) @ tc_HOL_Obool ) ) ) @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ ( tc_fun @ A @ B ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ A @ A ) @ tc_HOL_Obool ) @ ( tc_prod @ ( tc_fun @ ( tc_prod @ A @ A ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ A @ A ) @ tc_HOL_Obool ) ) ) ) @ ( tc_fun @ ( tc_fun @ ( tc_fun @ A @ B ) @ ( tc_fun @ ( tc_prod @ A @ A ) @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ A @ B ) @ ( tc_prod @ ( tc_fun @ ( tc_prod @ A @ A ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ A @ A ) @ tc_HOL_Obool ) ) ) ) @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) ) @ ( c_COMBS @ ( tc_fun @ A @ B ) @ ( tc_fun @ ( tc_prod @ A @ A ) @ tc_HOL_Obool ) @ ( tc_prod @ ( tc_fun @ ( tc_prod @ A @ A ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ A @ A ) @ tc_HOL_Obool ) ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ ( tc_fun @ A @ B ) @ ( tc_fun @ ( tc_prod @ A @ A ) @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ A @ B ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ A @ A ) @ tc_HOL_Obool ) @ ( tc_prod @ ( tc_fun @ ( tc_prod @ A @ A ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ A @ A ) @ tc_HOL_Obool ) ) ) ) @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_COMBB @ ( tc_fun @ ( tc_prod @ A @ A ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ A @ A ) @ tc_HOL_Obool ) @ ( tc_prod @ ( tc_fun @ ( tc_prod @ A @ A ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ A @ A ) @ tc_HOL_Obool ) ) ) @ ( tc_fun @ A @ B ) ) @ ( c_Product__Type_OPair @ ( tc_fun @ ( tc_prod @ A @ A ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ A @ A ) @ tc_HOL_Obool ) ) ) ) @ ( c_Relation_Oinv__image @ B @ A ) ) ) ) ) @ ( c_Relation_Oinv__image @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[437]) ).

thf(340,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Lattices_Oab__semigroup__idem__mult @ C )
     => ( ! [D: $i,E: $i] :
            ( ( hAPP @ B @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ D ) @ E ) )
            = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ ( hAPP @ B @ D ) ) @ ( hAPP @ B @ E ) ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ A ) )
         => ( ( A
             != ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) )
           => ( ( hAPP @ B @ ( hAPP @ ( c_Finite__Set_Ofold1 @ C @ ( c_Groups_Otimes__class_Otimes @ C ) ) @ A ) )
              = ( hAPP @ ( c_Finite__Set_Ofold1 @ C @ ( c_Groups_Otimes__class_Otimes @ C ) ) @ ( hAPP @ ( c_Set_Oimage @ C @ C @ B ) @ A ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_hom__fold1__commute) ).

thf(3614,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Lattices_Oab__semigroup__idem__mult @ C )
     => ( ! [D: $i,E: $i] :
            ( ( hAPP @ B @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ D ) @ E ) )
            = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ ( hAPP @ B @ D ) ) @ ( hAPP @ B @ E ) ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ A ) )
         => ( ( A
             != ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) )
           => ( ( hAPP @ B @ ( hAPP @ ( c_Finite__Set_Ofold1 @ C @ ( c_Groups_Otimes__class_Otimes @ C ) ) @ A ) )
              = ( hAPP @ ( c_Finite__Set_Ofold1 @ C @ ( c_Groups_Otimes__class_Otimes @ C ) ) @ ( hAPP @ ( c_Set_Oimage @ C @ C @ B ) @ A ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[340]) ).

thf(108,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D
        = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ B ) ) @ A ) )
     => ( ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ C ) )
           => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ A ) )
              & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ A ) @ C ) ) ) )
          & ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ C ) )
           => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ C ) @ A ) )
              & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) ) ) ) )
       => ( hBOOL @ ( hAPP @ ( c_Divides_Odivmod__int__rel @ D @ C ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ tc_Int_Oint @ tc_Int_Oint ) @ B ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_divmod__int__relI) ).

thf(2785,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D
        = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ B ) ) @ A ) )
     => ( ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ C ) )
           => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ A ) )
              & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ A ) @ C ) ) ) )
          & ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ C ) )
           => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ C ) @ A ) )
              & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) ) ) ) )
       => ( hBOOL @ ( hAPP @ ( c_Divides_Odivmod__int__rel @ D @ C ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ tc_Int_Oint @ tc_Int_Oint ) @ B ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[108]) ).

thf(295,axiom,
    ! [A: $i,B: $i] :
      ( ( c_List_Osublist @ B @ A @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) )
      = ( c_List_Olist_ONil @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_sublist__empty) ).

thf(3451,plain,
    ! [A: $i,B: $i] :
      ( ( c_List_Osublist @ B @ A @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) )
      = ( c_List_Olist_ONil @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[295]) ).

thf(252,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ A ) @ B ) )
      = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Int_Oint ) @ A ) @ ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ A @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zmult__div__cancel) ).

thf(3298,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ A ) @ B ) )
      = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Int_Oint ) @ A ) @ ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[252]) ).

thf(623,axiom,
    ! [A: $i,B: $i] :
      ( ( ( hAPP @ ( c_Set_OCollect @ B ) @ A )
        = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) )
    <=> ! [C: $i] :
          ~ ( hBOOL @ ( hAPP @ A @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Collect__empty__eq) ).

thf(4652,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( hAPP @ ( c_Set_OCollect @ B ) @ A )
          = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) )
       => ! [C: $i] :
            ~ ( hBOOL @ ( hAPP @ A @ C ) ) )
      & ( ! [C: $i] :
            ~ ( hBOOL @ ( hAPP @ A @ C ) )
       => ( ( hAPP @ ( c_Set_OCollect @ B ) @ A )
          = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[623]) ).

thf(2351,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Equiv__Relations_Oequivp @ B @ A )
    <=> ! [C: $i,D: $i] :
          ( ( hBOOL @ ( hAPP @ ( hAPP @ A @ C ) @ D ) )
        <=> ( ( hAPP @ A @ C )
            = ( hAPP @ A @ D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_equivp__def) ).

thf(11640,plain,
    ! [A: $i,B: $i] :
      ( ( ( c_Equiv__Relations_Oequivp @ B @ A )
       => ! [C: $i,D: $i] :
            ( ( ( hBOOL @ ( hAPP @ ( hAPP @ A @ C ) @ D ) )
             => ( ( hAPP @ A @ C )
                = ( hAPP @ A @ D ) ) )
            & ( ( ( hAPP @ A @ C )
                = ( hAPP @ A @ D ) )
             => ( hBOOL @ ( hAPP @ ( hAPP @ A @ C ) @ D ) ) ) ) )
      & ( ! [C: $i,D: $i] :
            ( ( ( hBOOL @ ( hAPP @ ( hAPP @ A @ C ) @ D ) )
             => ( ( hAPP @ A @ C )
                = ( hAPP @ A @ D ) ) )
            & ( ( ( hAPP @ A @ C )
                = ( hAPP @ A @ D ) )
             => ( hBOOL @ ( hAPP @ ( hAPP @ A @ C ) @ D ) ) ) )
       => ( c_Equiv__Relations_Oequivp @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2351]) ).

thf(1171,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Nat_Onat ) @ B ) @ A ) ) ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ ( c_Divides_Odiv__class_Omod @ tc_Nat_Onat @ B @ A ) ) ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_div__mult1__eq) ).

thf(6817,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Nat_Onat ) @ B ) @ A ) ) ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ ( c_Divides_Odiv__class_Omod @ tc_Nat_Onat @ B @ A ) ) ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1171]) ).

thf(1626,axiom,
    ! [A: $i,B: $i] :
      ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_not__add__less2) ).

thf(8782,plain,
    ! [A: $i,B: $i] :
      ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1626]) ).

thf(303,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_List_Odistinct @ B ) @ A ) )
    <=> ! [C: $i] :
          ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ C ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ B ) ) @ A ) ) )
         => ! [D: $i] :
              ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ D ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ B ) ) @ A ) ) )
             => ( ( C != D )
               => ( ( hAPP @ ( hAPP @ ( c_List_Onth @ B ) @ A ) @ C )
                 != ( hAPP @ ( hAPP @ ( c_List_Onth @ B ) @ A ) @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_distinct__conv__nth) ).

thf(3483,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( c_List_Odistinct @ B ) @ A ) )
       => ! [C: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ C ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ B ) ) @ A ) ) )
           => ! [D: $i] :
                ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ D ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ B ) ) @ A ) ) )
               => ( ( C != D )
                 => ( ( hAPP @ ( hAPP @ ( c_List_Onth @ B ) @ A ) @ C )
                   != ( hAPP @ ( hAPP @ ( c_List_Onth @ B ) @ A ) @ D ) ) ) ) ) )
      & ( ! [C: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ C ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ B ) ) @ A ) ) )
           => ! [D: $i] :
                ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ D ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ B ) ) @ A ) ) )
               => ( ( C != D )
                 => ( ( hAPP @ ( hAPP @ ( c_List_Onth @ B ) @ A ) @ C )
                   != ( hAPP @ ( hAPP @ ( c_List_Onth @ B ) @ A ) @ D ) ) ) ) )
       => ( hBOOL @ ( hAPP @ ( c_List_Odistinct @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[303]) ).

thf(435,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( c_Set_Oimage @ tc_Int_Oint @ tc_Int_Oint @ ( hAPP @ ( hAPP @ ( c_COMBC @ tc_Int_Oint @ tc_Int_Oint @ tc_Int_Oint ) @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) ) @ B ) ) @ ( hAPP @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Int_Oint ) @ A ) @ B ) ) )
      = ( hAPP @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Int_Oint @ B ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_image__add__int__atLeastLessThan) ).

thf(3971,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( c_Set_Oimage @ tc_Int_Oint @ tc_Int_Oint @ ( hAPP @ ( hAPP @ ( c_COMBC @ tc_Int_Oint @ tc_Int_Oint @ tc_Int_Oint ) @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) ) @ B ) ) @ ( hAPP @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Int_Oint @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Int_Oint ) @ A ) @ B ) ) )
      = ( hAPP @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Int_Oint @ B ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[435]) ).

thf(2146,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ ( hAPP @ c_Nat_OSuc @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_le__SucI) ).

thf(10825,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ ( hAPP @ c_Nat_OSuc @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2146]) ).

thf(331,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Fun_Oinj__on @ E @ D @ C @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ E @ tc_HOL_Obool ) ) )
     => ( ( ( hAPP @ ( c_Set_Oimage @ E @ D @ C ) @ B )
          = ( hAPP @ ( c_Set_Oimage @ E @ D @ C ) @ A ) )
      <=> ( B = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_inj__image__eq__iff) ).

thf(3583,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Fun_Oinj__on @ E @ D @ C @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ E @ tc_HOL_Obool ) ) )
     => ( ( ( ( hAPP @ ( c_Set_Oimage @ E @ D @ C ) @ B )
            = ( hAPP @ ( c_Set_Oimage @ E @ D @ C ) @ A ) )
         => ( B = A ) )
        & ( ( B = A )
         => ( ( hAPP @ ( c_Set_Oimage @ E @ D @ C ) @ B )
            = ( hAPP @ ( c_Set_Oimage @ E @ D @ C ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[331]) ).

thf(1893,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ c_Com_Ocom_OBODY @ D )
     != ( c_Com_Ocom_OCall @ C @ B @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_com_Osimps_I62_J) ).

thf(9862,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ c_Com_Ocom_OBODY @ D )
     != ( c_Com_Ocom_OCall @ C @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1893]) ).

thf(2348,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ B ) @ A ) )
     => ( ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ B ) @ A )
        = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_insert__absorb) ).

thf(11630,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ B ) @ A ) )
     => ( ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ B ) @ A )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2348]) ).

thf(999,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ ( hAPP @ ( c_Set_OCollect @ C ) @ B ) ) )
        | ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ ( hAPP @ ( c_Set_OCollect @ C ) @ A ) ) ) )
     => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ ( hAPP @ ( c_Set_OCollect @ C ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ C @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ C ) @ c_fconj ) @ B ) ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_finite__Collect__conjI) ).

thf(6044,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ ( hAPP @ ( c_Set_OCollect @ C ) @ B ) ) )
        | ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ ( hAPP @ ( c_Set_OCollect @ C ) @ A ) ) ) )
     => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ ( hAPP @ ( c_Set_OCollect @ C ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ C @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ C ) @ c_fconj ) @ B ) ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[999]) ).

thf(1317,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ B ) )
     => ( ! [C: $i,D: $i] :
            ( ( hBOOL @ ( hAPP @ A @ C ) )
          <=> ( hBOOL @ ( hAPP @ A @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Int_Oint ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ D ) @ B ) ) ) ) )
       => ( ? [C: $i] : ( hBOOL @ ( hAPP @ A @ C ) )
        <=> ? [C: $i] :
              ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Int_Oint ) @ C ) @ ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Int_Oint @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) @ B ) ) )
              & ( hBOOL @ ( hAPP @ A @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_periodic__finite__ex) ).

thf(7450,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ B ) )
     => ( ! [C: $i,D: $i] :
            ( ( ( hBOOL @ ( hAPP @ A @ C ) )
             => ( hBOOL @ ( hAPP @ A @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Int_Oint ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ D ) @ B ) ) ) ) )
            & ( ( hBOOL @ ( hAPP @ A @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Int_Oint ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ D ) @ B ) ) ) )
             => ( hBOOL @ ( hAPP @ A @ C ) ) ) )
       => ( ( ? [C: $i] : ( hBOOL @ ( hAPP @ A @ C ) )
           => ? [C: $i] :
                ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Int_Oint ) @ C ) @ ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Int_Oint @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) @ B ) ) )
                & ( hBOOL @ ( hAPP @ A @ C ) ) ) )
          & ( ? [C: $i] :
                ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Int_Oint ) @ C ) @ ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Int_Oint @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) @ B ) ) )
                & ( hBOOL @ ( hAPP @ A @ C ) ) )
           => ? [C: $i] : ( hBOOL @ ( hAPP @ A @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1317]) ).

thf(1166,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Transitive__Closure_Ortrancl @ B @ ( hAPP @ ( c_Relation_Oconverse @ B @ B ) @ A ) )
      = ( hAPP @ ( c_Relation_Oconverse @ B @ B ) @ ( c_Transitive__Closure_Ortrancl @ B @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_rtrancl__converse) ).

thf(6802,plain,
    ! [A: $i,B: $i] :
      ( ( c_Transitive__Closure_Ortrancl @ B @ ( hAPP @ ( c_Relation_Oconverse @ B @ B ) @ A ) )
      = ( hAPP @ ( c_Relation_Oconverse @ B @ B ) @ ( c_Transitive__Closure_Ortrancl @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1166]) ).

thf(646,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( D
          = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) )
       => ( ( hAPP @ ( c_Set_Oimage @ ( tc_prod @ B @ C ) @ B @ ( c_Product__Type_Ofst @ B @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ B @ C ) @ A ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ C @ tc_HOL_Obool ) @ B ) @ D ) ) )
          = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) )
      & ( ( D
         != ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) )
       => ( ( hAPP @ ( c_Set_Oimage @ ( tc_prod @ B @ C ) @ B @ ( c_Product__Type_Ofst @ B @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ B @ C ) @ A ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ C @ tc_HOL_Obool ) @ B ) @ D ) ) )
          = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_fst__image__times) ).

thf(4726,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( D
          = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) )
       => ( ( hAPP @ ( c_Set_Oimage @ ( tc_prod @ B @ C ) @ B @ ( c_Product__Type_Ofst @ B @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ B @ C ) @ A ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ C @ tc_HOL_Obool ) @ B ) @ D ) ) )
          = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) )
      & ( ( D
         != ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) )
       => ( ( hAPP @ ( c_Set_Oimage @ ( tc_prod @ B @ C ) @ B @ ( c_Product__Type_Ofst @ B @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ B @ C ) @ A ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ C @ tc_HOL_Obool ) @ B ) @ D ) ) )
          = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[646]) ).

thf(720,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Relation_Osingle__valued @ E @ D @ ( hAPP @ ( c_Relation_Oconverse @ D @ E ) @ C ) )
     => ( ( hAPP @ ( c_Relation_OImage @ D @ E @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Relation_OImage @ D @ E @ C ) @ B ) ) @ ( hAPP @ ( c_Relation_OImage @ D @ E @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Image__Int__eq) ).

thf(4993,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Relation_Osingle__valued @ E @ D @ ( hAPP @ ( c_Relation_Oconverse @ D @ E ) @ C ) )
     => ( ( hAPP @ ( c_Relation_OImage @ D @ E @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Relation_OImage @ D @ E @ C ) @ B ) ) @ ( hAPP @ ( c_Relation_OImage @ D @ E @ C ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[720]) ).

thf(2027,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i,I: $i,J: $i,K: $i,L: $i] :
      ( ( c_Com_Ocom_Ocom__case @ L @ K @ J @ I @ H @ G @ F @ E @ D @ ( c_Com_Ocom_OLocal @ C @ B @ A ) )
      = ( hAPP @ ( hAPP @ ( hAPP @ I @ C ) @ B ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_com_Osimps_I66_J) ).

thf(10380,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i,I: $i,J: $i,K: $i,L: $i] :
      ( ( c_Com_Ocom_Ocom__case @ L @ K @ J @ I @ H @ G @ F @ E @ D @ ( c_Com_Ocom_OLocal @ C @ B @ A ) )
      = ( hAPP @ ( hAPP @ ( hAPP @ I @ C ) @ B ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2027]) ).

thf(1112,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Fun_Oinj__on @ E @ D @ C @ B )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Set_Oimage @ E @ D @ C ) @ B ) ) @ A ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( c_Finite__Set_Ocard @ E ) @ B ) ) @ ( hAPP @ ( c_Finite__Set_Ocard @ D ) @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_card__inj__on__le) ).

thf(6568,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Fun_Oinj__on @ E @ D @ C @ B )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Set_Oimage @ E @ D @ C ) @ B ) ) @ A ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( c_Finite__Set_Ocard @ E ) @ B ) ) @ ( hAPP @ ( c_Finite__Set_Ocard @ D ) @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1112]) ).

thf(1133,axiom,
    ! [A: $i] :
      ( ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat )
     != ( hAPP @ c_Nat_OSuc @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Zero__neq__Suc) ).

thf(6692,plain,
    ! [A: $i] :
      ( ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat )
     != ( hAPP @ c_Nat_OSuc @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1133]) ).

thf(144,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ B ) @ A ) ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) )
       => ( ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ B @ A )
          = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mod__pos__neg__trivial) ).

thf(2928,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ B ) @ A ) ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) )
       => ( ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ B @ A )
          = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[144]) ).

thf(1595,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( c_Equiv__Relations_Oequiv @ G @ F @ E )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_fun @ G @ tc_HOL_Obool ) ) @ D ) @ ( hAPP @ ( hAPP @ ( c_Equiv__Relations_Oquotient @ G ) @ F ) @ E ) ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_fun @ G @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Equiv__Relations_Oquotient @ G ) @ F ) @ E ) ) )
         => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ G ) @ B ) @ D ) )
           => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ G ) @ A ) @ C ) )
             => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ G @ G ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ G @ G ) @ B ) @ A ) ) @ E ) )
               => ( D = C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_quotient__eqI) ).

thf(8652,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( c_Equiv__Relations_Oequiv @ G @ F @ E )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_fun @ G @ tc_HOL_Obool ) ) @ D ) @ ( hAPP @ ( hAPP @ ( c_Equiv__Relations_Oquotient @ G ) @ F ) @ E ) ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_fun @ G @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Equiv__Relations_Oquotient @ G ) @ F ) @ E ) ) )
         => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ G ) @ B ) @ D ) )
           => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ G ) @ A ) @ C ) )
             => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ G @ G ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ G @ G ) @ B ) @ A ) ) @ E ) )
               => ( D = C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1595]) ).

thf(1934,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Wellfounded_Owf @ B @ A )
     => ( c_Wellfounded_OwfP @ B @ ( c_FunDef_Oin__rel @ B @ B @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_wf__in__rel) ).

thf(10004,plain,
    ! [A: $i,B: $i] :
      ( ( c_Wellfounded_Owf @ B @ A )
     => ( c_Wellfounded_OwfP @ B @ ( c_FunDef_Oin__rel @ B @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1934]) ).

thf(846,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Typedef_Otype__definition @ E @ D @ C @ B @ A )
     => ( ( hAPP @ ( c_Set_Oimage @ E @ D @ C ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ E @ tc_HOL_Obool ) ) )
        = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_type__definition_ORep__range) ).

thf(5459,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Typedef_Otype__definition @ E @ D @ C @ B @ A )
     => ( ( hAPP @ ( c_Set_Oimage @ E @ D @ C ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ E @ tc_HOL_Obool ) ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[846]) ).

thf(2137,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) )
     => ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) )
          & ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ A ) @ B ) ) )
        | ( B = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_dvd_Ole__imp__less__or__eq) ).

thf(10792,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) )
     => ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) )
          & ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ A ) @ B ) ) )
        | ( B = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2137]) ).

thf(1497,axiom,
    ! [A: $i,B: $i] :
      ( ( c_List_Olist_Olist__size @ B @ A @ ( c_List_Olist_ONil @ B ) )
      = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_list_Osize_I1_J) ).

thf(8198,plain,
    ! [A: $i,B: $i] :
      ( ( c_List_Olist_Olist__size @ B @ A @ ( c_List_Olist_ONil @ B ) )
      = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1497]) ).

thf(986,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ D @ C @ B ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ ( tc_prod @ D @ C ) ) @ ( tc_fun @ C @ B ) @ D ) @ ( hAPP @ ( c_COMBB @ ( tc_prod @ D @ C ) @ B @ C ) @ A ) ) @ ( c_Product__Type_OPair @ D @ C ) ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_split__eta) ).

thf(5987,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ D @ C @ B ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ ( tc_prod @ D @ C ) ) @ ( tc_fun @ C @ B ) @ D ) @ ( hAPP @ ( c_COMBB @ ( tc_prod @ D @ C ) @ B @ C ) @ A ) ) @ ( c_Product__Type_OPair @ D @ C ) ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[986]) ).

thf(1712,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( c_Divides_Odiv__class_Omod @ tc_Nat_Onat @ C @ B ) ) @ A )
      = ( c_Divides_Odiv__class_Omod @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mod__mult__distrib) ).

thf(9143,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( c_Divides_Odiv__class_Omod @ tc_Nat_Onat @ C @ B ) ) @ A )
      = ( c_Divides_Odiv__class_Omod @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1712]) ).

thf(398,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ A ) @ B ) ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) )
      <=> ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_pos__imp__zdiv__neg__iff) ).

thf(3838,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ B ) )
     => ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ A ) @ B ) ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) ) )
        & ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ A ) @ B ) ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[398]) ).

thf(1480,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ C @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Nat_Onat ) @ B ) @ A ) ) )
    <=> ( ( ( A
            = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
         => ( hBOOL @ ( hAPP @ C @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) )
        & ( ( A
           != ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
         => ! [D: $i,E: $i] :
              ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ E ) @ A ) )
             => ( ( B
                  = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ A ) @ D ) ) @ E ) )
               => ( hBOOL @ ( hAPP @ C @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_split__div) ).

thf(8084,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP @ C @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Nat_Onat ) @ B ) @ A ) ) )
       => ( ( ( A
              = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
           => ( hBOOL @ ( hAPP @ C @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) )
          & ( ( A
             != ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
           => ! [D: $i,E: $i] :
                ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ E ) @ A ) )
               => ( ( B
                    = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ A ) @ D ) ) @ E ) )
                 => ( hBOOL @ ( hAPP @ C @ D ) ) ) ) ) ) )
      & ( ( ( ( A
              = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
           => ( hBOOL @ ( hAPP @ C @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) )
          & ( ( A
             != ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
           => ! [D: $i,E: $i] :
                ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ E ) @ A ) )
               => ( ( B
                    = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ A ) @ D ) ) @ E ) )
                 => ( hBOOL @ ( hAPP @ C @ D ) ) ) ) ) )
       => ( hBOOL @ ( hAPP @ C @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Nat_Onat ) @ B ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1480]) ).

thf(1438,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Wellfounded_Owf @ C @ B )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ C @ C @ C ) @ B ) @ A ) ) @ B ) )
       => ( c_FunDef_Oreduction__pair @ C @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) ) @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_reduction__pairI) ).

thf(7911,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Wellfounded_Owf @ C @ B )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ C @ C @ C ) @ B ) @ A ) ) @ B ) )
       => ( c_FunDef_Oreduction__pair @ C @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1438]) ).

thf(214,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Omap @ ( tc_prod @ D @ tc_Nat_Onat ) @ D ) @ ( c_Product__Type_Ofst @ D @ tc_Nat_Onat ) ) @ ( hAPP @ ( c_List_Ofilter @ ( tc_prod @ D @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_Nat_Onat @ tc_HOL_Obool @ ( tc_prod @ D @ tc_Nat_Onat ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_Nat_Onat @ tc_Nat_Onat @ ( tc_prod @ D @ tc_Nat_Onat ) ) @ c_Nat_OSuc ) @ ( c_Product__Type_Osnd @ D @ tc_Nat_Onat ) ) ) ) @ ( hAPP @ ( hAPP @ ( c_List_Ozip @ D @ tc_Nat_Onat ) @ B ) @ A ) ) )
      = ( hAPP @ ( hAPP @ ( c_List_Omap @ ( tc_prod @ D @ tc_Nat_Onat ) @ D ) @ ( c_Product__Type_Ofst @ D @ tc_Nat_Onat ) ) @ ( hAPP @ ( c_List_Ofilter @ ( tc_prod @ D @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_Nat_Onat @ tc_HOL_Obool @ ( tc_prod @ D @ tc_Nat_Onat ) ) @ C ) @ ( c_Product__Type_Osnd @ D @ tc_Nat_Onat ) ) ) @ ( hAPP @ ( hAPP @ ( c_List_Ozip @ D @ tc_Nat_Onat ) @ B ) @ ( hAPP @ ( hAPP @ ( c_List_Omap @ tc_Nat_Onat @ tc_Nat_Onat ) @ c_Nat_OSuc ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_sublist__shift__lemma__Suc) ).

thf(3178,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Omap @ ( tc_prod @ D @ tc_Nat_Onat ) @ D ) @ ( c_Product__Type_Ofst @ D @ tc_Nat_Onat ) ) @ ( hAPP @ ( c_List_Ofilter @ ( tc_prod @ D @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_Nat_Onat @ tc_HOL_Obool @ ( tc_prod @ D @ tc_Nat_Onat ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_Nat_Onat @ tc_Nat_Onat @ ( tc_prod @ D @ tc_Nat_Onat ) ) @ c_Nat_OSuc ) @ ( c_Product__Type_Osnd @ D @ tc_Nat_Onat ) ) ) ) @ ( hAPP @ ( hAPP @ ( c_List_Ozip @ D @ tc_Nat_Onat ) @ B ) @ A ) ) )
      = ( hAPP @ ( hAPP @ ( c_List_Omap @ ( tc_prod @ D @ tc_Nat_Onat ) @ D ) @ ( c_Product__Type_Ofst @ D @ tc_Nat_Onat ) ) @ ( hAPP @ ( c_List_Ofilter @ ( tc_prod @ D @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_Nat_Onat @ tc_HOL_Obool @ ( tc_prod @ D @ tc_Nat_Onat ) ) @ C ) @ ( c_Product__Type_Osnd @ D @ tc_Nat_Onat ) ) ) @ ( hAPP @ ( hAPP @ ( c_List_Ozip @ D @ tc_Nat_Onat ) @ B ) @ ( hAPP @ ( hAPP @ ( c_List_Omap @ tc_Nat_Onat @ tc_Nat_Onat ) @ c_Nat_OSuc ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[214]) ).

thf(1984,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_List_Odistinct @ E ) @ ( hAPP @ ( hAPP @ ( c_List_Omap @ ( tc_prod @ E @ D ) @ E ) @ ( c_Product__Type_Ofst @ E @ D ) ) @ C ) ) )
     => ( ( ( hAPP @ ( c_Option_Ooption_OSome @ D ) @ B )
          = ( hAPP @ ( c_Map_Omap__of @ E @ D @ C ) @ A ) )
      <=> ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ D ) @ A ) @ B ) ) @ ( hAPP @ ( c_List_Oset @ ( tc_prod @ E @ D ) ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Some__eq__map__of__iff) ).

thf(10226,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_List_Odistinct @ E ) @ ( hAPP @ ( hAPP @ ( c_List_Omap @ ( tc_prod @ E @ D ) @ E ) @ ( c_Product__Type_Ofst @ E @ D ) ) @ C ) ) )
     => ( ( ( ( hAPP @ ( c_Option_Ooption_OSome @ D ) @ B )
            = ( hAPP @ ( c_Map_Omap__of @ E @ D @ C ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ D ) @ A ) @ B ) ) @ ( hAPP @ ( c_List_Oset @ ( tc_prod @ E @ D ) ) @ C ) ) ) )
        & ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ D ) @ A ) @ B ) ) @ ( hAPP @ ( c_List_Oset @ ( tc_prod @ E @ D ) ) @ C ) ) )
         => ( ( hAPP @ ( c_Option_Ooption_OSome @ D ) @ B )
            = ( hAPP @ ( c_Map_Omap__of @ E @ D @ C ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1984]) ).

thf(1555,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_List_Olist_Olist__size @ D @ C @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ D ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ C @ B ) ) @ ( c_List_Olist_Olist__size @ D @ C @ A ) ) ) @ ( hAPP @ c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_list_Osize_I2_J) ).

thf(8423,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_List_Olist_Olist__size @ D @ C @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ D ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ C @ B ) ) @ ( c_List_Olist_Olist__size @ D @ C @ A ) ) ) @ ( hAPP @ c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1555]) ).

thf(1516,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) )
     => ( ( c_Divides_Odivmod__nat @ B @ A )
        = ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ tc_Nat_Onat @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_divmod__nat__base) ).

thf(8280,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) )
     => ( ( c_Divides_Odivmod__nat @ B @ A )
        = ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ tc_Nat_Onat @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1516]) ).

thf(2239,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Sum__Type_OProjr @ C @ B @ ( hAPP @ ( c_Sum__Type_OInr @ B @ C ) @ A ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Projr_Osimps) ).

thf(11173,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Sum__Type_OProjr @ C @ B @ ( hAPP @ ( c_Sum__Type_OInr @ B @ C ) @ A ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2239]) ).

thf(1965,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Omap @ E @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ E ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ D ) @ ( hAPP @ C @ B ) ) @ ( hAPP @ ( hAPP @ ( c_List_Omap @ E @ D ) @ C ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_map_Osimps_I2_J) ).

thf(10147,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Omap @ E @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ E ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ D ) @ ( hAPP @ C @ B ) ) @ ( hAPP @ ( hAPP @ ( c_List_Omap @ E @ D ) @ C ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1965]) ).

thf(1472,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_List_Odistinct @ C ) @ B ) )
     => ( ( hAPP @ ( c_List_Oset @ C ) @ ( c_List_Oremove1 @ C @ A @ B ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_List_Oset @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_set__remove1__eq) ).

thf(8052,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_List_Odistinct @ C ) @ B ) )
     => ( ( hAPP @ ( c_List_Oset @ C ) @ ( c_List_Oremove1 @ C @ A @ B ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_List_Oset @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1472]) ).

thf(706,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ ( c_Complete__Lattice_OSup__class_OSup @ ( tc_fun @ C @ tc_HOL_Obool ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ A ) @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Int__Union) ).

thf(4933,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ ( c_Complete__Lattice_OSup__class_OSup @ ( tc_fun @ C @ tc_HOL_Obool ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ A ) @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[706]) ).

thf(1050,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ C @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Nat_Onat ) @ B ) @ A ) ) )
    <=> ( ( ( A
            = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
          & ( hBOOL @ ( hAPP @ C @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) )
        | ? [D: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ A ) @ D ) ) @ B ) )
            & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ A ) @ ( hAPP @ c_Nat_OSuc @ D ) ) ) )
            & ( hBOOL @ ( hAPP @ C @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_split__div_H) ).

thf(6255,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP @ C @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Nat_Onat ) @ B ) @ A ) ) )
       => ( ( ( A
              = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
            & ( hBOOL @ ( hAPP @ C @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) )
          | ? [D: $i] :
              ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ A ) @ D ) ) @ B ) )
              & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ A ) @ ( hAPP @ c_Nat_OSuc @ D ) ) ) )
              & ( hBOOL @ ( hAPP @ C @ D ) ) ) ) )
      & ( ( ( ( A
              = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
            & ( hBOOL @ ( hAPP @ C @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) )
          | ? [D: $i] :
              ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ A ) @ D ) ) @ B ) )
              & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ A ) @ ( hAPP @ c_Nat_OSuc @ D ) ) ) )
              & ( hBOOL @ ( hAPP @ C @ D ) ) ) )
       => ( hBOOL @ ( hAPP @ C @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Nat_Onat ) @ B ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1050]) ).

thf(682,axiom,
    hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_int__0__less__1) ).

thf(4844,plain,
    hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[682]) ).

thf(743,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ A ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ A ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zmult__zless__mono2) ).

thf(5057,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ A ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ A ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[743]) ).

thf(2252,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_FunDef_Oreduction__pair @ D @ C )
     => ( c_FunDef_Oreduction__pair @ B @ ( hAPP @ ( hAPP @ ( c_FunDef_Orp__inv__image @ D @ B ) @ C ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_rp__inv__image__rp) ).

thf(11220,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_FunDef_Oreduction__pair @ D @ C )
     => ( c_FunDef_Oreduction__pair @ B @ ( hAPP @ ( hAPP @ ( c_FunDef_Orp__inv__image @ D @ B ) @ C ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2252]) ).

thf(931,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( E
          = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) )
       => ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ D @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ E ) @ B ) ) @ A )
          = A ) )
      & ( ( E
         != ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) )
       => ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ D @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ E ) @ B ) ) @ A )
          = ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ D @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ E ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ D ) @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) @ B ) ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_UN__extend__simps_I2_J) ).

thf(5742,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( E
          = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) )
       => ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ D @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ E ) @ B ) ) @ A )
          = A ) )
      & ( ( E
         != ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) )
       => ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ D @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ E ) @ B ) ) @ A )
          = ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ D @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ E ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ D ) @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) @ B ) ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[931]) ).

thf(936,axiom,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ tc_Int_Oint ) @ ( c_SetInterval_Oord__class_OgreaterThanAtMost @ tc_Int_Oint @ B @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_finite__greaterThanAtMost__int) ).

thf(5775,plain,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ tc_Int_Oint ) @ ( c_SetInterval_Oord__class_OgreaterThanAtMost @ tc_Int_Oint @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[936]) ).

thf(1169,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( c_Relation_ORange @ B @ B ) @ ( c_Transitive__Closure_Otrancl @ B @ A ) )
      = ( hAPP @ ( c_Relation_ORange @ B @ B ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_trancl__range) ).

thf(6811,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( c_Relation_ORange @ B @ B ) @ ( c_Transitive__Closure_Otrancl @ B @ A ) )
      = ( hAPP @ ( c_Relation_ORange @ B @ B ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1169]) ).

thf(974,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) @ B ) ) @ A ) )
     => ( ( hAPP @ ( hAPP @ ( c_List_Odrop @ C ) @ A ) @ B )
        = ( c_List_Olist_ONil @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_drop__all) ).

thf(5931,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) @ B ) ) @ A ) )
     => ( ( hAPP @ ( hAPP @ ( c_List_Odrop @ C ) @ A ) @ B )
        = ( c_List_Olist_ONil @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[974]) ).

thf(90,axiom,
    ! [A: $i,B: $i] :
      ( ( c_List_Otranspose @ B @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ ( tc_List_Olist @ B ) ) @ ( c_List_Olist_ONil @ B ) ) @ A ) )
      = ( c_List_Otranspose @ B @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_transpose_Osimps_I2_J) ).

thf(2725,plain,
    ! [A: $i,B: $i] :
      ( ( c_List_Otranspose @ B @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ ( tc_List_Olist @ B ) ) @ ( c_List_Olist_ONil @ B ) ) @ A ) )
      = ( c_List_Otranspose @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[90]) ).

thf(1079,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( c_Map_Omap__add @ F @ E @ D @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ F @ ( tc_Option_Ooption @ E ) ) @ C ) @ B ) @ ( hAPP @ ( c_Option_Ooption_OSome @ E ) @ A ) ) )
      = ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ F @ ( tc_Option_Ooption @ E ) ) @ ( c_Map_Omap__add @ F @ E @ D @ C ) ) @ B ) @ ( hAPP @ ( c_Option_Ooption_OSome @ E ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_map__add__upd) ).

thf(6377,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( c_Map_Omap__add @ F @ E @ D @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ F @ ( tc_Option_Ooption @ E ) ) @ C ) @ B ) @ ( hAPP @ ( c_Option_Ooption_OSome @ E ) @ A ) ) )
      = ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ F @ ( tc_Option_Ooption @ E ) ) @ ( c_Map_Omap__add @ F @ E @ D @ C ) ) @ B ) @ ( hAPP @ ( c_Option_Ooption_OSome @ E ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1079]) ).

thf(840,axiom,
    ! [A: $i,B: $i] :
      ( ( ( hAPP @ B @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
        = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
     => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ tc_Nat_Onat @ tc_Nat_Onat ) @ B ) @ ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Nat_Onat @ ( hAPP @ c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ tc_Nat_Onat @ tc_Nat_Onat ) @ B ) @ ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_setsum__shift__lb__Suc0__0) ).

thf(5442,plain,
    ! [A: $i,B: $i] :
      ( ( ( hAPP @ B @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
        = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
     => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ tc_Nat_Onat @ tc_Nat_Onat ) @ B ) @ ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Nat_Onat @ ( hAPP @ c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ tc_Nat_Onat @ tc_Nat_Onat ) @ B ) @ ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[840]) ).

thf(344,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Oab__group__add @ D )
     => ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ C ) @ B ) )
         => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ tc_Nat_Onat @ D ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ tc_Nat_Onat @ D @ D ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ D @ ( tc_fun @ D @ D ) @ tc_Nat_Onat ) @ ( c_Groups_Ominus__class_Ominus @ D ) ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_Nat_Onat @ D @ tc_Nat_Onat ) @ A ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ tc_Nat_Onat @ tc_Nat_Onat ) @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) ) ) ) @ ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Nat_Onat @ C @ B ) )
            = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ D ) @ ( hAPP @ A @ C ) ) @ ( hAPP @ A @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) ) ) ) )
        & ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ C ) @ B ) )
         => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ tc_Nat_Onat @ D ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ tc_Nat_Onat @ D @ D ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ D @ ( tc_fun @ D @ D ) @ tc_Nat_Onat ) @ ( c_Groups_Ominus__class_Ominus @ D ) ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_Nat_Onat @ D @ tc_Nat_Onat ) @ A ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ tc_Nat_Onat @ tc_Nat_Onat ) @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) ) ) ) @ ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Nat_Onat @ C @ B ) )
            = ( c_Groups_Ozero__class_Ozero @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_setsum__natinterval__difff) ).

thf(3627,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Oab__group__add @ D )
     => ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ C ) @ B ) )
         => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ tc_Nat_Onat @ D ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ tc_Nat_Onat @ D @ D ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ D @ ( tc_fun @ D @ D ) @ tc_Nat_Onat ) @ ( c_Groups_Ominus__class_Ominus @ D ) ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_Nat_Onat @ D @ tc_Nat_Onat ) @ A ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ tc_Nat_Onat @ tc_Nat_Onat ) @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) ) ) ) @ ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Nat_Onat @ C @ B ) )
            = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ D ) @ ( hAPP @ A @ C ) ) @ ( hAPP @ A @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) ) ) ) )
        & ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ C ) @ B ) )
         => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ tc_Nat_Onat @ D ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ tc_Nat_Onat @ D @ D ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ D @ ( tc_fun @ D @ D ) @ tc_Nat_Onat ) @ ( c_Groups_Ominus__class_Ominus @ D ) ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_Nat_Onat @ D @ tc_Nat_Onat ) @ A ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ tc_Nat_Onat @ tc_Nat_Onat ) @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) ) ) ) @ ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Nat_Onat @ C @ B ) )
            = ( c_Groups_Ozero__class_Ozero @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[344]) ).

thf(2209,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Predicate_ORangeP @ D @ C @ B ) @ A ) )
    <=> ? [E: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ B @ E ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_RangeP_Oequations) ).

thf(11041,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( c_Predicate_ORangeP @ D @ C @ B ) @ A ) )
       => ? [E: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ B @ E ) @ A ) ) )
      & ( ? [E: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ B @ E ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( c_Predicate_ORangeP @ D @ C @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2209]) ).

thf(46,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Groups_Ocomm__monoid__add @ E )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ C ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ B ) )
         => ( ! [F: $i] :
                ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ F ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) ) )
               => ( ( hAPP @ A @ F )
                  = ( c_Groups_Ozero__class_Ozero @ E ) ) )
           => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) )
              = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ E ) @ ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ A ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ A ) @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_setsum_Ounion__inter__neutral) ).

thf(2550,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Groups_Ocomm__monoid__add @ E )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ C ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ B ) )
         => ( ! [F: $i] :
                ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ F ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) ) )
               => ( ( hAPP @ A @ F )
                  = ( c_Groups_Ozero__class_Ozero @ E ) ) )
           => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) )
              = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ E ) @ ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ A ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ A ) @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[46]) ).

thf(22,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Divides_Odivmod__int__rel @ F @ E ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ tc_Int_Oint @ tc_Int_Oint ) @ D ) @ C ) ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Divides_Odivmod__int__rel @ F @ E ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ tc_Int_Oint @ tc_Int_Oint ) @ B ) @ A ) ) )
       => ( ( E
           != ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) )
         => ( D = B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_unique__quotient) ).

thf(2481,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Divides_Odivmod__int__rel @ F @ E ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ tc_Int_Oint @ tc_Int_Oint ) @ D ) @ C ) ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Divides_Odivmod__int__rel @ F @ E ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ tc_Int_Oint @ tc_Int_Oint ) @ B ) @ A ) ) )
       => ( ( E
           != ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) )
         => ( D = B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[22]) ).

thf(1217,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C = B )
     => ~ ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ A @ tc_HOL_Obool ) ) @ C ) @ B ) )
         => ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ A @ tc_HOL_Obool ) ) @ B ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_equalityE) ).

thf(7002,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( C = B )
     => ~ ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ A @ tc_HOL_Obool ) ) @ C ) @ B ) )
         => ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ A @ tc_HOL_Obool ) ) @ B ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1217]) ).

thf(1521,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( c_List_Oset @ tc_Nat_Onat ) @ ( c_List_Oupt @ B @ A ) )
      = ( hAPP @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Nat_Onat @ B ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_set__upt) ).

thf(8308,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( c_List_Oset @ tc_Nat_Onat ) @ ( c_List_Oupt @ B @ A ) )
      = ( hAPP @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Nat_Onat @ B ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1521]) ).

thf(62,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Ocomm__monoid__add @ D )
     => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ tc_Nat_Onat @ D ) @ C ) @ ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Nat_Onat @ ( hAPP @ c_Nat_OSuc @ B ) @ ( hAPP @ c_Nat_OSuc @ A ) ) )
        = ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ tc_Nat_Onat @ D ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_Nat_Onat @ D @ tc_Nat_Onat ) @ C ) @ c_Nat_OSuc ) ) @ ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Nat_Onat @ B @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_setsum__shift__bounds__cl__Suc__ivl) ).

thf(2620,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Ocomm__monoid__add @ D )
     => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ tc_Nat_Onat @ D ) @ C ) @ ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Nat_Onat @ ( hAPP @ c_Nat_OSuc @ B ) @ ( hAPP @ c_Nat_OSuc @ A ) ) )
        = ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ tc_Nat_Onat @ D ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_Nat_Onat @ D @ tc_Nat_Onat ) @ C ) @ c_Nat_OSuc ) ) @ ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Nat_Onat @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[62]) ).

thf(1938,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( c_Sum__Type_OInl @ D @ C ) @ B )
     != ( hAPP @ ( c_Sum__Type_OInr @ C @ D ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Inl__not__Inr) ).

thf(10020,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( c_Sum__Type_OInl @ D @ C ) @ B )
     != ( hAPP @ ( c_Sum__Type_OInr @ C @ D ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1938]) ).

thf(1445,axiom,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ A ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ A @ tc_HOL_Obool ) ) ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ ( hAPP @ ( c_Finite__Set_Ocard @ A ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ A @ tc_HOL_Obool ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_finite__UNIV__card__ge__0) ).

thf(7929,plain,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ A ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ A @ tc_HOL_Obool ) ) ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ ( hAPP @ ( c_Finite__Set_Ocard @ A ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ A @ tc_HOL_Obool ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1445]) ).

thf(2153,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ D ) @ ( hAPP @ ( c_List_Oset @ E ) @ C ) ) )
       => ( ( hAPP @ ( c_Map_Omap__of @ E @ B @ ( hAPP @ ( hAPP @ ( c_List_Ozip @ E @ B ) @ C ) @ ( hAPP @ ( hAPP @ ( c_List_Omap @ E @ B ) @ A ) @ C ) ) ) @ D )
          = ( hAPP @ ( c_Option_Ooption_OSome @ B ) @ ( hAPP @ A @ D ) ) ) )
      & ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ D ) @ ( hAPP @ ( c_List_Oset @ E ) @ C ) ) )
       => ( ( hAPP @ ( c_Map_Omap__of @ E @ B @ ( hAPP @ ( hAPP @ ( c_List_Ozip @ E @ B ) @ C ) @ ( hAPP @ ( hAPP @ ( c_List_Omap @ E @ B ) @ A ) @ C ) ) ) @ D )
          = ( c_Option_Ooption_ONone @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_map__of__zip__map) ).

thf(10852,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ D ) @ ( hAPP @ ( c_List_Oset @ E ) @ C ) ) )
       => ( ( hAPP @ ( c_Map_Omap__of @ E @ B @ ( hAPP @ ( hAPP @ ( c_List_Ozip @ E @ B ) @ C ) @ ( hAPP @ ( hAPP @ ( c_List_Omap @ E @ B ) @ A ) @ C ) ) ) @ D )
          = ( hAPP @ ( c_Option_Ooption_OSome @ B ) @ ( hAPP @ A @ D ) ) ) )
      & ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ D ) @ ( hAPP @ ( c_List_Oset @ E ) @ C ) ) )
       => ( ( hAPP @ ( c_Map_Omap__of @ E @ B @ ( hAPP @ ( hAPP @ ( c_List_Ozip @ E @ B ) @ C ) @ ( hAPP @ ( hAPP @ ( c_List_Omap @ E @ B ) @ A ) @ C ) ) ) @ D )
          = ( c_Option_Ooption_ONone @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2153]) ).

thf(2064,axiom,
    ! [A: $i,B: $i] :
      ( ( B
       != ( c_Option_Ooption_ONone @ A ) )
     => ~ ! [C: $i] :
            ( B
           != ( hAPP @ ( c_Option_Ooption_OSome @ A ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_option_Oexhaust) ).

thf(10514,plain,
    ! [A: $i,B: $i] :
      ( ( B
       != ( c_Option_Ooption_ONone @ A ) )
     => ~ ! [C: $i] :
            ( B
           != ( hAPP @ ( c_Option_Ooption_OSome @ A ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2064]) ).

thf(1864,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ F ) @ E ) @ D ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ B ) @ ( hAPP @ A @ E ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ F @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ F @ C ) @ E ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ F @ C ) @ D ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_SigmaI) ).

thf(9741,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ F ) @ E ) @ D ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ B ) @ ( hAPP @ A @ E ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ F @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ F @ C ) @ E ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ F @ C ) @ D ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1864]) ).

thf(1890,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ B ) ) @ ( hAPP @ c_Nat_OSuc @ A ) ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Suc__less__SucD) ).

thf(9841,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ B ) ) @ ( hAPP @ c_Nat_OSuc @ A ) ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1890]) ).

thf(1321,axiom,
    ! [A: $i,B: $i] :
      ( ( ( hAPP @ c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) )
    <=> ( ( ( B
            = ( hAPP @ c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) )
          & ( A
            = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) )
        | ( ( B
            = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
          & ( A
            = ( hAPP @ c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_one__is__add) ).

thf(7469,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( hAPP @ c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
          = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) )
       => ( ( ( B
              = ( hAPP @ c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) )
            & ( A
              = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) )
          | ( ( B
              = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
            & ( A
              = ( hAPP @ c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) ) ) )
      & ( ( ( ( B
              = ( hAPP @ c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) )
            & ( A
              = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) )
          | ( ( B
              = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
            & ( A
              = ( hAPP @ c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) ) )
       => ( ( hAPP @ c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
          = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1321]) ).

thf(1040,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ ( hAPP @ ( c_Finite__Set_Ocard @ B ) @ A ) ) )
    <=> ( ( A
         != ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) )
        & ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_card__gt__0__iff) ).

thf(6223,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ ( hAPP @ ( c_Finite__Set_Ocard @ B ) @ A ) ) )
       => ( ( A
           != ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) )
          & ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ B ) @ A ) ) ) )
      & ( ( ( A
           != ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) )
          & ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ B ) @ A ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ ( hAPP @ ( c_Finite__Set_Ocard @ B ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1040]) ).

thf(361,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ B ) )
     => ( ( hAPP @ ( c_Finite__Set_Ocard @ tc_Nat_Onat ) @ ( hAPP @ ( c_Set_OCollect @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ tc_Nat_Onat @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ tc_Nat_Onat ) @ c_fconj ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( c_member @ tc_Nat_Onat ) ) @ B ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) ) @ ( hAPP @ c_Nat_OSuc @ A ) ) ) ) )
       != ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_card__less) ).

thf(3687,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ B ) )
     => ( ( hAPP @ ( c_Finite__Set_Ocard @ tc_Nat_Onat ) @ ( hAPP @ ( c_Set_OCollect @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ tc_Nat_Onat @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ tc_Nat_Onat ) @ c_fconj ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( c_member @ tc_Nat_Onat ) ) @ B ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) ) @ ( hAPP @ c_Nat_OSuc @ A ) ) ) ) )
       != ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[361]) ).

thf(389,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ E @ D ) @ C ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ E @ ( hAPP @ ( c_Product__Type_Osnd @ B @ A ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ B @ A ) @ C ) @ D ) ) ) @ ( hAPP @ ( c_Product__Type_Ofst @ B @ A ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ B @ A ) @ C ) @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_exI__realizer) ).

thf(3806,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ E @ D ) @ C ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ E @ ( hAPP @ ( c_Product__Type_Osnd @ B @ A ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ B @ A ) @ C ) @ D ) ) ) @ ( hAPP @ ( c_Product__Type_Ofst @ B @ A ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ B @ A ) @ C ) @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[389]) ).

thf(472,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Lattices_Oab__semigroup__idem__mult @ C )
     => ( ( B
         != ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ B ) )
         => ( ( hAPP @ ( c_Finite__Set_Ofold1 @ C @ ( c_Groups_Otimes__class_Otimes @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ A ) @ B ) )
            = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ A ) @ ( hAPP @ ( c_Finite__Set_Ofold1 @ C @ ( c_Groups_Otimes__class_Otimes @ C ) ) @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_fold1__insert__idem) ).

thf(4085,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Lattices_Oab__semigroup__idem__mult @ C )
     => ( ( B
         != ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ B ) )
         => ( ( hAPP @ ( c_Finite__Set_Ofold1 @ C @ ( c_Groups_Otimes__class_Otimes @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ A ) @ B ) )
            = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ A ) @ ( hAPP @ ( c_Finite__Set_Ofold1 @ C @ ( c_Groups_Otimes__class_Otimes @ C ) ) @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[472]) ).

thf(208,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ D ) @ C ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ ( hAPP @ A @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ E @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_UN__upper) ).

thf(3159,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ D ) @ C ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ ( hAPP @ A @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ E @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ C ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[208]) ).

thf(1774,axiom,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Nat_Onat ) @ B ) @ A ) ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_div__le__dividend) ).

thf(9371,plain,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Nat_Onat ) @ B ) @ A ) ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1774]) ).

thf(356,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Int_Oint ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Int_Oint ) @ B ) @ A ) ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Int_Oint ) @ C ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Int_Oint ) @ C ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zdvd__zdiffD) ).

thf(3674,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Int_Oint ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Int_Oint ) @ B ) @ A ) ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Int_Oint ) @ C ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Int_Oint ) @ C ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[356]) ).

thf(675,axiom,
    hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Nat__Transfer_Otransfer__nat__int__function__closures_I5_J) ).

thf(4827,plain,
    hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[675]) ).

thf(145,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ E @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ E @ ( tc_fun @ D @ tc_HOL_Obool ) @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ D @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ D @ tc_HOL_Obool ) @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ E ) @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) @ B ) ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ E @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_UN__simps_I6_J) ).

thf(2931,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ E @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ E @ ( tc_fun @ D @ tc_HOL_Obool ) @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ D @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ D @ tc_HOL_Obool ) @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ E ) @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) @ B ) ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ E @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[145]) ).

thf(1910,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Com_Ocom_OAss @ E @ D )
     != ( c_Com_Ocom_OCall @ C @ B @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_com_Osimps_I32_J) ).

thf(9916,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Com_Ocom_OAss @ E @ D )
     != ( c_Com_Ocom_OCall @ C @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1910]) ).

thf(1728,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ C ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ C ) @ B ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Nat_Onat ) @ A ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Nat_Onat ) @ A ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_div__le__mono2) ).

thf(9197,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ C ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ C ) @ B ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Nat_Onat ) @ A ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Nat_Onat ) @ A ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1728]) ).

thf(430,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Int_Oint @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ B ) @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ) ) @ A )
      = ( c_SetInterval_Oord__class_OgreaterThanLessThan @ tc_Int_Oint @ B @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_atLeastPlusOneLessThan__greaterThanLessThan__int) ).

thf(3953,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Int_Oint @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ B ) @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ) ) @ A )
      = ( c_SetInterval_Oord__class_OgreaterThanLessThan @ tc_Int_Oint @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[430]) ).

thf(235,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Equiv__Relations_Oquotient @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ B ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Relation_OImage @ C @ C @ A ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ B ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) ) ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ ( tc_fun @ C @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_singleton__quotient) ).

thf(3241,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Equiv__Relations_Oquotient @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ B ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Relation_OImage @ C @ C @ A ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ B ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) ) ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ ( tc_fun @ C @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[235]) ).

thf(1293,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ B ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) ) ) @ A ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ B ) @ C ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ B ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_diff__single__insert) ).

thf(7324,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ B ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) ) ) @ A ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ B ) @ C ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ B ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1293]) ).

thf(169,axiom,
    ! [A: $i,B: $i] :
      ( B
      = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ B ) @ A ) ) ) @ ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ B @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zmod__zdiv__equality) ).

thf(3016,plain,
    ! [A: $i,B: $i] :
      ( B
      = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ B ) @ A ) ) ) @ ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[169]) ).

thf(1995,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ C ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( c_Finite__Set_Ocard @ B ) @ ( hAPP @ ( c_Set_Oimage @ D @ B @ A ) @ C ) ) ) @ ( hAPP @ ( c_Finite__Set_Ocard @ D ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_card__image__le) ).

thf(10264,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ C ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( c_Finite__Set_Ocard @ B ) @ ( hAPP @ ( c_Set_Oimage @ D @ B @ A ) @ C ) ) ) @ ( hAPP @ ( c_Finite__Set_Ocard @ D ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1995]) ).

thf(1700,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ E ) @ D ) )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ D ) @ C ) ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ E ) @ C ) ) @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ D ) @ E ) ) @ C ) ) @ B ) ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nat__diff__add__eq1) ).

thf(9106,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ E ) @ D ) )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ D ) @ C ) ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ E ) @ C ) ) @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ D ) @ E ) ) @ C ) ) @ B ) ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1700]) ).

thf(1856,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_List_Odistinct @ D ) @ ( hAPP @ ( hAPP @ ( c_List_Omap @ C @ D ) @ B ) @ A ) ) )
    <=> ( ( hBOOL @ ( hAPP @ ( c_List_Odistinct @ C ) @ A ) )
        & ( c_Fun_Oinj__on @ C @ D @ B @ ( hAPP @ ( c_List_Oset @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_distinct__map) ).

thf(9698,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( c_List_Odistinct @ D ) @ ( hAPP @ ( hAPP @ ( c_List_Omap @ C @ D ) @ B ) @ A ) ) )
       => ( ( hBOOL @ ( hAPP @ ( c_List_Odistinct @ C ) @ A ) )
          & ( c_Fun_Oinj__on @ C @ D @ B @ ( hAPP @ ( c_List_Oset @ C ) @ A ) ) ) )
      & ( ( ( hBOOL @ ( hAPP @ ( c_List_Odistinct @ C ) @ A ) )
          & ( c_Fun_Oinj__on @ C @ D @ B @ ( hAPP @ ( c_List_Oset @ C ) @ A ) ) )
       => ( hBOOL @ ( hAPP @ ( c_List_Odistinct @ D ) @ ( hAPP @ ( hAPP @ ( c_List_Omap @ C @ D ) @ B ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1856]) ).

thf(1773,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ A ) ) )
    <=> ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nat__add__left__cancel__le) ).

thf(9366,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ A ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ A ) ) )
      & ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1773]) ).

thf(1947,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ ( tc_prod @ B @ B ) ) @ ( c_Transitive__Closure_Otrancl @ B @ A ) ) )
    <=> ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ ( tc_prod @ B @ B ) ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_finite__trancl) ).

thf(10063,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ ( tc_prod @ B @ B ) ) @ ( c_Transitive__Closure_Otrancl @ B @ A ) ) )
       => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ ( tc_prod @ B @ B ) ) @ A ) ) )
      & ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ ( tc_prod @ B @ B ) ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ ( tc_prod @ B @ B ) ) @ ( c_Transitive__Closure_Otrancl @ B @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1947]) ).

thf(2340,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ D ) @ ( hAPP @ ( c_Set_Ovimage @ E @ C @ B ) @ A ) ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ ( hAPP @ B @ D ) ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_vimageD) ).

thf(11605,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ D ) @ ( hAPP @ ( c_Set_Ovimage @ E @ C @ B ) @ A ) ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ ( hAPP @ B @ D ) ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2340]) ).

thf(2182,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( c_Sum__Type_OInr @ D @ C ) @ B )
     != ( hAPP @ ( c_Sum__Type_OInl @ C @ D ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Inr__not__Inl) ).

thf(10930,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( c_Sum__Type_OInr @ D @ C ) @ B )
     != ( hAPP @ ( c_Sum__Type_OInl @ C @ D ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2182]) ).

thf(1009,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hAPP @ ( c_Set_Oimage @ F @ E @ D ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ C @ ( tc_fun @ F @ tc_HOL_Obool ) ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ C @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ F @ tc_HOL_Obool ) @ ( tc_fun @ E @ tc_HOL_Obool ) @ C ) @ ( c_Set_Oimage @ F @ E @ D ) ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_image__UN) ).

thf(6100,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hAPP @ ( c_Set_Oimage @ F @ E @ D ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ C @ ( tc_fun @ F @ tc_HOL_Obool ) ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ C @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ F @ tc_HOL_Obool ) @ ( tc_fun @ E @ tc_HOL_Obool ) @ C ) @ ( c_Set_Oimage @ F @ E @ D ) ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1009]) ).

thf(926,axiom,
    ! [A: $i,B: $i,C: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Un__upper1) ).

thf(5725,plain,
    ! [A: $i,B: $i,C: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[926]) ).

thf(2140,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Otake @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_List_Odrop @ D ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_List_Odrop @ D ) @ B ) @ ( hAPP @ ( hAPP @ ( c_List_Otake @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ B ) ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_take__drop) ).

thf(10803,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Otake @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_List_Odrop @ D ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_List_Odrop @ D ) @ B ) @ ( hAPP @ ( hAPP @ ( c_List_Otake @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ B ) ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2140]) ).

thf(203,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( c_Relation_ODomain @ B @ B ) @ ( c_Transitive__Closure_Ortrancl @ B @ A ) )
      = ( c_Orderings_Otop__class_Otop @ ( tc_fun @ B @ tc_HOL_Obool ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Domain__rtrancl) ).

thf(3142,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( c_Relation_ODomain @ B @ B ) @ ( c_Transitive__Closure_Ortrancl @ B @ A ) )
      = ( c_Orderings_Otop__class_Otop @ ( tc_fun @ B @ tc_HOL_Obool ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[203]) ).

thf(41,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( class_Groups_Ocomm__monoid__add @ G )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ F ) @ E ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ C ) )
         => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ F @ G ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ F @ ( tc_fun @ D @ tc_HOL_Obool ) @ G ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ D @ G ) @ ( tc_fun @ ( tc_fun @ D @ tc_HOL_Obool ) @ G ) @ F ) @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ G ) ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ D @ tc_HOL_Obool ) @ ( tc_fun @ D @ tc_HOL_Obool ) @ F ) @ ( c_Set_OCollect @ D ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ D @ tc_HOL_Obool ) @ ( tc_fun @ D @ tc_HOL_Obool ) @ F ) @ ( hAPP @ ( c_COMBS @ D @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ D ) @ c_fconj ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ ( tc_fun @ D @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( c_member @ D ) ) @ C ) ) ) ) @ A ) ) ) ) @ E )
            = ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ G ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ D @ ( tc_fun @ F @ tc_HOL_Obool ) @ G ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ F @ G ) @ ( tc_fun @ ( tc_fun @ F @ tc_HOL_Obool ) @ G ) @ D ) @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ F @ G ) ) @ ( hAPP @ ( c_COMBC @ F @ D @ G ) @ B ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ F @ tc_HOL_Obool ) @ ( tc_fun @ F @ tc_HOL_Obool ) @ D ) @ ( c_Set_OCollect @ F ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ F @ tc_HOL_Obool ) @ ( tc_fun @ F @ tc_HOL_Obool ) @ D ) @ ( hAPP @ ( c_COMBS @ F @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ F ) @ c_fconj ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ F @ ( tc_fun @ F @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( c_member @ F ) ) @ E ) ) ) ) @ ( hAPP @ ( c_COMBC @ F @ D @ tc_HOL_Obool ) @ A ) ) ) ) ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_setsum__setsum__restrict) ).

thf(2535,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( class_Groups_Ocomm__monoid__add @ G )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ F ) @ E ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ C ) )
         => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ F @ G ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ F @ ( tc_fun @ D @ tc_HOL_Obool ) @ G ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ D @ G ) @ ( tc_fun @ ( tc_fun @ D @ tc_HOL_Obool ) @ G ) @ F ) @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ G ) ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ D @ tc_HOL_Obool ) @ ( tc_fun @ D @ tc_HOL_Obool ) @ F ) @ ( c_Set_OCollect @ D ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ D @ tc_HOL_Obool ) @ ( tc_fun @ D @ tc_HOL_Obool ) @ F ) @ ( hAPP @ ( c_COMBS @ D @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ D ) @ c_fconj ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ ( tc_fun @ D @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( c_member @ D ) ) @ C ) ) ) ) @ A ) ) ) ) @ E )
            = ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ G ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ D @ ( tc_fun @ F @ tc_HOL_Obool ) @ G ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ F @ G ) @ ( tc_fun @ ( tc_fun @ F @ tc_HOL_Obool ) @ G ) @ D ) @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ F @ G ) ) @ ( hAPP @ ( c_COMBC @ F @ D @ G ) @ B ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ F @ tc_HOL_Obool ) @ ( tc_fun @ F @ tc_HOL_Obool ) @ D ) @ ( c_Set_OCollect @ F ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ F @ tc_HOL_Obool ) @ ( tc_fun @ F @ tc_HOL_Obool ) @ D ) @ ( hAPP @ ( c_COMBS @ F @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ F ) @ c_fconj ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ F @ ( tc_fun @ F @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( c_member @ F ) ) @ E ) ) ) ) @ ( hAPP @ ( c_COMBC @ F @ D @ tc_HOL_Obool ) @ A ) ) ) ) ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[41]) ).

thf(1601,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ C ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_diff__mult__distrib) ).

thf(8676,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ C ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1601]) ).

thf(2118,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Com_Ocom_OAss @ E @ D )
     != ( c_Com_Ocom_OLocal @ C @ B @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_com_Osimps_I22_J) ).

thf(10707,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Com_Ocom_OAss @ E @ D )
     != ( c_Com_Ocom_OLocal @ C @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2118]) ).

thf(946,axiom,
    ! [A: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zadd__0__right) ).

thf(5817,plain,
    ! [A: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[946]) ).

thf(1685,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( ( hAPP @ G @ F )
        = ( hAPP @ ( c_Option_Ooption_OSome @ E ) @ D ) )
     => ( ( c_Map_Omap__comp @ E @ C @ B @ A @ G @ F )
        = ( hAPP @ A @ D ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_map__comp__simps_I2_J) ).

thf(9033,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( ( hAPP @ G @ F )
        = ( hAPP @ ( c_Option_Ooption_OSome @ E ) @ D ) )
     => ( ( c_Map_Omap__comp @ E @ C @ B @ A @ G @ F )
        = ( hAPP @ A @ D ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1685]) ).

thf(1340,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ D ) @ C ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ tc_Nat_Onat @ tc_Nat_Onat ) @ D ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ tc_Nat_Onat @ tc_Nat_Onat ) @ C ) @ A ) ) ) @ c_FunDef_Opair__less ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_pair__lessI1) ).

thf(7558,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ D ) @ C ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ tc_Nat_Onat @ tc_Nat_Onat ) @ D ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ tc_Nat_Onat @ tc_Nat_Onat ) @ C ) @ A ) ) ) @ c_FunDef_Opair__less ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1340]) ).

thf(909,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( hAPP @ ( c_Product__Type_Osnd @ E @ D ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ D ) @ C ) @ B ) )
        = A )
     => ( B = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_snd__eqD) ).

thf(5667,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( hAPP @ ( c_Product__Type_Osnd @ E @ D ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ D ) @ C ) @ B ) )
        = A )
     => ( B = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[909]) ).

thf(1298,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ F @ E @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ F @ E ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ C @ B ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_split__conv) ).

thf(7339,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ F @ E @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ F @ E ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ C @ B ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1298]) ).

thf(500,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ F @ E ) @ D ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ C @ E ) @ D ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ B @ ( tc_fun @ A @ tc_HOL_Obool ) ) ) @ F ) @ C ) @ E ) @ D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_inf2I) ).

thf(4196,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ F @ E ) @ D ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ C @ E ) @ D ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ B @ ( tc_fun @ A @ tc_HOL_Obool ) ) ) @ F ) @ C ) @ E ) @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[500]) ).

thf(1065,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( c_Relation_ORange @ C @ B ) @ A )
      = ( hAPP @ ( c_Relation_ODomain @ B @ C ) @ ( hAPP @ ( c_Relation_Oconverse @ C @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Range__def) ).

thf(6325,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( c_Relation_ORange @ C @ B ) @ A )
      = ( hAPP @ ( c_Relation_ODomain @ B @ C ) @ ( hAPP @ ( c_Relation_Oconverse @ C @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1065]) ).

thf(165,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( E
          = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) )
       => ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ D @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ E ) @ A ) )
          = B ) )
      & ( ( E
         != ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) )
       => ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ D @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ E ) @ A ) )
          = ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ D @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ E ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) @ D ) @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_UN__extend__simps_I3_J) ).

thf(2999,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( E
          = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) )
       => ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ D @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ E ) @ A ) )
          = B ) )
      & ( ( E
         != ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) )
       => ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ D @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ E ) @ A ) )
          = ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ D @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ E ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) @ D ) @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[165]) ).

thf(263,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ B ) )
     => ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ C ) @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Int__insert__right__if1) ).

thf(3348,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ B ) )
     => ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ C ) @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[263]) ).

thf(750,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( c_Set_Oimage @ C @ B @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) )
      = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_image__empty) ).

thf(5089,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( c_Set_Oimage @ C @ B @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) )
      = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[750]) ).

thf(465,axiom,
    ! [A: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ A )
      = ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zdiv__zero) ).

thf(4067,plain,
    ! [A: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ A )
      = ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[465]) ).

thf(1440,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( c_Finite__Set_Ocard @ tc_Nat_Onat ) @ ( c_SetInterval_Oord__class_OgreaterThanAtMost @ tc_Nat_Onat @ B @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ A ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_card__greaterThanAtMost) ).

thf(7916,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( c_Finite__Set_Ocard @ tc_Nat_Onat ) @ ( c_SetInterval_Oord__class_OgreaterThanAtMost @ tc_Nat_Onat @ B @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ A ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1440]) ).

thf(761,axiom,
    ! [A: $i] :
      ( ( c_Option_Oset @ A @ ( c_Option_Ooption_ONone @ A ) )
      = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ A @ tc_HOL_Obool ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Option_Oset_Osimps_I1_J) ).

thf(5139,plain,
    ! [A: $i] :
      ( ( c_Option_Oset @ A @ ( c_Option_Ooption_ONone @ A ) )
      = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ A @ tc_HOL_Obool ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[761]) ).

thf(416,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hAPP @ ( c_Set_Ovimage @ F @ ( tc_prod @ E @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ E @ D ) @ B ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ D @ tc_HOL_Obool ) @ E ) @ A ) ) )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ F @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Set_Ovimage @ F @ E @ ( hAPP @ ( c_Fun_Ocomp @ ( tc_prod @ E @ D ) @ E @ F @ ( c_Product__Type_Ofst @ E @ D ) ) @ C ) ) @ B ) ) @ ( hAPP @ ( c_Set_Ovimage @ F @ D @ ( hAPP @ ( c_Fun_Ocomp @ ( tc_prod @ E @ D ) @ D @ F @ ( c_Product__Type_Osnd @ E @ D ) ) @ C ) ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_vimage__Times) ).

thf(3908,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hAPP @ ( c_Set_Ovimage @ F @ ( tc_prod @ E @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ E @ D ) @ B ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ D @ tc_HOL_Obool ) @ E ) @ A ) ) )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ F @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Set_Ovimage @ F @ E @ ( hAPP @ ( c_Fun_Ocomp @ ( tc_prod @ E @ D ) @ E @ F @ ( c_Product__Type_Ofst @ E @ D ) ) @ C ) ) @ B ) ) @ ( hAPP @ ( c_Set_Ovimage @ F @ D @ ( hAPP @ ( c_Fun_Ocomp @ ( tc_prod @ E @ D ) @ D @ F @ ( c_Product__Type_Osnd @ E @ D ) ) @ C ) ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[416]) ).

thf(1268,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Relation_Orefl__on @ C @ B @ A )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ C @ C @ C ) @ ( hAPP @ ( c_Relation_Oconverse @ C @ C ) @ A ) ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_refl__on__comp__subset) ).

thf(7232,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Relation_Orefl__on @ C @ B @ A )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ C @ C @ C ) @ ( hAPP @ ( c_Relation_Oconverse @ C @ C ) @ A ) ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1268]) ).

thf(1903,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( hAPP @ ( c_Map_Omap__of @ E @ D @ C ) @ B )
        = ( hAPP @ ( c_Option_Ooption_OSome @ D ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ D ) @ B ) @ A ) ) @ ( hAPP @ ( c_List_Oset @ ( tc_prod @ E @ D ) ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_map__of__SomeD) ).

thf(9894,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( hAPP @ ( c_Map_Omap__of @ E @ D @ C ) @ B )
        = ( hAPP @ ( c_Option_Ooption_OSome @ D ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ D ) @ B ) @ A ) ) @ ( hAPP @ ( c_List_Oset @ ( tc_prod @ E @ D ) ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1903]) ).

thf(1429,axiom,
    ! [A: $i] :
      ( ( A
       != ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_gr0I) ).

thf(7884,plain,
    ! [A: $i] :
      ( ( A
       != ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1429]) ).

thf(2060,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ B ) )
     => ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ A ) @ B ) )
       => ( ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ A ) @ B ) )
          = ( hAPP @ c_Nat_OSuc @ ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_card__insert__disjoint) ).

thf(10499,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ B ) )
     => ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ A ) @ B ) )
       => ( ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ A ) @ B ) )
          = ( hAPP @ c_Nat_OSuc @ ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2060]) ).

thf(2286,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Sum__Type_OProjl @ C @ B @ ( hAPP @ ( c_Sum__Type_OInl @ C @ B ) @ A ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Projl_Osimps) ).

thf(11361,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Sum__Type_OProjl @ C @ B @ ( hAPP @ ( c_Sum__Type_OInl @ C @ B ) @ A ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2286]) ).

thf(1811,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( c_Relation_ORange @ E @ D ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ ( tc_prod @ E @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ D ) @ C ) @ B ) ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ B ) @ ( hAPP @ ( c_Relation_ORange @ E @ D ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Range__insert) ).

thf(9503,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( c_Relation_ORange @ E @ D ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ ( tc_prod @ E @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ D ) @ C ) @ B ) ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ B ) @ ( hAPP @ ( c_Relation_ORange @ E @ D ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1811]) ).

thf(1160,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( c_Relation_ODomain @ B @ B ) @ ( c_Relation_OId__on @ B @ A ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Domain__Id__on) ).

thf(6785,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( c_Relation_ODomain @ B @ B ) @ ( c_Relation_OId__on @ B @ A ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1160]) ).

thf(31,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Omap @ D @ C ) @ B ) @ ( c_List_Oconcat @ D @ A ) )
      = ( c_List_Oconcat @ C @ ( hAPP @ ( hAPP @ ( c_List_Omap @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ C ) ) @ ( hAPP @ ( c_List_Omap @ D @ C ) @ B ) ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_map__concat) ).

thf(2506,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Omap @ D @ C ) @ B ) @ ( c_List_Oconcat @ D @ A ) )
      = ( c_List_Oconcat @ C @ ( hAPP @ ( hAPP @ ( c_List_Omap @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ C ) ) @ ( hAPP @ ( c_List_Omap @ D @ C ) @ B ) ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[31]) ).

thf(896,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ F @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ D ) @ ( hAPP @ ( c_Fun_Ocomp @ C @ ( tc_fun @ E @ tc_HOL_Obool ) @ F @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ C @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Set_Oimage @ F @ C @ A ) @ D ) ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_UN__o) ).

thf(5619,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ F @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ D ) @ ( hAPP @ ( c_Fun_Ocomp @ C @ ( tc_fun @ E @ tc_HOL_Obool ) @ F @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ C @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Set_Oimage @ F @ C @ A ) @ D ) ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[896]) ).

thf(1227,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ A ) ) )
    <=> ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ B ) )
        & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nat__0__less__mult__iff) ).

thf(7049,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ A ) ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ B ) )
          & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ A ) ) ) )
      & ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ B ) )
          & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ A ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1227]) ).

thf(2408,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] : ( c_Natural_Oevaln @ ( c_Com_Ocom_OAss @ D @ C ) @ B @ A @ ( hAPP @ ( hAPP @ ( hAPP @ c_Natural_Oupdate @ B ) @ D ) @ ( hAPP @ C @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_evaln_Oequations_I2_J) ).

thf(11887,plain,
    ! [A: $i,B: $i,C: $i,D: $i] : ( c_Natural_Oevaln @ ( c_Com_Ocom_OAss @ D @ C ) @ B @ A @ ( hAPP @ ( hAPP @ ( hAPP @ c_Natural_Oupdate @ B ) @ D ) @ ( hAPP @ C @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2408]) ).

thf(1255,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ B ) @ A ) ) )
    <=> ~ ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) )
            & ~ ( hBOOL @ ( hAPP @ C @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) )
          | ? [D: $i] :
              ( ( B
                = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ D ) )
              & ~ ( hBOOL @ ( hAPP @ C @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nat__diff__split__asm) ).

thf(7154,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ B ) @ A ) ) )
       => ~ ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) )
              & ~ ( hBOOL @ ( hAPP @ C @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) )
            | ? [D: $i] :
                ( ( B
                  = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ D ) )
                & ~ ( hBOOL @ ( hAPP @ C @ D ) ) ) ) )
      & ( ~ ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) )
              & ~ ( hBOOL @ ( hAPP @ C @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) )
            | ? [D: $i] :
                ( ( B
                  = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ D ) )
                & ~ ( hBOOL @ ( hAPP @ C @ D ) ) ) )
       => ( hBOOL @ ( hAPP @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ B ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1255]) ).

thf(2409,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ E @ D ) @ C ) )
     => ( hBOOL @ ( hAPP @ ( c_Predicate_ORangeP @ B @ A @ E ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_RangeP_Ointros) ).

thf(11889,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ E @ D ) @ C ) )
     => ( hBOOL @ ( hAPP @ ( c_Predicate_ORangeP @ B @ A @ E ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2409]) ).

thf(649,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) @ ( hAPP @ ( hAPP @ ( c_List_Odrop @ C ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) @ A ) ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_length__drop) ).

thf(4741,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) @ ( hAPP @ ( hAPP @ ( c_List_Odrop @ C ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) @ A ) ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[649]) ).

thf(151,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Relation_Oconverse @ C @ C ) @ B ) ) @ ( c_Transitive__Closure_Ortrancl @ C @ B ) )
        = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) ) )
     => ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ C @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ C @ C ) @ A ) @ A ) ) @ ( c_Transitive__Closure_Otrancl @ C @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_irrefl__tranclI) ).

thf(2952,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Relation_Oconverse @ C @ C ) @ B ) ) @ ( c_Transitive__Closure_Ortrancl @ C @ B ) )
        = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) ) )
     => ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ C @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ C @ C ) @ A ) @ A ) ) @ ( c_Transitive__Closure_Otrancl @ C @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[151]) ).

thf(592,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( c_Set_Ovimage @ D @ C @ B ) @ ( hAPP @ ( c_Set_OCollect @ C ) @ A ) )
      = ( hAPP @ ( c_Set_OCollect @ D ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ C @ tc_HOL_Obool @ D ) @ A ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_vimage__Collect__eq) ).

thf(4529,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( c_Set_Ovimage @ D @ C @ B ) @ ( hAPP @ ( c_Set_OCollect @ C ) @ A ) )
      = ( hAPP @ ( c_Set_OCollect @ D ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ C @ tc_HOL_Obool @ D ) @ A ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[592]) ).

thf(30,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_List_Oconcat @ C @ ( hAPP @ ( hAPP @ ( c_List_Oappend @ ( tc_List_Olist @ C ) ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_List_Oappend @ C ) @ ( c_List_Oconcat @ C @ B ) ) @ ( c_List_Oconcat @ C @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_concat__append) ).

thf(2503,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_List_Oconcat @ C @ ( hAPP @ ( hAPP @ ( c_List_Oappend @ ( tc_List_Olist @ C ) ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_List_Oappend @ C ) @ ( c_List_Oconcat @ C @ B ) ) @ ( c_List_Oconcat @ C @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[30]) ).

thf(460,axiom,
    ! [A: $i] :
      ( ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ A @ A )
      = ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zmod__self) ).

thf(4052,plain,
    ! [A: $i] :
      ( ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ A @ A )
      = ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[460]) ).

thf(2206,axiom,
    ! [A: $i,B: $i] :
      ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ c_fequal @ B ) @ A ) )
      | ( B = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_c__fequal__1) ).

thf(11030,plain,
    ! [A: $i,B: $i] :
      ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ c_fequal @ B ) @ A ) )
      | ( B = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2206]) ).

thf(525,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( class_Groups_Ocomm__monoid__mult @ F )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ E ) @ D ) )
       => ( ! [G: $i] :
              ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ G ) @ D ) )
             => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ ( hAPP @ B @ G ) ) ) )
         => ( ( hAPP @ ( hAPP @ ( hAPP @ ( c_Finite__Set_Ofold__image @ F @ E @ ( c_Groups_Otimes__class_Otimes @ F ) ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ E @ ( tc_fun @ C @ tc_HOL_Obool ) @ F ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ E @ F @ ( tc_fun @ ( tc_fun @ C @ tc_HOL_Obool ) @ F ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ F ) @ ( tc_fun @ F @ ( tc_fun @ ( tc_fun @ C @ tc_HOL_Obool ) @ F ) ) @ E ) @ ( c_Finite__Set_Ofold__image @ F @ C @ ( c_Groups_Otimes__class_Otimes @ F ) ) ) @ A ) ) @ ( c_Groups_Oone__class_Oone @ F ) ) ) @ B ) ) @ ( c_Groups_Oone__class_Oone @ F ) ) @ D )
            = ( hAPP @ ( hAPP @ ( hAPP @ ( c_Finite__Set_Ofold__image @ F @ ( tc_prod @ E @ C ) @ ( c_Groups_Otimes__class_Otimes @ F ) ) @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ E @ C @ F ) @ A ) ) @ ( c_Groups_Oone__class_Oone @ F ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ E @ C ) @ D ) @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_fold__image__Sigma) ).

thf(4292,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( class_Groups_Ocomm__monoid__mult @ F )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ E ) @ D ) )
       => ( ! [G: $i] :
              ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ G ) @ D ) )
             => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ ( hAPP @ B @ G ) ) ) )
         => ( ( hAPP @ ( hAPP @ ( hAPP @ ( c_Finite__Set_Ofold__image @ F @ E @ ( c_Groups_Otimes__class_Otimes @ F ) ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ E @ ( tc_fun @ C @ tc_HOL_Obool ) @ F ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ E @ F @ ( tc_fun @ ( tc_fun @ C @ tc_HOL_Obool ) @ F ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ F ) @ ( tc_fun @ F @ ( tc_fun @ ( tc_fun @ C @ tc_HOL_Obool ) @ F ) ) @ E ) @ ( c_Finite__Set_Ofold__image @ F @ C @ ( c_Groups_Otimes__class_Otimes @ F ) ) ) @ A ) ) @ ( c_Groups_Oone__class_Oone @ F ) ) ) @ B ) ) @ ( c_Groups_Oone__class_Oone @ F ) ) @ D )
            = ( hAPP @ ( hAPP @ ( hAPP @ ( c_Finite__Set_Ofold__image @ F @ ( tc_prod @ E @ C ) @ ( c_Groups_Otimes__class_Otimes @ F ) ) @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ E @ C @ F ) @ A ) ) @ ( c_Groups_Oone__class_Oone @ F ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ E @ C ) @ D ) @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[525]) ).

thf(1005,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i] :
          ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ C ) )
         => ( ( hBOOL @ ( hAPP @ B @ C ) )
          <=> ( hBOOL @ ( hAPP @ A @ C ) ) ) )
     => ( ( hAPP @ ( c_Set_OCollect @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ tc_Int_Oint @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ tc_Int_Oint ) @ c_fconj ) @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) ) ) @ B ) )
        = ( hAPP @ ( c_Set_OCollect @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ tc_Int_Oint @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ tc_Int_Oint ) @ c_fconj ) @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) ) ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_transfer__nat__int__set__cong) ).

thf(6076,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i] :
          ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ C ) )
         => ( ( ( hBOOL @ ( hAPP @ B @ C ) )
             => ( hBOOL @ ( hAPP @ A @ C ) ) )
            & ( ( hBOOL @ ( hAPP @ A @ C ) )
             => ( hBOOL @ ( hAPP @ B @ C ) ) ) ) )
     => ( ( hAPP @ ( c_Set_OCollect @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ tc_Int_Oint @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ tc_Int_Oint ) @ c_fconj ) @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) ) ) @ B ) )
        = ( hAPP @ ( c_Set_OCollect @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ tc_Int_Oint @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ tc_Int_Oint ) @ c_fconj ) @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) ) ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1005]) ).

thf(1246,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Odrop @ B ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ A )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_drop__0) ).

thf(7122,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Odrop @ B ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ A )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1246]) ).

thf(2377,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Hoare__Mirabelle_Otriple__valid @ C @ ( hAPP @ c_Nat_OSuc @ B ) @ A )
     => ( c_Hoare__Mirabelle_Otriple__valid @ C @ B @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_triple__valid__Suc) ).

thf(11758,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Hoare__Mirabelle_Otriple__valid @ C @ ( hAPP @ c_Nat_OSuc @ B ) @ A )
     => ( c_Hoare__Mirabelle_Otriple__valid @ C @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2377]) ).

thf(402,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ ( tc_fun @ D @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ ( tc_fun @ D @ tc_HOL_Obool ) @ ( tc_fun @ D @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ D @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ D @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( c_Complete__Lattice_OSup__class_OSup @ ( tc_fun @ D @ tc_HOL_Obool ) @ B ) ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_UN__extend__simps_I8_J) ).

thf(3849,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ ( tc_fun @ D @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ ( tc_fun @ D @ tc_HOL_Obool ) @ ( tc_fun @ D @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ D @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ D @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( c_Complete__Lattice_OSup__class_OSup @ ( tc_fun @ D @ tc_HOL_Obool ) @ B ) ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[402]) ).

thf(1921,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i,I: $i,J: $i,K: $i,L: $i] :
      ( ( c_Com_Ocom_Ocom__rec @ L @ K @ J @ I @ H @ G @ F @ E @ D @ ( c_Com_Ocom_OCond @ C @ B @ A ) )
      = ( hAPP @ ( hAPP @ ( hAPP @ ( hAPP @ ( hAPP @ G @ C ) @ B ) @ A ) @ ( c_Com_Ocom_Ocom__rec @ L @ K @ J @ I @ H @ G @ F @ E @ D @ B ) ) @ ( c_Com_Ocom_Ocom__rec @ L @ K @ J @ I @ H @ G @ F @ E @ D @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_com_Orecs_I5_J) ).

thf(9958,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i,I: $i,J: $i,K: $i,L: $i] :
      ( ( c_Com_Ocom_Ocom__rec @ L @ K @ J @ I @ H @ G @ F @ E @ D @ ( c_Com_Ocom_OCond @ C @ B @ A ) )
      = ( hAPP @ ( hAPP @ ( hAPP @ ( hAPP @ ( hAPP @ G @ C ) @ B ) @ A ) @ ( c_Com_Ocom_Ocom__rec @ L @ K @ J @ I @ H @ G @ F @ E @ D @ B ) ) @ ( c_Com_Ocom_Ocom__rec @ L @ K @ J @ I @ H @ G @ F @ E @ D @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1921]) ).

thf(407,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Ozip @ F @ E ) @ ( hAPP @ ( hAPP @ ( c_List_Omap @ D @ F ) @ C ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_List_Omap @ ( tc_prod @ D @ E ) @ ( tc_prod @ F @ E ) ) @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ D @ E @ ( tc_prod @ F @ E ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ F @ ( tc_fun @ E @ ( tc_prod @ F @ E ) ) @ D ) @ ( c_Product__Type_OPair @ F @ E ) ) @ C ) ) ) @ ( hAPP @ ( hAPP @ ( c_List_Ozip @ D @ E ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zip__map1) ).

thf(3881,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Ozip @ F @ E ) @ ( hAPP @ ( hAPP @ ( c_List_Omap @ D @ F ) @ C ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_List_Omap @ ( tc_prod @ D @ E ) @ ( tc_prod @ F @ E ) ) @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ D @ E @ ( tc_prod @ F @ E ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ F @ ( tc_fun @ E @ ( tc_prod @ F @ E ) ) @ D ) @ ( c_Product__Type_OPair @ F @ E ) ) @ C ) ) ) @ ( hAPP @ ( hAPP @ ( c_List_Ozip @ D @ E ) @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[407]) ).

thf(907,axiom,
    ! [A: $i] :
      ( ( c_Complete__Lattice_OSup__class_OSup @ ( tc_fun @ A @ tc_HOL_Obool ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ ( tc_fun @ A @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) )
      = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ A @ tc_HOL_Obool ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Union__empty) ).

thf(5661,plain,
    ! [A: $i] :
      ( ( c_Complete__Lattice_OSup__class_OSup @ ( tc_fun @ A @ tc_HOL_Obool ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ ( tc_fun @ A @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) )
      = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ A @ tc_HOL_Obool ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[907]) ).

thf(564,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ D @ C ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ A ) @ D ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_sup1I2) ).

thf(4426,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ D @ C ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ A ) @ D ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[564]) ).

thf(504,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Lattices_Oab__semigroup__idem__mult @ B )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ B ) @ A ) @ A )
        = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__idem) ).

thf(4207,plain,
    ! [A: $i,B: $i] :
      ( ( class_Lattices_Oab__semigroup__idem__mult @ B )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ B ) @ A ) @ A )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[504]) ).

thf(2274,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ B ) @ A ) ) )
    <=> ( ( C = B )
        | ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_insert__iff) ).

thf(11316,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ B ) @ A ) ) )
       => ( ( C = B )
          | ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ A ) ) ) )
      & ( ( ( C = B )
          | ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ A ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ B ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2274]) ).

thf(874,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ B ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) )
        = ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) )
     => ( B = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_singleton__inject) ).

thf(5538,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ B ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) )
        = ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) )
     => ( B = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[874]) ).

thf(114,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ A ) )
       => ( ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A )
            = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) )
         => ( ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) )
            = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ B ) ) @ ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_card__Un__disjoint) ).

thf(2816,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ A ) )
       => ( ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A )
            = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) )
         => ( ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) )
            = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ B ) ) @ ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[114]) ).

thf(1063,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( c_Finite__Set_Ofolding__image__simple @ G @ F @ E @ D @ C @ B )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ F ) @ A ) )
       => ( ( hAPP @ B @ A )
          = ( hAPP @ ( hAPP @ ( hAPP @ ( c_Finite__Set_Ofold__image @ G @ F @ E ) @ C ) @ D ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_folding__image__simple_Oeq__fold__g) ).

thf(6320,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( c_Finite__Set_Ofolding__image__simple @ G @ F @ E @ D @ C @ B )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ F ) @ A ) )
       => ( ( hAPP @ B @ A )
          = ( hAPP @ ( hAPP @ ( hAPP @ ( c_Finite__Set_Ofold__image @ G @ F @ E ) @ C ) @ D ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1063]) ).

thf(553,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Transitive__Closure_Ortrancl @ B @ A )
      = ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ tc_Nat_Onat @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) ) @ ( c_Nat_Ocompow @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_rtrancl__is__UN__rel__pow) ).

thf(4394,plain,
    ! [A: $i,B: $i] :
      ( ( c_Transitive__Closure_Ortrancl @ B @ A )
      = ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ tc_Nat_Onat @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) ) @ ( c_Nat_Ocompow @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[553]) ).

thf(486,axiom,
    ! [A: $i,B: $i,C: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ C ) @ ( tc_List_Olist @ C ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ C ) @ ( tc_List_Olist @ C ) ) @ B ) @ B ) ) @ ( c_List_Olistrel @ C @ ( c_Transitive__Closure_Ortrancl @ C @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_listrel__rtrancl__refl) ).

thf(4136,plain,
    ! [A: $i,B: $i,C: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ C ) @ ( tc_List_Olist @ C ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ C ) @ ( tc_List_Olist @ C ) ) @ B ) @ B ) ) @ ( c_List_Olistrel @ C @ ( c_Transitive__Closure_Ortrancl @ C @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[486]) ).

thf(60,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Groups_Ocomm__monoid__add @ E )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ D ) @ C ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ C ) @ B ) )
         => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ E ) @ ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ tc_Nat_Onat @ E ) @ A ) @ ( hAPP @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Nat_Onat @ D ) @ C ) ) ) @ ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ tc_Nat_Onat @ E ) @ A ) @ ( hAPP @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Nat_Onat @ C ) @ B ) ) )
            = ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ tc_Nat_Onat @ E ) @ A ) @ ( hAPP @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Nat_Onat @ D ) @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_setsum__add__nat__ivl) ).

thf(2614,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Groups_Ocomm__monoid__add @ E )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ D ) @ C ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ C ) @ B ) )
         => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ E ) @ ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ tc_Nat_Onat @ E ) @ A ) @ ( hAPP @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Nat_Onat @ D ) @ C ) ) ) @ ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ tc_Nat_Onat @ E ) @ A ) @ ( hAPP @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Nat_Onat @ C ) @ B ) ) )
            = ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ tc_Nat_Onat @ E ) @ A ) @ ( hAPP @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Nat_Onat @ D ) @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[60]) ).

thf(1334,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ C ) )
     => ! [D: $i] :
          ( ! [E: $i] :
              ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Int_Oint ) @ E ) @ ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Int_Oint @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) @ C ) ) )
             => ! [F: $i] :
                  ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Int_Oint ) @ F ) @ B ) )
                 => ( D
                   != ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ F ) @ E ) ) ) )
         => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ D ) @ A ) )
           => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Int_Oint ) @ D ) @ C ) ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_bset_I5_J) ).

thf(7526,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ C ) )
     => ! [D: $i] :
          ( ! [E: $i] :
              ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Int_Oint ) @ E ) @ ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Int_Oint @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) @ C ) ) )
             => ! [F: $i] :
                  ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Int_Oint ) @ F ) @ B ) )
                 => ( D
                   != ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ F ) @ E ) ) ) )
         => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ D ) @ A ) )
           => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Int_Oint ) @ D ) @ C ) ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1334]) ).

thf(425,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( c_Set_Ovimage @ E @ D @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Set_Ovimage @ E @ D @ C ) @ B ) ) @ ( hAPP @ ( c_Set_Ovimage @ E @ D @ C ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_vimage__Un) ).

thf(3936,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( c_Set_Ovimage @ E @ D @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Set_Ovimage @ E @ D @ C ) @ B ) ) @ ( hAPP @ ( c_Set_Ovimage @ E @ D @ C ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[425]) ).

thf(1244,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ E ) @ D ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ D ) @ C ) ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ E ) @ C ) ) @ A ) ) )
      <=> ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ D ) @ E ) ) @ C ) ) @ B ) ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nat__less__add__iff1) ).

thf(7117,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ E ) @ D ) )
     => ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ D ) @ C ) ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ E ) @ C ) ) @ A ) ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ D ) @ E ) ) @ C ) ) @ B ) ) @ A ) ) )
        & ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ D ) @ E ) ) @ C ) ) @ B ) ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ D ) @ C ) ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ E ) @ C ) ) @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1244]) ).

thf(1090,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ C @ ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ B @ A ) ) )
    <=> ( ( ( A
            = ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) )
         => ( hBOOL @ ( hAPP @ C @ B ) ) )
        & ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ A ) )
         => ! [D: $i,E: $i] :
              ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ E ) )
                & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ E ) @ A ) )
                & ( B
                  = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ A ) @ D ) ) @ E ) ) )
             => ( hBOOL @ ( hAPP @ C @ E ) ) ) )
        & ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) )
         => ! [D: $i,E: $i] :
              ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ A ) @ E ) )
                & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ E ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) )
                & ( B
                  = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ A ) @ D ) ) @ E ) ) )
             => ( hBOOL @ ( hAPP @ C @ E ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_split__zmod) ).

thf(6417,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP @ C @ ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ B @ A ) ) )
       => ( ( ( A
              = ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) )
           => ( hBOOL @ ( hAPP @ C @ B ) ) )
          & ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ A ) )
           => ! [D: $i,E: $i] :
                ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ E ) )
                  & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ E ) @ A ) )
                  & ( B
                    = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ A ) @ D ) ) @ E ) ) )
               => ( hBOOL @ ( hAPP @ C @ E ) ) ) )
          & ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) )
           => ! [D: $i,E: $i] :
                ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ A ) @ E ) )
                  & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ E ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) )
                  & ( B
                    = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ A ) @ D ) ) @ E ) ) )
               => ( hBOOL @ ( hAPP @ C @ E ) ) ) ) ) )
      & ( ( ( ( A
              = ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) )
           => ( hBOOL @ ( hAPP @ C @ B ) ) )
          & ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ A ) )
           => ! [D: $i,E: $i] :
                ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ E ) )
                  & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ E ) @ A ) )
                  & ( B
                    = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ A ) @ D ) ) @ E ) ) )
               => ( hBOOL @ ( hAPP @ C @ E ) ) ) )
          & ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) )
           => ! [D: $i,E: $i] :
                ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ A ) @ E ) )
                  & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ E ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) )
                  & ( B
                    = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ A ) @ D ) ) @ E ) ) )
               => ( hBOOL @ ( hAPP @ C @ E ) ) ) ) )
       => ( hBOOL @ ( hAPP @ C @ ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ B @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1090]) ).

thf(1851,axiom,
    ! [A: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ A ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_le__refl) ).

thf(9673,plain,
    ! [A: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ A ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1851]) ).

thf(2094,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_List_Odistinct @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( c_List_Odistinct @ C ) @ ( c_List_Oremove1 @ C @ A @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_distinct__remove1) ).

thf(10632,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_List_Odistinct @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( c_List_Odistinct @ C ) @ ( c_List_Oremove1 @ C @ A @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2094]) ).

thf(1287,axiom,
    ! [A: $i,B: $i] :
      ( ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ B ) @ A )
        = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
    <=> ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_diff__is__0__eq) ).

thf(7296,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ B ) @ A )
          = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ A ) ) )
      & ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ A ) )
       => ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ B ) @ A )
          = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1287]) ).

thf(1862,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( c_Com_Ocom_OAss @ D @ C )
        = ( c_Com_Ocom_OAss @ B @ A ) )
    <=> ( ( D = B )
        & ( C = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_com_Osimps_I1_J) ).

thf(9727,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( ( c_Com_Ocom_OAss @ D @ C )
          = ( c_Com_Ocom_OAss @ B @ A ) )
       => ( ( D = B )
          & ( C = A ) ) )
      & ( ( ( D = B )
          & ( C = A ) )
       => ( ( c_Com_Ocom_OAss @ D @ C )
          = ( c_Com_Ocom_OAss @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1862]) ).

thf(962,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hBOOL @ ( hAPP @ F @ E ) )
     => ( ( hBOOL @ ( hAPP @ D @ C ) )
       => ( ( hBOOL @ ( hAPP @ F @ ( hAPP @ ( c_Product__Type_Ofst @ B @ A ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ B @ A ) @ E ) @ C ) ) ) )
          & ( hBOOL @ ( hAPP @ D @ ( hAPP @ ( c_Product__Type_Osnd @ B @ A ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ B @ A ) @ E ) @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_conjI__realizer) ).

thf(5881,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hBOOL @ ( hAPP @ F @ E ) )
     => ( ( hBOOL @ ( hAPP @ D @ C ) )
       => ( ( hBOOL @ ( hAPP @ F @ ( hAPP @ ( c_Product__Type_Ofst @ B @ A ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ B @ A ) @ E ) @ C ) ) ) )
          & ( hBOOL @ ( hAPP @ D @ ( hAPP @ ( c_Product__Type_Osnd @ B @ A ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ B @ A ) @ E ) @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[962]) ).

thf(1053,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ C ) @ A ) ) )
      <=> ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_subset__insert) ).

thf(6276,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ B ) )
     => ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ C ) @ A ) ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) ) )
        & ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ C ) @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1053]) ).

thf(73,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ E @ D @ C ) @ B ) @ A )
      = ( hAPP @ ( c_Set_OCollect @ ( tc_prod @ E @ C ) ) @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ E @ C @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( tc_fun @ C @ tc_HOL_Obool ) @ E ) @ ( hAPP @ ( c_COMBB @ ( tc_fun @ D @ tc_HOL_Obool ) @ tc_HOL_Obool @ C ) @ ( c_HOL_OEx @ D ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ E @ ( tc_fun @ C @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( tc_fun @ C @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ ( tc_fun @ D @ tc_HOL_Obool ) @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ C @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( tc_fun @ C @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) @ E ) @ ( c_COMBB @ ( tc_fun @ D @ tc_HOL_Obool ) @ ( tc_fun @ D @ tc_HOL_Obool ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ D @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ D @ tc_HOL_Obool ) @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ E ) @ ( c_COMBS @ D @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ D @ tc_HOL_Obool ) @ ( tc_fun @ D @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ E ) @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ D ) @ c_fconj ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ E @ ( tc_fun @ ( tc_prod @ E @ D ) @ tc_HOL_Obool ) @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ D @ ( tc_fun @ ( tc_fun @ ( tc_prod @ E @ D ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ E @ D ) @ tc_HOL_Obool ) @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ E ) @ ( c_COMBC @ D @ ( tc_fun @ ( tc_prod @ E @ D ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ D @ ( tc_prod @ E @ D ) ) @ ( tc_fun @ D @ ( tc_fun @ ( tc_fun @ ( tc_prod @ E @ D ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ E ) @ ( hAPP @ ( c_COMBB @ ( tc_prod @ E @ D ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ E @ D ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ D ) @ ( c_member @ ( tc_prod @ E @ D ) ) ) ) @ ( c_Product__Type_OPair @ E @ D ) ) ) ) @ B ) ) ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ C @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ D @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ ( c_COMBC @ D @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ D @ ( tc_prod @ D @ C ) ) @ ( tc_fun @ D @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( c_COMBB @ ( tc_prod @ D @ C ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ D ) @ ( c_member @ ( tc_prod @ D @ C ) ) ) ) @ ( hAPP @ ( c_COMBC @ D @ C @ ( tc_prod @ D @ C ) ) @ ( c_Product__Type_OPair @ D @ C ) ) ) ) ) @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_rel__comp__def) ).

thf(2665,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ E @ D @ C ) @ B ) @ A )
      = ( hAPP @ ( c_Set_OCollect @ ( tc_prod @ E @ C ) ) @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ E @ C @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( tc_fun @ C @ tc_HOL_Obool ) @ E ) @ ( hAPP @ ( c_COMBB @ ( tc_fun @ D @ tc_HOL_Obool ) @ tc_HOL_Obool @ C ) @ ( c_HOL_OEx @ D ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ E @ ( tc_fun @ C @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( tc_fun @ C @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ ( tc_fun @ D @ tc_HOL_Obool ) @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ C @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( tc_fun @ C @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) @ E ) @ ( c_COMBB @ ( tc_fun @ D @ tc_HOL_Obool ) @ ( tc_fun @ D @ tc_HOL_Obool ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ D @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ D @ tc_HOL_Obool ) @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ E ) @ ( c_COMBS @ D @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ D @ tc_HOL_Obool ) @ ( tc_fun @ D @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ E ) @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ D ) @ c_fconj ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ E @ ( tc_fun @ ( tc_prod @ E @ D ) @ tc_HOL_Obool ) @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ D @ ( tc_fun @ ( tc_fun @ ( tc_prod @ E @ D ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ E @ D ) @ tc_HOL_Obool ) @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ E ) @ ( c_COMBC @ D @ ( tc_fun @ ( tc_prod @ E @ D ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ D @ ( tc_prod @ E @ D ) ) @ ( tc_fun @ D @ ( tc_fun @ ( tc_fun @ ( tc_prod @ E @ D ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ E ) @ ( hAPP @ ( c_COMBB @ ( tc_prod @ E @ D ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ E @ D ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ D ) @ ( c_member @ ( tc_prod @ E @ D ) ) ) ) @ ( c_Product__Type_OPair @ E @ D ) ) ) ) @ B ) ) ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ C @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ D @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ ( c_COMBC @ D @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ D @ ( tc_prod @ D @ C ) ) @ ( tc_fun @ D @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( c_COMBB @ ( tc_prod @ D @ C ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ D ) @ ( c_member @ ( tc_prod @ D @ C ) ) ) ) @ ( hAPP @ ( c_COMBC @ D @ C @ ( tc_prod @ D @ C ) ) @ ( c_Product__Type_OPair @ D @ C ) ) ) ) ) @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[73]) ).

thf(731,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i,I: $i,J: $i,K: $i] :
      ( ( c_Equiv__Relations_Oequiv @ K @ J @ I )
     => ( ( c_Equiv__Relations_Oequiv @ H @ G @ F )
       => ( ( c_Equiv__Relations_Ocongruent2 @ K @ H @ ( tc_fun @ E @ tc_HOL_Obool ) @ I @ F @ D )
         => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_fun @ K @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Equiv__Relations_Oquotient @ K ) @ J ) @ I ) ) )
           => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_fun @ H @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Equiv__Relations_Oquotient @ H ) @ G ) @ F ) ) )
             => ( ! [L: $i] :
                    ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ K ) @ L ) @ J ) )
                   => ! [M: $i] :
                        ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ H ) @ M ) @ G ) )
                       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ D @ L ) @ M ) ) @ A ) ) ) )
               => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ K @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ H @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( tc_fun @ E @ tc_HOL_Obool ) @ K ) @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ H @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ B ) ) @ D ) ) ) @ A ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_UN__equiv__class__type2) ).

thf(5025,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i,I: $i,J: $i,K: $i] :
      ( ( c_Equiv__Relations_Oequiv @ K @ J @ I )
     => ( ( c_Equiv__Relations_Oequiv @ H @ G @ F )
       => ( ( c_Equiv__Relations_Ocongruent2 @ K @ H @ ( tc_fun @ E @ tc_HOL_Obool ) @ I @ F @ D )
         => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_fun @ K @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Equiv__Relations_Oquotient @ K ) @ J ) @ I ) ) )
           => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_fun @ H @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Equiv__Relations_Oquotient @ H ) @ G ) @ F ) ) )
             => ( ! [L: $i] :
                    ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ K ) @ L ) @ J ) )
                   => ! [M: $i] :
                        ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ H ) @ M ) @ G ) )
                       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ D @ L ) @ M ) ) @ A ) ) ) )
               => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ K @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ H @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( tc_fun @ E @ tc_HOL_Obool ) @ K ) @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ H @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ B ) ) @ D ) ) ) @ A ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[731]) ).

thf(1974,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) )
     => ( ( ( hAPP @ c_Nat_OSuc @ B )
         != A )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ B ) ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Suc__lessI) ).

thf(10191,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) )
     => ( ( ( hAPP @ c_Nat_OSuc @ B )
         != A )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ B ) ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1974]) ).

thf(670,axiom,
    ! [A: $i,B: $i,C: $i] : ( c_Fun_Oinj__on @ ( tc_prod @ C @ B ) @ ( tc_prod @ B @ C ) @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ C @ B @ ( tc_prod @ B @ C ) ) @ ( hAPP @ ( c_COMBC @ B @ C @ ( tc_prod @ B @ C ) ) @ ( c_Product__Type_OPair @ B @ C ) ) ) @ A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_swap__inj__on) ).

thf(4813,plain,
    ! [A: $i,B: $i,C: $i] : ( c_Fun_Oinj__on @ ( tc_prod @ C @ B ) @ ( tc_prod @ B @ C ) @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ C @ B @ ( tc_prod @ B @ C ) ) @ ( hAPP @ ( c_COMBC @ B @ C @ ( tc_prod @ B @ C ) ) @ ( c_Product__Type_OPair @ B @ C ) ) ) @ A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[670]) ).

thf(2079,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_Wellfounded_Owf @ D @ C )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ D ) @ B ) @ A ) ) @ C ) )
       => ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ D ) @ A ) @ B ) ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_wf__asym) ).

thf(10589,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_Wellfounded_Owf @ D @ C )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ D ) @ B ) @ A ) ) @ C ) )
       => ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ D ) @ A ) @ B ) ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2079]) ).

thf(1745,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hAPP @ ( hAPP @ ( c_List_Oappend @ C ) @ B ) @ A )
        = A )
    <=> ( B
        = ( c_List_Olist_ONil @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_append__self__conv2) ).

thf(9259,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( hAPP @ ( hAPP @ ( c_List_Oappend @ C ) @ B ) @ A )
          = A )
       => ( B
          = ( c_List_Olist_ONil @ C ) ) )
      & ( ( B
          = ( c_List_Olist_ONil @ C ) )
       => ( ( hAPP @ ( hAPP @ ( c_List_Oappend @ C ) @ B ) @ A )
          = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1745]) ).

thf(250,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ C ) @ ( hAPP @ ( c_List_Ofoldl @ tc_Nat_Onat @ tc_Nat_Onat @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_start__le__sum) ).

thf(3293,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ C ) @ ( hAPP @ ( c_List_Ofoldl @ tc_Nat_Onat @ tc_Nat_Onat @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[250]) ).

thf(1848,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( hAPP @ F @ E )
        = ( hAPP @ ( c_Option_Ooption_OSome @ D ) @ C ) )
     => ( ( hAPP @ ( c_Map_Omap__add @ B @ D @ A @ F ) @ E )
        = ( hAPP @ ( c_Option_Ooption_OSome @ D ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_map__add__find__right) ).

thf(9651,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( hAPP @ F @ E )
        = ( hAPP @ ( c_Option_Ooption_OSome @ D ) @ C ) )
     => ( ( hAPP @ ( c_Map_Omap__add @ B @ D @ A @ F ) @ E )
        = ( hAPP @ ( c_Option_Ooption_OSome @ D ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1848]) ).

thf(611,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ E ) ) @ D )
        = ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) @ B ) )
     => ( ( ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ E ) ) @ A )
          = ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( c_List_Odistinct @ C ) @ B ) )
         => ( ( ( c_Map_Omap__of @ C @ E @ ( hAPP @ ( hAPP @ ( c_List_Ozip @ C @ E ) @ B ) @ D ) )
              = ( c_Map_Omap__of @ C @ E @ ( hAPP @ ( hAPP @ ( c_List_Ozip @ C @ E ) @ B ) @ A ) ) )
           => ( D = A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_map__of__zip__inject) ).

thf(4598,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ E ) ) @ D )
        = ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) @ B ) )
     => ( ( ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ E ) ) @ A )
          = ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( c_List_Odistinct @ C ) @ B ) )
         => ( ( ( c_Map_Omap__of @ C @ E @ ( hAPP @ ( hAPP @ ( c_List_Ozip @ C @ E ) @ B ) @ D ) )
              = ( c_Map_Omap__of @ C @ E @ ( hAPP @ ( hAPP @ ( c_List_Ozip @ C @ E ) @ B ) @ A ) ) )
           => ( D = A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[611]) ).

thf(209,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ C ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ B ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ D @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ C ) @ A ) ) )
      <=> ! [E: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ E ) @ C ) )
           => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ B ) @ ( hAPP @ A @ E ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_finite__UN) ).

thf(3161,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ C ) )
     => ( ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ B ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ D @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ C ) @ A ) ) )
         => ! [E: $i] :
              ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ E ) @ C ) )
             => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ B ) @ ( hAPP @ A @ E ) ) ) ) )
        & ( ! [E: $i] :
              ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ E ) @ C ) )
             => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ B ) @ ( hAPP @ A @ E ) ) ) )
         => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ B ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ D @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ C ) @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[209]) ).

thf(1983,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( c_List_Oset @ D ) @ ( hAPP @ ( hAPP @ ( c_List_Omap @ C @ D ) @ B ) @ A ) )
      = ( hAPP @ ( c_Set_Oimage @ C @ D @ B ) @ ( hAPP @ ( c_List_Oset @ C ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_set__map) ).

thf(10223,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( c_List_Oset @ D ) @ ( hAPP @ ( hAPP @ ( c_List_Omap @ C @ D ) @ B ) @ A ) )
      = ( hAPP @ ( c_Set_Oimage @ C @ D @ B ) @ ( hAPP @ ( c_List_Oset @ C ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1983]) ).

thf(1002,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hAPP @ ( c_Relation_ODomain @ C @ B ) @ A )
        = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) )
    <=> ( A
        = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ ( tc_prod @ C @ B ) @ tc_HOL_Obool ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Domain__empty__iff) ).

thf(6056,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( hAPP @ ( c_Relation_ODomain @ C @ B ) @ A )
          = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) )
       => ( A
          = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ ( tc_prod @ C @ B ) @ tc_HOL_Obool ) ) ) )
      & ( ( A
          = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ ( tc_prod @ C @ B ) @ tc_HOL_Obool ) ) )
       => ( ( hAPP @ ( c_Relation_ODomain @ C @ B ) @ A )
          = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1002]) ).

thf(494,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) )
     != ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ B ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_empty__not__insert) ).

thf(4177,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) )
     != ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ B ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[494]) ).

thf(65,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Map_Odom @ E @ D @ ( hAPP @ ( hAPP @ ( c_COMBS @ E @ ( tc_Option_Ooption @ D ) @ ( tc_Option_Ooption @ D ) ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ E @ ( tc_Option_Ooption @ D ) @ ( tc_fun @ ( tc_Option_Ooption @ D ) @ ( tc_Option_Ooption @ D ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ ( tc_Option_Ooption @ D ) @ ( tc_fun @ ( tc_Option_Ooption @ D ) @ ( tc_Option_Ooption @ D ) ) ) @ E ) @ ( c_If @ ( tc_Option_Ooption @ D ) ) ) @ C ) ) @ B ) ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( c_Map_Odom @ E @ D @ B ) ) @ ( hAPP @ ( c_Set_OCollect @ E ) @ C ) ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( c_Map_Odom @ E @ D @ A ) ) @ ( hAPP @ ( c_Set_OCollect @ E ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ tc_HOL_Obool @ E ) @ c_fNot ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_dom__if) ).

thf(2629,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Map_Odom @ E @ D @ ( hAPP @ ( hAPP @ ( c_COMBS @ E @ ( tc_Option_Ooption @ D ) @ ( tc_Option_Ooption @ D ) ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ E @ ( tc_Option_Ooption @ D ) @ ( tc_fun @ ( tc_Option_Ooption @ D ) @ ( tc_Option_Ooption @ D ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ ( tc_Option_Ooption @ D ) @ ( tc_fun @ ( tc_Option_Ooption @ D ) @ ( tc_Option_Ooption @ D ) ) ) @ E ) @ ( c_If @ ( tc_Option_Ooption @ D ) ) ) @ C ) ) @ B ) ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( c_Map_Odom @ E @ D @ B ) ) @ ( hAPP @ ( c_Set_OCollect @ E ) @ C ) ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( c_Map_Odom @ E @ D @ A ) ) @ ( hAPP @ ( c_Set_OCollect @ E ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ tc_HOL_Obool @ E ) @ c_fNot ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[65]) ).

thf(912,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( c_Relation_OField @ B ) @ ( c_Complete__Lattice_OSup__class_OSup @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) @ A ) )
      = ( c_Complete__Lattice_OSup__class_OSup @ ( tc_fun @ B @ tc_HOL_Obool ) @ ( hAPP @ ( c_Set_Oimage @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) @ ( tc_fun @ B @ tc_HOL_Obool ) @ ( c_Relation_OField @ B ) ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Field__Union) ).

thf(5678,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( c_Relation_OField @ B ) @ ( c_Complete__Lattice_OSup__class_OSup @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) @ A ) )
      = ( c_Complete__Lattice_OSup__class_OSup @ ( tc_fun @ B @ tc_HOL_Obool ) @ ( hAPP @ ( c_Set_Oimage @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) @ ( tc_fun @ B @ tc_HOL_Obool ) @ ( c_Relation_OField @ B ) ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[912]) ).

thf(770,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ A ) )
       => ( ( ? [D: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ D ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) ) )
           => ( ( c_Groups_Oone__class_Oone @ tc_Nat_Onat )
              = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) )
         => ( ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) )
            = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ B ) ) @ ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_card_Ounion__inter__neutral) ).

thf(5165,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ A ) )
       => ( ( ? [D: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ D ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) ) )
           => ( ( c_Groups_Oone__class_Oone @ tc_Nat_Onat )
              = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) )
         => ( ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) )
            = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ B ) ) @ ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[770]) ).

thf(1100,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ B ) )
     => ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ A ) @ B ) )
         => ( ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) ) )
            = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ B ) ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) ) )
        & ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ A ) @ B ) )
         => ( ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) ) )
            = ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_card__Diff__singleton__if) ).

thf(6517,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ B ) )
     => ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ A ) @ B ) )
         => ( ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) ) )
            = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ B ) ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) ) )
        & ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ A ) @ B ) )
         => ( ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) ) )
            = ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1100]) ).

thf(2360,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( hAPP @ ( hAPP @ ( c_List_Omap @ E @ D ) @ C ) @ B )
        = ( hAPP @ ( hAPP @ ( c_List_Omap @ E @ D ) @ A ) @ B ) )
    <=> ! [F: $i] :
          ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ F ) @ ( hAPP @ ( c_List_Oset @ E ) @ B ) ) )
         => ( ( hAPP @ C @ F )
            = ( hAPP @ A @ F ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_map__eq__conv) ).

thf(11689,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( ( hAPP @ ( hAPP @ ( c_List_Omap @ E @ D ) @ C ) @ B )
          = ( hAPP @ ( hAPP @ ( c_List_Omap @ E @ D ) @ A ) @ B ) )
       => ! [F: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ F ) @ ( hAPP @ ( c_List_Oset @ E ) @ B ) ) )
           => ( ( hAPP @ C @ F )
              = ( hAPP @ A @ F ) ) ) )
      & ( ! [F: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ F ) @ ( hAPP @ ( c_List_Oset @ E ) @ B ) ) )
           => ( ( hAPP @ C @ F )
              = ( hAPP @ A @ F ) ) )
       => ( ( hAPP @ ( hAPP @ ( c_List_Omap @ E @ D ) @ C ) @ B )
          = ( hAPP @ ( hAPP @ ( c_List_Omap @ E @ D ) @ A ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2360]) ).

thf(601,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( c_List_Opartition @ E @ D @ C )
        = ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ E ) @ ( tc_List_Olist @ E ) ) @ B ) @ A ) )
     => ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_List_Oset @ E ) @ B ) ) @ ( hAPP @ ( c_List_Oset @ E ) @ A ) )
        = ( hAPP @ ( c_List_Oset @ E ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_partition__set) ).

thf(4566,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( c_List_Opartition @ E @ D @ C )
        = ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ E ) @ ( tc_List_Olist @ E ) ) @ B ) @ A ) )
     => ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_List_Oset @ E ) @ B ) ) @ ( hAPP @ ( c_List_Oset @ E ) @ A ) )
        = ( hAPP @ ( c_List_Oset @ E ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[601]) ).

thf(1476,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ B ) ) @ A )
      = ( hAPP @ c_Nat_OSuc @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_add__Suc) ).

thf(8067,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ B ) ) @ A )
      = ( hAPP @ c_Nat_OSuc @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1476]) ).

thf(971,axiom,
    ! [A: $i] :
      ( ( c_Complete__Lattice_OSup__class_OSup @ ( tc_fun @ A @ tc_HOL_Obool ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ ( tc_fun @ A @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) )
      = ( c_Orderings_Otop__class_Otop @ ( tc_fun @ A @ tc_HOL_Obool ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Union__UNIV) ).

thf(5916,plain,
    ! [A: $i] :
      ( ( c_Complete__Lattice_OSup__class_OSup @ ( tc_fun @ A @ tc_HOL_Obool ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ ( tc_fun @ A @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) )
      = ( c_Orderings_Otop__class_Otop @ ( tc_fun @ A @ tc_HOL_Obool ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[971]) ).

thf(123,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( c_List_Olexn @ C @ B ) @ ( hAPP @ c_Nat_OSuc @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ ( tc_prod @ ( tc_List_Olist @ C ) @ ( tc_List_Olist @ C ) ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Set_Oimage @ ( tc_prod @ ( tc_prod @ C @ ( tc_List_Olist @ C ) ) @ ( tc_prod @ C @ ( tc_List_Olist @ C ) ) ) @ ( tc_prod @ ( tc_List_Olist @ C ) @ ( tc_List_Olist @ C ) ) @ ( c_Product__Type_Omap__pair @ ( tc_prod @ C @ ( tc_List_Olist @ C ) ) @ ( tc_List_Olist @ C ) @ ( tc_prod @ C @ ( tc_List_Olist @ C ) ) @ ( tc_List_Olist @ C ) @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ C @ ( tc_List_Olist @ C ) @ ( tc_List_Olist @ C ) ) @ ( c_List_Olist_OCons @ C ) ) @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ C @ ( tc_List_Olist @ C ) @ ( tc_List_Olist @ C ) ) @ ( c_List_Olist_OCons @ C ) ) ) ) @ ( c_Wellfounded_Olex__prod @ C @ ( tc_List_Olist @ C ) @ B @ ( hAPP @ ( c_List_Olexn @ C @ B ) @ A ) ) ) ) @ ( hAPP @ ( c_Set_OCollect @ ( tc_prod @ ( tc_List_Olist @ C ) @ ( tc_List_Olist @ C ) ) ) @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ ( tc_List_Olist @ C ) @ ( tc_List_Olist @ C ) @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ ( tc_List_Olist @ C ) @ ( tc_fun @ ( tc_List_Olist @ C ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_List_Olist @ C ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ ( tc_List_Olist @ C ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_List_Olist @ C ) @ tc_HOL_Obool ) ) @ ( tc_List_Olist @ C ) ) @ ( c_COMBB @ tc_HOL_Obool @ tc_HOL_Obool @ ( tc_List_Olist @ C ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( tc_List_Olist @ C ) ) @ c_fconj ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ ( tc_List_Olist @ C ) @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_Nat_Onat @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_List_Olist @ C ) ) @ c_fequal ) @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) ) ) @ ( hAPP @ c_Nat_OSuc @ A ) ) ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ ( tc_List_Olist @ C ) @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_Nat_Onat @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_List_Olist @ C ) ) @ c_fequal ) @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) ) ) @ ( hAPP @ c_Nat_OSuc @ A ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_lexn_Osimps_I2_J) ).

thf(2846,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( c_List_Olexn @ C @ B ) @ ( hAPP @ c_Nat_OSuc @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ ( tc_prod @ ( tc_List_Olist @ C ) @ ( tc_List_Olist @ C ) ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Set_Oimage @ ( tc_prod @ ( tc_prod @ C @ ( tc_List_Olist @ C ) ) @ ( tc_prod @ C @ ( tc_List_Olist @ C ) ) ) @ ( tc_prod @ ( tc_List_Olist @ C ) @ ( tc_List_Olist @ C ) ) @ ( c_Product__Type_Omap__pair @ ( tc_prod @ C @ ( tc_List_Olist @ C ) ) @ ( tc_List_Olist @ C ) @ ( tc_prod @ C @ ( tc_List_Olist @ C ) ) @ ( tc_List_Olist @ C ) @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ C @ ( tc_List_Olist @ C ) @ ( tc_List_Olist @ C ) ) @ ( c_List_Olist_OCons @ C ) ) @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ C @ ( tc_List_Olist @ C ) @ ( tc_List_Olist @ C ) ) @ ( c_List_Olist_OCons @ C ) ) ) ) @ ( c_Wellfounded_Olex__prod @ C @ ( tc_List_Olist @ C ) @ B @ ( hAPP @ ( c_List_Olexn @ C @ B ) @ A ) ) ) ) @ ( hAPP @ ( c_Set_OCollect @ ( tc_prod @ ( tc_List_Olist @ C ) @ ( tc_List_Olist @ C ) ) ) @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ ( tc_List_Olist @ C ) @ ( tc_List_Olist @ C ) @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ ( tc_List_Olist @ C ) @ ( tc_fun @ ( tc_List_Olist @ C ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_List_Olist @ C ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ ( tc_List_Olist @ C ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_List_Olist @ C ) @ tc_HOL_Obool ) ) @ ( tc_List_Olist @ C ) ) @ ( c_COMBB @ tc_HOL_Obool @ tc_HOL_Obool @ ( tc_List_Olist @ C ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( tc_List_Olist @ C ) ) @ c_fconj ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ ( tc_List_Olist @ C ) @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_Nat_Onat @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_List_Olist @ C ) ) @ c_fequal ) @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) ) ) @ ( hAPP @ c_Nat_OSuc @ A ) ) ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ ( tc_List_Olist @ C ) @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_Nat_Onat @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_List_Olist @ C ) ) @ c_fequal ) @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) ) ) @ ( hAPP @ c_Nat_OSuc @ A ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[123]) ).

thf(922,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_Equiv__Relations_Ocongruent @ D @ C @ B @ A )
    <=> ! [E: $i] :
          ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ E ) @ B ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ D @ D @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ ( tc_fun @ D @ C ) @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ D @ C ) @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ D ) @ ( c_COMBB @ C @ tc_HOL_Obool @ D ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ C @ ( tc_fun @ C @ tc_HOL_Obool ) @ D ) @ c_fequal ) @ A ) ) ) @ A ) ) @ E ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_congruent__def) ).

thf(5710,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( c_Equiv__Relations_Ocongruent @ D @ C @ B @ A )
       => ! [E: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ E ) @ B ) )
           => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ D @ D @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ ( tc_fun @ D @ C ) @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ D @ C ) @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ D ) @ ( c_COMBB @ C @ tc_HOL_Obool @ D ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ C @ ( tc_fun @ C @ tc_HOL_Obool ) @ D ) @ c_fequal ) @ A ) ) ) @ A ) ) @ E ) ) ) )
      & ( ! [E: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ E ) @ B ) )
           => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ D @ D @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ ( tc_fun @ D @ C ) @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ D @ C ) @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ D ) @ ( c_COMBB @ C @ tc_HOL_Obool @ D ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ C @ ( tc_fun @ C @ tc_HOL_Obool ) @ D ) @ c_fequal ) @ A ) ) ) @ A ) ) @ E ) ) )
       => ( c_Equiv__Relations_Ocongruent @ D @ C @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[922]) ).

thf(2136,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Com_Ocom_OSemi @ B @ A )
     != c_Com_Ocom_OSKIP ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_com_Osimps_I13_J) ).

thf(10788,plain,
    ! [A: $i,B: $i] :
      ( ( c_Com_Ocom_OSemi @ B @ A )
     != c_Com_Ocom_OSKIP ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2136]) ).

thf(1736,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_List_Obutlast @ C @ ( hAPP @ ( hAPP @ ( c_List_Oappend @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ C ) @ A ) @ ( c_List_Olist_ONil @ C ) ) ) )
      = B ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_butlast__snoc) ).

thf(9232,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_List_Obutlast @ C @ ( hAPP @ ( hAPP @ ( c_List_Oappend @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ C ) @ A ) @ ( c_List_Olist_ONil @ C ) ) ) )
      = B ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1736]) ).

thf(904,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ D @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Nat_Ocompow @ C @ C ) @ B ) @ D ) @ A ) )
      = ( hAPP @ ( hAPP @ ( hAPP @ ( c_Nat_Ocompow @ C @ C ) @ B ) @ D ) @ ( hAPP @ D @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_funpow__swap1) ).

thf(5652,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ D @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Nat_Ocompow @ C @ C ) @ B ) @ D ) @ A ) )
      = ( hAPP @ ( hAPP @ ( hAPP @ ( c_Nat_Ocompow @ C @ C ) @ B ) @ D ) @ ( hAPP @ D @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[904]) ).

thf(1749,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) @ C ) @ B ) ) @ ( c_Transitive__Closure_Ortrancl @ ( tc_List_Olist @ D ) @ ( c_List_Olistrel1 @ D @ A ) ) ) )
     => ( ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ D ) ) @ C )
        = ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ D ) ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_rtrancl__listrel1__eq__len) ).

thf(9276,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) @ C ) @ B ) ) @ ( c_Transitive__Closure_Ortrancl @ ( tc_List_Olist @ D ) @ ( c_List_Olistrel1 @ D @ A ) ) ) )
     => ( ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ D ) ) @ C )
        = ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ D ) ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1749]) ).

thf(542,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ E @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ C ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ E @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ E @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_UN__Un) ).

thf(4355,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ E @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ C ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ E @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ E @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[542]) ).

thf(942,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ G @ F @ E ) @ ( hAPP @ ( c_Fun_Ocomp @ D @ ( tc_fun @ F @ E ) @ G @ C ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ C @ ( hAPP @ B @ ( hAPP @ ( c_Product__Type_Ofst @ G @ F ) @ A ) ) ) @ ( hAPP @ ( c_Product__Type_Osnd @ G @ F ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_split__comp) ).

thf(5807,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ G @ F @ E ) @ ( hAPP @ ( c_Fun_Ocomp @ D @ ( tc_fun @ F @ E ) @ G @ C ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ C @ ( hAPP @ B @ ( hAPP @ ( c_Product__Type_Ofst @ G @ F ) @ A ) ) ) @ ( hAPP @ ( c_Product__Type_Osnd @ G @ F ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[942]) ).

thf(989,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ D @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( c_Complete__Lattice_OSup__class_OSup @ ( tc_fun @ D @ tc_HOL_Obool ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ ( tc_fun @ D @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ ( tc_fun @ D @ tc_HOL_Obool ) @ ( tc_fun @ D @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ D @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_UN__simps_I8_J) ).

thf(6008,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ D @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( c_Complete__Lattice_OSup__class_OSup @ ( tc_fun @ D @ tc_HOL_Obool ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ ( tc_fun @ D @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ ( tc_fun @ D @ tc_HOL_Obool ) @ ( tc_fun @ D @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ D @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[989]) ).

thf(1570,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) ) @ B ) @ c_Int_OInteg ) )
     => ( ! [C: $i] : ( hBOOL @ ( hAPP @ A @ ( hAPP @ c_Int_ORep__Integ @ C ) ) )
       => ( hBOOL @ ( hAPP @ A @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Rep__Integ__induct) ).

thf(8477,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) ) @ B ) @ c_Int_OInteg ) )
     => ( ! [C: $i] : ( hBOOL @ ( hAPP @ A @ ( hAPP @ c_Int_ORep__Integ @ C ) ) )
       => ( hBOOL @ ( hAPP @ A @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1570]) ).

thf(1126,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Wellfounded_Oacyclic @ C @ B )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) ) @ A ) @ B ) )
       => ( c_Wellfounded_Oacyclic @ C @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_acyclic__subset) ).

thf(6664,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Wellfounded_Oacyclic @ C @ B )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) ) @ A ) @ B ) )
       => ( c_Wellfounded_Oacyclic @ C @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1126]) ).

thf(1140,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_List_Olist_ONil @ C )
     != ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ C ) @ B ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_list_Osimps_I2_J) ).

thf(6719,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_List_Olist_ONil @ C )
     != ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ C ) @ B ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1140]) ).

thf(535,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( c_Set_OCollect @ B ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ B @ B @ tc_HOL_Obool ) @ c_fequal ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ B ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_singleton__conv) ).

thf(4336,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( c_Set_OCollect @ B ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ B @ B @ tc_HOL_Obool ) @ c_fequal ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ B ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[535]) ).

thf(717,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ D ) @ C ) )
     => ( ( hAPP @ ( c_Set_Oimage @ E @ B @ ( hAPP @ ( c_COMBK @ B @ E ) @ A ) ) @ C )
        = ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ B ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_image__constant) ).

thf(4986,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ D ) @ C ) )
     => ( ( hAPP @ ( c_Set_Oimage @ E @ B @ ( hAPP @ ( c_COMBK @ B @ E ) @ A ) ) @ C )
        = ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ B ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[717]) ).

thf(1730,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( hAPP @ ( c_Product__Type_Ofst @ G @ F ) @ ( hAPP @ ( c_Product__Type_Omap__pair @ E @ G @ D @ F @ C @ B ) @ A ) )
      = ( hAPP @ C @ ( hAPP @ ( c_Product__Type_Ofst @ E @ D ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_fst__map__pair) ).

thf(9202,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( hAPP @ ( c_Product__Type_Ofst @ G @ F ) @ ( hAPP @ ( c_Product__Type_Omap__pair @ E @ G @ D @ F @ C @ B ) @ A ) )
      = ( hAPP @ C @ ( hAPP @ ( c_Product__Type_Ofst @ E @ D ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1730]) ).

thf(1019,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( E
          = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) )
       => ( ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ D @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ E ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ D ) @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) @ B ) ) @ A ) )
          = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) )
      & ( ( E
         != ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) )
       => ( ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ D @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ E ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ D ) @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) @ B ) ) @ A ) )
          = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ D @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ E ) @ B ) ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_UN__simps_I2_J) ).

thf(6132,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( E
          = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) )
       => ( ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ D @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ E ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ D ) @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) @ B ) ) @ A ) )
          = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) )
      & ( ( E
         != ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) )
       => ( ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ D @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ E ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ D ) @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) @ B ) ) @ A ) )
          = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ D @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ E ) @ B ) ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1019]) ).

thf(1232,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( C
          = ( c_List_Olist_ONil @ B ) )
       => ( ( c_List_Obutlast @ B @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ B ) @ A ) @ C ) )
          = ( c_List_Olist_ONil @ B ) ) )
      & ( ( C
         != ( c_List_Olist_ONil @ B ) )
       => ( ( c_List_Obutlast @ B @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ B ) @ A ) @ C ) )
          = ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ B ) @ A ) @ ( c_List_Obutlast @ B @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_butlast_Osimps_I2_J) ).

thf(7070,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( C
          = ( c_List_Olist_ONil @ B ) )
       => ( ( c_List_Obutlast @ B @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ B ) @ A ) @ C ) )
          = ( c_List_Olist_ONil @ B ) ) )
      & ( ( C
         != ( c_List_Olist_ONil @ B ) )
       => ( ( c_List_Obutlast @ B @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ B ) @ A ) @ C ) )
          = ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ B ) @ A ) @ ( c_List_Obutlast @ B @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1232]) ).

thf(458,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( c_Product__Type_Omap__pair @ F @ E @ D @ C @ B @ A )
      = ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ F @ D @ ( tc_prod @ E @ C ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ F @ ( tc_fun @ D @ C ) @ ( tc_fun @ D @ ( tc_prod @ E @ C ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ ( tc_prod @ E @ C ) ) @ ( tc_fun @ ( tc_fun @ D @ C ) @ ( tc_fun @ D @ ( tc_prod @ E @ C ) ) ) @ F ) @ ( c_COMBB @ C @ ( tc_prod @ E @ C ) @ D ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ E @ ( tc_fun @ C @ ( tc_prod @ E @ C ) ) @ F ) @ ( c_Product__Type_OPair @ E @ C ) ) @ B ) ) ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_map__pair__def) ).

thf(4045,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( c_Product__Type_Omap__pair @ F @ E @ D @ C @ B @ A )
      = ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ F @ D @ ( tc_prod @ E @ C ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ F @ ( tc_fun @ D @ C ) @ ( tc_fun @ D @ ( tc_prod @ E @ C ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ ( tc_prod @ E @ C ) ) @ ( tc_fun @ ( tc_fun @ D @ C ) @ ( tc_fun @ D @ ( tc_prod @ E @ C ) ) ) @ F ) @ ( c_COMBB @ C @ ( tc_prod @ E @ C ) @ D ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ E @ ( tc_fun @ C @ ( tc_prod @ E @ C ) ) @ F ) @ ( c_Product__Type_OPair @ E @ C ) ) @ B ) ) ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[458]) ).

thf(1976,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Natural_Oevalc @ c_Com_Ocom_OSKIP ) @ B ) @ A ) )
     => ( A = B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_evalc__elim__cases_I1_J) ).

thf(10196,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Natural_Oevalc @ c_Com_Ocom_OSKIP ) @ B ) @ A ) )
     => ( A = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1976]) ).

thf(1405,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ C ) @ B ) )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ B ) @ C ) ) @ A )
        = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_add__diff__assoc2) ).

thf(7801,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ C ) @ B ) )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ B ) @ C ) ) @ A )
        = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1405]) ).

thf(759,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ D ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) @ B ) ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ D ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) @ A ) ) )
       => ( ! [E: $i] :
              ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ E ) @ D ) )
             => ( ( hAPP @ ( hAPP @ ( c_List_Onth @ C ) @ B ) @ E )
                = ( hAPP @ ( hAPP @ ( c_List_Onth @ C ) @ A ) @ E ) ) )
         => ( ( hAPP @ ( hAPP @ ( c_List_Otake @ C ) @ D ) @ B )
            = ( hAPP @ ( hAPP @ ( c_List_Otake @ C ) @ D ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nth__take__lemma) ).

thf(5125,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ D ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) @ B ) ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ D ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) @ A ) ) )
       => ( ! [E: $i] :
              ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ E ) @ D ) )
             => ( ( hAPP @ ( hAPP @ ( c_List_Onth @ C ) @ B ) @ E )
                = ( hAPP @ ( hAPP @ ( c_List_Onth @ C ) @ A ) @ E ) ) )
         => ( ( hAPP @ ( hAPP @ ( c_List_Otake @ C ) @ D ) @ B )
            = ( hAPP @ ( hAPP @ ( c_List_Otake @ C ) @ D ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[759]) ).

thf(1567,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ C ) @ B ) )
     => ( ( c_List_Oupt @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) )
        = ( hAPP @ ( hAPP @ ( c_List_Oappend @ tc_Nat_Onat ) @ ( c_List_Oupt @ C @ B ) ) @ ( c_List_Oupt @ B @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_upt__add__eq__append) ).

thf(8466,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ C ) @ B ) )
     => ( ( c_List_Oupt @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) )
        = ( hAPP @ ( hAPP @ ( c_List_Oappend @ tc_Nat_Onat ) @ ( c_List_Oupt @ C @ B ) ) @ ( c_List_Oupt @ B @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1567]) ).

thf(1488,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ F @ ( tc_fun @ E @ tc_HOL_Obool ) ) ) @ D ) @ C ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ D @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ C @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_predicate2D) ).

thf(8163,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ F @ ( tc_fun @ E @ tc_HOL_Obool ) ) ) @ D ) @ C ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ D @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ C @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1488]) ).

thf(2066,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Relation_Osingle__valued @ C @ B @ A )
    <=> ! [D: $i,E: $i] :
          ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ C @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ C @ B ) @ D ) @ E ) ) @ A ) )
         => ! [F: $i] :
              ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ C @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ C @ B ) @ D ) @ F ) ) @ A ) )
             => ( E = F ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_single__valued__def) ).

thf(10521,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( c_Relation_Osingle__valued @ C @ B @ A )
       => ! [D: $i,E: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ C @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ C @ B ) @ D ) @ E ) ) @ A ) )
           => ! [F: $i] :
                ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ C @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ C @ B ) @ D ) @ F ) ) @ A ) )
               => ( E = F ) ) ) )
      & ( ! [D: $i,E: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ C @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ C @ B ) @ D ) @ E ) ) @ A ) )
           => ! [F: $i] :
                ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ C @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ C @ B ) @ D ) @ F ) ) @ A ) )
               => ( E = F ) ) )
       => ( c_Relation_Osingle__valued @ C @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2066]) ).

thf(548,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i,I: $i,J: $i] :
      ( ( ( hAPP @ ( c_Set_Oimage @ J @ I @ H ) @ G )
        = F )
     => ( ( ( hAPP @ ( c_Set_Oimage @ E @ D @ C ) @ B )
          = A )
       => ( ( hAPP @ ( c_Set_Oimage @ ( tc_prod @ J @ E ) @ ( tc_prod @ I @ D ) @ ( c_Product__Type_Omap__pair @ J @ I @ E @ D @ H @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ J @ E ) @ G ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ E @ tc_HOL_Obool ) @ J ) @ B ) ) )
          = ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ I @ D ) @ F ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ D @ tc_HOL_Obool ) @ I ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_map__pair__surj__on) ).

thf(4377,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i,I: $i,J: $i] :
      ( ( ( hAPP @ ( c_Set_Oimage @ J @ I @ H ) @ G )
        = F )
     => ( ( ( hAPP @ ( c_Set_Oimage @ E @ D @ C ) @ B )
          = A )
       => ( ( hAPP @ ( c_Set_Oimage @ ( tc_prod @ J @ E ) @ ( tc_prod @ I @ D ) @ ( c_Product__Type_Omap__pair @ J @ I @ E @ D @ H @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ J @ E ) @ G ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ E @ tc_HOL_Obool ) @ J ) @ B ) ) )
          = ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ I @ D ) @ F ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ D @ tc_HOL_Obool ) @ I ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[548]) ).

thf(1951,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_Com_Ocom_OSemi @ D @ C )
     != ( c_Com_Ocom_OAss @ B @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_com_Osimps_I25_J) ).

thf(10080,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_Com_Ocom_OSemi @ D @ C )
     != ( c_Com_Ocom_OAss @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1951]) ).

thf(652,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( c_List_Oset @ B ) @ A )
      = ( hAPP @ ( c_Set_OCollect @ B ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ tc_HOL_Obool @ B ) @ ( c_HOL_OEx @ tc_Nat_Onat ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ B @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ tc_Nat_Onat @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) @ B ) @ ( c_COMBS @ tc_Nat_Onat @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ tc_Nat_Onat @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ tc_Nat_Onat ) @ c_fconj ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ B @ ( tc_fun @ tc_Nat_Onat @ B ) @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ B @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ tc_Nat_Onat @ B ) @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) @ B ) @ ( c_COMBB @ B @ tc_HOL_Obool @ tc_Nat_Onat ) ) @ c_fequal ) ) @ ( hAPP @ ( c_List_Onth @ B ) @ A ) ) ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ B ) ) @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_set__conv__nth) ).

thf(4749,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( c_List_Oset @ B ) @ A )
      = ( hAPP @ ( c_Set_OCollect @ B ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ tc_HOL_Obool @ B ) @ ( c_HOL_OEx @ tc_Nat_Onat ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ B @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ tc_Nat_Onat @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) @ B ) @ ( c_COMBS @ tc_Nat_Onat @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ tc_Nat_Onat @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ tc_Nat_Onat ) @ c_fconj ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ B @ ( tc_fun @ tc_Nat_Onat @ B ) @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ B @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ tc_Nat_Onat @ B ) @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) @ B ) @ ( c_COMBB @ B @ tc_HOL_Obool @ tc_Nat_Onat ) ) @ c_fequal ) ) @ ( hAPP @ ( c_List_Onth @ B ) @ A ) ) ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ B ) ) @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[652]) ).

thf(2371,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i,I: $i,J: $i,K: $i] :
      ( ( c_Com_Ocom_Ocom__case @ K @ J @ I @ H @ G @ F @ E @ D @ C @ ( c_Com_Ocom_OWhile @ B @ A ) )
      = ( hAPP @ ( hAPP @ E @ B ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_com_Osimps_I69_J) ).

thf(11738,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i,I: $i,J: $i,K: $i] :
      ( ( c_Com_Ocom_Ocom__case @ K @ J @ I @ H @ G @ F @ E @ D @ C @ ( c_Com_Ocom_OWhile @ B @ A ) )
      = ( hAPP @ ( hAPP @ E @ B ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2371]) ).

thf(1656,axiom,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ ( hAPP @ c_Nat_OSuc @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__add__Suc2) ).

thf(8905,plain,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ ( hAPP @ c_Nat_OSuc @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1656]) ).

thf(1116,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Fun_Oinj__on @ E @ D @ C @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ E @ tc_HOL_Obool ) ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( c_Set_Oimage @ E @ D @ C ) @ A ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Set_Ovimage @ E @ D @ C ) @ B ) ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_vimage__subsetI) ).

thf(6577,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Fun_Oinj__on @ E @ D @ C @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ E @ tc_HOL_Obool ) ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( c_Set_Oimage @ E @ D @ C ) @ A ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Set_Ovimage @ E @ D @ C ) @ B ) ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1116]) ).

thf(2107,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ F @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ F @ E ) @ D ) @ C ) ) @ ( hAPP @ ( c_List_Oset @ ( tc_prod @ F @ E ) ) @ ( hAPP @ ( hAPP @ ( c_List_Ozip @ F @ E ) @ B ) @ A ) ) ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ C ) @ ( hAPP @ ( c_List_Oset @ E ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_set__zip__rightD) ).

thf(10672,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ F @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ F @ E ) @ D ) @ C ) ) @ ( hAPP @ ( c_List_Oset @ ( tc_prod @ F @ E ) ) @ ( hAPP @ ( hAPP @ ( c_List_Ozip @ F @ E ) @ B ) @ A ) ) ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ C ) @ ( hAPP @ ( c_List_Oset @ E ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2107]) ).

thf(2138,axiom,
    ! [A: $i,B: $i] :
      ( c_Com_Ocom_OSKIP
     != ( c_Com_Ocom_OAss @ B @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_com_Osimps_I8_J) ).

thf(10797,plain,
    ! [A: $i,B: $i] :
      ( c_Com_Ocom_OSKIP
     != ( c_Com_Ocom_OAss @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2138]) ).

thf(1876,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_Fun_Oinj__on @ D @ C @ B @ A )
     => ( ( hAPP @ ( c_Set_Oimage @ C @ D @ ( c_Fun_Othe__inv__into @ D @ C @ A @ B ) ) @ ( hAPP @ ( c_Set_Oimage @ D @ C @ B ) @ A ) )
        = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_the__inv__into__onto) ).

thf(9794,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_Fun_Oinj__on @ D @ C @ B @ A )
     => ( ( hAPP @ ( c_Set_Oimage @ C @ D @ ( c_Fun_Othe__inv__into @ D @ C @ A @ B ) ) @ ( hAPP @ ( c_Set_Oimage @ D @ C @ B ) @ A ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1876]) ).

thf(2276,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( hAPP @ E @ D )
        = ( hAPP @ ( c_Option_Ooption_OSome @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ A ) @ D ) @ ( c_Map_Odom @ A @ C @ E ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_domI) ).

thf(11327,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( hAPP @ E @ D )
        = ( hAPP @ ( c_Option_Ooption_OSome @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ A ) @ D ) @ ( c_Map_Odom @ A @ C @ E ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2276]) ).

thf(2382,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( c_Natural_Oevaln @ ( c_Com_Ocom_OCond @ F @ E @ D ) @ C @ B @ A )
    <=> ( ( ( hBOOL @ ( hAPP @ F @ C ) )
          & ( c_Natural_Oevaln @ E @ C @ B @ A ) )
        | ( ~ ( hBOOL @ ( hAPP @ F @ C ) )
          & ( c_Natural_Oevaln @ D @ C @ B @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_evaln_Oequations_I5_J) ).

thf(11770,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( c_Natural_Oevaln @ ( c_Com_Ocom_OCond @ F @ E @ D ) @ C @ B @ A )
       => ( ( ( hBOOL @ ( hAPP @ F @ C ) )
            & ( c_Natural_Oevaln @ E @ C @ B @ A ) )
          | ( ~ ( hBOOL @ ( hAPP @ F @ C ) )
            & ( c_Natural_Oevaln @ D @ C @ B @ A ) ) ) )
      & ( ( ( ( hBOOL @ ( hAPP @ F @ C ) )
            & ( c_Natural_Oevaln @ E @ C @ B @ A ) )
          | ( ~ ( hBOOL @ ( hAPP @ F @ C ) )
            & ( c_Natural_Oevaln @ D @ C @ B @ A ) ) )
       => ( c_Natural_Oevaln @ ( c_Com_Ocom_OCond @ F @ E @ D ) @ C @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2382]) ).

thf(2112,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_Finite__Set_Ofolding__one @ D @ C @ B )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ A ) )
       => ( ( hAPP @ B @ A )
          = ( hAPP @ ( c_Finite__Set_Ofold1 @ D @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_folding__one_Oeq__fold) ).

thf(10685,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_Finite__Set_Ofolding__one @ D @ C @ B )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ A ) )
       => ( ( hAPP @ B @ A )
          = ( hAPP @ ( c_Finite__Set_Ofold1 @ D @ C ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2112]) ).

thf(3071,plain,
    ~ ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ tc_Nat_Onat ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) ) ),
    inference(polarity_switch,[status(thm)],[3070]) ).

thf(11905,plain,
    ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ tc_Nat_Onat ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) ) )
   != ( hBOOL @ c_fTrue ) ),
    inference(paramod_ordered,[status(thm)],[11368,3071]) ).

thf(11909,plain,
    ( ( hAPP @ ( c_Finite__Set_Ofinite @ tc_Nat_Onat ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) )
   != c_fTrue ),
    inference(simp,[status(thm)],[11905]) ).

thf(799,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ F @ ( tc_fun @ E @ tc_HOL_Obool ) ) ) @ D ) @ C ) @ B ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ D @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_inf2D1) ).

thf(5294,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ F @ ( tc_fun @ E @ tc_HOL_Obool ) ) ) @ D ) @ C ) @ B ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ D @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[799]) ).

thf(2070,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Relation_Ototal__on @ C @ B @ A )
    <=> ! [D: $i] :
          ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ D ) @ B ) )
         => ! [E: $i] :
              ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ E ) @ B ) )
             => ( ( D != E )
               => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ C @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ C @ C ) @ D ) @ E ) ) @ A ) )
                  | ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ C @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ C @ C ) @ E ) @ D ) ) @ A ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_total__on__def) ).

thf(10540,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( c_Relation_Ototal__on @ C @ B @ A )
       => ! [D: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ D ) @ B ) )
           => ! [E: $i] :
                ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ E ) @ B ) )
               => ( ( D != E )
                 => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ C @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ C @ C ) @ D ) @ E ) ) @ A ) )
                    | ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ C @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ C @ C ) @ E ) @ D ) ) @ A ) ) ) ) ) ) )
      & ( ! [D: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ D ) @ B ) )
           => ! [E: $i] :
                ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ E ) @ B ) )
               => ( ( D != E )
                 => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ C @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ C @ C ) @ D ) @ E ) ) @ A ) )
                    | ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ C @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ C @ C ) @ E ) @ D ) ) @ A ) ) ) ) ) )
       => ( c_Relation_Ototal__on @ C @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2070]) ).

thf(2309,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ B ) )
     => ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ A ) )
       => ( ( ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ C ) @ B )
            = ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ C ) @ A ) )
        <=> ( B = A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_insert__ident) ).

thf(11460,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ B ) )
     => ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ A ) )
       => ( ( ( ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ C ) @ B )
              = ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ C ) @ A ) )
           => ( B = A ) )
          & ( ( B = A )
           => ( ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ C ) @ B )
              = ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ C ) @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2309]) ).

thf(1204,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hAPP @ ( hAPP @ ( c_List_Oappend @ C ) @ B ) @ A )
        = ( c_List_Olist_ONil @ C ) )
    <=> ( ( B
          = ( c_List_Olist_ONil @ C ) )
        & ( A
          = ( c_List_Olist_ONil @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_append__is__Nil__conv) ).

thf(6945,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( hAPP @ ( hAPP @ ( c_List_Oappend @ C ) @ B ) @ A )
          = ( c_List_Olist_ONil @ C ) )
       => ( ( B
            = ( c_List_Olist_ONil @ C ) )
          & ( A
            = ( c_List_Olist_ONil @ C ) ) ) )
      & ( ( ( B
            = ( c_List_Olist_ONil @ C ) )
          & ( A
            = ( c_List_Olist_ONil @ C ) ) )
       => ( ( hAPP @ ( hAPP @ ( c_List_Oappend @ C ) @ B ) @ A )
          = ( c_List_Olist_ONil @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1204]) ).

thf(788,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zmult__assoc) ).

thf(5250,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[788]) ).

thf(2055,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ A ) )
    <=> ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) )
        | ( B = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_le__eq__less__or__eq) ).

thf(10476,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ A ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) )
          | ( B = A ) ) )
      & ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) )
          | ( B = A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2055]) ).

thf(2330,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_New__Random__Sequence_Opos__not__random__dseq @ D @ C @ B @ A )
      = ( c_New__DSequence_Opos__not__seq @ ( hAPP @ ( hAPP @ ( hAPP @ D @ C ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_pos__not__random__dseq__def) ).

thf(11562,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_New__Random__Sequence_Opos__not__random__dseq @ D @ C @ B @ A )
      = ( c_New__DSequence_Opos__not__seq @ ( hAPP @ ( hAPP @ ( hAPP @ D @ C ) @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2330]) ).

thf(827,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i] :
          ( ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ tc_Nat_Onat @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ E ) ) @ C )
          = ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ tc_Nat_Onat @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ E ) @ B ) ) ) @ A ) )
     => ( ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ tc_Nat_Onat @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) ) @ C )
        = ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ tc_Nat_Onat @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_UN__finite2__eq) ).

thf(5399,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i] :
          ( ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ tc_Nat_Onat @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ E ) ) @ C )
          = ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ tc_Nat_Onat @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Nat_Onat @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ E ) @ B ) ) ) @ A ) )
     => ( ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ tc_Nat_Onat @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) ) @ C )
        = ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ tc_Nat_Onat @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[827]) ).

thf(1502,axiom,
    ! [A: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ A ) @ A )
      = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_diff__self__eq__0) ).

thf(8218,plain,
    ! [A: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ A ) @ A )
      = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1502]) ).

thf(1946,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_Com_Ocom_OAss @ D @ C )
     != ( c_Com_Ocom_OSemi @ B @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_com_Osimps_I24_J) ).

thf(10059,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_Com_Ocom_OAss @ D @ C )
     != ( c_Com_Ocom_OSemi @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1946]) ).

thf(1858,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Wellfounded_Oacyclic @ B @ A )
    <=> ! [C: $i] :
          ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ B @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ B @ B ) @ C ) @ C ) ) @ ( c_Transitive__Closure_Otrancl @ B @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_acyclic__def) ).

thf(9710,plain,
    ! [A: $i,B: $i] :
      ( ( ( c_Wellfounded_Oacyclic @ B @ A )
       => ! [C: $i] :
            ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ B @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ B @ B ) @ C ) @ C ) ) @ ( c_Transitive__Closure_Otrancl @ B @ A ) ) ) )
      & ( ! [C: $i] :
            ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ B @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ B @ B ) @ C ) @ C ) ) @ ( c_Transitive__Closure_Otrancl @ B @ A ) ) )
       => ( c_Wellfounded_Oacyclic @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1858]) ).

thf(710,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ A ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ B @ tc_HOL_Obool ) ) )
      = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Diff__UNIV) ).

thf(4950,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ A ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ B @ tc_HOL_Obool ) ) )
      = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[710]) ).

thf(11937,plain,
    ( ( hBOOL @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
    | ( sk4 != sk3 )
    | ( ( hAPP @ ( c_Nat_Osize__class_Osize @ tc_HOL_Obool ) @ c_fTrue )
     != ( hAPP @ v_b @ sk4 ) ) ),
    inference(paramod_ordered,[status(thm)],[3503,11893]) ).

thf(11989,plain,
    ( ( hBOOL @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
    | ( sk4 != sk3 )
    | ( ( c_Nat_Osize__class_Osize @ tc_HOL_Obool )
     != v_b )
    | ( sk4 != c_fTrue ) ),
    inference(simp,[status(thm)],[11937]) ).

thf(352,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Equiv__Relations_Oquotient @ D ) @ B ) @ A ) ) )
     => ~ ! [E: $i] :
            ( ( C
              = ( hAPP @ ( c_Relation_OImage @ D @ D @ A ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ E ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) ) )
           => ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ E ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_quotientE) ).

thf(3659,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Equiv__Relations_Oquotient @ D ) @ B ) @ A ) ) )
     => ~ ! [E: $i] :
            ( ( C
              = ( hAPP @ ( c_Relation_OImage @ D @ D @ A ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ E ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) ) )
           => ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ E ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[352]) ).

thf(230,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Int__Diff) ).

thf(3228,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[230]) ).

thf(622,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( C
          = ( c_List_Olist_ONil @ ( tc_List_Olist @ B ) ) )
       => ( A
          = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) )
     => ( ! [D: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ D ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ ( tc_List_Olist @ B ) ) ) @ C ) ) )
           => ( ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ B ) ) @ ( hAPP @ ( hAPP @ ( c_List_Onth @ ( tc_List_Olist @ B ) ) @ C ) @ D ) )
              = A ) )
       => ( ( c_List_Otranspose @ B @ C )
          = ( hAPP @ ( hAPP @ ( c_List_Omap @ tc_Nat_Onat @ ( tc_List_Olist @ B ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ ( tc_List_Olist @ tc_Nat_Onat ) @ ( tc_List_Olist @ B ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ tc_Nat_Onat @ B ) @ ( tc_fun @ ( tc_List_Olist @ tc_Nat_Onat ) @ ( tc_List_Olist @ B ) ) @ tc_Nat_Onat ) @ ( c_List_Omap @ tc_Nat_Onat @ B ) ) @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ tc_Nat_Onat @ B ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_List_Olist @ B ) @ ( tc_fun @ tc_Nat_Onat @ B ) @ tc_Nat_Onat ) @ ( c_List_Onth @ B ) ) @ ( hAPP @ ( c_List_Onth @ ( tc_List_Olist @ B ) ) @ C ) ) ) ) ) @ ( c_List_Oupt @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ ( tc_List_Olist @ B ) ) ) @ C ) ) ) ) @ ( c_List_Oupt @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_transpose__rectangle) ).

thf(4641,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( C
          = ( c_List_Olist_ONil @ ( tc_List_Olist @ B ) ) )
       => ( A
          = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) )
     => ( ! [D: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ D ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ ( tc_List_Olist @ B ) ) ) @ C ) ) )
           => ( ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ B ) ) @ ( hAPP @ ( hAPP @ ( c_List_Onth @ ( tc_List_Olist @ B ) ) @ C ) @ D ) )
              = A ) )
       => ( ( c_List_Otranspose @ B @ C )
          = ( hAPP @ ( hAPP @ ( c_List_Omap @ tc_Nat_Onat @ ( tc_List_Olist @ B ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ ( tc_List_Olist @ tc_Nat_Onat ) @ ( tc_List_Olist @ B ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ tc_Nat_Onat @ B ) @ ( tc_fun @ ( tc_List_Olist @ tc_Nat_Onat ) @ ( tc_List_Olist @ B ) ) @ tc_Nat_Onat ) @ ( c_List_Omap @ tc_Nat_Onat @ B ) ) @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ tc_Nat_Onat @ B ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_List_Olist @ B ) @ ( tc_fun @ tc_Nat_Onat @ B ) @ tc_Nat_Onat ) @ ( c_List_Onth @ B ) ) @ ( hAPP @ ( c_List_Onth @ ( tc_List_Olist @ B ) ) @ C ) ) ) ) ) @ ( c_List_Oupt @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ ( tc_List_Olist @ B ) ) ) @ C ) ) ) ) @ ( c_List_Oupt @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[622]) ).

thf(574,axiom,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ) @ A ) )
    <=> ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_int__one__le__iff__zero__less) ).

thf(4469,plain,
    ! [A: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ A ) ) )
      & ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[574]) ).

thf(1679,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_Map_Omap__of @ D @ C @ ( hAPP @ ( hAPP @ ( c_List_Omap @ D @ ( tc_prod @ D @ C ) ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ D @ C @ ( tc_prod @ D @ C ) ) @ ( c_Product__Type_OPair @ D @ C ) ) @ B ) ) @ A ) )
      = ( c_Map_Orestrict__map @ D @ C @ ( hAPP @ ( c_Fun_Ocomp @ C @ ( tc_Option_Ooption @ C ) @ D @ ( c_Option_Ooption_OSome @ C ) ) @ B ) @ ( hAPP @ ( c_List_Oset @ D ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_map__of__map__restrict) ).

thf(9005,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_Map_Omap__of @ D @ C @ ( hAPP @ ( hAPP @ ( c_List_Omap @ D @ ( tc_prod @ D @ C ) ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ D @ C @ ( tc_prod @ D @ C ) ) @ ( c_Product__Type_OPair @ D @ C ) ) @ B ) ) @ A ) )
      = ( c_Map_Orestrict__map @ D @ C @ ( hAPP @ ( c_Fun_Ocomp @ C @ ( tc_Option_Ooption @ C ) @ D @ ( c_Option_Ooption_OSome @ C ) ) @ B ) @ ( hAPP @ ( c_List_Oset @ D ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1679]) ).

thf(1535,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Nat_Onat ) @ A ) @ B ) ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_div__less__dividend) ).

thf(8350,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Nat_Onat ) @ A ) @ B ) ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1535]) ).

thf(1358,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) @ C ) @ B ) ) @ ( c_List_Olistrel @ D @ A ) ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) @ C ) @ B ) ) @ ( c_Transitive__Closure_Ortrancl @ ( tc_List_Olist @ D ) @ ( c_List_Olistrel1 @ D @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_rtrancl__listrel1__if__listrel) ).

thf(7618,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) @ C ) @ B ) ) @ ( c_List_Olistrel @ D @ A ) ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) @ C ) @ B ) ) @ ( c_Transitive__Closure_Ortrancl @ ( tc_List_Olist @ D ) @ ( c_List_Olistrel1 @ D @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1358]) ).

thf(2232,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_New__Random__Sequence_Oneg__single @ E ) @ D ) @ C ) @ B ) @ A )
      = ( c_New__DSequence_Oneg__single @ E @ D ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_neg__single__def) ).

thf(11137,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_New__Random__Sequence_Oneg__single @ E ) @ D ) @ C ) @ B ) @ A )
      = ( c_New__DSequence_Oneg__single @ E @ D ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2232]) ).

thf(1539,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_Wellfounded_Owf @ D @ C )
     => ( c_Wellfounded_Owf @ B @ ( hAPP @ ( hAPP @ ( c_Relation_Oinv__image @ D @ B ) @ C ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_wf__inv__image) ).

thf(8364,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_Wellfounded_Owf @ D @ C )
     => ( c_Wellfounded_Owf @ B @ ( hAPP @ ( hAPP @ ( c_Relation_Oinv__image @ D @ B ) @ C ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1539]) ).

thf(2261,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Finite__Set_Ofolding__one__idem @ E @ D @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ E ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ A ) @ B ) )
         => ( ( hAPP @ ( hAPP @ D @ A ) @ ( hAPP @ C @ B ) )
            = ( hAPP @ C @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_folding__one__idem_Oin__idem) ).

thf(11252,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Finite__Set_Ofolding__one__idem @ E @ D @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ E ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ A ) @ B ) )
         => ( ( hAPP @ ( hAPP @ D @ A ) @ ( hAPP @ C @ B ) )
            = ( hAPP @ C @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2261]) ).

thf(1968,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( C
     != ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ B ) @ A ) @ C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_not__Cons__self) ).

thf(10159,plain,
    ! [A: $i,B: $i,C: $i] :
      ( C
     != ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ B ) @ A ) @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1968]) ).

thf(338,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( c_Finite__Set_Ofold1 @ C @ B ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_fold1__singleton) ).

thf(3608,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( c_Finite__Set_Ofold1 @ C @ B ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[338]) ).

thf(1859,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Natural_Oevalc @ ( c_Com_Ocom_OSemi @ D @ C ) ) @ B ) @ A ) )
     => ~ ! [E: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Natural_Oevalc @ D ) @ B ) @ E ) )
           => ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Natural_Oevalc @ C ) @ E ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_evalc__elim__cases_I4_J) ).

thf(9715,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Natural_Oevalc @ ( c_Com_Ocom_OSemi @ D @ C ) ) @ B ) @ A ) )
     => ~ ! [E: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Natural_Oevalc @ D ) @ B ) @ E ) )
           => ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Natural_Oevalc @ C ) @ E ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1859]) ).

thf(1228,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ B ) @ A ) )
    <=> ? [C: $i,D: $i] :
          ( A
          = ( hAPP @ ( c_Set_Oimage @ tc_Nat_Onat @ B @ D ) @ ( hAPP @ ( c_Set_OCollect @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_finite__conv__nat__seg__image) ).

thf(7055,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ B ) @ A ) )
       => ? [C: $i,D: $i] :
            ( A
            = ( hAPP @ ( c_Set_Oimage @ tc_Nat_Onat @ B @ D ) @ ( hAPP @ ( c_Set_OCollect @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) ) @ C ) ) ) ) )
      & ( ? [C: $i,D: $i] :
            ( A
            = ( hAPP @ ( c_Set_Oimage @ tc_Nat_Onat @ B @ D ) @ ( hAPP @ ( c_Set_OCollect @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) ) @ C ) ) ) )
       => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1228]) ).

thf(1614,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Divides_Odivmod__nat @ B @ A )
      = ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ tc_Nat_Onat @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Nat_Onat ) @ B ) @ A ) ) @ ( c_Divides_Odiv__class_Omod @ tc_Nat_Onat @ B @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_divmod__nat__div__mod) ).

thf(8744,plain,
    ! [A: $i,B: $i] :
      ( ( c_Divides_Odivmod__nat @ B @ A )
      = ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ tc_Nat_Onat @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Nat_Onat ) @ B ) @ A ) ) @ ( c_Divides_Odiv__class_Omod @ tc_Nat_Onat @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1614]) ).

thf(1022,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Lattices_Oab__semigroup__idem__mult @ C )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__left__idem) ).

thf(6146,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Lattices_Oab__semigroup__idem__mult @ C )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1022]) ).

thf(1468,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_Map_Omap__upds @ D @ C @ B @ ( c_List_Olist_ONil @ D ) @ A )
      = B ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_map__upds__Nil1) ).

thf(8039,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_Map_Omap__upds @ D @ C @ B @ ( c_List_Olist_ONil @ D ) @ A )
      = B ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1468]) ).

thf(81,axiom,
    ! [A: $i,B: $i] :
      ( ( ( c_List_Otranspose @ B @ A )
        = ( c_List_Olist_ONil @ ( tc_List_Olist @ B ) ) )
    <=> ! [C: $i] :
          ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_List_Olist @ B ) ) @ C ) @ ( hAPP @ ( c_List_Oset @ ( tc_List_Olist @ B ) ) @ A ) ) )
         => ( C
            = ( c_List_Olist_ONil @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_transpose__empty) ).

thf(2689,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( c_List_Otranspose @ B @ A )
          = ( c_List_Olist_ONil @ ( tc_List_Olist @ B ) ) )
       => ! [C: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_List_Olist @ B ) ) @ C ) @ ( hAPP @ ( c_List_Oset @ ( tc_List_Olist @ B ) ) @ A ) ) )
           => ( C
              = ( c_List_Olist_ONil @ B ) ) ) )
      & ( ! [C: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_List_Olist @ B ) ) @ C ) @ ( hAPP @ ( c_List_Oset @ ( tc_List_Olist @ B ) ) @ A ) ) )
           => ( C
              = ( c_List_Olist_ONil @ B ) ) )
       => ( ( c_List_Otranspose @ B @ A )
          = ( c_List_Olist_ONil @ ( tc_List_Olist @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[81]) ).

thf(294,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ C @ C @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Nat_Ocompow @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) @ A ) @ B ) )
      = ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ C @ C @ C ) @ ( hAPP @ ( hAPP @ ( c_Nat_Ocompow @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) @ A ) @ B ) ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_rel__pow__commute) ).

thf(3448,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ C @ C @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Nat_Ocompow @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) @ A ) @ B ) )
      = ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ C @ C @ C ) @ ( hAPP @ ( hAPP @ ( c_Nat_Ocompow @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) @ A ) @ B ) ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[294]) ).

thf(2205,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ B ) @ A ) ) )
     => ( ( C != B )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_insertE) ).

thf(11027,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ B ) @ A ) ) )
     => ( ( C != B )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2205]) ).

thf(1192,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Wellfounded_Owf @ B @ A )
     => ( c_Wellfounded_Oacyclic @ B @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_wf__acyclic) ).

thf(6912,plain,
    ! [A: $i,B: $i] :
      ( ( c_Wellfounded_Owf @ B @ A )
     => ( c_Wellfounded_Oacyclic @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1192]) ).

thf(1645,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Divides_Odiv__class_Omod @ tc_Nat_Onat @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ ( c_Divides_Odiv__class_Omod @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Nat_Onat ) @ C ) @ B ) @ A ) ) ) @ ( c_Divides_Odiv__class_Omod @ tc_Nat_Onat @ C @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mod__mult2__eq) ).

thf(8873,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Divides_Odiv__class_Omod @ tc_Nat_Onat @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ ( c_Divides_Odiv__class_Omod @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Nat_Onat ) @ C ) @ B ) @ A ) ) ) @ ( c_Divides_Odiv__class_Omod @ tc_Nat_Onat @ C @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1645]) ).

thf(2021,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ B ) ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Suc__leD) ).

thf(10365,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ B ) ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2021]) ).

thf(1070,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ A ) @ C ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ A ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_subsetD) ).

thf(6352,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ A ) @ C ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ A ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1070]) ).

thf(689,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( c_Set_Oimage @ E @ D @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Set_Oimage @ E @ D @ C ) @ B ) ) @ ( hAPP @ ( c_Set_Oimage @ E @ D @ C ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_image__Un) ).

thf(4875,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( c_Set_Oimage @ E @ D @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Set_Oimage @ E @ D @ C ) @ B ) ) @ ( hAPP @ ( c_Set_Oimage @ E @ D @ C ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[689]) ).

thf(1793,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( c_Com_Ocom_OCond @ F @ E @ D )
     != ( c_Com_Ocom_OCall @ C @ B @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_com_Osimps_I56_J) ).

thf(9445,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( c_Com_Ocom_OCond @ F @ E @ D )
     != ( c_Com_Ocom_OCall @ C @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1793]) ).

thf(170,axiom,
    ! [A: $i] :
      ( ( c_Transitive__Closure_Otrancl @ A @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ ( tc_prod @ A @ A ) @ tc_HOL_Obool ) ) )
      = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ ( tc_prod @ A @ A ) @ tc_HOL_Obool ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_trancl__empty) ).

thf(3019,plain,
    ! [A: $i] :
      ( ( c_Transitive__Closure_Otrancl @ A @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ ( tc_prod @ A @ A ) @ tc_HOL_Obool ) ) )
      = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ ( tc_prod @ A @ A ) @ tc_HOL_Obool ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[170]) ).

thf(914,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ B ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Divides_Otransfer__nat__int__function__closures_I1_J) ).

thf(5684,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ B ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[914]) ).

thf(558,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Typedef_Otype__definition @ E @ D @ C @ B @ A )
     => ( ( hAPP @ ( c_Set_Oimage @ D @ E @ B ) @ A )
        = ( c_Orderings_Otop__class_Otop @ ( tc_fun @ E @ tc_HOL_Obool ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_type__definition_OAbs__image) ).

thf(4409,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Typedef_Otype__definition @ E @ D @ C @ B @ A )
     => ( ( hAPP @ ( c_Set_Oimage @ D @ E @ B ) @ A )
        = ( c_Orderings_Otop__class_Otop @ ( tc_fun @ E @ tc_HOL_Obool ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[558]) ).

thf(50,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Groups_Oab__group__add @ E )
     => ( ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ E ) @ D ) @ C )
          = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ E ) @ B ) @ A ) )
       => ( ( D = C )
        <=> ( B = A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_diff__eq__diff__eq) ).

thf(2569,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Groups_Oab__group__add @ E )
     => ( ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ E ) @ D ) @ C )
          = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ E ) @ B ) @ A ) )
       => ( ( ( D = C )
           => ( B = A ) )
          & ( ( B = A )
           => ( D = C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[50]) ).

thf(2004,axiom,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ c_Com_OWT @ ( c_Com_Ocom_OAss @ B @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_WT_Oequations_I2_J) ).

thf(10307,plain,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ c_Com_OWT @ ( c_Com_Ocom_OAss @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2004]) ).

thf(1307,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Odrop @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_List_Odrop @ D ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_List_Odrop @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ B ) ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_drop__drop) ).

thf(7388,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Odrop @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_List_Odrop @ D ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_List_Odrop @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ B ) ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1307]) ).

thf(568,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) ) )
    <=> ( B = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_singleton__iff) ).

thf(4437,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) ) )
       => ( B = A ) )
      & ( ( B = A )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[568]) ).

thf(1475,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Equiv__Relations_Oequiv @ E @ D @ C )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ B ) @ A ) ) @ C ) )
      <=> ( ( ( hAPP @ ( c_Relation_OImage @ E @ E @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ B ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ E @ tc_HOL_Obool ) ) ) )
            = ( hAPP @ ( c_Relation_OImage @ E @ E @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ E @ tc_HOL_Obool ) ) ) ) )
          & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ B ) @ D ) )
          & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ A ) @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_equiv__class__eq__iff) ).

thf(8060,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Equiv__Relations_Oequiv @ E @ D @ C )
     => ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ B ) @ A ) ) @ C ) )
         => ( ( ( hAPP @ ( c_Relation_OImage @ E @ E @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ B ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ E @ tc_HOL_Obool ) ) ) )
              = ( hAPP @ ( c_Relation_OImage @ E @ E @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ E @ tc_HOL_Obool ) ) ) ) )
            & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ B ) @ D ) )
            & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ A ) @ D ) ) ) )
        & ( ( ( ( hAPP @ ( c_Relation_OImage @ E @ E @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ B ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ E @ tc_HOL_Obool ) ) ) )
              = ( hAPP @ ( c_Relation_OImage @ E @ E @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ E @ tc_HOL_Obool ) ) ) ) )
            & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ B ) @ D ) )
            & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ A ) @ D ) ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ B ) @ A ) ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1475]) ).

thf(1548,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ D ) @ C ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ tc_Nat_Onat @ tc_Nat_Onat ) @ D ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ tc_Nat_Onat @ tc_Nat_Onat ) @ C ) @ A ) ) ) @ c_FunDef_Opair__leq ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_pair__leqI1) ).

thf(8396,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ D ) @ C ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ tc_Nat_Onat @ tc_Nat_Onat ) @ D ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ tc_Nat_Onat @ tc_Nat_Onat ) @ C ) @ A ) ) ) @ c_FunDef_Opair__leq ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1548]) ).

thf(837,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( hAPP @ ( c_Set_Oimage @ F @ E @ D ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ F @ tc_HOL_Obool ) ) )
        = ( c_Orderings_Otop__class_Otop @ ( tc_fun @ E @ tc_HOL_Obool ) ) )
     => ( ( ( hAPP @ ( c_Set_Oimage @ C @ B @ A ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ C @ tc_HOL_Obool ) ) )
          = ( c_Orderings_Otop__class_Otop @ ( tc_fun @ B @ tc_HOL_Obool ) ) )
       => ( ( hAPP @ ( c_Set_Oimage @ ( tc_prod @ F @ C ) @ ( tc_prod @ E @ B ) @ ( c_Product__Type_Omap__pair @ F @ E @ C @ B @ D @ A ) ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ ( tc_prod @ F @ C ) @ tc_HOL_Obool ) ) )
          = ( c_Orderings_Otop__class_Otop @ ( tc_fun @ ( tc_prod @ E @ B ) @ tc_HOL_Obool ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_map__pair__surj) ).

thf(5431,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( hAPP @ ( c_Set_Oimage @ F @ E @ D ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ F @ tc_HOL_Obool ) ) )
        = ( c_Orderings_Otop__class_Otop @ ( tc_fun @ E @ tc_HOL_Obool ) ) )
     => ( ( ( hAPP @ ( c_Set_Oimage @ C @ B @ A ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ C @ tc_HOL_Obool ) ) )
          = ( c_Orderings_Otop__class_Otop @ ( tc_fun @ B @ tc_HOL_Obool ) ) )
       => ( ( hAPP @ ( c_Set_Oimage @ ( tc_prod @ F @ C ) @ ( tc_prod @ E @ B ) @ ( c_Product__Type_Omap__pair @ F @ E @ C @ B @ D @ A ) ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ ( tc_prod @ F @ C ) @ tc_HOL_Obool ) ) )
          = ( c_Orderings_Otop__class_Otop @ ( tc_fun @ ( tc_prod @ E @ B ) @ tc_HOL_Obool ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[837]) ).

thf(346,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ C ) )
     => ( ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ A ) @ C ) )
        = ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ B ) @ A ) ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zdiv__zmult2__eq) ).

thf(3642,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ C ) )
     => ( ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ A ) @ C ) )
        = ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ B ) @ A ) ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[346]) ).

thf(1575,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ E @ D @ tc_HOL_Obool ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ D ) @ B ) @ A ) ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ C @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_splitD) ).

thf(8498,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ E @ D @ tc_HOL_Obool ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ D ) @ B ) @ A ) ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ C @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1575]) ).

thf(2044,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ G @ F ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ G @ F ) @ E ) @ D ) ) @ C ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ B ) @ ( hAPP @ ( hAPP @ A @ E ) @ D ) ) @ ( hAPP @ ( c_Set_Oimage @ ( tc_prod @ G @ F ) @ B @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ G @ F @ B ) @ A ) ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_pair__imageI) ).

thf(10439,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ G @ F ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ G @ F ) @ E ) @ D ) ) @ C ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ B ) @ ( hAPP @ ( hAPP @ A @ E ) @ D ) ) @ ( hAPP @ ( c_Set_Oimage @ ( tc_prod @ G @ F ) @ B @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ G @ F @ B ) @ A ) ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2044]) ).

thf(833,axiom,
    ! [A: $i,B: $i] : ( c_Relation_Ototal__on @ B @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_total__on__empty) ).

thf(5420,plain,
    ! [A: $i,B: $i] : ( c_Relation_Ototal__on @ B @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[833]) ).

thf(1211,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ D ) @ C ) )
     => ( ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ C )
          = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ D ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__add__eq__less) ).

thf(6981,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ D ) @ C ) )
     => ( ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ C )
          = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ D ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1211]) ).

thf(1279,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ C ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ B ) @ C ) )
       => ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ B ) @ A ) )
         => ( ( hAPP @ ( c_Finite__Set_Ocard @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ B ) @ A ) ) )
            = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ ( hAPP @ ( c_Finite__Set_Ocard @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ A ) ) ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_card__Diff__insert) ).

thf(7264,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ C ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ B ) @ C ) )
       => ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ B ) @ A ) )
         => ( ( hAPP @ ( c_Finite__Set_Ocard @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ B ) @ A ) ) )
            = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ ( hAPP @ ( c_Finite__Set_Ocard @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ A ) ) ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1279]) ).

thf(1325,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( c_Finite__Set_Ocard @ tc_Nat_Onat ) @ ( hAPP @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Nat_Onat @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ A ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_card__atLeastLessThan) ).

thf(7494,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( c_Finite__Set_Ocard @ tc_Nat_Onat ) @ ( hAPP @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Nat_Onat @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ A ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1325]) ).

thf(498,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ F ) @ E ) ) @ D )
        = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ B ) ) @ A ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ B ) ) @ A ) ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ A ) @ C ) )
         => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ D ) )
           => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ C ) )
             => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ C ) @ F ) )
               => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ E ) @ B ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zdiv__mono2__lemma) ).

thf(4190,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ F ) @ E ) ) @ D )
        = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ B ) ) @ A ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ B ) ) @ A ) ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ A ) @ C ) )
         => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ D ) )
           => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ C ) )
             => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ C ) @ F ) )
               => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ E ) @ B ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[498]) ).

thf(393,axiom,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ B ) @ A ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_iso__tuple__UNIV__I) ).

thf(3818,plain,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ B ) @ A ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[393]) ).

thf(196,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ C ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zadd__assoc) ).

thf(3114,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ C ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[196]) ).

thf(244,axiom,
    ! [A: $i,B: $i] :
      ( ( ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ B @ A )
        = ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) )
    <=> ? [C: $i] :
          ( B
          = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ A ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zmod__eq__0__iff) ).

thf(3273,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ B @ A )
          = ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) )
       => ? [C: $i] :
            ( B
            = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ A ) @ C ) ) )
      & ( ? [C: $i] :
            ( B
            = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ A ) @ C ) )
       => ( ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ B @ A )
          = ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[244]) ).

thf(261,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D
       != ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) )
     => ( ? [E: $i] :
            ( ( c_Fun_Oinj__on @ C @ B @ E @ D )
            & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Set_Oimage @ C @ B @ E ) @ D ) ) @ A ) ) )
      <=> ? [E: $i] :
            ( ( hAPP @ ( c_Set_Oimage @ B @ C @ E ) @ A )
            = D ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_inj__on__iff__surj) ).

thf(3334,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D
       != ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) )
     => ( ( ? [E: $i] :
              ( ( c_Fun_Oinj__on @ C @ B @ E @ D )
              & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Set_Oimage @ C @ B @ E ) @ D ) ) @ A ) ) )
         => ? [E: $i] :
              ( ( hAPP @ ( c_Set_Oimage @ B @ C @ E ) @ A )
              = D ) )
        & ( ? [E: $i] :
              ( ( hAPP @ ( c_Set_Oimage @ B @ C @ E ) @ A )
              = D )
         => ? [E: $i] :
              ( ( c_Fun_Oinj__on @ C @ B @ E @ D )
              & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Set_Oimage @ C @ B @ E ) @ D ) ) @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[261]) ).

thf(1272,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ G @ F @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ D ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ G @ F ) @ C ) @ B ) ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( hAPP @ D @ C ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_splitD_H) ).

thf(7243,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ G @ F @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ D ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ G @ F ) @ C ) @ B ) ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( hAPP @ D @ C ) @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1272]) ).

thf(445,axiom,
    ! [A: $i,B: $i] :
      ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ B ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_empty__iff) ).

thf(4003,plain,
    ! [A: $i,B: $i] :
      ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ B ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[445]) ).

thf(1929,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_trans__le__add1) ).

thf(9986,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1929]) ).

thf(1725,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ E ) ) @ D )
        = ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ A ) @ ( hAPP @ ( c_List_Oset @ E ) @ D ) ) )
      <=> ? [F: $i] :
            ( ( hAPP @ ( c_Map_Omap__of @ E @ C @ ( hAPP @ ( hAPP @ ( c_List_Ozip @ E @ C ) @ D ) @ B ) ) @ A )
            = ( hAPP @ ( c_Option_Ooption_OSome @ C ) @ F ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_map__of__zip__is__Some) ).

thf(9182,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ E ) ) @ D )
        = ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) @ B ) )
     => ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ A ) @ ( hAPP @ ( c_List_Oset @ E ) @ D ) ) )
         => ? [F: $i] :
              ( ( hAPP @ ( c_Map_Omap__of @ E @ C @ ( hAPP @ ( hAPP @ ( c_List_Ozip @ E @ C ) @ D ) @ B ) ) @ A )
              = ( hAPP @ ( c_Option_Ooption_OSome @ C ) @ F ) ) )
        & ( ? [F: $i] :
              ( ( hAPP @ ( c_Map_Omap__of @ E @ C @ ( hAPP @ ( hAPP @ ( c_List_Ozip @ E @ C ) @ D ) @ B ) ) @ A )
              = ( hAPP @ ( c_Option_Ooption_OSome @ C ) @ F ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ A ) @ ( hAPP @ ( c_List_Oset @ E ) @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1725]) ).

thf(933,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Int_Oint ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ A ) ) ) )
    <=> ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Int_Oint ) @ C ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zdvd__reduce) ).

thf(5753,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Int_Oint ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ A ) ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Int_Oint ) @ C ) @ B ) ) )
      & ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Int_Oint ) @ C ) @ B ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Int_Oint ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[933]) ).

thf(597,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( E
          = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) )
       => ( ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ D @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ E ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) @ D ) @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) ) @ A ) )
          = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) )
      & ( ( E
         != ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) )
       => ( ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ D @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ E ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) @ D ) @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) ) @ A ) )
          = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ D @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ E ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_UN__simps_I3_J) ).

thf(4549,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( E
          = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) )
       => ( ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ D @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ E ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) @ D ) @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) ) @ A ) )
          = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) )
      & ( ( E
         != ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) )
       => ( ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ D @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ E ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) @ D ) @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) ) @ A ) )
          = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ D @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ E ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[597]) ).

thf(2121,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( c_Big__Operators_Ocomm__monoid__big @ G @ F @ E @ D @ C )
     => ( ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ F ) @ B ) )
         => ( ( hAPP @ ( hAPP @ C @ A ) @ B )
            = ( hAPP @ ( hAPP @ ( hAPP @ ( c_Finite__Set_Ofold__image @ G @ F @ E ) @ A ) @ D ) @ B ) ) )
        & ( ~ ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ F ) @ B ) )
         => ( ( hAPP @ ( hAPP @ C @ A ) @ B )
            = D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_comm__monoid__big_OF__eq) ).

thf(10717,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( c_Big__Operators_Ocomm__monoid__big @ G @ F @ E @ D @ C )
     => ( ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ F ) @ B ) )
         => ( ( hAPP @ ( hAPP @ C @ A ) @ B )
            = ( hAPP @ ( hAPP @ ( hAPP @ ( c_Finite__Set_Ofold__image @ G @ F @ E ) @ A ) @ D ) @ B ) ) )
        & ( ~ ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ F ) @ B ) )
         => ( ( hAPP @ ( hAPP @ C @ A ) @ B )
            = D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2121]) ).

thf(2238,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Predicate_OPowp @ C @ B ) @ A ) )
    <=> ! [D: $i] :
          ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ D ) @ A ) )
         => ( hBOOL @ ( hAPP @ B @ D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Powp__def) ).

thf(11166,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( c_Predicate_OPowp @ C @ B ) @ A ) )
       => ! [D: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ D ) @ A ) )
           => ( hBOOL @ ( hAPP @ B @ D ) ) ) )
      & ( ! [D: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ D ) @ A ) )
           => ( hBOOL @ ( hAPP @ B @ D ) ) )
       => ( hBOOL @ ( hAPP @ ( c_Predicate_OPowp @ C @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2238]) ).

thf(1371,axiom,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) ) @ ( c_Relation_OId__on @ B @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ B @ B ) @ A ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ B @ tc_HOL_Obool ) @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Id__on__subset__Times) ).

thf(7667,plain,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) ) @ ( c_Relation_OId__on @ B @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ B @ B ) @ A ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ B @ tc_HOL_Obool ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1371]) ).

thf(302,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ! [D: $i] :
          ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ D ) @ ( hAPP @ ( c_List_Oset @ C ) @ B ) ) )
         => ( hBOOL @ ( hAPP @ A @ D ) ) )
    <=> ! [D: $i] :
          ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ D ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) @ B ) ) )
         => ( hBOOL @ ( hAPP @ A @ ( hAPP @ ( hAPP @ ( c_List_Onth @ C ) @ B ) @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_all__set__conv__all__nth) ).

thf(3475,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ! [D: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ D ) @ ( hAPP @ ( c_List_Oset @ C ) @ B ) ) )
           => ( hBOOL @ ( hAPP @ A @ D ) ) )
       => ! [D: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ D ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) @ B ) ) )
           => ( hBOOL @ ( hAPP @ A @ ( hAPP @ ( hAPP @ ( c_List_Onth @ C ) @ B ) @ D ) ) ) ) )
      & ( ! [D: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ D ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) @ B ) ) )
           => ( hBOOL @ ( hAPP @ A @ ( hAPP @ ( hAPP @ ( c_List_Onth @ C ) @ B ) @ D ) ) ) )
       => ! [D: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ D ) @ ( hAPP @ ( c_List_Oset @ C ) @ B ) ) )
           => ( hBOOL @ ( hAPP @ A @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[302]) ).

thf(1404,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ B ) @ A ) )
     => ( ( ? [C: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ B ) @ C ) @ A ) )
         => ( ( c_Groups_Oone__class_Oone @ tc_Nat_Onat )
            = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) )
       => ( ( hAPP @ ( c_Finite__Set_Ocard @ B ) @ A )
          = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_card_Oneutral) ).

thf(7796,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ B ) @ A ) )
     => ( ( ? [C: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ B ) @ C ) @ A ) )
         => ( ( c_Groups_Oone__class_Oone @ tc_Nat_Onat )
            = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) )
       => ( ( hAPP @ ( c_Finite__Set_Ocard @ B ) @ A )
          = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1404]) ).

thf(1459,axiom,
    ! [A: $i,B: $i] :
      ( ( c_HOL_OThe @ B @ ( hAPP @ c_fequal @ A ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_the__sym__eq__trivial) ).

thf(7988,plain,
    ! [A: $i,B: $i] :
      ( ( c_HOL_OThe @ B @ ( hAPP @ c_fequal @ A ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1459]) ).

thf(173,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( c_Relation_OImage @ C @ B @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) )
      = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Image__empty) ).

thf(3026,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( c_Relation_OImage @ C @ B @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) )
      = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[173]) ).

thf(658,axiom,
    ! [A: $i] :
      ( ( hAPP @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Nat_Onat @ A ) @ ( hAPP @ c_Nat_OSuc @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ tc_Nat_Onat ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_atLeastLessThan__singleton) ).

thf(4768,plain,
    ! [A: $i] :
      ( ( hAPP @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Nat_Onat @ A ) @ ( hAPP @ c_Nat_OSuc @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ tc_Nat_Onat ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[658]) ).

thf(1882,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ D ) @ C ) ) @ ( c_Transitive__Closure_Ortrancl @ E @ B ) ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ C ) @ A ) ) @ B ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ D ) @ A ) ) @ ( c_Transitive__Closure_Otrancl @ E @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_rtrancl__into__trancl1) ).

thf(9809,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ D ) @ C ) ) @ ( c_Transitive__Closure_Ortrancl @ E @ B ) ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ C ) @ A ) ) @ B ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ D ) @ A ) ) @ ( c_Transitive__Closure_Otrancl @ E @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1882]) ).

thf(880,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ B ) ) @ A ) )
     => ( ( hAPP @ ( hAPP @ ( c_List_Onth @ tc_Nat_Onat ) @ ( c_List_Oupt @ C @ A ) ) @ B )
        = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nth__upt) ).

thf(5560,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ B ) ) @ A ) )
     => ( ( hAPP @ ( hAPP @ ( c_List_Onth @ tc_Nat_Onat ) @ ( c_List_Oupt @ C @ A ) ) @ B )
        = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[880]) ).

thf(1338,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( E
          = ( c_Option_Ooption_ONone @ D ) )
       => ( ( c_Map_Odom @ C @ D @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ C @ ( tc_Option_Ooption @ D ) ) @ B ) @ A ) @ E ) )
          = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( c_Map_Odom @ C @ D @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) ) ) )
      & ( ( E
         != ( c_Option_Ooption_ONone @ D ) )
       => ( ( c_Map_Odom @ C @ D @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ C @ ( tc_Option_Ooption @ D ) ) @ B ) @ A ) @ E ) )
          = ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ A ) @ ( c_Map_Odom @ C @ D @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_dom__fun__upd) ).

thf(7543,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( E
          = ( c_Option_Ooption_ONone @ D ) )
       => ( ( c_Map_Odom @ C @ D @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ C @ ( tc_Option_Ooption @ D ) ) @ B ) @ A ) @ E ) )
          = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( c_Map_Odom @ C @ D @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) ) ) )
      & ( ( E
         != ( c_Option_Ooption_ONone @ D ) )
       => ( ( c_Map_Odom @ C @ D @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ C @ ( tc_Option_Ooption @ D ) ) @ B ) @ A ) @ E ) )
          = ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ A ) @ ( c_Map_Odom @ C @ D @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1338]) ).

thf(1097,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Transitive__Closure_Otrancl @ B @ ( hAPP @ ( c_Relation_Oconverse @ B @ B ) @ A ) )
      = ( hAPP @ ( c_Relation_Oconverse @ B @ B ) @ ( c_Transitive__Closure_Otrancl @ B @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_trancl__converse) ).

thf(6508,plain,
    ! [A: $i,B: $i] :
      ( ( c_Transitive__Closure_Otrancl @ B @ ( hAPP @ ( c_Relation_Oconverse @ B @ B ) @ A ) )
      = ( hAPP @ ( c_Relation_Oconverse @ B @ B ) @ ( c_Transitive__Closure_Otrancl @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1097]) ).

thf(1427,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_le__add__diff) ).

thf(7879,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1427]) ).

thf(1578,axiom,
    ! [A: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ A ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nat__mult__1__right) ).

thf(8509,plain,
    ! [A: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ A ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1578]) ).

thf(387,axiom,
    ! [A: $i,B: $i] : ( c_Wellfounded_Owf @ B @ ( hAPP @ ( c_Wellfounded_Omeasure @ B ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_wf__measure) ).

thf(3801,plain,
    ! [A: $i,B: $i] : ( c_Wellfounded_Owf @ B @ ( hAPP @ ( c_Wellfounded_Omeasure @ B ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[387]) ).

thf(1694,axiom,
    ! [A: $i] :
      ( ( hAPP @ ( c_SetInterval_Oord__class_OatMost @ tc_Nat_Onat ) @ A )
      = ( hAPP @ ( c_List_Oset @ tc_Nat_Onat ) @ ( c_List_Oupt @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_atMost__upto) ).

thf(9082,plain,
    ! [A: $i] :
      ( ( hAPP @ ( c_SetInterval_Oord__class_OatMost @ tc_Nat_Onat ) @ A )
      = ( hAPP @ ( c_List_Oset @ tc_Nat_Onat ) @ ( c_List_Oupt @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1694]) ).

thf(766,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Wellfounded_Oaccp @ C @ ( hAPP @ ( hAPP @ ( c_COMBC @ C @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ ( tc_fun @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ C ) @ ( c_COMBC @ C @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ ( tc_prod @ C @ C ) ) @ ( tc_fun @ C @ ( tc_fun @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( c_COMBB @ ( tc_prod @ C @ C ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ C ) @ ( c_member @ ( tc_prod @ C @ C ) ) ) ) @ ( c_Product__Type_OPair @ C @ C ) ) ) ) @ B ) ) @ A ) )
    <=> ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ A ) @ ( c_Wellfounded_Oacc @ C @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_accp__acc__eq) ).

thf(5152,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( c_Wellfounded_Oaccp @ C @ ( hAPP @ ( hAPP @ ( c_COMBC @ C @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ ( tc_fun @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ C ) @ ( c_COMBC @ C @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ ( tc_prod @ C @ C ) ) @ ( tc_fun @ C @ ( tc_fun @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( c_COMBB @ ( tc_prod @ C @ C ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ C ) @ ( c_member @ ( tc_prod @ C @ C ) ) ) ) @ ( c_Product__Type_OPair @ C @ C ) ) ) ) @ B ) ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ A ) @ ( c_Wellfounded_Oacc @ C @ B ) ) ) )
      & ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ A ) @ ( c_Wellfounded_Oacc @ C @ B ) ) )
       => ( hBOOL @ ( hAPP @ ( c_Wellfounded_Oaccp @ C @ ( hAPP @ ( hAPP @ ( c_COMBC @ C @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ ( tc_fun @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ C ) @ ( c_COMBC @ C @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ ( tc_prod @ C @ C ) ) @ ( tc_fun @ C @ ( tc_fun @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( c_COMBB @ ( tc_prod @ C @ C ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ C ) @ ( c_member @ ( tc_prod @ C @ C ) ) ) ) @ ( c_Product__Type_OPair @ C @ C ) ) ) ) @ B ) ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[766]) ).

thf(1528,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Ozip @ C @ B ) @ A ) @ ( c_List_Olist_ONil @ B ) )
      = ( c_List_Olist_ONil @ ( tc_prod @ C @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zip_Osimps_I1_J) ).

thf(8327,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Ozip @ C @ B ) @ A ) @ ( c_List_Olist_ONil @ B ) )
      = ( c_List_Olist_ONil @ ( tc_prod @ C @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1528]) ).

thf(1484,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ C ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Int_Oint ) @ B ) @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ) ) @ A ) )
       => ( ! [D: $i] :
              ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Int_Oint ) @ D ) @ ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Int_Oint @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) @ C ) ) )
             => ! [E: $i] :
                  ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Int_Oint ) @ E ) @ A ) )
                 => ( B
                   != ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ E ) @ D ) ) ) )
         => ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Int_Oint ) @ B ) @ C )
            = B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_bset_I3_J) ).

thf(8148,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ C ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Int_Oint ) @ B ) @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ) ) @ A ) )
       => ( ! [D: $i] :
              ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Int_Oint ) @ D ) @ ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Int_Oint @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) @ C ) ) )
             => ! [E: $i] :
                  ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Int_Oint ) @ E ) @ A ) )
                 => ( B
                   != ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ E ) @ D ) ) ) )
         => ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Int_Oint ) @ B ) @ C )
            = B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1484]) ).

thf(396,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Fun_Oinj__on @ E @ D @ C @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ E @ tc_HOL_Obool ) ) )
     => ( ( hAPP @ ( c_Set_Oimage @ E @ D @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ B ) @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Set_Oimage @ E @ D @ C ) @ B ) ) @ ( hAPP @ ( c_Set_Oimage @ E @ D @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_image__Int) ).

thf(3828,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Fun_Oinj__on @ E @ D @ C @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ E @ tc_HOL_Obool ) ) )
     => ( ( hAPP @ ( c_Set_Oimage @ E @ D @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ B ) @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Set_Oimage @ E @ D @ C ) @ B ) ) @ ( hAPP @ ( c_Set_Oimage @ E @ D @ C ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[396]) ).

thf(2225,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ F ) @ E ) @ ( c_SetInterval_Oord_OatLeastLessThan @ F @ D @ C @ B @ A ) ) )
    <=> ( ( hBOOL @ ( hAPP @ ( hAPP @ D @ B ) @ E ) )
        & ( hBOOL @ ( hAPP @ ( hAPP @ C @ E ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_ord_OatLeastLessThan__iff) ).

thf(11109,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ F ) @ E ) @ ( c_SetInterval_Oord_OatLeastLessThan @ F @ D @ C @ B @ A ) ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ D @ B ) @ E ) )
          & ( hBOOL @ ( hAPP @ ( hAPP @ C @ E ) @ A ) ) ) )
      & ( ( ( hBOOL @ ( hAPP @ ( hAPP @ D @ B ) @ E ) )
          & ( hBOOL @ ( hAPP @ ( hAPP @ C @ E ) @ A ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ F ) @ E ) @ ( c_SetInterval_Oord_OatLeastLessThan @ F @ D @ C @ B @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2225]) ).

thf(1698,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ B ) )
     => ( ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ A ) ) @ B )
        = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_div__mult__self1__is__m) ).

thf(9093,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ B ) )
     => ( ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ A ) ) @ B )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1698]) ).

thf(283,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_Map_Odom @ D @ C @ ( c_Map_Orestrict__map @ D @ C @ B @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( c_Map_Odom @ D @ C @ B ) ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_dom__restrict) ).

thf(3412,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_Map_Odom @ D @ C @ ( c_Map_Orestrict__map @ D @ C @ B @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( c_Map_Odom @ D @ C @ B ) ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[283]) ).

thf(326,axiom,
    ! [A: $i] :
      ( ( A
       != ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) )
     => ( ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ A ) @ A )
        = ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zdiv__self) ).

thf(3564,plain,
    ! [A: $i] :
      ( ( A
       != ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) )
     => ( ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ A ) @ A )
        = ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[326]) ).

thf(267,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ B @ A ) ) @ A )
      = ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ B ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zmod__simps_I3_J) ).

thf(3359,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ B @ A ) ) @ A )
      = ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ B ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[267]) ).

thf(648,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
      ( ( c_Finite__Set_Ofolding__image__simple @ H @ G @ F @ E @ D @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ G ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ G ) @ A ) )
         => ( ( hAPP @ ( hAPP @ F @ ( hAPP @ C @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ G @ tc_HOL_Obool ) ) @ B ) @ A ) ) ) @ ( hAPP @ C @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ G @ tc_HOL_Obool ) ) @ B ) @ A ) ) )
            = ( hAPP @ ( hAPP @ F @ ( hAPP @ C @ B ) ) @ ( hAPP @ C @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_folding__image__simple_Ounion__inter) ).

thf(4737,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
      ( ( c_Finite__Set_Ofolding__image__simple @ H @ G @ F @ E @ D @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ G ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ G ) @ A ) )
         => ( ( hAPP @ ( hAPP @ F @ ( hAPP @ C @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ G @ tc_HOL_Obool ) ) @ B ) @ A ) ) ) @ ( hAPP @ C @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ G @ tc_HOL_Obool ) ) @ B ) @ A ) ) )
            = ( hAPP @ ( hAPP @ F @ ( hAPP @ C @ B ) ) @ ( hAPP @ C @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[648]) ).

thf(1025,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ B ) )
     => ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ A ) @ B ) )
         => ( ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Nat_Onat ) @ A ) @ B )
            = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) )
        & ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ A ) @ B ) )
         => ( ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Nat_Onat ) @ A ) @ B )
            = ( hAPP @ c_Nat_OSuc @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ A ) @ B ) ) @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_div__if) ).

thf(6159,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ B ) )
     => ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ A ) @ B ) )
         => ( ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Nat_Onat ) @ A ) @ B )
            = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) )
        & ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ A ) @ B ) )
         => ( ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Nat_Onat ) @ A ) @ B )
            = ( hAPP @ c_Nat_OSuc @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ A ) @ B ) ) @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1025]) ).

thf(1923,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ D ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ E @ D ) @ B ) @ A ) ) )
     => ~ ! [F: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ F ) @ B ) )
           => ! [G: $i] :
                ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ G ) @ ( hAPP @ A @ F ) ) )
               => ( C
                 != ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ D ) @ F ) @ G ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_SigmaE) ).

thf(9963,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ D ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ E @ D ) @ B ) @ A ) ) )
     => ~ ! [F: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ F ) @ B ) )
           => ! [G: $i] :
                ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ G ) @ ( hAPP @ A @ F ) ) )
               => ( C
                 != ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ D ) @ F ) @ G ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1923]) ).

thf(1067,axiom,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ A ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_subset__refl) ).

thf(6330,plain,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ A ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1067]) ).

thf(1881,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ D ) @ C ) ) @ ( c_Transitive__Closure_Otrancl @ E @ B ) ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ C ) @ A ) ) @ ( c_Transitive__Closure_Ortrancl @ E @ B ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ D ) @ A ) ) @ ( c_Transitive__Closure_Otrancl @ E @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_trancl__rtrancl__trancl) ).

thf(9807,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ D ) @ C ) ) @ ( c_Transitive__Closure_Otrancl @ E @ B ) ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ C ) @ A ) ) @ ( c_Transitive__Closure_Ortrancl @ E @ B ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ D ) @ A ) ) @ ( c_Transitive__Closure_Otrancl @ E @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1881]) ).

thf(1218,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ D ) @ C ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Set_Ovimage @ B @ E @ A ) @ D ) ) @ ( hAPP @ ( c_Set_Ovimage @ B @ E @ A ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_vimage__mono) ).

thf(7009,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ D ) @ C ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Set_Ovimage @ B @ E @ A ) @ D ) ) @ ( hAPP @ ( c_Set_Ovimage @ B @ E @ A ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1218]) ).

thf(794,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ E @ D ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ C ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ ( tc_prod @ E @ D ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ E @ D ) @ C ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ E @ D ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Sigma__Int__distrib1) ).

thf(5277,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ E @ D ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ C ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ ( tc_prod @ E @ D ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ E @ D ) @ C ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ E @ D ) @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[794]) ).

thf(745,axiom,
    ! [A: $i,B: $i] :
      ( ( ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ B ) ) @ A )
        = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
    <=> ( A
        = ( c_List_Olist_ONil @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_length__0__conv) ).

thf(5062,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ B ) ) @ A )
          = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
       => ( A
          = ( c_List_Olist_ONil @ B ) ) )
      & ( ( A
          = ( c_List_Olist_ONil @ B ) )
       => ( ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ B ) ) @ A )
          = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[745]) ).

thf(911,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ F @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Set_Oimage @ D @ F @ C ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ D @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ F @ ( tc_fun @ E @ tc_HOL_Obool ) @ D ) @ A ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_UN__simps_I10_J) ).

thf(5675,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ F @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Set_Oimage @ D @ F @ C ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ D @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ F @ ( tc_fun @ E @ tc_HOL_Obool ) @ D ) @ A ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[911]) ).

thf(2313,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( c_Product__Type_Oprod_Oprod__rec @ F @ E @ D @ C @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ F @ E ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ C @ B ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_prod_Orecs) ).

thf(11486,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( c_Product__Type_Oprod_Oprod__rec @ F @ E @ D @ C @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ F @ E ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ C @ B ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2313]) ).

thf(1772,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( c_Relation_Osingle__valued @ F @ E @ D )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ F @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ F @ E ) @ C ) @ B ) ) @ D ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ F @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ F @ E ) @ C ) @ A ) ) @ D ) )
         => ( B = A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_single__valuedD) ).

thf(9363,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( c_Relation_Osingle__valued @ F @ E @ D )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ F @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ F @ E ) @ C ) @ B ) ) @ D ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ F @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ F @ E ) @ C ) @ A ) ) @ D ) )
         => ( B = A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1772]) ).

thf(1602,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ B @ B @ B ) @ A ) @ ( c_Transitive__Closure_Ortrancl @ B @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ B @ B @ B ) @ ( c_Transitive__Closure_Ortrancl @ B @ A ) ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_r__comp__rtrancl__eq) ).

thf(8679,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ B @ B @ B ) @ A ) @ ( c_Transitive__Closure_Ortrancl @ B @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ B @ B @ B ) @ ( c_Transitive__Closure_Ortrancl @ B @ A ) ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1602]) ).

thf(2322,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ D ) @ C ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ B ) @ ( hAPP @ A @ D ) ) @ ( hAPP @ ( c_Set_Oimage @ E @ B @ A ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_imageI) ).

thf(11518,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ D ) @ C ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ B ) @ ( hAPP @ A @ D ) ) @ ( hAPP @ ( c_Set_Oimage @ E @ B @ A ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2322]) ).

thf(807,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ ( tc_prod @ D @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ D @ C ) @ B ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ C @ tc_HOL_Obool ) @ D ) @ A ) ) ) )
     => ( ( B
         != ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) )
       => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_finite__cartesian__productD2) ).

thf(5319,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ ( tc_prod @ D @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ D @ C ) @ B ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ C @ tc_HOL_Obool ) @ D ) @ A ) ) ) )
     => ( ( B
         != ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) )
       => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[807]) ).

thf(2284,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_Com_Ocom_OWhile @ D @ C )
     != ( c_Com_Ocom_OAss @ B @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_com_Osimps_I29_J) ).

thf(11354,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_Com_Ocom_OWhile @ D @ C )
     != ( c_Com_Ocom_OAss @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2284]) ).

thf(780,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_FunDef_Oreduction__pair @ D @ C )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( c_Product__Type_Ofst @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) ) @ C ) ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) ) @ A ) @ ( hAPP @ ( c_Product__Type_Osnd @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) ) @ C ) ) )
         => ( ( c_Wellfounded_Owf @ D @ A )
           => ( c_Wellfounded_Owf @ D @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) ) @ B ) @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_reduction__pair__lemma) ).

thf(5208,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_FunDef_Oreduction__pair @ D @ C )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( c_Product__Type_Ofst @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) ) @ C ) ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) ) @ A ) @ ( hAPP @ ( c_Product__Type_Osnd @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) ) @ C ) ) )
         => ( ( c_Wellfounded_Owf @ D @ A )
           => ( c_Wellfounded_Owf @ D @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) ) @ B ) @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[780]) ).

thf(1136,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ B )
        = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ A ) )
    <=> ( ( B = A )
        | ( C
          = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__cancel1) ).

thf(6701,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ B )
          = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ A ) )
       => ( ( B = A )
          | ( C
            = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) )
      & ( ( ( B = A )
          | ( C
            = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) )
       => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ B )
          = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1136]) ).

thf(1701,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ C ) )
     => ( ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Nat_Onat ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nat__mult__div__cancel1) ).

thf(9109,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ C ) )
     => ( ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Nat_Onat ) @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1701]) ).

thf(1149,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ C ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ B ) ) @ A ) ) )
     => ( ( c_List_Obutlast @ B @ ( hAPP @ ( hAPP @ ( c_List_Otake @ B ) @ C ) @ A ) )
        = ( hAPP @ ( hAPP @ ( c_List_Otake @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ C ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_butlast__take) ).

thf(6746,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ C ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ B ) ) @ A ) ) )
     => ( ( c_List_Obutlast @ B @ ( hAPP @ ( hAPP @ ( c_List_Otake @ B ) @ C ) @ A ) )
        = ( hAPP @ ( hAPP @ ( c_List_Otake @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ C ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1149]) ).

thf(29,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( c_List_Ofilter @ C @ B ) @ ( c_List_Oconcat @ C @ A ) )
      = ( c_List_Oconcat @ C @ ( hAPP @ ( hAPP @ ( c_List_Omap @ ( tc_List_Olist @ C ) @ ( tc_List_Olist @ C ) ) @ ( c_List_Ofilter @ C @ B ) ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_filter__concat) ).

thf(2500,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( c_List_Ofilter @ C @ B ) @ ( c_List_Oconcat @ C @ A ) )
      = ( c_List_Oconcat @ C @ ( hAPP @ ( hAPP @ ( c_List_Omap @ ( tc_List_Olist @ C ) @ ( tc_List_Olist @ C ) ) @ ( c_List_Ofilter @ C @ B ) ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[29]) ).

thf(1426,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ A ) )
     => ( ( c_Divides_Odiv__class_Omod @ tc_Nat_Onat @ A @ B )
        = ( c_Divides_Odiv__class_Omod @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ A ) @ B ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_le__mod__geq) ).

thf(7876,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ A ) )
     => ( ( c_Divides_Odiv__class_Omod @ tc_Nat_Onat @ A @ B )
        = ( c_Divides_Odiv__class_Omod @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ A ) @ B ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1426]) ).

thf(1805,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ B ) ) @ A ) )
    <=> ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Suc__le__eq) ).

thf(9486,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ B ) ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) ) )
      & ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ B ) ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1805]) ).

thf(1808,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_termination__basic__simps_I5_J) ).

thf(9495,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1808]) ).

thf(1130,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ E ) @ D ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ D ) @ C ) ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ E ) @ C ) ) @ A ) ) )
      <=> ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ D ) @ E ) ) @ C ) ) @ B ) ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nat__le__add__iff1) ).

thf(6679,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ E ) @ D ) )
     => ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ D ) @ C ) ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ E ) @ C ) ) @ A ) ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ D ) @ E ) ) @ C ) ) @ B ) ) @ A ) ) )
        & ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ D ) @ E ) ) @ C ) ) @ B ) ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ D ) @ C ) ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ E ) @ C ) ) @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1130]) ).

thf(1402,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ A ) @ C ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ C ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ B ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_diff__less__mono) ).

thf(7779,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ A ) @ C ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ C ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ B ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1402]) ).

thf(2375,axiom,
    ! [A: $i,B: $i] :
      ( ( ( hAPP @ c_Int_ORep__Integ @ B )
        = ( hAPP @ c_Int_ORep__Integ @ A ) )
    <=> ( B = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Rep__Integ__inject) ).

thf(11749,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( hAPP @ c_Int_ORep__Integ @ B )
          = ( hAPP @ c_Int_ORep__Integ @ A ) )
       => ( B = A ) )
      & ( ( B = A )
       => ( ( hAPP @ c_Int_ORep__Integ @ B )
          = ( hAPP @ c_Int_ORep__Integ @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2375]) ).

thf(1189,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
      ( ( ( hAPP @ ( c_Fun_Ocomp @ H @ G @ F @ E ) @ D )
        = ( hAPP @ ( c_Fun_Ocomp @ C @ G @ F @ B ) @ A ) )
     => ! [I: $i] :
          ( ( hAPP @ E @ ( hAPP @ D @ I ) )
          = ( hAPP @ B @ ( hAPP @ A @ I ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_o__eq__elim) ).

thf(6899,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
      ( ( ( hAPP @ ( c_Fun_Ocomp @ H @ G @ F @ E ) @ D )
        = ( hAPP @ ( c_Fun_Ocomp @ C @ G @ F @ B ) @ A ) )
     => ! [I: $i] :
          ( ( hAPP @ E @ ( hAPP @ D @ I ) )
          = ( hAPP @ B @ ( hAPP @ A @ I ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1189]) ).

thf(284,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ A ) @ A )
      = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Diff__cancel) ).

thf(3415,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ A ) @ A )
      = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[284]) ).

thf(783,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A )
      = ( c_Complete__Lattice_OSup__class_OSup @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ ( tc_fun @ C @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Un__eq__Union) ).

thf(5223,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A )
      = ( c_Complete__Lattice_OSup__class_OSup @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ ( tc_fun @ C @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[783]) ).

thf(485,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ D ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) @ ( c_List_OtakeWhile @ C @ B @ A ) ) ) )
     => ( ( hAPP @ ( hAPP @ ( c_List_Onth @ C ) @ ( c_List_OtakeWhile @ C @ B @ A ) ) @ D )
        = ( hAPP @ ( hAPP @ ( c_List_Onth @ C ) @ A ) @ D ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_takeWhile__nth) ).

thf(4133,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ D ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) @ ( c_List_OtakeWhile @ C @ B @ A ) ) ) )
     => ( ( hAPP @ ( hAPP @ ( c_List_Onth @ C ) @ ( c_List_OtakeWhile @ C @ B @ A ) ) @ D )
        = ( hAPP @ ( hAPP @ ( c_List_Onth @ C ) @ A ) @ D ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[485]) ).

thf(1442,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ C @ A ) )
       => ( hBOOL @ ( hAPP @ B @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_predicate1D) ).

thf(7921,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ C @ A ) )
       => ( hBOOL @ ( hAPP @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1442]) ).

thf(1852,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ c_Com_Ocom_OBODY @ D )
     != ( c_Com_Ocom_OLocal @ C @ B @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_com_Osimps_I41_J) ).

thf(9675,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ c_Com_Ocom_OBODY @ D )
     != ( c_Com_Ocom_OLocal @ C @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1852]) ).

thf(1441,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Divides_Odivmod__nat__rel @ E @ D ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ tc_Nat_Onat @ tc_Nat_Onat ) @ C ) @ B ) ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ D ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( c_Divides_Odivmod__nat__rel @ E @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ D ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ tc_Nat_Onat @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Nat_Onat ) @ C ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ D ) @ ( c_Divides_Odiv__class_Omod @ tc_Nat_Onat @ C @ A ) ) ) @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_divmod__nat__rel__mult2__eq) ).

thf(7919,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Divides_Odivmod__nat__rel @ E @ D ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ tc_Nat_Onat @ tc_Nat_Onat ) @ C ) @ B ) ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ D ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( c_Divides_Odivmod__nat__rel @ E @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ D ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ tc_Nat_Onat @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Nat_Onat ) @ C ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ D ) @ ( c_Divides_Odiv__class_Omod @ tc_Nat_Onat @ C @ A ) ) ) @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1441]) ).

thf(1810,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ B ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_insert__absorb2) ).

thf(9500,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ B ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1810]) ).

thf(618,axiom,
    ! [A: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ) @ A ) ) @ A )
     != ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_odd__nonzero) ).

thf(4624,plain,
    ! [A: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ) @ A ) ) @ A )
     != ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[618]) ).

thf(1299,axiom,
    ! [A: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ A ) ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_diff__Suc__1) ).

thf(7342,plain,
    ! [A: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ A ) ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1299]) ).

thf(1062,axiom,
    ! [A: $i,B: $i] : ( c_Relation_Orefl__on @ B @ A @ ( c_Relation_OId__on @ B @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_refl__on__Id__on) ).

thf(6318,plain,
    ! [A: $i,B: $i] : ( c_Relation_Orefl__on @ B @ A @ ( c_Relation_OId__on @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1062]) ).

thf(266,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A )
      = ( hAPP @ ( c_Set_OCollect @ C ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ C @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ C ) @ c_fconj ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ C @ ( tc_fun @ C @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( c_member @ C ) ) @ B ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ tc_HOL_Obool @ C ) @ c_fNot ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ C @ ( tc_fun @ C @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( c_member @ C ) ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_set__diff__eq) ).

thf(3356,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A )
      = ( hAPP @ ( c_Set_OCollect @ C ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ C @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ C ) @ c_fconj ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ C @ ( tc_fun @ C @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( c_member @ C ) ) @ B ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ tc_HOL_Obool @ C ) @ c_fNot ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ C @ ( tc_fun @ C @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( c_member @ C ) ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[266]) ).

thf(2294,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Natural_Oevaln @ ( c_Com_Ocom_OSemi @ E @ D ) @ C @ B @ A )
    <=> ? [F: $i] :
          ( ( c_Natural_Oevaln @ E @ C @ B @ F )
          & ( c_Natural_Oevaln @ D @ F @ B @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_evaln_Oequations_I4_J) ).

thf(11386,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( c_Natural_Oevaln @ ( c_Com_Ocom_OSemi @ E @ D ) @ C @ B @ A )
       => ? [F: $i] :
            ( ( c_Natural_Oevaln @ E @ C @ B @ F )
            & ( c_Natural_Oevaln @ D @ F @ B @ A ) ) )
      & ( ? [F: $i] :
            ( ( c_Natural_Oevaln @ E @ C @ B @ F )
            & ( c_Natural_Oevaln @ D @ F @ B @ A ) )
       => ( c_Natural_Oevaln @ ( c_Com_Ocom_OSemi @ E @ D ) @ C @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2294]) ).

thf(2037,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( c_Sum__Type_OInl @ D @ C ) @ B )
     != ( hAPP @ ( c_Sum__Type_OInr @ C @ D ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_sum_Osimps_I3_J) ).

thf(10415,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( c_Sum__Type_OInl @ D @ C ) @ B )
     != ( hAPP @ ( c_Sum__Type_OInr @ C @ D ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2037]) ).

thf(1274,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_set__rev__mp) ).

thf(7251,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1274]) ).

thf(1158,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) ) )
     => ( ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ B ) ) @ ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_card__Diff__subset__Int) ).

thf(6775,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) ) )
     => ( ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ B ) ) @ ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1158]) ).

thf(1231,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ B ) )
     => ( ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ A ) @ B ) ) @ B )
        = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_div__mult__self__is__m) ).

thf(7067,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ B ) )
     => ( ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ A ) @ B ) ) @ B )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1231]) ).

thf(2214,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Fun_Oinj__on @ E @ D @ C @ B )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ A ) @ B ) )
       => ( ( hAPP @ ( c_Fun_Othe__inv__into @ E @ D @ B @ C ) @ ( hAPP @ C @ A ) )
          = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_the__inv__into__f__f) ).

thf(11054,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Fun_Oinj__on @ E @ D @ C @ B )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ A ) @ B ) )
       => ( ( hAPP @ ( c_Fun_Othe__inv__into @ E @ D @ B @ C ) @ ( hAPP @ C @ A ) )
          = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2214]) ).

thf(1740,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Transitive__Closure_Otrancl @ B @ A )
      = ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ B @ B @ B ) @ A ) @ ( c_Transitive__Closure_Ortrancl @ B @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_trancl__unfold__left) ).

thf(9247,plain,
    ! [A: $i,B: $i] :
      ( ( c_Transitive__Closure_Otrancl @ B @ A )
      = ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ B @ B @ B ) @ A ) @ ( c_Transitive__Closure_Ortrancl @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1740]) ).

thf(122,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Finite__Set_Ofold @ D @ C @ B ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_fold__empty) ).

thf(2843,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Finite__Set_Ofold @ D @ C @ B ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[122]) ).

thf(1308,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) )
     => ( A
       != ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_gr__implies__not0) ).

thf(7391,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) )
     => ( A
       != ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1308]) ).

thf(1744,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ A ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_subset__insertI2) ).

thf(9257,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ A ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1744]) ).

thf(86,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Groups_Ocomm__monoid__mult @ E )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ C ) )
       => ( ( hAPP @ ( hAPP @ ( hAPP @ ( c_Finite__Set_Ofold__image @ E @ D @ ( c_Groups_Otimes__class_Otimes @ E ) ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ D @ E @ E ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ E @ ( tc_fun @ E @ E ) @ D ) @ ( c_Groups_Otimes__class_Otimes @ E ) ) @ B ) ) @ A ) ) @ ( c_Groups_Oone__class_Oone @ E ) ) @ C )
          = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Finite__Set_Ofold__image @ E @ D @ ( c_Groups_Otimes__class_Otimes @ E ) ) @ B ) @ ( c_Groups_Oone__class_Oone @ E ) ) @ C ) ) @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Finite__Set_Ofold__image @ E @ D @ ( c_Groups_Otimes__class_Otimes @ E ) ) @ A ) @ ( c_Groups_Oone__class_Oone @ E ) ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_fold__image__distrib) ).

thf(2711,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Groups_Ocomm__monoid__mult @ E )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ C ) )
       => ( ( hAPP @ ( hAPP @ ( hAPP @ ( c_Finite__Set_Ofold__image @ E @ D @ ( c_Groups_Otimes__class_Otimes @ E ) ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ D @ E @ E ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ E @ ( tc_fun @ E @ E ) @ D ) @ ( c_Groups_Otimes__class_Otimes @ E ) ) @ B ) ) @ A ) ) @ ( c_Groups_Oone__class_Oone @ E ) ) @ C )
          = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Finite__Set_Ofold__image @ E @ D @ ( c_Groups_Otimes__class_Otimes @ E ) ) @ B ) @ ( c_Groups_Oone__class_Oone @ E ) ) @ C ) ) @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Finite__Set_Ofold__image @ E @ D @ ( c_Groups_Otimes__class_Otimes @ E ) ) @ A ) @ ( c_Groups_Oone__class_Oone @ E ) ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[86]) ).

thf(719,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ F @ F ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ F @ F ) @ E ) @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Nat_Ocompow @ ( tc_prod @ F @ F ) @ tc_HOL_Obool ) @ C ) @ B ) ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ F @ F ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ F @ F ) @ D ) @ A ) ) @ B ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ F @ F ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ F @ F ) @ E ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Nat_Ocompow @ ( tc_prod @ F @ F ) @ tc_HOL_Obool ) @ ( hAPP @ c_Nat_OSuc @ C ) ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_rel__pow__Suc__I) ).

thf(4991,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ F @ F ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ F @ F ) @ E ) @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Nat_Ocompow @ ( tc_prod @ F @ F ) @ tc_HOL_Obool ) @ C ) @ B ) ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ F @ F ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ F @ F ) @ D ) @ A ) ) @ B ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ F @ F ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ F @ F ) @ E ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Nat_Ocompow @ ( tc_prod @ F @ F ) @ tc_HOL_Obool ) @ ( hAPP @ c_Nat_OSuc @ C ) ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[719]) ).

thf(1038,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Relation_Ototal__on @ C @ B @ ( hAPP @ ( c_Relation_Oconverse @ C @ C ) @ A ) )
    <=> ( c_Relation_Ototal__on @ C @ B @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_total__on__converse) ).

thf(6213,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( c_Relation_Ototal__on @ C @ B @ ( hAPP @ ( c_Relation_Oconverse @ C @ C ) @ A ) )
       => ( c_Relation_Ototal__on @ C @ B @ A ) )
      & ( ( c_Relation_Ototal__on @ C @ B @ A )
       => ( c_Relation_Ototal__on @ C @ B @ ( hAPP @ ( c_Relation_Oconverse @ C @ C ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1038]) ).

thf(1940,axiom,
    ! [A: $i,B: $i] :
      ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ ( hAPP @ c_Nat_OSuc @ A ) ) )
      <=> ( B = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_not__less__less__Suc__eq) ).

thf(10028,plain,
    ! [A: $i,B: $i] :
      ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) )
     => ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ ( hAPP @ c_Nat_OSuc @ A ) ) )
         => ( B = A ) )
        & ( ( B = A )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ ( hAPP @ c_Nat_OSuc @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1940]) ).

thf(1524,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ A ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ A ) ) @ ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ B ) ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ A ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_card__psubset) ).

thf(8317,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ A ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ A ) ) @ ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ B ) ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ A ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1524]) ).

thf(1583,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Ozip @ E @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_List_Oappend @ D ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_List_Oappend @ ( tc_prod @ E @ D ) ) @ ( hAPP @ ( hAPP @ ( c_List_Ozip @ E @ D ) @ ( hAPP @ ( hAPP @ ( c_List_Otake @ E ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ D ) ) @ B ) ) @ C ) ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_List_Ozip @ E @ D ) @ ( hAPP @ ( hAPP @ ( c_List_Odrop @ E ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ D ) ) @ B ) ) @ C ) ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zip__append2) ).

thf(8602,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Ozip @ E @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_List_Oappend @ D ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_List_Oappend @ ( tc_prod @ E @ D ) ) @ ( hAPP @ ( hAPP @ ( c_List_Ozip @ E @ D ) @ ( hAPP @ ( hAPP @ ( c_List_Otake @ E ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ D ) ) @ B ) ) @ C ) ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_List_Ozip @ E @ D ) @ ( hAPP @ ( hAPP @ ( c_List_Odrop @ E ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ D ) ) @ B ) ) @ C ) ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1583]) ).

thf(1494,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ D ) @ ( hAPP @ ( c_Set_Oimage @ C @ E @ B ) @ A ) ) )
    <=> ? [F: $i] :
          ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ F ) @ A ) )
          & ( D
            = ( hAPP @ ( c_Set_Oimage @ C @ E @ B ) @ F ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_subset__image__iff) ).

thf(8185,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ D ) @ ( hAPP @ ( c_Set_Oimage @ C @ E @ B ) @ A ) ) )
       => ? [F: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ F ) @ A ) )
            & ( D
              = ( hAPP @ ( c_Set_Oimage @ C @ E @ B ) @ F ) ) ) )
      & ( ? [F: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ F ) @ A ) )
            & ( D
              = ( hAPP @ ( c_Set_Oimage @ C @ E @ B ) @ F ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ D ) @ ( hAPP @ ( c_Set_Oimage @ C @ E @ B ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1494]) ).

thf(1288,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( hAPP @ E @ D )
        = C )
     => ( ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ B @ A ) @ E ) @ D ) @ C )
        = E ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_fun__upd__idem) ).

thf(7303,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( hAPP @ E @ D )
        = C )
     => ( ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ B @ A ) @ E ) @ D ) @ C )
        = E ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1288]) ).

thf(2199,axiom,
    ! [A: $i,B: $i] :
      ( ( B
       != ( c_List_Olist_ONil @ A ) )
     => ( ( hBOOL @ ( hAPP @ ( c_List_Odistinct @ A ) @ B ) )
       => ( hBOOL @ ( hAPP @ ( c_List_Odistinct @ A ) @ ( c_List_Obutlast @ A @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_distinct__butlast) ).

thf(11005,plain,
    ! [A: $i,B: $i] :
      ( ( B
       != ( c_List_Olist_ONil @ A ) )
     => ( ( hBOOL @ ( hAPP @ ( c_List_Odistinct @ A ) @ B ) )
       => ( hBOOL @ ( hAPP @ ( c_List_Odistinct @ A ) @ ( c_List_Obutlast @ A @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2199]) ).

thf(566,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( hAPP @ ( hAPP @ ( c_List_Omap @ E @ D ) @ C ) @ B )
        = ( hAPP @ ( hAPP @ ( c_List_Omap @ E @ D ) @ C ) @ A ) )
     => ( ( c_Fun_Oinj__on @ E @ D @ C @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ E @ tc_HOL_Obool ) ) )
       => ( B = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_map__injective) ).

thf(4431,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( hAPP @ ( hAPP @ ( c_List_Omap @ E @ D ) @ C ) @ B )
        = ( hAPP @ ( hAPP @ ( c_List_Omap @ E @ D ) @ C ) @ A ) )
     => ( ( c_Fun_Oinj__on @ E @ D @ C @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ E @ tc_HOL_Obool ) ) )
       => ( B = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[566]) ).

thf(621,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hAPP @ ( hAPP @ ( c_Equiv__Relations_Oquotient @ C ) @ B ) @ A )
        = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ ( tc_fun @ C @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) )
    <=> ( B
        = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_quotient__is__empty) ).

thf(4634,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( hAPP @ ( hAPP @ ( c_Equiv__Relations_Oquotient @ C ) @ B ) @ A )
          = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ ( tc_fun @ C @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) )
       => ( B
          = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) )
      & ( ( B
          = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) )
       => ( ( hAPP @ ( hAPP @ ( c_Equiv__Relations_Oquotient @ C ) @ B ) @ A )
          = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ ( tc_fun @ C @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[621]) ).

thf(784,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( D
          = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) )
       => ( ( hAPP @ ( c_Set_Oimage @ C @ B @ ( hAPP @ ( c_COMBK @ B @ C ) @ A ) ) @ D )
          = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) )
      & ( ( D
         != ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) )
       => ( ( hAPP @ ( c_Set_Oimage @ C @ B @ ( hAPP @ ( c_COMBK @ B @ C ) @ A ) ) @ D )
          = ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ B ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_image__constant__conv) ).

thf(5226,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( D
          = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) )
       => ( ( hAPP @ ( c_Set_Oimage @ C @ B @ ( hAPP @ ( c_COMBK @ B @ C ) @ A ) ) @ D )
          = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) )
      & ( ( D
         != ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) )
       => ( ( hAPP @ ( c_Set_Oimage @ C @ B @ ( hAPP @ ( c_COMBK @ B @ C ) @ A ) ) @ D )
          = ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ B ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[784]) ).

thf(636,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_fun @ C @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ B ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( c_Complete__Lattice_OSup__class_OSup @ ( tc_fun @ C @ tc_HOL_Obool ) @ B ) ) @ ( c_Complete__Lattice_OSup__class_OSup @ ( tc_fun @ C @ tc_HOL_Obool ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Union__mono) ).

thf(4695,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_fun @ C @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ B ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( c_Complete__Lattice_OSup__class_OSup @ ( tc_fun @ C @ tc_HOL_Obool ) @ B ) ) @ ( c_Complete__Lattice_OSup__class_OSup @ ( tc_fun @ C @ tc_HOL_Obool ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[636]) ).

thf(2392,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( c_Finite__Set_Ofolding__image__simple__idem @ G @ F @ E @ D @ C @ B )
     => ( ( hAPP @ ( hAPP @ E @ A ) @ A )
        = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_folding__image__simple__idem_Oidem) ).

thf(11825,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( c_Finite__Set_Ofolding__image__simple__idem @ G @ F @ E @ D @ C @ B )
     => ( ( hAPP @ ( hAPP @ E @ A ) @ A )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2392]) ).

thf(172,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ C ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ B ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ C ) @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zdiv__mono2__neg) ).

thf(3024,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ C ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ B ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ C ) @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[172]) ).

thf(1985,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ D ) @ C ) @ B ) ) @ ( c_Transitive__Closure_Ortrancl @ D @ A ) ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ D ) @ B ) @ C ) ) @ ( c_Transitive__Closure_Ortrancl @ D @ ( hAPP @ ( c_Relation_Oconverse @ D @ D ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_rtrancl__converseI) ).

thf(10231,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ D ) @ C ) @ B ) ) @ ( c_Transitive__Closure_Ortrancl @ D @ A ) ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ D ) @ B ) @ C ) ) @ ( c_Transitive__Closure_Ortrancl @ D @ ( hAPP @ ( c_Relation_Oconverse @ D @ D ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1985]) ).

thf(197,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ E @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ E @ ( tc_fun @ D @ tc_HOL_Obool ) @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ D @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ D @ tc_HOL_Obool ) @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ E ) @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) @ B ) ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( c_Complete__Lattice_OSup__class_OSup @ ( tc_fun @ D @ tc_HOL_Obool ) @ ( hAPP @ ( c_Set_Oimage @ E @ ( tc_fun @ D @ tc_HOL_Obool ) @ B ) @ C ) ) ) @ ( c_Complete__Lattice_OSup__class_OSup @ ( tc_fun @ D @ tc_HOL_Obool ) @ ( hAPP @ ( c_Set_Oimage @ E @ ( tc_fun @ D @ tc_HOL_Obool ) @ A ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Un__Union__image) ).

thf(3117,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ E @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ E @ ( tc_fun @ D @ tc_HOL_Obool ) @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ D @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ D @ tc_HOL_Obool ) @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ E ) @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) @ B ) ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( c_Complete__Lattice_OSup__class_OSup @ ( tc_fun @ D @ tc_HOL_Obool ) @ ( hAPP @ ( c_Set_Oimage @ E @ ( tc_fun @ D @ tc_HOL_Obool ) @ B ) @ C ) ) ) @ ( c_Complete__Lattice_OSup__class_OSup @ ( tc_fun @ D @ tc_HOL_Obool ) @ ( hAPP @ ( c_Set_Oimage @ E @ ( tc_fun @ D @ tc_HOL_Obool ) @ A ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[197]) ).

thf(150,axiom,
    ! [A: $i] :
      ( ( hAPP @ ( c_List_Oset @ A ) @ ( c_List_Olist_ONil @ A ) )
      = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ A @ tc_HOL_Obool ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_List_Oset_Osimps_I1_J) ).

thf(2949,plain,
    ! [A: $i] :
      ( ( hAPP @ ( c_List_Oset @ A ) @ ( c_List_Olist_ONil @ A ) )
      = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ A @ tc_HOL_Obool ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[150]) ).

thf(89,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Ocomm__monoid__add @ D )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ C ) @ B ) )
       => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ tc_Nat_Onat @ D ) @ A ) @ ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Nat_Onat @ C @ B ) )
          = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ ( hAPP @ A @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ tc_Nat_Onat @ D ) @ A ) @ ( c_SetInterval_Oord__class_OgreaterThanAtMost @ tc_Nat_Onat @ C @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_setsum__head) ).

thf(2722,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Ocomm__monoid__add @ D )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ C ) @ B ) )
       => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ tc_Nat_Onat @ D ) @ A ) @ ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Nat_Onat @ C @ B ) )
          = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ ( hAPP @ A @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ tc_Nat_Onat @ D ) @ A ) @ ( c_SetInterval_Oord__class_OgreaterThanAtMost @ tc_Nat_Onat @ C @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[89]) ).

thf(1966,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ A ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ A ) @ ( hAPP @ c_Nat_OSuc @ B ) ) )
      <=> ( A = B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_le__less__Suc__eq) ).

thf(10150,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ A ) )
     => ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ A ) @ ( hAPP @ c_Nat_OSuc @ B ) ) )
         => ( A = B ) )
        & ( ( A = B )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ A ) @ ( hAPP @ c_Nat_OSuc @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1966]) ).

thf(821,axiom,
    ! [A: $i,B: $i] :
      ( ( ( hAPP @ ( c_Finite__Set_Ocard @ B ) @ ( hAPP @ ( c_List_Oset @ B ) @ A ) )
        = ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ B ) ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( c_List_Odistinct @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_card__distinct) ).

thf(5364,plain,
    ! [A: $i,B: $i] :
      ( ( ( hAPP @ ( c_Finite__Set_Ocard @ B ) @ ( hAPP @ ( c_List_Oset @ B ) @ A ) )
        = ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ B ) ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( c_List_Odistinct @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[821]) ).

thf(91,axiom,
    ! [A: $i,B: $i] :
      ( ( c_List_Oconcat @ B @ A )
      = ( hAPP @ ( c_List_Ofoldl @ ( tc_List_Olist @ B ) @ ( tc_List_Olist @ B ) @ ( c_List_Oappend @ B ) @ ( c_List_Olist_ONil @ B ) ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_concat__conv__foldl) ).

thf(2728,plain,
    ! [A: $i,B: $i] :
      ( ( c_List_Oconcat @ B @ A )
      = ( hAPP @ ( c_List_Ofoldl @ ( tc_List_Olist @ B ) @ ( tc_List_Olist @ B ) @ ( c_List_Oappend @ B ) @ ( c_List_Olist_ONil @ B ) ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[91]) ).

thf(2164,axiom,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) )
        & ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ A ) @ B ) ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_dvd_Oless__imp__le) ).

thf(10884,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) )
        & ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ A ) @ B ) ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2164]) ).

thf(1843,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Omap @ D @ C ) @ B ) @ ( c_List_Obutlast @ D @ A ) )
      = ( c_List_Obutlast @ C @ ( hAPP @ ( hAPP @ ( c_List_Omap @ D @ C ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_map__butlast) ).

thf(9637,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Omap @ D @ C ) @ B ) @ ( c_List_Obutlast @ D @ A ) )
      = ( c_List_Obutlast @ C @ ( hAPP @ ( hAPP @ ( c_List_Omap @ D @ C ) @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1843]) ).

thf(1313,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( ( hAPP @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ G @ ( tc_Option_Ooption @ F ) ) @ E ) @ D ) @ ( hAPP @ ( c_Option_Ooption_OSome @ F ) @ C ) ) @ B )
        = ( hAPP @ ( c_Option_Ooption_OSome @ F ) @ A ) )
    <=> ( ( ( B = D )
          & ( C = A ) )
        | ( ( B != D )
          & ( ( hAPP @ E @ B )
            = ( hAPP @ ( c_Option_Ooption_OSome @ F ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_map__upd__Some__unfold) ).

thf(7426,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( ( ( hAPP @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ G @ ( tc_Option_Ooption @ F ) ) @ E ) @ D ) @ ( hAPP @ ( c_Option_Ooption_OSome @ F ) @ C ) ) @ B )
          = ( hAPP @ ( c_Option_Ooption_OSome @ F ) @ A ) )
       => ( ( ( B = D )
            & ( C = A ) )
          | ( ( B != D )
            & ( ( hAPP @ E @ B )
              = ( hAPP @ ( c_Option_Ooption_OSome @ F ) @ A ) ) ) ) )
      & ( ( ( ( B = D )
            & ( C = A ) )
          | ( ( B != D )
            & ( ( hAPP @ E @ B )
              = ( hAPP @ ( c_Option_Ooption_OSome @ F ) @ A ) ) ) )
       => ( ( hAPP @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ G @ ( tc_Option_Ooption @ F ) ) @ E ) @ D ) @ ( hAPP @ ( c_Option_Ooption_OSome @ F ) @ C ) ) @ B )
          = ( hAPP @ ( c_Option_Ooption_OSome @ F ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1313]) ).

thf(537,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_Map_Odom @ D @ C @ ( c_Map_Omap__add @ D @ C @ B @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( c_Map_Odom @ D @ C @ A ) ) @ ( c_Map_Odom @ D @ C @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_dom__map__add) ).

thf(4342,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_Map_Odom @ D @ C @ ( c_Map_Omap__add @ D @ C @ B @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( c_Map_Odom @ D @ C @ A ) ) @ ( c_Map_Odom @ D @ C @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[537]) ).

thf(1203,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) ) @ B ) @ A ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ C @ C @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) ) @ ( c_Transitive__Closure_Otrancl @ C @ B ) ) @ A ) ) @ B ) ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) ) @ ( c_Transitive__Closure_Otrancl @ C @ B ) ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_trancl__Int__subset) ).

thf(6943,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) ) @ B ) @ A ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ C @ C @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) ) @ ( c_Transitive__Closure_Otrancl @ C @ B ) ) @ A ) ) @ B ) ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) ) @ ( c_Transitive__Closure_Otrancl @ C @ B ) ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1203]) ).

thf(2043,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( hAPP @ ( c_List_Oset @ D ) @ ( c_List_Obutlast @ D @ B ) ) ) )
        | ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( hAPP @ ( c_List_Oset @ D ) @ ( c_List_Obutlast @ D @ A ) ) ) ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( hAPP @ ( c_List_Oset @ D ) @ ( c_List_Obutlast @ D @ ( hAPP @ ( hAPP @ ( c_List_Oappend @ D ) @ B ) @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_in__set__butlast__appendI) ).

thf(10436,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( hAPP @ ( c_List_Oset @ D ) @ ( c_List_Obutlast @ D @ B ) ) ) )
        | ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( hAPP @ ( c_List_Oset @ D ) @ ( c_List_Obutlast @ D @ A ) ) ) ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( hAPP @ ( c_List_Oset @ D ) @ ( c_List_Obutlast @ D @ ( hAPP @ ( hAPP @ ( c_List_Oappend @ D ) @ B ) @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2043]) ).

thf(923,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Groups_Oab__semigroup__mult @ E )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofold__graph @ E @ E @ ( c_Groups_Otimes__class_Otimes @ E ) @ D @ C ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ A ) @ C ) )
         => ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ D ) @ C ) )
           => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofold__graph @ E @ E @ ( c_Groups_Otimes__class_Otimes @ E ) @ A @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ E @ tc_HOL_Obool ) ) ) ) ) ) @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_fold__graph__permute__diff) ).

thf(5717,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Groups_Oab__semigroup__mult @ E )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofold__graph @ E @ E @ ( c_Groups_Otimes__class_Otimes @ E ) @ D @ C ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ A ) @ C ) )
         => ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ D ) @ C ) )
           => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofold__graph @ E @ E @ ( c_Groups_Otimes__class_Otimes @ E ) @ A @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ E @ tc_HOL_Obool ) ) ) ) ) ) @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[923]) ).

thf(680,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( c_Relation_ODomain @ C @ B ) @ ( c_Complete__Lattice_OSup__class_OSup @ ( tc_fun @ ( tc_prod @ C @ B ) @ tc_HOL_Obool ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ ( tc_fun @ ( tc_prod @ C @ B ) @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ A ) @ ( c_Relation_ODomain @ C @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Domain__Union) ).

thf(4838,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( c_Relation_ODomain @ C @ B ) @ ( c_Complete__Lattice_OSup__class_OSup @ ( tc_fun @ ( tc_prod @ C @ B ) @ tc_HOL_Obool ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ ( tc_fun @ ( tc_prod @ C @ B ) @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ A ) @ ( c_Relation_ODomain @ C @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[680]) ).

thf(1866,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ D ) @ C ) @ B ) ) @ ( c_Transitive__Closure_Ortrancl @ D @ A ) ) )
    <=> ( ( C = B )
        | ( ( C != B )
          & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ D ) @ C ) @ B ) ) @ ( c_Transitive__Closure_Otrancl @ D @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_rtrancl__eq__or__trancl) ).

thf(9747,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ D ) @ C ) @ B ) ) @ ( c_Transitive__Closure_Ortrancl @ D @ A ) ) )
       => ( ( C = B )
          | ( ( C != B )
            & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ D ) @ C ) @ B ) ) @ ( c_Transitive__Closure_Otrancl @ D @ A ) ) ) ) ) )
      & ( ( ( C = B )
          | ( ( C != B )
            & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ D ) @ C ) @ B ) ) @ ( c_Transitive__Closure_Otrancl @ D @ A ) ) ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ D ) @ C ) @ B ) ) @ ( c_Transitive__Closure_Ortrancl @ D @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1866]) ).

thf(136,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ D ) ) @ C )
        = ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ B ) ) @ A ) )
     => ( ( hAPP @ ( hAPP @ ( c_List_Omap @ ( tc_prod @ D @ B ) @ B ) @ ( c_Product__Type_Osnd @ D @ B ) ) @ ( hAPP @ ( hAPP @ ( c_List_Ozip @ D @ B ) @ C ) @ A ) )
        = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_map__snd__zip) ).

thf(2892,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ D ) ) @ C )
        = ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ B ) ) @ A ) )
     => ( ( hAPP @ ( hAPP @ ( c_List_Omap @ ( tc_prod @ D @ B ) @ B ) @ ( c_Product__Type_Osnd @ D @ B ) ) @ ( hAPP @ ( hAPP @ ( c_List_Ozip @ D @ B ) @ C ) @ A ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[136]) ).

thf(1098,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( c_Fun_Oinj__on @ F @ E @ D @ C )
     => ( ( c_Fun_Oinj__on @ E @ B @ A @ ( hAPP @ ( c_Set_Oimage @ F @ E @ D ) @ C ) )
      <=> ( c_Fun_Oinj__on @ F @ B @ ( hAPP @ ( c_Fun_Ocomp @ E @ B @ F @ A ) @ D ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_comp__inj__on__iff) ).

thf(6511,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( c_Fun_Oinj__on @ F @ E @ D @ C )
     => ( ( ( c_Fun_Oinj__on @ E @ B @ A @ ( hAPP @ ( c_Set_Oimage @ F @ E @ D ) @ C ) )
         => ( c_Fun_Oinj__on @ F @ B @ ( hAPP @ ( c_Fun_Ocomp @ E @ B @ F @ A ) @ D ) @ C ) )
        & ( ( c_Fun_Oinj__on @ F @ B @ ( hAPP @ ( c_Fun_Ocomp @ E @ B @ F @ A ) @ D ) @ C )
         => ( c_Fun_Oinj__on @ E @ B @ A @ ( hAPP @ ( c_Set_Oimage @ F @ E @ D ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1098]) ).

thf(418,axiom,
    ! [A: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ) @ A )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zmult__1) ).

thf(3914,plain,
    ! [A: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ) @ A )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[418]) ).

thf(1849,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( c_Com_Ocom_OCond @ F @ E @ D )
        = ( c_Com_Ocom_OCond @ C @ B @ A ) )
    <=> ( ( F = C )
        & ( E = B )
        & ( D = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_com_Osimps_I4_J) ).

thf(9654,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( ( c_Com_Ocom_OCond @ F @ E @ D )
          = ( c_Com_Ocom_OCond @ C @ B @ A ) )
       => ( ( F = C )
          & ( E = B )
          & ( D = A ) ) )
      & ( ( ( F = C )
          & ( E = B )
          & ( D = A ) )
       => ( ( c_Com_Ocom_OCond @ F @ E @ D )
          = ( c_Com_Ocom_OCond @ C @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1849]) ).

thf(314,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( ( hAPP @ ( c_Int_Onumber__class_Onumber__of @ tc_Nat_Onat ) @ D )
          = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
       => ( ( hAPP @ ( hAPP @ ( c_List_Odrop @ C ) @ ( hAPP @ ( c_Int_Onumber__class_Onumber__of @ tc_Nat_Onat ) @ D ) ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ C ) @ B ) @ A ) )
          = ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ C ) @ B ) @ A ) ) )
      & ( ( ( hAPP @ ( c_Int_Onumber__class_Onumber__of @ tc_Nat_Onat ) @ D )
         != ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
       => ( ( hAPP @ ( hAPP @ ( c_List_Odrop @ C ) @ ( hAPP @ ( c_Int_Onumber__class_Onumber__of @ tc_Nat_Onat ) @ D ) ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ C ) @ B ) @ A ) )
          = ( hAPP @ ( hAPP @ ( c_List_Odrop @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ ( hAPP @ ( c_Int_Onumber__class_Onumber__of @ tc_Nat_Onat ) @ D ) ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_drop__Cons__number__of) ).

thf(3520,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( ( hAPP @ ( c_Int_Onumber__class_Onumber__of @ tc_Nat_Onat ) @ D )
          = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
       => ( ( hAPP @ ( hAPP @ ( c_List_Odrop @ C ) @ ( hAPP @ ( c_Int_Onumber__class_Onumber__of @ tc_Nat_Onat ) @ D ) ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ C ) @ B ) @ A ) )
          = ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ C ) @ B ) @ A ) ) )
      & ( ( ( hAPP @ ( c_Int_Onumber__class_Onumber__of @ tc_Nat_Onat ) @ D )
         != ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
       => ( ( hAPP @ ( hAPP @ ( c_List_Odrop @ C ) @ ( hAPP @ ( c_Int_Onumber__class_Onumber__of @ tc_Nat_Onat ) @ D ) ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ C ) @ B ) @ A ) )
          = ( hAPP @ ( hAPP @ ( c_List_Odrop @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ ( hAPP @ ( c_Int_Onumber__class_Onumber__of @ tc_Nat_Onat ) @ D ) ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[314]) ).

thf(11902,plain,
    ( ( hBOOL @ c_fFalse )
   != ( hBOOL @ c_fTrue ) ),
    inference(paramod_ordered,[status(thm)],[11368,11422]) ).

thf(11906,plain,
    c_fFalse != c_fTrue,
    inference(simp,[status(thm)],[11902]) ).

thf(1641,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i,I: $i] :
      ( ( ( hAPP @ ( c_Fun_Ocomp @ I @ H @ G @ F ) @ E )
        = ( hAPP @ ( c_Fun_Ocomp @ D @ H @ G @ C ) @ B ) )
     => ( ( hAPP @ F @ ( hAPP @ E @ A ) )
        = ( hAPP @ C @ ( hAPP @ B @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_o__eq__dest) ).

thf(8861,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i,I: $i] :
      ( ( ( hAPP @ ( c_Fun_Ocomp @ I @ H @ G @ F ) @ E )
        = ( hAPP @ ( c_Fun_Ocomp @ D @ H @ G @ C ) @ B ) )
     => ( ( hAPP @ F @ ( hAPP @ E @ A ) )
        = ( hAPP @ C @ ( hAPP @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1641]) ).

thf(166,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( hAPP @ ( hAPP @ ( c_Finite__Set_Ofold__image @ E @ D @ C ) @ B ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_fold__image__empty) ).

thf(3007,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( hAPP @ ( hAPP @ ( c_Finite__Set_Ofold__image @ E @ D @ C ) @ B ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[166]) ).

thf(1104,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) )
      = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_diff__add__0) ).

thf(6531,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) )
      = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1104]) ).

thf(1972,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( c_Sum__Type_OInr @ D @ C ) @ B )
     != ( hAPP @ ( c_Sum__Type_OInl @ C @ D ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_sum_Osimps_I4_J) ).

thf(10178,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( c_Sum__Type_OInr @ D @ C ) @ B )
     != ( hAPP @ ( c_Sum__Type_OInl @ C @ D ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1972]) ).

thf(1323,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ G @ F @ E ) @ ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ G @ D @ F ) @ C ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ G @ D @ E ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ D @ F @ E ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_O__assoc) ).

thf(7488,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ G @ F @ E ) @ ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ G @ D @ F ) @ C ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ G @ D @ E ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ D @ F @ E ) @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1323]) ).

thf(2216,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ B ) @ A ) )
    <=> ( hBOOL @ ( hAPP @ A @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mem__def) ).

thf(11059,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ A @ B ) ) )
      & ( ( hBOOL @ ( hAPP @ A @ B ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2216]) ).

thf(2110,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ A ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ A ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_diff__le__mono2) ).

thf(10680,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ A ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ A ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2110]) ).

thf(949,axiom,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ tc_Int_Oint ) @ ( c_SetInterval_Oord__class_OgreaterThanLessThan @ tc_Int_Oint @ B @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_finite__greaterThanLessThan__int) ).

thf(5841,plain,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ tc_Int_Oint ) @ ( c_SetInterval_Oord__class_OgreaterThanLessThan @ tc_Int_Oint @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[949]) ).

thf(560,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( c_Set_Oimage @ ( tc_prod @ C @ B ) @ B @ ( c_Product__Type_Osnd @ C @ B ) ) @ A )
      = ( hAPP @ ( c_Relation_ORange @ C @ B ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_snd__eq__Range) ).

thf(4414,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( c_Set_Oimage @ ( tc_prod @ C @ B ) @ B @ ( c_Product__Type_Osnd @ C @ B ) ) @ A )
      = ( hAPP @ ( c_Relation_ORange @ C @ B ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[560]) ).

thf(1905,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofold__graph @ F @ E @ D @ C @ B ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ F ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_fold__graph__imp__finite) ).

thf(9902,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofold__graph @ F @ E @ D @ C @ B ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ F ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1905]) ).

thf(248,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Int_Oint ) @ B ) @ A ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zdvd__imp__le) ).

thf(3288,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Int_Oint ) @ B ) @ A ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[248]) ).

thf(1659,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( c_Fun_Oinj__on @ F @ E @ ( hAPP @ ( c_Fun_Ocomp @ D @ E @ F @ C ) @ B ) @ A )
     => ( c_Fun_Oinj__on @ D @ E @ C @ ( hAPP @ ( c_Set_Oimage @ F @ D @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_inj__on__imageI) ).

thf(8914,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( c_Fun_Oinj__on @ F @ E @ ( hAPP @ ( c_Fun_Ocomp @ D @ E @ F @ C ) @ B ) @ A )
     => ( c_Fun_Oinj__on @ D @ E @ C @ ( hAPP @ ( c_Set_Oimage @ F @ D @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1659]) ).

thf(2289,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
      ( ( c_Finite__Set_Ofolding__image__simple__idem @ H @ G @ F @ E @ D @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ G ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ G ) @ A ) @ B ) )
         => ( ( hAPP @ ( hAPP @ F @ ( hAPP @ D @ A ) ) @ ( hAPP @ C @ B ) )
            = ( hAPP @ C @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_folding__image__simple__idem_Oin__idem) ).

thf(11369,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
      ( ( c_Finite__Set_Ofolding__image__simple__idem @ H @ G @ F @ E @ D @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ G ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ G ) @ A ) @ B ) )
         => ( ( hAPP @ ( hAPP @ F @ ( hAPP @ D @ A ) ) @ ( hAPP @ C @ B ) )
            = ( hAPP @ C @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2289]) ).

thf(1269,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Otake @ B ) @ A ) @ ( c_List_Olist_ONil @ B ) )
      = ( c_List_Olist_ONil @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_take__Nil) ).

thf(7234,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Otake @ B ) @ A ) @ ( c_List_Olist_ONil @ B ) )
      = ( c_List_Olist_ONil @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1269]) ).

thf(1163,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ F ) @ E ) @ D ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ C @ F ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ C @ F ) @ B ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ F @ tc_HOL_Obool ) @ C ) @ D ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ C @ F ) @ A ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ F @ tc_HOL_Obool ) @ C ) @ D ) ) ) )
      <=> ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Times__subset__cancel2) ).

thf(6793,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ F ) @ E ) @ D ) )
     => ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ C @ F ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ C @ F ) @ B ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ F @ tc_HOL_Obool ) @ C ) @ D ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ C @ F ) @ A ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ F @ tc_HOL_Obool ) @ C ) @ D ) ) ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) ) )
        & ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ C @ F ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ C @ F ) @ B ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ F @ tc_HOL_Obool ) @ C ) @ D ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ C @ F ) @ A ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ F @ tc_HOL_Obool ) @ C ) @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1163]) ).

thf(129,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ D @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ C ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ D @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ ( tc_fun @ E @ tc_HOL_Obool ) @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ E @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ E @ tc_HOL_Obool ) @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ D ) @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ E @ tc_HOL_Obool ) ) ) @ B ) ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_UN__extend__simps_I6_J) ).

thf(2865,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ D @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ C ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ D @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ ( tc_fun @ E @ tc_HOL_Obool ) @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ E @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ E @ tc_HOL_Obool ) @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ D ) @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ E @ tc_HOL_Obool ) ) ) @ B ) ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[129]) ).

thf(379,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C = B )
    <=> ( ( ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ A ) ) @ C )
          = ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ A ) ) @ B ) )
        & ! [D: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ D ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ A ) ) @ C ) ) )
           => ( ( hAPP @ ( hAPP @ ( c_List_Onth @ A ) @ C ) @ D )
              = ( hAPP @ ( hAPP @ ( c_List_Onth @ A ) @ B ) @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_list__eq__iff__nth__eq) ).

thf(3757,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( C = B )
       => ( ( ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ A ) ) @ C )
            = ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ A ) ) @ B ) )
          & ! [D: $i] :
              ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ D ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ A ) ) @ C ) ) )
             => ( ( hAPP @ ( hAPP @ ( c_List_Onth @ A ) @ C ) @ D )
                = ( hAPP @ ( hAPP @ ( c_List_Onth @ A ) @ B ) @ D ) ) ) ) )
      & ( ( ( ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ A ) ) @ C )
            = ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ A ) ) @ B ) )
          & ! [D: $i] :
              ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ D ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ A ) ) @ C ) ) )
             => ( ( hAPP @ ( hAPP @ ( c_List_Onth @ A ) @ C ) @ D )
                = ( hAPP @ ( hAPP @ ( c_List_Onth @ A ) @ B ) @ D ) ) ) )
       => ( C = B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[379]) ).

thf(280,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ C @ ( tc_fun @ C @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( c_member @ C ) ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ C @ ( tc_fun @ C @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( c_member @ C ) ) @ A ) ) )
    <=> ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_pred__subset__eq) ).

thf(3401,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ C @ ( tc_fun @ C @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( c_member @ C ) ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ C @ ( tc_fun @ C @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( c_member @ C ) ) @ A ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) ) )
      & ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ C @ ( tc_fun @ C @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( c_member @ C ) ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ C @ ( tc_fun @ C @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( c_member @ C ) ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[280]) ).

thf(1899,axiom,
    ! [A: $i,B: $i] :
      ( ( B != A )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) )
          & ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ A ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_dvd_Oneq__le__trans) ).

thf(9876,plain,
    ! [A: $i,B: $i] :
      ( ( B != A )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) )
          & ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ A ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1899]) ).

thf(2240,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ~ ( hBOOL @ ( hAPP @ F @ E ) )
     => ( ( c_Natural_Oevaln @ D @ E @ C @ B )
       => ( c_Natural_Oevaln @ ( c_Com_Ocom_OCond @ F @ A @ D ) @ E @ C @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_evaln_OIfFalse) ).

thf(11176,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ~ ( hBOOL @ ( hAPP @ F @ E ) )
     => ( ( c_Natural_Oevaln @ D @ E @ C @ B )
       => ( c_Natural_Oevaln @ ( c_Com_Ocom_OCond @ F @ A @ D ) @ E @ C @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2240]) ).

thf(2308,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Natural_Oevalc @ ( c_Com_Ocom_OWhile @ D @ C ) ) @ B ) @ A ) )
     => ( ( ( A = B )
         => ( hBOOL @ ( hAPP @ D @ B ) ) )
       => ~ ( ( hBOOL @ ( hAPP @ D @ B ) )
           => ! [E: $i] :
                ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Natural_Oevalc @ C ) @ B ) @ E ) )
               => ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Natural_Oevalc @ ( c_Com_Ocom_OWhile @ D @ C ) ) @ E ) @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_evalc__WHILE__case) ).

thf(11450,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Natural_Oevalc @ ( c_Com_Ocom_OWhile @ D @ C ) ) @ B ) @ A ) )
     => ( ( ( A = B )
         => ( hBOOL @ ( hAPP @ D @ B ) ) )
       => ~ ( ( hBOOL @ ( hAPP @ D @ B ) )
           => ! [E: $i] :
                ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Natural_Oevalc @ C ) @ B ) @ E ) )
               => ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Natural_Oevalc @ ( c_Com_Ocom_OWhile @ D @ C ) ) @ E ) @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2308]) ).

thf(1175,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ? [E: $i,F: $i] :
          ( D
          = ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ C ) @ E ) @ F ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ C ) @ ( tc_List_Olist @ C ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ C ) @ ( tc_List_Olist @ C ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_List_Oappend @ C ) @ B ) @ D ) ) ) @ ( c_List_Olexord @ C @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_lexord__append__rightI) ).

thf(6835,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ? [E: $i,F: $i] :
          ( D
          = ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ C ) @ E ) @ F ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ C ) @ ( tc_List_Olist @ C ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ C ) @ ( tc_List_Olist @ C ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_List_Oappend @ C ) @ B ) @ D ) ) ) @ ( c_List_Olexord @ C @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1175]) ).

thf(2169,axiom,
    ! [A: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ A ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_le__cube) ).

thf(10899,plain,
    ! [A: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ A ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2169]) ).

thf(101,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( c_List_Olexn @ B @ A ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
      = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ ( tc_prod @ ( tc_List_Olist @ B ) @ ( tc_List_Olist @ B ) ) @ tc_HOL_Obool ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_lexn_Osimps_I1_J) ).

thf(2760,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( c_List_Olexn @ B @ A ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
      = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ ( tc_prod @ ( tc_List_Olist @ B ) @ ( tc_List_Olist @ B ) ) @ tc_HOL_Obool ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[101]) ).

thf(1056,axiom,
    ! [A: $i,B: $i] :
      ( ( ( c_List_Oupt @ B @ A )
        = ( c_List_Olist_ONil @ tc_Nat_Onat ) )
    <=> ( ( A
          = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
        | ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ A ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_upt__eq__Nil__conv) ).

thf(6286,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( c_List_Oupt @ B @ A )
          = ( c_List_Olist_ONil @ tc_Nat_Onat ) )
       => ( ( A
            = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
          | ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ A ) @ B ) ) ) )
      & ( ( ( A
            = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
          | ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ A ) @ B ) ) )
       => ( ( c_List_Oupt @ B @ A )
          = ( c_List_Olist_ONil @ tc_Nat_Onat ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1056]) ).

thf(1816,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ! [F: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ ( hAPP @ D @ ( hAPP @ c_Nat_OSuc @ F ) ) ) @ ( hAPP @ D @ F ) ) ) @ ( c_Transitive__Closure_Ortrancl @ E @ C ) ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ ( hAPP @ D @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) ) ) @ ( hAPP @ D @ B ) ) ) @ ( c_Transitive__Closure_Ortrancl @ E @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_sequence__trans) ).

thf(9532,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ! [F: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ ( hAPP @ D @ ( hAPP @ c_Nat_OSuc @ F ) ) ) @ ( hAPP @ D @ F ) ) ) @ ( c_Transitive__Closure_Ortrancl @ E @ C ) ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ ( hAPP @ D @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) ) ) @ ( hAPP @ D @ B ) ) ) @ ( c_Transitive__Closure_Ortrancl @ E @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1816]) ).

thf(1674,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ A ) @ B ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ A ) ) @ ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_psubset__card__mono) ).

thf(8973,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ A ) @ B ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ A ) ) @ ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1674]) ).

thf(524,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ C ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ B ) ) @ A ) ) )
     => ( ( hAPP @ ( hAPP @ ( c_List_Otake @ B ) @ C ) @ ( c_List_Obutlast @ B @ A ) )
        = ( hAPP @ ( hAPP @ ( c_List_Otake @ B ) @ C ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_take__butlast) ).

thf(4289,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ C ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ B ) ) @ A ) ) )
     => ( ( hAPP @ ( hAPP @ ( c_List_Otake @ B ) @ C ) @ ( c_List_Obutlast @ B @ A ) )
        = ( hAPP @ ( hAPP @ ( c_List_Otake @ B ) @ C ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[524]) ).

thf(1688,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ B ) ) @ A ) )
    <=> ( hBOOL @ ( c_Nat_Onat_Onat__case @ tc_HOL_Obool @ c_fFalse @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__eq__nat_Osimps_I2_J) ).

thf(9046,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ B ) ) @ A ) )
       => ( hBOOL @ ( c_Nat_Onat_Onat__case @ tc_HOL_Obool @ c_fFalse @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ A ) ) )
      & ( ( hBOOL @ ( c_Nat_Onat_Onat__case @ tc_HOL_Obool @ c_fFalse @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ B ) ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1688]) ).

thf(2401,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( c_List_Olistrelp @ F @ E @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ F ) @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ F ) @ B ) @ A ) )
    <=> ( ( hBOOL @ ( hAPP @ ( hAPP @ E @ D ) @ B ) )
        & ( c_List_Olistrelp @ F @ E @ C @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_listrelp_Oequations_I2_J) ).

thf(11862,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( c_List_Olistrelp @ F @ E @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ F ) @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ F ) @ B ) @ A ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ E @ D ) @ B ) )
          & ( c_List_Olistrelp @ F @ E @ C @ A ) ) )
      & ( ( ( hBOOL @ ( hAPP @ ( hAPP @ E @ D ) @ B ) )
          & ( c_List_Olistrelp @ F @ E @ C @ A ) )
       => ( c_List_Olistrelp @ F @ E @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ F ) @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ F ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2401]) ).

thf(960,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Predicate_ORangeP @ D @ C @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ D ) @ ( c_COMBC @ C @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ ( tc_prod @ D @ C ) ) @ ( tc_fun @ C @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ D ) @ ( hAPP @ ( c_COMBB @ ( tc_prod @ D @ C ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ C ) @ ( c_member @ ( tc_prod @ D @ C ) ) ) ) @ ( c_Product__Type_OPair @ D @ C ) ) ) ) @ B ) ) @ A ) )
    <=> ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ A ) @ ( hAPP @ ( c_Relation_ORange @ D @ C ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_RangeP__Range__eq) ).

thf(5873,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( c_Predicate_ORangeP @ D @ C @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ D ) @ ( c_COMBC @ C @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ ( tc_prod @ D @ C ) ) @ ( tc_fun @ C @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ D ) @ ( hAPP @ ( c_COMBB @ ( tc_prod @ D @ C ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ C ) @ ( c_member @ ( tc_prod @ D @ C ) ) ) ) @ ( c_Product__Type_OPair @ D @ C ) ) ) ) @ B ) ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ A ) @ ( hAPP @ ( c_Relation_ORange @ D @ C ) @ B ) ) ) )
      & ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ A ) @ ( hAPP @ ( c_Relation_ORange @ D @ C ) @ B ) ) )
       => ( hBOOL @ ( hAPP @ ( c_Predicate_ORangeP @ D @ C @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ D ) @ ( c_COMBC @ C @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ ( tc_prod @ D @ C ) ) @ ( tc_fun @ C @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ D ) @ ( hAPP @ ( c_COMBB @ ( tc_prod @ D @ C ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ C ) @ ( c_member @ ( tc_prod @ D @ C ) ) ) ) @ ( c_Product__Type_OPair @ D @ C ) ) ) ) @ B ) ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[960]) ).

thf(1750,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Int_Oint ) @ D ) @ C ) )
     => ! [E: $i] :
          ( ! [F: $i] :
              ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Int_Oint ) @ F ) @ ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Int_Oint @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) @ C ) ) )
             => ! [G: $i] :
                  ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Int_Oint ) @ G ) @ B ) )
                 => ( E
                   != ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ G ) @ F ) ) ) )
         => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Int_Oint ) @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ E ) @ A ) ) )
           => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Int_Oint ) @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Int_Oint ) @ E ) @ C ) ) @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_bset_I9_J) ).

thf(9279,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Int_Oint ) @ D ) @ C ) )
     => ! [E: $i] :
          ( ! [F: $i] :
              ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Int_Oint ) @ F ) @ ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Int_Oint @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) @ C ) ) )
             => ! [G: $i] :
                  ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Int_Oint ) @ G ) @ B ) )
                 => ( E
                   != ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ G ) @ F ) ) ) )
         => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Int_Oint ) @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ E ) @ A ) ) )
           => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Int_Oint ) @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Int_Oint ) @ E ) @ C ) ) @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1750]) ).

thf(64,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Ocomm__monoid__add @ B )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ B ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ B ) )
        = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_add_Ocomm__neutral) ).

thf(2626,plain,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Ocomm__monoid__add @ B )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ B ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ B ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[64]) ).

thf(487,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( c_Relation_ODomain @ D @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Relation_ODomain @ D @ C ) @ B ) ) @ ( hAPP @ ( c_Relation_ODomain @ D @ C ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Domain__Un__eq) ).

thf(4138,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( c_Relation_ODomain @ D @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Relation_ODomain @ D @ C ) @ B ) ) @ ( hAPP @ ( c_Relation_ODomain @ D @ C ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[487]) ).

thf(1178,axiom,
    ! [A: $i,B: $i] :
      ( ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A )
        = ( hAPP @ c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) )
    <=> ( ( ( B
            = ( hAPP @ c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) )
          & ( A
            = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) )
        | ( ( B
            = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
          & ( A
            = ( hAPP @ c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_add__is__1) ).

thf(6846,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A )
          = ( hAPP @ c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) )
       => ( ( ( B
              = ( hAPP @ c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) )
            & ( A
              = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) )
          | ( ( B
              = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
            & ( A
              = ( hAPP @ c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) ) ) )
      & ( ( ( ( B
              = ( hAPP @ c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) )
            & ( A
              = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) )
          | ( ( B
              = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
            & ( A
              = ( hAPP @ c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) ) )
       => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A )
          = ( hAPP @ c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1178]) ).

thf(332,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ F @ E ) @ D ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ C @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) @ F ) @ A ) @ E ) @ D ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_sup2I1) ).

thf(3589,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ F @ E ) @ D ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ C @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) @ F ) @ A ) @ E ) @ D ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[332]) ).

thf(533,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( E
          = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) )
       => ( ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ D @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ E ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) @ D ) @ ( hAPP @ ( c_Set_Oinsert @ C ) @ B ) ) @ A ) )
          = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) )
      & ( ( E
         != ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) )
       => ( ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ D @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ E ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) @ D ) @ ( hAPP @ ( c_Set_Oinsert @ C ) @ B ) ) @ A ) )
          = ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ D @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ E ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_UN__simps_I1_J) ).

thf(4325,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( E
          = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) )
       => ( ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ D @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ E ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) @ D ) @ ( hAPP @ ( c_Set_Oinsert @ C ) @ B ) ) @ A ) )
          = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) )
      & ( ( E
         != ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) )
       => ( ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ D @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ E ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) @ D ) @ ( hAPP @ ( c_Set_Oinsert @ C ) @ B ) ) @ A ) )
          = ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ D @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ E ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[533]) ).

thf(1153,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ D ) @ ( hAPP @ ( c_Relation_OImage @ C @ E @ B ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) ) ) )
    <=> ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ C @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ C @ E ) @ A ) @ D ) ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Image__singleton__iff) ).

thf(6758,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ D ) @ ( hAPP @ ( c_Relation_OImage @ C @ E @ B ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ C @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ C @ E ) @ A ) @ D ) ) @ B ) ) )
      & ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ C @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ C @ E ) @ A ) @ D ) ) @ B ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ D ) @ ( hAPP @ ( c_Relation_OImage @ C @ E @ B ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1153]) ).

thf(186,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ E @ D ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ C ) @ B ) ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ D @ tc_HOL_Obool ) @ E ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ ( tc_prod @ E @ D ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ E @ D ) @ C ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ D @ tc_HOL_Obool ) @ E ) @ A ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ E @ D ) @ B ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ D @ tc_HOL_Obool ) @ E ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Times__Int__distrib1) ).

thf(3067,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ E @ D ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ C ) @ B ) ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ D @ tc_HOL_Obool ) @ E ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ ( tc_prod @ E @ D ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ E @ D ) @ C ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ D @ tc_HOL_Obool ) @ E ) @ A ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ E @ D ) @ B ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ D @ tc_HOL_Obool ) @ E ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[186]) ).

thf(2159,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ F @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ F @ E ) @ D ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ F @ E ) @ B ) @ A ) ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ F ) @ D ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_SigmaD1) ).

thf(10873,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ F @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ F @ E ) @ D ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ F @ E ) @ B ) @ A ) ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ F ) @ D ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2159]) ).

thf(2141,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i,I: $i,J: $i,K: $i,L: $i] :
      ( ( c_Com_Ocom_Ocom__case @ L @ K @ J @ I @ H @ G @ F @ E @ D @ ( c_Com_Ocom_OCall @ C @ B @ A ) )
      = ( hAPP @ ( hAPP @ ( hAPP @ D @ C ) @ B ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_com_Osimps_I71_J) ).

thf(10806,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i,I: $i,J: $i,K: $i,L: $i] :
      ( ( c_Com_Ocom_Ocom__case @ L @ K @ J @ I @ H @ G @ F @ E @ D @ ( c_Com_Ocom_OCall @ C @ B @ A ) )
      = ( hAPP @ ( hAPP @ ( hAPP @ D @ C ) @ B ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2141]) ).

thf(2082,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] : ( c_Hoare__Mirabelle_Otriple__valid @ D @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Hoare__Mirabelle_Otriple_Otriple @ D ) @ C ) @ ( hAPP @ c_Com_Ocom_OBODY @ B ) ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Body__triple__valid__0) ).

thf(10595,plain,
    ! [A: $i,B: $i,C: $i,D: $i] : ( c_Hoare__Mirabelle_Otriple__valid @ D @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Hoare__Mirabelle_Otriple_Otriple @ D ) @ C ) @ ( hAPP @ c_Com_Ocom_OBODY @ B ) ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2082]) ).

thf(997,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ A @ B ) ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_pos__mod__bound) ).

thf(6041,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ A @ B ) ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[997]) ).

thf(1867,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( c_Finite__Set_Ofun__left__comm @ G @ F @ E )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofold__graph @ G @ F @ E @ D @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ G ) @ C ) @ B ) ) @ A ) )
       => ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ G ) @ C ) @ B ) )
         => ~ ! [H: $i] :
                ( ( A
                  = ( hAPP @ ( hAPP @ E @ C ) @ H ) )
               => ~ ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofold__graph @ G @ F @ E @ D @ B ) @ H ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_fun__left__comm_Ofold__graph__insertE) ).

thf(9760,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( c_Finite__Set_Ofun__left__comm @ G @ F @ E )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofold__graph @ G @ F @ E @ D @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ G ) @ C ) @ B ) ) @ A ) )
       => ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ G ) @ C ) @ B ) )
         => ~ ! [H: $i] :
                ( ( A
                  = ( hAPP @ ( hAPP @ E @ C ) @ H ) )
               => ~ ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofold__graph @ G @ F @ E @ D @ B ) @ H ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1867]) ).

thf(1842,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( hAPP @ ( c_Map_Omap__add @ E @ D @ C @ B ) @ A )
        = ( c_Option_Ooption_ONone @ D ) )
    <=> ( ( ( hAPP @ B @ A )
          = ( c_Option_Ooption_ONone @ D ) )
        & ( ( hAPP @ C @ A )
          = ( c_Option_Ooption_ONone @ D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_map__add__None) ).

thf(9628,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( ( hAPP @ ( c_Map_Omap__add @ E @ D @ C @ B ) @ A )
          = ( c_Option_Ooption_ONone @ D ) )
       => ( ( ( hAPP @ B @ A )
            = ( c_Option_Ooption_ONone @ D ) )
          & ( ( hAPP @ C @ A )
            = ( c_Option_Ooption_ONone @ D ) ) ) )
      & ( ( ( ( hAPP @ B @ A )
            = ( c_Option_Ooption_ONone @ D ) )
          & ( ( hAPP @ C @ A )
            = ( c_Option_Ooption_ONone @ D ) ) )
       => ( ( hAPP @ ( c_Map_Omap__add @ E @ D @ C @ B ) @ A )
          = ( c_Option_Ooption_ONone @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1842]) ).

thf(656,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ F ) @ E ) @ D ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ B ) @ ( hAPP @ A @ E ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ F @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ D ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_UN__I) ).

thf(4763,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ F ) @ E ) @ D ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ B ) @ ( hAPP @ A @ E ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ F @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ D ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[656]) ).

thf(1435,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( hAPP @ E @ D )
        = ( c_Option_Ooption_ONone @ C ) )
     => ( ( c_Map_Oran @ B @ C @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ B @ ( tc_Option_Ooption @ C ) ) @ E ) @ D ) @ ( hAPP @ ( c_Option_Ooption_OSome @ C ) @ A ) ) )
        = ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ A ) @ ( c_Map_Oran @ B @ C @ E ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_ran__map__upd) ).

thf(7904,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( hAPP @ E @ D )
        = ( c_Option_Ooption_ONone @ C ) )
     => ( ( c_Map_Oran @ B @ C @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ B @ ( tc_Option_Ooption @ C ) ) @ E ) @ D ) @ ( hAPP @ ( c_Option_Ooption_OSome @ C ) @ A ) ) )
        = ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ A ) @ ( c_Map_Oran @ B @ C @ E ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1435]) ).

thf(1874,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( c_Com_Ocom_OSKIP
     != ( c_Com_Ocom_OLocal @ C @ B @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_com_Osimps_I10_J) ).

thf(9780,plain,
    ! [A: $i,B: $i,C: $i] :
      ( c_Com_Ocom_OSKIP
     != ( c_Com_Ocom_OLocal @ C @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1874]) ).

thf(1790,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ A ) @ B ) )
       => ( B = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_dvd__antisym) ).

thf(9438,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ A ) @ B ) )
       => ( B = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1790]) ).

thf(937,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ ( c_Complete__Lattice_OSup__class_OSup @ ( tc_fun @ C @ tc_HOL_Obool ) @ B ) ) )
       => ( ! [D: $i] :
              ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ D ) @ B ) )
             => ( ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ D )
                = A ) )
         => ( ! [D: $i] :
                ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ D ) @ B ) )
               => ! [E: $i] :
                    ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ E ) @ B ) )
                   => ( ( D != E )
                     => ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ D ) @ E )
                        = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) ) ) )
           => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ A ) @ ( hAPP @ ( c_Finite__Set_Ocard @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) )
              = ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ ( c_Complete__Lattice_OSup__class_OSup @ ( tc_fun @ C @ tc_HOL_Obool ) @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_card__partition) ).

thf(5777,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ ( c_Complete__Lattice_OSup__class_OSup @ ( tc_fun @ C @ tc_HOL_Obool ) @ B ) ) )
       => ( ! [D: $i] :
              ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ D ) @ B ) )
             => ( ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ D )
                = A ) )
         => ( ! [D: $i] :
                ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ D ) @ B ) )
               => ! [E: $i] :
                    ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ E ) @ B ) )
                   => ( ( D != E )
                     => ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ D ) @ E )
                        = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) ) ) )
           => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ A ) @ ( hAPP @ ( c_Finite__Set_Ocard @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) )
              = ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ ( c_Complete__Lattice_OSup__class_OSup @ ( tc_fun @ C @ tc_HOL_Obool ) @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[937]) ).

thf(2134,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ C ) @ A ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ B ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ A ) @ C ) ) )
        <=> ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_le__diff__iff) ).

thf(10774,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ C ) @ A ) )
       => ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ B ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ A ) @ C ) ) )
           => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ A ) ) )
          & ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ A ) )
           => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ B ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ A ) @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2134]) ).

thf(1082,axiom,
    ! [A: $i] :
      ( ( hAPP @ ( c_Finite__Set_Ocard @ tc_Nat_Onat ) @ ( hAPP @ ( c_SetInterval_Oord__class_OatMost @ tc_Nat_Onat ) @ A ) )
      = ( hAPP @ c_Nat_OSuc @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_card__atMost) ).

thf(6390,plain,
    ! [A: $i] :
      ( ( hAPP @ ( c_Finite__Set_Ocard @ tc_Nat_Onat ) @ ( hAPP @ ( c_SetInterval_Oord__class_OatMost @ tc_Nat_Onat ) @ A ) )
      = ( hAPP @ c_Nat_OSuc @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1082]) ).

thf(405,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( hAPP @ ( c_Int_Onumber__class_Onumber__of @ tc_Int_Oint ) @ B ) ) @ ( hAPP @ ( c_Int_Onumber__class_Onumber__of @ tc_Int_Oint ) @ A ) ) )
    <=> ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__number__of__int__code) ).

thf(3874,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( hAPP @ ( c_Int_Onumber__class_Onumber__of @ tc_Int_Oint ) @ B ) ) @ ( hAPP @ ( c_Int_Onumber__class_Onumber__of @ tc_Int_Oint ) @ A ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ B ) @ A ) ) )
      & ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( hAPP @ ( c_Int_Onumber__class_Onumber__of @ tc_Int_Oint ) @ B ) ) @ ( hAPP @ ( c_Int_Onumber__class_Onumber__of @ tc_Int_Oint ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[405]) ).

thf(863,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ ( tc_List_Olist @ C ) ) @ ( hAPP @ ( c_Set_OCollect @ ( tc_List_Olist @ C ) ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ ( tc_List_Olist @ C ) @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( tc_List_Olist @ C ) ) @ c_fconj ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ ( tc_List_Olist @ C ) @ ( tc_fun @ C @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ C @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( tc_List_Olist @ C ) ) @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) @ ( c_List_Oset @ C ) ) ) @ B ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ ( tc_List_Olist @ C ) @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_Nat_Onat @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_List_Olist @ C ) ) @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) ) @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) ) ) @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_finite__lists__length__le) ).

thf(5510,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ ( tc_List_Olist @ C ) ) @ ( hAPP @ ( c_Set_OCollect @ ( tc_List_Olist @ C ) ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ ( tc_List_Olist @ C ) @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( tc_List_Olist @ C ) ) @ c_fconj ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ ( tc_List_Olist @ C ) @ ( tc_fun @ C @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ C @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( tc_List_Olist @ C ) ) @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) @ ( c_List_Oset @ C ) ) ) @ B ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ ( tc_List_Olist @ C ) @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_Nat_Onat @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_List_Olist @ C ) ) @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) ) @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) ) ) @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[863]) ).

thf(1977,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ D ) @ C ) @ B ) ) @ ( c_Relation_OId__on @ D @ A ) ) )
    <=> ( ( C = B )
        & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Id__on__iff) ).

thf(10199,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ D ) @ C ) @ B ) ) @ ( c_Relation_OId__on @ D @ A ) ) )
       => ( ( C = B )
          & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ A ) ) ) )
      & ( ( ( C = B )
          & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ A ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ D ) @ C ) @ B ) ) @ ( c_Relation_OId__on @ D @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1977]) ).

thf(1967,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_List_Obutlast @ C @ ( hAPP @ ( hAPP @ ( c_List_Odrop @ C ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_List_Odrop @ C ) @ B ) @ ( c_List_Obutlast @ C @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_butlast__drop) ).

thf(10156,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_List_Obutlast @ C @ ( hAPP @ ( hAPP @ ( c_List_Odrop @ C ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_List_Odrop @ C ) @ B ) @ ( c_List_Obutlast @ C @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1967]) ).

thf(1396,axiom,
    ! [A: $i] :
      ( ( c_Orderings_Otop__class_Otop @ ( tc_fun @ ( tc_Option_Ooption @ A ) @ tc_HOL_Obool ) )
      = ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ ( tc_Option_Ooption @ A ) ) @ ( c_Option_Ooption_ONone @ A ) ) @ ( hAPP @ ( c_Set_Oimage @ A @ ( tc_Option_Ooption @ A ) @ ( c_Option_Ooption_OSome @ A ) ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ A @ tc_HOL_Obool ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_UNIV__option__conv) ).

thf(7765,plain,
    ! [A: $i] :
      ( ( c_Orderings_Otop__class_Otop @ ( tc_fun @ ( tc_Option_Ooption @ A ) @ tc_HOL_Obool ) )
      = ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ ( tc_Option_Ooption @ A ) ) @ ( c_Option_Ooption_ONone @ A ) ) @ ( hAPP @ ( c_Set_Oimage @ A @ ( tc_Option_Ooption @ A ) @ ( c_Option_Ooption_OSome @ A ) ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ A @ tc_HOL_Obool ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1396]) ).

thf(975,axiom,
    ! [A: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ A )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zadd__0) ).

thf(5934,plain,
    ! [A: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ A )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[975]) ).

thf(1635,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D
        = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ B ) ) @ A ) )
     => ( ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ C ) )
           => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ A ) )
              & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ A ) @ C ) ) ) )
          & ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ C ) )
           => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ C ) @ A ) )
              & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) ) ) ) )
       => ( ( C
           != ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) )
         => ( ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ D ) @ C )
            = B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_divmod__int__rel__div__eq) ).

thf(8825,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D
        = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ B ) ) @ A ) )
     => ( ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ C ) )
           => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ A ) )
              & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ A ) @ C ) ) ) )
          & ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ C ) )
           => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ C ) @ A ) )
              & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) ) ) ) )
       => ( ( C
           != ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) )
         => ( ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ D ) @ C )
            = B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1635]) ).

thf(662,axiom,
    ! [A: $i,B: $i] :
      ( ! [C: $i] :
          ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( c_Relation_OImage @ B @ B @ A ) @ C ) ) )
         => ( C
            = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) )
     => ( c_Wellfounded_Owf @ B @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_wfI__pf) ).

thf(4780,plain,
    ! [A: $i,B: $i] :
      ( ! [C: $i] :
          ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( c_Relation_OImage @ B @ B @ A ) @ C ) ) )
         => ( C
            = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) )
     => ( c_Wellfounded_Owf @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[662]) ).

thf(59,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Divides_Odivmod__int__rel @ E @ D ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ tc_Int_Oint @ tc_Int_Oint ) @ C ) @ B ) ) )
     => ( ( D
         != ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) )
       => ( hBOOL @ ( hAPP @ ( c_Divides_Odivmod__int__rel @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ A ) @ E ) @ D ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ tc_Int_Oint @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ A ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ A ) @ B ) ) @ D ) ) ) @ ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ A ) @ B ) @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zmult1__lemma) ).

thf(2611,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Divides_Odivmod__int__rel @ E @ D ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ tc_Int_Oint @ tc_Int_Oint ) @ C ) @ B ) ) )
     => ( ( D
         != ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) )
       => ( hBOOL @ ( hAPP @ ( c_Divides_Odivmod__int__rel @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ A ) @ E ) @ D ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ tc_Int_Oint @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ A ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ A ) @ B ) ) @ D ) ) ) @ ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ A ) @ B ) @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[59]) ).

thf(422,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_List_Odistinct @ B ) @ A ) )
     => ( ( hAPP @ ( c_Finite__Set_Ocard @ B ) @ ( hAPP @ ( c_List_Oset @ B ) @ A ) )
        = ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ B ) ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_distinct__card) ).

thf(3928,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_List_Odistinct @ B ) @ A ) )
     => ( ( hAPP @ ( c_Finite__Set_Ocard @ B ) @ ( hAPP @ ( c_List_Oset @ B ) @ A ) )
        = ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ B ) ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[422]) ).

thf(469,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ C ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ D @ D ) @ C ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ D @ tc_HOL_Obool ) @ D ) @ C ) ) ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Equiv__Relations_Oquotient @ D ) @ C ) @ B ) ) )
         => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_finite__equiv__class) ).

thf(4078,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ C ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ D @ D ) @ C ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ D @ tc_HOL_Obool ) @ D ) @ C ) ) ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Equiv__Relations_Oquotient @ D ) @ C ) @ B ) ) )
         => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[469]) ).

thf(804,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ F @ ( tc_fun @ E @ tc_HOL_Obool ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ F @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ E @ ( tc_fun @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ F ) @ ( c_COMBC @ E @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ E @ ( tc_prod @ F @ E ) ) @ ( tc_fun @ E @ ( tc_fun @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ F ) @ ( hAPP @ ( c_COMBB @ ( tc_prod @ F @ E ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ E ) @ ( c_member @ ( tc_prod @ F @ E ) ) ) ) @ ( c_Product__Type_OPair @ F @ E ) ) ) ) @ D ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ F @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ E @ ( tc_fun @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ F ) @ ( c_COMBC @ E @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ E @ ( tc_prod @ F @ E ) ) @ ( tc_fun @ E @ ( tc_fun @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ F ) @ ( hAPP @ ( c_COMBB @ ( tc_prod @ F @ E ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ E ) @ ( c_member @ ( tc_prod @ F @ E ) ) ) ) @ ( c_Product__Type_OPair @ F @ E ) ) ) ) @ C ) ) @ B ) @ A ) )
    <=> ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ F @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ F @ E ) @ B ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) ) @ D ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_inf__Int__eq2) ).

thf(5308,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ F @ ( tc_fun @ E @ tc_HOL_Obool ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ F @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ E @ ( tc_fun @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ F ) @ ( c_COMBC @ E @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ E @ ( tc_prod @ F @ E ) ) @ ( tc_fun @ E @ ( tc_fun @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ F ) @ ( hAPP @ ( c_COMBB @ ( tc_prod @ F @ E ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ E ) @ ( c_member @ ( tc_prod @ F @ E ) ) ) ) @ ( c_Product__Type_OPair @ F @ E ) ) ) ) @ D ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ F @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ E @ ( tc_fun @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ F ) @ ( c_COMBC @ E @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ E @ ( tc_prod @ F @ E ) ) @ ( tc_fun @ E @ ( tc_fun @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ F ) @ ( hAPP @ ( c_COMBB @ ( tc_prod @ F @ E ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ E ) @ ( c_member @ ( tc_prod @ F @ E ) ) ) ) @ ( c_Product__Type_OPair @ F @ E ) ) ) ) @ C ) ) @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ F @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ F @ E ) @ B ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) ) @ D ) @ C ) ) ) )
      & ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ F @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ F @ E ) @ B ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) ) @ D ) @ C ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ F @ ( tc_fun @ E @ tc_HOL_Obool ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ F @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ E @ ( tc_fun @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ F ) @ ( c_COMBC @ E @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ E @ ( tc_prod @ F @ E ) ) @ ( tc_fun @ E @ ( tc_fun @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ F ) @ ( hAPP @ ( c_COMBB @ ( tc_prod @ F @ E ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ E ) @ ( c_member @ ( tc_prod @ F @ E ) ) ) ) @ ( c_Product__Type_OPair @ F @ E ) ) ) ) @ D ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ F @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ E @ ( tc_fun @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ F ) @ ( c_COMBC @ E @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ E @ ( tc_prod @ F @ E ) ) @ ( tc_fun @ E @ ( tc_fun @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ F ) @ ( hAPP @ ( c_COMBB @ ( tc_prod @ F @ E ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ E ) @ ( c_member @ ( tc_prod @ F @ E ) ) ) ) @ ( c_Product__Type_OPair @ F @ E ) ) ) ) @ C ) ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[804]) ).

thf(96,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Ocomm__monoid__add @ D )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ C ) @ B ) )
       => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ tc_Nat_Onat @ D ) @ A ) @ ( hAPP @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Nat_Onat @ C ) @ B ) )
          = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ ( hAPP @ A @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ tc_Nat_Onat @ D ) @ A ) @ ( hAPP @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Nat_Onat @ ( hAPP @ c_Nat_OSuc @ C ) ) @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_setsum__head__upt__Suc) ).

thf(2743,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Ocomm__monoid__add @ D )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ C ) @ B ) )
       => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ tc_Nat_Onat @ D ) @ A ) @ ( hAPP @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Nat_Onat @ C ) @ B ) )
          = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ ( hAPP @ A @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ tc_Nat_Onat @ D ) @ A ) @ ( hAPP @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Nat_Onat @ ( hAPP @ c_Nat_OSuc @ C ) ) @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[96]) ).

thf(1504,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ B ) @ A ) ) ) )
    <=> ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_finite__Diff__insert) ).

thf(8228,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ B ) @ A ) ) ) )
       => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ A ) ) ) )
      & ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ A ) ) )
       => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ B ) @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1504]) ).

thf(446,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ A ) ) )
     => ( ( C
         != ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Int_Oint ) @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zdvd__mult__cancel) ).

thf(4006,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ A ) ) )
     => ( ( C
         != ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Int_Oint ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[446]) ).

thf(1148,axiom,
    ! [A: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ A )
      = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_diff__0__eq__0) ).

thf(6743,plain,
    ! [A: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ A )
      = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1148]) ).

thf(1711,axiom,
    ! [A: $i,B: $i,C: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_List_Oset @ C ) @ ( hAPP @ ( c_List_Ofilter @ C @ B ) @ A ) ) ) @ ( hAPP @ ( c_List_Oset @ C ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_filter__is__subset) ).

thf(9141,plain,
    ! [A: $i,B: $i,C: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_List_Oset @ C ) @ ( hAPP @ ( c_List_Ofilter @ C @ B ) @ A ) ) ) @ ( hAPP @ ( c_List_Oset @ C ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1711]) ).

thf(2181,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_List_Odistinct @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( c_List_Odistinct @ C ) @ ( hAPP @ ( hAPP @ ( c_List_Otake @ C ) @ A ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_distinct__take) ).

thf(10928,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_List_Odistinct @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( c_List_Odistinct @ C ) @ ( hAPP @ ( hAPP @ ( c_List_Otake @ C ) @ A ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2181]) ).

thf(1455,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ C ) @ B ) ) @ A ) )
    <=> ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ A ) )
        & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_insert__subset) ).

thf(7974,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ C ) @ B ) ) @ A ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ A ) )
          & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) ) ) )
      & ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ A ) )
          & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ C ) @ B ) ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1455]) ).

thf(1894,axiom,
    hBOOL @ ( hAPP @ c_Com_OWT @ c_Com_Ocom_OSKIP ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_WT_OSkip) ).

thf(9866,plain,
    hBOOL @ ( hAPP @ c_Com_OWT @ c_Com_Ocom_OSKIP ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1894]) ).

thf(182,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Un__left__commute) ).

thf(3056,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[182]) ).

thf(854,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i] :
          ( ( hBOOL @ ( hAPP @ D @ E ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ C @ ( hAPP @ B @ E ) ) ) @ ( hAPP @ C @ E ) ) ) )
     => ( c_Wellfounded_Owf @ A @ ( hAPP @ ( c_Set_OCollect @ ( tc_prod @ A @ A ) ) @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ A @ A @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ A @ tc_HOL_Obool ) @ ( tc_fun @ A @ tc_HOL_Obool ) @ A ) @ ( hAPP @ ( c_COMBS @ A @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ A ) @ c_fconj ) @ D ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ A @ ( tc_fun @ A @ A ) @ ( tc_fun @ A @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ A @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ A @ A ) @ ( tc_fun @ A @ tc_HOL_Obool ) ) @ A ) @ ( c_COMBB @ A @ tc_HOL_Obool @ A ) ) @ c_fequal ) ) @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_wf__if__measure) ).

thf(5485,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i] :
          ( ( hBOOL @ ( hAPP @ D @ E ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ C @ ( hAPP @ B @ E ) ) ) @ ( hAPP @ C @ E ) ) ) )
     => ( c_Wellfounded_Owf @ A @ ( hAPP @ ( c_Set_OCollect @ ( tc_prod @ A @ A ) ) @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ A @ A @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ A @ tc_HOL_Obool ) @ ( tc_fun @ A @ tc_HOL_Obool ) @ A ) @ ( hAPP @ ( c_COMBS @ A @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ A ) @ c_fconj ) @ D ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ A @ ( tc_fun @ A @ A ) @ ( tc_fun @ A @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ A @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ A @ A ) @ ( tc_fun @ A @ tc_HOL_Obool ) ) @ A ) @ ( c_COMBB @ A @ tc_HOL_Obool @ A ) ) @ c_fequal ) ) @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[854]) ).

thf(1619,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( class_Groups_Oab__semigroup__mult @ G )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ F ) @ E ) )
       => ( ( c_Fun_Oinj__on @ F @ D @ C @ E )
         => ( ( hAPP @ ( hAPP @ ( hAPP @ ( c_Finite__Set_Ofold__image @ G @ D @ ( c_Groups_Otimes__class_Otimes @ G ) ) @ B ) @ A ) @ ( hAPP @ ( c_Set_Oimage @ F @ D @ C ) @ E ) )
            = ( hAPP @ ( hAPP @ ( hAPP @ ( c_Finite__Set_Ofold__image @ G @ F @ ( c_Groups_Otimes__class_Otimes @ G ) ) @ ( hAPP @ ( c_Fun_Ocomp @ D @ G @ F @ B ) @ C ) ) @ A ) @ E ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_fold__image__reindex) ).

thf(8761,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( class_Groups_Oab__semigroup__mult @ G )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ F ) @ E ) )
       => ( ( c_Fun_Oinj__on @ F @ D @ C @ E )
         => ( ( hAPP @ ( hAPP @ ( hAPP @ ( c_Finite__Set_Ofold__image @ G @ D @ ( c_Groups_Otimes__class_Otimes @ G ) ) @ B ) @ A ) @ ( hAPP @ ( c_Set_Oimage @ F @ D @ C ) @ E ) )
            = ( hAPP @ ( hAPP @ ( hAPP @ ( c_Finite__Set_Ofold__image @ G @ F @ ( c_Groups_Otimes__class_Otimes @ G ) ) @ ( hAPP @ ( c_Fun_Ocomp @ D @ G @ F @ B ) @ C ) ) @ A ) @ E ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1619]) ).

thf(1813,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( ( c_Sum__Type_Osum_Osum__case @ G @ F @ E @ D @ C )
        = ( c_Sum__Type_Osum_Osum__case @ G @ F @ E @ B @ A ) )
     => ~ ( ( D = B )
         => ( C != A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_sum__case__inject) ).

thf(9515,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( ( c_Sum__Type_Osum_Osum__case @ G @ F @ E @ D @ C )
        = ( c_Sum__Type_Osum_Osum__case @ G @ F @ E @ B @ A ) )
     => ~ ( ( D = B )
         => ( C != A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1813]) ).

thf(1315,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_n__less__n__mult__m) ).

thf(7446,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1315]) ).

thf(2293,axiom,
    ! [A: $i,B: $i] :
      ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ c_fconj @ B ) @ A ) )
      | ( hBOOL @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_c__fconj__3) ).

thf(11383,plain,
    ! [A: $i,B: $i] :
      ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ c_fconj @ B ) @ A ) )
      | ( hBOOL @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2293]) ).

thf(1568,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ C ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Int_Oint ) @ B ) @ A ) )
       => ! [D: $i] :
            ( ! [E: $i] :
                ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Int_Oint ) @ E ) @ ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Int_Oint @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) @ C ) ) )
               => ! [F: $i] :
                    ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Int_Oint ) @ F ) @ A ) )
                   => ( D
                     != ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Int_Oint ) @ F ) @ E ) ) ) )
           => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ D ) @ B ) )
             => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ D ) @ C ) ) @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_aset_I5_J) ).

thf(8469,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ C ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Int_Oint ) @ B ) @ A ) )
       => ! [D: $i] :
            ( ! [E: $i] :
                ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Int_Oint ) @ E ) @ ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Int_Oint @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) @ C ) ) )
               => ! [F: $i] :
                    ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Int_Oint ) @ F ) @ A ) )
                   => ( D
                     != ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Int_Oint ) @ F ) @ E ) ) ) )
           => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ D ) @ B ) )
             => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ D ) @ C ) ) @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1568]) ).

thf(463,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ B ) )
        | ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ A ) ) )
     => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_finite__Int) ).

thf(4061,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ B ) )
        | ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ A ) ) )
     => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[463]) ).

thf(1170,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ E ) @ D ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ E ) @ C ) ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ D ) @ C ) ) @ A ) ) )
      <=> ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ D ) @ E ) ) @ C ) ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nat__le__add__iff2) ).

thf(6814,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ E ) @ D ) )
     => ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ E ) @ C ) ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ D ) @ C ) ) @ A ) ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ D ) @ E ) ) @ C ) ) @ A ) ) ) )
        & ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ D ) @ E ) ) @ C ) ) @ A ) ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ E ) @ C ) ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ D ) @ C ) ) @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1170]) ).

thf(354,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Int_Oint ) @ C ) @ ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ B @ A ) ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Int_Oint ) @ C ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Int_Oint ) @ C ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zdvd__zmod__imp__zdvd) ).

thf(3665,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Int_Oint ) @ C ) @ ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ B @ A ) ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Int_Oint ) @ C ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Int_Oint ) @ C ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[354]) ).

thf(1137,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ D ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ C ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ C ) @ B ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ B ) @ ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ A @ D ) ) ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ B ) @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zmult2__lemma__aux4) ).

thf(6711,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ D ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ C ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ C ) @ B ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ B ) @ ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ A @ D ) ) ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ B ) @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1137]) ).

thf(1142,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Divides_Odivmod__nat__rel @ G @ F ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ tc_Nat_Onat @ tc_Nat_Onat ) @ E ) @ D ) ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Divides_Odivmod__nat__rel @ C @ F ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ tc_Nat_Onat @ tc_Nat_Onat ) @ B ) @ A ) ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ F ) )
         => ( hBOOL @ ( hAPP @ ( c_Divides_Odivmod__nat__rel @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ G ) @ C ) @ F ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ tc_Nat_Onat @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ E ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ D ) @ A ) ) @ F ) ) ) @ ( c_Divides_Odiv__class_Omod @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ D ) @ A ) @ F ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_divmod__nat__rel__add1__eq) ).

thf(6726,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Divides_Odivmod__nat__rel @ G @ F ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ tc_Nat_Onat @ tc_Nat_Onat ) @ E ) @ D ) ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Divides_Odivmod__nat__rel @ C @ F ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ tc_Nat_Onat @ tc_Nat_Onat ) @ B ) @ A ) ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ F ) )
         => ( hBOOL @ ( hAPP @ ( c_Divides_Odivmod__nat__rel @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ G ) @ C ) @ F ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ tc_Nat_Onat @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ E ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ D ) @ A ) ) @ F ) ) ) @ ( c_Divides_Odiv__class_Omod @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ D ) @ A ) @ F ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1142]) ).

thf(161,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Groups_Oab__semigroup__mult @ E )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofold__graph @ E @ E @ ( c_Groups_Otimes__class_Otimes @ E ) @ D @ C ) @ B ) )
       => ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ D ) @ C ) )
         => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofold__graph @ E @ E @ ( c_Groups_Otimes__class_Otimes @ E ) @ A @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ D ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ A ) @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_fold__graph__insert__swap) ).

thf(2988,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Groups_Oab__semigroup__mult @ E )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofold__graph @ E @ E @ ( c_Groups_Otimes__class_Otimes @ E ) @ D @ C ) @ B ) )
       => ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ D ) @ C ) )
         => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofold__graph @ E @ E @ ( c_Groups_Otimes__class_Otimes @ E ) @ A @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ D ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ A ) @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[161]) ).

thf(1489,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Omap @ C @ B ) @ A ) @ ( c_List_Olist_ONil @ C ) )
      = ( c_List_Olist_ONil @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_map_Osimps_I1_J) ).

thf(8166,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Omap @ C @ B ) @ A ) @ ( c_List_Olist_ONil @ C ) )
      = ( c_List_Olist_ONil @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1489]) ).

thf(725,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ A ) @ B ) )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Un__Diff__cancel) ).

thf(5008,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ A ) @ B ) )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[725]) ).

thf(40,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Groups_Ocomm__monoid__add @ E )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) ) )
       => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ tc_Nat_Onat @ E ) @ B ) @ ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Nat_Onat @ D @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ A ) ) )
          = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ E ) @ ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ tc_Nat_Onat @ E ) @ B ) @ ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Nat_Onat @ D @ C ) ) ) @ ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ tc_Nat_Onat @ E ) @ B ) @ ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ A ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_setsum__ub__add__nat) ).

thf(2532,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Groups_Ocomm__monoid__add @ E )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) ) )
       => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ tc_Nat_Onat @ E ) @ B ) @ ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Nat_Onat @ D @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ A ) ) )
          = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ E ) @ ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ tc_Nat_Onat @ E ) @ B ) @ ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Nat_Onat @ D @ C ) ) ) @ ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ tc_Nat_Onat @ E ) @ B ) @ ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ A ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[40]) ).

thf(632,axiom,
    ! [A: $i] :
      ( ( c_Orderings_Obot__class_Obot @ ( tc_fun @ A @ tc_HOL_Obool ) )
      = ( hAPP @ ( c_Set_OCollect @ A ) @ ( hAPP @ ( c_COMBK @ tc_HOL_Obool @ A ) @ c_fFalse ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_empty__def) ).

thf(4683,plain,
    ! [A: $i] :
      ( ( c_Orderings_Obot__class_Obot @ ( tc_fun @ A @ tc_HOL_Obool ) )
      = ( hAPP @ ( c_Set_OCollect @ A ) @ ( hAPP @ ( c_COMBK @ tc_HOL_Obool @ A ) @ c_fFalse ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[632]) ).

thf(816,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Divides_Odiv__class_Omod @ tc_Nat_Onat @ B @ A )
      = ( hAPP @ ( c_Product__Type_Osnd @ tc_Nat_Onat @ tc_Nat_Onat ) @ ( c_Divides_Odivmod__nat @ B @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mod__nat__def) ).

thf(5351,plain,
    ! [A: $i,B: $i] :
      ( ( c_Divides_Odiv__class_Omod @ tc_Nat_Onat @ B @ A )
      = ( hAPP @ ( c_Product__Type_Osnd @ tc_Nat_Onat @ tc_Nat_Onat ) @ ( c_Divides_Odivmod__nat @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[816]) ).

thf(457,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( c_Product__Type_Ofst @ ( tc_List_Olist @ C ) @ ( tc_List_Olist @ C ) ) @ ( c_List_Opartition @ C @ B @ A ) )
      = ( hAPP @ ( c_List_Ofilter @ C @ B ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_partition__filter1) ).

thf(4042,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( c_Product__Type_Ofst @ ( tc_List_Olist @ C ) @ ( tc_List_Olist @ C ) ) @ ( c_List_Opartition @ C @ B @ A ) )
      = ( hAPP @ ( c_List_Ofilter @ C @ B ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[457]) ).

thf(944,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ ( c_Complete__Lattice_OSup__class_OSup @ ( tc_fun @ C @ tc_HOL_Obool ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Union__upper) ).

thf(5813,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ ( c_Complete__Lattice_OSup__class_OSup @ ( tc_fun @ C @ tc_HOL_Obool ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[944]) ).

thf(861,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ G @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ F @ ( tc_fun @ G @ tc_HOL_Obool ) ) @ E ) @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ C @ ( tc_fun @ G @ tc_HOL_Obool ) ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ F @ ( tc_fun @ G @ tc_HOL_Obool ) ) @ E ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ ( tc_fun @ G @ tc_HOL_Obool ) ) @ ( tc_fun @ G @ tc_HOL_Obool ) @ F ) @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ C @ ( tc_fun @ G @ tc_HOL_Obool ) ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ F @ ( tc_fun @ C @ ( tc_fun @ G @ tc_HOL_Obool ) ) @ ( tc_fun @ C @ ( tc_fun @ G @ tc_HOL_Obool ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ ( tc_fun @ G @ tc_HOL_Obool ) @ ( tc_fun @ G @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ C @ ( tc_fun @ G @ tc_HOL_Obool ) ) @ ( tc_fun @ C @ ( tc_fun @ G @ tc_HOL_Obool ) ) ) @ F ) @ ( c_COMBB @ ( tc_fun @ G @ tc_HOL_Obool ) @ ( tc_fun @ G @ tc_HOL_Obool ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ G @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ G @ tc_HOL_Obool ) @ ( tc_fun @ G @ tc_HOL_Obool ) ) @ F ) @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ G @ tc_HOL_Obool ) ) ) @ D ) ) ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Int__UN__distrib2) ).

thf(5505,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ G @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ F @ ( tc_fun @ G @ tc_HOL_Obool ) ) @ E ) @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ C @ ( tc_fun @ G @ tc_HOL_Obool ) ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ F @ ( tc_fun @ G @ tc_HOL_Obool ) ) @ E ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ ( tc_fun @ G @ tc_HOL_Obool ) ) @ ( tc_fun @ G @ tc_HOL_Obool ) @ F ) @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ C @ ( tc_fun @ G @ tc_HOL_Obool ) ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ F @ ( tc_fun @ C @ ( tc_fun @ G @ tc_HOL_Obool ) ) @ ( tc_fun @ C @ ( tc_fun @ G @ tc_HOL_Obool ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ ( tc_fun @ G @ tc_HOL_Obool ) @ ( tc_fun @ G @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ C @ ( tc_fun @ G @ tc_HOL_Obool ) ) @ ( tc_fun @ C @ ( tc_fun @ G @ tc_HOL_Obool ) ) ) @ F ) @ ( c_COMBB @ ( tc_fun @ G @ tc_HOL_Obool ) @ ( tc_fun @ G @ tc_HOL_Obool ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ G @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ G @ tc_HOL_Obool ) @ ( tc_fun @ G @ tc_HOL_Obool ) ) @ F ) @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ G @ tc_HOL_Obool ) ) ) @ D ) ) ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[861]) ).

thf(1388,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ ( tc_prod @ B @ B ) ) @ A ) )
     => ( ( c_Wellfounded_Oacyclic @ B @ A )
       => ( c_Wellfounded_Owf @ B @ ( hAPP @ ( c_Relation_Oconverse @ B @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_finite__acyclic__wf__converse) ).

thf(7732,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ ( tc_prod @ B @ B ) ) @ A ) )
     => ( ( c_Wellfounded_Oacyclic @ B @ A )
       => ( c_Wellfounded_Owf @ B @ ( hAPP @ ( c_Relation_Oconverse @ B @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1388]) ).

thf(1448,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ E ) @ D ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( c_Set_Oimage @ E @ C @ A ) @ D ) ) )
       => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_finite__surj) ).

thf(7950,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ E ) @ D ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( c_Set_Oimage @ E @ C @ A ) @ D ) ) )
       => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1448]) ).

thf(1537,axiom,
    ( ( c_Groups_Oone__class_Oone @ tc_Nat_Onat )
    = ( hAPP @ c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_One__nat__def) ).

thf(8356,plain,
    ( ( c_Groups_Oone__class_Oone @ tc_Nat_Onat )
    = ( hAPP @ c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1537]) ).

thf(12,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
      ( ( class_Groups_Ocomm__monoid__add @ H )
     => ( ( c_Fun_Oinj__on @ G @ F @ E @ D )
       => ( ( C
            = ( hAPP @ ( c_Set_Oimage @ G @ F @ E ) @ D ) )
         => ( ! [I: $i] :
                ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ G ) @ I ) @ D ) )
               => ( ( hAPP @ B @ I )
                  = ( hAPP @ A @ ( hAPP @ E @ I ) ) ) )
           => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ F @ H ) @ A ) @ C )
              = ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ G @ H ) @ B ) @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_setsum__reindex__cong) ).

thf(2443,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
      ( ( class_Groups_Ocomm__monoid__add @ H )
     => ( ( c_Fun_Oinj__on @ G @ F @ E @ D )
       => ( ( C
            = ( hAPP @ ( c_Set_Oimage @ G @ F @ E ) @ D ) )
         => ( ! [I: $i] :
                ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ G ) @ I ) @ D ) )
               => ( ( hAPP @ B @ I )
                  = ( hAPP @ A @ ( hAPP @ E @ I ) ) ) )
           => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ F @ H ) @ A ) @ C )
              = ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ G @ H ) @ B ) @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[12]) ).

thf(1222,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( c_Map_Omap__of @ C @ B @ ( c_List_Olist_ONil @ ( tc_prod @ C @ B ) ) ) @ A )
      = ( c_Option_Ooption_ONone @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_map__of_Osimps_I1_J) ).

thf(7034,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( c_Map_Omap__of @ C @ B @ ( c_List_Olist_ONil @ ( tc_prod @ C @ B ) ) ) @ A )
      = ( c_Option_Ooption_ONone @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1222]) ).

thf(898,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) ) @ ( c_Transitive__Closure_Ortrancl @ B @ A ) ) @ ( c_Relation_OId @ B ) )
      = ( c_Transitive__Closure_Ortrancl @ B @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_rtrancl__reflcl__absorb) ).

thf(5632,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) ) @ ( c_Transitive__Closure_Ortrancl @ B @ A ) ) @ ( c_Relation_OId @ B ) )
      = ( c_Transitive__Closure_Ortrancl @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[898]) ).

thf(481,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ B ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ A ) @ B ) ) )
      <=> ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_neg__imp__zdiv__nonneg__iff) ).

thf(4121,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ B ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) )
     => ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ A ) @ B ) ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) ) )
        & ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ A ) @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[481]) ).

thf(1531,axiom,
    ! [A: $i] :
      ( ( hAPP @ ( c_SetInterval_Oord__class_OlessThan @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ A ) )
      = ( hAPP @ ( c_SetInterval_Oord__class_OatMost @ tc_Nat_Onat ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_lessThan__Suc__atMost) ).

thf(8337,plain,
    ! [A: $i] :
      ( ( hAPP @ ( c_SetInterval_Oord__class_OlessThan @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ A ) )
      = ( hAPP @ ( c_SetInterval_Oord__class_OatMost @ tc_Nat_Onat ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1531]) ).

thf(417,axiom,
    ! [A: $i,B: $i] :
      ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_not__psubset__empty) ).

thf(3911,plain,
    ! [A: $i,B: $i] :
      ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[417]) ).

thf(1190,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ F ) @ E ) @ D ) )
       => ( ( c_Map_Orestrict__map @ F @ C @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ F @ ( tc_Option_Ooption @ C ) ) @ B ) @ E ) @ A ) @ D )
          = ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ F @ ( tc_Option_Ooption @ C ) ) @ ( c_Map_Orestrict__map @ F @ C @ B @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ F @ tc_HOL_Obool ) ) @ D ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ F ) @ E ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ F @ tc_HOL_Obool ) ) ) ) ) ) @ E ) @ A ) ) )
      & ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ F ) @ E ) @ D ) )
       => ( ( c_Map_Orestrict__map @ F @ C @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ F @ ( tc_Option_Ooption @ C ) ) @ B ) @ E ) @ A ) @ D )
          = ( c_Map_Orestrict__map @ F @ C @ B @ D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_restrict__fun__upd) ).

thf(6903,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ F ) @ E ) @ D ) )
       => ( ( c_Map_Orestrict__map @ F @ C @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ F @ ( tc_Option_Ooption @ C ) ) @ B ) @ E ) @ A ) @ D )
          = ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ F @ ( tc_Option_Ooption @ C ) ) @ ( c_Map_Orestrict__map @ F @ C @ B @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ F @ tc_HOL_Obool ) ) @ D ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ F ) @ E ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ F @ tc_HOL_Obool ) ) ) ) ) ) @ E ) @ A ) ) )
      & ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ F ) @ E ) @ D ) )
       => ( ( c_Map_Orestrict__map @ F @ C @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ F @ ( tc_Option_Ooption @ C ) ) @ B ) @ E ) @ A ) @ D )
          = ( c_Map_Orestrict__map @ F @ C @ B @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1190]) ).

thf(1962,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ ( tc_prod @ C @ B ) ) @ ( hAPP @ ( c_Relation_Oconverse @ B @ C ) @ A ) ) )
    <=> ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ ( tc_prod @ B @ C ) ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_finite__converse) ).

thf(10135,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ ( tc_prod @ C @ B ) ) @ ( hAPP @ ( c_Relation_Oconverse @ B @ C ) @ A ) ) )
       => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ ( tc_prod @ B @ C ) ) @ A ) ) )
      & ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ ( tc_prod @ B @ C ) ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ ( tc_prod @ C @ B ) ) @ ( hAPP @ ( c_Relation_Oconverse @ B @ C ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1962]) ).

thf(1503,axiom,
    ! [A: $i,B: $i] :
      ( ( ( c_Divides_Odiv__class_Omod @ tc_Nat_Onat @ B @ A )
        = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
    <=> ? [C: $i] :
          ( B
          = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ A ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mod__eq__0__iff) ).

thf(8221,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( c_Divides_Odiv__class_Omod @ tc_Nat_Onat @ B @ A )
          = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
       => ? [C: $i] :
            ( B
            = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ A ) @ C ) ) )
      & ( ? [C: $i] :
            ( B
            = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ A ) @ C ) )
       => ( ( c_Divides_Odiv__class_Omod @ tc_Nat_Onat @ B @ A )
          = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1503]) ).

thf(1569,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ B ) )
      = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ C ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_diff__cancel2) ).

thf(8474,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ B ) )
      = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ C ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1569]) ).

thf(1625,axiom,
    ! [A: $i,B: $i,C: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Diff__subset) ).

thf(8780,plain,
    ! [A: $i,B: $i,C: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1625]) ).

thf(1879,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ D ) @ C ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ tc_Nat_Onat @ tc_Nat_Onat ) @ D ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ tc_Nat_Onat @ tc_Nat_Onat ) @ C ) @ A ) ) ) @ c_FunDef_Opair__less ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_pair__lessI2) ).

thf(9802,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ D ) @ C ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ tc_Nat_Onat @ tc_Nat_Onat ) @ D ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ tc_Nat_Onat @ tc_Nat_Onat ) @ C ) @ A ) ) ) @ c_FunDef_Opair__less ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1879]) ).

thf(395,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ B ) @ A )
      = ( hAPP @ ( c_Set_OCollect @ C ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ C @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ C ) @ c_fdisj ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ C @ C @ tc_HOL_Obool ) @ c_fequal ) @ B ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ C @ ( tc_fun @ C @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( c_member @ C ) ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_insert__compr) ).

thf(3825,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ B ) @ A )
      = ( hAPP @ ( c_Set_OCollect @ C ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ C @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ C ) @ c_fdisj ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ C @ C @ tc_HOL_Obool ) @ c_fequal ) @ B ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ C @ ( tc_fun @ C @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( c_member @ C ) ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[395]) ).

thf(1464,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ E ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ D ) @ C ) ) )
     => ( ( hAPP @ ( hAPP @ ( c_List_Onth @ B ) @ ( hAPP @ ( hAPP @ ( c_List_Omap @ tc_Nat_Onat @ B ) @ A ) @ ( c_List_Oupt @ C @ D ) ) ) @ E )
        = ( hAPP @ A @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ E ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nth__map__upt) ).

thf(8011,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ E ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ D ) @ C ) ) )
     => ( ( hAPP @ ( hAPP @ ( c_List_Onth @ B ) @ ( hAPP @ ( hAPP @ ( c_List_Omap @ tc_Nat_Onat @ B ) @ A ) @ ( c_List_Oupt @ C @ D ) ) ) @ E )
        = ( hAPP @ A @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ E ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1464]) ).

thf(1187,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( hAPP @ E @ D )
        = ( hAPP @ ( c_Option_Ooption_OSome @ C ) @ B ) )
     => ( ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ A @ ( tc_Option_Ooption @ C ) ) @ E ) @ D ) @ ( hAPP @ ( c_Option_Ooption_OSome @ C ) @ B ) )
        = E ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_map__upd__triv) ).

thf(6893,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( hAPP @ E @ D )
        = ( hAPP @ ( c_Option_Ooption_OSome @ C ) @ B ) )
     => ( ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ A @ ( tc_Option_Ooption @ C ) ) @ E ) @ D ) @ ( hAPP @ ( c_Option_Ooption_OSome @ C ) @ B ) )
        = E ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1187]) ).

thf(1226,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Int_Oint ) @ D ) @ C ) )
     => ! [E: $i] :
          ( ! [F: $i] :
              ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Int_Oint ) @ F ) @ ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Int_Oint @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) @ C ) ) )
             => ! [G: $i] :
                  ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Int_Oint ) @ G ) @ B ) )
                 => ( E
                   != ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ G ) @ F ) ) ) )
         => ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Int_Oint ) @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ E ) @ A ) ) )
           => ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Int_Oint ) @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Int_Oint ) @ E ) @ C ) ) @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_bset_I10_J) ).

thf(7044,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Int_Oint ) @ D ) @ C ) )
     => ! [E: $i] :
          ( ! [F: $i] :
              ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Int_Oint ) @ F ) @ ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Int_Oint @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) @ C ) ) )
             => ! [G: $i] :
                  ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Int_Oint ) @ G ) @ B ) )
                 => ( E
                   != ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ G ) @ F ) ) ) )
         => ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Int_Oint ) @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ E ) @ A ) ) )
           => ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Int_Oint ) @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Int_Oint ) @ E ) @ C ) ) @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1226]) ).

thf(2052,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ D ) @ C ) ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ C ) @ A ) ) @ ( c_Transitive__Closure_Otrancl @ E @ B ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ D ) @ A ) ) @ ( c_Transitive__Closure_Otrancl @ E @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_trancl__into__trancl2) ).

thf(10468,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ D ) @ C ) ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ C ) @ A ) ) @ ( c_Transitive__Closure_Otrancl @ E @ B ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ D ) @ A ) ) @ ( c_Transitive__Closure_Otrancl @ E @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2052]) ).

thf(2103,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Divides_Odivmod__nat__rel @ D @ C ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ tc_Nat_Onat @ tc_Nat_Onat ) @ B ) @ A ) ) )
     => ( ( c_Divides_Odiv__class_Omod @ tc_Nat_Onat @ D @ C )
        = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mod__eq) ).

thf(10659,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Divides_Odivmod__nat__rel @ D @ C ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ tc_Nat_Onat @ tc_Nat_Onat ) @ B ) @ A ) ) )
     => ( ( c_Divides_Odiv__class_Omod @ tc_Nat_Onat @ D @ C )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2103]) ).

thf(1914,axiom,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( c_Divides_Odiv__class_Omod @ tc_Nat_Onat @ B @ A ) ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mod__less__eq__dividend) ).

thf(9934,plain,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( c_Divides_Odiv__class_Omod @ tc_Nat_Onat @ B @ A ) ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1914]) ).

thf(192,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Fun_Oinj__on @ ( tc_List_Olist @ C ) @ ( tc_List_Olist @ B ) @ ( hAPP @ ( c_List_Omap @ C @ B ) @ A ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ ( tc_List_Olist @ C ) @ tc_HOL_Obool ) ) )
     => ( c_Fun_Oinj__on @ C @ B @ A @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_inj__mapD) ).

thf(3104,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Fun_Oinj__on @ ( tc_List_Olist @ C ) @ ( tc_List_Olist @ B ) @ ( hAPP @ ( c_List_Omap @ C @ B ) @ A ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ ( tc_List_Olist @ C ) @ tc_HOL_Obool ) ) )
     => ( c_Fun_Oinj__on @ C @ B @ A @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[192]) ).

thf(1731,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( c_Equiv__Relations_Oequiv @ G @ F @ E )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_fun @ G @ tc_HOL_Obool ) ) @ D ) @ ( hAPP @ ( hAPP @ ( c_Equiv__Relations_Oquotient @ G ) @ F ) @ E ) ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_fun @ G @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Equiv__Relations_Oquotient @ G ) @ F ) @ E ) ) )
         => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ G ) @ B ) @ D ) )
           => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ G ) @ A ) @ C ) )
             => ( ( D = C )
              <=> ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ G @ G ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ G @ G ) @ B ) @ A ) ) @ E ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_quotient__eq__iff) ).

thf(9205,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( c_Equiv__Relations_Oequiv @ G @ F @ E )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_fun @ G @ tc_HOL_Obool ) ) @ D ) @ ( hAPP @ ( hAPP @ ( c_Equiv__Relations_Oquotient @ G ) @ F ) @ E ) ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_fun @ G @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Equiv__Relations_Oquotient @ G ) @ F ) @ E ) ) )
         => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ G ) @ B ) @ D ) )
           => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ G ) @ A ) @ C ) )
             => ( ( ( D = C )
                 => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ G @ G ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ G @ G ) @ B ) @ A ) ) @ E ) ) )
                & ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ G @ G ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ G @ G ) @ B ) @ A ) ) @ E ) )
                 => ( D = C ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1731]) ).

thf(1798,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( hAPP @ ( c_Sum__Type_OInl @ D @ C ) @ B )
        = ( hAPP @ ( c_Sum__Type_OInl @ D @ C ) @ A ) )
     => ( B = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Inl__inject) ).

thf(9459,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( hAPP @ ( c_Sum__Type_OInl @ D @ C ) @ B )
        = ( hAPP @ ( c_Sum__Type_OInl @ D @ C ) @ A ) )
     => ( B = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1798]) ).

thf(2099,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Wellfounded_OwfP @ C @ B )
     => ( hBOOL @ ( hAPP @ ( c_Wellfounded_Oaccp @ C @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_accp__wfPD) ).

thf(10648,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Wellfounded_OwfP @ C @ B )
     => ( hBOOL @ ( hAPP @ ( c_Wellfounded_Oaccp @ C @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2099]) ).

thf(2397,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( c_Typedef_Otype__definition @ G @ F @ E @ D @ C )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ F ) @ B ) @ C ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ F ) @ A ) @ C ) )
         => ( ( ( hAPP @ D @ B )
              = ( hAPP @ D @ A ) )
          <=> ( B = A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_type__definition_OAbs__inject) ).

thf(11847,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( c_Typedef_Otype__definition @ G @ F @ E @ D @ C )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ F ) @ B ) @ C ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ F ) @ A ) @ C ) )
         => ( ( ( ( hAPP @ D @ B )
                = ( hAPP @ D @ A ) )
             => ( B = A ) )
            & ( ( B = A )
             => ( ( hAPP @ D @ B )
                = ( hAPP @ D @ A ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2397]) ).

thf(1191,axiom,
    ! [A: $i,B: $i,C: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_List_Oset @ C ) @ B ) ) @ ( hAPP @ ( c_List_Oset @ C ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ C ) @ A ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_set__subset__Cons) ).

thf(6910,plain,
    ! [A: $i,B: $i,C: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_List_Oset @ C ) @ B ) ) @ ( hAPP @ ( c_List_Oset @ C ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ C ) @ A ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1191]) ).

thf(1591,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( hAPP @ ( hAPP @ ( c_List_Oappend @ D ) @ C ) @ B )
        = ( hAPP @ ( hAPP @ ( c_List_Oappend @ D ) @ C ) @ A ) )
    <=> ( B = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_same__append__eq) ).

thf(8634,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( ( hAPP @ ( hAPP @ ( c_List_Oappend @ D ) @ C ) @ B )
          = ( hAPP @ ( hAPP @ ( c_List_Oappend @ D ) @ C ) @ A ) )
       => ( B = A ) )
      & ( ( B = A )
       => ( ( hAPP @ ( hAPP @ ( c_List_Oappend @ D ) @ C ) @ B )
          = ( hAPP @ ( hAPP @ ( c_List_Oappend @ D ) @ C ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1591]) ).

thf(325,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( c_Relation_ORange @ B @ B ) @ ( c_Transitive__Closure_Ortrancl @ B @ A ) )
      = ( c_Orderings_Otop__class_Otop @ ( tc_fun @ B @ tc_HOL_Obool ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Range__rtrancl) ).

thf(3561,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( c_Relation_ORange @ B @ B ) @ ( c_Transitive__Closure_Ortrancl @ B @ A ) )
      = ( c_Orderings_Otop__class_Otop @ ( tc_fun @ B @ tc_HOL_Obool ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[325]) ).

thf(58,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_List_Opartition @ D @ C @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ D ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) @ ( tc_prod @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ ( tc_List_Olist @ D ) @ ( tc_fun @ ( tc_List_Olist @ D ) @ ( tc_prod @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) ) @ ( tc_fun @ ( tc_List_Olist @ D ) @ ( tc_prod @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ ( tc_List_Olist @ D ) @ ( tc_fun @ ( tc_prod @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) @ ( tc_prod @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) ) ) @ ( tc_fun @ ( tc_fun @ ( tc_List_Olist @ D ) @ ( tc_prod @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) ) @ ( tc_fun @ ( tc_List_Olist @ D ) @ ( tc_prod @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) ) ) @ ( tc_List_Olist @ D ) ) @ ( c_COMBS @ ( tc_List_Olist @ D ) @ ( tc_prod @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) @ ( tc_prod @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ ( tc_List_Olist @ D ) @ ( tc_prod @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) ) @ ( tc_fun @ ( tc_List_Olist @ D ) @ ( tc_fun @ ( tc_prod @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) @ ( tc_prod @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) ) ) @ ( tc_List_Olist @ D ) ) @ ( hAPP @ ( c_COMBB @ ( tc_prod @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) @ ( tc_fun @ ( tc_prod @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) @ ( tc_prod @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) ) @ ( tc_List_Olist @ D ) ) @ ( hAPP @ ( c_If @ ( tc_prod @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) ) @ ( hAPP @ C @ B ) ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_List_Olist @ D ) @ ( tc_fun @ ( tc_List_Olist @ D ) @ ( tc_prod @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) ) @ ( tc_List_Olist @ D ) ) @ ( c_Product__Type_OPair @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) ) @ ( hAPP @ ( c_List_Olist_OCons @ D ) @ B ) ) ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ ( tc_List_Olist @ D ) @ ( tc_fun @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) @ ( tc_fun @ ( tc_List_Olist @ D ) @ ( tc_prod @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ ( tc_List_Olist @ D ) @ ( tc_prod @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) ) @ ( tc_fun @ ( tc_fun @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) @ ( tc_fun @ ( tc_List_Olist @ D ) @ ( tc_prod @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) ) ) @ ( tc_List_Olist @ D ) ) @ ( c_COMBB @ ( tc_List_Olist @ D ) @ ( tc_prod @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) @ ( tc_List_Olist @ D ) ) ) @ ( c_Product__Type_OPair @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) ) ) @ ( hAPP @ ( c_List_Olist_OCons @ D ) @ B ) ) ) ) @ ( c_List_Opartition @ D @ C @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_partition_Osimps_I2_J) ).

thf(2608,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_List_Opartition @ D @ C @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ D ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) @ ( tc_prod @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ ( tc_List_Olist @ D ) @ ( tc_fun @ ( tc_List_Olist @ D ) @ ( tc_prod @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) ) @ ( tc_fun @ ( tc_List_Olist @ D ) @ ( tc_prod @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ ( tc_List_Olist @ D ) @ ( tc_fun @ ( tc_prod @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) @ ( tc_prod @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) ) ) @ ( tc_fun @ ( tc_fun @ ( tc_List_Olist @ D ) @ ( tc_prod @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) ) @ ( tc_fun @ ( tc_List_Olist @ D ) @ ( tc_prod @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) ) ) @ ( tc_List_Olist @ D ) ) @ ( c_COMBS @ ( tc_List_Olist @ D ) @ ( tc_prod @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) @ ( tc_prod @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ ( tc_List_Olist @ D ) @ ( tc_prod @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) ) @ ( tc_fun @ ( tc_List_Olist @ D ) @ ( tc_fun @ ( tc_prod @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) @ ( tc_prod @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) ) ) @ ( tc_List_Olist @ D ) ) @ ( hAPP @ ( c_COMBB @ ( tc_prod @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) @ ( tc_fun @ ( tc_prod @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) @ ( tc_prod @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) ) @ ( tc_List_Olist @ D ) ) @ ( hAPP @ ( c_If @ ( tc_prod @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) ) @ ( hAPP @ C @ B ) ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_List_Olist @ D ) @ ( tc_fun @ ( tc_List_Olist @ D ) @ ( tc_prod @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) ) @ ( tc_List_Olist @ D ) ) @ ( c_Product__Type_OPair @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) ) @ ( hAPP @ ( c_List_Olist_OCons @ D ) @ B ) ) ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ ( tc_List_Olist @ D ) @ ( tc_fun @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) @ ( tc_fun @ ( tc_List_Olist @ D ) @ ( tc_prod @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ ( tc_List_Olist @ D ) @ ( tc_prod @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) ) @ ( tc_fun @ ( tc_fun @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) @ ( tc_fun @ ( tc_List_Olist @ D ) @ ( tc_prod @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) ) ) @ ( tc_List_Olist @ D ) ) @ ( c_COMBB @ ( tc_List_Olist @ D ) @ ( tc_prod @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) @ ( tc_List_Olist @ D ) ) ) @ ( c_Product__Type_OPair @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) ) ) @ ( hAPP @ ( c_List_Olist_OCons @ D ) @ B ) ) ) ) @ ( c_List_Opartition @ D @ C @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[58]) ).

thf(8,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( c_List_Oset @ B ) @ ( c_List_Oconcat @ B @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ ( tc_List_Olist @ B ) @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_List_Oset @ ( tc_List_Olist @ B ) ) @ A ) ) @ ( c_List_Oset @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_set__concat) ).

thf(2431,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( c_List_Oset @ B ) @ ( c_List_Oconcat @ B @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ ( tc_List_Olist @ B ) @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_List_Oset @ ( tc_List_Olist @ B ) ) @ A ) ) @ ( c_List_Oset @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[8]) ).

thf(1973,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ F @ E ) @ D ) @ C )
        = ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ F @ E ) @ B ) @ A ) )
    <=> ( ( D = B )
        & ( C = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Pair__eq) ).

thf(10182,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ F @ E ) @ D ) @ C )
          = ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ F @ E ) @ B ) @ A ) )
       => ( ( D = B )
          & ( C = A ) ) )
      & ( ( ( D = B )
          & ( C = A ) )
       => ( ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ F @ E ) @ D ) @ C )
          = ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ F @ E ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1973]) ).

thf(825,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ A ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ A ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zadd__left__mono) ).

thf(5394,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ A ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ A ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[825]) ).

thf(312,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ B ) @ A ) )
      | ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ A ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zle__linear) ).

thf(3516,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ B ) @ A ) )
      | ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ A ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[312]) ).

thf(581,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ F @ ( tc_fun @ E @ tc_HOL_Obool ) ) ) @ D ) @ C ) @ B ) @ A ) )
     => ~ ( ( hBOOL @ ( hAPP @ ( hAPP @ D @ B ) @ A ) )
         => ~ ( hBOOL @ ( hAPP @ ( hAPP @ C @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_inf2E) ).

thf(4495,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ F @ ( tc_fun @ E @ tc_HOL_Obool ) ) ) @ D ) @ C ) @ B ) @ A ) )
     => ~ ( ( hBOOL @ ( hAPP @ ( hAPP @ D @ B ) @ A ) )
         => ~ ( hBOOL @ ( hAPP @ ( hAPP @ C @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[581]) ).

thf(640,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i,I: $i,J: $i] :
      ( ( c_Equiv__Relations_Oequiv @ J @ I @ H )
     => ( ( c_Equiv__Relations_Oequiv @ G @ F @ E )
       => ( ( c_Equiv__Relations_Ocongruent2 @ J @ G @ ( tc_fun @ D @ tc_HOL_Obool ) @ H @ E @ C )
         => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ J ) @ B ) @ I ) )
           => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ G ) @ A ) @ F ) )
             => ( ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ J @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Relation_OImage @ J @ J @ H ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ J ) @ B ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ J @ tc_HOL_Obool ) ) ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ G @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( tc_fun @ D @ tc_HOL_Obool ) @ J ) @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ G @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Relation_OImage @ G @ G @ E ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ G ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ G @ tc_HOL_Obool ) ) ) ) ) ) @ C ) )
                = ( hAPP @ ( hAPP @ C @ B ) @ A ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_UN__equiv__class2) ).

thf(4705,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i,I: $i,J: $i] :
      ( ( c_Equiv__Relations_Oequiv @ J @ I @ H )
     => ( ( c_Equiv__Relations_Oequiv @ G @ F @ E )
       => ( ( c_Equiv__Relations_Ocongruent2 @ J @ G @ ( tc_fun @ D @ tc_HOL_Obool ) @ H @ E @ C )
         => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ J ) @ B ) @ I ) )
           => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ G ) @ A ) @ F ) )
             => ( ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ J @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Relation_OImage @ J @ J @ H ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ J ) @ B ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ J @ tc_HOL_Obool ) ) ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ G @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( tc_fun @ D @ tc_HOL_Obool ) @ J ) @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ G @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Relation_OImage @ G @ G @ E ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ G ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ G @ tc_HOL_Obool ) ) ) ) ) ) @ C ) )
                = ( hAPP @ ( hAPP @ C @ B ) @ A ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[640]) ).

thf(1956,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ B ) ) @ ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ A ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_card__insert__le) ).

thf(10104,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ B ) ) @ ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ A ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1956]) ).

thf(1245,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) ) @ B ) @ A ) )
     => ( ( c_Relation_Osingle__valued @ D @ C @ A )
       => ( c_Relation_Osingle__valued @ D @ C @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_single__valued__subset) ).

thf(7120,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) ) @ B ) @ A ) )
     => ( ( c_Relation_Osingle__valued @ D @ C @ A )
       => ( c_Relation_Osingle__valued @ D @ C @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1245]) ).

thf(1120,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ A ) @ C ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ A ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_psubsetD) ).

thf(6618,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ A ) @ C ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ A ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1120]) ).

thf(1247,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ C ) ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ C ) ) @ A ) ) )
    <=> ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Suc__mult__less__cancel1) ).

thf(7125,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ C ) ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ C ) ) @ A ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) ) )
      & ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ C ) ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ C ) ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1247]) ).

thf(270,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ C ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zadd__left__commute) ).

thf(3368,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ C ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[270]) ).

thf(691,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ E @ D ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ C ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ ( tc_prod @ E @ D ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ E @ D ) @ C ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ E @ D ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Sigma__Un__distrib1) ).

thf(4881,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ E @ D ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ C ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ ( tc_prod @ E @ D ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ E @ D ) @ C ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ E @ D ) @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[691]) ).

thf(1794,axiom,
    ! [A: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ A ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_dvd_Oorder__refl) ).

thf(9449,plain,
    ! [A: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ A ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1794]) ).

thf(591,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] : ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofold__graph @ D @ C @ B @ A @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_fold__graph_OemptyI) ).

thf(4527,plain,
    ! [A: $i,B: $i,C: $i,D: $i] : ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofold__graph @ D @ C @ B @ A @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[591]) ).

thf(2416,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ~ ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ ( hAPP @ ( c_Set_OCollect @ D ) @ C ) ) )
      | ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ B ) @ ( hAPP @ ( c_Set_OCollect @ B ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ D @ tc_HOL_Obool ) @ tc_HOL_Obool @ B ) @ ( c_HOL_OEx @ D ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ B @ ( tc_fun @ D @ tc_HOL_Obool ) @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ D @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ D @ tc_HOL_Obool ) @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ ( c_COMBS @ D @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ D @ tc_HOL_Obool ) @ ( tc_fun @ D @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ D ) @ c_fconj ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ B @ ( tc_fun @ D @ B ) @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ B @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ D @ B ) @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ ( c_COMBB @ B @ tc_HOL_Obool @ D ) ) @ c_fequal ) ) @ A ) ) ) ) @ C ) ) ) ) ) ),
    inference(cnf,[status(esa)],[2415]) ).

thf(2092,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ A ) )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ A ) @ B ) )
        = B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_diff__diff__cancel) ).

thf(10627,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ A ) )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ A ) @ B ) )
        = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2092]) ).

thf(1165,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ C ) )
     => ( ( C
          = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ A ) ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ B ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ A ) @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_self__quotient__aux2) ).

thf(6799,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ C ) )
     => ( ( C
          = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ A ) ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ B ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ A ) @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1165]) ).

thf(2221,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Natural_Oevaln @ ( c_Com_Ocom_OWhile @ E @ D ) @ C @ B @ A )
     => ( ( ( A = C )
         => ( hBOOL @ ( hAPP @ E @ C ) ) )
       => ~ ( ( hBOOL @ ( hAPP @ E @ C ) )
           => ! [F: $i] :
                ( ( c_Natural_Oevaln @ D @ C @ B @ F )
               => ~ ( c_Natural_Oevaln @ ( c_Com_Ocom_OWhile @ E @ D ) @ F @ B @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_evaln__WHILE__case) ).

thf(11091,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Natural_Oevaln @ ( c_Com_Ocom_OWhile @ E @ D ) @ C @ B @ A )
     => ( ( ( A = C )
         => ( hBOOL @ ( hAPP @ E @ C ) ) )
       => ~ ( ( hBOOL @ ( hAPP @ E @ C ) )
           => ! [F: $i] :
                ( ( c_Natural_Oevaln @ D @ C @ B @ F )
               => ~ ( c_Natural_Oevaln @ ( c_Com_Ocom_OWhile @ E @ D ) @ F @ B @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2221]) ).

thf(1724,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oab__semigroup__mult @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ B ) )
       => ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ A ) @ B ) )
         => ( ( hAPP @ ( c_Finite__Set_Ofold1 @ C @ ( c_Groups_Otimes__class_Otimes @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ A ) @ B ) )
            = ( hAPP @ ( hAPP @ ( c_Finite__Set_Ofold @ C @ C @ ( c_Groups_Otimes__class_Otimes @ C ) ) @ A ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_fold1__eq__fold) ).

thf(9179,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oab__semigroup__mult @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ B ) )
       => ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ A ) @ B ) )
         => ( ( hAPP @ ( c_Finite__Set_Ofold1 @ C @ ( c_Groups_Otimes__class_Otimes @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ A ) @ B ) )
            = ( hAPP @ ( hAPP @ ( c_Finite__Set_Ofold @ C @ C @ ( c_Groups_Otimes__class_Otimes @ C ) ) @ A ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1724]) ).

thf(698,axiom,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ B ) ) @ ( c_List_Oremdups @ B @ A ) ) ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ B ) ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_length__remdups__leq) ).

thf(4900,plain,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ B ) ) @ ( c_List_Oremdups @ B @ A ) ) ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ B ) ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[698]) ).

thf(146,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ B ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) )
     => ( ( ( hAPP @ ( c_Finite__Set_Ocard @ B ) @ A )
          = ( hAPP @ ( c_Finite__Set_Ocard @ B ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) )
       => ( A
          = ( c_Orderings_Otop__class_Otop @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_card__eq__UNIV__imp__eq__UNIV) ).

thf(2934,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ B ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) )
     => ( ( ( hAPP @ ( c_Finite__Set_Ocard @ B ) @ A )
          = ( hAPP @ ( c_Finite__Set_Ocard @ B ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) )
       => ( A
          = ( c_Orderings_Otop__class_Otop @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[146]) ).

thf(1950,axiom,
    ! [A: $i] :
      ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ A ) ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Suc__n__not__le__n) ).

thf(10077,plain,
    ! [A: $i] :
      ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ A ) ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1950]) ).

thf(11918,plain,
    ( ( sk4 != sk3 )
    | ( ( c_Finite__Set_Ofinite @ tc_Int_Oint )
     != v_b )
    | ( ( c_Orderings_Otop__class_Otop @ ( tc_fun @ tc_Int_Oint @ tc_HOL_Obool ) )
     != sk4 ) ),
    inference(simp,[status(thm)],[11898]) ).

thf(973,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) )
     => ( ! [D: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ D ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) ) )
           => ! [E: $i] :
                ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ E ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) ) )
               => ( ( D != E )
                 => ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ D ) @ E )
                    = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) ) ) )
       => ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ ( c_Complete__Lattice_OSup__class_OSup @ ( tc_fun @ C @ tc_HOL_Obool ) @ A ) )
          = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_insert__partition) ).

thf(5922,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) )
     => ( ! [D: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ D ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) ) )
           => ! [E: $i] :
                ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ E ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) ) )
               => ( ( D != E )
                 => ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ D ) @ E )
                    = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) ) ) )
       => ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ ( c_Complete__Lattice_OSup__class_OSup @ ( tc_fun @ C @ tc_HOL_Obool ) @ A ) )
          = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[973]) ).

thf(1071,axiom,
    ! [A: $i] :
      ( ( c_Divides_Odiv__class_Omod @ tc_Nat_Onat @ A @ ( hAPP @ c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) )
      = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mod__1) ).

thf(6354,plain,
    ! [A: $i] :
      ( ( c_Divides_Odiv__class_Omod @ tc_Nat_Onat @ A @ ( hAPP @ c_Nat_OSuc @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) )
      = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1071]) ).

thf(1717,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( c_Finite__Set_Ofun__left__comm @ F @ E @ D )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofold__graph @ F @ E @ D @ C @ B ) @ A ) )
       => ( ( hAPP @ ( hAPP @ ( c_Finite__Set_Ofold @ F @ E @ D ) @ C ) @ B )
          = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_fun__left__comm_Ofold__equality) ).

thf(9160,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( c_Finite__Set_Ofun__left__comm @ F @ E @ D )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofold__graph @ F @ E @ D @ C @ B ) @ A ) )
       => ( ( hAPP @ ( hAPP @ ( c_Finite__Set_Ofold @ F @ E @ D ) @ C ) @ B )
          = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1717]) ).

thf(769,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( c_Set_Ovimage @ C @ B @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) )
      = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_vimage__empty) ).

thf(5162,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( c_Set_Ovimage @ C @ B @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) )
      = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[769]) ).

thf(872,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ B ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ A ) ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ A ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_UnCI) ).

thf(5532,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ B ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ A ) ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ A ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[872]) ).

thf(991,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Int__assoc) ).

thf(6018,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[991]) ).

thf(1089,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ C ) )
     => ! [D: $i] :
          ( ! [E: $i] :
              ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Int_Oint ) @ E ) @ ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Int_Oint @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) @ C ) ) )
             => ! [F: $i] :
                  ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Int_Oint ) @ F ) @ B ) )
                 => ( D
                   != ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Int_Oint ) @ F ) @ E ) ) ) )
         => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ A ) @ D ) )
           => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ D ) @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_aset_I8_J) ).

thf(6412,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ C ) )
     => ! [D: $i] :
          ( ! [E: $i] :
              ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Int_Oint ) @ E ) @ ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Int_Oint @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) @ C ) ) )
             => ! [F: $i] :
                  ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Int_Oint ) @ F ) @ B ) )
                 => ( D
                   != ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Int_Oint ) @ F ) @ E ) ) ) )
         => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ A ) @ D ) )
           => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ D ) @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1089]) ).

thf(1920,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Odrop @ C ) @ B ) @ ( c_List_Obutlast @ C @ A ) )
      = ( c_List_Obutlast @ C @ ( hAPP @ ( hAPP @ ( c_List_Odrop @ C ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_drop__butlast) ).

thf(9955,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Odrop @ C ) @ B ) @ ( c_List_Obutlast @ C @ A ) )
      = ( c_List_Obutlast @ C @ ( hAPP @ ( hAPP @ ( c_List_Odrop @ C ) @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1920]) ).

thf(930,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Ozip @ F @ E ) @ D ) @ ( hAPP @ ( hAPP @ ( c_List_Omap @ C @ E ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_List_Omap @ ( tc_prod @ F @ C ) @ ( tc_prod @ F @ E ) ) @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ F @ C @ ( tc_prod @ F @ E ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ F @ ( tc_fun @ C @ E ) @ ( tc_fun @ C @ ( tc_prod @ F @ E ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ E @ ( tc_prod @ F @ E ) ) @ ( tc_fun @ ( tc_fun @ C @ E ) @ ( tc_fun @ C @ ( tc_prod @ F @ E ) ) ) @ F ) @ ( c_COMBB @ E @ ( tc_prod @ F @ E ) @ C ) ) @ ( c_Product__Type_OPair @ F @ E ) ) ) @ B ) ) ) @ ( hAPP @ ( hAPP @ ( c_List_Ozip @ F @ C ) @ D ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zip__map2) ).

thf(5739,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Ozip @ F @ E ) @ D ) @ ( hAPP @ ( hAPP @ ( c_List_Omap @ C @ E ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_List_Omap @ ( tc_prod @ F @ C ) @ ( tc_prod @ F @ E ) ) @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ F @ C @ ( tc_prod @ F @ E ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ F @ ( tc_fun @ C @ E ) @ ( tc_fun @ C @ ( tc_prod @ F @ E ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ E @ ( tc_prod @ F @ E ) ) @ ( tc_fun @ ( tc_fun @ C @ E ) @ ( tc_fun @ C @ ( tc_prod @ F @ E ) ) ) @ F ) @ ( c_COMBB @ E @ ( tc_prod @ F @ E ) @ C ) ) @ ( c_Product__Type_OPair @ F @ E ) ) ) @ B ) ) ) @ ( hAPP @ ( hAPP @ ( c_List_Ozip @ F @ C ) @ D ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[930]) ).

thf(1819,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( c_Relation_ODomain @ E @ D ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ ( tc_prod @ E @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ D ) @ C ) @ B ) ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ C ) @ ( hAPP @ ( c_Relation_ODomain @ E @ D ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Domain__insert) ).

thf(9542,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( c_Relation_ODomain @ E @ D ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ ( tc_prod @ E @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ D ) @ C ) @ B ) ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ C ) @ ( hAPP @ ( c_Relation_ODomain @ E @ D ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1819]) ).

thf(1703,axiom,
    ! [A: $i,B: $i] :
      ( ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ B ) @ A )
        = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
     => ( ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ A ) @ B )
          = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
       => ( B = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_diffs0__imp__equal) ).

thf(9115,plain,
    ! [A: $i,B: $i] :
      ( ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ B ) @ A )
        = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
     => ( ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ A ) @ B )
          = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
       => ( B = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1703]) ).

thf(149,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Wellfounded_OwfP @ B @ ( hAPP @ ( hAPP @ ( c_COMBC @ B @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ B @ ( tc_fun @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ B ) @ ( c_COMBC @ B @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ B @ ( tc_prod @ B @ B ) ) @ ( tc_fun @ B @ ( tc_fun @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( c_COMBB @ ( tc_prod @ B @ B ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ B ) @ ( c_member @ ( tc_prod @ B @ B ) ) ) ) @ ( c_Product__Type_OPair @ B @ B ) ) ) ) @ A ) )
    <=> ( c_Wellfounded_Owf @ B @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_wfP__wf__eq) ).

thf(2944,plain,
    ! [A: $i,B: $i] :
      ( ( ( c_Wellfounded_OwfP @ B @ ( hAPP @ ( hAPP @ ( c_COMBC @ B @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ B @ ( tc_fun @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ B ) @ ( c_COMBC @ B @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ B @ ( tc_prod @ B @ B ) ) @ ( tc_fun @ B @ ( tc_fun @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( c_COMBB @ ( tc_prod @ B @ B ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ B ) @ ( c_member @ ( tc_prod @ B @ B ) ) ) ) @ ( c_Product__Type_OPair @ B @ B ) ) ) ) @ A ) )
       => ( c_Wellfounded_Owf @ B @ A ) )
      & ( ( c_Wellfounded_Owf @ B @ A )
       => ( c_Wellfounded_OwfP @ B @ ( hAPP @ ( hAPP @ ( c_COMBC @ B @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ B @ ( tc_fun @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ B ) @ ( c_COMBC @ B @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ B @ ( tc_prod @ B @ B ) ) @ ( tc_fun @ B @ ( tc_fun @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( c_COMBB @ ( tc_prod @ B @ B ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ B ) @ ( c_member @ ( tc_prod @ B @ B ) ) ) ) @ ( c_Product__Type_OPair @ B @ B ) ) ) ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[149]) ).

thf(1642,axiom,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ ( c_Divides_Odivmod__nat__rel @ B @ A ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ tc_Nat_Onat @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Nat_Onat ) @ B ) @ A ) ) @ ( c_Divides_Odiv__class_Omod @ tc_Nat_Onat @ B @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_divmod__nat__rel) ).

thf(8865,plain,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ ( c_Divides_Odivmod__nat__rel @ B @ A ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ tc_Nat_Onat @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Nat_Onat ) @ B ) @ A ) ) @ ( c_Divides_Odiv__class_Omod @ tc_Nat_Onat @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1642]) ).

thf(2135,axiom,
    ! [A: $i,B: $i] :
      ( ( B = A )
    <=> ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) )
        & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ A ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_dvd_Oeq__iff) ).

thf(10777,plain,
    ! [A: $i,B: $i] :
      ( ( ( B = A )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) )
          & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ A ) @ B ) ) ) )
      & ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) )
          & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ A ) @ B ) ) )
       => ( B = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2135]) ).

thf(1376,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Wellfounded_Oacyclic @ B @ ( hAPP @ ( c_Relation_Oconverse @ B @ B ) @ A ) )
    <=> ( c_Wellfounded_Oacyclic @ B @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_acyclic__converse) ).

thf(7680,plain,
    ! [A: $i,B: $i] :
      ( ( ( c_Wellfounded_Oacyclic @ B @ ( hAPP @ ( c_Relation_Oconverse @ B @ B ) @ A ) )
       => ( c_Wellfounded_Oacyclic @ B @ A ) )
      & ( ( c_Wellfounded_Oacyclic @ B @ A )
       => ( c_Wellfounded_Oacyclic @ B @ ( hAPP @ ( c_Relation_Oconverse @ B @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1376]) ).

thf(1577,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ C ) ) @ B ) ) @ ( hAPP @ c_Nat_OSuc @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ C ) @ B ) ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Suc__diff__diff) ).

thf(8506,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ C ) ) @ B ) ) @ ( hAPP @ c_Nat_OSuc @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ C ) @ B ) ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1577]) ).

thf(1979,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_dvd_Oorder__trans) ).

thf(10210,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ C ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1979]) ).

thf(1860,axiom,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) )
        & ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ A ) @ B ) ) )
     => ~ ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ A ) @ B ) )
          & ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_dvd_Oless__asym_H) ).

thf(9718,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) )
        & ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ A ) @ B ) ) )
     => ~ ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ A ) @ B ) )
          & ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1860]) ).

thf(215,axiom,
    ! [A: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ A ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zle__refl) ).

thf(3181,plain,
    ! [A: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ A ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[215]) ).

thf(1276,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ C ) )
     => ( ( c_Equiv__Relations_Oequiv @ D @ C @ B )
       => ( ! [E: $i] :
              ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ E ) @ ( hAPP @ ( hAPP @ ( c_Equiv__Relations_Oquotient @ D ) @ C ) @ B ) ) )
             => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ A ) @ ( hAPP @ ( c_Finite__Set_Ocard @ D ) @ E ) ) ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ A ) @ ( hAPP @ ( c_Finite__Set_Ocard @ D ) @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_equiv__imp__dvd__card) ).

thf(7256,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ C ) )
     => ( ( c_Equiv__Relations_Oequiv @ D @ C @ B )
       => ( ! [E: $i] :
              ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ E ) @ ( hAPP @ ( hAPP @ ( c_Equiv__Relations_Oquotient @ D ) @ C ) @ B ) ) )
             => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ A ) @ ( hAPP @ ( c_Finite__Set_Ocard @ D ) @ E ) ) ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ A ) @ ( hAPP @ ( c_Finite__Set_Ocard @ D ) @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1276]) ).

thf(685,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ D ) @ ( hAPP @ ( c_Set_Ovimage @ E @ C @ B ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) ) ) )
    <=> ( ( hAPP @ B @ D )
        = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_vimage__singleton__eq) ).

thf(4855,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ D ) @ ( hAPP @ ( c_Set_Ovimage @ E @ C @ B ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) ) ) )
       => ( ( hAPP @ B @ D )
          = A ) )
      & ( ( ( hAPP @ B @ D )
          = A )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ D ) @ ( hAPP @ ( c_Set_Ovimage @ E @ C @ B ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[685]) ).

thf(2383,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Wellfounded_Oaccp @ C @ B ) @ A ) )
    <=> ! [D: $i] :
          ( ( hBOOL @ ( hAPP @ ( hAPP @ B @ D ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( c_Wellfounded_Oaccp @ C @ B ) @ D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_accp_Oequations) ).

thf(11780,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( c_Wellfounded_Oaccp @ C @ B ) @ A ) )
       => ! [D: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ B @ D ) @ A ) )
           => ( hBOOL @ ( hAPP @ ( c_Wellfounded_Oaccp @ C @ B ) @ D ) ) ) )
      & ( ! [D: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ B @ D ) @ A ) )
           => ( hBOOL @ ( hAPP @ ( c_Wellfounded_Oaccp @ C @ B ) @ D ) ) )
       => ( hBOOL @ ( hAPP @ ( c_Wellfounded_Oaccp @ C @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2383]) ).

thf(1764,axiom,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ B ) @ A ) ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_diff__le__self) ).

thf(9338,plain,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ B ) @ A ) ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1764]) ).

thf(1541,axiom,
    ! [A: $i] :
      ( ( hAPP @ c_Nat_OSuc @ A )
     != ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nat_Osimps_I3_J) ).

thf(8369,plain,
    ! [A: $i] :
      ( ( hAPP @ c_Nat_OSuc @ A )
     != ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1541]) ).

thf(1942,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( hAPP @ ( c_Map_Omap__of @ D @ C @ B ) @ A )
        = ( c_Option_Ooption_ONone @ C ) )
    <=> ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ A ) @ ( hAPP @ ( c_Set_Oimage @ ( tc_prod @ D @ C ) @ D @ ( c_Product__Type_Ofst @ D @ C ) ) @ ( hAPP @ ( c_List_Oset @ ( tc_prod @ D @ C ) ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_map__of__eq__None__iff) ).

thf(10040,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( ( hAPP @ ( c_Map_Omap__of @ D @ C @ B ) @ A )
          = ( c_Option_Ooption_ONone @ C ) )
       => ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ A ) @ ( hAPP @ ( c_Set_Oimage @ ( tc_prod @ D @ C ) @ D @ ( c_Product__Type_Ofst @ D @ C ) ) @ ( hAPP @ ( c_List_Oset @ ( tc_prod @ D @ C ) ) @ B ) ) ) ) )
      & ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ A ) @ ( hAPP @ ( c_Set_Oimage @ ( tc_prod @ D @ C ) @ D @ ( c_Product__Type_Ofst @ D @ C ) ) @ ( hAPP @ ( c_List_Oset @ ( tc_prod @ D @ C ) ) @ B ) ) ) )
       => ( ( hAPP @ ( c_Map_Omap__of @ D @ C @ B ) @ A )
          = ( c_Option_Ooption_ONone @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1942]) ).

thf(1802,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( c_Nitpick_Opair__box_OPairBox @ F @ E @ D @ C )
        = ( c_Nitpick_Opair__box_OPairBox @ F @ E @ B @ A ) )
    <=> ( ( D = B )
        & ( C = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_pair__box_Oinject) ).

thf(9470,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( ( c_Nitpick_Opair__box_OPairBox @ F @ E @ D @ C )
          = ( c_Nitpick_Opair__box_OPairBox @ F @ E @ B @ A ) )
       => ( ( D = B )
          & ( C = A ) ) )
      & ( ( ( D = B )
          & ( C = A ) )
       => ( ( c_Nitpick_Opair__box_OPairBox @ F @ E @ D @ C )
          = ( c_Nitpick_Opair__box_OPairBox @ F @ E @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1802]) ).

thf(304,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( c_Relation_OImage @ C @ C @ ( c_Relation_OId__on @ C @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Image__Id__on) ).

thf(3497,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( c_Relation_OImage @ C @ C @ ( c_Relation_OId__on @ C @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[304]) ).

thf(406,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ E ) @ ( tc_List_Olist @ E ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ E ) @ ( tc_List_Olist @ E ) ) @ D ) @ C ) ) @ ( c_List_Olistrel @ E @ ( c_Transitive__Closure_Ortrancl @ E @ B ) ) ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ E ) @ ( tc_List_Olist @ E ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ E ) @ ( tc_List_Olist @ E ) ) @ C ) @ A ) ) @ ( c_List_Olistrel @ E @ ( c_Transitive__Closure_Ortrancl @ E @ B ) ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ E ) @ ( tc_List_Olist @ E ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ E ) @ ( tc_List_Olist @ E ) ) @ D ) @ A ) ) @ ( c_List_Olistrel @ E @ ( c_Transitive__Closure_Ortrancl @ E @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_listrel__rtrancl__trans) ).

thf(3879,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ E ) @ ( tc_List_Olist @ E ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ E ) @ ( tc_List_Olist @ E ) ) @ D ) @ C ) ) @ ( c_List_Olistrel @ E @ ( c_Transitive__Closure_Ortrancl @ E @ B ) ) ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ E ) @ ( tc_List_Olist @ E ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ E ) @ ( tc_List_Olist @ E ) ) @ C ) @ A ) ) @ ( c_List_Olistrel @ E @ ( c_Transitive__Closure_Ortrancl @ E @ B ) ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ E ) @ ( tc_List_Olist @ E ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ E ) @ ( tc_List_Olist @ E ) ) @ D ) @ A ) ) @ ( c_List_Olistrel @ E @ ( c_Transitive__Closure_Ortrancl @ E @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[406]) ).

thf(1576,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ E ) ) @ D )
        = ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) @ B ) )
     => ( ( ( hAPP @ ( c_Map_Omap__of @ E @ C @ ( hAPP @ ( hAPP @ ( c_List_Ozip @ E @ C ) @ D ) @ B ) ) @ A )
          = ( c_Option_Ooption_ONone @ C ) )
      <=> ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ A ) @ ( hAPP @ ( c_List_Oset @ E ) @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_map__of__zip__is__None) ).

thf(8501,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ E ) ) @ D )
        = ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) @ B ) )
     => ( ( ( ( hAPP @ ( c_Map_Omap__of @ E @ C @ ( hAPP @ ( hAPP @ ( c_List_Ozip @ E @ C ) @ D ) @ B ) ) @ A )
            = ( c_Option_Ooption_ONone @ C ) )
         => ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ A ) @ ( hAPP @ ( c_List_Oset @ E ) @ D ) ) ) )
        & ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ A ) @ ( hAPP @ ( c_List_Oset @ E ) @ D ) ) )
         => ( ( hAPP @ ( c_Map_Omap__of @ E @ C @ ( hAPP @ ( hAPP @ ( c_List_Ozip @ E @ C ) @ D ) @ B ) ) @ A )
            = ( c_Option_Ooption_ONone @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1576]) ).

thf(70,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( class_Groups_Ocomm__monoid__add @ F )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ E ) @ D ) )
       => ( ! [G: $i] :
              ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ G ) @ D ) )
             => ! [H: $i] :
                  ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ H ) @ D ) )
                 => ( ( G != H )
                   => ( ( ( hAPP @ C @ G )
                        = ( hAPP @ C @ H ) )
                     => ( ( hAPP @ B @ ( hAPP @ C @ G ) )
                        = ( c_Groups_Ozero__class_Ozero @ F ) ) ) ) ) )
         => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ A @ F ) @ B ) @ ( hAPP @ ( c_Set_Oimage @ E @ A @ C ) @ D ) )
            = ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ E @ F ) @ ( hAPP @ ( c_Fun_Ocomp @ A @ F @ E @ B ) @ C ) ) @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_setsum__reindex__nonzero) ).

thf(2648,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( class_Groups_Ocomm__monoid__add @ F )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ E ) @ D ) )
       => ( ! [G: $i] :
              ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ G ) @ D ) )
             => ! [H: $i] :
                  ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ H ) @ D ) )
                 => ( ( G != H )
                   => ( ( ( hAPP @ C @ G )
                        = ( hAPP @ C @ H ) )
                     => ( ( hAPP @ B @ ( hAPP @ C @ G ) )
                        = ( c_Groups_Ozero__class_Ozero @ F ) ) ) ) ) )
         => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ A @ F ) @ B ) @ ( hAPP @ ( c_Set_Oimage @ E @ A @ C ) @ D ) )
            = ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ E @ F ) @ ( hAPP @ ( c_Fun_Ocomp @ A @ F @ E @ B ) @ C ) ) @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[70]) ).

thf(1907,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Com_Ocom_OCond @ E @ D @ C )
     != ( c_Com_Ocom_OSemi @ B @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_com_Osimps_I45_J) ).

thf(9906,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Com_Ocom_OCond @ E @ D @ C )
     != ( c_Com_Ocom_OSemi @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1907]) ).

thf(608,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Nat_Ocompow @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ D @ D @ D ) @ ( hAPP @ ( hAPP @ ( c_Nat_Ocompow @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) @ C ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Nat_Ocompow @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_rel__pow__add) ).

thf(4591,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Nat_Ocompow @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ D @ D @ D ) @ ( hAPP @ ( hAPP @ ( c_Nat_Ocompow @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) @ C ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Nat_Ocompow @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[608]) ).

thf(1789,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( c_Com_Ocom_OLocal @ F @ E @ D )
     != ( c_Com_Ocom_OCond @ C @ B @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_com_Osimps_I36_J) ).

thf(9434,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( c_Com_Ocom_OLocal @ F @ E @ D )
     != ( c_Com_Ocom_OCond @ C @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1789]) ).

thf(2127,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ B ) ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Suc__le__lessD) ).

thf(10738,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ B ) ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2127]) ).

thf(17,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Groups_Ocomm__monoid__add @ E )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ C ) )
       => ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ B ) @ C ) )
           => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ E @ E ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ D @ E @ ( tc_fun @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ E @ ( tc_fun @ E @ E ) ) @ D ) @ ( c_If @ E ) ) @ ( hAPP @ c_fequal @ B ) ) ) @ A ) ) @ ( c_Groups_Ozero__class_Ozero @ E ) ) ) @ C )
              = ( hAPP @ A @ B ) ) )
          & ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ B ) @ C ) )
           => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ E @ E ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ D @ E @ ( tc_fun @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ E @ ( tc_fun @ E @ E ) ) @ D ) @ ( c_If @ E ) ) @ ( hAPP @ c_fequal @ B ) ) ) @ A ) ) @ ( c_Groups_Ozero__class_Ozero @ E ) ) ) @ C )
              = ( c_Groups_Ozero__class_Ozero @ E ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_setsum__delta_H) ).

thf(2462,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Groups_Ocomm__monoid__add @ E )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ C ) )
       => ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ B ) @ C ) )
           => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ E @ E ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ D @ E @ ( tc_fun @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ E @ ( tc_fun @ E @ E ) ) @ D ) @ ( c_If @ E ) ) @ ( hAPP @ c_fequal @ B ) ) ) @ A ) ) @ ( c_Groups_Ozero__class_Ozero @ E ) ) ) @ C )
              = ( hAPP @ A @ B ) ) )
          & ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ B ) @ C ) )
           => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ E @ E ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ D @ E @ ( tc_fun @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ E @ ( tc_fun @ E @ E ) ) @ D ) @ ( c_If @ E ) ) @ ( hAPP @ c_fequal @ B ) ) ) @ A ) ) @ ( c_Groups_Ozero__class_Ozero @ E ) ) ) @ C )
              = ( c_Groups_Ozero__class_Ozero @ E ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[17]) ).

thf(736,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ B @ A ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ A @ tc_HOL_Obool ) @ B ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ A @ tc_HOL_Obool ) ) ) )
      = ( c_Orderings_Otop__class_Otop @ ( tc_fun @ ( tc_prod @ B @ A ) @ tc_HOL_Obool ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_UNIV__Times__UNIV) ).

thf(5040,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ B @ A ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ A @ tc_HOL_Obool ) @ B ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ A @ tc_HOL_Obool ) ) ) )
      = ( c_Orderings_Otop__class_Otop @ ( tc_fun @ ( tc_prod @ B @ A ) @ tc_HOL_Obool ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[736]) ).

thf(660,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A )
      = ( hAPP @ ( c_Set_OCollect @ C ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ C @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ C ) @ c_fconj ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ C @ ( tc_fun @ C @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( c_member @ C ) ) @ B ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ C @ ( tc_fun @ C @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( c_member @ C ) ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Int__def) ).

thf(4774,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A )
      = ( hAPP @ ( c_Set_OCollect @ C ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ C @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ C ) @ c_fconj ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ C @ ( tc_fun @ C @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( c_member @ C ) ) @ B ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ C @ ( tc_fun @ C @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( c_member @ C ) ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[660]) ).

thf(527,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ A ) @ B ) )
      = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Diff__Int2) ).

thf(4299,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ A ) @ B ) )
      = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[527]) ).

thf(1224,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_finite__Diff) ).

thf(7039,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1224]) ).

thf(1391,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Nat_Onat ) @ B ) @ A ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ B ) @ ( c_Divides_Odiv__class_Omod @ tc_Nat_Onat @ B @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_div__mod__equality_H) ).

thf(7747,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Nat_Onat ) @ B ) @ A ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ B ) @ ( c_Divides_Odiv__class_Omod @ tc_Nat_Onat @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1391]) ).

thf(2352,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hBOOL @ ( hAPP @ F @ E ) )
     => ( ( c_Natural_Oevaln @ D @ E @ C @ B )
       => ( ( c_Natural_Oevaln @ ( c_Com_Ocom_OWhile @ F @ D ) @ B @ C @ A )
         => ( c_Natural_Oevaln @ ( c_Com_Ocom_OWhile @ F @ D ) @ E @ C @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_evaln_OWhileTrue) ).

thf(11658,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hBOOL @ ( hAPP @ F @ E ) )
     => ( ( c_Natural_Oevaln @ D @ E @ C @ B )
       => ( ( c_Natural_Oevaln @ ( c_Com_Ocom_OWhile @ F @ D ) @ B @ C @ A )
         => ( c_Natural_Oevaln @ ( c_Com_Ocom_OWhile @ F @ D ) @ E @ C @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2352]) ).

thf(1554,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i,I: $i,J: $i,K: $i] :
      ( ( hAPP @ ( c_Product__Type_Omap__pair @ K @ J @ I @ H @ G @ F ) @ ( hAPP @ ( c_Product__Type_Omap__pair @ E @ K @ D @ I @ C @ B ) @ A ) )
      = ( hAPP @ ( c_Product__Type_Omap__pair @ E @ J @ D @ H @ ( hAPP @ ( c_Fun_Ocomp @ K @ J @ E @ G ) @ C ) @ ( hAPP @ ( c_Fun_Ocomp @ I @ H @ D @ F ) @ B ) ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_map__pair_Ocompositionality) ).

thf(8420,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i,I: $i,J: $i,K: $i] :
      ( ( hAPP @ ( c_Product__Type_Omap__pair @ K @ J @ I @ H @ G @ F ) @ ( hAPP @ ( c_Product__Type_Omap__pair @ E @ K @ D @ I @ C @ B ) @ A ) )
      = ( hAPP @ ( c_Product__Type_Omap__pair @ E @ J @ D @ H @ ( hAPP @ ( c_Fun_Ocomp @ K @ J @ E @ G ) @ C ) @ ( hAPP @ ( c_Fun_Ocomp @ I @ H @ D @ F ) @ B ) ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1554]) ).

thf(1310,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Equiv__Relations_Oequiv @ C @ B @ A )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ C @ C ) @ B ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ C @ tc_HOL_Obool ) @ C ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_equiv__type) ).

thf(7399,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Equiv__Relations_Oequiv @ C @ B @ A )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ C @ C ) @ B ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ C @ tc_HOL_Obool ) @ C ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1310]) ).

thf(412,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ C ) @ ( c_Transitive__Closure_Ortrancl @ D @ B ) ) )
        | ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ C ) @ ( c_Transitive__Closure_Ortrancl @ D @ A ) ) ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ C ) @ ( c_Transitive__Closure_Ortrancl @ D @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) ) @ B ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_in__rtrancl__UnI) ).

thf(3896,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ C ) @ ( c_Transitive__Closure_Ortrancl @ D @ B ) ) )
        | ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ C ) @ ( c_Transitive__Closure_Ortrancl @ D @ A ) ) ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ C ) @ ( c_Transitive__Closure_Ortrancl @ D @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) ) @ B ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[412]) ).

thf(551,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ B ) )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ A ) @ B )
        = ( hAPP @ ( hAPP @ ( c_Finite__Set_Ofold @ C @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( c_COMBC @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ C @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( c_Set_Oinsert @ C ) ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) ) ) @ A ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_minus__fold__remove) ).

thf(4387,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ B ) )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ A ) @ B )
        = ( hAPP @ ( hAPP @ ( c_Finite__Set_Ofold @ C @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( c_COMBC @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ C @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( c_Set_Oinsert @ C ) ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) ) ) @ A ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[551]) ).

thf(2299,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ D ) @ ( c_Map_Odom @ E @ C @ B ) ) )
     => ( ( hAPP @ ( c_Map_Omap__add @ E @ C @ A @ B ) @ D )
        = ( hAPP @ A @ D ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_map__add__dom__app__simps_I3_J) ).

thf(11405,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ D ) @ ( c_Map_Odom @ E @ C @ B ) ) )
     => ( ( hAPP @ ( c_Map_Omap__add @ E @ C @ A @ B ) @ D )
        = ( hAPP @ A @ D ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2299]) ).

thf(51,axiom,
    class_Groups_Ocomm__monoid__mult @ tc_Nat_Onat,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',arity_Nat__Onat__Groups_Ocomm__monoid__mult) ).

thf(2576,plain,
    class_Groups_Ocomm__monoid__mult @ tc_Nat_Onat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[51]) ).

thf(576,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ D @ C ) )
     => ( ( hBOOL @ ( hAPP @ B @ C ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ A @ tc_HOL_Obool ) ) @ D ) @ B ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_inf1I) ).

thf(4478,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ D @ C ) )
     => ( ( hBOOL @ ( hAPP @ B @ C ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ A @ tc_HOL_Obool ) ) @ D ) @ B ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[576]) ).

thf(1430,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) )
     => ( ( c_Divides_Odiv__class_Omod @ tc_Nat_Onat @ B @ A )
        = B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mod__less) ).

thf(7887,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) )
     => ( ( c_Divides_Odiv__class_Omod @ tc_Nat_Onat @ B @ A )
        = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1430]) ).

thf(385,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ E @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) @ A ) )
    <=> ? [F: $i] :
          ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ F ) @ C ) )
          & ( hBOOL @ ( hAPP @ ( hAPP @ B @ F ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_SUP1__iff) ).

thf(3793,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ E @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) @ A ) )
       => ? [F: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ F ) @ C ) )
            & ( hBOOL @ ( hAPP @ ( hAPP @ B @ F ) @ A ) ) ) )
      & ( ? [F: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ F ) @ C ) )
            & ( hBOOL @ ( hAPP @ ( hAPP @ B @ F ) @ A ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ E @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[385]) ).

thf(78,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ C ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ ( tc_List_Olist @ B ) ) ) @ ( c_List_Otranspose @ B @ A ) ) ) )
     => ( ( hAPP @ ( hAPP @ ( c_List_Onth @ ( tc_List_Olist @ B ) ) @ ( c_List_Otranspose @ B @ A ) ) @ C )
        = ( hAPP @ ( hAPP @ ( c_List_Omap @ ( tc_List_Olist @ B ) @ B ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ ( tc_List_Olist @ B ) @ tc_Nat_Onat @ B ) @ ( c_List_Onth @ B ) ) @ C ) ) @ ( hAPP @ ( c_List_Ofilter @ ( tc_List_Olist @ B ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_Nat_Onat @ tc_HOL_Obool @ ( tc_List_Olist @ B ) ) @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ C ) ) @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ B ) ) ) ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nth__transpose) ).

thf(2680,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ C ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ ( tc_List_Olist @ B ) ) ) @ ( c_List_Otranspose @ B @ A ) ) ) )
     => ( ( hAPP @ ( hAPP @ ( c_List_Onth @ ( tc_List_Olist @ B ) ) @ ( c_List_Otranspose @ B @ A ) ) @ C )
        = ( hAPP @ ( hAPP @ ( c_List_Omap @ ( tc_List_Olist @ B ) @ B ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ ( tc_List_Olist @ B ) @ tc_Nat_Onat @ B ) @ ( c_List_Onth @ B ) ) @ C ) ) @ ( hAPP @ ( c_List_Ofilter @ ( tc_List_Olist @ B ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_Nat_Onat @ tc_HOL_Obool @ ( tc_List_Olist @ B ) ) @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ C ) ) @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ B ) ) ) ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[78]) ).

thf(1221,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ B )
        = ( hAPP @ c_Nat_OSuc @ A ) )
    <=> ? [D: $i,E: $i] :
          ( ( B
            = ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ D ) @ E ) )
          & ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ D ) @ E ) )
          & ( ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ E )
            = A )
          & ( ( A
              = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
           => ( E
              = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_card__Suc__eq) ).

thf(7017,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ B )
          = ( hAPP @ c_Nat_OSuc @ A ) )
       => ? [D: $i,E: $i] :
            ( ( B
              = ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ D ) @ E ) )
            & ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ D ) @ E ) )
            & ( ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ E )
              = A )
            & ( ( A
                = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
             => ( E
                = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) ) ) )
      & ( ? [D: $i,E: $i] :
            ( ( B
              = ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ D ) @ E ) )
            & ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ D ) @ E ) )
            & ( ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ E )
              = A )
            & ( ( A
                = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
             => ( E
                = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) ) )
       => ( ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ B )
          = ( hAPP @ c_Nat_OSuc @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1221]) ).

thf(747,axiom,
    ! [A: $i,B: $i] :
      ( ( ( hAPP @ ( c_List_Ofoldl @ tc_Nat_Onat @ tc_Nat_Onat @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A )
        = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
    <=> ( ( B
          = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
        & ! [C: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Nat_Onat ) @ C ) @ ( hAPP @ ( c_List_Oset @ tc_Nat_Onat ) @ A ) ) )
           => ( C
              = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_sum__eq__0__conv) ).

thf(5072,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( hAPP @ ( c_List_Ofoldl @ tc_Nat_Onat @ tc_Nat_Onat @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A )
          = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
       => ( ( B
            = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
          & ! [C: $i] :
              ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Nat_Onat ) @ C ) @ ( hAPP @ ( c_List_Oset @ tc_Nat_Onat ) @ A ) ) )
             => ( C
                = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) ) )
      & ( ( ( B
            = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
          & ! [C: $i] :
              ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Nat_Onat ) @ C ) @ ( hAPP @ ( c_List_Oset @ tc_Nat_Onat ) @ A ) ) )
             => ( C
                = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) )
       => ( ( hAPP @ ( c_List_Ofoldl @ tc_Nat_Onat @ tc_Nat_Onat @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A )
          = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[747]) ).

thf(264,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Int_Oint ) @ ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ C @ B ) ) @ A ) @ B )
      = ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Int_Oint ) @ C ) @ A ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zdiff__zmod__left) ).

thf(3351,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Int_Oint ) @ ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ C @ B ) ) @ A ) @ B )
      = ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Int_Oint ) @ C ) @ A ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[264]) ).

thf(2359,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Com_Ocom_OWhile @ E @ D )
     != ( c_Com_Ocom_OCall @ C @ B @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_com_Osimps_I60_J) ).

thf(11685,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Com_Ocom_OWhile @ E @ D )
     != ( c_Com_Ocom_OCall @ C @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2359]) ).

thf(139,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Otake @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_List_Oappend @ D ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_List_Oappend @ D ) @ ( hAPP @ ( hAPP @ ( c_List_Otake @ D ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_List_Otake @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ C ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ D ) ) @ B ) ) ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_take__append) ).

thf(2902,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Otake @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_List_Oappend @ D ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_List_Oappend @ D ) @ ( hAPP @ ( hAPP @ ( c_List_Otake @ D ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_List_Otake @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ C ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ D ) ) @ B ) ) ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[139]) ).

thf(1332,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( c_Relation_ORange @ B @ B ) @ ( c_Relation_OId__on @ B @ A ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Range__Id__on) ).

thf(7520,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( c_Relation_ORange @ B @ B ) @ ( c_Relation_OId__on @ B @ A ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1332]) ).

thf(5590,plain,
    ( ( c_Orderings_Obot__class_Obot @ tc_Nat_Onat )
    = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ),
    inference(lifteq,[status(thm)],[5589]) ).

thf(1732,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ A ) ) )
    <=> ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ C ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__le__cancel1) ).

thf(9211,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ A ) ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ C ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ A ) ) ) )
      & ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ C ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ A ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1732]) ).

thf(2180,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Com_Ocom_OLocal @ E @ D @ C )
     != ( c_Com_Ocom_OSemi @ B @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_com_Osimps_I34_J) ).

thf(10924,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Com_Ocom_OLocal @ E @ D @ C )
     != ( c_Com_Ocom_OSemi @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2180]) ).

thf(347,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ E @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ E @ ( tc_fun @ D @ tc_HOL_Obool ) @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ D @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ D @ tc_HOL_Obool ) @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ E ) @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) @ B ) ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ E @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ E @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_UN__Un__distrib) ).

thf(3645,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ E @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ E @ ( tc_fun @ D @ tc_HOL_Obool ) @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ D @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ D @ tc_HOL_Obool ) @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ E ) @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) @ B ) ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ E @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ E @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[347]) ).

thf(1511,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( c_Fun_Ocomp @ ( tc_List_Olist @ E ) @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ C ) @ ( hAPP @ ( c_List_Omap @ E @ D ) @ B ) ) @ ( hAPP @ ( c_List_Omap @ C @ E ) @ A ) )
      = ( hAPP @ ( c_List_Omap @ C @ D ) @ ( hAPP @ ( c_Fun_Ocomp @ E @ D @ C @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_List_Omap_Ocomp) ).

thf(8267,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( c_Fun_Ocomp @ ( tc_List_Olist @ E ) @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ C ) @ ( hAPP @ ( c_List_Omap @ E @ D ) @ B ) ) @ ( hAPP @ ( c_List_Omap @ C @ E ) @ A ) )
      = ( hAPP @ ( c_List_Omap @ C @ D ) @ ( hAPP @ ( c_Fun_Ocomp @ E @ D @ C @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1511]) ).

thf(1345,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Finite__Set_Ofolding__one @ E @ D @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ E ) @ B ) )
       => ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ A ) @ B ) )
         => ( ( B
             != ( c_Orderings_Obot__class_Obot @ ( tc_fun @ E @ tc_HOL_Obool ) ) )
           => ( ( hAPP @ C @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ A ) @ B ) )
              = ( hAPP @ ( hAPP @ D @ A ) @ ( hAPP @ C @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_folding__one_Oinsert) ).

thf(7574,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Finite__Set_Ofolding__one @ E @ D @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ E ) @ B ) )
       => ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ A ) @ B ) )
         => ( ( B
             != ( c_Orderings_Obot__class_Obot @ ( tc_fun @ E @ tc_HOL_Obool ) ) )
           => ( ( hAPP @ C @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ A ) @ B ) )
              = ( hAPP @ ( hAPP @ D @ A ) @ ( hAPP @ C @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1345]) ).

thf(910,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( c_List_Ofilter @ D @ C ) @ ( hAPP @ ( c_List_Ofilter @ D @ B ) @ A ) )
      = ( hAPP @ ( c_List_Ofilter @ D @ ( hAPP @ ( hAPP @ ( c_COMBS @ D @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ D ) @ c_fconj ) @ B ) ) @ C ) ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_filter__filter) ).

thf(5672,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( c_List_Ofilter @ D @ C ) @ ( hAPP @ ( c_List_Ofilter @ D @ B ) @ A ) )
      = ( hAPP @ ( c_List_Ofilter @ D @ ( hAPP @ ( hAPP @ ( c_COMBS @ D @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ D ) @ c_fconj ) @ B ) ) @ C ) ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[910]) ).

thf(1768,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Com_Ocom_OSemi @ E @ D )
     != ( c_Com_Ocom_OCond @ C @ B @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_com_Osimps_I44_J) ).

thf(9350,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Com_Ocom_OSemi @ E @ D )
     != ( c_Com_Ocom_OCond @ C @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1768]) ).

thf(1901,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ B ) ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_add__leD2) ).

thf(9887,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ B ) ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1901]) ).

thf(27,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Divides_Odivmod__int__rel @ D @ C ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ tc_Int_Oint @ tc_Int_Oint ) @ B ) @ A ) ) )
     => ( ( C
         != ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) )
       => ( ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ D @ C )
          = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_divmod__int__rel__mod) ).

thf(2494,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Divides_Odivmod__int__rel @ D @ C ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ tc_Int_Oint @ tc_Int_Oint ) @ B ) @ A ) ) )
     => ( ( C
         != ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) )
       => ( ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ D @ C )
          = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[27]) ).

thf(1181,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ C ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) )
       => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ tc_Nat_Onat ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ C ) )
          = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ tc_Nat_Onat ) @ A ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ tc_Nat_Onat ) @ A ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_setsum__diff__nat) ).

thf(6871,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ C ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) )
       => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ tc_Nat_Onat ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ C ) )
          = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ tc_Nat_Onat ) @ A ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ tc_Nat_Onat ) @ A ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1181]) ).

thf(2326,axiom,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ ( c_List_Odistinct @ B ) @ ( c_List_Oremdups @ B @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_distinct__remdups) ).

thf(11537,plain,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ ( c_List_Odistinct @ B ) @ ( c_List_Oremdups @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2326]) ).

thf(2292,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ B )
      | ( hBOOL @ ( hAPP @ ( hAPP @ c_fimplies @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_c__fimplies__1) ).

thf(11381,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ B )
      | ( hBOOL @ ( hAPP @ ( hAPP @ c_fimplies @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2292]) ).

thf(63,axiom,
    ! [A: $i] :
      ( ( c_List_Otranspose @ A @ ( c_List_Olist_ONil @ ( tc_List_Olist @ A ) ) )
      = ( c_List_Olist_ONil @ ( tc_List_Olist @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_transpose_Osimps_I1_J) ).

thf(2623,plain,
    ! [A: $i] :
      ( ( c_List_Otranspose @ A @ ( c_List_Olist_ONil @ ( tc_List_Olist @ A ) ) )
      = ( c_List_Olist_ONil @ ( tc_List_Olist @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[63]) ).

thf(2018,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Com_Ocom_OSemi @ E @ D )
     != ( c_Com_Ocom_OCall @ C @ B @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_com_Osimps_I50_J) ).

thf(10357,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Com_Ocom_OSemi @ E @ D )
     != ( c_Com_Ocom_OCall @ C @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2018]) ).

thf(1406,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ A ) )
     => ( ( c_List_Oupt @ B @ ( hAPP @ c_Nat_OSuc @ A ) )
        = ( hAPP @ ( hAPP @ ( c_List_Oappend @ tc_Nat_Onat ) @ ( c_List_Oupt @ B @ A ) ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ tc_Nat_Onat ) @ A ) @ ( c_List_Olist_ONil @ tc_Nat_Onat ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_upt__Suc__append) ).

thf(7804,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ A ) )
     => ( ( c_List_Oupt @ B @ ( hAPP @ c_Nat_OSuc @ A ) )
        = ( hAPP @ ( hAPP @ ( c_List_Oappend @ tc_Nat_Onat ) @ ( c_List_Oupt @ B @ A ) ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ tc_Nat_Onat ) @ A ) @ ( c_List_Olist_ONil @ tc_Nat_Onat ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1406]) ).

thf(687,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Otake @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ D ) @ B ) @ A ) )
      = ( c_Nat_Onat_Onat__case @ ( tc_List_Olist @ D ) @ ( c_List_Olist_ONil @ D ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) @ tc_Nat_Onat ) @ ( hAPP @ ( c_List_Olist_OCons @ D ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) @ ( c_List_Otake @ D ) ) @ A ) ) @ C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_take__Cons) ).

thf(4869,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Otake @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ D ) @ B ) @ A ) )
      = ( c_Nat_Onat_Onat__case @ ( tc_List_Olist @ D ) @ ( c_List_Olist_ONil @ D ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) @ tc_Nat_Onat ) @ ( hAPP @ ( c_List_Olist_OCons @ D ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) @ ( c_List_Otake @ D ) ) @ A ) ) @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[687]) ).

thf(1093,axiom,
    c_Wellfounded_Owf @ tc_Nat_Onat @ c_Wellfounded_Opred__nat,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_wf__pred__nat) ).

thf(6499,plain,
    c_Wellfounded_Owf @ tc_Nat_Onat @ c_Wellfounded_Opred__nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1093]) ).

thf(1250,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( c_Finite__Set_Ocard @ ( tc_prod @ D @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ D @ C ) @ B ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ C @ tc_HOL_Obool ) @ D ) @ A ) ) )
      = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( hAPP @ ( c_Finite__Set_Ocard @ D ) @ B ) ) @ ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_card__cartesian__product) ).

thf(7135,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( c_Finite__Set_Ocard @ ( tc_prod @ D @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ D @ C ) @ B ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ C @ tc_HOL_Obool ) @ D ) @ A ) ) )
      = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( hAPP @ ( c_Finite__Set_Ocard @ D ) @ B ) ) @ ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1250]) ).

thf(813,axiom,
    ! [A: $i,B: $i] :
      ( ( ( c_Orderings_Obot__class_Obot @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) )
        = ( c_SetInterval_Oord_OatLeastLessThan @ tc_Nat_Onat @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ tc_Nat_Onat @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ tc_Nat_Onat @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) @ tc_Nat_Onat ) @ ( c_COMBS @ tc_Nat_Onat @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ tc_Nat_Onat @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ tc_Nat_Onat ) @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ tc_Nat_Onat ) @ c_fconj ) ) @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ tc_Nat_Onat ) @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ tc_HOL_Obool @ tc_Nat_Onat ) @ c_fNot ) ) @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) ) ) ) @ B @ A ) )
    <=> ~ ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) )
          & ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ A ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_dvd_OatLeastLessThan__empty__iff2) ).

thf(5336,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( c_Orderings_Obot__class_Obot @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) )
          = ( c_SetInterval_Oord_OatLeastLessThan @ tc_Nat_Onat @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ tc_Nat_Onat @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ tc_Nat_Onat @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) @ tc_Nat_Onat ) @ ( c_COMBS @ tc_Nat_Onat @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ tc_Nat_Onat @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ tc_Nat_Onat ) @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ tc_Nat_Onat ) @ c_fconj ) ) @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ tc_Nat_Onat ) @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ tc_HOL_Obool @ tc_Nat_Onat ) @ c_fNot ) ) @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) ) ) ) @ B @ A ) )
       => ~ ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) )
            & ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ A ) @ B ) ) ) )
      & ( ~ ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) )
            & ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ A ) @ B ) ) )
       => ( ( c_Orderings_Obot__class_Obot @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) )
          = ( c_SetInterval_Oord_OatLeastLessThan @ tc_Nat_Onat @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ tc_Nat_Onat @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ tc_Nat_Onat @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) @ tc_Nat_Onat ) @ ( c_COMBS @ tc_Nat_Onat @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ tc_Nat_Onat @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ tc_Nat_Onat ) @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ tc_Nat_Onat ) @ c_fconj ) ) @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ tc_Nat_Onat ) @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ tc_HOL_Obool @ tc_Nat_Onat ) @ c_fNot ) ) @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) ) ) ) @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[813]) ).

thf(106,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hAPP @ ( c_Set_Oimage @ F @ E @ ( hAPP @ ( hAPP @ ( c_COMBS @ F @ E @ E ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ F @ E @ ( tc_fun @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ E @ ( tc_fun @ E @ E ) ) @ F ) @ ( c_If @ E ) ) @ D ) ) @ C ) ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Set_Oimage @ F @ E @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ F @ tc_HOL_Obool ) ) @ A ) @ ( hAPP @ ( c_Set_OCollect @ F ) @ D ) ) ) ) @ ( hAPP @ ( c_Set_Oimage @ F @ E @ B ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ F @ tc_HOL_Obool ) ) @ A ) @ ( hAPP @ ( c_Set_OCollect @ F ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ tc_HOL_Obool @ F ) @ c_fNot ) @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_if__image__distrib) ).

thf(2779,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hAPP @ ( c_Set_Oimage @ F @ E @ ( hAPP @ ( hAPP @ ( c_COMBS @ F @ E @ E ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ F @ E @ ( tc_fun @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ E @ ( tc_fun @ E @ E ) ) @ F ) @ ( c_If @ E ) ) @ D ) ) @ C ) ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Set_Oimage @ F @ E @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ F @ tc_HOL_Obool ) ) @ A ) @ ( hAPP @ ( c_Set_OCollect @ F ) @ D ) ) ) ) @ ( hAPP @ ( c_Set_Oimage @ F @ E @ B ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ F @ tc_HOL_Obool ) ) @ A ) @ ( hAPP @ ( c_Set_OCollect @ F ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ tc_HOL_Obool @ F ) @ c_fNot ) @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[106]) ).

thf(617,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( c_Predicate_Opred__comp @ G @ F @ E @ ( hAPP @ ( hAPP @ ( c_COMBC @ G @ ( tc_fun @ ( tc_prod @ G @ F ) @ tc_HOL_Obool ) @ ( tc_fun @ F @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ F @ ( tc_fun @ ( tc_fun @ ( tc_prod @ G @ F ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ G @ F ) @ tc_HOL_Obool ) @ ( tc_fun @ F @ tc_HOL_Obool ) ) @ G ) @ ( c_COMBC @ F @ ( tc_fun @ ( tc_prod @ G @ F ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ F @ ( tc_prod @ G @ F ) ) @ ( tc_fun @ F @ ( tc_fun @ ( tc_fun @ ( tc_prod @ G @ F ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ G ) @ ( hAPP @ ( c_COMBB @ ( tc_prod @ G @ F ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ G @ F ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ F ) @ ( c_member @ ( tc_prod @ G @ F ) ) ) ) @ ( c_Product__Type_OPair @ G @ F ) ) ) ) @ D ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ F @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ E @ ( tc_fun @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ F ) @ ( c_COMBC @ E @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ E @ ( tc_prod @ F @ E ) ) @ ( tc_fun @ E @ ( tc_fun @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ F ) @ ( hAPP @ ( c_COMBB @ ( tc_prod @ F @ E ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ E ) @ ( c_member @ ( tc_prod @ F @ E ) ) ) ) @ ( c_Product__Type_OPair @ F @ E ) ) ) ) @ C ) @ B @ A )
    <=> ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ G @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ G @ E ) @ B ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ G @ F @ E ) @ D ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_pred__comp__rel__comp__eq) ).

thf(4619,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( ( c_Predicate_Opred__comp @ G @ F @ E @ ( hAPP @ ( hAPP @ ( c_COMBC @ G @ ( tc_fun @ ( tc_prod @ G @ F ) @ tc_HOL_Obool ) @ ( tc_fun @ F @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ F @ ( tc_fun @ ( tc_fun @ ( tc_prod @ G @ F ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ G @ F ) @ tc_HOL_Obool ) @ ( tc_fun @ F @ tc_HOL_Obool ) ) @ G ) @ ( c_COMBC @ F @ ( tc_fun @ ( tc_prod @ G @ F ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ F @ ( tc_prod @ G @ F ) ) @ ( tc_fun @ F @ ( tc_fun @ ( tc_fun @ ( tc_prod @ G @ F ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ G ) @ ( hAPP @ ( c_COMBB @ ( tc_prod @ G @ F ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ G @ F ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ F ) @ ( c_member @ ( tc_prod @ G @ F ) ) ) ) @ ( c_Product__Type_OPair @ G @ F ) ) ) ) @ D ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ F @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ E @ ( tc_fun @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ F ) @ ( c_COMBC @ E @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ E @ ( tc_prod @ F @ E ) ) @ ( tc_fun @ E @ ( tc_fun @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ F ) @ ( hAPP @ ( c_COMBB @ ( tc_prod @ F @ E ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ E ) @ ( c_member @ ( tc_prod @ F @ E ) ) ) ) @ ( c_Product__Type_OPair @ F @ E ) ) ) ) @ C ) @ B @ A )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ G @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ G @ E ) @ B ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ G @ F @ E ) @ D ) @ C ) ) ) )
      & ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ G @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ G @ E ) @ B ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ G @ F @ E ) @ D ) @ C ) ) )
       => ( c_Predicate_Opred__comp @ G @ F @ E @ ( hAPP @ ( hAPP @ ( c_COMBC @ G @ ( tc_fun @ ( tc_prod @ G @ F ) @ tc_HOL_Obool ) @ ( tc_fun @ F @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ F @ ( tc_fun @ ( tc_fun @ ( tc_prod @ G @ F ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ G @ F ) @ tc_HOL_Obool ) @ ( tc_fun @ F @ tc_HOL_Obool ) ) @ G ) @ ( c_COMBC @ F @ ( tc_fun @ ( tc_prod @ G @ F ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ F @ ( tc_prod @ G @ F ) ) @ ( tc_fun @ F @ ( tc_fun @ ( tc_fun @ ( tc_prod @ G @ F ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ G ) @ ( hAPP @ ( c_COMBB @ ( tc_prod @ G @ F ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ G @ F ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ F ) @ ( c_member @ ( tc_prod @ G @ F ) ) ) ) @ ( c_Product__Type_OPair @ G @ F ) ) ) ) @ D ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ F @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ E @ ( tc_fun @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ F ) @ ( c_COMBC @ E @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ E @ ( tc_prod @ F @ E ) ) @ ( tc_fun @ E @ ( tc_fun @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ F ) @ ( hAPP @ ( c_COMBB @ ( tc_prod @ F @ E ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ E ) @ ( c_member @ ( tc_prod @ F @ E ) ) ) ) @ ( c_Product__Type_OPair @ F @ E ) ) ) ) @ C ) @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[617]) ).

thf(2249,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D = C )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ B ) @ D ) @ A ) )
      <=> ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ B ) @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_eqelem__imp__iff) ).

thf(11208,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D = C )
     => ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ B ) @ D ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ B ) @ C ) @ A ) ) )
        & ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ B ) @ C ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ B ) @ D ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2249]) ).

thf(1759,axiom,
    ! [A: $i,B: $i,C: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ C @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ C @ C ) @ B ) @ B ) ) @ ( c_Transitive__Closure_Ortrancl @ C @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_rtrancl_Ortrancl__refl) ).

thf(9321,plain,
    ! [A: $i,B: $i,C: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ C @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ C @ C ) @ B ) @ B ) ) @ ( c_Transitive__Closure_Ortrancl @ C @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1759]) ).

thf(775,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ C ) )
     => ( ! [D: $i] :
            ( ( hBOOL @ ( hAPP @ B @ D ) )
           => ( hBOOL @ ( hAPP @ B @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Int_Oint ) @ D ) @ C ) ) ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ A ) )
         => ! [D: $i] :
              ( ( hBOOL @ ( hAPP @ B @ D ) )
             => ( hBOOL @ ( hAPP @ B @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Int_Oint ) @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ A ) @ C ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_decr__mult__lemma) ).

thf(5187,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ C ) )
     => ( ! [D: $i] :
            ( ( hBOOL @ ( hAPP @ B @ D ) )
           => ( hBOOL @ ( hAPP @ B @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Int_Oint ) @ D ) @ C ) ) ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ A ) )
         => ! [D: $i] :
              ( ( hBOOL @ ( hAPP @ B @ D ) )
             => ( hBOOL @ ( hAPP @ B @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Int_Oint ) @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ A ) @ C ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[775]) ).

thf(2148,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_Relation_Orefl__on @ D @ C @ B )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ A ) @ C ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ D ) @ A ) @ A ) ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_refl__onD) ).

thf(10838,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_Relation_Orefl__on @ D @ C @ B )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ A ) @ C ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ D ) @ A ) @ A ) ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2148]) ).

thf(2334,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ B ) @ ( hAPP @ c_fequal @ A ) ) )
    <=> ( B = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_eq__mem) ).

thf(11580,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ B ) @ ( hAPP @ c_fequal @ A ) ) )
       => ( B = A ) )
      & ( ( B = A )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ B ) @ ( hAPP @ c_fequal @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2334]) ).

thf(1302,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ B ) )
       => ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ C ) @ A ) ) @ B )
          = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ A ) @ B ) ) )
      & ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ B ) )
       => ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ C ) @ A ) ) @ B )
          = ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ A ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_insert__Diff__if) ).

thf(7355,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ B ) )
       => ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ C ) @ A ) ) @ B )
          = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ A ) @ B ) ) )
      & ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ B ) )
       => ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ C ) @ A ) ) @ B )
          = ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ A ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1302]) ).

thf(1814,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ F @ E ) @ D ) @ C )
        = ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ F @ E ) @ B ) @ A ) )
     => ~ ( ( D = B )
         => ( C != A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Pair__inject) ).

thf(9523,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ F @ E ) @ D ) @ C )
        = ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ F @ E ) @ B ) @ A ) )
     => ~ ( ( D = B )
         => ( C != A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1814]) ).

thf(681,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hAPP @ ( hAPP @ ( hAPP @ ( c_COMBC @ F @ E @ D ) @ C ) @ B ) @ A )
      = ( hAPP @ ( hAPP @ C @ A ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_c__COMBC__1) ).

thf(4841,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hAPP @ ( hAPP @ ( hAPP @ ( c_COMBC @ F @ E @ D ) @ C ) @ B ) @ A )
      = ( hAPP @ ( hAPP @ C @ A ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[681]) ).

thf(2280,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ B ) )
       => ( C = A ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ A ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_insertCI) ).

thf(11341,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ B ) )
       => ( C = A ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ A ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2280]) ).

thf(546,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Int_Oint ) @ ( hAPP @ ( c_Int_Onumber__class_Onumber__of @ tc_Int_Oint ) @ B ) ) @ ( hAPP @ ( c_Int_Onumber__class_Onumber__of @ tc_Int_Oint ) @ A ) ) )
    <=> ( ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ ( hAPP @ ( c_Int_Onumber__class_Onumber__of @ tc_Int_Oint ) @ A ) @ ( hAPP @ ( c_Int_Onumber__class_Onumber__of @ tc_Int_Oint ) @ B ) )
        = ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zdvd__iff__zmod__eq__0__number__of) ).

thf(4367,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Int_Oint ) @ ( hAPP @ ( c_Int_Onumber__class_Onumber__of @ tc_Int_Oint ) @ B ) ) @ ( hAPP @ ( c_Int_Onumber__class_Onumber__of @ tc_Int_Oint ) @ A ) ) )
       => ( ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ ( hAPP @ ( c_Int_Onumber__class_Onumber__of @ tc_Int_Oint ) @ A ) @ ( hAPP @ ( c_Int_Onumber__class_Onumber__of @ tc_Int_Oint ) @ B ) )
          = ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) )
      & ( ( ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ ( hAPP @ ( c_Int_Onumber__class_Onumber__of @ tc_Int_Oint ) @ A ) @ ( hAPP @ ( c_Int_Onumber__class_Onumber__of @ tc_Int_Oint ) @ B ) )
          = ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Int_Oint ) @ ( hAPP @ ( c_Int_Onumber__class_Onumber__of @ tc_Int_Oint ) @ B ) ) @ ( hAPP @ ( c_Int_Onumber__class_Onumber__of @ tc_Int_Oint ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[546]) ).

thf(441,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Map_Omap__comp @ E @ D @ C @ ( hAPP @ ( c_COMBK @ ( tc_Option_Ooption @ D ) @ E ) @ ( c_Option_Ooption_ONone @ D ) ) @ B @ A )
      = ( c_Option_Ooption_ONone @ D ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_map__comp__empty_I2_J) ).

thf(3992,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Map_Omap__comp @ E @ D @ C @ ( hAPP @ ( c_COMBK @ ( tc_Option_Ooption @ D ) @ E ) @ ( c_Option_Ooption_ONone @ D ) ) @ B @ A )
      = ( c_Option_Ooption_ONone @ D ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[441]) ).

thf(2373,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ B ) @ ( hAPP @ ( c_List_Oset @ C ) @ A ) ) )
     => ( ( c_List_Oremove1 @ C @ B @ A )
        = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_remove1__idem) ).

thf(11744,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ B ) @ ( hAPP @ ( c_List_Oset @ C ) @ A ) ) )
     => ( ( c_List_Oremove1 @ C @ B @ A )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2373]) ).

thf(274,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ B ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ A ) @ B ) )
       => ( ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ B ) @ A )
          = ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_div__neg__neg__trivial) ).

thf(3379,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ B ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ A ) @ B ) )
       => ( ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ B ) @ A )
          = ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[274]) ).

thf(1367,axiom,
    ! [A: $i,B: $i] :
      ( ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A )
        = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
    <=> ( ( B
          = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
        & ( A
          = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_add__is__0) ).

thf(7649,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A )
          = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
       => ( ( B
            = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
          & ( A
            = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) )
      & ( ( ( B
            = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
          & ( A
            = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) )
       => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A )
          = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1367]) ).

thf(818,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) @ ( c_List_Osublist @ C @ B @ A ) )
      = ( hAPP @ ( c_Finite__Set_Ocard @ tc_Nat_Onat ) @ ( hAPP @ ( c_Set_OCollect @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ tc_Nat_Onat @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ tc_Nat_Onat ) @ c_fconj ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) @ B ) ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( c_member @ tc_Nat_Onat ) ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_length__sublist) ).

thf(5357,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) @ ( c_List_Osublist @ C @ B @ A ) )
      = ( hAPP @ ( c_Finite__Set_Ocard @ tc_Nat_Onat ) @ ( hAPP @ ( c_Set_OCollect @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ tc_Nat_Onat @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ tc_Nat_Onat ) @ c_fconj ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) @ B ) ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( c_member @ tc_Nat_Onat ) ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[818]) ).

thf(1234,axiom,
    ! [A: $i,B: $i] :
      ( ( ( hAPP @ ( c_Finite__Set_Ocard @ B ) @ A )
        = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
    <=> ( ( A
          = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) )
        | ~ ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_card__eq__0__iff) ).

thf(7081,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( hAPP @ ( c_Finite__Set_Ocard @ B ) @ A )
          = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
       => ( ( A
            = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) )
          | ~ ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ B ) @ A ) ) ) )
      & ( ( ( A
            = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) )
          | ~ ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ B ) @ A ) ) )
       => ( ( hAPP @ ( c_Finite__Set_Ocard @ B ) @ A )
          = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1234]) ).

thf(2073,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ C ) ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__trans__Suc) ).

thf(10569,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ c_Nat_OSuc @ C ) ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2073]) ).

thf(1823,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ c_Com_OWT @ ( hAPP @ c_Com_Ocom_OBODY @ C ) ) )
     => ( hBOOL @ ( hAPP @ c_Com_OWT @ ( c_Com_Ocom_OCall @ B @ C @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_WT_OCall) ).

thf(9551,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ c_Com_OWT @ ( hAPP @ c_Com_Ocom_OBODY @ C ) ) )
     => ( hBOOL @ ( hAPP @ c_Com_OWT @ ( c_Com_Ocom_OCall @ B @ C @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1823]) ).

thf(702,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ C ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ B ) ) @ A ) ) )
     => ( ( hAPP @ ( hAPP @ ( c_List_Otake @ B ) @ ( hAPP @ c_Nat_OSuc @ C ) ) @ A )
        = ( hAPP @ ( hAPP @ ( c_List_Oappend @ B ) @ ( hAPP @ ( hAPP @ ( c_List_Otake @ B ) @ C ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ B ) @ ( hAPP @ ( hAPP @ ( c_List_Onth @ B ) @ A ) @ C ) ) @ ( c_List_Olist_ONil @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_take__Suc__conv__app__nth) ).

thf(4917,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ C ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ B ) ) @ A ) ) )
     => ( ( hAPP @ ( hAPP @ ( c_List_Otake @ B ) @ ( hAPP @ c_Nat_OSuc @ C ) ) @ A )
        = ( hAPP @ ( hAPP @ ( c_List_Oappend @ B ) @ ( hAPP @ ( hAPP @ ( c_List_Otake @ B ) @ C ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ B ) @ ( hAPP @ ( hAPP @ ( c_List_Onth @ B ) @ A ) @ C ) ) @ ( c_List_Olist_ONil @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[702]) ).

thf(79,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Divides_Odivmod__int__rel @ F @ E ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ tc_Int_Oint @ tc_Int_Oint ) @ D ) @ C ) ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Divides_Odivmod__int__rel @ F @ E ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ tc_Int_Oint @ tc_Int_Oint ) @ B ) @ A ) ) )
       => ( ( E
           != ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) )
         => ( C = A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_unique__remainder) ).

thf(2683,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Divides_Odivmod__int__rel @ F @ E ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ tc_Int_Oint @ tc_Int_Oint ) @ D ) @ C ) ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Divides_Odivmod__int__rel @ F @ E ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ tc_Int_Oint @ tc_Int_Oint ) @ B ) @ A ) ) )
       => ( ( E
           != ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) )
         => ( C = A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[79]) ).

thf(1500,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_Fun_Oinj__on @ D @ C @ B @ ( c_Complete__Lattice_OSup__class_OSup @ ( tc_fun @ D @ tc_HOL_Obool ) @ ( hAPP @ ( c_Set_Oimage @ ( tc_List_Olist @ D ) @ ( tc_fun @ D @ tc_HOL_Obool ) @ ( c_List_Oset @ D ) ) @ A ) ) )
     => ( c_Fun_Oinj__on @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ C ) @ ( hAPP @ ( c_List_Omap @ D @ C ) @ B ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_inj__on__mapI) ).

thf(8209,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_Fun_Oinj__on @ D @ C @ B @ ( c_Complete__Lattice_OSup__class_OSup @ ( tc_fun @ D @ tc_HOL_Obool ) @ ( hAPP @ ( c_Set_Oimage @ ( tc_List_Olist @ D ) @ ( tc_fun @ D @ tc_HOL_Obool ) @ ( c_List_Oset @ D ) ) @ A ) ) )
     => ( c_Fun_Oinj__on @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ C ) @ ( hAPP @ ( c_List_Omap @ D @ C ) @ B ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1500]) ).

thf(1460,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ C ) @ B ) )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ A ) @ ( hAPP @ c_Nat_OSuc @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ B ) @ C ) ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ C ) ) @ ( hAPP @ c_Nat_OSuc @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_diff__Suc__diff__eq1) ).

thf(7991,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ C ) @ B ) )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ A ) @ ( hAPP @ c_Nat_OSuc @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ B ) @ C ) ) )
        = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ C ) ) @ ( hAPP @ c_Nat_OSuc @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1460]) ).

thf(1477,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( C
          = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
       => ( ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ A ) )
          = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) )
      & ( ( C
         != ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
       => ( ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ A ) )
          = ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Nat_Onat ) @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nat__mult__div__cancel__disj) ).

thf(8070,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( C
          = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
       => ( ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ A ) )
          = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) )
      & ( ( C
         != ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
       => ( ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ A ) )
          = ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Nat_Onat ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1477]) ).

thf(1206,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
      ( ( c_Finite__Set_Ofolding__image__simple__idem @ H @ G @ F @ E @ D @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ G ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ G @ tc_HOL_Obool ) ) @ A ) @ B ) )
         => ( ( hAPP @ ( hAPP @ F @ ( hAPP @ C @ A ) ) @ ( hAPP @ C @ B ) )
            = ( hAPP @ C @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_folding__image__simple__idem_Osubset__idem) ).

thf(6957,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
      ( ( c_Finite__Set_Ofolding__image__simple__idem @ H @ G @ F @ E @ D @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ G ) @ B ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ G @ tc_HOL_Obool ) ) @ A ) @ B ) )
         => ( ( hAPP @ ( hAPP @ F @ ( hAPP @ C @ A ) ) @ ( hAPP @ C @ B ) )
            = ( hAPP @ C @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1206]) ).

thf(1650,axiom,
    ! [A: $i] :
      ( ( hAPP @ c_Nat_OSuc @ A )
      = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Suc__eq__plus1__left) ).

thf(8886,plain,
    ! [A: $i] :
      ( ( hAPP @ c_Nat_OSuc @ A )
      = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1650]) ).

thf(628,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( c_Finite__Set_Ofolding__image__simple @ F @ E @ D @ C @ B @ A )
     => ( ( hAPP @ A @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ E @ tc_HOL_Obool ) ) )
        = C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_folding__image__simple_Oempty) ).

thf(4670,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( c_Finite__Set_Ofolding__image__simple @ F @ E @ D @ C @ B @ A )
     => ( ( hAPP @ A @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ E @ tc_HOL_Obool ) ) )
        = C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[628]) ).

thf(310,axiom,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ tc_Int_Oint ) @ ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Int_Oint @ B @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_finite__atLeastAtMost__int) ).

thf(3511,plain,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ tc_Int_Oint ) @ ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Int_Oint @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[310]) ).

thf(1362,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_psubset__trans) ).

thf(7634,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1362]) ).

thf(1423,axiom,
    ! [A: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Nat_Oadd__0__right) ).

thf(7868,plain,
    ! [A: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1423]) ).

thf(443,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ E @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ E @ ( tc_fun @ D @ tc_HOL_Obool ) @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ D @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ D @ tc_HOL_Obool ) @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ E ) @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) @ B ) ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ ( tc_prod @ E @ D ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ E @ D ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ E @ D ) @ C ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Sigma__Diff__distrib2) ).

thf(3997,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ E @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ E @ ( tc_fun @ D @ tc_HOL_Obool ) @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ D @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ D @ tc_HOL_Obool ) @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ E ) @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) @ B ) ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ ( tc_prod @ E @ D ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ E @ D ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ E @ D ) @ C ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[443]) ).

thf(1252,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Fun_Oinj__on @ E @ D @ C @ B )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ A ) @ B ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Set_Oimage @ E @ D @ C ) @ A ) ) @ ( hAPP @ ( c_Set_Oimage @ E @ D @ C ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_inj__on__strict__subset) ).

thf(7140,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Fun_Oinj__on @ E @ D @ C @ B )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ A ) @ B ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Set_Oimage @ E @ D @ C ) @ A ) ) @ ( hAPP @ ( c_Set_Oimage @ E @ D @ C ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1252]) ).

thf(1495,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ C ) @ A ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ B ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ A ) @ C ) ) )
        <=> ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__diff__iff) ).

thf(8193,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ C ) @ A ) )
       => ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ B ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ A ) @ C ) ) )
           => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) ) )
          & ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) )
           => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ B ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ A ) @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1495]) ).

thf(1331,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ C ) @ B ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ A @ B ) @ C ) ) )
     => ( ( ( C = B )
         => ( hBOOL @ ( hAPP @ ( hAPP @ A @ B ) @ C ) ) )
       => ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ C ) )
           => ( hBOOL @ ( hAPP @ ( hAPP @ A @ B ) @ C ) ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ A @ B ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nat__less__cases) ).

thf(7510,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ C ) @ B ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ A @ B ) @ C ) ) )
     => ( ( ( C = B )
         => ( hBOOL @ ( hAPP @ ( hAPP @ A @ B ) @ C ) ) )
       => ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ C ) )
           => ( hBOOL @ ( hAPP @ ( hAPP @ A @ B ) @ C ) ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ A @ B ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1331]) ).

thf(88,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Map_Oran @ C @ B @ A )
      = ( hAPP @ ( c_Set_OCollect @ B ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ tc_HOL_Obool ) @ tc_HOL_Obool @ B ) @ ( c_HOL_OEx @ C ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_Option_Ooption @ B ) @ ( tc_fun @ C @ tc_HOL_Obool ) @ B ) @ ( hAPP @ ( c_COMBC @ C @ ( tc_Option_Ooption @ B ) @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_Option_Ooption @ B ) @ ( tc_fun @ ( tc_Option_Ooption @ B ) @ tc_HOL_Obool ) @ C ) @ c_fequal ) @ A ) ) ) @ ( c_Option_Ooption_OSome @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_ran__def) ).

thf(2719,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Map_Oran @ C @ B @ A )
      = ( hAPP @ ( c_Set_OCollect @ B ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ tc_HOL_Obool ) @ tc_HOL_Obool @ B ) @ ( c_HOL_OEx @ C ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_Option_Ooption @ B ) @ ( tc_fun @ C @ tc_HOL_Obool ) @ B ) @ ( hAPP @ ( c_COMBC @ C @ ( tc_Option_Ooption @ B ) @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_Option_Ooption @ B ) @ ( tc_fun @ ( tc_Option_Ooption @ B ) @ tc_HOL_Obool ) @ C ) @ c_fequal ) @ A ) ) ) @ ( c_Option_Ooption_OSome @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[88]) ).

thf(372,axiom,
    ! [A: $i] : ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ A @ tc_HOL_Obool ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_finite_OemptyI) ).

thf(3733,plain,
    ! [A: $i] : ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ A @ tc_HOL_Obool ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[372]) ).

thf(2233,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ F @ E ) @ D ) )
     => ( ( c_List_Olistrelp @ C @ F @ B @ A )
       => ( c_List_Olistrelp @ C @ F @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ C ) @ E ) @ B ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ C ) @ D ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_listrelp_OCons) ).

thf(11140,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ F @ E ) @ D ) )
     => ( ( c_List_Olistrelp @ C @ F @ B @ A )
       => ( c_List_Olistrelp @ C @ F @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ C ) @ E ) @ B ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ C ) @ D ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2233]) ).

thf(53,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Ocomm__monoid__add @ B )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ B ) @ ( c_Groups_Ozero__class_Ozero @ B ) ) @ A )
        = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_add__0) ).

thf(2587,plain,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Ocomm__monoid__add @ B )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ B ) @ ( c_Groups_Ozero__class_Ozero @ B ) ) @ A )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[53]) ).

thf(1390,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Odrop @ ( tc_prod @ E @ D ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_List_Ozip @ E @ D ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_List_Ozip @ E @ D ) @ ( hAPP @ ( hAPP @ ( c_List_Odrop @ E ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_List_Odrop @ D ) @ C ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_drop__zip) ).

thf(7744,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Odrop @ ( tc_prod @ E @ D ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_List_Ozip @ E @ D ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_List_Ozip @ E @ D ) @ ( hAPP @ ( hAPP @ ( c_List_Odrop @ E ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_List_Odrop @ D ) @ C ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1390]) ).

thf(1869,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ~ ( hBOOL @ ( hAPP @ E @ D ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Natural_Oevalc @ C ) @ D ) @ B ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Natural_Oevalc @ ( c_Com_Ocom_OCond @ E @ A @ C ) ) @ D ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_evalc_OIfFalse) ).

thf(9767,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ~ ( hBOOL @ ( hAPP @ E @ D ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Natural_Oevalc @ C ) @ D ) @ B ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Natural_Oevalc @ ( c_Com_Ocom_OCond @ E @ A @ C ) ) @ D ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1869]) ).

thf(1519,axiom,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ ( tc_List_Olist @ B ) @ ( tc_List_Olist @ B ) ) @ tc_HOL_Obool ) ) @ ( c_List_Olistrel @ B @ A ) ) @ ( c_Transitive__Closure_Ortrancl @ ( tc_List_Olist @ B ) @ ( c_List_Olistrel1 @ B @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_listrel__subset__rtrancl__listrel1) ).

thf(8296,plain,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ ( tc_List_Olist @ B ) @ ( tc_List_Olist @ B ) ) @ tc_HOL_Obool ) ) @ ( c_List_Olistrel @ B @ A ) ) @ ( c_Transitive__Closure_Ortrancl @ ( tc_List_Olist @ B ) @ ( c_List_Olistrel1 @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1519]) ).

thf(2358,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_FunDef_Oin__rel @ E @ D @ C ) @ B ) @ A ) )
    <=> ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ D ) @ B ) @ A ) ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_in__rel__def) ).

thf(11680,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_FunDef_Oin__rel @ E @ D @ C ) @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ D ) @ B ) @ A ) ) @ C ) ) )
      & ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ D ) @ B ) @ A ) ) @ C ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_FunDef_Oin__rel @ E @ D @ C ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2358]) ).

thf(1110,axiom,
    ! [A: $i,B: $i,C: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_List_Oset @ C ) @ ( c_List_Oremove1 @ C @ B @ A ) ) ) @ ( hAPP @ ( c_List_Oset @ C ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_set__remove1__subset) ).

thf(6563,plain,
    ! [A: $i,B: $i,C: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_List_Oset @ C ) @ ( c_List_Oremove1 @ C @ B @ A ) ) ) @ ( hAPP @ ( c_List_Oset @ C ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1110]) ).

thf(1552,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ A ) @ B ) ) @ B ) )
      <=> ( A
          = ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_dvd__mult__cancel2) ).

thf(8412,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ B ) )
     => ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ A ) @ B ) ) @ B ) )
         => ( A
            = ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) )
        & ( ( A
            = ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ A ) @ B ) ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1552]) ).

thf(1758,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i,I: $i,J: $i] :
      ( ( c_Equiv__Relations_Ocongruent2 @ J @ I @ H @ G @ F @ E )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ J @ J ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ J @ J ) @ D ) @ C ) ) @ G ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ I @ I ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ I @ I ) @ B ) @ A ) ) @ F ) )
         => ( ( hAPP @ ( hAPP @ E @ D ) @ B )
            = ( hAPP @ ( hAPP @ E @ C ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_congruent2D) ).

thf(9318,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i,I: $i,J: $i] :
      ( ( c_Equiv__Relations_Ocongruent2 @ J @ I @ H @ G @ F @ E )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ J @ J ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ J @ J ) @ D ) @ C ) ) @ G ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ I @ I ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ I @ I ) @ B ) @ A ) ) @ F ) )
         => ( ( hAPP @ ( hAPP @ E @ D ) @ B )
            = ( hAPP @ ( hAPP @ E @ C ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1758]) ).

thf(610,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ E ) @ ( hAPP @ ( c_Set_Oimage @ D @ E @ C ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) ) )
     => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ B ) @ ( hAPP @ ( c_Set_Oimage @ D @ B @ ( hAPP @ ( hAPP @ ( c_COMBB @ E @ B @ D ) @ A ) @ C ) ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_finite__range__imageI) ).

thf(4596,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ E ) @ ( hAPP @ ( c_Set_Oimage @ D @ E @ C ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) ) )
     => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ B ) @ ( hAPP @ ( c_Set_Oimage @ D @ B @ ( hAPP @ ( hAPP @ ( c_COMBB @ E @ B @ D ) @ A ) @ C ) ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[610]) ).

thf(878,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ E @ D @ C ) @ B ) @ A )
      = ( hAPP @ ( hAPP @ B @ ( hAPP @ ( c_Product__Type_Ofst @ E @ D ) @ A ) ) @ ( hAPP @ ( c_Product__Type_Osnd @ E @ D ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_prod__case__beta) ).

thf(5554,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ E @ D @ C ) @ B ) @ A )
      = ( hAPP @ ( hAPP @ B @ ( hAPP @ ( c_Product__Type_Ofst @ E @ D ) @ A ) ) @ ( hAPP @ ( c_Product__Type_Osnd @ E @ D ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[878]) ).

thf(1197,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D
        = ( hAPP @ ( c_Set_Oimage @ tc_Nat_Onat @ C @ B ) @ ( hAPP @ ( c_Set_OCollect @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) ) @ A ) ) ) )
     => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ D ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nat__seg__image__imp__finite) ).

thf(6925,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D
        = ( hAPP @ ( c_Set_Oimage @ tc_Nat_Onat @ C @ B ) @ ( hAPP @ ( c_Set_OCollect @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) ) @ A ) ) ) )
     => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ D ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1197]) ).

thf(1277,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( c_Relation_Oconverse @ C @ B ) @ ( hAPP @ ( c_Relation_Oconverse @ B @ C ) @ A ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_converse__converse) ).

thf(7259,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( c_Relation_Oconverse @ C @ B ) @ ( hAPP @ ( c_Relation_Oconverse @ B @ C ) @ A ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1277]) ).

thf(2006,axiom,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) )
        & ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ A ) @ B ) ) )
     => ( A != B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_dvd_Oless__imp__not__eq2) ).

thf(10312,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) )
        & ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ A ) @ B ) ) )
     => ( A != B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2006]) ).

thf(439,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_Fun_Oinj__on @ D @ C @ B @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ D @ tc_HOL_Obool ) ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Set_Ovimage @ D @ C @ B ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) ) ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ ( c_HOL_OThe @ D @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ C @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ C @ ( tc_fun @ C @ tc_HOL_Obool ) @ D ) @ c_fequal ) @ B ) ) @ A ) ) ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_inj__vimage__singleton) ).

thf(3987,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_Fun_Oinj__on @ D @ C @ B @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ D @ tc_HOL_Obool ) ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Set_Ovimage @ D @ C @ B ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) ) ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ ( c_HOL_OThe @ D @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ C @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ C @ ( tc_fun @ C @ tc_HOL_Obool ) @ D ) @ c_fequal ) @ B ) ) @ A ) ) ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[439]) ).

thf(2306,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ D ) @ C ) )
       => ( ( hAPP @ ( c_Map_Orestrict__map @ E @ B @ A @ C ) @ D )
          = ( hAPP @ A @ D ) ) )
      & ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ D ) @ C ) )
       => ( ( hAPP @ ( c_Map_Orestrict__map @ E @ B @ A @ C ) @ D )
          = ( c_Option_Ooption_ONone @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_restrict__map__def) ).

thf(11437,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ D ) @ C ) )
       => ( ( hAPP @ ( c_Map_Orestrict__map @ E @ B @ A @ C ) @ D )
          = ( hAPP @ A @ D ) ) )
      & ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ D ) @ C ) )
       => ( ( hAPP @ ( c_Map_Orestrict__map @ E @ B @ A @ C ) @ D )
          = ( c_Option_Ooption_ONone @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2306]) ).

thf(241,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ B ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ C ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zdiv__mono1__neg) ).

thf(3264,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ B ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ C ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[241]) ).

thf(2046,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ ( hAPP @ c_Nat_OSuc @ A ) ) )
    <=> ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ A ) )
        | ( B
          = ( hAPP @ c_Nat_OSuc @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_le__Suc__eq) ).

thf(10443,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ ( hAPP @ c_Nat_OSuc @ A ) ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ A ) )
          | ( B
            = ( hAPP @ c_Nat_OSuc @ A ) ) ) )
      & ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ A ) )
          | ( B
            = ( hAPP @ c_Nat_OSuc @ A ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ ( hAPP @ c_Nat_OSuc @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2046]) ).

thf(36,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Divides_Odivmod__int__rel @ E @ D ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ tc_Int_Oint @ tc_Int_Oint ) @ C ) @ B ) ) )
     => ( ( D
         != ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( c_Divides_Odivmod__int__rel @ E @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ D ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ tc_Int_Oint @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ C ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ D ) @ ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ C @ A ) ) ) @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zmult2__lemma) ).

thf(2522,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Divides_Odivmod__int__rel @ E @ D ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ tc_Int_Oint @ tc_Int_Oint ) @ C ) @ B ) ) )
     => ( ( D
         != ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( c_Divides_Odivmod__int__rel @ E @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ D ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ tc_Int_Oint @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ C ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ D ) @ ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ C @ A ) ) ) @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[36]) ).

thf(2364,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_Equiv__Relations_Oequivp @ D @ C )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ C @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ C @ A ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_equivp__symp) ).

thf(11712,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_Equiv__Relations_Oequivp @ D @ C )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ C @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ C @ A ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2364]) ).

thf(1043,axiom,
    ! [A: $i,B: $i,C: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ A ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_subset__insertI) ).

thf(6237,plain,
    ! [A: $i,B: $i,C: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ A ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1043]) ).

thf(384,axiom,
    ! [A: $i] :
      ( ( hAPP @ ( c_Relation_ORange @ A @ A ) @ ( c_Relation_OId @ A ) )
      = ( c_Orderings_Otop__class_Otop @ ( tc_fun @ A @ tc_HOL_Obool ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Range__Id) ).

thf(3790,plain,
    ! [A: $i] :
      ( ( hAPP @ ( c_Relation_ORange @ A @ A ) @ ( c_Relation_OId @ A ) )
      = ( c_Orderings_Otop__class_Otop @ ( tc_fun @ A @ tc_HOL_Obool ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[384]) ).

thf(868,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_finite__UnI) ).

thf(5522,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[868]) ).

thf(963,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hAPP @ ( hAPP @ ( c_COMBC @ C @ ( tc_fun @ C @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( c_member @ C ) ) @ B )
        = ( hAPP @ ( hAPP @ ( c_COMBC @ C @ ( tc_fun @ C @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( c_member @ C ) ) @ A ) )
    <=> ( B = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_pred__equals__eq) ).

thf(5884,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( hAPP @ ( hAPP @ ( c_COMBC @ C @ ( tc_fun @ C @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( c_member @ C ) ) @ B )
          = ( hAPP @ ( hAPP @ ( c_COMBC @ C @ ( tc_fun @ C @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( c_member @ C ) ) @ A ) )
       => ( B = A ) )
      & ( ( B = A )
       => ( ( hAPP @ ( hAPP @ ( c_COMBC @ C @ ( tc_fun @ C @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( c_member @ C ) ) @ B )
          = ( hAPP @ ( hAPP @ ( c_COMBC @ C @ ( tc_fun @ C @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( c_member @ C ) ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[963]) ).

thf(1689,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( hAPP @ ( hAPP @ ( c_List_Oappend @ E ) @ D ) @ C )
        = ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ E ) @ B ) @ A ) )
    <=> ( ( ( D
            = ( c_List_Olist_ONil @ E ) )
          & ( C
            = ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ E ) @ B ) @ A ) ) )
        | ? [F: $i] :
            ( ( D
              = ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ E ) @ B ) @ F ) )
            & ( ( hAPP @ ( hAPP @ ( c_List_Oappend @ E ) @ F ) @ C )
              = A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_append__eq__Cons__conv) ).

thf(9051,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( ( hAPP @ ( hAPP @ ( c_List_Oappend @ E ) @ D ) @ C )
          = ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ E ) @ B ) @ A ) )
       => ( ( ( D
              = ( c_List_Olist_ONil @ E ) )
            & ( C
              = ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ E ) @ B ) @ A ) ) )
          | ? [F: $i] :
              ( ( D
                = ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ E ) @ B ) @ F ) )
              & ( ( hAPP @ ( hAPP @ ( c_List_Oappend @ E ) @ F ) @ C )
                = A ) ) ) )
      & ( ( ( ( D
              = ( c_List_Olist_ONil @ E ) )
            & ( C
              = ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ E ) @ B ) @ A ) ) )
          | ? [F: $i] :
              ( ( D
                = ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ E ) @ B ) @ F ) )
              & ( ( hAPP @ ( hAPP @ ( c_List_Oappend @ E ) @ F ) @ C )
                = A ) ) )
       => ( ( hAPP @ ( hAPP @ ( c_List_Oappend @ E ) @ D ) @ C )
          = ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ E ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1689]) ).

thf(625,axiom,
    ! [A: $i,B: $i,C: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ A ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Un__upper2) ).

thf(4663,plain,
    ! [A: $i,B: $i,C: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ A ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[625]) ).

thf(727,axiom,
    ! [A: $i,B: $i] : ( c_Fun_Oinj__on @ ( tc_fun @ B @ A ) @ ( tc_fun @ ( tc_prod @ B @ A ) @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ ( tc_prod @ B @ A ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ B @ A ) @ tc_HOL_Obool ) @ ( tc_fun @ B @ A ) ) @ ( c_Set_OCollect @ ( tc_prod @ B @ A ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ B @ ( tc_fun @ A @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_prod @ B @ A ) @ tc_HOL_Obool ) @ ( tc_fun @ B @ A ) ) @ ( c_Product__Type_Oprod_Oprod__case @ B @ A @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_COMBB @ A @ ( tc_fun @ A @ tc_HOL_Obool ) @ B ) @ ( hAPP @ ( c_COMBC @ A @ A @ tc_HOL_Obool ) @ c_fequal ) ) ) ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ ( tc_fun @ B @ A ) @ tc_HOL_Obool ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_inj__graph) ).

thf(5014,plain,
    ! [A: $i,B: $i] : ( c_Fun_Oinj__on @ ( tc_fun @ B @ A ) @ ( tc_fun @ ( tc_prod @ B @ A ) @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ ( tc_prod @ B @ A ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ B @ A ) @ tc_HOL_Obool ) @ ( tc_fun @ B @ A ) ) @ ( c_Set_OCollect @ ( tc_prod @ B @ A ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ B @ ( tc_fun @ A @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_prod @ B @ A ) @ tc_HOL_Obool ) @ ( tc_fun @ B @ A ) ) @ ( c_Product__Type_Oprod_Oprod__case @ B @ A @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_COMBB @ A @ ( tc_fun @ A @ tc_HOL_Obool ) @ B ) @ ( hAPP @ ( c_COMBC @ A @ A @ tc_HOL_Obool ) @ c_fequal ) ) ) ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ ( tc_fun @ B @ A ) @ tc_HOL_Obool ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[727]) ).

thf(362,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) @ A )
      = ( c_Orderings_Otop__class_Otop @ ( tc_fun @ B @ tc_HOL_Obool ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Un__UNIV__left) ).

thf(3690,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) @ A )
      = ( c_Orderings_Otop__class_Otop @ ( tc_fun @ B @ tc_HOL_Obool ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[362]) ).

thf(1691,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ C ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ B ) @ C ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ A ) @ C ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ ( c_Finite__Set_Ocard @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ B ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) ) ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) ) ) ) @ ( hAPP @ ( c_Finite__Set_Ocard @ D ) @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_card__Diff2__less) ).

thf(9070,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ C ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ B ) @ C ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ A ) @ C ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ ( c_Finite__Set_Ocard @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ B ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) ) ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) ) ) ) @ ( hAPP @ ( c_Finite__Set_Ocard @ D ) @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1691]) ).

thf(1799,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ D ) @ C ) ) @ ( c_Transitive__Closure_Ortrancl @ E @ B ) ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ C ) @ A ) ) @ ( c_Transitive__Closure_Otrancl @ E @ B ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ D ) @ A ) ) @ ( c_Transitive__Closure_Otrancl @ E @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_rtrancl__trancl__trancl) ).

thf(9463,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ D ) @ C ) ) @ ( c_Transitive__Closure_Ortrancl @ E @ B ) ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ C ) @ A ) ) @ ( c_Transitive__Closure_Otrancl @ E @ B ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ D ) @ A ) ) @ ( c_Transitive__Closure_Otrancl @ E @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1799]) ).

thf(787,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ B ) @ A ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ A ) @ B ) )
       => ( B = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zle__antisym) ).

thf(5247,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ B ) @ A ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ A ) @ B ) )
       => ( B = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[787]) ).

thf(11910,plain,
    ( ( hBOOL @ ( hAPP @ c_Com_OWT @ c_Com_Ocom_OSKIP ) )
   != ( hBOOL @ c_fFalse ) ),
    inference(paramod_ordered,[status(thm)],[9359,11422]) ).

thf(11914,plain,
    ( ( hAPP @ c_Com_OWT @ c_Com_Ocom_OSKIP )
   != c_fFalse ),
    inference(simp,[status(thm)],[11910]) ).

thf(1450,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C
        = ( hAPP @ ( hAPP @ ( c_List_Oappend @ B ) @ C ) @ A ) )
    <=> ( A
        = ( c_List_Olist_ONil @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_self__append__conv) ).

thf(7955,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( C
          = ( hAPP @ ( hAPP @ ( c_List_Oappend @ B ) @ C ) @ A ) )
       => ( A
          = ( c_List_Olist_ONil @ B ) ) )
      & ( ( A
          = ( c_List_Olist_ONil @ B ) )
       => ( C
          = ( hAPP @ ( hAPP @ ( c_List_Oappend @ B ) @ C ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1450]) ).

thf(1588,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ B ) ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ C ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_add__lessD1) ).

thf(8621,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ B ) ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ C ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1588]) ).

thf(300,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Map_Odom @ E @ D @ ( c_Fun_Ooverride__on @ E @ ( tc_Option_Ooption @ D ) @ C @ B @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( c_Map_Odom @ E @ D @ C ) ) @ ( hAPP @ ( c_Set_OCollect @ E ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ E @ ( tc_fun @ E @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( c_member @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ A ) @ ( c_Map_Odom @ E @ D @ B ) ) ) ) ) ) @ ( hAPP @ ( c_Set_OCollect @ E ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ E @ ( tc_fun @ E @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( c_member @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ A ) @ ( c_Map_Odom @ E @ D @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_dom__override__on) ).

thf(3469,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Map_Odom @ E @ D @ ( c_Fun_Ooverride__on @ E @ ( tc_Option_Ooption @ D ) @ C @ B @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( c_Map_Odom @ E @ D @ C ) ) @ ( hAPP @ ( c_Set_OCollect @ E ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ E @ ( tc_fun @ E @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( c_member @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ A ) @ ( c_Map_Odom @ E @ D @ B ) ) ) ) ) ) @ ( hAPP @ ( c_Set_OCollect @ E ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ E @ ( tc_fun @ E @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( c_member @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ A ) @ ( c_Map_Odom @ E @ D @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[300]) ).

thf(620,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ D ) @ C ) )
     => ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ ( hAPP @ A @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ E @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ C ) @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ E @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ C ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_UN__absorb) ).

thf(4631,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ D ) @ C ) )
     => ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ ( hAPP @ A @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ E @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ C ) @ A ) )
        = ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ E @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ C ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[620]) ).

thf(847,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ B ) ) @ A ) ) )
    <=> ( A
       != ( c_List_Olist_ONil @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_length__greater__0__conv) ).

thf(5462,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ B ) ) @ A ) ) )
       => ( A
         != ( c_List_Olist_ONil @ B ) ) )
      & ( ( A
         != ( c_List_Olist_ONil @ B ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ B ) ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[847]) ).

thf(1753,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_List_Oremove1 @ D @ C @ ( c_List_Oremove1 @ D @ B @ A ) )
      = ( c_List_Oremove1 @ D @ B @ ( c_List_Oremove1 @ D @ C @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_remove1__commute) ).

thf(9299,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_List_Oremove1 @ D @ C @ ( c_List_Oremove1 @ D @ B @ A ) )
      = ( c_List_Oremove1 @ D @ B @ ( c_List_Oremove1 @ D @ C @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1753]) ).

thf(823,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Relation_Orefl__on @ C @ B @ A )
    <=> ( ! [D: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ C @ C ) ) @ D ) @ A ) )
           => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ C @ C @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ C @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ C ) @ ( c_COMBB @ tc_HOL_Obool @ tc_HOL_Obool @ C ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ C ) @ c_fconj ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ C @ ( tc_fun @ C @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( c_member @ C ) ) @ B ) ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ C @ ( tc_fun @ C @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( c_member @ C ) ) @ B ) ) ) @ D ) ) )
        & ! [D: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ D ) @ B ) )
           => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ C @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ C @ C ) @ D ) @ D ) ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_refl__on__def_H) ).

thf(5378,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( c_Relation_Orefl__on @ C @ B @ A )
       => ( ! [D: $i] :
              ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ C @ C ) ) @ D ) @ A ) )
             => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ C @ C @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ C @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ C ) @ ( c_COMBB @ tc_HOL_Obool @ tc_HOL_Obool @ C ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ C ) @ c_fconj ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ C @ ( tc_fun @ C @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( c_member @ C ) ) @ B ) ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ C @ ( tc_fun @ C @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( c_member @ C ) ) @ B ) ) ) @ D ) ) )
          & ! [D: $i] :
              ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ D ) @ B ) )
             => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ C @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ C @ C ) @ D ) @ D ) ) @ A ) ) ) ) )
      & ( ( ! [D: $i] :
              ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ C @ C ) ) @ D ) @ A ) )
             => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ C @ C @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ C @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ C ) @ ( c_COMBB @ tc_HOL_Obool @ tc_HOL_Obool @ C ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ C ) @ c_fconj ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ C @ ( tc_fun @ C @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( c_member @ C ) ) @ B ) ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ C @ ( tc_fun @ C @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( c_member @ C ) ) @ B ) ) ) @ D ) ) )
          & ! [D: $i] :
              ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ D ) @ B ) )
             => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ C @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ C @ C ) @ D ) @ D ) ) @ A ) ) ) )
       => ( c_Relation_Orefl__on @ C @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[823]) ).

thf(755,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ B ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ A ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zmult__commute) ).

thf(5110,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ B ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ A ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[755]) ).

thf(1095,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Omap @ E @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_List_Oappend @ E ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_List_Oappend @ D ) @ ( hAPP @ ( hAPP @ ( c_List_Omap @ E @ D ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_List_Omap @ E @ D ) @ C ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_map__append) ).

thf(6503,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Omap @ E @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_List_Oappend @ E ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_List_Oappend @ D ) @ ( hAPP @ ( hAPP @ ( c_List_Omap @ E @ D ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_List_Omap @ E @ D ) @ C ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1095]) ).

thf(256,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ D ) ) @ C )
        = ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ B ) ) @ A ) )
     => ( ( hAPP @ ( hAPP @ ( c_List_Omap @ ( tc_prod @ D @ B ) @ D ) @ ( c_Product__Type_Ofst @ D @ B ) ) @ ( hAPP @ ( hAPP @ ( c_List_Ozip @ D @ B ) @ C ) @ A ) )
        = C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_map__fst__zip) ).

thf(3309,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ D ) ) @ C )
        = ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ B ) ) @ A ) )
     => ( ( hAPP @ ( hAPP @ ( c_List_Omap @ ( tc_prod @ D @ B ) @ D ) @ ( c_Product__Type_Ofst @ D @ B ) ) @ ( hAPP @ ( hAPP @ ( c_List_Ozip @ D @ B ) @ C ) @ A ) )
        = C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[256]) ).

thf(204,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ A ) )
     => ( ! [C: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ C ) @ A ) )
           => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ B ) @ C ) ) )
       => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ B ) @ ( c_Complete__Lattice_OSup__class_OSup @ ( tc_fun @ B @ tc_HOL_Obool ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_finite__Union) ).

thf(3145,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ A ) )
     => ( ! [C: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ C ) @ A ) )
           => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ B ) @ C ) ) )
       => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ B ) @ ( c_Complete__Lattice_OSup__class_OSup @ ( tc_fun @ B @ tc_HOL_Obool ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[204]) ).

thf(918,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ ( hAPP @ C @ B ) ) @ ( hAPP @ ( c_Set_Oimage @ A @ D @ C ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ A @ tc_HOL_Obool ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_rangeI) ).

thf(5700,plain,
    ! [A: $i,B: $i,C: $i,D: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ ( hAPP @ C @ B ) ) @ ( hAPP @ ( c_Set_Oimage @ A @ D @ C ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ A @ tc_HOL_Obool ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[918]) ).

thf(1399,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Wellfounded_OwfP @ C @ B )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) @ A ) @ B ) )
       => ( c_Wellfounded_OwfP @ C @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_wfP__subset) ).

thf(7772,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Wellfounded_OwfP @ C @ B )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) @ A ) @ B ) )
       => ( c_Wellfounded_OwfP @ C @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1399]) ).

thf(1629,axiom,
    ! [A: $i,B: $i,C: $i] : ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ ( tc_Option_Ooption @ C ) ) @ ( hAPP @ ( c_Set_Oimage @ B @ ( tc_Option_Ooption @ C ) @ ( c_Map_Omap__of @ B @ C @ A ) ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_finite__range__map__of) ).

thf(8790,plain,
    ! [A: $i,B: $i,C: $i] : ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ ( tc_Option_Ooption @ C ) ) @ ( hAPP @ ( c_Set_Oimage @ B @ ( tc_Option_Ooption @ C ) @ ( c_Map_Omap__of @ B @ C @ A ) ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1629]) ).

thf(431,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( c_Orderings_Obot__class_Obot @ ( tc_fun @ ( tc_fun @ C @ tc_HOL_Obool ) @ tc_HOL_Obool ) )
        = ( hAPP @ ( hAPP @ ( c_Equiv__Relations_Oquotient @ C ) @ B ) @ A ) )
    <=> ( B
        = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_quotient__is__empty2) ).

thf(3956,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( c_Orderings_Obot__class_Obot @ ( tc_fun @ ( tc_fun @ C @ tc_HOL_Obool ) @ tc_HOL_Obool ) )
          = ( hAPP @ ( hAPP @ ( c_Equiv__Relations_Oquotient @ C ) @ B ) @ A ) )
       => ( B
          = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) )
      & ( ( B
          = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) )
       => ( ( c_Orderings_Obot__class_Obot @ ( tc_fun @ ( tc_fun @ C @ tc_HOL_Obool ) @ tc_HOL_Obool ) )
          = ( hAPP @ ( hAPP @ ( c_Equiv__Relations_Oquotient @ C ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[431]) ).

thf(2367,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ c_Com_Ocom_OBODY @ C )
     != ( c_Com_Ocom_OWhile @ B @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_com_Osimps_I59_J) ).

thf(11727,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ c_Com_Ocom_OBODY @ C )
     != ( c_Com_Ocom_OWhile @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2367]) ).

thf(661,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( c_Set_Ovimage @ D @ C @ B ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ C @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ A ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ D @ tc_HOL_Obool ) @ C ) @ ( c_Set_Ovimage @ D @ C @ B ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ C @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( c_Set_Oinsert @ C ) ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_vimage__eq__UN) ).

thf(4777,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( c_Set_Ovimage @ D @ C @ B ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ C @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ A ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ D @ tc_HOL_Obool ) @ C ) @ ( c_Set_Ovimage @ D @ C @ B ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ C @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( c_Set_Oinsert @ C ) ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[661]) ).

thf(2337,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ B ) @ ( hAPP @ ( c_Set_OCollect @ C ) @ A ) ) )
    <=> ( hBOOL @ ( hAPP @ A @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mem__Collect__eq) ).

thf(11595,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ B ) @ ( hAPP @ ( c_Set_OCollect @ C ) @ A ) ) )
       => ( hBOOL @ ( hAPP @ A @ B ) ) )
      & ( ( hBOOL @ ( hAPP @ A @ B ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ B ) @ ( hAPP @ ( c_Set_OCollect @ C ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2337]) ).

thf(2142,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Relation_Orefl__on @ E @ D @ C )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ B ) @ A ) ) @ C ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ B ) @ D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_refl__onD1) ).

thf(10809,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Relation_Orefl__on @ E @ D @ C )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ B ) @ A ) ) @ C ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ B ) @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2142]) ).

thf(801,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oab__semigroup__mult @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ B ) )
       => ( ( B
           != ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) )
         => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ A ) )
           => ( ( A
               != ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) )
             => ( ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A )
                  = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) )
               => ( ( hAPP @ ( c_Finite__Set_Ofold1 @ C @ ( c_Groups_Otimes__class_Otimes @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) )
                  = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ ( hAPP @ ( c_Finite__Set_Ofold1 @ C @ ( c_Groups_Otimes__class_Otimes @ C ) ) @ B ) ) @ ( hAPP @ ( c_Finite__Set_Ofold1 @ C @ ( c_Groups_Otimes__class_Otimes @ C ) ) @ A ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_fold1__Un) ).

thf(5300,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Oab__semigroup__mult @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ B ) )
       => ( ( B
           != ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) )
         => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ A ) )
           => ( ( A
               != ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) )
             => ( ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A )
                  = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) )
               => ( ( hAPP @ ( c_Finite__Set_Ofold1 @ C @ ( c_Groups_Otimes__class_Otimes @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) )
                  = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ C ) @ ( hAPP @ ( c_Finite__Set_Ofold1 @ C @ ( c_Groups_Otimes__class_Otimes @ C ) ) @ B ) ) @ ( hAPP @ ( c_Finite__Set_Ofold1 @ C @ ( c_Groups_Otimes__class_Otimes @ C ) ) @ A ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[801]) ).

thf(1028,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ B @ B @ B ) @ ( c_Transitive__Closure_Ortrancl @ B @ A ) ) @ ( c_Transitive__Closure_Ortrancl @ B @ A ) )
      = ( c_Transitive__Closure_Ortrancl @ B @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_rtrancl__idemp__self__comp) ).

thf(6169,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ B @ B @ B ) @ ( c_Transitive__Closure_Ortrancl @ B @ A ) ) @ ( c_Transitive__Closure_Ortrancl @ B @ A ) )
      = ( c_Transitive__Closure_Ortrancl @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1028]) ).

thf(2186,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_Wellfounded_Owf @ D @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ ( tc_prod @ D @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ D ) @ C ) @ B ) ) @ A ) )
    <=> ( ( c_Wellfounded_Owf @ D @ A )
        & ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ D ) @ B ) @ C ) ) @ ( c_Transitive__Closure_Ortrancl @ D @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_wf__insert) ).

thf(10941,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( c_Wellfounded_Owf @ D @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ ( tc_prod @ D @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ D ) @ C ) @ B ) ) @ A ) )
       => ( ( c_Wellfounded_Owf @ D @ A )
          & ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ D ) @ B ) @ C ) ) @ ( c_Transitive__Closure_Ortrancl @ D @ A ) ) ) ) )
      & ( ( ( c_Wellfounded_Owf @ D @ A )
          & ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ D ) @ B ) @ C ) ) @ ( c_Transitive__Closure_Ortrancl @ D @ A ) ) ) )
       => ( c_Wellfounded_Owf @ D @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ ( tc_prod @ D @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ D ) @ C ) @ B ) ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2186]) ).

thf(654,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ F @ ( tc_fun @ E @ tc_HOL_Obool ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ F @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ E @ ( tc_fun @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ F ) @ ( c_COMBC @ E @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ E @ ( tc_prod @ F @ E ) ) @ ( tc_fun @ E @ ( tc_fun @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ F ) @ ( hAPP @ ( c_COMBB @ ( tc_prod @ F @ E ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ E ) @ ( c_member @ ( tc_prod @ F @ E ) ) ) ) @ ( c_Product__Type_OPair @ F @ E ) ) ) ) @ D ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ F @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ E @ ( tc_fun @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ F ) @ ( c_COMBC @ E @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ E @ ( tc_prod @ F @ E ) ) @ ( tc_fun @ E @ ( tc_fun @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ F ) @ ( hAPP @ ( c_COMBB @ ( tc_prod @ F @ E ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ E ) @ ( c_member @ ( tc_prod @ F @ E ) ) ) ) @ ( c_Product__Type_OPair @ F @ E ) ) ) ) @ C ) ) @ B ) @ A ) )
    <=> ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ F @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ F @ E ) @ B ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) ) @ D ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_sup__Un__eq2) ).

thf(4755,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ F @ ( tc_fun @ E @ tc_HOL_Obool ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ F @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ E @ ( tc_fun @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ F ) @ ( c_COMBC @ E @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ E @ ( tc_prod @ F @ E ) ) @ ( tc_fun @ E @ ( tc_fun @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ F ) @ ( hAPP @ ( c_COMBB @ ( tc_prod @ F @ E ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ E ) @ ( c_member @ ( tc_prod @ F @ E ) ) ) ) @ ( c_Product__Type_OPair @ F @ E ) ) ) ) @ D ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ F @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ E @ ( tc_fun @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ F ) @ ( c_COMBC @ E @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ E @ ( tc_prod @ F @ E ) ) @ ( tc_fun @ E @ ( tc_fun @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ F ) @ ( hAPP @ ( c_COMBB @ ( tc_prod @ F @ E ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ E ) @ ( c_member @ ( tc_prod @ F @ E ) ) ) ) @ ( c_Product__Type_OPair @ F @ E ) ) ) ) @ C ) ) @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ F @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ F @ E ) @ B ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) ) @ D ) @ C ) ) ) )
      & ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ F @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ F @ E ) @ B ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) ) @ D ) @ C ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ F @ ( tc_fun @ E @ tc_HOL_Obool ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ F @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ E @ ( tc_fun @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ F ) @ ( c_COMBC @ E @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ E @ ( tc_prod @ F @ E ) ) @ ( tc_fun @ E @ ( tc_fun @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ F ) @ ( hAPP @ ( c_COMBB @ ( tc_prod @ F @ E ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ E ) @ ( c_member @ ( tc_prod @ F @ E ) ) ) ) @ ( c_Product__Type_OPair @ F @ E ) ) ) ) @ D ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ F @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ E @ ( tc_fun @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ F ) @ ( c_COMBC @ E @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ E @ ( tc_prod @ F @ E ) ) @ ( tc_fun @ E @ ( tc_fun @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ F ) @ ( hAPP @ ( c_COMBB @ ( tc_prod @ F @ E ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ E ) @ ( c_member @ ( tc_prod @ F @ E ) ) ) ) @ ( c_Product__Type_OPair @ F @ E ) ) ) ) @ C ) ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[654]) ).

thf(751,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( c_Set_OCollect @ C ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ C @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ C ) @ c_fdisj ) @ B ) ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Set_OCollect @ C ) @ B ) ) @ ( hAPP @ ( c_Set_OCollect @ C ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Collect__disj__eq) ).

thf(5092,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( c_Set_OCollect @ C ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ C @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ C ) @ c_fdisj ) @ B ) ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Set_OCollect @ C ) @ B ) ) @ ( hAPP @ ( c_Set_OCollect @ C ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[751]) ).

thf(1260,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ C ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ B ) @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ) ) @ A ) )
       => ( ! [D: $i] :
              ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Int_Oint ) @ D ) @ ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Int_Oint @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) @ C ) ) )
             => ! [E: $i] :
                  ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Int_Oint ) @ E ) @ A ) )
                 => ( B
                   != ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Int_Oint ) @ E ) @ D ) ) ) )
         => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ B ) @ C )
            = B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_aset_I3_J) ).

thf(7184,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ C ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ B ) @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ) ) @ A ) )
       => ( ! [D: $i] :
              ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Int_Oint ) @ D ) @ ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Int_Oint @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) @ C ) ) )
             => ! [E: $i] :
                  ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Int_Oint ) @ E ) @ A ) )
                 => ( B
                   != ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Int_Oint ) @ E ) @ D ) ) ) )
         => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ B ) @ C )
            = B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1260]) ).

thf(715,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ D ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ C @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ C @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ E @ tc_HOL_Obool ) @ ( tc_fun @ E @ tc_HOL_Obool ) @ C ) @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ D ) ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Int__UN__distrib) ).

thf(4981,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ D ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ C @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ C @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ E @ tc_HOL_Obool ) @ ( tc_fun @ E @ tc_HOL_Obool ) @ C ) @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ D ) ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[715]) ).

thf(696,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) )
     => ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ A ) @ B )
        = B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Int__absorb1) ).

thf(4895,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) )
     => ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ A ) @ B )
        = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[696]) ).

thf(2114,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ C ) @ B ) )
        & ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ C ) ) )
     => ( ( B = A )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ C ) @ A ) )
          & ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ A ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_dvd_Oord__less__eq__trans) ).

thf(10691,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ C ) @ B ) )
        & ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ C ) ) )
     => ( ( B = A )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ C ) @ A ) )
          & ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ A ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2114]) ).

thf(523,axiom,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) ) @ A ) )
     => ( ( A
         != ( c_Orderings_Obot__class_Obot @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) ) ) @ A ) ) @ c_FunDef_Omax__strict ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_smax__emptyI) ).

thf(4286,plain,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) ) @ A ) )
     => ( ( A
         != ( c_Orderings_Obot__class_Obot @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) ) ) @ A ) ) @ c_FunDef_Omax__strict ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[523]) ).

thf(2246,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ D ) @ ( c_SetInterval_Oord_OgreaterThanLessThan @ E @ C @ B @ A ) ) )
    <=> ( ( hBOOL @ ( hAPP @ ( hAPP @ C @ B ) @ D ) )
        & ( hBOOL @ ( hAPP @ ( hAPP @ C @ D ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_ord_OgreaterThanLessThan__iff) ).

thf(11194,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ D ) @ ( c_SetInterval_Oord_OgreaterThanLessThan @ E @ C @ B @ A ) ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ C @ B ) @ D ) )
          & ( hBOOL @ ( hAPP @ ( hAPP @ C @ D ) @ A ) ) ) )
      & ( ( ( hBOOL @ ( hAPP @ ( hAPP @ C @ B ) @ D ) )
          & ( hBOOL @ ( hAPP @ ( hAPP @ C @ D ) @ A ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ D ) @ ( c_SetInterval_Oord_OgreaterThanLessThan @ E @ C @ B @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2246]) ).

thf(2398,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ ( hAPP @ D @ C ) ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ A ) @ C ) @ ( hAPP @ ( c_Set_Ovimage @ A @ E @ D ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_vimageI2) ).

thf(11854,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ ( hAPP @ D @ C ) ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ A ) @ C ) @ ( hAPP @ ( c_Set_Ovimage @ A @ E @ D ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2398]) ).

thf(1542,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hAPP @ ( c_Set_Ovimage @ F @ E @ ( hAPP @ ( c_Fun_Ocomp @ D @ E @ F @ C ) @ B ) ) @ A )
      = ( hAPP @ ( c_Set_Ovimage @ F @ D @ B ) @ ( hAPP @ ( c_Set_Ovimage @ D @ E @ C ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_vimage__compose) ).

thf(8373,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hAPP @ ( c_Set_Ovimage @ F @ E @ ( hAPP @ ( c_Fun_Ocomp @ D @ E @ F @ C ) @ B ) ) @ A )
      = ( hAPP @ ( c_Set_Ovimage @ F @ D @ B ) @ ( hAPP @ ( c_Set_Ovimage @ D @ E @ C ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1542]) ).

thf(324,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ E @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ E @ ( tc_fun @ D @ tc_HOL_Obool ) @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ D @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ D @ tc_HOL_Obool ) @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ E ) @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) @ B ) ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ ( tc_prod @ E @ D ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ E @ D ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ E @ D ) @ C ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Sigma__Un__distrib2) ).

thf(3558,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ E @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ E @ ( tc_fun @ D @ tc_HOL_Obool ) @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ D @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ D @ tc_HOL_Obool ) @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ E ) @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) @ B ) ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ ( tc_prod @ E @ D ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ E @ D ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ E @ D ) @ C ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[324]) ).

thf(987,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ H ) @ G ) @ F ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( hAPP @ E @ G ) @ D ) @ C ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ H @ ( tc_fun @ B @ ( tc_fun @ A @ tc_HOL_Obool ) ) ) @ F ) @ E ) @ D ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_SUP2__I) ).

thf(5990,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ H ) @ G ) @ F ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( hAPP @ E @ G ) @ D ) @ C ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ H @ ( tc_fun @ B @ ( tc_fun @ A @ tc_HOL_Obool ) ) ) @ F ) @ E ) @ D ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[987]) ).

thf(505,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( D
          = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
       => ( ( hAPP @ ( hAPP @ ( c_List_Onth @ C ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ C ) @ B ) @ A ) ) @ D )
          = B ) )
      & ( ( D
         != ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
       => ( ( hAPP @ ( hAPP @ ( c_List_Onth @ C ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ C ) @ B ) @ A ) ) @ D )
          = ( hAPP @ ( hAPP @ ( c_List_Onth @ C ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ D ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nth__Cons_H) ).

thf(4210,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( D
          = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
       => ( ( hAPP @ ( hAPP @ ( c_List_Onth @ C ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ C ) @ B ) @ A ) ) @ D )
          = B ) )
      & ( ( D
         != ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
       => ( ( hAPP @ ( hAPP @ ( c_List_Onth @ C ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ C ) @ B ) @ A ) ) @ D )
          = ( hAPP @ ( hAPP @ ( c_List_Onth @ C ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ D ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[505]) ).

thf(1708,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_psubset__subset__trans) ).

thf(9134,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1708]) ).

thf(575,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ B ) @ A )
     != ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_insert__not__empty) ).

thf(4474,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ B ) @ A )
     != ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[575]) ).

thf(2347,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ c_Com_OWT @ ( c_Com_Ocom_OWhile @ B @ A ) ) )
     => ( hBOOL @ ( hAPP @ c_Com_OWT @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_WTs__elim__cases_I6_J) ).

thf(11627,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ c_Com_OWT @ ( c_Com_Ocom_OWhile @ B @ A ) ) )
     => ( hBOOL @ ( hAPP @ c_Com_OWT @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2347]) ).

thf(97,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Groups_Ocomm__monoid__mult @ E )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ C ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ B ) )
         => ( ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B )
              = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) )
           => ( ( hAPP @ ( hAPP @ ( hAPP @ ( c_Finite__Set_Ofold__image @ E @ D @ ( c_Groups_Otimes__class_Otimes @ E ) ) @ A ) @ ( c_Groups_Oone__class_Oone @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) )
              = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Finite__Set_Ofold__image @ E @ D @ ( c_Groups_Otimes__class_Otimes @ E ) ) @ A ) @ ( c_Groups_Oone__class_Oone @ E ) ) @ C ) ) @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Finite__Set_Ofold__image @ E @ D @ ( c_Groups_Otimes__class_Otimes @ E ) ) @ A ) @ ( c_Groups_Oone__class_Oone @ E ) ) @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_fold__Un__disjoint) ).

thf(2746,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Groups_Ocomm__monoid__mult @ E )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ C ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ B ) )
         => ( ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B )
              = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) )
           => ( ( hAPP @ ( hAPP @ ( hAPP @ ( c_Finite__Set_Ofold__image @ E @ D @ ( c_Groups_Otimes__class_Otimes @ E ) ) @ A ) @ ( c_Groups_Oone__class_Oone @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) )
              = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ E ) @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Finite__Set_Ofold__image @ E @ D @ ( c_Groups_Otimes__class_Otimes @ E ) ) @ A ) @ ( c_Groups_Oone__class_Oone @ E ) ) @ C ) ) @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Finite__Set_Ofold__image @ E @ D @ ( c_Groups_Otimes__class_Otimes @ E ) ) @ A ) @ ( c_Groups_Oone__class_Oone @ E ) ) @ B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[97]) ).

thf(1123,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) ) @ D ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ F @ E ) @ C ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ E @ tc_HOL_Obool ) @ F ) @ B ) ) ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Relation_OImage @ F @ E @ D ) @ A ) ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Image__subset) ).

thf(6644,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) ) @ D ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ F @ E ) @ C ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ E @ tc_HOL_Obool ) @ F ) @ B ) ) ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Relation_OImage @ F @ E @ D ) @ A ) ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1123]) ).

thf(1873,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Wellfounded_Owf @ B @ A )
     => ( c_Wellfounded_Owf @ B @ ( c_Transitive__Closure_Otrancl @ B @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_wf__trancl) ).

thf(9778,plain,
    ! [A: $i,B: $i] :
      ( ( c_Wellfounded_Owf @ B @ A )
     => ( c_Wellfounded_Owf @ B @ ( c_Transitive__Closure_Otrancl @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1873]) ).

thf(1662,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) ) @ D ) @ C ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ F @ tc_HOL_Obool ) ) @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Relation_OImage @ F @ E @ D ) @ B ) ) @ ( hAPP @ ( c_Relation_OImage @ F @ E @ C ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Image__mono) ).

thf(8921,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ F @ E ) @ tc_HOL_Obool ) ) @ D ) @ C ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ F @ tc_HOL_Obool ) ) @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Relation_OImage @ F @ E @ D ) @ B ) ) @ ( hAPP @ ( c_Relation_OImage @ F @ E @ C ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1662]) ).

thf(729,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ E @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ E @ ( tc_fun @ D @ tc_HOL_Obool ) @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ D @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ D @ tc_HOL_Obool ) @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ E ) @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) @ B ) ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ E @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_UN__simps_I4_J) ).

thf(5019,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ E @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ E @ ( tc_fun @ D @ tc_HOL_Obool ) @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ D @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ D @ tc_HOL_Obool ) @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ E ) @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) @ B ) ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ E @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[729]) ).

thf(2316,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ G ) @ F ) @ E ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofold__graph @ G @ D @ C @ B @ E ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofold__graph @ G @ D @ C @ B @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ G ) @ F ) @ E ) ) @ ( hAPP @ ( hAPP @ C @ F ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_fold__graph_OinsertI) ).

thf(11494,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ G ) @ F ) @ E ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofold__graph @ G @ D @ C @ B @ E ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofold__graph @ G @ D @ C @ B @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ G ) @ F ) @ E ) ) @ ( hAPP @ ( hAPP @ C @ F ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2316]) ).

thf(1108,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Equiv__Relations_Oequiv @ E @ D @ C )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ B ) @ A ) ) @ C ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Relation_OImage @ E @ E @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ B ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ E @ tc_HOL_Obool ) ) ) ) ) @ ( hAPP @ ( c_Relation_OImage @ E @ E @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ E @ tc_HOL_Obool ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_equiv__class__subset) ).

thf(6558,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Equiv__Relations_Oequiv @ E @ D @ C )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ B ) @ A ) ) @ C ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Relation_OImage @ E @ E @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ B ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ E @ tc_HOL_Obool ) ) ) ) ) @ ( hAPP @ ( c_Relation_OImage @ E @ E @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ E @ tc_HOL_Obool ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1108]) ).

thf(1898,axiom,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ c_Com_OWT @ ( c_Com_Ocom_OAss @ B @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_WT_OAssign) ).

thf(9874,plain,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ c_Com_OWT @ ( c_Com_Ocom_OAss @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1898]) ).

thf(1580,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Omap @ F @ E ) @ D ) @ ( hAPP @ ( hAPP @ ( c_List_Omap @ C @ F ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_List_Omap @ C @ E ) @ ( hAPP @ ( c_Fun_Ocomp @ F @ E @ C @ D ) @ B ) ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_map__map) ).

thf(8516,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Omap @ F @ E ) @ D ) @ ( hAPP @ ( hAPP @ ( c_List_Omap @ C @ F ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_List_Omap @ C @ E ) @ ( hAPP @ ( c_Fun_Ocomp @ F @ E @ C @ D ) @ B ) ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1580]) ).

thf(2054,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ c_Com_OWT @ ( c_Com_Ocom_OCond @ C @ B @ A ) ) )
     => ~ ( ( hBOOL @ ( hAPP @ c_Com_OWT @ B ) )
         => ~ ( hBOOL @ ( hAPP @ c_Com_OWT @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_WTs__elim__cases_I5_J) ).

thf(10472,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ c_Com_OWT @ ( c_Com_Ocom_OCond @ C @ B @ A ) ) )
     => ~ ( ( hBOOL @ ( hAPP @ c_Com_OWT @ B ) )
         => ~ ( hBOOL @ ( hAPP @ c_Com_OWT @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2054]) ).

thf(2189,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( hBOOL @ ( hAPP @ D @ C ) )
       => ( ( c_List_OtakeWhile @ B @ D @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ B ) @ C ) @ A ) )
          = ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ B ) @ C ) @ ( c_List_OtakeWhile @ B @ D @ A ) ) ) )
      & ( ~ ( hBOOL @ ( hAPP @ D @ C ) )
       => ( ( c_List_OtakeWhile @ B @ D @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ B ) @ C ) @ A ) )
          = ( c_List_Olist_ONil @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_takeWhile_Osimps_I2_J) ).

thf(10956,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( hBOOL @ ( hAPP @ D @ C ) )
       => ( ( c_List_OtakeWhile @ B @ D @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ B ) @ C ) @ A ) )
          = ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ B ) @ C ) @ ( c_List_OtakeWhile @ B @ D @ A ) ) ) )
      & ( ~ ( hBOOL @ ( hAPP @ D @ C ) )
       => ( ( c_List_OtakeWhile @ B @ D @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ B ) @ C ) @ A ) )
          = ( c_List_Olist_ONil @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2189]) ).

thf(154,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ B ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ A ) @ B ) ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) )
      <=> ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_neg__imp__zdiv__neg__iff) ).

thf(2961,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ B ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) )
     => ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ A ) @ B ) ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ A ) ) )
        & ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ A ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ A ) @ B ) ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[154]) ).

thf(1515,axiom,
    ! [A: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) @ A )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nat__mult__1) ).

thf(8277,plain,
    ! [A: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) @ A )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1515]) ).

thf(1080,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ A ) @ C ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ A ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_set__mp) ).

thf(6380,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ A ) @ C ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ A ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1080]) ).

thf(888,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( c_Set_OCollect @ ( tc_prod @ D @ C ) ) @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ D @ C @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ D @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ D ) @ ( c_COMBB @ tc_HOL_Obool @ tc_HOL_Obool @ C ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ D ) @ c_fconj ) @ B ) ) ) @ A ) ) )
      = ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ D @ C ) @ ( hAPP @ ( c_Set_OCollect @ D ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) @ D ) @ ( c_Set_OCollect @ C ) ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_SetCompr__Sigma__eq) ).

thf(5586,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( c_Set_OCollect @ ( tc_prod @ D @ C ) ) @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ D @ C @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ D @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ D ) @ ( c_COMBB @ tc_HOL_Obool @ tc_HOL_Obool @ C ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ D ) @ c_fconj ) @ B ) ) ) @ A ) ) )
      = ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ D @ C ) @ ( hAPP @ ( c_Set_OCollect @ D ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) @ D ) @ ( c_Set_OCollect @ C ) ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[888]) ).

thf(260,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) @ C ) @ B ) ) @ ( c_List_Olistrel @ D @ A ) ) )
    <=> ( ( ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ D ) ) @ C )
          = ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ D ) ) @ B ) )
        & ! [E: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ E ) @ ( hAPP @ ( c_List_Oset @ ( tc_prod @ D @ D ) ) @ ( hAPP @ ( hAPP @ ( c_List_Ozip @ D @ D ) @ C ) @ B ) ) ) )
           => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ D @ D @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ D @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ D ) @ ( c_COMBC @ D @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ D @ ( tc_prod @ D @ D ) ) @ ( tc_fun @ D @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ D ) @ ( hAPP @ ( c_COMBB @ ( tc_prod @ D @ D ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ D ) @ ( c_member @ ( tc_prod @ D @ D ) ) ) ) @ ( c_Product__Type_OPair @ D @ D ) ) ) ) @ A ) ) @ E ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_listrel__iff__zip) ).

thf(3323,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) @ C ) @ B ) ) @ ( c_List_Olistrel @ D @ A ) ) )
       => ( ( ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ D ) ) @ C )
            = ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ D ) ) @ B ) )
          & ! [E: $i] :
              ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ E ) @ ( hAPP @ ( c_List_Oset @ ( tc_prod @ D @ D ) ) @ ( hAPP @ ( hAPP @ ( c_List_Ozip @ D @ D ) @ C ) @ B ) ) ) )
             => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ D @ D @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ D @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ D ) @ ( c_COMBC @ D @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ D @ ( tc_prod @ D @ D ) ) @ ( tc_fun @ D @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ D ) @ ( hAPP @ ( c_COMBB @ ( tc_prod @ D @ D ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ D ) @ ( c_member @ ( tc_prod @ D @ D ) ) ) ) @ ( c_Product__Type_OPair @ D @ D ) ) ) ) @ A ) ) @ E ) ) ) ) )
      & ( ( ( ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ D ) ) @ C )
            = ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ D ) ) @ B ) )
          & ! [E: $i] :
              ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ E ) @ ( hAPP @ ( c_List_Oset @ ( tc_prod @ D @ D ) ) @ ( hAPP @ ( hAPP @ ( c_List_Ozip @ D @ D ) @ C ) @ B ) ) ) )
             => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ D @ D @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ D @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ D ) @ ( c_COMBC @ D @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ D @ ( tc_prod @ D @ D ) ) @ ( tc_fun @ D @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) ) @ D ) @ ( hAPP @ ( c_COMBB @ ( tc_prod @ D @ D ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ D ) @ ( c_member @ ( tc_prod @ D @ D ) ) ) ) @ ( c_Product__Type_OPair @ D @ D ) ) ) ) @ A ) ) @ E ) ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) @ C ) @ B ) ) @ ( c_List_Olistrel @ D @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[260]) ).

thf(87,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Ocomm__monoid__add @ D )
     => ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ C ) @ B ) )
         => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ tc_Nat_Onat @ D ) @ A ) @ ( hAPP @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Nat_Onat @ B ) @ ( hAPP @ c_Nat_OSuc @ C ) ) )
            = ( c_Groups_Ozero__class_Ozero @ D ) ) )
        & ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ C ) @ B ) )
         => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ tc_Nat_Onat @ D ) @ A ) @ ( hAPP @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Nat_Onat @ B ) @ ( hAPP @ c_Nat_OSuc @ C ) ) )
            = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ tc_Nat_Onat @ D ) @ A ) @ ( hAPP @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Nat_Onat @ B ) @ C ) ) ) @ ( hAPP @ A @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_setsum__op__ivl__Suc) ).

thf(2714,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( class_Groups_Ocomm__monoid__add @ D )
     => ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ C ) @ B ) )
         => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ tc_Nat_Onat @ D ) @ A ) @ ( hAPP @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Nat_Onat @ B ) @ ( hAPP @ c_Nat_OSuc @ C ) ) )
            = ( c_Groups_Ozero__class_Ozero @ D ) ) )
        & ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ C ) @ B ) )
         => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ tc_Nat_Onat @ D ) @ A ) @ ( hAPP @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Nat_Onat @ B ) @ ( hAPP @ c_Nat_OSuc @ C ) ) )
            = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ D ) @ ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ tc_Nat_Onat @ D ) @ A ) @ ( hAPP @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Nat_Onat @ B ) @ C ) ) ) @ ( hAPP @ A @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[87]) ).

thf(288,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ B ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) ) ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Diff__insert2) ).

thf(3426,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ B ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) ) ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[288]) ).

thf(1988,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Wellfounded_Owf @ C @ B )
     => ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ C @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ C @ C ) @ A ) @ A ) ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_wf__not__refl) ).

thf(10237,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Wellfounded_Owf @ C @ B )
     => ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ C @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ C @ C ) @ A ) @ A ) ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1988]) ).

thf(1785,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i,I: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ I @ H ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ I @ H ) @ G ) @ F ) ) @ E ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ C ) @ ( hAPP @ B @ G ) ) @ ( hAPP @ A @ F ) ) ) @ ( hAPP @ ( c_Set_Oimage @ ( tc_prod @ I @ H ) @ ( tc_prod @ D @ C ) @ ( c_Product__Type_Omap__pair @ I @ D @ H @ C @ B @ A ) ) @ E ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_map__pair__imageI) ).

thf(9414,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i,I: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ I @ H ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ I @ H ) @ G ) @ F ) ) @ E ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ C ) @ ( hAPP @ B @ G ) ) @ ( hAPP @ A @ F ) ) ) @ ( hAPP @ ( c_Set_Oimage @ ( tc_prod @ I @ H ) @ ( tc_prod @ D @ C ) @ ( c_Product__Type_Omap__pair @ I @ D @ H @ C @ B @ A ) ) @ E ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1785]) ).

thf(1131,axiom,
    ! [A: $i,B: $i,C: $i] :
      ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ C ) @ ( tc_List_Olist @ C ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ C ) @ ( tc_List_Olist @ C ) ) @ ( c_List_Olist_ONil @ C ) ) @ B ) ) @ ( c_List_Olistrel1 @ C @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_not__Nil__listrel1) ).

thf(6682,plain,
    ! [A: $i,B: $i,C: $i] :
      ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ C ) @ ( tc_List_Olist @ C ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ C ) @ ( tc_List_Olist @ C ) ) @ ( c_List_Olist_ONil @ C ) ) @ B ) ) @ ( c_List_Olistrel1 @ C @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1131]) ).

thf(1593,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Relation_Osingle__valued @ E @ D @ C )
     => ( ( c_Relation_Osingle__valued @ D @ B @ A )
       => ( c_Relation_Osingle__valued @ E @ B @ ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ E @ D @ B ) @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_single__valued__rel__comp) ).

thf(8648,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Relation_Osingle__valued @ E @ D @ C )
     => ( ( c_Relation_Osingle__valued @ D @ B @ A )
       => ( c_Relation_Osingle__valued @ E @ B @ ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ E @ D @ B ) @ C ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1593]) ).

thf(499,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ C ) )
     => ( ( hAPP @ ( c_Set_Oimage @ D @ B @ A ) @ C )
        = ( hAPP @ ( hAPP @ ( hAPP @ ( c_Finite__Set_Ofold__image @ ( tc_fun @ B @ tc_HOL_Obool ) @ D @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ ( tc_fun @ B @ tc_HOL_Obool ) @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ B @ ( tc_fun @ ( tc_fun @ B @ tc_HOL_Obool ) @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ D ) @ ( c_Set_Oinsert @ B ) ) @ A ) ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_image__eq__fold__image) ).

thf(4193,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ C ) )
     => ( ( hAPP @ ( c_Set_Oimage @ D @ B @ A ) @ C )
        = ( hAPP @ ( hAPP @ ( hAPP @ ( c_Finite__Set_Ofold__image @ ( tc_fun @ B @ tc_HOL_Obool ) @ D @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ D @ ( tc_fun @ B @ tc_HOL_Obool ) @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ B @ ( tc_fun @ ( tc_fun @ B @ tc_HOL_Obool ) @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ D ) @ ( c_Set_Oinsert @ B ) ) @ A ) ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[499]) ).

thf(287,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ B ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ B ) @ A ) ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_div__nonpos__pos__le0) ).

thf(3424,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ B ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ B ) @ A ) ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[287]) ).

thf(701,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) @ B )
       != ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) @ A ) )
     => ( B != A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_neq__if__length__neq) ).

thf(4912,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) @ B )
       != ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) @ A ) )
     => ( B != A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[701]) ).

thf(1083,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Ozip @ E @ D ) @ ( hAPP @ ( hAPP @ ( c_List_Oappend @ E ) @ C ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_List_Oappend @ ( tc_prod @ E @ D ) ) @ ( hAPP @ ( hAPP @ ( c_List_Ozip @ E @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_List_Otake @ D ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ E ) ) @ C ) ) @ A ) ) ) @ ( hAPP @ ( hAPP @ ( c_List_Ozip @ E @ D ) @ B ) @ ( hAPP @ ( hAPP @ ( c_List_Odrop @ D ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ E ) ) @ C ) ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zip__append1) ).

thf(6393,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Ozip @ E @ D ) @ ( hAPP @ ( hAPP @ ( c_List_Oappend @ E ) @ C ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_List_Oappend @ ( tc_prod @ E @ D ) ) @ ( hAPP @ ( hAPP @ ( c_List_Ozip @ E @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_List_Otake @ D ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ E ) ) @ C ) ) @ A ) ) ) @ ( hAPP @ ( hAPP @ ( c_List_Ozip @ E @ D ) @ B ) @ ( hAPP @ ( hAPP @ ( c_List_Odrop @ D ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ E ) ) @ C ) ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1083]) ).

thf(1198,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ E ) @ D ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ E ) @ C ) ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ D ) @ C ) ) @ A ) ) )
      <=> ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ D ) @ E ) ) @ C ) ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nat__less__add__iff2) ).

thf(6929,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ E ) @ D ) )
     => ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ E ) @ C ) ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ D ) @ C ) ) @ A ) ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ D ) @ E ) ) @ C ) ) @ A ) ) ) )
        & ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ D ) @ E ) ) @ C ) ) @ A ) ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ E ) @ C ) ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ D ) @ C ) ) @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1198]) ).

thf(1913,axiom,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ A ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) )
    <=> ( A
        = ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nat__dvd__1__iff__1) ).

thf(9927,plain,
    ! [A: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ A ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) )
       => ( A
          = ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) )
      & ( ( A
          = ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ A ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1913]) ).

thf(1741,axiom,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ B ) @ A ) ) @ ( hAPP @ c_Nat_OSuc @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_diff__less__Suc) ).

thf(9250,plain,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ B ) @ A ) ) @ ( hAPP @ c_Nat_OSuc @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1741]) ).

thf(74,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Groups_Ocomm__monoid__add @ E )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ C ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ C ) )
         => ( ! [F: $i] :
                ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ F ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) ) )
               => ( ( hAPP @ A @ F )
                  = ( c_Groups_Ozero__class_Ozero @ E ) ) )
           => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ A ) @ C )
              = ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ A ) @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_setsum__mono__zero__right) ).

thf(2668,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Groups_Ocomm__monoid__add @ E )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ C ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ C ) )
         => ( ! [F: $i] :
                ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ F ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) ) )
               => ( ( hAPP @ A @ F )
                  = ( c_Groups_Ozero__class_Ozero @ E ) ) )
           => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ A ) @ C )
              = ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ A ) @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[74]) ).

thf(178,axiom,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) ) ) @ A ) ) @ c_FunDef_Omax__weak ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_wmax__emptyI) ).

thf(3045,plain,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) ) ) @ A ) ) @ c_FunDef_Omax__weak ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[178]) ).

thf(308,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ A @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_pos__mod__sign) ).

thf(3506,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ A @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[308]) ).

thf(773,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Set_Oimage @ D @ E @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Set_Oimage @ D @ E @ C ) @ B ) ) @ ( hAPP @ ( c_Set_Oimage @ D @ E @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_image__Int__subset) ).

thf(5176,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Set_Oimage @ D @ E @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Set_Oimage @ D @ E @ C ) @ B ) ) @ ( hAPP @ ( c_Set_Oimage @ D @ E @ C ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[773]) ).

thf(400,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ A ) @ B ) )
      = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Diff__disjoint) ).

thf(3843,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ A ) @ B ) )
      = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[400]) ).

thf(2056,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofold1Set @ E @ D @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ C ) @ B ) ) @ A ) )
     => ~ ! [F: $i,G: $i] :
            ( ( ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ C ) @ B )
              = ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ F ) @ G ) )
           => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofold__graph @ E @ E @ D @ F @ G ) @ A ) )
             => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ F ) @ G ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_insert__fold1SetE) ).

thf(10485,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofold1Set @ E @ D @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ C ) @ B ) ) @ A ) )
     => ~ ! [F: $i,G: $i] :
            ( ( ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ C ) @ B )
              = ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ F ) @ G ) )
           => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofold__graph @ E @ E @ D @ F @ G ) @ A ) )
             => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ F ) @ G ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2056]) ).

thf(1827,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ D ) @ C ) @ B ) ) @ ( c_Transitive__Closure_Ortrancl @ D @ A ) ) )
     => ( ( C = B )
        | ( ( C != B )
          & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ D ) @ C ) @ B ) ) @ ( c_Transitive__Closure_Otrancl @ D @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_rtranclD) ).

thf(9564,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ D ) @ C ) @ B ) ) @ ( c_Transitive__Closure_Ortrancl @ D @ A ) ) )
     => ( ( C = B )
        | ( ( C != B )
          & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ D ) @ C ) @ B ) ) @ ( c_Transitive__Closure_Otrancl @ D @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1827]) ).

thf(2017,axiom,
    ! [A: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Natural_Oevalc @ c_Com_Ocom_OSKIP ) @ A ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_evalc_Oequations_I1_J) ).

thf(10355,plain,
    ! [A: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Natural_Oevalc @ c_Com_Ocom_OSKIP ) @ A ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2017]) ).

thf(796,axiom,
    ! [A: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ A ) @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zmult__1__right) ).

thf(5283,plain,
    ! [A: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ A ) @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[796]) ).

thf(808,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hAPP @ ( c_Set_Ovimage @ F @ E @ D ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ C @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ C @ ( tc_fun @ F @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ E @ tc_HOL_Obool ) @ ( tc_fun @ F @ tc_HOL_Obool ) @ C ) @ ( c_Set_Ovimage @ F @ E @ D ) ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_vimage__UN) ).

thf(5322,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hAPP @ ( c_Set_Ovimage @ F @ E @ D ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ C @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ C @ ( tc_fun @ F @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ E @ tc_HOL_Obool ) @ ( tc_fun @ F @ tc_HOL_Obool ) @ C ) @ ( c_Set_Ovimage @ F @ E @ D ) ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[808]) ).

thf(1791,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__imp__le__nat) ).

thf(9441,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1791]) ).

thf(1737,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) ) )
    <=> ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ B ) )
        & ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Diff__iff) ).

thf(9235,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ B ) )
          & ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ A ) ) ) )
      & ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ B ) )
          & ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ A ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1737]) ).

thf(584,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ F ) ) @ E )
        = ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ D ) ) @ C ) )
     => ( ( c_Map_Omap__upds @ F @ D @ B @ E @ ( hAPP @ ( hAPP @ ( c_List_Oappend @ D ) @ C ) @ A ) )
        = ( c_Map_Omap__upds @ F @ D @ B @ E @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_fun__upds__append2__drop) ).

thf(4506,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ F ) ) @ E )
        = ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ D ) ) @ C ) )
     => ( ( c_Map_Omap__upds @ F @ D @ B @ E @ ( hAPP @ ( hAPP @ ( c_List_Oappend @ D ) @ C ) @ A ) )
        = ( c_Map_Omap__upds @ F @ D @ B @ E @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[584]) ).

thf(2151,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Map_Omap__add @ E @ D @ C @ ( c_Map_Omap__add @ E @ D @ B @ A ) )
      = ( c_Map_Omap__add @ E @ D @ ( c_Map_Omap__add @ E @ D @ C @ B ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_map__add__assoc) ).

thf(10845,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Map_Omap__add @ E @ D @ C @ ( c_Map_Omap__add @ E @ D @ B @ A ) )
      = ( c_Map_Omap__add @ E @ D @ ( c_Map_Omap__add @ E @ D @ C @ B ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2151]) ).

thf(1563,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ D ) @ C ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ D ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_add__less__mono) ).

thf(8453,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ D ) @ C ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ D ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1563]) ).

thf(1622,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) @ B ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( c_Wellfounded_Oaccp @ C @ A ) ) @ ( c_Wellfounded_Oaccp @ C @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_accp__subset) ).

thf(8773,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) @ B ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ ( c_Wellfounded_Oaccp @ C @ A ) ) @ ( c_Wellfounded_Oaccp @ C @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1622]) ).

thf(2314,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i,I: $i,J: $i,K: $i] :
      ( ( c_Com_Ocom_Ocom__rec @ K @ J @ I @ H @ G @ F @ E @ D @ C @ ( c_Com_Ocom_OWhile @ B @ A ) )
      = ( hAPP @ ( hAPP @ ( hAPP @ E @ B ) @ A ) @ ( c_Com_Ocom_Ocom__rec @ K @ J @ I @ H @ G @ F @ E @ D @ C @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_com_Orecs_I6_J) ).

thf(11489,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i,I: $i,J: $i,K: $i] :
      ( ( c_Com_Ocom_Ocom__rec @ K @ J @ I @ H @ G @ F @ E @ D @ C @ ( c_Com_Ocom_OWhile @ B @ A ) )
      = ( hAPP @ ( hAPP @ ( hAPP @ E @ B ) @ A ) @ ( c_Com_Ocom_Ocom__rec @ K @ J @ I @ H @ G @ F @ E @ D @ C @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2314]) ).

thf(1182,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ D ) @ C ) ) @ ( c_Transitive__Closure_Ortrancl @ E @ B ) ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ E @ E ) @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ E @ E ) @ A ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ E @ tc_HOL_Obool ) @ E ) @ A ) ) ) )
       => ( ( D = C )
          | ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ D ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_trancl__subset__Sigma__aux) ).

thf(6874,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ D ) @ C ) ) @ ( c_Transitive__Closure_Ortrancl @ E @ B ) ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ E @ E ) @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ E @ E ) @ A ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ E @ tc_HOL_Obool ) @ E ) @ A ) ) ) )
       => ( ( D = C )
          | ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ D ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1182]) ).

thf(1761,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ tc_Nat_Onat @ tc_Nat_Onat ) @ B ) @ A ) ) @ ( c_Transitive__Closure_Ortrancl @ tc_Nat_Onat @ c_Wellfounded_Opred__nat ) ) )
    <=> ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_pred__nat__trancl__eq__le) ).

thf(9325,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ tc_Nat_Onat @ tc_Nat_Onat ) @ B ) @ A ) ) @ ( c_Transitive__Closure_Ortrancl @ tc_Nat_Onat @ c_Wellfounded_Opred__nat ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ A ) ) )
      & ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ tc_Nat_Onat @ tc_Nat_Onat ) @ B ) @ A ) ) @ ( c_Transitive__Closure_Ortrancl @ tc_Nat_Onat @ c_Wellfounded_Opred__nat ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1761]) ).

thf(1821,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ D ) @ C ) ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ C ) @ A ) ) @ ( c_Transitive__Closure_Ortrancl @ E @ B ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ D ) @ A ) ) @ ( c_Transitive__Closure_Ortrancl @ E @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_converse__rtrancl__into__rtrancl) ).

thf(9547,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ D ) @ C ) ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ C ) @ A ) ) @ ( c_Transitive__Closure_Ortrancl @ E @ B ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ D ) @ A ) ) @ ( c_Transitive__Closure_Ortrancl @ E @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1821]) ).

thf(1608,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Divides_Odivmod__nat__rel @ C @ B ) @ A ) )
    <=> ( ( C
          = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( hAPP @ ( c_Product__Type_Ofst @ tc_Nat_Onat @ tc_Nat_Onat ) @ A ) ) @ B ) ) @ ( hAPP @ ( c_Product__Type_Osnd @ tc_Nat_Onat @ tc_Nat_Onat ) @ A ) ) )
        & ( ( B
            = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
         => ( ( hAPP @ ( c_Product__Type_Ofst @ tc_Nat_Onat @ tc_Nat_Onat ) @ A )
            = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) )
        & ( ( B
           != ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
         => ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ B ) )
             => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ ( hAPP @ ( c_Product__Type_Osnd @ tc_Nat_Onat @ tc_Nat_Onat ) @ A ) ) )
                & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ ( c_Product__Type_Osnd @ tc_Nat_Onat @ tc_Nat_Onat ) @ A ) ) @ B ) ) ) )
            & ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ B ) )
             => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ ( hAPP @ ( c_Product__Type_Osnd @ tc_Nat_Onat @ tc_Nat_Onat ) @ A ) ) )
                & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( c_Product__Type_Osnd @ tc_Nat_Onat @ tc_Nat_Onat ) @ A ) ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_divmod__nat__rel__def) ).

thf(8692,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( c_Divides_Odivmod__nat__rel @ C @ B ) @ A ) )
       => ( ( C
            = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( hAPP @ ( c_Product__Type_Ofst @ tc_Nat_Onat @ tc_Nat_Onat ) @ A ) ) @ B ) ) @ ( hAPP @ ( c_Product__Type_Osnd @ tc_Nat_Onat @ tc_Nat_Onat ) @ A ) ) )
          & ( ( B
              = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
           => ( ( hAPP @ ( c_Product__Type_Ofst @ tc_Nat_Onat @ tc_Nat_Onat ) @ A )
              = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) )
          & ( ( B
             != ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
           => ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ B ) )
               => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ ( hAPP @ ( c_Product__Type_Osnd @ tc_Nat_Onat @ tc_Nat_Onat ) @ A ) ) )
                  & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ ( c_Product__Type_Osnd @ tc_Nat_Onat @ tc_Nat_Onat ) @ A ) ) @ B ) ) ) )
              & ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ B ) )
               => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ ( hAPP @ ( c_Product__Type_Osnd @ tc_Nat_Onat @ tc_Nat_Onat ) @ A ) ) )
                  & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( c_Product__Type_Osnd @ tc_Nat_Onat @ tc_Nat_Onat ) @ A ) ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) ) ) ) ) ) )
      & ( ( ( C
            = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( hAPP @ ( c_Product__Type_Ofst @ tc_Nat_Onat @ tc_Nat_Onat ) @ A ) ) @ B ) ) @ ( hAPP @ ( c_Product__Type_Osnd @ tc_Nat_Onat @ tc_Nat_Onat ) @ A ) ) )
          & ( ( B
              = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
           => ( ( hAPP @ ( c_Product__Type_Ofst @ tc_Nat_Onat @ tc_Nat_Onat ) @ A )
              = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) )
          & ( ( B
             != ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
           => ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ B ) )
               => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ ( hAPP @ ( c_Product__Type_Osnd @ tc_Nat_Onat @ tc_Nat_Onat ) @ A ) ) )
                  & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ ( c_Product__Type_Osnd @ tc_Nat_Onat @ tc_Nat_Onat ) @ A ) ) @ B ) ) ) )
              & ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ B ) )
               => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ ( hAPP @ ( c_Product__Type_Osnd @ tc_Nat_Onat @ tc_Nat_Onat ) @ A ) ) )
                  & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( c_Product__Type_Osnd @ tc_Nat_Onat @ tc_Nat_Onat ) @ A ) ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) ) ) ) ) )
       => ( hBOOL @ ( hAPP @ ( c_Divides_Odivmod__nat__rel @ C @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1608]) ).

thf(236,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) )
     => ( ( c_SetInterval_Oord_OatLeastLessThan @ tc_Nat_Onat @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ tc_Nat_Onat @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ tc_Nat_Onat @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) @ tc_Nat_Onat ) @ ( c_COMBS @ tc_Nat_Onat @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ tc_Nat_Onat @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ tc_Nat_Onat ) @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ tc_Nat_Onat ) @ c_fconj ) ) @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ tc_Nat_Onat ) @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ tc_HOL_Obool @ tc_Nat_Onat ) @ c_fNot ) ) @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) ) ) ) @ A @ B )
        = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_dvd_OatLeastLessThan__empty) ).

thf(3244,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) )
     => ( ( c_SetInterval_Oord_OatLeastLessThan @ tc_Nat_Onat @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ tc_Nat_Onat @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ tc_Nat_Onat @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ ( tc_fun @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) @ tc_Nat_Onat ) @ ( c_COMBS @ tc_Nat_Onat @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ tc_Nat_Onat @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) ) @ tc_Nat_Onat ) @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ tc_Nat_Onat ) @ c_fconj ) ) @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ tc_Nat_Onat ) @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ tc_HOL_Obool @ tc_Nat_Onat ) @ c_fNot ) ) @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) ) ) ) @ A @ B )
        = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[236]) ).

thf(2257,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( c_New__DSequence_Oneg__single @ C @ B ) @ A )
      = ( c_Lazy__Sequence_Ohb__single @ C @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_New__DSequence_Oneg__single__def) ).

thf(11236,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( c_New__DSequence_Oneg__single @ C @ B ) @ A )
      = ( c_Lazy__Sequence_Ohb__single @ C @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2257]) ).

thf(704,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ F ) @ E ) @ D ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ C @ E ) @ B ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ F @ ( tc_fun @ A @ tc_HOL_Obool ) ) @ D ) @ C ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_SUP1__I) ).

thf(4923,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ F ) @ E ) @ D ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ C @ E ) @ B ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ F @ ( tc_fun @ A @ tc_HOL_Obool ) ) @ D ) @ C ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[704]) ).

thf(1156,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( c_Set_Oimage @ C @ C @ A ) @ B ) ) )
       => ( c_Fun_Oinj__on @ C @ C @ A @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_finite__surj__inj) ).

thf(6771,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( c_Set_Oimage @ C @ C @ A ) @ B ) ) )
       => ( c_Fun_Oinj__on @ C @ C @ A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1156]) ).

thf(978,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Wellfounded_Owf @ C @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) ) @ B ) @ A ) )
    <=> ( c_Wellfounded_Owf @ C @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ C @ C @ C ) @ B ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ C @ C @ C ) @ A ) @ B ) ) ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_wf__union__merge) ).

thf(5945,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( c_Wellfounded_Owf @ C @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) ) @ B ) @ A ) )
       => ( c_Wellfounded_Owf @ C @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ C @ C @ C ) @ B ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ C @ C @ C ) @ A ) @ B ) ) ) @ A ) ) )
      & ( ( c_Wellfounded_Owf @ C @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ C @ C @ C ) @ B ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ C @ C @ C ) @ A ) @ B ) ) ) @ A ) )
       => ( c_Wellfounded_Owf @ C @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[978]) ).

thf(859,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Int_Oint ) @ E ) @ D ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Int_Oint ) @ E ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ C ) @ B ) ) )
      <=> ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Int_Oint ) @ E ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ A ) @ D ) ) ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zdvd__period) ).

thf(5500,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Int_Oint ) @ E ) @ D ) )
     => ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Int_Oint ) @ E ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ C ) @ B ) ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Int_Oint ) @ E ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ A ) @ D ) ) ) @ B ) ) ) )
        & ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Int_Oint ) @ E ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ A ) @ D ) ) ) @ B ) ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Int_Oint ) @ E ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ C ) @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[859]) ).

thf(1565,axiom,
    ! [A: $i,B: $i] : ( c_Relation_Osingle__valued @ B @ B @ ( c_Relation_OId__on @ B @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_single__valued__Id__on) ).

thf(8460,plain,
    ! [A: $i,B: $i] : ( c_Relation_Osingle__valued @ B @ B @ ( c_Relation_OId__on @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1565]) ).

thf(1337,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Finite__Set_Ofolding__one @ E @ D @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ E ) @ B ) )
       => ( ( ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ E @ tc_HOL_Obool ) ) ) )
              = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ E @ tc_HOL_Obool ) ) )
           => ( ( hAPP @ C @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ A ) @ B ) )
              = A ) )
          & ( ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ E @ tc_HOL_Obool ) ) ) )
             != ( c_Orderings_Obot__class_Obot @ ( tc_fun @ E @ tc_HOL_Obool ) ) )
           => ( ( hAPP @ C @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ A ) @ B ) )
              = ( hAPP @ ( hAPP @ D @ A ) @ ( hAPP @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ E @ tc_HOL_Obool ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_folding__one_Oinsert__remove) ).

thf(7538,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Finite__Set_Ofolding__one @ E @ D @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ E ) @ B ) )
       => ( ( ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ E @ tc_HOL_Obool ) ) ) )
              = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ E @ tc_HOL_Obool ) ) )
           => ( ( hAPP @ C @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ A ) @ B ) )
              = A ) )
          & ( ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ E @ tc_HOL_Obool ) ) ) )
             != ( c_Orderings_Obot__class_Obot @ ( tc_fun @ E @ tc_HOL_Obool ) ) )
           => ( ( hAPP @ C @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ A ) @ B ) )
              = ( hAPP @ ( hAPP @ D @ A ) @ ( hAPP @ C @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ E @ tc_HOL_Obool ) ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1337]) ).

thf(268,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( c_Finite__Set_Ofolding__idem @ F @ E @ D @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ F ) @ B ) )
       => ( ( hAPP @ C @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ F ) @ A ) @ B ) )
          = ( hAPP @ ( c_Fun_Ocomp @ E @ E @ E @ ( hAPP @ C @ B ) ) @ ( hAPP @ D @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_folding__idem_Oinsert__idem) ).

thf(3362,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( c_Finite__Set_Ofolding__idem @ F @ E @ D @ C )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ F ) @ B ) )
       => ( ( hAPP @ C @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ F ) @ A ) @ B ) )
          = ( hAPP @ ( c_Fun_Ocomp @ E @ E @ E @ ( hAPP @ C @ B ) ) @ ( hAPP @ D @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[268]) ).

thf(1885,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( c_Wellfounded_Oacc @ D @ B ) ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ D ) @ A ) @ C ) ) @ B ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ A ) @ ( c_Wellfounded_Oacc @ D @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_acc__downward) ).

thf(9816,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( c_Wellfounded_Oacc @ D @ B ) ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ D @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ D @ D ) @ A ) @ C ) ) @ B ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ A ) @ ( c_Wellfounded_Oacc @ D @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1885]) ).

thf(92,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Groups_Ocomm__monoid__add @ E )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ C ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ B ) @ C ) )
         => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ A ) @ C )
            = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ E ) @ ( hAPP @ A @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ B ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_setsum_Oremove) ).

thf(2731,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Groups_Ocomm__monoid__add @ E )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ C ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ B ) @ C ) )
         => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ A ) @ C )
            = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ E ) @ ( hAPP @ A @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ B ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[92]) ).

thf(436,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( c_Relation_OImage @ D @ C @ B ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) )
      = ( hAPP @ ( c_Set_OCollect @ C ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ C @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_prod @ D @ C ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ C ) @ ( c_member @ ( tc_prod @ D @ C ) ) ) @ ( hAPP @ ( c_Product__Type_OPair @ D @ C ) @ A ) ) ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Image__singleton) ).

thf(3974,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( c_Relation_OImage @ D @ C @ B ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) )
      = ( hAPP @ ( c_Set_OCollect @ C ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ C @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_prod @ D @ C ) @ ( tc_fun @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ C ) @ ( c_member @ ( tc_prod @ D @ C ) ) ) @ ( hAPP @ ( c_Product__Type_OPair @ D @ C ) @ A ) ) ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[436]) ).

thf(2039,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Natural_Oevalc @ ( c_Com_Ocom_OSemi @ D @ C ) ) @ B ) @ A ) )
    <=> ? [E: $i] :
          ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Natural_Oevalc @ D ) @ B ) @ E ) )
          & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Natural_Oevalc @ C ) @ E ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_evalc_Oequations_I4_J) ).

thf(10421,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Natural_Oevalc @ ( c_Com_Ocom_OSemi @ D @ C ) ) @ B ) @ A ) )
       => ? [E: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Natural_Oevalc @ D ) @ B ) @ E ) )
            & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Natural_Oevalc @ C ) @ E ) @ A ) ) ) )
      & ( ? [E: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Natural_Oevalc @ D ) @ B ) @ E ) )
            & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Natural_Oevalc @ C ) @ E ) @ A ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Natural_Oevalc @ ( c_Com_Ocom_OSemi @ D @ C ) ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2039]) ).

thf(156,axiom,
    ! [A: $i,B: $i,C: $i] :
      ~ ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofold1Set @ C @ B @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_empty__fold1SetE) ).

thf(2966,plain,
    ! [A: $i,B: $i,C: $i] :
      ~ ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofold1Set @ C @ B @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[156]) ).

thf(1687,axiom,
    ! [A: $i,B: $i] :
      ( ( ( B
          = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
       => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A )
          = A ) )
      & ( ( B
         != ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
       => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A )
          = ( hAPP @ c_Nat_OSuc @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ B ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_add__eq__if) ).

thf(9038,plain,
    ! [A: $i,B: $i] :
      ( ( ( B
          = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
       => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A )
          = A ) )
      & ( ( B
         != ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
       => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A )
          = ( hAPP @ c_Nat_OSuc @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ B ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1687]) ).

thf(2075,axiom,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ tc_Nat_Onat ) @ ( c_SetInterval_Oord__class_OgreaterThanLessThan @ tc_Nat_Onat @ B @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_finite__greaterThanLessThan) ).

thf(10573,plain,
    ! [A: $i,B: $i] : ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ tc_Nat_Onat ) @ ( c_SetInterval_Oord__class_OgreaterThanLessThan @ tc_Nat_Onat @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2075]) ).

thf(1560,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) ) )
     => ~ ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ B ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_DiffE) ).

thf(8444,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) ) )
     => ~ ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ B ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1560]) ).

thf(1651,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ F @ ( tc_Option_Ooption @ E ) ) @ ( c_Map_Orestrict__map @ F @ E @ D @ C ) ) @ B ) @ A )
      = ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ F @ ( tc_Option_Ooption @ E ) ) @ ( c_Map_Orestrict__map @ F @ E @ D @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ F @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ F ) @ B ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ F @ tc_HOL_Obool ) ) ) ) ) ) @ B ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_fun__upd__restrict) ).

thf(8889,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ F @ ( tc_Option_Ooption @ E ) ) @ ( c_Map_Orestrict__map @ F @ E @ D @ C ) ) @ B ) @ A )
      = ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ F @ ( tc_Option_Ooption @ E ) ) @ ( c_Map_Orestrict__map @ F @ E @ D @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ F @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ F ) @ B ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ F @ tc_HOL_Obool ) ) ) ) ) ) @ B ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1651]) ).

thf(2235,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ C @ B ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ A ) @ B ) @ ( hAPP @ ( c_Set_OCollect @ A ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_CollectI) ).

thf(11144,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ C @ B ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ A ) @ B ) @ ( hAPP @ ( c_Set_OCollect @ A ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2235]) ).

thf(1853,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ E ) @ D ) @ C )
        = ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ E ) @ B ) @ A ) )
    <=> ( ( D = B )
        & ( C = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_list_Oinject) ).

thf(9679,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ E ) @ D ) @ C )
          = ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ E ) @ B ) @ A ) )
       => ( ( D = B )
          & ( C = A ) ) )
      & ( ( ( D = B )
          & ( C = A ) )
       => ( ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ E ) @ D ) @ C )
          = ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ E ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1853]) ).

thf(1347,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i] :
          ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ E ) @ ( hAPP @ ( c_List_Oset @ D ) @ C ) ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ B @ E ) ) @ ( hAPP @ A @ E ) ) ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( c_List_Olist_Olist__size @ D @ B @ C ) ) @ ( c_List_Olist_Olist__size @ D @ A @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_list__size__pointwise) ).

thf(7584,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ! [E: $i] :
          ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ E ) @ ( hAPP @ ( c_List_Oset @ D ) @ C ) ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ B @ E ) ) @ ( hAPP @ A @ E ) ) ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( c_List_Olist_Olist__size @ D @ B @ C ) ) @ ( c_List_Olist_Olist__size @ D @ A @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1347]) ).

thf(1266,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ H @ G ) @ tc_HOL_Obool ) ) @ F ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ H @ G ) @ E ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ G @ tc_HOL_Obool ) @ H ) @ D ) ) ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ G @ C ) @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ G @ C ) @ D ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ C @ tc_HOL_Obool ) @ G ) @ A ) ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ H @ C ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ H @ G @ C ) @ F ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ H @ C ) @ E ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ C @ tc_HOL_Obool ) @ H ) @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_rel__comp__subset__Sigma) ).

thf(7227,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ H @ G ) @ tc_HOL_Obool ) ) @ F ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ H @ G ) @ E ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ G @ tc_HOL_Obool ) @ H ) @ D ) ) ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ G @ C ) @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ G @ C ) @ D ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ C @ tc_HOL_Obool ) @ G ) @ A ) ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ H @ C ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ H @ G @ C ) @ F ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ H @ C ) @ E ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ C @ tc_HOL_Obool ) @ H ) @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1266]) ).

thf(666,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Fun_Oinj__on @ E @ D @ C @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_List_Oset @ E ) @ B ) ) @ ( hAPP @ ( c_List_Oset @ E ) @ A ) ) )
     => ( ( ( hAPP @ ( hAPP @ ( c_List_Omap @ E @ D ) @ C ) @ B )
          = ( hAPP @ ( hAPP @ ( c_List_Omap @ E @ D ) @ C ) @ A ) )
      <=> ( B = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_inj__on__map__eq__map) ).

thf(4799,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Fun_Oinj__on @ E @ D @ C @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_List_Oset @ E ) @ B ) ) @ ( hAPP @ ( c_List_Oset @ E ) @ A ) ) )
     => ( ( ( ( hAPP @ ( hAPP @ ( c_List_Omap @ E @ D ) @ C ) @ B )
            = ( hAPP @ ( hAPP @ ( c_List_Omap @ E @ D ) @ C ) @ A ) )
         => ( B = A ) )
        & ( ( B = A )
         => ( ( hAPP @ ( hAPP @ ( c_List_Omap @ E @ D ) @ C ) @ B )
            = ( hAPP @ ( hAPP @ ( c_List_Omap @ E @ D ) @ C ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[666]) ).

thf(2345,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Natural_Oevalc @ D ) @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Natural_Oevalc @ D ) @ C ) @ A ) )
       => ( A = B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_com__det) ).

thf(11621,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Natural_Oevalc @ D ) @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Natural_Oevalc @ D ) @ C ) @ A ) )
       => ( A = B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2345]) ).

thf(56,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Divides_Odivmod__int__rel @ G @ F ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ tc_Int_Oint @ tc_Int_Oint ) @ E ) @ D ) ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Divides_Odivmod__int__rel @ C @ F ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ tc_Int_Oint @ tc_Int_Oint ) @ B ) @ A ) ) )
       => ( ( F
           != ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) )
         => ( hBOOL @ ( hAPP @ ( c_Divides_Odivmod__int__rel @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ G ) @ C ) @ F ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ tc_Int_Oint @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ E ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ D ) @ A ) ) @ F ) ) ) @ ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ D ) @ A ) @ F ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zadd1__lemma) ).

thf(2600,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Divides_Odivmod__int__rel @ G @ F ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ tc_Int_Oint @ tc_Int_Oint ) @ E ) @ D ) ) )
     => ( ( hBOOL @ ( hAPP @ ( c_Divides_Odivmod__int__rel @ C @ F ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ tc_Int_Oint @ tc_Int_Oint ) @ B ) @ A ) ) )
       => ( ( F
           != ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) )
         => ( hBOOL @ ( hAPP @ ( c_Divides_Odivmod__int__rel @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ G ) @ C ) @ F ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ tc_Int_Oint @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ E ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ D ) @ A ) ) @ F ) ) ) @ ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ D ) @ A ) @ F ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[56]) ).

thf(337,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Transitive__Closure_Ortrancl @ C @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) ) @ ( c_Transitive__Closure_Ortrancl @ C @ B ) ) @ ( c_Transitive__Closure_Ortrancl @ C @ A ) ) )
      = ( c_Transitive__Closure_Ortrancl @ C @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_rtrancl__Un__rtrancl) ).

thf(3605,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Transitive__Closure_Ortrancl @ C @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) ) @ ( c_Transitive__Closure_Ortrancl @ C @ B ) ) @ ( c_Transitive__Closure_Ortrancl @ C @ A ) ) )
      = ( c_Transitive__Closure_Ortrancl @ C @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) ) @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[337]) ).

thf(713,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) ) )
    <=> ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ B ) )
        | ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Un__iff) ).

thf(4972,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ B ) )
          | ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ A ) ) ) )
      & ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ B ) )
          | ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ A ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[713]) ).

thf(2385,axiom,
    ! [A: $i] :
      ( ~ ( hBOOL @ ( hAPP @ c_fNot @ A ) )
      | ~ ( hBOOL @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',help_c__fNot__1) ).

thf(11794,plain,
    ! [A: $i] :
      ( ~ ( hBOOL @ ( hAPP @ c_fNot @ A ) )
      | ~ ( hBOOL @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2385]) ).

thf(2212,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Natural_Oevaln @ E @ D @ C @ B )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ C ) @ A ) )
       => ( c_Natural_Oevaln @ E @ D @ A @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_evaln__nonstrict) ).

thf(11050,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Natural_Oevaln @ E @ D @ C @ B )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ C ) @ A ) )
       => ( c_Natural_Oevaln @ E @ D @ A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2212]) ).

thf(900,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) @ A )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Int__UNIV__left) ).

thf(5637,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) @ A )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[900]) ).

thf(1214,axiom,
    ! [A: $i,B: $i] :
      ( ~ ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ B ) @ A ) )
     => ( ( hAPP @ ( c_Finite__Set_Ocard @ B ) @ A )
        = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_card__infinite) ).

thf(6993,plain,
    ! [A: $i,B: $i] :
      ( ~ ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ B ) @ A ) )
     => ( ( hAPP @ ( c_Finite__Set_Ocard @ B ) @ A )
        = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1214]) ).

thf(1992,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D = C )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ B ) @ D ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ B @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ B @ B ) @ D ) @ C ) ) @ ( c_Relation_OId__on @ B @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Id__on__eqI) ).

thf(10255,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D = C )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ B ) @ D ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ B @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ B @ B ) @ D ) @ C ) ) @ ( c_Relation_OId__on @ B @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1992]) ).

thf(2285,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( c_Nitpick_Opair__box_Opair__box__case @ F @ E @ D @ C @ ( c_Nitpick_Opair__box_OPairBox @ F @ E @ B @ A ) )
      = ( hAPP @ ( hAPP @ C @ B ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_pair__box_Osimps_I2_J) ).

thf(11358,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( c_Nitpick_Opair__box_Opair__box__case @ F @ E @ D @ C @ ( c_Nitpick_Opair__box_OPairBox @ F @ E @ B @ A ) )
      = ( hAPP @ ( hAPP @ C @ B ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2285]) ).

thf(1467,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Wellfounded_Oaccp @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ c_Nitpick_Onat__gcd__rel ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ tc_Nat_Onat @ tc_Nat_Onat ) @ C ) @ B ) ) )
     => ( ! [D: $i,E: $i] :
            ( ( hBOOL @ ( hAPP @ ( c_Wellfounded_Oaccp @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ c_Nitpick_Onat__gcd__rel ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ tc_Nat_Onat @ tc_Nat_Onat ) @ D ) @ E ) ) )
           => ( ( ( E
                 != ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
               => ( hBOOL @ ( hAPP @ ( hAPP @ A @ E ) @ ( c_Divides_Odiv__class_Omod @ tc_Nat_Onat @ D @ E ) ) ) )
             => ( hBOOL @ ( hAPP @ ( hAPP @ A @ D ) @ E ) ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ A @ C ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nat__gcd_Opinduct) ).

thf(8034,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Wellfounded_Oaccp @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ c_Nitpick_Onat__gcd__rel ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ tc_Nat_Onat @ tc_Nat_Onat ) @ C ) @ B ) ) )
     => ( ! [D: $i,E: $i] :
            ( ( hBOOL @ ( hAPP @ ( c_Wellfounded_Oaccp @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ c_Nitpick_Onat__gcd__rel ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ tc_Nat_Onat @ tc_Nat_Onat ) @ D ) @ E ) ) )
           => ( ( ( E
                 != ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) )
               => ( hBOOL @ ( hAPP @ ( hAPP @ A @ E ) @ ( c_Divides_Odiv__class_Omod @ tc_Nat_Onat @ D @ E ) ) ) )
             => ( hBOOL @ ( hAPP @ ( hAPP @ A @ D ) @ E ) ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ A @ C ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1467]) ).

thf(18,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( c_Relation_ODomain @ C @ B ) @ ( hAPP @ ( c_Set_OCollect @ ( tc_prod @ C @ B ) ) @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ C @ B @ tc_HOL_Obool ) @ A ) ) )
      = ( hAPP @ ( c_Set_OCollect @ C ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ B @ tc_HOL_Obool ) @ tc_HOL_Obool @ C ) @ ( c_HOL_OEx @ B ) ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Domain__Collect__split) ).

thf(2467,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( c_Relation_ODomain @ C @ B ) @ ( hAPP @ ( c_Set_OCollect @ ( tc_prod @ C @ B ) ) @ ( hAPP @ ( c_Product__Type_Oprod_Oprod__case @ C @ B @ tc_HOL_Obool ) @ A ) ) )
      = ( hAPP @ ( c_Set_OCollect @ C ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ ( tc_fun @ B @ tc_HOL_Obool ) @ tc_HOL_Obool @ C ) @ ( c_HOL_OEx @ B ) ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[18]) ).

thf(795,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_Nat_Onat_Onat__case @ D @ C @ B @ ( hAPP @ c_Nat_OSuc @ A ) )
      = ( hAPP @ B @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nat__case__Suc) ).

thf(5280,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_Nat_Onat_Onat__case @ D @ C @ B @ ( hAPP @ c_Nat_OSuc @ A ) )
      = ( hAPP @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[795]) ).

thf(2120,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_Com_Ocom_OLocal @ D @ C @ B )
     != ( hAPP @ c_Com_Ocom_OBODY @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_com_Osimps_I40_J) ).

thf(10713,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_Com_Ocom_OLocal @ D @ C @ B )
     != ( hAPP @ c_Com_Ocom_OBODY @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2120]) ).

thf(1357,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ C @ C ) @ A ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ C @ tc_HOL_Obool ) @ C ) @ A ) ) ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) ) @ ( c_Transitive__Closure_Otrancl @ C @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ C @ C ) @ A ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ C @ tc_HOL_Obool ) @ C ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_trancl__subset__Sigma) ).

thf(7616,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ C @ C ) @ A ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ C @ tc_HOL_Obool ) @ C ) @ A ) ) ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) ) @ ( c_Transitive__Closure_Otrancl @ C @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ C @ C ) @ A ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ C @ tc_HOL_Obool ) @ C ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1357]) ).

thf(913,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) @ ( hAPP @ ( hAPP @ ( c_List_Oappend @ C ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) @ B ) ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_length__append) ).

thf(5681,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) @ ( hAPP @ ( hAPP @ ( c_List_Oappend @ C ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) @ B ) ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[913]) ).

thf(1610,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ D ) @ C ) ) @ ( c_Transitive__Closure_Ortrancl @ E @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ ( tc_prod @ E @ E ) @ tc_HOL_Obool ) ) @ B ) @ A ) ) ) )
     => ( ! [F: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ F ) @ C ) ) @ ( c_Transitive__Closure_Ortrancl @ E @ B ) ) )
           => ! [G: $i] :
                ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ G ) @ F ) ) @ A ) )
               => ( G = F ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ D ) @ C ) ) @ ( c_Transitive__Closure_Ortrancl @ E @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_rtrancl__Un__separator__converseE) ).

thf(8730,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ D ) @ C ) ) @ ( c_Transitive__Closure_Ortrancl @ E @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ ( tc_prod @ E @ E ) @ tc_HOL_Obool ) ) @ B ) @ A ) ) ) )
     => ( ! [F: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ F ) @ C ) ) @ ( c_Transitive__Closure_Ortrancl @ E @ B ) ) )
           => ! [G: $i] :
                ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ G ) @ F ) ) @ A ) )
               => ( G = F ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ D ) @ C ) ) @ ( c_Transitive__Closure_Ortrancl @ E @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1610]) ).

thf(1783,axiom,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) )
        & ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ A ) @ B ) ) )
     => ~ ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ A ) @ B ) )
          & ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_dvd_Oless__asym) ).

thf(9402,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) )
        & ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ A ) @ B ) ) )
     => ~ ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ A ) @ B ) )
          & ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1783]) ).

thf(512,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ ( hAPP @ ( c_Set_OCollect @ C ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ C @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ C ) @ c_fdisj ) @ B ) ) @ A ) ) ) )
    <=> ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ ( hAPP @ ( c_Set_OCollect @ C ) @ B ) ) )
        & ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ ( hAPP @ ( c_Set_OCollect @ C ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_finite__Collect__disjI) ).

thf(4239,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ ( hAPP @ ( c_Set_OCollect @ C ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ C @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ C ) @ c_fdisj ) @ B ) ) @ A ) ) ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ ( hAPP @ ( c_Set_OCollect @ C ) @ B ) ) )
          & ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ ( hAPP @ ( c_Set_OCollect @ C ) @ A ) ) ) ) )
      & ( ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ ( hAPP @ ( c_Set_OCollect @ C ) @ B ) ) )
          & ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ ( hAPP @ ( c_Set_OCollect @ C ) @ A ) ) ) )
       => ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ ( hAPP @ ( c_Set_OCollect @ C ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ C @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ C ) @ c_fdisj ) @ B ) ) @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[512]) ).

thf(677,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ B ) @ A ) ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_div__nonneg__neg__le0) ).

thf(4831,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ A ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ B ) @ A ) ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[677]) ).

thf(738,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Wellfounded_Owf @ C @ ( hAPP @ ( hAPP @ ( c_Nat_Ocompow @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) @ B ) @ A ) )
     => ( c_Wellfounded_Owf @ C @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_wf__exp) ).

thf(5046,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Wellfounded_Owf @ C @ ( hAPP @ ( hAPP @ ( c_Nat_Ocompow @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) @ B ) @ A ) )
     => ( c_Wellfounded_Owf @ C @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[738]) ).

thf(2311,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_Com_Ocom_OSemi @ D @ C )
     != ( c_Com_Ocom_OWhile @ B @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_com_Osimps_I46_J) ).

thf(11477,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( c_Com_Ocom_OSemi @ D @ C )
     != ( c_Com_Ocom_OWhile @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2311]) ).

thf(2242,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ C @ C ) ) @ B ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ C @ C ) ) @ B ) @ ( c_Transitive__Closure_Ortrancl @ C @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_r__into__rtrancl) ).

thf(11182,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ C @ C ) ) @ B ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ C @ C ) ) @ B ) @ ( c_Transitive__Closure_Ortrancl @ C @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2242]) ).

thf(624,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ D ) ) @ C ) ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ D ) ) @ B ) ) )
     => ( C
       != ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ D ) @ A ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_impossible__Cons) ).

thf(4659,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ D ) ) @ C ) ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ D ) ) @ B ) ) )
     => ( C
       != ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ D ) @ A ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[624]) ).

thf(2161,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__le__mono1) ).

thf(10877,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ C ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ C ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2161]) ).

thf(2008,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ ( hAPP @ c_Nat_OSuc @ A ) ) )
     => ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ A ) )
       => ( B
          = ( hAPP @ c_Nat_OSuc @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_le__SucE) ).

thf(10323,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ ( hAPP @ c_Nat_OSuc @ A ) ) )
     => ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ A ) )
       => ( B
          = ( hAPP @ c_Nat_OSuc @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2008]) ).

thf(2068,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_le__trans) ).

thf(10535,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ C ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2068]) ).

thf(707,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Relation_ODomain @ D @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) ) @ B ) @ A ) ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Relation_ODomain @ D @ C ) @ B ) ) @ ( hAPP @ ( c_Relation_ODomain @ D @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Domain__Int__subset) ).

thf(4936,plain,
    ! [A: $i,B: $i,C: $i,D: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Relation_ODomain @ D @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) ) @ B ) @ A ) ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Relation_ODomain @ D @ C ) @ B ) ) @ ( hAPP @ ( c_Relation_ODomain @ D @ C ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[707]) ).

thf(2277,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D != C )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ B ) @ D ) @ ( hAPP @ ( c_List_Oset @ B ) @ ( c_List_Oremove1 @ B @ C @ A ) ) ) )
      <=> ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ B ) @ D ) @ ( hAPP @ ( c_List_Oset @ B ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_in__set__remove1) ).

thf(11330,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D != C )
     => ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ B ) @ D ) @ ( hAPP @ ( c_List_Oset @ B ) @ ( c_List_Oremove1 @ B @ C @ A ) ) ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ B ) @ D ) @ ( hAPP @ ( c_List_Oset @ B ) @ A ) ) ) )
        & ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ B ) @ D ) @ ( hAPP @ ( c_List_Oset @ B ) @ A ) ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ B ) @ D ) @ ( hAPP @ ( c_List_Oset @ B ) @ ( c_List_Oremove1 @ B @ C @ A ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2277]) ).

thf(1960,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( hBOOL @ ( hAPP @ D @ C ) )
       => ( ( hAPP @ ( c_List_Ofilter @ B @ D ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ B ) @ C ) @ A ) )
          = ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ B ) @ C ) @ ( hAPP @ ( c_List_Ofilter @ B @ D ) @ A ) ) ) )
      & ( ~ ( hBOOL @ ( hAPP @ D @ C ) )
       => ( ( hAPP @ ( c_List_Ofilter @ B @ D ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ B ) @ C ) @ A ) )
          = ( hAPP @ ( c_List_Ofilter @ B @ D ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_filter_Osimps_I2_J) ).

thf(10124,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( hBOOL @ ( hAPP @ D @ C ) )
       => ( ( hAPP @ ( c_List_Ofilter @ B @ D ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ B ) @ C ) @ A ) )
          = ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ B ) @ C ) @ ( hAPP @ ( c_List_Ofilter @ B @ D ) @ A ) ) ) )
      & ( ~ ( hBOOL @ ( hAPP @ D @ C ) )
       => ( ( hAPP @ ( c_List_Ofilter @ B @ D ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ B ) @ C ) @ A ) )
          = ( hAPP @ ( c_List_Ofilter @ B @ D ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1960]) ).

thf(2097,axiom,
    ! [A: $i] :
      ( ( hAPP @ c_Nat_OSuc @ A )
     != A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Suc__n__not__n) ).

thf(10642,plain,
    ! [A: $i] :
      ( ( hAPP @ c_Nat_OSuc @ A )
     != A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2097]) ).

thf(426,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ C ) @ ( tc_List_Olist @ C ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ C ) @ ( tc_List_Olist @ C ) ) @ B ) @ ( c_List_Olist_ONil @ C ) ) ) @ ( c_List_Olistrel @ C @ A ) ) )
     => ( B
        = ( c_List_Olist_ONil @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_listrel__Nil2) ).

thf(3939,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ C ) @ ( tc_List_Olist @ C ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ C ) @ ( tc_List_Olist @ C ) ) @ B ) @ ( c_List_Olist_ONil @ C ) ) ) @ ( c_List_Olistrel @ C @ A ) ) )
     => ( B
        = ( c_List_Olist_ONil @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[426]) ).

thf(964,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Transitive__Closure_Ortrancl @ B @ A )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) ) @ ( c_Relation_OId @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ B @ B @ B ) @ ( c_Transitive__Closure_Ortrancl @ B @ A ) ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_rtrancl__unfold) ).

thf(5891,plain,
    ! [A: $i,B: $i] :
      ( ( c_Transitive__Closure_Ortrancl @ B @ A )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) ) @ ( c_Relation_OId @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ B @ B @ B ) @ ( c_Transitive__Closure_Ortrancl @ B @ A ) ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[964]) ).

thf(817,axiom,
    ! [A: $i,B: $i] :
      ( ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Int_Oint @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ B ) @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ) @ A )
      = ( c_SetInterval_Oord__class_OgreaterThanAtMost @ tc_Int_Oint @ B @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_atLeastPlusOneAtMost__greaterThanAtMost__int) ).

thf(5354,plain,
    ! [A: $i,B: $i] :
      ( ( c_SetInterval_Oord__class_OatLeastAtMost @ tc_Int_Oint @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ B ) @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ) @ A )
      = ( c_SetInterval_Oord__class_OgreaterThanAtMost @ tc_Int_Oint @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[817]) ).

thf(1012,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ B @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ A ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ B @ ( tc_fun @ B @ tc_HOL_Obool ) @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ ( c_Set_Oinsert @ B ) ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_UN__singleton) ).

thf(6109,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ B @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ A ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ B @ ( tc_fun @ B @ tc_HOL_Obool ) @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ ( c_Set_Oinsert @ B ) ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1012]) ).

thf(1380,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) ) )
    <=> ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ B ) )
        | ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_add__gr__0) ).

thf(7694,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ B ) )
          | ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ A ) ) ) )
      & ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ B ) )
          | ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ A ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1380]) ).

thf(1892,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( c_Com_Ocom_OLocal @ F @ E @ D )
        = ( c_Com_Ocom_OLocal @ C @ B @ A ) )
    <=> ( ( F = C )
        & ( E = B )
        & ( D = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_com_Osimps_I2_J) ).

thf(9848,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( ( c_Com_Ocom_OLocal @ F @ E @ D )
          = ( c_Com_Ocom_OLocal @ C @ B @ A ) )
       => ( ( F = C )
          & ( E = B )
          & ( D = A ) ) )
      & ( ( ( F = C )
          & ( E = B )
          & ( D = A ) )
       => ( ( c_Com_Ocom_OLocal @ F @ E @ D )
          = ( c_Com_Ocom_OLocal @ C @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1892]) ).

thf(1159,axiom,
    ! [A: $i,B: $i] :
      ( ( ( ( hAPP @ c_Nat_OSuc @ ( c_Divides_Odiv__class_Omod @ tc_Nat_Onat @ B @ A ) )
          = A )
       => ( ( c_Divides_Odiv__class_Omod @ tc_Nat_Onat @ ( hAPP @ c_Nat_OSuc @ B ) @ A )
          = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) )
      & ( ( ( hAPP @ c_Nat_OSuc @ ( c_Divides_Odiv__class_Omod @ tc_Nat_Onat @ B @ A ) )
         != A )
       => ( ( c_Divides_Odiv__class_Omod @ tc_Nat_Onat @ ( hAPP @ c_Nat_OSuc @ B ) @ A )
          = ( hAPP @ c_Nat_OSuc @ ( c_Divides_Odiv__class_Omod @ tc_Nat_Onat @ B @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mod__Suc) ).

thf(6778,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( hAPP @ c_Nat_OSuc @ ( c_Divides_Odiv__class_Omod @ tc_Nat_Onat @ B @ A ) )
          = A )
       => ( ( c_Divides_Odiv__class_Omod @ tc_Nat_Onat @ ( hAPP @ c_Nat_OSuc @ B ) @ A )
          = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) )
      & ( ( ( hAPP @ c_Nat_OSuc @ ( c_Divides_Odiv__class_Omod @ tc_Nat_Onat @ B @ A ) )
         != A )
       => ( ( c_Divides_Odiv__class_Omod @ tc_Nat_Onat @ ( hAPP @ c_Nat_OSuc @ B ) @ A )
          = ( hAPP @ c_Nat_OSuc @ ( c_Divides_Odiv__class_Omod @ tc_Nat_Onat @ B @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1159]) ).

thf(82,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ B ) ) @ ( c_List_Oremdups @ B @ ( c_List_Oconcat @ B @ A ) ) )
      = ( hAPP @ ( c_Finite__Set_Ocard @ B ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ ( tc_List_Olist @ B ) @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_List_Oset @ ( tc_List_Olist @ B ) ) @ A ) ) @ ( c_List_Oset @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_length__remdups__concat) ).

thf(2699,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ B ) ) @ ( c_List_Oremdups @ B @ ( c_List_Oconcat @ B @ A ) ) )
      = ( hAPP @ ( c_Finite__Set_Ocard @ B ) @ ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ ( tc_List_Olist @ B ) @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_List_Oset @ ( tc_List_Olist @ B ) ) @ A ) ) @ ( c_List_Oset @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[82]) ).

thf(1646,axiom,
    ! [A: $i] : ( c_Relation_Osingle__valued @ A @ A @ ( c_Relation_OId @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_single__valued__Id) ).

thf(8876,plain,
    ! [A: $i] : ( c_Relation_Osingle__valued @ A @ A @ ( c_Relation_OId @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1646]) ).

thf(391,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Transitive__Closure_Otrancl @ B @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) ) @ A ) @ ( c_Relation_OId @ B ) ) )
      = ( c_Transitive__Closure_Ortrancl @ B @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_trancl__reflcl) ).

thf(3812,plain,
    ! [A: $i,B: $i] :
      ( ( c_Transitive__Closure_Otrancl @ B @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) ) @ A ) @ ( c_Relation_OId @ B ) ) )
      = ( c_Transitive__Closure_Ortrancl @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[391]) ).

thf(1257,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) @ C ) @ B ) ) @ ( c_List_Olistrel1 @ D @ A ) ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) @ C ) @ B ) ) @ ( c_List_Olistrel @ D @ ( c_Transitive__Closure_Ortrancl @ D @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_listrel__reflcl__if__listrel1) ).

thf(7176,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) @ C ) @ B ) ) @ ( c_List_Olistrel1 @ D @ A ) ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) @ C ) @ B ) ) @ ( c_List_Olistrel @ D @ ( c_Transitive__Closure_Ortrancl @ D @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1257]) ).

thf(23,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Ocomm__monoid__add @ C )
     => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ B @ C ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) )
        = ( c_Groups_Ozero__class_Ozero @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_setsum__empty) ).

thf(2484,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( class_Groups_Ocomm__monoid__add @ C )
     => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ B @ C ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) )
        = ( c_Groups_Ozero__class_Ozero @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[23]) ).

thf(2231,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
      ( ( c_Equiv__Relations_Oequiv @ H @ G @ F )
     => ( ( c_Equiv__Relations_Ocongruent2 @ H @ E @ D @ F @ C @ B )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ H ) @ A ) @ G ) )
         => ( c_Equiv__Relations_Ocongruent @ E @ D @ C @ ( hAPP @ B @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_congruent2__implies__congruent) ).

thf(11135,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
      ( ( c_Equiv__Relations_Oequiv @ H @ G @ F )
     => ( ( c_Equiv__Relations_Ocongruent2 @ H @ E @ D @ F @ C @ B )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ H ) @ A ) @ G ) )
         => ( c_Equiv__Relations_Ocongruent @ E @ D @ C @ ( hAPP @ B @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2231]) ).

thf(538,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ B ) @ C ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) )
     => ( hBOOL @ ( hAPP @ A @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_ball__empty) ).

thf(4345,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ B ) @ C ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) )
     => ( hBOOL @ ( hAPP @ A @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[538]) ).

thf(1017,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ B ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) )
     => ( ( c_Fun_Oinj__on @ B @ B @ A @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ B @ tc_HOL_Obool ) ) )
       => ( ( hAPP @ ( c_Set_Oimage @ B @ B @ A ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ B @ tc_HOL_Obool ) ) )
          = ( c_Orderings_Otop__class_Otop @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_finite__UNIV__inj__surj) ).

thf(6126,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ B ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) )
     => ( ( c_Fun_Oinj__on @ B @ B @ A @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ B @ tc_HOL_Obool ) ) )
       => ( ( hAPP @ ( c_Set_Oimage @ B @ B @ A ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ B @ tc_HOL_Obool ) ) )
          = ( c_Orderings_Otop__class_Otop @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1017]) ).

thf(519,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ tc_Nat_Onat ) ) @ ( c_List_Oupt @ B @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ A ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_length__upt) ).

thf(4271,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ tc_Nat_Onat ) ) @ ( c_List_Oupt @ B @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ A ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[519]) ).

thf(473,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Oab__semigroup__mult @ B )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ B ) @ A ) )
       => ( ( A
           != ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) )
         => ( ! [C: $i,D: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ B ) @ C ) @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ B ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ B ) @ D ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) ) ) )
           => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ B ) @ ( hAPP @ ( c_Finite__Set_Ofold1 @ B @ ( c_Groups_Otimes__class_Otimes @ B ) ) @ A ) ) @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_fold1__in) ).

thf(4088,plain,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Oab__semigroup__mult @ B )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ B ) @ A ) )
       => ( ( A
           != ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) )
         => ( ! [C: $i,D: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ B ) @ C ) @ D ) ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ B ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ B ) @ D ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) ) ) )
           => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ B ) @ ( hAPP @ ( c_Finite__Set_Ofold1 @ B @ ( c_Groups_Otimes__class_Otimes @ B ) ) @ A ) ) @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[473]) ).

thf(1273,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ B ) @ A ) ) )
    <=> ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ A ) ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__diff__conv) ).

thf(7246,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ B ) @ A ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ A ) ) @ B ) ) )
      & ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ C ) @ A ) ) @ B ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ B ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1273]) ).

thf(2290,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ c_Com_OWT @ ( c_Com_Ocom_OWhile @ B @ A ) ) )
    <=> ( hBOOL @ ( hAPP @ c_Com_OWT @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_WT_Oequations_I6_J) ).

thf(11373,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ c_Com_OWT @ ( c_Com_Ocom_OWhile @ B @ A ) ) )
       => ( hBOOL @ ( hAPP @ c_Com_OWT @ A ) ) )
      & ( ( hBOOL @ ( hAPP @ c_Com_OWT @ A ) )
       => ( hBOOL @ ( hAPP @ c_Com_OWT @ ( c_Com_Ocom_OWhile @ B @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2290]) ).

thf(1342,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( c_Finite__Set_Ofun__left__comm @ F @ E @ D )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ F ) @ C ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ F ) @ B ) @ C ) )
         => ( ( hAPP @ ( hAPP @ ( c_Finite__Set_Ofold @ F @ E @ D ) @ A ) @ C )
            = ( hAPP @ ( hAPP @ D @ B ) @ ( hAPP @ ( hAPP @ ( c_Finite__Set_Ofold @ F @ E @ D ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ F @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ F ) @ B ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ F @ tc_HOL_Obool ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_fun__left__comm_Ofold__rec) ).

thf(7563,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( c_Finite__Set_Ofun__left__comm @ F @ E @ D )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ F ) @ C ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ F ) @ B ) @ C ) )
         => ( ( hAPP @ ( hAPP @ ( c_Finite__Set_Ofold @ F @ E @ D ) @ A ) @ C )
            = ( hAPP @ ( hAPP @ D @ B ) @ ( hAPP @ ( hAPP @ ( c_Finite__Set_Ofold @ F @ E @ D ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ F @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ F ) @ B ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ F @ tc_HOL_Obool ) ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1342]) ).

thf(507,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ B ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ B ) @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ) ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zless__imp__add1__zle) ).

thf(4221,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ B ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ B ) @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ) ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[507]) ).

thf(849,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ D @ D @ D ) @ C ) @ B )
        = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) ) )
     => ( ( ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ D @ D @ D ) @ C ) @ A )
          = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) ) )
       => ( ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ D @ D @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) ) @ B ) @ A ) )
          = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_union__comp__emptyR) ).

thf(5473,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ D @ D @ D ) @ C ) @ B )
        = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) ) )
     => ( ( ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ D @ D @ D ) @ C ) @ A )
          = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) ) )
       => ( ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ D @ D @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) ) @ B ) @ A ) )
          = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ ( tc_prod @ D @ D ) @ tc_HOL_Obool ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[849]) ).

thf(2266,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ D ) @ C ) )
     => ( ( hAPP @ ( c_Map_Orestrict__map @ E @ B @ A @ C ) @ D )
        = ( hAPP @ A @ D ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_restrict__in) ).

thf(11276,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ D ) @ C ) )
     => ( ( hAPP @ ( c_Map_Orestrict__map @ E @ B @ A @ C ) @ D )
        = ( hAPP @ A @ D ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2266]) ).

thf(629,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( c_Set_Oimage @ tc_Nat_Onat @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ tc_Nat_Onat @ tc_Nat_Onat ) @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) ) @ C ) ) @ ( hAPP @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Nat_Onat @ B ) @ A ) )
      = ( hAPP @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_image__add__atLeastLessThan) ).

thf(4674,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( c_Set_Oimage @ tc_Nat_Onat @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ tc_Nat_Onat @ tc_Nat_Onat ) @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) ) @ C ) ) @ ( hAPP @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Nat_Onat @ B ) @ A ) )
      = ( hAPP @ ( c_SetInterval_Oord__class_OatLeastLessThan @ tc_Nat_Onat @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ B ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[629]) ).

thf(168,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Finite__Set_Ofolding__one__idem @ E @ D @ C )
     => ( ! [F: $i,G: $i] :
            ( ( hAPP @ B @ ( hAPP @ ( hAPP @ D @ F ) @ G ) )
            = ( hAPP @ ( hAPP @ D @ ( hAPP @ B @ F ) ) @ ( hAPP @ B @ G ) ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ E ) @ A ) )
         => ( ( A
             != ( c_Orderings_Obot__class_Obot @ ( tc_fun @ E @ tc_HOL_Obool ) ) )
           => ( ( hAPP @ B @ ( hAPP @ C @ A ) )
              = ( hAPP @ C @ ( hAPP @ ( c_Set_Oimage @ E @ E @ B ) @ A ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_folding__one__idem_Ohom__commute) ).

thf(3013,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Finite__Set_Ofolding__one__idem @ E @ D @ C )
     => ( ! [F: $i,G: $i] :
            ( ( hAPP @ B @ ( hAPP @ ( hAPP @ D @ F ) @ G ) )
            = ( hAPP @ ( hAPP @ D @ ( hAPP @ B @ F ) ) @ ( hAPP @ B @ G ) ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ E ) @ A ) )
         => ( ( A
             != ( c_Orderings_Obot__class_Obot @ ( tc_fun @ E @ tc_HOL_Obool ) ) )
           => ( ( hAPP @ B @ ( hAPP @ C @ A ) )
              = ( hAPP @ C @ ( hAPP @ ( c_Set_Oimage @ E @ E @ B ) @ A ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[168]) ).

thf(857,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Complete__Lattice_OSup__class_OSup @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ ( c_Complete__Lattice_OSup__class_OSup @ ( tc_fun @ C @ tc_HOL_Obool ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Union__insert) ).

thf(5494,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( c_Complete__Lattice_OSup__class_OSup @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ ( c_Complete__Lattice_OSup__class_OSup @ ( tc_fun @ C @ tc_HOL_Obool ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[857]) ).

thf(1875,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Natural_Oevalc @ ( c_Com_Ocom_OCond @ E @ D @ C ) ) @ B ) @ A ) )
    <=> ( ( ( hBOOL @ ( hAPP @ E @ B ) )
          & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Natural_Oevalc @ D ) @ B ) @ A ) ) )
        | ( ~ ( hBOOL @ ( hAPP @ E @ B ) )
          & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Natural_Oevalc @ C ) @ B ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_evalc_Oequations_I5_J) ).

thf(9784,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Natural_Oevalc @ ( c_Com_Ocom_OCond @ E @ D @ C ) ) @ B ) @ A ) )
       => ( ( ( hBOOL @ ( hAPP @ E @ B ) )
            & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Natural_Oevalc @ D ) @ B ) @ A ) ) )
          | ( ~ ( hBOOL @ ( hAPP @ E @ B ) )
            & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Natural_Oevalc @ C ) @ B ) @ A ) ) ) ) )
      & ( ( ( ( hBOOL @ ( hAPP @ E @ B ) )
            & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Natural_Oevalc @ D ) @ B ) @ A ) ) )
          | ( ~ ( hBOOL @ ( hAPP @ E @ B ) )
            & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Natural_Oevalc @ C ) @ B ) @ A ) ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Natural_Oevalc @ ( c_Com_Ocom_OCond @ E @ D @ C ) ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1875]) ).

thf(118,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ D ) @ C ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ D ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ C ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Un__mono) ).

thf(2832,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ D ) @ C ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ D ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ C ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[118]) ).

thf(647,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ F @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ D ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ C @ ( tc_fun @ E @ tc_HOL_Obool ) @ F ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ C @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Set_Oimage @ F @ C @ A ) @ D ) ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_UN__extend__simps_I10_J) ).

thf(4734,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ F @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ D ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ C @ ( tc_fun @ E @ tc_HOL_Obool ) @ F ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ C @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Set_Oimage @ F @ C @ A ) @ D ) ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[647]) ).

thf(1605,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) ) @ B ) @ ( c_Transitive__Closure_Ortrancl @ C @ A ) ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) ) @ ( c_Transitive__Closure_Ortrancl @ C @ B ) ) @ ( c_Transitive__Closure_Ortrancl @ C @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_rtrancl__subset__rtrancl) ).

thf(8685,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) ) @ B ) @ ( c_Transitive__Closure_Ortrancl @ C @ A ) ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ C @ C ) @ tc_HOL_Obool ) ) @ ( c_Transitive__Closure_Ortrancl @ C @ B ) ) @ ( c_Transitive__Closure_Ortrancl @ C @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1605]) ).

thf(1239,axiom,
    ! [A: $i,B: $i,C: $i] :
      ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ C ) @ ( tc_List_Olist @ C ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ C ) @ ( tc_List_Olist @ C ) ) @ B ) @ ( c_List_Olist_ONil @ C ) ) ) @ ( c_List_Olistrel1 @ C @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_not__listrel1__Nil) ).

thf(7103,plain,
    ! [A: $i,B: $i,C: $i] :
      ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_List_Olist @ C ) @ ( tc_List_Olist @ C ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_List_Olist @ C ) @ ( tc_List_Olist @ C ) ) @ B ) @ ( c_List_Olist_ONil @ C ) ) ) @ ( c_List_Olistrel1 @ C @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1239]) ).

thf(2028,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( hAPP @ ( c_Sum__Type_OInr @ D @ C ) @ B )
        = ( hAPP @ ( c_Sum__Type_OInr @ D @ C ) @ A ) )
    <=> ( B = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_sum_Osimps_I2_J) ).

thf(10383,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( ( hAPP @ ( c_Sum__Type_OInr @ D @ C ) @ B )
          = ( hAPP @ ( c_Sum__Type_OInr @ D @ C ) @ A ) )
       => ( B = A ) )
      & ( ( B = A )
       => ( ( hAPP @ ( c_Sum__Type_OInr @ D @ C ) @ B )
          = ( hAPP @ ( c_Sum__Type_OInr @ D @ C ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2028]) ).

thf(99,axiom,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Ocomm__monoid__mult @ B )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ B ) @ A ) @ ( c_Groups_Oone__class_Oone @ B ) )
        = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult_Ocomm__neutral) ).

thf(2754,plain,
    ! [A: $i,B: $i] :
      ( ( class_Groups_Ocomm__monoid__mult @ B )
     => ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ B ) @ A ) @ ( c_Groups_Oone__class_Oone @ B ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[99]) ).

thf(772,axiom,
    ! [A: $i] :
      ( ( c_Orderings_Otop__class_Otop @ ( tc_fun @ A @ tc_HOL_Obool ) )
     != ( c_Orderings_Obot__class_Obot @ ( tc_fun @ A @ tc_HOL_Obool ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_UNIV__not__empty) ).

thf(5172,plain,
    ! [A: $i] :
      ( ( c_Orderings_Otop__class_Otop @ ( tc_fun @ A @ tc_HOL_Obool ) )
     != ( c_Orderings_Obot__class_Obot @ ( tc_fun @ A @ tc_HOL_Obool ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[772]) ).

thf(429,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ C ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_insert__inter__insert) ).

thf(3950,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ C ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[429]) ).

thf(732,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ B ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ B ) @ A ) ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_div__neg__pos__less0) ).

thf(5029,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ B ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ B ) @ A ) ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[732]) ).

thf(297,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( c_Equiv__Relations_Oequiv @ F @ E @ D )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ F ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ F @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Relation_OImage @ F @ F @ D ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ F ) @ B ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ F @ tc_HOL_Obool ) ) ) ) ) @ ( hAPP @ ( c_Relation_OImage @ F @ F @ D ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ F ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ F @ tc_HOL_Obool ) ) ) ) ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ F @ F ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ F @ F ) @ B ) @ A ) ) @ D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_equiv__class__nondisjoint) ).

thf(3461,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( c_Equiv__Relations_Oequiv @ F @ E @ D )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ F ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ F @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Relation_OImage @ F @ F @ D ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ F ) @ B ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ F @ tc_HOL_Obool ) ) ) ) ) @ ( hAPP @ ( c_Relation_OImage @ F @ F @ D ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ F ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ F @ tc_HOL_Obool ) ) ) ) ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ F @ F ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ F @ F ) @ B ) @ A ) ) @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[297]) ).

thf(890,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( hAPP @ ( c_Set_Oimage @ E @ D @ C ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ E @ tc_HOL_Obool ) ) )
        = ( c_Orderings_Otop__class_Otop @ ( tc_fun @ D @ tc_HOL_Obool ) ) )
     => ( ( ( hAPP @ ( c_Set_Oimage @ D @ B @ A ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ D @ tc_HOL_Obool ) ) )
          = ( c_Orderings_Otop__class_Otop @ ( tc_fun @ B @ tc_HOL_Obool ) ) )
       => ( ( hAPP @ ( c_Set_Oimage @ E @ B @ ( hAPP @ ( c_Fun_Ocomp @ D @ B @ E @ A ) @ C ) ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ E @ tc_HOL_Obool ) ) )
          = ( c_Orderings_Otop__class_Otop @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_comp__surj) ).

thf(5591,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( hAPP @ ( c_Set_Oimage @ E @ D @ C ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ E @ tc_HOL_Obool ) ) )
        = ( c_Orderings_Otop__class_Otop @ ( tc_fun @ D @ tc_HOL_Obool ) ) )
     => ( ( ( hAPP @ ( c_Set_Oimage @ D @ B @ A ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ D @ tc_HOL_Obool ) ) )
          = ( c_Orderings_Otop__class_Otop @ ( tc_fun @ B @ tc_HOL_Obool ) ) )
       => ( ( hAPP @ ( c_Set_Oimage @ E @ B @ ( hAPP @ ( c_Fun_Ocomp @ D @ B @ E @ A ) @ C ) ) @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ E @ tc_HOL_Obool ) ) )
          = ( c_Orderings_Otop__class_Otop @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[890]) ).

thf(970,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Int__Un__distrib) ).

thf(5913,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[970]) ).

thf(2058,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i,I: $i,J: $i,K: $i] :
      ( ( c_Com_Ocom_Ocom__rec @ K @ J @ I @ H @ G @ F @ E @ D @ C @ ( c_Com_Ocom_OAss @ B @ A ) )
      = ( hAPP @ ( hAPP @ I @ B ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_com_Orecs_I2_J) ).

thf(10492,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i,I: $i,J: $i,K: $i] :
      ( ( c_Com_Ocom_Ocom__rec @ K @ J @ I @ H @ G @ F @ E @ D @ C @ ( c_Com_Ocom_OAss @ B @ A ) )
      = ( hAPP @ ( hAPP @ I @ B ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2058]) ).

thf(1207,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ G @ F ) @ tc_HOL_Obool ) ) @ E ) @ D ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ F @ C ) @ tc_HOL_Obool ) ) @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ G @ C ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ G @ F @ C ) @ E ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ G @ F @ C ) @ D ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_rel__comp__mono) ).

thf(6961,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ G @ F ) @ tc_HOL_Obool ) ) @ E ) @ D ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ F @ C ) @ tc_HOL_Obool ) ) @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ G @ C ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ G @ F @ C ) @ E ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ G @ F @ C ) @ D ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1207]) ).

thf(520,axiom,
    ! [A: $i,B: $i] :
      ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ B ) )
     => ( ( hAPP @ ( c_Finite__Set_Ocard @ tc_Nat_Onat ) @ ( hAPP @ ( c_Set_OCollect @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ tc_Nat_Onat @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ tc_Nat_Onat ) @ c_fconj ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_Nat_Onat @ ( tc_fun @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ tc_Nat_Onat ) @ ( c_member @ tc_Nat_Onat ) ) @ c_Nat_OSuc ) ) @ B ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) ) @ A ) ) ) )
        = ( hAPP @ ( c_Finite__Set_Ocard @ tc_Nat_Onat ) @ ( hAPP @ ( c_Set_OCollect @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ tc_Nat_Onat @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ tc_Nat_Onat ) @ c_fconj ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( c_member @ tc_Nat_Onat ) ) @ B ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) ) @ ( hAPP @ c_Nat_OSuc @ A ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_card__less__Suc2) ).

thf(4274,plain,
    ! [A: $i,B: $i] :
      ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ B ) )
     => ( ( hAPP @ ( c_Finite__Set_Ocard @ tc_Nat_Onat ) @ ( hAPP @ ( c_Set_OCollect @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ tc_Nat_Onat @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ tc_Nat_Onat ) @ c_fconj ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_Nat_Onat @ ( tc_fun @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ tc_Nat_Onat ) @ ( c_member @ tc_Nat_Onat ) ) @ c_Nat_OSuc ) ) @ B ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) ) @ A ) ) ) )
        = ( hAPP @ ( c_Finite__Set_Ocard @ tc_Nat_Onat ) @ ( hAPP @ ( c_Set_OCollect @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ tc_Nat_Onat @ tc_HOL_Obool @ tc_HOL_Obool ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ tc_HOL_Obool @ ( tc_fun @ tc_HOL_Obool @ tc_HOL_Obool ) @ tc_Nat_Onat ) @ c_fconj ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ ( tc_fun @ tc_Nat_Onat @ tc_HOL_Obool ) @ tc_HOL_Obool ) @ ( c_member @ tc_Nat_Onat ) ) @ B ) ) ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ tc_Nat_Onat @ tc_HOL_Obool ) @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) ) @ ( hAPP @ c_Nat_OSuc @ A ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[520]) ).

thf(1335,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ D ) @ C ) ) @ ( c_Transitive__Closure_Ortrancl @ E @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ ( tc_prod @ E @ E ) @ tc_HOL_Obool ) ) @ B ) @ A ) ) ) )
     => ( ! [F: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ D ) @ F ) ) @ ( c_Transitive__Closure_Ortrancl @ E @ B ) ) )
           => ! [G: $i] :
                ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ F ) @ G ) ) @ A ) )
               => ( F = G ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ D ) @ C ) ) @ ( c_Transitive__Closure_Ortrancl @ E @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_rtrancl__Un__separatorE) ).

thf(7531,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ D ) @ C ) ) @ ( c_Transitive__Closure_Ortrancl @ E @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ ( tc_prod @ E @ E ) @ tc_HOL_Obool ) ) @ B ) @ A ) ) ) )
     => ( ! [F: $i] :
            ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ D ) @ F ) ) @ ( c_Transitive__Closure_Ortrancl @ E @ B ) ) )
           => ! [G: $i] :
                ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ F ) @ G ) ) @ A ) )
               => ( F = G ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ D ) @ C ) ) @ ( c_Transitive__Closure_Ortrancl @ E @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1335]) ).

thf(639,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ C ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Un__insert__left) ).

thf(4702,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ C ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[639]) ).

thf(875,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( D
          = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) )
       => ( ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ C @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ D ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ B @ tc_HOL_Obool ) @ C ) @ A ) )
          = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) )
      & ( ( D
         != ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) )
       => ( ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ C @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ D ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ B @ tc_HOL_Obool ) @ C ) @ A ) )
          = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_UN__constant) ).

thf(5542,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( D
          = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) )
       => ( ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ C @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ D ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ B @ tc_HOL_Obool ) @ C ) @ A ) )
          = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) )
      & ( ( D
         != ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) )
       => ( ( hAPP @ ( hAPP @ ( c_Complete__Lattice_Ocomplete__lattice__class_OSUPR @ C @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ D ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ B @ tc_HOL_Obool ) @ C ) @ A ) )
          = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[875]) ).

thf(605,axiom,
    ! [A: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) ) ) ) @ c_FunDef_Omin__weak ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_wmin__emptyI) ).

thf(4584,plain,
    ! [A: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) ) ) ) @ c_FunDef_Omin__weak ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[605]) ).

thf(1259,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Oappend @ D ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ D ) @ C ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_List_Oappend @ D ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_append__Cons) ).

thf(7181,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Oappend @ D ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ D ) @ C ) @ B ) ) @ A )
      = ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_List_Oappend @ D ) @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1259]) ).

thf(1782,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hAPP @ ( c_Sum__Type_Osum_Osum__case @ F @ E @ D @ C @ B ) @ ( hAPP @ ( c_Sum__Type_OInl @ F @ D ) @ A ) )
      = ( hAPP @ C @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_sum_Osimps_I5_J) ).

thf(9399,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hAPP @ ( c_Sum__Type_Osum_Osum__case @ F @ E @ D @ C @ B ) @ ( hAPP @ ( c_Sum__Type_OInl @ F @ D ) @ A ) )
      = ( hAPP @ C @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1782]) ).

thf(2183,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( c_Fun_Oinj__on @ F @ E @ D @ C )
     => ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ B ) @ ( hAPP @ ( c_Set_Oimage @ F @ E @ D ) @ C ) ) )
       => ( c_Fun_Oinj__on @ F @ E @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ F @ E ) @ D ) @ A ) @ B ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_inj__on__fun__updI) ).

thf(10934,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( c_Fun_Oinj__on @ F @ E @ D @ C )
     => ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ B ) @ ( hAPP @ ( c_Set_Oimage @ F @ E @ D ) @ C ) ) )
       => ( c_Fun_Oinj__on @ F @ E @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ F @ E ) @ D ) @ A ) @ B ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2183]) ).

thf(1937,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ C @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ C @ C ) @ B ) @ A ) ) @ ( c_Relation_OId @ C ) ) )
    <=> ( B = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_pair__in__Id__conv) ).

thf(10013,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ C @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ C @ C ) @ B ) @ A ) ) @ ( c_Relation_OId @ C ) ) )
       => ( B = A ) )
      & ( ( B = A )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ C @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ C @ C ) @ B ) @ A ) ) @ ( c_Relation_OId @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1937]) ).

thf(2268,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D = C )
     => ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ B ) @ A ) @ D ) )
         => ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ B ) @ A ) @ C ) ) )
       => ~ ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ B ) @ A ) @ D ) )
           => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ B ) @ A ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_equalityCE) ).

thf(11281,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( D = C )
     => ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ B ) @ A ) @ D ) )
         => ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ B ) @ A ) @ C ) ) )
       => ~ ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ B ) @ A ) @ D ) )
           => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ B ) @ A ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2268]) ).

thf(226,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( ( hAPP @ ( hAPP @ ( c_List_Omap @ G @ F ) @ E ) @ D )
        = ( hAPP @ ( hAPP @ ( c_List_Omap @ C @ F ) @ B ) @ A ) )
     => ( ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ G ) ) @ D )
        = ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_map__eq__imp__length__eq) ).

thf(3212,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
      ( ( ( hAPP @ ( hAPP @ ( c_List_Omap @ G @ F ) @ E ) @ D )
        = ( hAPP @ ( hAPP @ ( c_List_Omap @ C @ F ) @ B ) @ A ) )
     => ( ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ G ) ) @ D )
        = ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[226]) ).

thf(1420,axiom,
    ! [A: $i,B: $i] :
      ( ( c_FunDef_Oreduction__pair @ B @ A )
    <=> ( ( c_Wellfounded_Owf @ B @ ( hAPP @ ( c_Product__Type_Ofst @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) ) @ A ) )
        & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ B @ B @ B ) @ ( hAPP @ ( c_Product__Type_Ofst @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) ) @ A ) ) @ ( hAPP @ ( c_Product__Type_Osnd @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) ) @ A ) ) ) @ ( hAPP @ ( c_Product__Type_Ofst @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_reduction__pair__def) ).

thf(7848,plain,
    ! [A: $i,B: $i] :
      ( ( ( c_FunDef_Oreduction__pair @ B @ A )
       => ( ( c_Wellfounded_Owf @ B @ ( hAPP @ ( c_Product__Type_Ofst @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) ) @ A ) )
          & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ B @ B @ B ) @ ( hAPP @ ( c_Product__Type_Ofst @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) ) @ A ) ) @ ( hAPP @ ( c_Product__Type_Osnd @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) ) @ A ) ) ) @ ( hAPP @ ( c_Product__Type_Ofst @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) ) @ A ) ) ) ) )
      & ( ( ( c_Wellfounded_Owf @ B @ ( hAPP @ ( c_Product__Type_Ofst @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) ) @ A ) )
          & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Relation_Orel__comp @ B @ B @ B ) @ ( hAPP @ ( c_Product__Type_Ofst @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) ) @ A ) ) @ ( hAPP @ ( c_Product__Type_Osnd @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) ) @ A ) ) ) @ ( hAPP @ ( c_Product__Type_Ofst @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) ) @ A ) ) ) )
       => ( c_FunDef_Oreduction__pair @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1420]) ).

thf(645,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ A ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Un__commute) ).

thf(4723,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ A ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[645]) ).

thf(1145,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) )
     => ( B != A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__not__refl3) ).

thf(6733,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ B ) @ A ) )
     => ( B != A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1145]) ).

thf(663,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ A ) )
    <=> ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ B ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_bot__empty__eq) ).

thf(4784,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ B ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) ) )
      & ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ B ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) )
       => ( hBOOL @ ( hAPP @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[663]) ).

thf(1613,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ F ) @ E ) @ D ) )
     => ( ( hAPP @ ( c_Set_Oimage @ F @ ( tc_Option_Ooption @ C ) @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ F @ ( tc_Option_Ooption @ C ) ) @ B ) @ E ) @ ( hAPP @ ( c_Option_Ooption_OSome @ C ) @ A ) ) ) @ D )
        = ( hAPP @ ( c_Set_Oimage @ F @ ( tc_Option_Ooption @ C ) @ B ) @ D ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_image__map__upd) ).

thf(8741,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ F ) @ E ) @ D ) )
     => ( ( hAPP @ ( c_Set_Oimage @ F @ ( tc_Option_Ooption @ C ) @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Fun_Ofun__upd @ F @ ( tc_Option_Ooption @ C ) ) @ B ) @ E ) @ ( hAPP @ ( c_Option_Ooption_OSome @ C ) @ A ) ) ) @ D )
        = ( hAPP @ ( c_Set_Oimage @ F @ ( tc_Option_Ooption @ C ) @ B ) @ D ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1613]) ).

thf(2325,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ F ) @ E ) @ D ) )
     => ( ( hAPP @ ( c_Fun_Ooverride__on @ F @ C @ B @ A @ D ) @ E )
        = ( hAPP @ A @ E ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_override__on__apply__in) ).

thf(11534,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ F ) @ E ) @ D ) )
     => ( ( hAPP @ ( c_Fun_Ooverride__on @ F @ C @ B @ A @ D ) @ E )
        = ( hAPP @ A @ E ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2325]) ).

thf(1212,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) ) @ ( c_Wellfounded_Ofinite__psubset @ C ) ) )
    <=> ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) )
        & ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_in__finite__psubset) ).

thf(6984,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) ) @ ( c_Wellfounded_Ofinite__psubset @ C ) ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) )
          & ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ A ) ) ) )
      & ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) )
          & ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ A ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_fun @ C @ tc_HOL_Obool ) @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ A ) ) @ ( c_Wellfounded_Ofinite__psubset @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1212]) ).

thf(1492,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ A ) @ B ) )
       => ( ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ B )
          = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) @ ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_card_Oremove) ).

thf(8179,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ A ) @ B ) )
       => ( ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ B )
          = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) @ ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1492]) ).

thf(2101,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( c_Set_OCollect @ B ) @ A )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Collect__def) ).

thf(10654,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( c_Set_OCollect @ B ) @ A )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2101]) ).

thf(1422,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( c_List_Olist_ONil @ D )
        = ( hAPP @ ( hAPP @ ( c_List_Omap @ C @ D ) @ B ) @ A ) )
    <=> ( A
        = ( c_List_Olist_ONil @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Nil__is__map__conv) ).

thf(7861,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( ( c_List_Olist_ONil @ D )
          = ( hAPP @ ( hAPP @ ( c_List_Omap @ C @ D ) @ B ) @ A ) )
       => ( A
          = ( c_List_Olist_ONil @ C ) ) )
      & ( ( A
          = ( c_List_Olist_ONil @ C ) )
       => ( ( c_List_Olist_ONil @ D )
          = ( hAPP @ ( hAPP @ ( c_List_Omap @ C @ D ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1422]) ).

thf(1285,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ C ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ B @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ A ) @ C ) ) @ ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ A @ C ) ) )
      <=> ! [D: $i,E: $i] :
            ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ E ) )
              & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ E ) @ C ) )
              & ( A
                = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ D ) ) @ E ) ) )
           => ( hBOOL @ ( hAPP @ ( hAPP @ B @ D ) @ E ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_split__pos__lemma) ).

thf(7284,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ C ) )
     => ( ( ( hBOOL @ ( hAPP @ ( hAPP @ B @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ A ) @ C ) ) @ ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ A @ C ) ) )
         => ! [D: $i,E: $i] :
              ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ E ) )
                & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ E ) @ C ) )
                & ( A
                  = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ D ) ) @ E ) ) )
             => ( hBOOL @ ( hAPP @ ( hAPP @ B @ D ) @ E ) ) ) )
        & ( ! [D: $i,E: $i] :
              ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ E ) )
                & ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ E ) @ C ) )
                & ( A
                  = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ C ) @ D ) ) @ E ) ) )
             => ( hBOOL @ ( hAPP @ ( hAPP @ B @ D ) @ E ) ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ B @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Int_Oint ) @ A ) @ C ) ) @ ( c_Divides_Odiv__class_Omod @ tc_Int_Oint @ A @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1285]) ).

thf(1353,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Divides_Odiv__class_Omod @ tc_Nat_Onat @ B @ A )
      = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Nat_Onat ) @ B ) @ A ) ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Divides_Omod__div__equality_H) ).

thf(7602,plain,
    ! [A: $i,B: $i] :
      ( ( c_Divides_Odiv__class_Omod @ tc_Nat_Onat @ B @ A )
      = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Divides_Odiv__class_Odiv @ tc_Nat_Onat ) @ B ) @ A ) ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1353]) ).

thf(2177,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ D ) @ ( hAPP @ ( c_List_Oset @ E ) @ C ) ) )
     => ( ~ ( hBOOL @ ( hAPP @ B @ D ) )
       => ( ( c_List_OtakeWhile @ E @ B @ ( hAPP @ ( hAPP @ ( c_List_Oappend @ E ) @ C ) @ A ) )
          = ( c_List_OtakeWhile @ E @ B @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_takeWhile__append1) ).

thf(10916,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ D ) @ ( hAPP @ ( c_List_Oset @ E ) @ C ) ) )
     => ( ~ ( hBOOL @ ( hAPP @ B @ D ) )
       => ( ( c_List_OtakeWhile @ E @ B @ ( hAPP @ ( hAPP @ ( c_List_Oappend @ E ) @ C ) @ A ) )
          = ( c_List_OtakeWhile @ E @ B @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2177]) ).

thf(883,axiom,
    ! [A: $i] : ( c_Relation_Orefl__on @ A @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ A @ tc_HOL_Obool ) ) @ ( c_Relation_OId @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_refl__Id) ).

thf(5569,plain,
    ! [A: $i] : ( c_Relation_Orefl__on @ A @ ( c_Orderings_Otop__class_Otop @ ( tc_fun @ A @ tc_HOL_Obool ) ) @ ( c_Relation_OId @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[883]) ).

thf(2259,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ B ) @ ( c_Option_Oset @ C @ A ) ) )
    <=> ( A
        = ( hAPP @ ( c_Option_Ooption_OSome @ C ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_elem__set) ).

thf(11241,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ B ) @ ( c_Option_Oset @ C @ A ) ) )
       => ( A
          = ( hAPP @ ( c_Option_Ooption_OSome @ C ) @ B ) ) )
      & ( ( A
          = ( hAPP @ ( c_Option_Ooption_OSome @ C ) @ B ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ B ) @ ( c_Option_Oset @ C @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2259]) ).

thf(653,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Nat_Ocompow @ D @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Nat_Ocompow @ D @ D ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Nat_Ocompow @ D @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ C ) ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_funpow__mult) ).

thf(4752,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Nat_Ocompow @ D @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_Nat_Ocompow @ D @ D ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Nat_Ocompow @ D @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ C ) ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[653]) ).

thf(1513,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ D ) @ C ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ D ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ C ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Diff__mono) ).

thf(8273,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ D ) @ C ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ D ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ C ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1513]) ).

thf(841,axiom,
    ! [A: $i] :
      ( ( hAPP @ ( c_Int_Onumber__class_Onumber__of @ tc_Int_Oint ) @ A )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_number__of__is__id) ).

thf(5445,plain,
    ! [A: $i] :
      ( ( hAPP @ ( c_Int_Onumber__class_Onumber__of @ tc_Int_Oint ) @ A )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[841]) ).

thf(1241,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Divides_Odiv__class_Omod @ tc_Nat_Onat @ ( hAPP @ c_Nat_OSuc @ B ) @ A )
      = ( c_Divides_Odiv__class_Omod @ tc_Nat_Onat @ ( hAPP @ c_Nat_OSuc @ ( c_Divides_Odiv__class_Omod @ tc_Nat_Onat @ B @ A ) ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mod__Suc__eq__Suc__mod) ).

thf(7108,plain,
    ! [A: $i,B: $i] :
      ( ( c_Divides_Odiv__class_Omod @ tc_Nat_Onat @ ( hAPP @ c_Nat_OSuc @ B ) @ A )
      = ( c_Divides_Odiv__class_Omod @ tc_Nat_Onat @ ( hAPP @ c_Nat_OSuc @ ( c_Divides_Odiv__class_Omod @ tc_Nat_Onat @ B @ A ) ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1241]) ).

thf(1473,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Divides_Odiv__class_Omod @ tc_Nat_Onat @ A @ B ) ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mod__less__divisor) ).

thf(8055,plain,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ B ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Divides_Odiv__class_Omod @ tc_Nat_Onat @ A @ B ) ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1473]) ).

thf(2307,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Hoare__Mirabelle_Opeek__and @ E @ D @ C ) @ B ) @ A ) )
    <=> ( ( hBOOL @ ( hAPP @ ( hAPP @ D @ B ) @ A ) )
        & ( hBOOL @ ( hAPP @ C @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_peek__and__def) ).

thf(11444,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Hoare__Mirabelle_Opeek__and @ E @ D @ C ) @ B ) @ A ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ D @ B ) @ A ) )
          & ( hBOOL @ ( hAPP @ C @ A ) ) ) )
      & ( ( ( hBOOL @ ( hAPP @ ( hAPP @ D @ B ) @ A ) )
          & ( hBOOL @ ( hAPP @ C @ A ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Hoare__Mirabelle_Opeek__and @ E @ D @ C ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2307]) ).

thf(1671,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) ) @ D ) @ C ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) ) @ B ) @ D ) ) @ c_FunDef_Opair__leq ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) ) @ A ) @ C ) ) @ c_FunDef_Omax__weak ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) ) @ B ) @ A ) ) @ C ) ) @ c_FunDef_Omax__weak ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_wmax__insertI) ).

thf(8966,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) ) @ D ) @ C ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) ) @ B ) @ D ) ) @ c_FunDef_Opair__leq ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) ) @ A ) @ C ) ) @ c_FunDef_Omax__weak ) )
         => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) @ ( tc_fun @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) @ tc_HOL_Obool ) ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ ( tc_prod @ tc_Nat_Onat @ tc_Nat_Onat ) ) @ B ) @ A ) ) @ C ) ) @ c_FunDef_Omax__weak ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1671]) ).

thf(2211,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( hAPP @ ( c_List_Oset @ D ) @ ( hAPP @ ( hAPP @ ( c_List_Otake @ D ) @ B ) @ A ) ) ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( hAPP @ ( c_List_Oset @ D ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_in__set__takeD) ).

thf(11048,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( hAPP @ ( c_List_Oset @ D ) @ ( hAPP @ ( hAPP @ ( c_List_Otake @ D ) @ B ) @ A ) ) ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ D ) @ C ) @ ( hAPP @ ( c_List_Oset @ D ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2211]) ).

thf(1115,axiom,
    ! [A: $i,B: $i] : ( c_Relation_Oirrefl @ B @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) ) @ A ) @ ( c_Relation_OId @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_irrefl__diff__Id) ).

thf(6575,plain,
    ! [A: $i,B: $i] : ( c_Relation_Oirrefl @ B @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ ( tc_prod @ B @ B ) @ tc_HOL_Obool ) ) @ A ) @ ( c_Relation_OId @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1115]) ).

thf(1280,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ A ) @ B ) )
       => ( ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) ) )
          = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ B ) ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_card__Diff__singleton) ).

thf(7267,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ C ) @ A ) @ B ) )
       => ( ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ ( tc_fun @ C @ tc_HOL_Obool ) ) @ B ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ C ) @ A ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) ) )
          = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ ( hAPP @ ( c_Finite__Set_Ocard @ C ) @ B ) ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1280]) ).

thf(563,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ B ) ) @ ( c_List_Obutlast @ B @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ B ) ) @ A ) ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_length__butlast) ).

thf(4423,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ B ) ) @ ( c_List_Obutlast @ B @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ B ) ) @ A ) ) @ ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[563]) ).

thf(14,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Groups_Ocomm__monoid__add @ E )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ C ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ B ) )
         => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ E ) @ ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) ) ) @ ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) ) )
            = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ E ) @ ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ A ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ A ) @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_setsum__Un__Int) ).

thf(2453,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Groups_Ocomm__monoid__add @ E )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ C ) )
       => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ D ) @ B ) )
         => ( ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ E ) @ ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__sup__class_Osup @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) ) ) @ ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ C ) @ B ) ) )
            = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ E ) @ ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ A ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ A ) @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[14]) ).

thf(1014,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) @ A )
      = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Int__empty__left) ).

thf(6119,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ B @ tc_HOL_Obool ) ) @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) ) @ A )
      = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ B @ tc_HOL_Obool ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1014]) ).

thf(1086,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ A ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ A ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__less__mono2) ).

thf(6404,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ C ) @ B ) )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ A ) @ C ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ A ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1086]) ).

thf(119,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Odrop @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_List_Oappend @ D ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_List_Oappend @ D ) @ ( hAPP @ ( hAPP @ ( c_List_Odrop @ D ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_List_Odrop @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ C ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ D ) ) @ B ) ) ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_drop__append) ).

thf(2834,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Odrop @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_List_Oappend @ D ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_List_Oappend @ D ) @ ( hAPP @ ( hAPP @ ( c_List_Odrop @ D ) @ C ) @ B ) ) @ ( hAPP @ ( hAPP @ ( c_List_Odrop @ D ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ C ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ D ) ) @ B ) ) ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[119]) ).

thf(908,axiom,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ ( hAPP @ ( c_Int_Onumber__class_Onumber__of @ tc_Int_Oint ) @ B ) ) @ ( hAPP @ ( c_Int_Onumber__class_Onumber__of @ tc_Int_Oint ) @ A ) )
      = ( hAPP @ ( c_Int_Onumber__class_Onumber__of @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_times__numeral__code_I5_J) ).

thf(5664,plain,
    ! [A: $i,B: $i] :
      ( ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ ( hAPP @ ( c_Int_Onumber__class_Onumber__of @ tc_Int_Oint ) @ B ) ) @ ( hAPP @ ( c_Int_Onumber__class_Onumber__of @ tc_Int_Oint ) @ A ) )
      = ( hAPP @ ( c_Int_Onumber__class_Onumber__of @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Int_Oint ) @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[908]) ).

thf(1176,axiom,
    ! [A: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ ( hAPP @ c_Nat_OSuc @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zero__less__Suc) ).

thf(6839,plain,
    ! [A: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) @ ( hAPP @ c_Nat_OSuc @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1176]) ).

thf(163,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( c_Set_Ovimage @ E @ D @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Set_Ovimage @ E @ D @ C ) @ B ) ) @ ( hAPP @ ( c_Set_Ovimage @ E @ D @ C ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_vimage__Int) ).

thf(2993,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( c_Set_Ovimage @ E @ D @ C ) @ ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ D @ tc_HOL_Obool ) ) @ B ) @ A ) )
      = ( hAPP @ ( hAPP @ ( c_Lattices_Osemilattice__inf__class_Oinf @ ( tc_fun @ E @ tc_HOL_Obool ) ) @ ( hAPP @ ( c_Set_Ovimage @ E @ D @ C ) @ B ) ) @ ( hAPP @ ( c_Set_Ovimage @ E @ D @ C ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[163]) ).

thf(2387,axiom,
    ! [A: $i,B: $i] :
      ( ( c_Nitpick_Orefl_H @ B @ A )
    <=> ! [C: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ B @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ B @ B ) @ C ) @ C ) ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Nitpick_Orefl_H__def) ).

thf(11799,plain,
    ! [A: $i,B: $i] :
      ( ( ( c_Nitpick_Orefl_H @ B @ A )
       => ! [C: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ B @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ B @ B ) @ C ) @ C ) ) @ A ) ) )
      & ( ! [C: $i] : ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ B @ B ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ B @ B ) @ C ) @ C ) ) @ A ) )
       => ( c_Nitpick_Orefl_H @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2387]) ).

thf(1933,axiom,
    ! [A: $i] : ( c_Fun_Oinj__on @ tc_Nat_Onat @ tc_Nat_Onat @ c_Nat_OSuc @ A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_inj__Suc) ).

thf(10002,plain,
    ! [A: $i] : ( c_Fun_Oinj__on @ tc_Nat_Onat @ tc_Nat_Onat @ c_Nat_OSuc @ A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1933]) ).

thf(322,axiom,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_le__imp__0__less) ).

thf(3553,plain,
    ! [A: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ A ) )
     => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) @ ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Int_Oint ) @ ( c_Groups_Oone__class_Oone @ tc_Int_Oint ) ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[322]) ).

thf(571,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ B ) @ A ) )
    <=> ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Int_Oint ) @ B ) @ A ) ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_less__bin__lemma) ).

thf(4459,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ B ) @ A ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Int_Oint ) @ B ) @ A ) ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) ) )
      & ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Int_Oint ) @ B ) @ A ) ) @ ( c_Groups_Ozero__class_Ozero @ tc_Int_Oint ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Int_Oint ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[571]) ).

thf(2130,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Relation_Osingle__valued @ E @ E @ D )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ C ) @ B ) ) @ ( c_Transitive__Closure_Ortrancl @ E @ D ) ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ C ) @ A ) ) @ ( c_Transitive__Closure_Ortrancl @ E @ D ) ) )
         => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ B ) @ A ) ) @ ( c_Transitive__Closure_Ortrancl @ E @ D ) ) )
            | ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ A ) @ B ) ) @ ( c_Transitive__Closure_Ortrancl @ E @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_single__valued__confluent) ).

thf(10750,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( c_Relation_Osingle__valued @ E @ E @ D )
     => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ C ) @ B ) ) @ ( c_Transitive__Closure_Ortrancl @ E @ D ) ) )
       => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ C ) @ A ) ) @ ( c_Transitive__Closure_Ortrancl @ E @ D ) ) )
         => ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ B ) @ A ) ) @ ( c_Transitive__Closure_Ortrancl @ E @ D ) ) )
            | ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ ( tc_prod @ E @ E ) ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ E ) @ A ) @ B ) ) @ ( c_Transitive__Closure_Ortrancl @ E @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2130]) ).

thf(2223,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ C @ B ) )
     => ( hBOOL @ ( hAPP @ C @ ( c_Hilbert__Choice_OEps @ A @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_someI) ).

thf(11103,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ C @ B ) )
     => ( hBOOL @ ( hAPP @ C @ ( c_Hilbert__Choice_OEps @ A @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2223]) ).

thf(179,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Ozip @ E @ D ) @ C ) @ ( c_List_OtakeWhile @ D @ B @ A ) )
      = ( c_List_OtakeWhile @ ( tc_prod @ E @ D ) @ ( hAPP @ ( c_Fun_Ocomp @ D @ tc_HOL_Obool @ ( tc_prod @ E @ D ) @ B ) @ ( c_Product__Type_Osnd @ E @ D ) ) @ ( hAPP @ ( hAPP @ ( c_List_Ozip @ E @ D ) @ C ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_zip__takeWhile__snd) ).

thf(3047,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Ozip @ E @ D ) @ C ) @ ( c_List_OtakeWhile @ D @ B @ A ) )
      = ( c_List_OtakeWhile @ ( tc_prod @ E @ D ) @ ( hAPP @ ( c_Fun_Ocomp @ D @ tc_HOL_Obool @ ( tc_prod @ E @ D ) @ B ) @ ( c_Product__Type_Osnd @ E @ D ) ) @ ( hAPP @ ( hAPP @ ( c_List_Ozip @ E @ D ) @ C ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[179]) ).

thf(19,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Groups_Ocomm__monoid__add @ E )
     => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ D @ E @ E ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ E @ ( tc_fun @ E @ E ) @ D ) @ ( c_Groups_Oplus__class_Oplus @ E ) ) @ C ) ) @ B ) ) @ A )
        = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ E ) @ ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ C ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_setsum__addf) ).

thf(2470,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( class_Groups_Ocomm__monoid__add @ E )
     => ( ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ ( hAPP @ ( hAPP @ ( c_COMBS @ D @ E @ E ) @ ( hAPP @ ( hAPP @ ( c_COMBB @ E @ ( tc_fun @ E @ E ) @ D ) @ ( c_Groups_Oplus__class_Oplus @ E ) ) @ C ) ) @ B ) ) @ A )
        = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ E ) @ ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ C ) @ A ) ) @ ( hAPP @ ( hAPP @ ( c_Big__Operators_Ocomm__monoid__add__class_Osetsum @ D @ E ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[19]) ).

thf(2133,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( hAPP @ ( c_Map_Omap__add @ F @ E @ D @ C ) @ B )
        = ( hAPP @ ( c_Option_Ooption_OSome @ E ) @ A ) )
     => ( ( ( hAPP @ C @ B )
          = ( hAPP @ ( c_Option_Ooption_OSome @ E ) @ A ) )
        | ( ( ( hAPP @ C @ B )
            = ( c_Option_Ooption_ONone @ E ) )
          & ( ( hAPP @ D @ B )
            = ( hAPP @ ( c_Option_Ooption_OSome @ E ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_map__add__SomeD) ).

thf(10768,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( ( hAPP @ ( c_Map_Omap__add @ F @ E @ D @ C ) @ B )
        = ( hAPP @ ( c_Option_Ooption_OSome @ E ) @ A ) )
     => ( ( ( hAPP @ C @ B )
          = ( hAPP @ ( c_Option_Ooption_OSome @ E ) @ A ) )
        | ( ( ( hAPP @ C @ B )
            = ( c_Option_Ooption_ONone @ E ) )
          & ( ( hAPP @ D @ B )
            = ( hAPP @ ( c_Option_Ooption_OSome @ E ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2133]) ).

thf(1551,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C
        = ( hAPP @ ( hAPP @ ( c_List_Oappend @ B ) @ A ) @ C ) )
    <=> ( A
        = ( c_List_Olist_ONil @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_self__append__conv2) ).

thf(8405,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( C
          = ( hAPP @ ( hAPP @ ( c_List_Oappend @ B ) @ A ) @ C ) )
       => ( A
          = ( c_List_Olist_ONil @ B ) ) )
      & ( ( A
          = ( c_List_Olist_ONil @ B ) )
       => ( C
          = ( hAPP @ ( hAPP @ ( c_List_Oappend @ B ) @ A ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1551]) ).

thf(1670,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( hAPP @ ( c_Product__Type_Ofst @ E @ D ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ D ) @ C ) @ B ) )
        = A )
     => ( C = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_fst__eqD) ).

thf(8961,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i] :
      ( ( ( hAPP @ ( c_Product__Type_Ofst @ E @ D ) @ ( hAPP @ ( hAPP @ ( c_Product__Type_OPair @ E @ D ) @ C ) @ B ) )
        = A )
     => ( C = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1670]) ).

thf(2269,axiom,
    ! [A: $i,B: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Nat_Onat ) @ B ) @ ( c_SetInterval_Oord_OatLeast @ tc_Nat_Onat @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ A ) ) )
    <=> ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ A ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_dvd_OatLeast__iff) ).

thf(11294,plain,
    ! [A: $i,B: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Nat_Onat ) @ B ) @ ( c_SetInterval_Oord_OatLeast @ tc_Nat_Onat @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ A ) ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ A ) @ B ) ) )
      & ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ A ) @ B ) )
       => ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ tc_Nat_Onat ) @ B ) @ ( c_SetInterval_Oord_OatLeast @ tc_Nat_Onat @ ( c_Rings_Odvd__class_Odvd @ tc_Nat_Onat ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2269]) ).

thf(1599,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( class_Groups_Oab__semigroup__mult @ F )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ E ) @ D ) )
       => ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ C ) @ D ) )
         => ( ( hAPP @ ( hAPP @ ( hAPP @ ( c_Finite__Set_Ofold__image @ F @ E @ ( c_Groups_Otimes__class_Otimes @ F ) ) @ B ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ C ) @ D ) )
            = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ F ) @ ( hAPP @ B @ C ) ) @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Finite__Set_Ofold__image @ F @ E @ ( c_Groups_Otimes__class_Otimes @ F ) ) @ B ) @ A ) @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_fold__image__insert) ).

thf(8670,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( class_Groups_Oab__semigroup__mult @ F )
     => ( ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofinite @ E ) @ D ) )
       => ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_member @ E ) @ C ) @ D ) )
         => ( ( hAPP @ ( hAPP @ ( hAPP @ ( c_Finite__Set_Ofold__image @ F @ E @ ( c_Groups_Otimes__class_Otimes @ F ) ) @ B ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Set_Oinsert @ E ) @ C ) @ D ) )
            = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ F ) @ ( hAPP @ B @ C ) ) @ ( hAPP @ ( hAPP @ ( hAPP @ ( c_Finite__Set_Ofold__image @ F @ E @ ( c_Groups_Otimes__class_Otimes @ F ) ) @ B ) @ A ) @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1599]) ).

thf(709,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ D @ C ) @ B ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ C @ tc_HOL_Obool ) @ D ) @ A ) )
        = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) ) )
    <=> ( ( B
          = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) )
        | ( A
          = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_Times__empty) ).

thf(4940,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ D @ C ) @ B ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ C @ tc_HOL_Obool ) @ D ) @ A ) )
          = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) ) )
       => ( ( B
            = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) )
          | ( A
            = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) ) )
      & ( ( ( B
            = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) )
          | ( A
            = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ C @ tc_HOL_Obool ) ) ) )
       => ( ( hAPP @ ( hAPP @ ( c_Product__Type_OSigma @ D @ C ) @ B ) @ ( hAPP @ ( c_COMBK @ ( tc_fun @ C @ tc_HOL_Obool ) @ D ) @ A ) )
          = ( c_Orderings_Obot__class_Obot @ ( tc_fun @ ( tc_prod @ D @ C ) @ tc_HOL_Obool ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[709]) ).

thf(885,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] : ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofold__graph @ D @ C @ B @ A @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_fold__graph_Oequations_I1_J) ).

thf(5574,plain,
    ! [A: $i,B: $i,C: $i,D: $i] : ( hBOOL @ ( hAPP @ ( c_Finite__Set_Ofold__graph @ D @ C @ B @ A @ ( c_Orderings_Obot__class_Obot @ ( tc_fun @ D @ tc_HOL_Obool ) ) ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[885]) ).

thf(1188,axiom,
    ! [A: $i,B: $i] :
      ( ( B
        = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ A ) )
     => ( ( A
          = ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) )
        | ( B
          = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_mult__eq__self__implies__10) ).

thf(6896,plain,
    ! [A: $i,B: $i] :
      ( ( B
        = ( hAPP @ ( hAPP @ ( c_Groups_Otimes__class_Otimes @ tc_Nat_Onat ) @ B ) @ A ) )
     => ( ( A
          = ( c_Groups_Oone__class_Oone @ tc_Nat_Onat ) )
        | ( B
          = ( c_Groups_Ozero__class_Ozero @ tc_Nat_Onat ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1188]) ).

thf(1186,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ C ) @ B ) )
     => ( ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ B ) @ C )
          = A )
      <=> ( B
          = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_le__imp__diff__is__add) ).

thf(6886,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless__eq @ tc_Nat_Onat ) @ C ) @ B ) )
     => ( ( ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ B ) @ C )
            = A )
         => ( B
            = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ C ) ) )
        & ( ( B
            = ( hAPP @ ( hAPP @ ( c_Groups_Oplus__class_Oplus @ tc_Nat_Onat ) @ A ) @ C ) )
         => ( ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ B ) @ C )
            = A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[1186]) ).

thf(2069,axiom,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hAPP @ ( c_Sum__Type_Osum_Osum__case @ F @ E @ D @ C @ B ) @ ( hAPP @ ( c_Sum__Type_OInr @ D @ F ) @ A ) )
      = ( hAPP @ B @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_sum_Osimps_I6_J) ).

thf(10537,plain,
    ! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
      ( ( hAPP @ ( c_Sum__Type_Osum_Osum__case @ F @ E @ D @ C @ B ) @ ( hAPP @ ( c_Sum__Type_OInr @ D @ F ) @ A ) )
      = ( hAPP @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2069]) ).

thf(176,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ D ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) @ B ) ) )
       => ( ( hAPP @ ( hAPP @ ( c_List_Onth @ C ) @ ( hAPP @ ( hAPP @ ( c_List_Oappend @ C ) @ B ) @ A ) ) @ D )
          = ( hAPP @ ( hAPP @ ( c_List_Onth @ C ) @ B ) @ D ) ) )
      & ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ D ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) @ B ) ) )
       => ( ( hAPP @ ( hAPP @ ( c_List_Onth @ C ) @ ( hAPP @ ( hAPP @ ( c_List_Oappend @ C ) @ B ) @ A ) ) @ D )
          = ( hAPP @ ( hAPP @ ( c_List_Onth @ C ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ D ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_nth__append) ).

thf(3035,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ D ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) @ B ) ) )
       => ( ( hAPP @ ( hAPP @ ( c_List_Onth @ C ) @ ( hAPP @ ( hAPP @ ( c_List_Oappend @ C ) @ B ) @ A ) ) @ D )
          = ( hAPP @ ( hAPP @ ( c_List_Onth @ C ) @ B ) @ D ) ) )
      & ( ~ ( hBOOL @ ( hAPP @ ( hAPP @ ( c_Orderings_Oord__class_Oless @ tc_Nat_Onat ) @ D ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) @ B ) ) )
       => ( ( hAPP @ ( hAPP @ ( c_List_Onth @ C ) @ ( hAPP @ ( hAPP @ ( c_List_Oappend @ C ) @ B ) @ A ) ) @ D )
          = ( hAPP @ ( hAPP @ ( c_List_Onth @ C ) @ A ) @ ( hAPP @ ( hAPP @ ( c_Groups_Ominus__class_Ominus @ tc_Nat_Onat ) @ D ) @ ( hAPP @ ( c_Nat_Osize__class_Osize @ ( tc_List_Olist @ C ) ) @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[176]) ).

thf(328,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Odrop @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ D ) @ B ) @ A ) )
      = ( c_Nat_Onat_Onat__case @ ( tc_List_Olist @ D ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ D ) @ B ) @ A ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) @ ( c_List_Odrop @ D ) ) @ A ) @ C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_drop__Cons) ).

thf(3575,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( hAPP @ ( hAPP @ ( c_List_Odrop @ D ) @ C ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ D ) @ B ) @ A ) )
      = ( c_Nat_Onat_Onat__case @ ( tc_List_Olist @ D ) @ ( hAPP @ ( hAPP @ ( c_List_Olist_OCons @ D ) @ B ) @ A ) @ ( hAPP @ ( hAPP @ ( c_COMBC @ tc_Nat_Onat @ ( tc_List_Olist @ D ) @ ( tc_List_Olist @ D ) ) @ ( c_List_Odrop @ D ) ) @ A ) @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[328]) ).

thf(15362,plain,
    $false,
    inference(e,[status(thm)],[10822,5810,5116,10452,11504,10363,6374,3172,5686,9982,2822,9131,8618,4201,8930,4560,5422,4094,6167,8398,2889,10259,10373,7890,3053,4464,4824,7854,4909,7569,7691,5260,9648,5950,3167,8114,2412,6283,7100,10495,8283,3581,4852,10743,2595,3969,8843,5205,6172,4015,2544,11898,6954,9091,7396,9305,10901,11527,4166,6808,2940,6883,2676,4711,8169,10886,4747,3382,4532,11008,9241,9342,5454,6791,11352,9020,6063,2512,11350,9621,3072,8883,8450,7596,8044,6918,5820,4339,4075,9163,9373,11880,4311,3905,6106,4515,5069,11571,9604,2776,2659,10277,8402,10912,9769,3153,7322,9704,2590,5297,8682,4697,6796,10711,5878,3098,10705,10262,6323,7253,8447,3108,5405,2837,7781,2854,5237,6031,5529,7208,9152,8366,5614,9818,6971,3648,8204,4543,6522,5425,5550,4234,5037,8182,9644,4403,3439,3120,3429,7203,10433,9127,5589,2686,4575,11399,4097,3942,11664,10869,4884,3365,2740,4026,8981,7824,7115,4110,3175,8057,8747,10346,11303,8667,10864,8519,10049,9813,7669,5316,7345,2434,8298,7467,3781,9218,3693,11160,4058,11239,2459,10827,6241,5005,6996,10170,7871,6336,9961,5178,5708,8650,7980,11549,11756,6877,5192,9481,5412,10859,10662,11335,5903,3466,9946,7605,9625,9443,10336,10427,6570,3236,6914,3545,5977,6251,7064,2491,8898,2763,6268,11834,8325,3502,2985,10599,10908,5518,11907,7499,11568,4266,11699,11613,3513,11273,7734,5195,11146,5476,9096,9994,7240,3947,4124,7326,5056,4198,11731,9971,9465,10698,8487,8330,4607,8025,4760,7761,2825,4665,2819,11481,5750,9394,2531,7178,7311,10248,5306,4692,9988,3148,4984,3820,5553,3204,5953,3442,5765,10903,10326,3221,4546,7620,10286,6766,2603,2916,4524,9172,7277,3029,9287,7279,10305,11844,7262,4563,7701,9067,10630,11421,8426,8996,3131,4956,4070,7401,10997,6547,7822,5910,6723,11299,3061,11087,2708,10722,6749,6066,3617,10221,8785,6506,8787,9360,8919,11178,11200,5150,6676,4130,6315,4349,5130,5160,3280,11812,6713,10010,3749,4023,5640,2488,2897,9668,6332,9938,11416,3803,4428,3855,8880,8753,7011,11760,8389,6396,4358,10665,10702,8770,8907,2420,6049,6253,11232,3463,3374,3385,9800,5868,5605,4862,5802,8819,6149,10891,6310,9843,6579,4492,5622,11378,3919,3216,6365,3084,11115,11600,5052,9529,11741,6923,9199,8615,10650,3312,5851,7728,3902,3657,11991,7560,5942,9869,2520,4141,10555,5727,8632,9497,4390,3602,4400,8738,4771,3231,10682,4302,7464,8352,8161,5617,4253,6382,3258,9266,8870,9580,5974,10913,11314,9593,5084,4297,5333,7986,7677,9457,5805,4686,9309,2456,10309,10419,10818,3887,3966,11064,5526,4173,3934,4018,9330,10871,10623,2996,10591,7969,3295,2782,8158,7818,3527,9557,6189,2880,6534,7376,9811,7845,11676,4815,11215,3621,2925,6841,5650,8246,2555,3290,8479,5360,10734,8196,8664,3771,4810,10602,7899,10404,4718,9347,10233,8892,8939,3406,3389,2964,5059,7707,2437,3537,10670,9897,6648,11325,8001,4055,5448,4033,7931,7613,3189,6685,11265,5210,6755,11119,6740,7381,3421,9255,7807,9167,11346,5101,3685,8823,9743,7336,3808,11806,4364,7909,2979,4996,7645,6932,2772,8347,3676,5846,5897,11885,9187,10390,6021,11408,2809,10106,10801,10977,9764,4892,9923,7688,6103,4588,9272,7460,5328,4628,9589,6210,2982,7170,4082,9584,8289,9738,3899,9720,4604,8382,8418,3995,11069,2737,2905,10216,8867,9026,7952,7674,8975,11660,3151,10725,5157,3503,10811,11897,6560,3591,3301,10593,8655,8201,6183,6406,11255,2473,11395,9340,3567,10875,8233,3720,6731,8078,9221,8243,9968,4480,9136,6047,6296,11079,3852,7037,7217,5697,3186,8795,6588,9797,11672,5142,4557,7937,5291,5034,10947,6307,9252,5519,10639,7078,4100,9688,3550,11499,4999,3418,9384,9323,5122,11410,4925,9881,3679,6200,3735,11520,3261,4920,4499,11592,9036,2673,9549,2705,8343,4466,6124,8211,5611,8265,8778,6122,2428,4016,6328,6550,6820,2541,11768,4397,5002,5794,4434,8756,11037,6371,10990,5579,6528,5512,4878,6773,4989,7229,7881,5349,7994,5266,11587,3022,5815,7834,11217,8856,6990,5719,10657,8608,6565,2969,4091,3752,10409,4521,3989,3371,5470,2937,4204,3884,10102,7640,10394,10084,11565,6496,4821,5396,6696,2414,10441,4836,5594,4160,2528,6232,11191,8878,3207,3682,9993,3286,9158,5843,6620,6582,8623,9455,7842,8270,6043,9586,10583,6143,4187,4979,5566,2958,7097,3307,3271,7237,6646,8810,5391,8255,5702,3980,10634,5937,10939,8832,10756,5655,10502,10490,9190,10462,8942,3198,8386,10369,3578,7902,11492,3535,5113,10367,7501,10006,4417,5658,3050,6402,10235,2547,5858,10615,6666,11624,11702,3667,11602,11101,4080,3931,11206,4680,5330,5515,3799,2769,8358,3445,8155,6090,10897,7777,9608,10266,9977,6360,8429,8916,5497,11234,10131,6728,3731,4808,5313,7750,8462,7281,9493,10848,8314,7142,3831,2840,5145,2515,3254,5098,2990,6940,9804,5270,2801,7205,7962,4512,2895,11715,4248,10413,6028,7623,11762,4486,2577,5635,9773,4789,4048,8319,9244,10646,6235,4845,5428,8014,9936,7610,11057,2509,5362,3518,3963,7593,5919,10298,3632,10430,11189,4012,9333,10882,9904,7874,8117,6218,11877,11019,4817,4898,11364,11229,9085,7577,8792,6299,4320,2798,4333,6862,9872,7091,2877,4518,8835,5031,11860,7774,9640,5694,9910,7631,9397,3846,10120,5705,7717,3210,7111,9387,11403,11496,5992,6363,3728,2868,9426,8687,10914,2419,4421,3010,5253,4938,10620,5263,10936,4406,4744,8132,6164,11046,8275,5213,3032,6763,6573,8968,5865,10688,6978,3723,10163,7404,10239,4444,6625,3555,10752,6935,10353,9429,6271,7523,10094,7293,7448,11248,9825,11269,3126,7485,8690,7913,4902,10919,10575,6752,5576,11129,3611,8673,11608,11023,2450,3654,7927,9359,5481,4384,6097,3787,9149,11052,11142,7014,3059,6805,2487,7138,3354,8764,6999,7966,7508,6788,5234,5900,6129,7587,2851,3540,2497,6239,10455,2566,10518,10207,9112,8439,10401,11184,9017,8896,10122,10047,10879,9990,4231,2813,4347,8455,8775,10302,3247,3409,2862,9598,4953,3064,11610,7826,10075,4689,10270,6186,7333,9942,11873,3201,4223,10894,3215,4361,5479,8527,8512,6011,4572,5170,4163,10143,3454,10212,7661,10465,10406,10510,6514,3755,4535,8174,9534,8846,2749,6409,5557,4184,4352,10597,7770,9468,9913,6963,11856,2702,10338,8482,3624,11368,5402,4889,3183,10740,9176,8080,4765,2883,9269,11787,10377,9008,8923,6716,10470,8049,5571,11544,5434,8393,6279,3321,5737,8955,9169,8004,2440,4448,8262,10951,9284,8945,5417,4036,10090,3508,2645,6038,11539,3841,7972,2617,6869,2662,7106,9139,4308,2423,7133,7444,2909,6192,6368,4872,3042,5855,11016,5043,3343,9545,10862,3707,8322,5483,10840,2413,5722,5524,11837,11747,9416,4708,10571,11033,8641,10607,11710,11638,2556,4116,4613,6824,4594,5969,5451,6699,3890,5508,9577,7831,4412,4256,8728,7753,5048,11427,10993,6243,7599,7491,3738,7636,9404,3375,2734,5016,4503,5095,3111,5647,8311,9451,3696,6492,10625,4489,5758,9974,9072,11011,3396,6140,5456,10963,11804,11124,5535,9617,4144,10068,4000,3387,10582,7714,7590,4268,3070,10194,8909,11633,3319,3234,3925,2859,11105,3504,10906,4283,5488,11279,7497,6881,2525,2849,10334,3472,11809,3283,4677,3266,10329,8894,8684,10242,7041,7704,2806,5984,8332,8993,4381,7763,7828,5799,9820,5503,9952,5022,7815,3599,10611,4127,6937,9506,7347,7671,7658,10024,6248,11618,9483,7191,10034,7270,4796,6357,5011,3726,7983,6673,3651,11466,7719,11502,3208,3663,4218,10348,11423,3594,5086,5148,9538,7094,3123,7924,5190,10728,3436,7145,7536,7062,10974,3304,9354,11765,10361,4887,7315,10176,4245,9089,8376,10505,9307,10140,10056,3532,9224,6525,9390,2679,2415,10457,10565,8436,4716,10980,6024,8042,11222,9560,3101,10842,5870,9623,10921,10532,4228,2538,11002,10910,2478,7130,2642,7643,11117,3815,6920,3238,6179,11508,9867,2637,5325,9377,4305,7810,7907,5848,8970,10279,10677,10675,6500,4828,3779,7379,9771,6197,6112,4699,5303,5827,5853,2955,11735,5491,4601,3106,11733,9118,9920,11829,10316,6832,5054,7330,4805,4586,9570,11338,3710,5407,4616,9146,9015,5346,4073,8485,4667,6154,3500,3138,4374,7362,6737,5972,11718,9491,11883,2871,4509,9890,11306,7307,4107,9174,4028,8340,3893,5563,9302,7504,5367,8735,6087,8605,9129,11868,7768,3747,8958,11669,2632,10397,10375,4009,5730,4833,5894,11796,10995,4569,9079,11893,4064,9830,7566,4818,11392,9642,9691,5439,7820,7685,7892,5286,8750,4277,2757,7664,3165,4181,7551,10732,3192,9553,3977,2410,3614,2785,3451,3298,4652,11640,6817,8782,3483,3971,10825,3583,9862,11630,6044,7450,6802,4726,4993,10380,6568,6692,2928,8652,10004,5459,10792,8198,5987,9143,3838,8084,7911,3178,10226,8423,8280,11173,10147,8052,4933,6255,4844,5057,11220,5742,5775,6811,5931,2725,6377,5442,3627,11041,2550,2481,7002,8308,2620,10020,7929,10852,10514,9741,3917,9841,7469,6223,3687,3806,4085,3159,9371,3674,4827,2931,9916,9197,3953,3241,7324,3016,10264,9106,9698,9366,10063,11605,10930,6100,5725,10803,3142,2535,8676,10707,5817,9033,7558,5667,7339,4196,6325,2999,3348,5089,4067,7916,5139,3908,7232,9894,7884,10499,11361,9503,6785,2506,5619,7049,11887,7154,11889,4741,2952,4529,2503,4052,11030,4292,6076,7122,11758,3849,9958,3881,5661,4426,4207,11316,5538,2816,6320,4394,4136,2614,7526,3936,7117,6417,9673,10632,7296,9727,5881,6276,2665,5025,10191,4813,10589,9259,3293,9651,4598,3161,10223,6056,4177,2629,5678,5165,6517,11689,4566,8067,5916,2846,5710,10788,9232,5652,9276,4355,5807,6008,8477,6664,6719,4336,4986,9202,6132,7070,4045,10196,7801,5125,8466,8163,10521,4377,10080,4749,11738,8905,6577,10672,10797,9794,11327,11770,10685,11909,5294,10540,11460,6945,5250,10476,11562,5399,8218,10059,9710,4950,11989,3659,3228,4641,4469,9005,8350,7618,11137,8364,11252,10159,3608,9715,7055,8744,6146,8039,2689,3448,11027,6912,8873,10365,6352,4875,9445,3019,5684,4409,2569,10307,7388,4437,8060,8396,5431,3642,8498,10439,5420,6981,7264,7494,4190,3818,3114,3273,3334,7243,4003,9986,9182,5753,4549,10717,11166,7667,3475,7796,7988,3026,4768,9809,5560,7543,6508,7879,8509,3801,9082,5152,8327,8148,3828,11109,9093,3412,3564,3359,4737,6159,9963,6330,9807,7009,5277,5062,5675,11486,9363,3071,8679,11518,5319,11354,5208,6701,9109,6746,2411,2500,7876,9486,9495,6679,7779,11749,6899,3415,5223,4133,7921,9675,7919,9500,4624,7342,6318,3356,11386,10415,7251,6775,7067,11054,9247,2843,7391,9257,2711,4991,6213,10028,8317,8602,8185,7303,11005,4431,4634,5226,4695,11825,3024,10231,3117,2949,2722,10150,5364,2728,10884,9637,7426,4342,6943,10436,5717,4838,9747,2892,6511,3914,9654,3520,11906,8861,3007,6531,10178,7488,11059,10680,5841,4414,9902,3288,8914,11369,7234,6793,2865,3757,3401,9876,11176,11450,6835,10899,2760,6286,9532,8973,4289,9046,11862,5873,9279,2626,4138,6846,3589,4325,6758,3067,10873,10806,10595,6041,9760,9628,4763,7904,9780,9438,5777,10774,6390,3874,5510,10199,10156,7765,5934,8825,4780,2611,3928,4078,5308,2743,8228,4006,6743,9141,10928,7974,9866,3056,5485,8761,9515,7446,11383,8469,4061,6814,3665,6711,6726,2988,8166,5008,2532,4683,5351,4042,5813,5505,7732,7950,8356,2443,7034,5632,4121,8337,3911,6903,10135,8221,8474,8780,9802,3825,8011,6893,7044,10468,10659,9934,3104,9205,9459,10648,11847,6910,8634,3561,2608,2431,10182,5394,3516,4495,4705,10104,7120,6618,7125,3368,4881,9449,4527,2416,10627,6799,11091,9179,4900,2934,10077,11918,5922,6354,9160,5162,5532,6018,6412,9955,5739,9542,9115,2944,8865,10777,7680,8506,10210,9718,3181,7256,4855,11780,9338,8369,10040,9470,3497,3879,8501,2648,9906,4591,9434,10738,2462,5040,4774,4299,7039,7747,11658,8420,7399,3896,4387,11405,2576,4478,7887,3793,2680,7017,5072,3351,11685,2902,7520,5590,9211,10924,3645,8267,7574,5672,9350,9887,2494,6871,11537,11381,2623,10357,7804,4869,6499,7135,5336,2779,4619,11208,9321,5187,10838,11580,7355,9523,4841,11341,4367,3992,11744,3379,7649,5357,7081,10569,9551,4917,2683,8209,7991,8070,6957,8886,4670,3511,7634,7868,3997,7140,8193,7510,2719,3733,11140,2587,7744,9767,8296,11680,6563,8412,9318,4596,5554,6925,7259,10312,3987,11437,3264,10443,2522,11712,6237,3790,5522,5884,9051,4663,5014,3690,9070,9463,5247,11914,7955,8621,3469,4631,5462,9299,5378,5110,6503,3309,3145,5700,7772,8790,3956,11727,4777,11595,10809,5300,6169,10941,4755,5092,7184,4981,4895,10691,4286,11194,11854,8373,3558,5990,4210,9134,4474,11627,2746,6644,9778,8921,5019,11494,6558,2417,9874,8516,10472,10956,2961,8277,6380,5586,3323,2714,3426,10237,9414,6682,8648,11422,4193,3424,4912,6393,6929,9927,9250,2668,3045,3506,5176,3843,10485,9564,10355,5283,5322,9441,9235,4506,10845,8453,8773,11489,6874,9325,9547,8692,3244,11236,4923,6771,5945,5500,8460,7538,3362,9816,2731,3974,10421,2966,9038,10573,8444,8889,11144,9679,7584,7227,4799,11621,2600,3605,4972,11794,11050,5637,6993,10255,11358,8034,2467,5280,10713,7616,5681,8730,9402,4239,4831,5046,11477,11182,4659,10877,10323,10535,4936,11330,10124,10642,3939,5891,5354,6109,7694,9848,6778,2699,8876,3812,7176,2484,11135,4345,6126,4271,3918,4088,7246,11373,7563,4221,5473,11276,4674,3013,5494,9784,2832,4734,8685,7103,10383,2754,5172,3950,5029,3461,5591,5913,10492,6961,4274,7531,4702,5542,4584,7181,9399,10934,10013,11281,3212,7848,4723,6733,4784,8741,11534,6984,8179,10654,7861,7284,7602,10916,5569,11241,4752,8273,5445,7108,8055,11444,8966,11048,6575,7267,4423,2453,6119,6404,2834,5664,6839,2993,11799,10002,3553,4459,10750,11103,3047,2470,10768,8405,8961,11294,8670,4940,5574,6896,6886,10537,3035,3575]) ).

%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.14  % Problem  : SWW325+1 : TPTP v8.2.0. Released v5.2.0.
% 0.07/0.14  % Command  : run_Leo-III %s %d THM
% 0.13/0.35  % Computer : n003.cluster.edu
% 0.13/0.35  % Model    : x86_64 x86_64
% 0.13/0.35  % CPU      : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.35  % Memory   : 8042.1875MB
% 0.13/0.35  % OS       : Linux 3.10.0-693.el7.x86_64
% 0.13/0.35  % CPULimit : 300
% 0.13/0.35  % WCLimit  : 300
% 0.13/0.35  % DateTime : Wed Jun 19 04:16:40 EDT 2024
% 0.13/0.35  % CPUTime  : 
% 1.06/0.92  % [INFO] 	 Parsing problem /export/starexec/sandbox2/benchmark/theBenchmark.p ... 
% 2.88/1.54  % [INFO] 	 Parsing done (609ms). 
% 2.94/1.55  % [INFO] 	 Running in sequential loop mode. 
% 3.37/1.79  % [INFO] 	 eprover registered as external prover. 
% 3.37/1.79  % [INFO] 	 Scanning for conjecture ... 
% 4.97/2.28  % [INFO] 	 Found a conjecture (or negated_conjecture) and 5205 axioms. Running axiom selection ... 
% 7.71/3.53  % [INFO] 	 Axiom selection finished. Selected 2407 axioms (removed 2798 axioms). 
% 11.00/4.81  % [INFO] 	 Problem is first-order (TPTP FOF). 
% 11.15/4.86  % [INFO] 	 Type checking passed. 
% 11.15/4.87  % [CONFIG] 	 Using configuration: timeout(300) with strategy<name(default),share(1.0),primSubst(3),sos(false),unifierCount(4),uniDepth(8),boolExt(true),choice(true),renaming(true),funcspec(false), domConstr(0),specialInstances(39),restrictUniAttempts(true),termOrdering(CPO)>.  Searching for refutation ... 
% 129.43/49.05  % External prover 'e' found a proof!
% 129.43/49.05  % [INFO] 	 Killing All external provers ... 
% 129.43/49.05  % Time passed: 48496ms (effective reasoning time: 47496ms)
% 129.43/49.05  % Solved by strategy<name(default),share(1.0),primSubst(3),sos(false),unifierCount(4),uniDepth(8),boolExt(true),choice(true),renaming(true),funcspec(false), domConstr(0),specialInstances(39),restrictUniAttempts(true),termOrdering(CPO)>
% 129.48/49.07  % Axioms used in derivation (2407): fact_While, fact_wf__inv__image, fact_wf__not__refl, fact_setsum__shift__lb__Suc0__0, fact_SUP1__I, fact_rtrancl__Int__subset, fact_Int__insert__left__if1, fact_card__partition, fact_Nat_Oadd__0__right, fact_self__quotient, fact_converseI, fact_dvd_Oord__less__eq__trans, fact_insert__partition, fact_wmax__insertI, fact_sum_Osimps_I5_J, fact_atLeastLessThan__upt, fact_insert__compr, fact_Un__absorb1, fact_Sigma__Diff__distrib2, fact_fold__graph__insert__swap, fact_le__SucI, fact_notin__set__remove1, fact_drop__Cons__number__of, fact_finite__greaterThanLessThan__int, fact_UN__simps_I3_J, fact_set__drop__subset, fact_singleton__iff, fact_wmin__emptyI, fact_divmod__nat__zero, fact_single__valued__Id__on, fact_nat__gcd_Opinduct, fact_divmod__nat__rel__divmod__nat, fact_append__Nil2, fact_com_Osimps_I56_J, fact_insertCI, fact_Diff__insert__absorb, fact_concat_Osimps_I1_J, fact_wfP__SUP, fact_card__image__le, fact_divmod__nat__base, fact_nth__Cons__0, fact_folding__image__simple__idem_Osubset__idem, fact_rel__pow__Suc__I, fact_evalc__evaln, fact_nat_Osimps_I3_J, fact_wf__union__merge, fact_le__imp__diff__is__add, fact_Union__upper, fact_divmod__nat__rel__mult1__eq, fact_set__empty__eq, fact_evalc_Oequations_I8_J, fact_wf, fact_vimage__insert, fact_ImageE, fact_inj__graph, fact_dvd_Oless__trans, fact_length__filter__less, fact_folding__one_Osingleton, fact_lessThan__Suc__atMost, fact_finite__vimageI, fact_wf__not__sym, fact_finite__UN__I, fact_rel__pow__E2, fact_com__det, fact_lexn_Osimps_I1_J, fact_refl__onD2, fact_Int__lower1, fact_setsum__shift__lb__Suc0__0__upt, fact_drop__butlast, fact_sequence__trans, fact_converse__converse, fact_card__atLeastAtMost, fact_zdvd__zmod, fact_restrict__map__def, fact_finite_Oequations_I1_J, fact_fold1Set_Ointros, fact_sup1I1, fact_incr__mult__lemma, fact_com_Osimps_I8_J, fact_Sigma__Un__distrib1, fact_map__upds__twist, fact_map__upd__Some__unfold, fact_finite__lists__length__le, fact_diff__0__eq__0, fact_fun__upds__append__drop, fact_zle__refl, fact_inj__comp, fact_rel__pow__E, fact_Field__empty, fact_Sigma__def, fact_length__0__conv, fact_Domain__Id, fact_take__drop, fact_le__eq__less__or__eq, fact_UnI1, fact_map_Osimps_I1_J, fact_trans__less__add1, fact_comp__inj__on__iff, fact_less__eq__nat_Osimps_I2_J, fact_list__size__pointwise, fact_accp_Oequations, fact_q__pos__lemma, fact_setsum__Un__nat, fact_card__Diff__subset__Int, fact_map__fst__zip, fact_subset__UNIV, fact_map__pair__inj__on, fact_inf2I, fact_finite__greaterThanAtMost, fact_Range__Id, fact_notin__set__sublistI, fact_com_Osimps_I71_J, fact_lexord__append__leftI, fact_finite_OemptyI, fact_list__eq__iff__nth__eq, fact_dvd_Oeq__iff, fact_nat__add__left__commute, fact_set__append, fact_nth__upt, fact_image__constant__conv, fact_zdiv__eq__0__iff, fact_Range__Int__subset, fact_zdvd__not__zless, fact_Image__Int__eq, fact_pair__lessI2, fact_smax__insertI, fact_finite__dom__map__of, fact_listrel1E, fact_inj__on__diff__nat, fact_map__upds__apply__nontin, fact_map__add__SomeD, fact_insert__iff, fact_zip__append1, fact_atLeast__upt, fact_mod__mult2__eq, fact_image__insert, fact_setsum__empty, fact_diff__add__inverse, fact_inj__Some, fact_ran__restrictD, fact_map__add__dom__app__simps_I3_J, fact_minus__fold__remove, fact_fst__eq__Domain, fact_card__gt__0__iff, fact_zdvd__antisym__nonneg, fact_evalc_OIfFalse, fact_evaln__elim__cases_I5_J, fact_fst__map__pair, fact_nat__less__cases, fact_drop__upt, fact_one__is__add, fact_diff__le__mono2, fact_listrel__Cons1, fact_zip__Nil, fact_termination__basic__simps_I5_J, fact_insert__subset, fact_odd__nonzero, fact_SUP1__iff, fact_zip__takeWhile__snd, fact_dvd_Oorder__trans, fact_zdiff__zmod__right, fact_injD, fact_set__take__subset__set__take, fact_folding__image__simple__idem_Oin__idem, fact_com_Osimps_I17_J, fact_rtranclD, fact_set__empty2, fact_dvd_Oless__asym_H, fact_divmod__nat__eq, fact_Inr__not__Inl, fact_in__listrel1__converse, fact_map__add__Some__iff, fact_less__imp__le__nat, fact_UNION__empty__conv_I1_J, fact_drop__eq__Nil, fact_sum__case__weak__cong, fact_card__Diff__subset, fact_setsum__Un__zero, fact_pred__nat__trancl__eq__le, fact_rel__comp__distrib, fact_setsum_Oreindex, fact_equiv__imp__dvd__card, fact_dom__map__upds, fact_le__imp__0__less, fact_add__leD1, fact_Times__Diff__distrib1, fact_converse__UNION, fact_Un__mono, fact_finite__Collect__subsets, fact_com_Osimps_I50_J, fact_com_Osimps_I40_J, fact_WTs__elim__cases_I3_J, fact_n__less__m__mult__n, fact_fold__image__Un__one, fact_Transitive__Closure_Otrancl__into__trancl, fact_fun__left__comm_Ofold__equality, fact_Suc__times__mod__eq, fact_prod_Orecs, fact_union__comp__emptyR, fact_UN__finite__subset, fact_mod__eq, fact_com_Osimps_I22_J, fact_Suc__mono, fact_concat__eq__Nil__conv, fact_Int__UNIV__right, fact_distinct__remdups, fact_self__append__conv, fact_O__assoc, fact_less__nat__zero__code, fact_wmin__insertI, fact_List_Omap_Ocompositionality, fact_mono__Field, fact_nat_Osize_I4_J, fact_setsum__diff__nat, fact_divmod__int__rel__mod, fact_finite__lists__length__eq, fact_equivp__transp, fact_setsum__setsum__restrict, fact_com_Osimps_I31_J, fact_add__less__mono, fact_UN__simps_I10_J, fact_zadd__commute, fact_setsum__atMost__Suc, fact_image__Int, fact_com_Osimps_I37_J, fact_IdI, fact_List_Omap_Ocomp, fact_com_Osimps_I69_J, fact_lexn__length, fact_map__of_Osimps_I2_J, fact_evalc__elim__cases_I1_J, fact_diff__Suc__less, fact_insertI1, fact_rel__comp__distrib2, fact_mult__0, fact_vimage__UNIV, fact_wf__no__loop, fact_WT_Oequations_I5_J, fact_zip__Cons__Cons, fact_Un__empty__left, fact_sup2I2, fact_divmod__int__relI, fact_pair__box_Oinject, fact_folding__one_Ounion__inter, fact_com_Osimps_I10_J, fact_folding__image__simple_Oeq__fold__g, fact_option_Osimps_I2_J, fact_com_Osimps_I46_J, fact_image__add__atLeastLessThan, fact_gr0__conv__Suc, fact_o__eq__elim, fact_zdvd__mult__cancel, fact_SigmaE2, fact_UNIV__not__empty, fact_zip__map1, fact_Range__rtrancl, fact_image__constant, fact_Un__insert__left, fact_triple_Osize_I2_J, fact_dvd_Oless__imp__not__less, fact_div__mod__equality_H, fact_Int__Union, fact_comm__monoid__big_Oinfinite, fact_pred__nat__def, fact_div__nonpos__pos__le0, fact_com_Osimps_I24_J, fact_rtrancl_Ortrancl__refl, fact_inj__image__mem__iff, fact_map__butlast, fact_evalc__elim__cases_I4_J, fact_prod__eqI, fact_zero__less__diff, fact_remove1_Osimps_I1_J, fact_card_Oempty, fact_acc_Osimps, fact_zmult2__lemma__aux4, fact_atLeastLessThanSuc, fact_setsum__insert, fact_finite__range__imageI, fact_diff__diff__cancel, fact_evalc_Oequations_I1_J, fact_equiv__class__eq, fact_funpow__add, fact_map__of__append, fact_nat__add__right__cancel, fact_distinct__append, fact_Some__eq__map__of__iff, fact_com_Osimps_I2_J, fact_set__subset__Cons, fact_aset_I3_J, fact_quotient__diff1, fact_inj__on__the__inv__into, fact_finite__Collect__bounded__ex, fact_div__le__dividend, fact_fold1__eq__fold, fact_listrel__subset__rtrancl__listrel1, fact_listrelp_Oequations_I2_J, fact_takeWhile__eq__all__conv, fact_WTs__elim__cases_I8_J, fact_Ex__inj__on__UNION__Sigma, fact_mod__pos__pos__trivial, fact_setsum__commute, fact_dvd_Oord__eq__less__trans, fact_the__inv__f__f, fact_take__upt, fact_dvd_OatLeastLessThan__empty__iff2, fact_dvd_OgreaterThan__iff, fact_Int__left__commute, fact_empty__not__insert, fact_UN__extend__simps_I5_J, fact_image__vimage__eq, fact_mult__le__mono2, help_c__fconj__3, fact_zmod__zdiv__equality, fact_com_Osimps_I60_J, fact_Suc__diff__diff, fact_image__Int__subset, fact_evaln_OAssign, fact_CollectI, fact_rtrancl__empty, fact_Union__Int__subset, fact_Suc__le__mono, fact_Image__UN, fact_option_Oexhaust, fact_comp__inj__on, fact_aset_I10_J, fact_Cons__acc__listrel1I, fact_image__UN, fact_int__0__less__1, fact_rtrancl__listrel1__if__listrel, fact_set__take__subset, fact_r__into__trancl_H, fact_mod__less, fact_rel__compI, fact_empty__upd__none, fact_Int__UN__distrib, fact_setsum__lessThan__Suc, fact_Un__UNIV__left, fact_filter__empty__conv, fact_setsum__multicount, fact_atLeastPlusOneLessThan__greaterThanLessThan__int, fact_infinite__UNIV__int, help_c__fequal__2, fact_UNION__empty__conv_I2_J, fact_mult__eq__self__implies__10, fact_Image__eq__UN, fact_wf__Int1, fact_le__neq__implies__less, fact_tranclD2, fact_wf__less, fact_Projl_Osimps, fact_distinct__zipI2, fact_divmod__nat__rel__unique, fact_expand__Suc, fact_sublist__shift__lemma, fact_map__add__upd, fact_dvd__diffD, fact_com_Osimps_I27_J, fact_sum_Osimps_I3_J, fact_snd__eqD, fact_filter__concat, fact_zle__linear, fact_append__eq__conv__conj, fact_triple__valid__def2, fact_neg__single__def, fact_zmult__div__cancel, fact_com_Osimps_I59_J, fact_setsum__reindex, fact_SUP__UN__eq2, fact_pred__equals__eq2, fact_setsum__multicount__gen, fact_nat__add__left__cancel__less, fact_map__pair__def, fact_finite__UnionD, fact_nth__Cons__Suc, help_c__fFalse__1, fact_setsum__eq__Suc0__iff, fact_evaln__Suc, fact_o__apply, fact_max__rpair__set, fact_fold1__Un2, fact_pair__leq__def, fact_swap__inj__on, fact_rtrancl__mono, fact_rtrancl__subset__rtrancl, fact_com_Orecs_I1_J, fact_mod__mult__distrib2, fact_inj__on__Un, fact_dom__map__add, fact_aset_I6_J, fact_nat__mod__eq__lemma, help_c__COMBC__1, fact_div__if, fact_Divides_Omod__div__equality_H, fact_self__remainder, fact_fun__left__comm_Ofun__left__comm, fact_finite__atLeastZeroLessThan__int, fact_inj__on__imageI2, fact_append1__eq__conv, fact_UN__equiv__class__type, fact_insert__not__empty, fact_fun__upd__None__restrict, fact_take__Suc__conv__app__nth, fact_setsum__0, fact_inv__image__def, fact_dvd_OatLeastAtMost__singleton, fact_prod_Osimps_I2_J, fact_Field__Union, fact_smax__emptyI, fact_mod__if, fact_nth__take, fact_div__1, fact_image__image, fact_rtrancl__converse, fact_Int__insert__right, fact_Sigma__empty1, fact_peek__and__def, fact_finite__SigmaI, fact_type__definition_OAbs__inject, fact_inj__on__fun__updI, fact_exE__realizer, fact_zadd__left__mono, fact_exI__realizer, fact_dvd_OlessThan__iff, fact_surj__image__vimage__eq, fact_mod__mult__self4, fact_number__of__is__id, arity_Int__Oint__Rings_Olinordered__semiring__1__strict, fact_listrel__iff__nth, fact_split__paired__All, fact_restrict__map__insert, fact_Un__Diff__Int, fact_append__is__Nil__conv, fact_Diff__subset, fact_distinct_Osimps_I2_J, fact_card__eq__UNIV__imp__eq__UNIV, fact_upt__eq__Nil__conv, fact_Field__Un, fact_less__eq, fact_sup__Un__eq2, fact_aset_I4_J, fact_acyclic__converse, fact_setsum__Un__disjoint, fact_rp__inv__image__def, fact_zdvd__mult__div__cancel, fact_dvd__reduce, fact_list_Osimps_I3_J, fact_split__div_H, fact_Range__empty, fact_rev__ImageI, fact_com_Orecs_I4_J, fact_fun__left__comm_Ofold__graph__insertE, fact_upt__conv__Cons, fact_finite__Collect__conjI, fact_nth__map, fact_fold1__in, fact_inf1D2, fact_zdiv__zero, fact_map__add__upds, fact_concat__injective, fact_evalc__elim__cases_I2_J, fact_com_Osimps_I62_J, fact_folding__image__simple_Oremove, fact_image__Un, fact_lessThan__0, fact_acc_OaccI, fact_transpose_Osimps_I2_J, fact_atMost__upto, fact_equalityD2, fact_split__div__lemma, fact_finite__cartesian__productD1, fact_butlast__take, fact_image__Suc__atLeastLessThan, fact_converse__Int, fact_finite__atLeastLessThan, fact_bot__empty__eq, fact_image__compose, fact_WT_OLocal, fact_reduction__pairI, fact_finite__nat__set__iff__bounded__le, fact_inf2D1, fact_takeWhile__nth, fact_zdvd__zmod__imp__zdvd, fact_remove1__filter__not, fact_map__upd__nonempty, fact_self__quotient__aux1, fact_Nil__eq__concat__conv, fact_rel__pow__add, fact_fold__graph_OemptyI, fact_rel__comp__mono, fact_mod__pos__neg__trivial, fact_setsum__reindex__nonzero, fact_snd__eq__Range, arity_Int__Oint__Groups_Ocomm__monoid__add, fact_setsum_OF__eq, fact_concat__eq__concat__iff, fact_greaterThanAtMost__upt, fact_single__valued__Id, fact_type__definition_ORep__range, fact_set__diff__eq, fact_map__add__dom__app__simps_I2_J, fact_nonneg1__imp__zdiv__pos__iff, fact_gr0I, fact_predicate2D, fact_fold__graph__imp__finite, fact_add__eq__if, fact_pigeonhole__infinite, fact_folding__one__idem_Oin__idem, fact_listrel_ONil, fact_fun__left__comm_Ofold__graph__fold, fact_finite__UnI, fact_triple_Oinject, fact_single__valued__rel__comp, fact_singleton__inject, fact_take__Cons__number__of, fact_setsum__Un__Int, fact_inj__on__map__add__dom, fact_set__remove1__subset, fact_Divides_Otransfer__nat__int__function__closures_I2_J, fact_emptyE, fact_acc__subset, fact_DiffD2, fact_finite__Prod__UNIV, fact_set__takeWhileD, fact_map__of__is__SomeD, fact_Int__empty__left, fact_setsum__Un, fact_UN__simps_I1_J, fact_neg__mod__sign, fact_setsum_Ounion__inter__neutral, fact_folding__one_Oinsert, fact_list_Osize_I4_J, fact_remove1__idem, fact_divmod__nat__rel__def, fact_Diff__idemp, fact_nat__dvd__not__less, fact_setsum_Oinfinite, fact_Collect__split, fact_restrict__restrict, fact_diff__le__self, fact_listrel_OCons, fact_UN__equiv__class__inject, fact_upt__Suc, fact_mult__eq__1__iff, fact_set__empty, fact_override__on__def, fact_folding__image__simple__idem_Oinsert__idem, fact_WT_Oequations_I2_J, fact_lessThan__Suc__eq__insert__0, fact_set__sublist, fact_pos__imp__zdiv__neg__iff, fact_le__iff__add, fact_wf__measure, fact_diff__commute, fact_list_Osize_I2_J, fact_setsum__diff1, fact_neq__if__length__neq, fact_map__add__empty, fact_partition__filter1, help_c__COMBS__1, fact_dom__fun__upd, fact_diff__Suc__eq__diff__pred, fact_less__eq__number__of__int__code, fact_UN__Un__distrib, fact_split__twice, fact_listrel1__subset__listrel, fact_Int__absorb2, fact_r__comp__rtrancl__eq, fact_Diff__subset__conv, fact_less__add__eq__less, fact_nat__mult__1__right, fact_singleton__conv, fact_wf__irrefl, fact_mod__1, fact_com_Osimps_I14_J, fact_finite__trancl, fact_eq__iff__diff__eq__0, fact_ran__def, fact_dvd_OatLeastatMost__subset__iff, fact_Image__Un, fact_ran__distinct, fact_setsum__head__upt__Suc, fact_Union__quotient, fact_fold1Set_Oequations, fact_fun__upd__same, fact_add__lessD1, fact_add__le__mono1, fact_UN__extend__simps_I1_J, fact_Int__Diff, fact_Eps__split__eq, fact_less__eq__nat_Osimps_I1_J, fact_DiffE, fact_com_Osimps_I55_J, fact_quotient__is__empty, fact_inf__Int__eq, fact_folding__one__idem_Ounion__idem, fact_Domain__empty__iff, fact_lessI, fact_finite__Un, fact_setsum__cartesian__product, fact_someI__ex, fact_mem__splitE, fact_partition__filter__conv, fact_zdiv__zmult1__eq, fact_in__mono, fact_image__eq__fold__image, fact_dvd_Oantisym__conv, fact_com_Osimps_I53_J, fact_Domain__Un__eq, fact_termination__basic__simps_I4_J, fact_wf__lex__prod, fact_Int__Un__distrib, fact_equiv__class__subset, fact_map__inj__on, fact_zdvd__reduce, fact_rtrancl__subset, fact_inj__on__iff__card__le, fact_dvd__mult__cancel2, fact_card__image, fact_nat_Oinject, fact_neg__mod__bound, fact_insert__Collect, fact_fun__upds__append2__drop, fact_union__fold__insert, fact_mem__splitI, fact_Int__assoc, fact_rtrancl__unfold, fact_finite__Collect__not, fact_not__add__less2, fact_less__or__eq__imp__le, fact_rtrancl__listrel1__ConsI1, fact_option_Osimps_I3_J, fact_fold__image__def, fact_list_Osimps_I2_J, fact_card_Oinsert__remove, fact_image__add__int__atLeastLessThan, fact_in__measure, fact_diff__card__le__card__Diff, fact_dvd_Oeq__refl, fact_refl__reflcl, fact_int__div__less__self, fact_div__neg__neg__trivial, fact_Suc__less__SucD, fact_aset_I9_J, fact_com_Osimps_I3_J, fact_top1I, fact_rev__image__eqI, fact_zadd__strict__right__mono, fact_congruent2__implies__congruent, fact_zdiv__zadd1__eq, fact_q__neg__lemma, fact_zadd__zless__mono, fact_split__div, fact_listrelp__listrel__eq, fact_set__minus__filter__out, fact_quotient__eq__iff, fact_funpow__swap1, fact_dvd_Oord__eq__le__trans, fact_nat__eq__add__iff2, fact_com_Osimps_I20_J, fact_dom__map__of__zip, fact_nth__length__takeWhile, fact_domI, fact_finite__acyclic__wf__converse, fact_rel__comp__UNION__distrib, fact_mult__less__mono1, fact_com_Osimps_I5_J, fact_folding__one_Oeq__fold, fact_subset__singletonD, fact_ball__empty, fact_Field__converse, fact_ord_OatMost__iff, fact_le__div__geq, fact_Suc__leI, fact_folding__one_Oeq__fold_H, fact_wfE__pf, fact_subset__iff__psubset__eq, fact_finite__Collect__less__nat, fact_div__nonneg__neg__le0, fact_folding__one__idem_Oeq__fold__idem_H, fact_single__valued__subset, fact_in__finite__psubset, fact_inj__on__contraD, fact_Id__on__iff, fact_Diff__insert, fact_map__comp__empty_I1_J, fact_insert__is__Un, fact_wfP__def, fact_split__zdiv, fact_pos__imp__zdiv__pos__iff, fact_unique__quotient__lemma__neg, fact_set__map, fact_diff__less, fact_convex__bound__lt, fact_UNIV__I, fact_evaln_Oequations_I2_J, fact_refl__onD, fact_distinct__takeWhile, fact_distinct__remove1, fact_map__zip__map, fact_not__less__less__Suc__eq, fact_vimage__eq, fact_comp__funpow, fact_finite__imageD, fact_converseE, fact_nat__mult__1, fact_comp__cong, fact_take__Cons, fact_Image__subset, fact_fold1__singleton, fact_append__listrel1I, fact_diff__is__0__eq_H, fact_elem__set, fact_Nat__Transfer_Otransfer__nat__int__function__closures_I6_J, fact_option_Osize_I3_J, fact_trancl__converse, fact_converse__inv__image, fact_com_Osimps_I66_J, fact_atLeastSucAtMost__greaterThanAtMost, fact_map__add__upd__left, fact_wf__subset, fact_rtrancl__Un__separator__converseE, fact_restrict__out, fact_div__mult2__eq, fact_pos__empty__def, fact_Rep__Integ__inject, help_c__fconj__2, fact_reflpE, fact_le__add2, fact_insert__compr__raw, fact_minus__nat_Odiff__0, fact_dvd_Oless__imp__neq, fact_finite__conv__nat__seg__image, fact_Int__insert__left, fact_Domain__insert, fact_ord_OgreaterThan__iff, fact_zadd__assoc, fact_pos__zmult__eq__1__iff, fact_inj__on__strict__subset, fact_distinct__take, fact_setsum_Odistrib, fact_divmod__int__rel__div__mod, fact_map__of_Osimps_I1_J, fact_com_Osimps_I23_J, fact_rtrancl__reflcl, fact_com_Osimps_I13_J, fact_map__upd__upds__conv__if, fact_pair__box_Osize_I1_J, 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fact_folding__image__simple__idem_Oidem, fact_eval__eq, fact_zdiv__mono2__neg__lemma, fact_map__of__map__restrict, fact_vimage__compose, fact_lexord__append__rightI, fact_ord_OatLeast__iff, fact_zadd1__lemma, fact_UN__lessThan__UNIV, fact_distinct__upt, fact_self__append__conv2, fact_rel__comp__empty2, fact_converse__rel__comp, fact_add__Suc__shift, fact_zdvd__iff__zmod__eq__0__number__of, fact_doubleton__eq__iff, fact_map__of__eq__dom, fact_less__SucI, fact_Suc__not__Zero, fact_insert__Diff1, fact_evalc_Oequations_I7_J, fact_quotient__def, fact_atLeast0AtMost, fact_butlast_Osimps_I1_J, fact_same__append__eq, fact_Suc__lessI, fact_add__Suc__right, fact_foldl__conv__concat, fact_folding__idem_Oidem__left__comp, fact_trancl__reflcl, fact_converse__Id, fact_com_Osimps_I51_J, fact_mult__1, fact_map_Osimps_I2_J, fact_relpow_Osimps_I2_J, fact_evaln__elim__cases_I1_J, fact_setsum__eq__0__iff, fact_listrel__iff__zip, fact_Field__def, fact_transfer__nat__int__set__cong, fact_less__bin__lemma, fact_rel__pow__commute, help_c__fdisj__3, fact_inj__on__iff__surj, fact_splitD_H, fact_Image__Id, fact_evaln_Oequations_I1_J, fact_Rep__Integ__cases, fact_Inl__not__Inr, fact_bot__nat__def, fact_mod__mult__distrib, fact_trancl__domain, fact_com_Osimps_I65_J, fact_Un__assoc, fact_rtrancl__power, fact_distinct__sublistI, fact_fold1__Un, fact_lexord__irreflexive, fact_listrelp_ONil, fact_evaln__evalc, fact_single__valuedD, fact_Sigma__empty2, fact_quotient__empty, fact_imageI, fact_disjoint__iff__not__equal
% 129.48/49.07  % No. of inferences in proof: 4856
% 129.48/49.07  % SZS status Theorem for /export/starexec/sandbox2/benchmark/theBenchmark.p : 48496 ms resp. 47496 ms w/o parsing
% 134.17/51.04  % SZS output start Refutation for /export/starexec/sandbox2/benchmark/theBenchmark.p
% See solution above
% 134.17/51.06  % [INFO] 	 Killing All external provers ... 
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