TPTP Problem File: SWW472+6.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : SWW472+6 : TPTP v8.2.0. Released v5.3.0.
% Domain : Software Verification
% Problem : Hoare's Logic with Procedures line 327, 500 axioms selected
% Version : Especial.
% English :
% Refs : [BN10] Boehme & Nipkow (2010), Sledgehammer: Judgement Day
% : [Bla11] Blanchette (2011), Email to Geoff Sutcliffe
% Source : [Bla11]
% Names : hoare_500_fofpt_l327 [Bla11]
% Status : Theorem
% Rating : 1.00 v5.3.0
% Syntax : Number of formulae : 617 ( 119 unt; 0 def)
% Number of atoms : 1782 ( 488 equ)
% Maximal formula atoms : 9 ( 2 avg)
% Number of connectives : 1342 ( 177 ~; 37 |; 60 &)
% ( 78 <=>; 990 =>; 0 <=; 0 <~>)
% Maximal formula depth : 17 ( 7 avg)
% Maximal term depth : 21 ( 2 avg)
% Number of predicates : 30 ( 29 usr; 0 prp; 1-2 aty)
% Number of functors : 68 ( 68 usr; 19 con; 0-4 aty)
% Number of variables : 2146 (2126 !; 20 ?)
% SPC : FOF_THM_RFO_SEQ
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-08-09 14:19:52
% : Encoded with polymorphic tags.
%------------------------------------------------------------------------------
%----Explicit typings (63)
fof(tsy_c_Big__Operators_Ocomm__monoid__add__class_Osetsum_res,axiom,
! [X_a,X_b] :
( comm_monoid_add(X_b)
=> ti(fun(fun(X_a,X_b),fun(fun(X_a,bool),X_b)),big_co1399186613setsum(X_a,X_b)) = big_co1399186613setsum(X_a,X_b) ) ).
fof(tsy_c_Big__Operators_Ocomm__monoid__mult__class_Osetprod_res,axiom,
! [X_a,X_b] :
( comm_monoid_mult(X_b)
=> ti(fun(fun(X_a,X_b),fun(fun(X_a,bool),X_b)),big_co302546470etprod(X_a,X_b)) = big_co302546470etprod(X_a,X_b) ) ).
fof(tsy_c_Big__Operators_Olattice_OInf__fin_res,axiom,
! [X_a] :
( linorder(X_a)
=> ti(fun(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,bool),X_a)),big_Inf_fin(X_a)) = big_Inf_fin(X_a) ) ).
fof(tsy_c_Big__Operators_Olattice_OSup__fin_res,axiom,
! [X_a] :
( linorder(X_a)
=> ti(fun(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,bool),X_a)),big_Sup_fin(X_a)) = big_Sup_fin(X_a) ) ).
fof(tsy_c_Big__Operators_Olinorder__class_OMax_res,axiom,
! [X_a] :
( linorder(X_a)
=> ti(fun(fun(X_a,bool),X_a),big_linorder_Max(X_a)) = big_linorder_Max(X_a) ) ).
fof(tsy_c_Big__Operators_Olinorder__class_OMin_res,axiom,
! [X_a] :
( linorder(X_a)
=> ti(fun(fun(X_a,bool),X_a),big_linorder_Min(X_a)) = big_linorder_Min(X_a) ) ).
fof(tsy_c_Big__Operators_Osemilattice__big_res,axiom,
! [X_a] : ti(fun(fun(X_a,fun(X_a,X_a)),fun(fun(fun(X_a,bool),X_a),bool)),big_semilattice_big(X_a)) = big_semilattice_big(X_a) ).
fof(tsy_c_COMBB_res,axiom,
! [X_b,X_c,X_a] : ti(fun(fun(X_b,X_c),fun(fun(X_a,X_b),fun(X_a,X_c))),combb(X_b,X_c,X_a)) = combb(X_b,X_c,X_a) ).
fof(tsy_c_COMBC_res,axiom,
! [X_a,X_b,X_c] : ti(fun(fun(X_a,fun(X_b,X_c)),fun(X_b,fun(X_a,X_c))),combc(X_a,X_b,X_c)) = combc(X_a,X_b,X_c) ).
fof(tsy_c_COMBK_res,axiom,
! [X_a,X_b] : ti(fun(X_a,fun(X_b,X_a)),combk(X_a,X_b)) = combk(X_a,X_b) ).
fof(tsy_c_COMBS_res,axiom,
! [X_a,X_b,X_c] : ti(fun(fun(X_a,fun(X_b,X_c)),fun(fun(X_a,X_b),fun(X_a,X_c))),combs(X_a,X_b,X_c)) = combs(X_a,X_b,X_c) ).
fof(tsy_c_Com_Ocom_OSKIP_res,axiom,
ti(com,skip) = skip ).
fof(tsy_c_Com_Ocom_OSemi_res,axiom,
ti(fun(com,fun(com,com)),semi) = semi ).
fof(tsy_c_Com_Ocom_Ocom__case_res,axiom,
! [X_a] : ti(fun(X_a,fun(fun(vname,fun(fun(state,nat),X_a)),fun(fun(loc,fun(fun(state,nat),fun(com,X_a))),fun(fun(com,fun(com,X_a)),fun(fun(fun(state,bool),fun(com,fun(com,X_a))),fun(fun(fun(state,bool),fun(com,X_a)),fun(fun(pname,X_a),fun(fun(vname,fun(pname,fun(fun(state,nat),X_a))),fun(com,X_a))))))))),com_case(X_a)) = com_case(X_a) ).
fof(tsy_c_Finite__Set_Ocomp__fun__commute_res,axiom,
! [X_a,X_b] : ti(fun(fun(X_a,fun(X_b,X_b)),bool),finite100568337ommute(X_a,X_b)) = finite100568337ommute(X_a,X_b) ).
fof(tsy_c_Finite__Set_Ocomp__fun__idem_res,axiom,
! [X_a,X_b] : ti(fun(fun(X_a,fun(X_b,X_b)),bool),finite_comp_fun_idem(X_a,X_b)) = finite_comp_fun_idem(X_a,X_b) ).
fof(tsy_c_Finite__Set_Ofinite_res,axiom,
! [X_a] : ti(fun(fun(X_a,bool),bool),finite_finite_1(X_a)) = finite_finite_1(X_a) ).
fof(tsy_c_Finite__Set_Ofold_res,axiom,
! [X_a,X_b] : ti(fun(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(fun(X_a,bool),X_b))),finite_fold(X_a,X_b)) = finite_fold(X_a,X_b) ).
fof(tsy_c_Finite__Set_Ofold1_res,axiom,
! [X_a] : ti(fun(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,bool),X_a)),finite_fold1(X_a)) = finite_fold1(X_a) ).
fof(tsy_c_Finite__Set_Ofold1Set_res,axiom,
! [X_a] : ti(fun(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,bool),fun(X_a,bool))),finite_fold1Set(X_a)) = finite_fold1Set(X_a) ).
fof(tsy_c_Finite__Set_Ofold__graph_res,axiom,
! [X_a,X_b] : ti(fun(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(fun(X_a,bool),fun(X_b,bool)))),finite_fold_graph(X_a,X_b)) = finite_fold_graph(X_a,X_b) ).
fof(tsy_c_Finite__Set_Ofolding__image__simple_res,axiom,
! [X_a,X_b] : ti(fun(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(fun(X_b,X_a),fun(fun(fun(X_b,bool),X_a),bool)))),finite1357897459simple(X_a,X_b)) = finite1357897459simple(X_a,X_b) ).
fof(tsy_c_Finite__Set_Ofolding__image__simple__idem_res,axiom,
! [X_a,X_b] : ti(fun(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(fun(X_b,X_a),fun(fun(fun(X_b,bool),X_a),bool)))),finite908156982e_idem(X_a,X_b)) = finite908156982e_idem(X_a,X_b) ).
fof(tsy_c_Finite__Set_Ofolding__one_res,axiom,
! [X_a] : ti(fun(fun(X_a,fun(X_a,X_a)),fun(fun(fun(X_a,bool),X_a),bool)),finite_folding_one(X_a)) = finite_folding_one(X_a) ).
fof(tsy_c_Finite__Set_Ofolding__one__idem_res,axiom,
! [X_a] : ti(fun(fun(X_a,fun(X_a,X_a)),fun(fun(fun(X_a,bool),X_a),bool)),finite2073411215e_idem(X_a)) = finite2073411215e_idem(X_a) ).
fof(tsy_c_Groups_Ominus__class_Ominus_res,axiom,
! [X_a] :
( minus(X_a)
=> ti(fun(X_a,fun(X_a,X_a)),minus_minus(X_a)) = minus_minus(X_a) ) ).
fof(tsy_c_Groups_Oplus__class_Oplus_0_res,axiom,
! [X_a] :
( cancel_semigroup_add(X_a)
=> ti(fun(X_a,fun(X_a,X_a)),plus_plus(X_a)) = plus_plus(X_a) ) ).
fof(tsy_c_Groups_Oplus__class_Oplus_1_res,axiom,
! [X_a] :
( ab_semigroup_add(X_a)
=> ti(fun(X_a,fun(X_a,X_a)),plus_plus(X_a)) = plus_plus(X_a) ) ).
fof(tsy_c_Groups_Otimes__class_Otimes_0_res,axiom,
! [X_a] :
( semiring(X_a)
=> ti(fun(X_a,fun(X_a,X_a)),times_times(X_a)) = times_times(X_a) ) ).
fof(tsy_c_Groups_Otimes__class_Otimes_1_res,axiom,
! [X_a] :
( ab_semigroup_mult(X_a)
=> ti(fun(X_a,fun(X_a,X_a)),times_times(X_a)) = times_times(X_a) ) ).
fof(tsy_c_HOL_Oundefined_res,axiom,
! [X_a] : ti(X_a,undefined(X_a)) = undefined(X_a) ).
fof(tsy_c_Hoare__Mirabelle__nrugjuseim_OMGT_res,hypothesis,
ti(fun(com,hoare_1656922687triple(state)),hoare_Mirabelle_MGT) = hoare_Mirabelle_MGT ).
fof(tsy_c_Hoare__Mirabelle__nrugjuseim_Ohoare__derivs_res,axiom,
! [X_a] : ti(fun(fun(hoare_1656922687triple(X_a),bool),fun(fun(hoare_1656922687triple(X_a),bool),bool)),hoare_279057269derivs(X_a)) = hoare_279057269derivs(X_a) ).
fof(tsy_c_Hoare__Mirabelle__nrugjuseim_Ohoare__valids_res,axiom,
! [X_a] : ti(fun(fun(hoare_1656922687triple(X_a),bool),fun(fun(hoare_1656922687triple(X_a),bool),bool)),hoare_2027193591valids(X_a)) = hoare_2027193591valids(X_a) ).
fof(tsy_c_Hoare__Mirabelle__nrugjuseim_Otriple_Otriple_res,axiom,
! [X_a] : ti(fun(fun(X_a,fun(state,bool)),fun(com,fun(fun(X_a,fun(state,bool)),hoare_1656922687triple(X_a)))),hoare_246368825triple(X_a)) = hoare_246368825triple(X_a) ).
fof(tsy_c_Hoare__Mirabelle__nrugjuseim_Otriple_Otriple__case_res,axiom,
! [X_b,X_a] : ti(fun(fun(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),X_a))),fun(hoare_1656922687triple(X_b),X_a)),hoare_1312322281e_case(X_b,X_a)) = hoare_1312322281e_case(X_b,X_a) ).
fof(tsy_c_Hoare__Mirabelle__nrugjuseim_Otriple_Otriple__rec_res,axiom,
! [X_b,X_a] : ti(fun(fun(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),X_a))),fun(hoare_1656922687triple(X_b),X_a)),hoare_1632998903le_rec(X_b,X_a)) = hoare_1632998903le_rec(X_b,X_a) ).
fof(tsy_c_Nitpick_Ofold__graph_H_res,axiom,
! [X_a,X_b] : ti(fun(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(fun(X_a,bool),fun(X_b,bool)))),fold_graph(X_a,X_b)) = fold_graph(X_a,X_b) ).
fof(tsy_c_Orderings_Obot__class_Obot_res,axiom,
! [X_a] :
( bot(X_a)
=> ti(X_a,bot_bot(X_a)) = bot_bot(X_a) ) ).
fof(tsy_c_Orderings_Oord_Omax_res,axiom,
! [X_a] :
( ord(X_a)
=> ti(fun(fun(X_a,fun(X_a,bool)),fun(X_a,fun(X_a,X_a))),max(X_a)) = max(X_a) ) ).
fof(tsy_c_Orderings_Oord_Omin_res,axiom,
! [X_a] :
( ord(X_a)
=> ti(fun(fun(X_a,fun(X_a,bool)),fun(X_a,fun(X_a,X_a))),min(X_a)) = min(X_a) ) ).
fof(tsy_c_Orderings_Oord__class_Oless_res,axiom,
! [X_a] :
( ord(X_a)
=> ti(fun(X_a,fun(X_a,bool)),ord_less(X_a)) = ord_less(X_a) ) ).
fof(tsy_c_Orderings_Oord__class_Oless__eq_res,axiom,
! [X_b] :
( ord(X_b)
=> ti(fun(X_b,fun(X_b,bool)),ord_less_eq(X_b)) = ord_less_eq(X_b) ) ).
fof(tsy_c_Orderings_Oord__class_Omax_res,axiom,
! [X_a] :
( ord(X_a)
=> ti(fun(X_a,fun(X_a,X_a)),ord_max(X_a)) = ord_max(X_a) ) ).
fof(tsy_c_Orderings_Oord__class_Omin_res,axiom,
! [X_a] :
( ord(X_a)
=> ti(fun(X_a,fun(X_a,X_a)),ord_min(X_a)) = ord_min(X_a) ) ).
fof(tsy_c_Set_OCollect_res,axiom,
! [X_a] : ti(fun(fun(X_a,bool),fun(X_a,bool)),collect(X_a)) = collect(X_a) ).
fof(tsy_c_Set_Oinsert_res,axiom,
! [X_a] : ti(fun(X_a,fun(fun(X_a,bool),fun(X_a,bool))),insert(X_a)) = insert(X_a) ).
fof(tsy_c_Set_Othe__elem_res,axiom,
! [X_a] : ti(fun(fun(X_a,bool),X_a),the_elem(X_a)) = the_elem(X_a) ).
fof(tsy_c_fFalse_res,axiom,
ti(bool,fFalse) = fFalse ).
fof(tsy_c_fNot_res,axiom,
ti(fun(bool,bool),fNot) = fNot ).
fof(tsy_c_fTrue_res,axiom,
ti(bool,fTrue) = fTrue ).
fof(tsy_c_fconj_res,axiom,
ti(fun(bool,fun(bool,bool)),fconj) = fconj ).
fof(tsy_c_fdisj_res,axiom,
ti(fun(bool,fun(bool,bool)),fdisj) = fdisj ).
fof(tsy_c_fequal_res,axiom,
! [X_a] : ti(fun(X_a,fun(X_a,bool)),fequal(X_a)) = fequal(X_a) ).
fof(tsy_c_fimplies_res,axiom,
ti(fun(bool,fun(bool,bool)),fimplies) = fimplies ).
fof(tsy_c_hAPP_arg1,axiom,
! [X_a,X_c,B_1_1,B_2_1] : hAPP(X_a,X_c,ti(fun(X_a,X_c),B_1_1),B_2_1) = hAPP(X_a,X_c,B_1_1,B_2_1) ).
fof(tsy_c_hAPP_arg2,axiom,
! [X_a,X_c,B_1_1,B_2_1] : hAPP(X_a,X_c,B_1_1,ti(X_a,B_2_1)) = hAPP(X_a,X_c,B_1_1,B_2_1) ).
fof(tsy_c_hAPP_res,axiom,
! [X_c,X_a,B_1_1,B_2_1] : ti(X_c,hAPP(X_a,X_c,B_1_1,B_2_1)) = hAPP(X_a,X_c,B_1_1,B_2_1) ).
fof(tsy_c_hBOOL_arg1,hypothesis,
! [B_1_1] :
( hBOOL(ti(bool,B_1_1))
<=> hBOOL(B_1_1) ) ).
fof(tsy_c_member_res,axiom,
! [X_a] : ti(fun(X_a,fun(fun(X_a,bool),bool)),member(X_a)) = member(X_a) ).
fof(tsy_v_P_res,hypothesis,
ti(fun(state,fun(state,bool)),p) = p ).
fof(tsy_v_Q_res,hypothesis,
ti(fun(state,fun(state,bool)),q) = q ).
fof(tsy_v_c_res,hypothesis,
ti(com,c) = c ).
%----Relevant facts (500)
fof(fact_0_empty,axiom,
! [X_a,G_2] : hBOOL(hAPP(fun(hoare_1656922687triple(X_a),bool),bool,hAPP(fun(hoare_1656922687triple(X_a),bool),fun(fun(hoare_1656922687triple(X_a),bool),bool),hoare_279057269derivs(X_a),G_2),bot_bot(fun(hoare_1656922687triple(X_a),bool)))) ).
fof(fact_1_triple_Oinject,axiom,
! [X_a,Fun1_1,Com_1,Fun2_1,Fun1_2,Com_2,Fun2_2] :
( hAPP(fun(X_a,fun(state,bool)),hoare_1656922687triple(X_a),hAPP(com,fun(fun(X_a,fun(state,bool)),hoare_1656922687triple(X_a)),hAPP(fun(X_a,fun(state,bool)),fun(com,fun(fun(X_a,fun(state,bool)),hoare_1656922687triple(X_a))),hoare_246368825triple(X_a),Fun1_1),Com_1),Fun2_1) = hAPP(fun(X_a,fun(state,bool)),hoare_1656922687triple(X_a),hAPP(com,fun(fun(X_a,fun(state,bool)),hoare_1656922687triple(X_a)),hAPP(fun(X_a,fun(state,bool)),fun(com,fun(fun(X_a,fun(state,bool)),hoare_1656922687triple(X_a))),hoare_246368825triple(X_a),Fun1_2),Com_2),Fun2_2)
<=> ( Fun1_1 = Fun1_2
& Com_1 = Com_2
& Fun2_1 = Fun2_2 ) ) ).
fof(fact_2_hoare__sound,axiom,
! [X_a,G_2,Ts] :
( hBOOL(hAPP(fun(hoare_1656922687triple(X_a),bool),bool,hAPP(fun(hoare_1656922687triple(X_a),bool),fun(fun(hoare_1656922687triple(X_a),bool),bool),hoare_279057269derivs(X_a),G_2),Ts))
=> hBOOL(hAPP(fun(hoare_1656922687triple(X_a),bool),bool,hAPP(fun(hoare_1656922687triple(X_a),bool),fun(fun(hoare_1656922687triple(X_a),bool),bool),hoare_2027193591valids(X_a),G_2),Ts)) ) ).
fof(fact_3_cut,axiom,
! [X_a,G_2,G_1,Ts] :
( hBOOL(hAPP(fun(hoare_1656922687triple(X_a),bool),bool,hAPP(fun(hoare_1656922687triple(X_a),bool),fun(fun(hoare_1656922687triple(X_a),bool),bool),hoare_279057269derivs(X_a),G_1),Ts))
=> ( hBOOL(hAPP(fun(hoare_1656922687triple(X_a),bool),bool,hAPP(fun(hoare_1656922687triple(X_a),bool),fun(fun(hoare_1656922687triple(X_a),bool),bool),hoare_279057269derivs(X_a),G_2),G_1))
=> hBOOL(hAPP(fun(hoare_1656922687triple(X_a),bool),bool,hAPP(fun(hoare_1656922687triple(X_a),bool),fun(fun(hoare_1656922687triple(X_a),bool),bool),hoare_279057269derivs(X_a),G_2),Ts)) ) ) ).
fof(fact_4_hoare__derivs_Oinsert,axiom,
! [X_a,Ts,G_2,T_3] :
( hBOOL(hAPP(fun(hoare_1656922687triple(X_a),bool),bool,hAPP(fun(hoare_1656922687triple(X_a),bool),fun(fun(hoare_1656922687triple(X_a),bool),bool),hoare_279057269derivs(X_a),G_2),hAPP(fun(hoare_1656922687triple(X_a),bool),fun(hoare_1656922687triple(X_a),bool),hAPP(hoare_1656922687triple(X_a),fun(fun(hoare_1656922687triple(X_a),bool),fun(hoare_1656922687triple(X_a),bool)),insert(hoare_1656922687triple(X_a)),T_3),bot_bot(fun(hoare_1656922687triple(X_a),bool)))))
=> ( hBOOL(hAPP(fun(hoare_1656922687triple(X_a),bool),bool,hAPP(fun(hoare_1656922687triple(X_a),bool),fun(fun(hoare_1656922687triple(X_a),bool),bool),hoare_279057269derivs(X_a),G_2),Ts))
=> hBOOL(hAPP(fun(hoare_1656922687triple(X_a),bool),bool,hAPP(fun(hoare_1656922687triple(X_a),bool),fun(fun(hoare_1656922687triple(X_a),bool),bool),hoare_279057269derivs(X_a),G_2),hAPP(fun(hoare_1656922687triple(X_a),bool),fun(hoare_1656922687triple(X_a),bool),hAPP(hoare_1656922687triple(X_a),fun(fun(hoare_1656922687triple(X_a),bool),fun(hoare_1656922687triple(X_a),bool)),insert(hoare_1656922687triple(X_a)),T_3),Ts))) ) ) ).
fof(fact_5_derivs__insertD,axiom,
! [X_a,G_2,T_3,Ts] :
( hBOOL(hAPP(fun(hoare_1656922687triple(X_a),bool),bool,hAPP(fun(hoare_1656922687triple(X_a),bool),fun(fun(hoare_1656922687triple(X_a),bool),bool),hoare_279057269derivs(X_a),G_2),hAPP(fun(hoare_1656922687triple(X_a),bool),fun(hoare_1656922687triple(X_a),bool),hAPP(hoare_1656922687triple(X_a),fun(fun(hoare_1656922687triple(X_a),bool),fun(hoare_1656922687triple(X_a),bool)),insert(hoare_1656922687triple(X_a)),T_3),Ts)))
=> ( hBOOL(hAPP(fun(hoare_1656922687triple(X_a),bool),bool,hAPP(fun(hoare_1656922687triple(X_a),bool),fun(fun(hoare_1656922687triple(X_a),bool),bool),hoare_279057269derivs(X_a),G_2),hAPP(fun(hoare_1656922687triple(X_a),bool),fun(hoare_1656922687triple(X_a),bool),hAPP(hoare_1656922687triple(X_a),fun(fun(hoare_1656922687triple(X_a),bool),fun(hoare_1656922687triple(X_a),bool)),insert(hoare_1656922687triple(X_a)),T_3),bot_bot(fun(hoare_1656922687triple(X_a),bool)))))
& hBOOL(hAPP(fun(hoare_1656922687triple(X_a),bool),bool,hAPP(fun(hoare_1656922687triple(X_a),bool),fun(fun(hoare_1656922687triple(X_a),bool),bool),hoare_279057269derivs(X_a),G_2),Ts)) ) ) ).
fof(fact_6_conseq2,axiom,
! [X_a,Qa,G_2,Pa,Ca,Q_2] :
( hBOOL(hAPP(fun(hoare_1656922687triple(X_a),bool),bool,hAPP(fun(hoare_1656922687triple(X_a),bool),fun(fun(hoare_1656922687triple(X_a),bool),bool),hoare_279057269derivs(X_a),G_2),hAPP(fun(hoare_1656922687triple(X_a),bool),fun(hoare_1656922687triple(X_a),bool),hAPP(hoare_1656922687triple(X_a),fun(fun(hoare_1656922687triple(X_a),bool),fun(hoare_1656922687triple(X_a),bool)),insert(hoare_1656922687triple(X_a)),hAPP(fun(X_a,fun(state,bool)),hoare_1656922687triple(X_a),hAPP(com,fun(fun(X_a,fun(state,bool)),hoare_1656922687triple(X_a)),hAPP(fun(X_a,fun(state,bool)),fun(com,fun(fun(X_a,fun(state,bool)),hoare_1656922687triple(X_a))),hoare_246368825triple(X_a),Pa),Ca),Q_2)),bot_bot(fun(hoare_1656922687triple(X_a),bool)))))
=> ( ! [Z_3,S] :
( hBOOL(hAPP(state,bool,hAPP(X_a,fun(state,bool),Q_2,Z_3),S))
=> hBOOL(hAPP(state,bool,hAPP(X_a,fun(state,bool),Qa,Z_3),S)) )
=> hBOOL(hAPP(fun(hoare_1656922687triple(X_a),bool),bool,hAPP(fun(hoare_1656922687triple(X_a),bool),fun(fun(hoare_1656922687triple(X_a),bool),bool),hoare_279057269derivs(X_a),G_2),hAPP(fun(hoare_1656922687triple(X_a),bool),fun(hoare_1656922687triple(X_a),bool),hAPP(hoare_1656922687triple(X_a),fun(fun(hoare_1656922687triple(X_a),bool),fun(hoare_1656922687triple(X_a),bool)),insert(hoare_1656922687triple(X_a)),hAPP(fun(X_a,fun(state,bool)),hoare_1656922687triple(X_a),hAPP(com,fun(fun(X_a,fun(state,bool)),hoare_1656922687triple(X_a)),hAPP(fun(X_a,fun(state,bool)),fun(com,fun(fun(X_a,fun(state,bool)),hoare_1656922687triple(X_a))),hoare_246368825triple(X_a),Pa),Ca),Qa)),bot_bot(fun(hoare_1656922687triple(X_a),bool))))) ) ) ).
fof(fact_7_conseq1,axiom,
! [X_a,Pa,G_2,P_2,Ca,Qa] :
( hBOOL(hAPP(fun(hoare_1656922687triple(X_a),bool),bool,hAPP(fun(hoare_1656922687triple(X_a),bool),fun(fun(hoare_1656922687triple(X_a),bool),bool),hoare_279057269derivs(X_a),G_2),hAPP(fun(hoare_1656922687triple(X_a),bool),fun(hoare_1656922687triple(X_a),bool),hAPP(hoare_1656922687triple(X_a),fun(fun(hoare_1656922687triple(X_a),bool),fun(hoare_1656922687triple(X_a),bool)),insert(hoare_1656922687triple(X_a)),hAPP(fun(X_a,fun(state,bool)),hoare_1656922687triple(X_a),hAPP(com,fun(fun(X_a,fun(state,bool)),hoare_1656922687triple(X_a)),hAPP(fun(X_a,fun(state,bool)),fun(com,fun(fun(X_a,fun(state,bool)),hoare_1656922687triple(X_a))),hoare_246368825triple(X_a),P_2),Ca),Qa)),bot_bot(fun(hoare_1656922687triple(X_a),bool)))))
=> ( ! [Z_3,S] :
( hBOOL(hAPP(state,bool,hAPP(X_a,fun(state,bool),Pa,Z_3),S))
=> hBOOL(hAPP(state,bool,hAPP(X_a,fun(state,bool),P_2,Z_3),S)) )
=> hBOOL(hAPP(fun(hoare_1656922687triple(X_a),bool),bool,hAPP(fun(hoare_1656922687triple(X_a),bool),fun(fun(hoare_1656922687triple(X_a),bool),bool),hoare_279057269derivs(X_a),G_2),hAPP(fun(hoare_1656922687triple(X_a),bool),fun(hoare_1656922687triple(X_a),bool),hAPP(hoare_1656922687triple(X_a),fun(fun(hoare_1656922687triple(X_a),bool),fun(hoare_1656922687triple(X_a),bool)),insert(hoare_1656922687triple(X_a)),hAPP(fun(X_a,fun(state,bool)),hoare_1656922687triple(X_a),hAPP(com,fun(fun(X_a,fun(state,bool)),hoare_1656922687triple(X_a)),hAPP(fun(X_a,fun(state,bool)),fun(com,fun(fun(X_a,fun(state,bool)),hoare_1656922687triple(X_a))),hoare_246368825triple(X_a),Pa),Ca),Qa)),bot_bot(fun(hoare_1656922687triple(X_a),bool))))) ) ) ).
fof(fact_8_insertE,axiom,
! [X_a,A_1,B,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_1),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),B),A_2)))
=> ( ti(X_a,A_1) != ti(X_a,B)
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_1),A_2)) ) ) ).
fof(fact_9_insertCI,axiom,
! [X_a,B,A_1,B_1] :
( ( ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_1),B_1))
=> ti(X_a,A_1) = ti(X_a,B) )
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_1),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),B),B_1))) ) ).
fof(fact_10_conseq12,axiom,
! [X_a,Qa,Pa,G_2,P_2,Ca,Q_2] :
( hBOOL(hAPP(fun(hoare_1656922687triple(X_a),bool),bool,hAPP(fun(hoare_1656922687triple(X_a),bool),fun(fun(hoare_1656922687triple(X_a),bool),bool),hoare_279057269derivs(X_a),G_2),hAPP(fun(hoare_1656922687triple(X_a),bool),fun(hoare_1656922687triple(X_a),bool),hAPP(hoare_1656922687triple(X_a),fun(fun(hoare_1656922687triple(X_a),bool),fun(hoare_1656922687triple(X_a),bool)),insert(hoare_1656922687triple(X_a)),hAPP(fun(X_a,fun(state,bool)),hoare_1656922687triple(X_a),hAPP(com,fun(fun(X_a,fun(state,bool)),hoare_1656922687triple(X_a)),hAPP(fun(X_a,fun(state,bool)),fun(com,fun(fun(X_a,fun(state,bool)),hoare_1656922687triple(X_a))),hoare_246368825triple(X_a),P_2),Ca),Q_2)),bot_bot(fun(hoare_1656922687triple(X_a),bool)))))
=> ( ! [Z_3,S] :
( hBOOL(hAPP(state,bool,hAPP(X_a,fun(state,bool),Pa,Z_3),S))
=> ! [S_1] :
( ! [Z_4] :
( hBOOL(hAPP(state,bool,hAPP(X_a,fun(state,bool),P_2,Z_4),S))
=> hBOOL(hAPP(state,bool,hAPP(X_a,fun(state,bool),Q_2,Z_4),S_1)) )
=> hBOOL(hAPP(state,bool,hAPP(X_a,fun(state,bool),Qa,Z_3),S_1)) ) )
=> hBOOL(hAPP(fun(hoare_1656922687triple(X_a),bool),bool,hAPP(fun(hoare_1656922687triple(X_a),bool),fun(fun(hoare_1656922687triple(X_a),bool),bool),hoare_279057269derivs(X_a),G_2),hAPP(fun(hoare_1656922687triple(X_a),bool),fun(hoare_1656922687triple(X_a),bool),hAPP(hoare_1656922687triple(X_a),fun(fun(hoare_1656922687triple(X_a),bool),fun(hoare_1656922687triple(X_a),bool)),insert(hoare_1656922687triple(X_a)),hAPP(fun(X_a,fun(state,bool)),hoare_1656922687triple(X_a),hAPP(com,fun(fun(X_a,fun(state,bool)),hoare_1656922687triple(X_a)),hAPP(fun(X_a,fun(state,bool)),fun(com,fun(fun(X_a,fun(state,bool)),hoare_1656922687triple(X_a))),hoare_246368825triple(X_a),Pa),Ca),Qa)),bot_bot(fun(hoare_1656922687triple(X_a),bool))))) ) ) ).
fof(fact_11_emptyE,axiom,
! [X_a,A_1] : ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_1),bot_bot(fun(X_a,bool)))) ).
fof(fact_12_triple_Orecs,axiom,
! [X_b,X_a,F1,Fun1_1,Com_1,Fun2_1] : hAPP(hoare_1656922687triple(X_b),X_a,hAPP(fun(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),X_a))),fun(hoare_1656922687triple(X_b),X_a),hoare_1632998903le_rec(X_b,X_a),F1),hAPP(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b),hAPP(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b)),hAPP(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b))),hoare_246368825triple(X_b),Fun1_1),Com_1),Fun2_1)) = hAPP(fun(X_b,fun(state,bool)),X_a,hAPP(com,fun(fun(X_b,fun(state,bool)),X_a),hAPP(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),X_a)),F1,Fun1_1),Com_1),Fun2_1) ).
fof(fact_13_triple_Osimps_I2_J,axiom,
! [X_b,X_a,F1,Fun1_1,Com_1,Fun2_1] : hAPP(hoare_1656922687triple(X_b),X_a,hAPP(fun(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),X_a))),fun(hoare_1656922687triple(X_b),X_a),hoare_1312322281e_case(X_b,X_a),F1),hAPP(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b),hAPP(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b)),hAPP(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b))),hoare_246368825triple(X_b),Fun1_1),Com_1),Fun2_1)) = hAPP(fun(X_b,fun(state,bool)),X_a,hAPP(com,fun(fun(X_b,fun(state,bool)),X_a),hAPP(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),X_a)),F1,Fun1_1),Com_1),Fun2_1) ).
fof(fact_14_empty__not__insert,axiom,
! [X_a,A_1,A_2] : bot_bot(fun(X_a,bool)) != hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),A_1),A_2) ).
fof(fact_15_insert__not__empty,axiom,
! [X_a,A_1,A_2] : hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),A_1),A_2) != bot_bot(fun(X_a,bool)) ).
fof(fact_16_singleton__iff,axiom,
! [X_a,B,A_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),B),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),A_1),bot_bot(fun(X_a,bool)))))
<=> ti(X_a,B) = ti(X_a,A_1) ) ).
fof(fact_17_doubleton__eq__iff,axiom,
! [X_a,A_1,B,Ca,D] :
( hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),A_1),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),B),bot_bot(fun(X_a,bool)))) = hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),Ca),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),D),bot_bot(fun(X_a,bool))))
<=> ( ( ti(X_a,A_1) = ti(X_a,Ca)
& ti(X_a,B) = ti(X_a,D) )
| ( ti(X_a,A_1) = ti(X_a,D)
& ti(X_a,B) = ti(X_a,Ca) ) ) ) ).
fof(fact_18_equals0D,axiom,
! [X_a,A_1,A_2] :
( ti(fun(X_a,bool),A_2) = bot_bot(fun(X_a,bool))
=> ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_1),A_2)) ) ).
fof(fact_19_Collect__empty__eq,axiom,
! [X_a,Pa] :
( hAPP(fun(X_a,bool),fun(X_a,bool),collect(X_a),Pa) = bot_bot(fun(X_a,bool))
<=> ! [X_1] : ~ hBOOL(hAPP(X_a,bool,Pa,X_1)) ) ).
fof(fact_20_empty__iff,axiom,
! [X_a,Ca] : ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Ca),bot_bot(fun(X_a,bool)))) ).
fof(fact_21_empty__Collect__eq,axiom,
! [X_a,Pa] :
( bot_bot(fun(X_a,bool)) = hAPP(fun(X_a,bool),fun(X_a,bool),collect(X_a),Pa)
<=> ! [X_1] : ~ hBOOL(hAPP(X_a,bool,Pa,X_1)) ) ).
fof(fact_22_ex__in__conv,axiom,
! [X_a,A_2] :
( ? [X_1] : hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),A_2))
<=> ti(fun(X_a,bool),A_2) != bot_bot(fun(X_a,bool)) ) ).
fof(fact_23_all__not__in__conv,axiom,
! [X_a,A_2] :
( ! [X_1] : ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),A_2))
<=> ti(fun(X_a,bool),A_2) = bot_bot(fun(X_a,bool)) ) ).
fof(fact_24_empty__def,axiom,
! [X_a] : bot_bot(fun(X_a,bool)) = hAPP(fun(X_a,bool),fun(X_a,bool),collect(X_a),hAPP(bool,fun(X_a,bool),combk(bool,X_a),fFalse)) ).
fof(fact_25_insert__absorb,axiom,
! [X_a,A_1,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_1),A_2))
=> hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),A_1),A_2) = ti(fun(X_a,bool),A_2) ) ).
fof(fact_26_insertI2,axiom,
! [X_a,B,A_1,B_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_1),B_1))
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_1),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),B),B_1))) ) ).
fof(fact_27_insert__ident,axiom,
! [X_a,B_1,X_2,A_2] :
( ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),A_2))
=> ( ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),B_1))
=> ( hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_2),A_2) = hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_2),B_1)
<=> ti(fun(X_a,bool),A_2) = ti(fun(X_a,bool),B_1) ) ) ) ).
fof(fact_28_insert__code,axiom,
! [X_a,Y_2,A_2,X_2] :
( hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),Y_2),A_2),X_2))
<=> ( ti(X_a,Y_2) = ti(X_a,X_2)
| hBOOL(hAPP(X_a,bool,A_2,X_2)) ) ) ).
fof(fact_29_insert__iff,axiom,
! [X_a,A_1,B,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_1),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),B),A_2)))
<=> ( ti(X_a,A_1) = ti(X_a,B)
| hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_1),A_2)) ) ) ).
fof(fact_30_insert__commute,axiom,
! [X_a,X_2,Y_2,A_2] : hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_2),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),Y_2),A_2)) = hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),Y_2),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_2),A_2)) ).
fof(fact_31_insert__absorb2,axiom,
! [X_a,X_2,A_2] : hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_2),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_2),A_2)) = hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_2),A_2) ).
fof(fact_32_insert__Collect,axiom,
! [X_a,A_1,Pa] : hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),A_1),hAPP(fun(X_a,bool),fun(X_a,bool),collect(X_a),Pa)) = hAPP(fun(X_a,bool),fun(X_a,bool),collect(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,fun(bool,bool)),fun(fun(X_a,bool),fun(X_a,bool)),combs(X_a,bool,bool),hAPP(fun(X_a,bool),fun(X_a,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_a,bool),fun(X_a,fun(bool,bool))),combb(bool,fun(bool,bool),X_a),fimplies),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(bool,bool),fun(fun(X_a,bool),fun(X_a,bool)),combb(bool,bool,X_a),fNot),hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,fun(X_a,bool)),fun(X_a,fun(X_a,bool)),combc(X_a,X_a,bool),fequal(X_a)),A_1)))),Pa)) ).
fof(fact_33_insert__compr,axiom,
! [X_a,A_1,B_1] : hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),A_1),B_1) = hAPP(fun(X_a,bool),fun(X_a,bool),collect(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,fun(bool,bool)),fun(fun(X_a,bool),fun(X_a,bool)),combs(X_a,bool,bool),hAPP(fun(X_a,bool),fun(X_a,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_a,bool),fun(X_a,fun(bool,bool))),combb(bool,fun(bool,bool),X_a),fdisj),hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,fun(X_a,bool)),fun(X_a,fun(X_a,bool)),combc(X_a,X_a,bool),fequal(X_a)),A_1))),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,fun(fun(X_a,bool),bool)),fun(fun(X_a,bool),fun(X_a,bool)),combc(X_a,fun(X_a,bool),bool),member(X_a)),B_1))) ).
fof(fact_34_insertI1,axiom,
! [X_a,A_1,B_1] : hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_1),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),A_1),B_1))) ).
fof(fact_35_singleton__inject,axiom,
! [X_a,A_1,B] :
( hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),A_1),bot_bot(fun(X_a,bool))) = hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),B),bot_bot(fun(X_a,bool)))
=> ti(X_a,A_1) = ti(X_a,B) ) ).
fof(fact_36_singletonE,axiom,
! [X_a,B,A_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),B),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),A_1),bot_bot(fun(X_a,bool)))))
=> ti(X_a,B) = ti(X_a,A_1) ) ).
fof(fact_37_the__elem__eq,axiom,
! [X_a,X_2] : hAPP(fun(X_a,bool),X_a,the_elem(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_2),bot_bot(fun(X_a,bool)))) = ti(X_a,X_2) ).
fof(fact_38_bot__apply,axiom,
! [X_b,X_a] :
( bot(X_a)
=> ! [X_2] : hAPP(X_b,X_a,bot_bot(fun(X_b,X_a)),X_2) = bot_bot(X_a) ) ).
fof(fact_39_bot__fun__def,axiom,
! [X_a,X_b] :
( bot(X_b)
=> ! [X_1] : hAPP(X_a,X_b,bot_bot(fun(X_a,X_b)),X_1) = bot_bot(X_b) ) ).
fof(fact_40_hoare__derivs_OSkip,axiom,
! [X_a,G_2,Pa] : hBOOL(hAPP(fun(hoare_1656922687triple(X_a),bool),bool,hAPP(fun(hoare_1656922687triple(X_a),bool),fun(fun(hoare_1656922687triple(X_a),bool),bool),hoare_279057269derivs(X_a),G_2),hAPP(fun(hoare_1656922687triple(X_a),bool),fun(hoare_1656922687triple(X_a),bool),hAPP(hoare_1656922687triple(X_a),fun(fun(hoare_1656922687triple(X_a),bool),fun(hoare_1656922687triple(X_a),bool)),insert(hoare_1656922687triple(X_a)),hAPP(fun(X_a,fun(state,bool)),hoare_1656922687triple(X_a),hAPP(com,fun(fun(X_a,fun(state,bool)),hoare_1656922687triple(X_a)),hAPP(fun(X_a,fun(state,bool)),fun(com,fun(fun(X_a,fun(state,bool)),hoare_1656922687triple(X_a))),hoare_246368825triple(X_a),Pa),skip),Pa)),bot_bot(fun(hoare_1656922687triple(X_a),bool))))) ).
fof(fact_41_Comp,axiom,
! [X_a,D,R_1,G_2,Pa,Ca,Qa] :
( hBOOL(hAPP(fun(hoare_1656922687triple(X_a),bool),bool,hAPP(fun(hoare_1656922687triple(X_a),bool),fun(fun(hoare_1656922687triple(X_a),bool),bool),hoare_279057269derivs(X_a),G_2),hAPP(fun(hoare_1656922687triple(X_a),bool),fun(hoare_1656922687triple(X_a),bool),hAPP(hoare_1656922687triple(X_a),fun(fun(hoare_1656922687triple(X_a),bool),fun(hoare_1656922687triple(X_a),bool)),insert(hoare_1656922687triple(X_a)),hAPP(fun(X_a,fun(state,bool)),hoare_1656922687triple(X_a),hAPP(com,fun(fun(X_a,fun(state,bool)),hoare_1656922687triple(X_a)),hAPP(fun(X_a,fun(state,bool)),fun(com,fun(fun(X_a,fun(state,bool)),hoare_1656922687triple(X_a))),hoare_246368825triple(X_a),Pa),Ca),Qa)),bot_bot(fun(hoare_1656922687triple(X_a),bool)))))
=> ( hBOOL(hAPP(fun(hoare_1656922687triple(X_a),bool),bool,hAPP(fun(hoare_1656922687triple(X_a),bool),fun(fun(hoare_1656922687triple(X_a),bool),bool),hoare_279057269derivs(X_a),G_2),hAPP(fun(hoare_1656922687triple(X_a),bool),fun(hoare_1656922687triple(X_a),bool),hAPP(hoare_1656922687triple(X_a),fun(fun(hoare_1656922687triple(X_a),bool),fun(hoare_1656922687triple(X_a),bool)),insert(hoare_1656922687triple(X_a)),hAPP(fun(X_a,fun(state,bool)),hoare_1656922687triple(X_a),hAPP(com,fun(fun(X_a,fun(state,bool)),hoare_1656922687triple(X_a)),hAPP(fun(X_a,fun(state,bool)),fun(com,fun(fun(X_a,fun(state,bool)),hoare_1656922687triple(X_a))),hoare_246368825triple(X_a),Qa),D),R_1)),bot_bot(fun(hoare_1656922687triple(X_a),bool)))))
=> hBOOL(hAPP(fun(hoare_1656922687triple(X_a),bool),bool,hAPP(fun(hoare_1656922687triple(X_a),bool),fun(fun(hoare_1656922687triple(X_a),bool),bool),hoare_279057269derivs(X_a),G_2),hAPP(fun(hoare_1656922687triple(X_a),bool),fun(hoare_1656922687triple(X_a),bool),hAPP(hoare_1656922687triple(X_a),fun(fun(hoare_1656922687triple(X_a),bool),fun(hoare_1656922687triple(X_a),bool)),insert(hoare_1656922687triple(X_a)),hAPP(fun(X_a,fun(state,bool)),hoare_1656922687triple(X_a),hAPP(com,fun(fun(X_a,fun(state,bool)),hoare_1656922687triple(X_a)),hAPP(fun(X_a,fun(state,bool)),fun(com,fun(fun(X_a,fun(state,bool)),hoare_1656922687triple(X_a))),hoare_246368825triple(X_a),Pa),hAPP(com,com,hAPP(com,fun(com,com),semi,Ca),D)),R_1)),bot_bot(fun(hoare_1656922687triple(X_a),bool))))) ) ) ).
fof(fact_42_triple_Oexhaust,axiom,
! [X_a,Y_2] :
~ ! [Fun1,Com,Fun2] : Y_2 != hAPP(fun(X_a,fun(state,bool)),hoare_1656922687triple(X_a),hAPP(com,fun(fun(X_a,fun(state,bool)),hoare_1656922687triple(X_a)),hAPP(fun(X_a,fun(state,bool)),fun(com,fun(fun(X_a,fun(state,bool)),hoare_1656922687triple(X_a))),hoare_246368825triple(X_a),Fun1),Com),Fun2) ).
fof(fact_43_Set_Oset__insert,axiom,
! [X_a,X_2,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),A_2))
=> ~ ! [B_4] :
( ti(fun(X_a,bool),A_2) = hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_2),B_4)
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),B_4)) ) ) ).
fof(fact_44_mk__disjoint__insert,axiom,
! [X_a,A_1,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_1),A_2))
=> ? [B_4] :
( ti(fun(X_a,bool),A_2) = hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),A_1),B_4)
& ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_1),B_4)) ) ) ).
fof(fact_45_equals0I,axiom,
! [X_a,A_2] :
( ! [Y_1] : ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Y_1),A_2))
=> ti(fun(X_a,bool),A_2) = bot_bot(fun(X_a,bool)) ) ).
fof(fact_46_conseq,axiom,
! [X_a,Qa,G_2,Ca,Pa] :
( ! [Z_3,S] :
( hBOOL(hAPP(state,bool,hAPP(X_a,fun(state,bool),Pa,Z_3),S))
=> ? [P_1,Q_1] :
( hBOOL(hAPP(fun(hoare_1656922687triple(X_a),bool),bool,hAPP(fun(hoare_1656922687triple(X_a),bool),fun(fun(hoare_1656922687triple(X_a),bool),bool),hoare_279057269derivs(X_a),G_2),hAPP(fun(hoare_1656922687triple(X_a),bool),fun(hoare_1656922687triple(X_a),bool),hAPP(hoare_1656922687triple(X_a),fun(fun(hoare_1656922687triple(X_a),bool),fun(hoare_1656922687triple(X_a),bool)),insert(hoare_1656922687triple(X_a)),hAPP(fun(X_a,fun(state,bool)),hoare_1656922687triple(X_a),hAPP(com,fun(fun(X_a,fun(state,bool)),hoare_1656922687triple(X_a)),hAPP(fun(X_a,fun(state,bool)),fun(com,fun(fun(X_a,fun(state,bool)),hoare_1656922687triple(X_a))),hoare_246368825triple(X_a),P_1),Ca),Q_1)),bot_bot(fun(hoare_1656922687triple(X_a),bool)))))
& ! [S_1] :
( ! [Z_4] :
( hBOOL(hAPP(state,bool,hAPP(X_a,fun(state,bool),P_1,Z_4),S))
=> hBOOL(hAPP(state,bool,hAPP(X_a,fun(state,bool),Q_1,Z_4),S_1)) )
=> hBOOL(hAPP(state,bool,hAPP(X_a,fun(state,bool),Qa,Z_3),S_1)) ) ) )
=> hBOOL(hAPP(fun(hoare_1656922687triple(X_a),bool),bool,hAPP(fun(hoare_1656922687triple(X_a),bool),fun(fun(hoare_1656922687triple(X_a),bool),bool),hoare_279057269derivs(X_a),G_2),hAPP(fun(hoare_1656922687triple(X_a),bool),fun(hoare_1656922687triple(X_a),bool),hAPP(hoare_1656922687triple(X_a),fun(fun(hoare_1656922687triple(X_a),bool),fun(hoare_1656922687triple(X_a),bool)),insert(hoare_1656922687triple(X_a)),hAPP(fun(X_a,fun(state,bool)),hoare_1656922687triple(X_a),hAPP(com,fun(fun(X_a,fun(state,bool)),hoare_1656922687triple(X_a)),hAPP(fun(X_a,fun(state,bool)),fun(com,fun(fun(X_a,fun(state,bool)),hoare_1656922687triple(X_a))),hoare_246368825triple(X_a),Pa),Ca),Qa)),bot_bot(fun(hoare_1656922687triple(X_a),bool))))) ) ).
fof(fact_47_nonempty__iff,axiom,
! [X_a,A_2] :
( ti(fun(X_a,bool),A_2) != bot_bot(fun(X_a,bool))
<=> ? [X_1,B_4] :
( ti(fun(X_a,bool),A_2) = hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_1),B_4)
& ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),B_4)) ) ) ).
fof(fact_48_com_Osimps_I13_J,axiom,
! [Com1_2,Com2_2] : hAPP(com,com,hAPP(com,fun(com,com),semi,Com1_2),Com2_2) != skip ).
fof(fact_49_com_Osimps_I12_J,axiom,
! [Com1_2,Com2_2] : skip != hAPP(com,com,hAPP(com,fun(com,com),semi,Com1_2),Com2_2) ).
fof(fact_50_com_Osimps_I3_J,axiom,
! [Com1,Com2,Com1_1,Com2_1] :
( hAPP(com,com,hAPP(com,fun(com,com),semi,Com1),Com2) = hAPP(com,com,hAPP(com,fun(com,com),semi,Com1_1),Com2_1)
<=> ( Com1 = Com1_1
& Com2 = Com2_1 ) ) ).
fof(fact_51_fold1Set__sing,axiom,
! [X_a,F,A_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,bool),fun(X_a,bool)),finite_fold1Set(X_a),F),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),A_1),bot_bot(fun(X_a,bool)))),B))
<=> ti(X_a,A_1) = ti(X_a,B) ) ).
fof(fact_52_folding__one_Osingleton,axiom,
! [X_a,X_2,F,F_1] :
( hBOOL(hAPP(fun(fun(X_a,bool),X_a),bool,hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(fun(X_a,bool),X_a),bool),finite_folding_one(X_a),F),F_1))
=> hAPP(fun(X_a,bool),X_a,F_1,hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_2),bot_bot(fun(X_a,bool)))) = ti(X_a,X_2) ) ).
fof(fact_53_bot__empty__eq,axiom,
! [X_a,X_1] :
( hBOOL(hAPP(X_a,bool,bot_bot(fun(X_a,bool)),X_1))
<=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),bot_bot(fun(X_a,bool)))) ) ).
fof(fact_54_fold1__singleton,axiom,
! [X_a,F,A_1] : hAPP(fun(X_a,bool),X_a,hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,bool),X_a),finite_fold1(X_a),F),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),A_1),bot_bot(fun(X_a,bool)))) = ti(X_a,A_1) ).
fof(fact_55_fold1__singleton__def,axiom,
! [X_a,A_1,G,F] :
( G = hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,bool),X_a),finite_fold1(X_a),F)
=> hAPP(fun(X_a,bool),X_a,G,hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),A_1),bot_bot(fun(X_a,bool)))) = ti(X_a,A_1) ) ).
fof(fact_56_empty__fold1SetE,axiom,
! [X_a,F,X_2] : ~ hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,bool),fun(X_a,bool)),finite_fold1Set(X_a),F),bot_bot(fun(X_a,bool))),X_2)) ).
fof(fact_57_fold1Set__nonempty,axiom,
! [X_a,F,A_2,X_2] :
( hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,bool),fun(X_a,bool)),finite_fold1Set(X_a),F),A_2),X_2))
=> ti(fun(X_a,bool),A_2) != bot_bot(fun(X_a,bool)) ) ).
fof(fact_58_fold1Set_Ointros,axiom,
! [X_a,F,A_1,A_2,X_2] :
( hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(fun(X_a,bool),fun(X_a,bool))),finite_fold_graph(X_a,X_a),F),A_1),A_2),X_2))
=> ( ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_1),A_2))
=> hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,bool),fun(X_a,bool)),finite_fold1Set(X_a),F),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),A_1),A_2)),X_2)) ) ) ).
fof(fact_59_folding__one_Oinsert,axiom,
! [X_a,X_2,A_2,F,F_1] :
( hBOOL(hAPP(fun(fun(X_a,bool),X_a),bool,hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(fun(X_a,bool),X_a),bool),finite_folding_one(X_a),F),F_1))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> ( ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),A_2))
=> ( ti(fun(X_a,bool),A_2) != bot_bot(fun(X_a,bool))
=> hAPP(fun(X_a,bool),X_a,F_1,hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_2),A_2)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),F,X_2),hAPP(fun(X_a,bool),X_a,F_1,A_2)) ) ) ) ) ).
fof(fact_60_folding__one_Oeq__fold,axiom,
! [X_a,A_2,F,F_1] :
( hBOOL(hAPP(fun(fun(X_a,bool),X_a),bool,hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(fun(X_a,bool),X_a),bool),finite_folding_one(X_a),F),F_1))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> hAPP(fun(X_a,bool),X_a,F_1,A_2) = hAPP(fun(X_a,bool),X_a,hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,bool),X_a),finite_fold1(X_a),F),A_2) ) ) ).
fof(fact_61_folding__one_Oclosed,axiom,
! [X_a,A_2,F,F_1] :
( hBOOL(hAPP(fun(fun(X_a,bool),X_a),bool,hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(fun(X_a,bool),X_a),bool),finite_folding_one(X_a),F),F_1))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> ( ti(fun(X_a,bool),A_2) != bot_bot(fun(X_a,bool))
=> ( ! [X_1,Y_1] : hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),F,X_1),Y_1)),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_1),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),Y_1),bot_bot(fun(X_a,bool))))))
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),hAPP(fun(X_a,bool),X_a,F_1,A_2)),A_2)) ) ) ) ) ).
fof(fact_62_com_Osimps_I64_J,axiom,
! [X_a,F1,F2,F3,F4,F5,F6,F7,F8] : hAPP(com,X_a,hAPP(fun(vname,fun(pname,fun(fun(state,nat),X_a))),fun(com,X_a),hAPP(fun(pname,X_a),fun(fun(vname,fun(pname,fun(fun(state,nat),X_a))),fun(com,X_a)),hAPP(fun(fun(state,bool),fun(com,X_a)),fun(fun(pname,X_a),fun(fun(vname,fun(pname,fun(fun(state,nat),X_a))),fun(com,X_a))),hAPP(fun(fun(state,bool),fun(com,fun(com,X_a))),fun(fun(fun(state,bool),fun(com,X_a)),fun(fun(pname,X_a),fun(fun(vname,fun(pname,fun(fun(state,nat),X_a))),fun(com,X_a)))),hAPP(fun(com,fun(com,X_a)),fun(fun(fun(state,bool),fun(com,fun(com,X_a))),fun(fun(fun(state,bool),fun(com,X_a)),fun(fun(pname,X_a),fun(fun(vname,fun(pname,fun(fun(state,nat),X_a))),fun(com,X_a))))),hAPP(fun(loc,fun(fun(state,nat),fun(com,X_a))),fun(fun(com,fun(com,X_a)),fun(fun(fun(state,bool),fun(com,fun(com,X_a))),fun(fun(fun(state,bool),fun(com,X_a)),fun(fun(pname,X_a),fun(fun(vname,fun(pname,fun(fun(state,nat),X_a))),fun(com,X_a)))))),hAPP(fun(vname,fun(fun(state,nat),X_a)),fun(fun(loc,fun(fun(state,nat),fun(com,X_a))),fun(fun(com,fun(com,X_a)),fun(fun(fun(state,bool),fun(com,fun(com,X_a))),fun(fun(fun(state,bool),fun(com,X_a)),fun(fun(pname,X_a),fun(fun(vname,fun(pname,fun(fun(state,nat),X_a))),fun(com,X_a))))))),hAPP(X_a,fun(fun(vname,fun(fun(state,nat),X_a)),fun(fun(loc,fun(fun(state,nat),fun(com,X_a))),fun(fun(com,fun(com,X_a)),fun(fun(fun(state,bool),fun(com,fun(com,X_a))),fun(fun(fun(state,bool),fun(com,X_a)),fun(fun(pname,X_a),fun(fun(vname,fun(pname,fun(fun(state,nat),X_a))),fun(com,X_a)))))))),com_case(X_a),F1),F2),F3),F4),F5),F6),F7),F8),skip) = ti(X_a,F1) ).
fof(fact_63_com_Osimps_I67_J,axiom,
! [X_a,F1,F2,F3,F4,F5,F6,F7,F8,Com1,Com2] : hAPP(com,X_a,hAPP(fun(vname,fun(pname,fun(fun(state,nat),X_a))),fun(com,X_a),hAPP(fun(pname,X_a),fun(fun(vname,fun(pname,fun(fun(state,nat),X_a))),fun(com,X_a)),hAPP(fun(fun(state,bool),fun(com,X_a)),fun(fun(pname,X_a),fun(fun(vname,fun(pname,fun(fun(state,nat),X_a))),fun(com,X_a))),hAPP(fun(fun(state,bool),fun(com,fun(com,X_a))),fun(fun(fun(state,bool),fun(com,X_a)),fun(fun(pname,X_a),fun(fun(vname,fun(pname,fun(fun(state,nat),X_a))),fun(com,X_a)))),hAPP(fun(com,fun(com,X_a)),fun(fun(fun(state,bool),fun(com,fun(com,X_a))),fun(fun(fun(state,bool),fun(com,X_a)),fun(fun(pname,X_a),fun(fun(vname,fun(pname,fun(fun(state,nat),X_a))),fun(com,X_a))))),hAPP(fun(loc,fun(fun(state,nat),fun(com,X_a))),fun(fun(com,fun(com,X_a)),fun(fun(fun(state,bool),fun(com,fun(com,X_a))),fun(fun(fun(state,bool),fun(com,X_a)),fun(fun(pname,X_a),fun(fun(vname,fun(pname,fun(fun(state,nat),X_a))),fun(com,X_a)))))),hAPP(fun(vname,fun(fun(state,nat),X_a)),fun(fun(loc,fun(fun(state,nat),fun(com,X_a))),fun(fun(com,fun(com,X_a)),fun(fun(fun(state,bool),fun(com,fun(com,X_a))),fun(fun(fun(state,bool),fun(com,X_a)),fun(fun(pname,X_a),fun(fun(vname,fun(pname,fun(fun(state,nat),X_a))),fun(com,X_a))))))),hAPP(X_a,fun(fun(vname,fun(fun(state,nat),X_a)),fun(fun(loc,fun(fun(state,nat),fun(com,X_a))),fun(fun(com,fun(com,X_a)),fun(fun(fun(state,bool),fun(com,fun(com,X_a))),fun(fun(fun(state,bool),fun(com,X_a)),fun(fun(pname,X_a),fun(fun(vname,fun(pname,fun(fun(state,nat),X_a))),fun(com,X_a)))))))),com_case(X_a),F1),F2),F3),F4),F5),F6),F7),F8),hAPP(com,com,hAPP(com,fun(com,com),semi,Com1),Com2)) = hAPP(com,X_a,hAPP(com,fun(com,X_a),F4,Com1),Com2) ).
fof(fact_64_insert__fold1SetE,axiom,
! [X_a,F,A_1,X_3,X_2] :
( hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,bool),fun(X_a,bool)),finite_fold1Set(X_a),F),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),A_1),X_3)),X_2))
=> ~ ! [A_5,A_4] :
( hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),A_1),X_3) = hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),A_5),A_4)
=> ( hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(fun(X_a,bool),fun(X_a,bool))),finite_fold_graph(X_a,X_a),F),A_5),A_4),X_2))
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_5),A_4)) ) ) ) ).
fof(fact_65_subset__singletonD,axiom,
! [X_a,A_2,X_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_2),bot_bot(fun(X_a,bool)))))
=> ( ti(fun(X_a,bool),A_2) = bot_bot(fun(X_a,bool))
| ti(fun(X_a,bool),A_2) = hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_2),bot_bot(fun(X_a,bool))) ) ) ).
fof(fact_66_folding__image__simple_Oempty,axiom,
! [X_b,X_a,F,Z_1,G,F_1] :
( hBOOL(hAPP(fun(fun(X_b,bool),X_a),bool,hAPP(fun(X_b,X_a),fun(fun(fun(X_b,bool),X_a),bool),hAPP(X_a,fun(fun(X_b,X_a),fun(fun(fun(X_b,bool),X_a),bool)),hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(fun(X_b,X_a),fun(fun(fun(X_b,bool),X_a),bool))),finite1357897459simple(X_a,X_b),F),Z_1),G),F_1))
=> hAPP(fun(X_b,bool),X_a,F_1,bot_bot(fun(X_b,bool))) = ti(X_a,Z_1) ) ).
fof(fact_67_order__refl,axiom,
! [X_a] :
( preorder(X_a)
=> ! [X] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),X)) ) ).
fof(fact_68_subsetD,axiom,
! [X_a,Ca,A_2,B_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Ca),A_2))
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Ca),B_1)) ) ) ).
fof(fact_69_equalityI,axiom,
! [X_a,A_2,B_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),B_1),A_2))
=> ti(fun(X_a,bool),A_2) = ti(fun(X_a,bool),B_1) ) ) ).
fof(fact_70_finite_OemptyI,axiom,
! [X_a] : hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),bot_bot(fun(X_a,bool)))) ).
fof(fact_71_finite_OinsertI,axiom,
! [X_a,A_1,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),A_1),A_2))) ) ).
fof(fact_72_empty__subsetI,axiom,
! [X_a,A_2] : hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),bot_bot(fun(X_a,bool))),A_2)) ).
fof(fact_73_rev__predicate1D,axiom,
! [X_a,Qa,Pa,X_2] :
( hBOOL(hAPP(X_a,bool,Pa,X_2))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),Pa),Qa))
=> hBOOL(hAPP(X_a,bool,Qa,X_2)) ) ) ).
fof(fact_74_predicate1D,axiom,
! [X_a,X_2,Pa,Qa] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),Pa),Qa))
=> ( hBOOL(hAPP(X_a,bool,Pa,X_2))
=> hBOOL(hAPP(X_a,bool,Qa,X_2)) ) ) ).
fof(fact_75_ext,axiom,
! [X_a,X_b,F,G] :
( ! [X_1] : hAPP(X_a,X_b,F,X_1) = hAPP(X_a,X_b,G,X_1)
=> ti(fun(X_a,X_b),F) = ti(fun(X_a,X_b),G) ) ).
fof(fact_76_mem__def,axiom,
! [X_a,X_2,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),A_2))
<=> hBOOL(hAPP(X_a,bool,A_2,X_2)) ) ).
fof(fact_77_Collect__def,axiom,
! [X_a,Pa] : hAPP(fun(X_a,bool),fun(X_a,bool),collect(X_a),Pa) = ti(fun(X_a,bool),Pa) ).
fof(fact_78_subset__refl,axiom,
! [X_a,A_2] : hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),A_2)) ).
fof(fact_79_set__eq__subset,axiom,
! [X_a,A_2,B_1] :
( ti(fun(X_a,bool),A_2) = ti(fun(X_a,bool),B_1)
<=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1))
& hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),B_1),A_2)) ) ) ).
fof(fact_80_equalityD1,axiom,
! [X_a,A_2,B_1] :
( ti(fun(X_a,bool),A_2) = ti(fun(X_a,bool),B_1)
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1)) ) ).
fof(fact_81_equalityD2,axiom,
! [X_a,A_2,B_1] :
( ti(fun(X_a,bool),A_2) = ti(fun(X_a,bool),B_1)
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),B_1),A_2)) ) ).
fof(fact_82_in__mono,axiom,
! [X_a,X_2,A_2,B_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),A_2))
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),B_1)) ) ) ).
fof(fact_83_set__rev__mp,axiom,
! [X_a,B_1,X_2,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),A_2))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1))
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),B_1)) ) ) ).
fof(fact_84_set__mp,axiom,
! [X_a,X_2,A_2,B_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),A_2))
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),B_1)) ) ) ).
fof(fact_85_subset__trans,axiom,
! [X_a,C,A_2,B_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),B_1),C))
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),C)) ) ) ).
fof(fact_86_equalityE,axiom,
! [X_a,A_2,B_1] :
( ti(fun(X_a,bool),A_2) = ti(fun(X_a,bool),B_1)
=> ~ ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1))
=> ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),B_1),A_2)) ) ) ).
fof(fact_87_finite,axiom,
! [X_a] :
( finite_finite(X_a)
=> ! [A_2] : hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2)) ) ).
fof(fact_88_finite__code,axiom,
! [X_a] :
( finite_finite(X_a)
=> ! [A_2] : hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2)) ) ).
fof(fact_89_le__fun__def,axiom,
! [X_a,X_b] :
( ord(X_b)
=> ! [F,G] :
( hBOOL(hAPP(fun(X_a,X_b),bool,hAPP(fun(X_a,X_b),fun(fun(X_a,X_b),bool),ord_less_eq(fun(X_a,X_b)),F),G))
<=> ! [X_1] : hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(X_a,X_b,F,X_1)),hAPP(X_a,X_b,G,X_1))) ) ) ).
fof(fact_90_linorder__linear,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X,Y] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),Y))
| hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),X)) ) ) ).
fof(fact_91_order__eq__iff,axiom,
! [X_a] :
( order(X_a)
=> ! [X_2,Y_2] :
( ti(X_a,X_2) = ti(X_a,Y_2)
<=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_2),Y_2))
& hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y_2),X_2)) ) ) ) ).
fof(fact_92_rev__finite__subset,axiom,
! [X_a,A_2,B_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),B_1))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1))
=> hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2)) ) ) ).
fof(fact_93_order__eq__refl,axiom,
! [X_a] :
( preorder(X_a)
=> ! [X,Y] :
( X = Y
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),Y)) ) ) ).
fof(fact_94_le__funD,axiom,
! [X_a,X_b] :
( ord(X_b)
=> ! [X_2,F,G] :
( hBOOL(hAPP(fun(X_a,X_b),bool,hAPP(fun(X_a,X_b),fun(fun(X_a,X_b),bool),ord_less_eq(fun(X_a,X_b)),F),G))
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(X_a,X_b,F,X_2)),hAPP(X_a,X_b,G,X_2))) ) ) ).
fof(fact_95_order__antisym__conv,axiom,
! [X_a] :
( order(X_a)
=> ! [Y_2,X_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y_2),X_2))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_2),Y_2))
<=> ti(X_a,X_2) = ti(X_a,Y_2) ) ) ) ).
fof(fact_96_finite__subset,axiom,
! [X_a,A_2,B_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),B_1))
=> hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2)) ) ) ).
fof(fact_97_ord__eq__le__trans,axiom,
! [X_a] :
( ord(X_a)
=> ! [C_1,A_3,B_3] :
( ti(X_a,A_3) = ti(X_a,B_3)
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B_3),C_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_3),C_1)) ) ) ) ).
fof(fact_98_xt1_I3_J,axiom,
! [X_a] :
( order(X_a)
=> ! [C_1,A_3,B_3] :
( ti(X_a,A_3) = ti(X_a,B_3)
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C_1),B_3))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C_1),A_3)) ) ) ) ).
fof(fact_99_ord__le__eq__trans,axiom,
! [X_a] :
( ord(X_a)
=> ! [C_1,A_3,B_3] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_3),B_3))
=> ( ti(X_a,B_3) = ti(X_a,C_1)
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_3),C_1)) ) ) ) ).
fof(fact_100_xt1_I4_J,axiom,
! [X_a] :
( order(X_a)
=> ! [C_1,B_3,A_3] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B_3),A_3))
=> ( ti(X_a,B_3) = ti(X_a,C_1)
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C_1),A_3)) ) ) ) ).
fof(fact_101_order__antisym,axiom,
! [X_a] :
( order(X_a)
=> ! [X,Y] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),Y))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),X))
=> ti(X_a,X) = ti(X_a,Y) ) ) ) ).
fof(fact_102_order__trans,axiom,
! [X_a] :
( preorder(X_a)
=> ! [Z,X,Y] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),Y))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),Z))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),Z)) ) ) ) ).
fof(fact_103_xt1_I5_J,axiom,
! [X_a] :
( order(X_a)
=> ! [Y,X] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),X))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),Y))
=> ti(X_a,X) = ti(X_a,Y) ) ) ) ).
fof(fact_104_xt1_I6_J,axiom,
! [X_a] :
( order(X_a)
=> ! [Z,Y,X] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),X))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Z),Y))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Z),X)) ) ) ) ).
fof(fact_105_le__funE,axiom,
! [X_a,X_b] :
( ord(X_b)
=> ! [X_2,F,G] :
( hBOOL(hAPP(fun(X_a,X_b),bool,hAPP(fun(X_a,X_b),fun(fun(X_a,X_b),bool),ord_less_eq(fun(X_a,X_b)),F),G))
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(X_a,X_b,F,X_2)),hAPP(X_a,X_b,G,X_2))) ) ) ).
fof(fact_106_linorder__le__cases,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X,Y] :
( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),Y))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),X)) ) ) ).
fof(fact_107_folding__image__simple_Oinsert,axiom,
! [X_b,X_a,X_2,A_2,F,Z_1,G,F_1] :
( hBOOL(hAPP(fun(fun(X_b,bool),X_a),bool,hAPP(fun(X_b,X_a),fun(fun(fun(X_b,bool),X_a),bool),hAPP(X_a,fun(fun(X_b,X_a),fun(fun(fun(X_b,bool),X_a),bool)),hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(fun(X_b,X_a),fun(fun(fun(X_b,bool),X_a),bool))),finite1357897459simple(X_a,X_b),F),Z_1),G),F_1))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_2))
=> ( ~ hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_2),A_2))
=> hAPP(fun(X_b,bool),X_a,F_1,hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_2),A_2)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),F,hAPP(X_b,X_a,G,X_2)),hAPP(fun(X_b,bool),X_a,F_1,A_2)) ) ) ) ).
fof(fact_108_bot__least,axiom,
! [X_a] :
( bot(X_a)
=> ! [A_3] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),bot_bot(X_a)),A_3)) ) ).
fof(fact_109_bot__unique,axiom,
! [X_a] :
( bot(X_a)
=> ! [A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),bot_bot(X_a)))
<=> ti(X_a,A_1) = bot_bot(X_a) ) ) ).
fof(fact_110_le__bot,axiom,
! [X_a] :
( bot(X_a)
=> ! [A_3] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_3),bot_bot(X_a)))
=> ti(X_a,A_3) = bot_bot(X_a) ) ) ).
fof(fact_111_subset__empty,axiom,
! [X_a,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),bot_bot(fun(X_a,bool))))
<=> ti(fun(X_a,bool),A_2) = bot_bot(fun(X_a,bool)) ) ).
fof(fact_112_subset__insertI,axiom,
! [X_a,B_1,A_1] : hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),B_1),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),A_1),B_1))) ).
fof(fact_113_insert__subset,axiom,
! [X_a,X_2,A_2,B_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_2),A_2)),B_1))
<=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),B_1))
& hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1)) ) ) ).
fof(fact_114_subset__insert,axiom,
! [X_a,B_1,X_2,A_2] :
( ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),A_2))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_2),B_1)))
<=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1)) ) ) ).
fof(fact_115_subset__insertI2,axiom,
! [X_a,B,A_2,B_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1))
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),B),B_1))) ) ).
fof(fact_116_insert__mono,axiom,
! [X_a,A_1,C,D_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),C),D_2))
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),A_1),C)),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),A_1),D_2))) ) ).
fof(fact_117_finite__insert,axiom,
! [X_a,A_1,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),A_1),A_2)))
<=> hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2)) ) ).
fof(fact_118_asm,axiom,
! [X_a,Ts,G_2] :
( hBOOL(hAPP(fun(hoare_1656922687triple(X_a),bool),bool,hAPP(fun(hoare_1656922687triple(X_a),bool),fun(fun(hoare_1656922687triple(X_a),bool),bool),ord_less_eq(fun(hoare_1656922687triple(X_a),bool)),Ts),G_2))
=> hBOOL(hAPP(fun(hoare_1656922687triple(X_a),bool),bool,hAPP(fun(hoare_1656922687triple(X_a),bool),fun(fun(hoare_1656922687triple(X_a),bool),bool),hoare_279057269derivs(X_a),G_2),Ts)) ) ).
fof(fact_119_weaken,axiom,
! [X_a,Ts,G_2,Ts_1] :
( hBOOL(hAPP(fun(hoare_1656922687triple(X_a),bool),bool,hAPP(fun(hoare_1656922687triple(X_a),bool),fun(fun(hoare_1656922687triple(X_a),bool),bool),hoare_279057269derivs(X_a),G_2),Ts_1))
=> ( hBOOL(hAPP(fun(hoare_1656922687triple(X_a),bool),bool,hAPP(fun(hoare_1656922687triple(X_a),bool),fun(fun(hoare_1656922687triple(X_a),bool),bool),ord_less_eq(fun(hoare_1656922687triple(X_a),bool)),Ts),Ts_1))
=> hBOOL(hAPP(fun(hoare_1656922687triple(X_a),bool),bool,hAPP(fun(hoare_1656922687triple(X_a),bool),fun(fun(hoare_1656922687triple(X_a),bool),bool),hoare_279057269derivs(X_a),G_2),Ts)) ) ) ).
fof(fact_120_thin,axiom,
! [X_a,G_2,G_1,Ts] :
( hBOOL(hAPP(fun(hoare_1656922687triple(X_a),bool),bool,hAPP(fun(hoare_1656922687triple(X_a),bool),fun(fun(hoare_1656922687triple(X_a),bool),bool),hoare_279057269derivs(X_a),G_1),Ts))
=> ( hBOOL(hAPP(fun(hoare_1656922687triple(X_a),bool),bool,hAPP(fun(hoare_1656922687triple(X_a),bool),fun(fun(hoare_1656922687triple(X_a),bool),bool),ord_less_eq(fun(hoare_1656922687triple(X_a),bool)),G_1),G_2))
=> hBOOL(hAPP(fun(hoare_1656922687triple(X_a),bool),bool,hAPP(fun(hoare_1656922687triple(X_a),bool),fun(fun(hoare_1656922687triple(X_a),bool),bool),hoare_279057269derivs(X_a),G_2),Ts)) ) ) ).
fof(fact_121_fold__graph_OemptyI,axiom,
! [X_b,X_a,F,Z_1] : hBOOL(hAPP(X_b,bool,hAPP(fun(X_a,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_a,bool),fun(X_b,bool)),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(fun(X_a,bool),fun(X_b,bool))),finite_fold_graph(X_a,X_b),F),Z_1),bot_bot(fun(X_a,bool))),Z_1)) ).
fof(fact_122_empty__fold__graphE,axiom,
! [X_b,X_a,F,Z_1,X_2] :
( hBOOL(hAPP(X_b,bool,hAPP(fun(X_a,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_a,bool),fun(X_b,bool)),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(fun(X_a,bool),fun(X_b,bool))),finite_fold_graph(X_a,X_b),F),Z_1),bot_bot(fun(X_a,bool))),X_2))
=> ti(X_b,X_2) = ti(X_b,Z_1) ) ).
fof(fact_123_fold__graph_OinsertI,axiom,
! [X_b,X_a,F,Z_1,Y_2,X_2,A_2] :
( ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),A_2))
=> ( hBOOL(hAPP(X_b,bool,hAPP(fun(X_a,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_a,bool),fun(X_b,bool)),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(fun(X_a,bool),fun(X_b,bool))),finite_fold_graph(X_a,X_b),F),Z_1),A_2),Y_2))
=> hBOOL(hAPP(X_b,bool,hAPP(fun(X_a,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_a,bool),fun(X_b,bool)),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(fun(X_a,bool),fun(X_b,bool))),finite_fold_graph(X_a,X_b),F),Z_1),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_2),A_2)),hAPP(X_b,X_b,hAPP(X_a,fun(X_b,X_b),F,X_2),Y_2))) ) ) ).
fof(fact_124_finite__subset__induct,axiom,
! [X_a,Pa,A_2,F_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),F_1))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),F_1),A_2))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,Pa,bot_bot(fun(X_a,bool))))
=> ( ! [A_5,F_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),F_2))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_5),A_2))
=> ( ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_5),F_2))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,Pa,F_2))
=> hBOOL(hAPP(fun(X_a,bool),bool,Pa,hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),A_5),F_2))) ) ) ) )
=> hBOOL(hAPP(fun(X_a,bool),bool,Pa,F_1)) ) ) ) ) ).
fof(fact_125_finite__nonempty__imp__fold1Set,axiom,
! [X_a,F,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> ( ti(fun(X_a,bool),A_2) != bot_bot(fun(X_a,bool))
=> ? [X1] : hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,bool),fun(X_a,bool)),finite_fold1Set(X_a),F),A_2),X1)) ) ) ).
fof(fact_126_subsetI,axiom,
! [X_a,B_1,A_2] :
( ! [X_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),A_2))
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),B_1)) )
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1)) ) ).
fof(fact_127_finite_Osimps,axiom,
! [X_a,A_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_1))
<=> ( ti(fun(X_a,bool),A_1) = bot_bot(fun(X_a,bool))
| ? [A_4,A_5] :
( ti(fun(X_a,bool),A_1) = hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),A_5),A_4)
& hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_4)) ) ) ) ).
fof(fact_128_finite__induct,axiom,
! [X_a,Pa,F_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),F_1))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,Pa,bot_bot(fun(X_a,bool))))
=> ( ! [X_1,F_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),F_2))
=> ( ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),F_2))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,Pa,F_2))
=> hBOOL(hAPP(fun(X_a,bool),bool,Pa,hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_1),F_2))) ) ) )
=> hBOOL(hAPP(fun(X_a,bool),bool,Pa,F_1)) ) ) ) ).
fof(fact_129_folding__image__simple_Oneutral,axiom,
! [X_b,X_a,A_2,F,Z_1,G,F_1] :
( hBOOL(hAPP(fun(fun(X_b,bool),X_a),bool,hAPP(fun(X_b,X_a),fun(fun(fun(X_b,bool),X_a),bool),hAPP(X_a,fun(fun(X_b,X_a),fun(fun(fun(X_b,bool),X_a),bool)),hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(fun(X_b,X_a),fun(fun(fun(X_b,bool),X_a),bool))),finite1357897459simple(X_a,X_b),F),Z_1),G),F_1))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_2))
=> ( ! [X_1] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),A_2))
=> hAPP(X_b,X_a,G,X_1) = ti(X_a,Z_1) )
=> hAPP(fun(X_b,bool),X_a,F_1,A_2) = ti(X_a,Z_1) ) ) ) ).
fof(fact_130_finite__imp__fold__graph,axiom,
! [X_b,X_a,F,Z_1,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> ? [X1] : hBOOL(hAPP(X_b,bool,hAPP(fun(X_a,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_a,bool),fun(X_b,bool)),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(fun(X_a,bool),fun(X_b,bool))),finite_fold_graph(X_a,X_b),F),Z_1),A_2),X1)) ) ).
fof(fact_131_le__funI,axiom,
! [X_a,X_b] :
( ord(X_b)
=> ! [F,G] :
( ! [X_1] : hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(X_a,X_b,F,X_1)),hAPP(X_a,X_b,G,X_1)))
=> hBOOL(hAPP(fun(X_a,X_b),bool,hAPP(fun(X_a,X_b),fun(fun(X_a,X_b),bool),ord_less_eq(fun(X_a,X_b)),F),G)) ) ) ).
fof(fact_132_fold1Set_Osimps,axiom,
! [X_a,F,A1,A2] :
( hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,bool),fun(X_a,bool)),finite_fold1Set(X_a),F),A1),A2))
<=> ? [A_5,A_4,X_1] :
( ti(fun(X_a,bool),A1) = hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),A_5),A_4)
& ti(X_a,A2) = ti(X_a,X_1)
& hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(fun(X_a,bool),fun(X_a,bool))),finite_fold_graph(X_a,X_a),F),A_5),A_4),X_1))
& ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_5),A_4)) ) ) ).
fof(fact_133_fold__graph_Osimps,axiom,
! [X_b,X_a,F,Z_1,A1,A2] :
( hBOOL(hAPP(X_b,bool,hAPP(fun(X_a,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_a,bool),fun(X_b,bool)),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(fun(X_a,bool),fun(X_b,bool))),finite_fold_graph(X_a,X_b),F),Z_1),A1),A2))
<=> ( ( ti(fun(X_a,bool),A1) = bot_bot(fun(X_a,bool))
& ti(X_b,A2) = ti(X_b,Z_1) )
| ? [X_1,A_4,Y_1] :
( ti(fun(X_a,bool),A1) = hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_1),A_4)
& ti(X_b,A2) = hAPP(X_b,X_b,hAPP(X_a,fun(X_b,X_b),F,X_1),Y_1)
& ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),A_4))
& hBOOL(hAPP(X_b,bool,hAPP(fun(X_a,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_a,bool),fun(X_b,bool)),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(fun(X_a,bool),fun(X_b,bool))),finite_fold_graph(X_a,X_b),F),Z_1),A_4),Y_1)) ) ) ) ).
fof(fact_134_folding__one__idem_Osubset__idem,axiom,
! [X_a,B_1,A_2,F,F_1] :
( hBOOL(hAPP(fun(fun(X_a,bool),X_a),bool,hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(fun(X_a,bool),X_a),bool),finite2073411215e_idem(X_a),F),F_1))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> ( ti(fun(X_a,bool),B_1) != bot_bot(fun(X_a,bool))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),B_1),A_2))
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),F,hAPP(fun(X_a,bool),X_a,F_1,B_1)),hAPP(fun(X_a,bool),X_a,F_1,A_2)) = hAPP(fun(X_a,bool),X_a,F_1,A_2) ) ) ) ) ).
fof(fact_135_folding__one__idem_Oinsert__idem,axiom,
! [X_a,X_2,A_2,F,F_1] :
( hBOOL(hAPP(fun(fun(X_a,bool),X_a),bool,hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(fun(X_a,bool),X_a),bool),finite2073411215e_idem(X_a),F),F_1))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> ( ti(fun(X_a,bool),A_2) != bot_bot(fun(X_a,bool))
=> hAPP(fun(X_a,bool),X_a,F_1,hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_2),A_2)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),F,X_2),hAPP(fun(X_a,bool),X_a,F_1,A_2)) ) ) ) ).
fof(fact_136_finite__ne__induct,axiom,
! [X_a,Pa,F_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),F_1))
=> ( ti(fun(X_a,bool),F_1) != bot_bot(fun(X_a,bool))
=> ( ! [X_1] : hBOOL(hAPP(fun(X_a,bool),bool,Pa,hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_1),bot_bot(fun(X_a,bool)))))
=> ( ! [X_1,F_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),F_2))
=> ( ti(fun(X_a,bool),F_2) != bot_bot(fun(X_a,bool))
=> ( ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),F_2))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,Pa,F_2))
=> hBOOL(hAPP(fun(X_a,bool),bool,Pa,hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_1),F_2))) ) ) ) )
=> hBOOL(hAPP(fun(X_a,bool),bool,Pa,F_1)) ) ) ) ) ).
fof(fact_137_folding__image__simple__idem_Osubset__idem,axiom,
! [X_b,X_a,B_1,A_2,F,Z_1,G,F_1] :
( hBOOL(hAPP(fun(fun(X_b,bool),X_a),bool,hAPP(fun(X_b,X_a),fun(fun(fun(X_b,bool),X_a),bool),hAPP(X_a,fun(fun(X_b,X_a),fun(fun(fun(X_b,bool),X_a),bool)),hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(fun(X_b,X_a),fun(fun(fun(X_b,bool),X_a),bool))),finite908156982e_idem(X_a,X_b),F),Z_1),G),F_1))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_2))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),B_1),A_2))
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),F,hAPP(fun(X_b,bool),X_a,F_1,B_1)),hAPP(fun(X_b,bool),X_a,F_1,A_2)) = hAPP(fun(X_b,bool),X_a,F_1,A_2) ) ) ) ).
fof(fact_138_folding__image__simple__idem_Oinsert__idem,axiom,
! [X_b,X_a,X_2,A_2,F,Z_1,G,F_1] :
( hBOOL(hAPP(fun(fun(X_b,bool),X_a),bool,hAPP(fun(X_b,X_a),fun(fun(fun(X_b,bool),X_a),bool),hAPP(X_a,fun(fun(X_b,X_a),fun(fun(fun(X_b,bool),X_a),bool)),hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(fun(X_b,X_a),fun(fun(fun(X_b,bool),X_a),bool))),finite908156982e_idem(X_a,X_b),F),Z_1),G),F_1))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_2))
=> hAPP(fun(X_b,bool),X_a,F_1,hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_2),A_2)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),F,hAPP(X_b,X_a,G,X_2)),hAPP(fun(X_b,bool),X_a,F_1,A_2)) ) ) ).
fof(fact_139_Collect__mono,axiom,
! [X_a,Qa,Pa] :
( ! [X_1] :
( hBOOL(hAPP(X_a,bool,Pa,X_1))
=> hBOOL(hAPP(X_a,bool,Qa,X_1)) )
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),hAPP(fun(X_a,bool),fun(X_a,bool),collect(X_a),Pa)),hAPP(fun(X_a,bool),fun(X_a,bool),collect(X_a),Qa))) ) ).
fof(fact_140_folding__image__simple__idem_Oidem,axiom,
! [X_b,X_a,X_2,F,Z_1,G,F_1] :
( hBOOL(hAPP(fun(fun(X_b,bool),X_a),bool,hAPP(fun(X_b,X_a),fun(fun(fun(X_b,bool),X_a),bool),hAPP(X_a,fun(fun(X_b,X_a),fun(fun(fun(X_b,bool),X_a),bool)),hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(fun(X_b,X_a),fun(fun(fun(X_b,bool),X_a),bool))),finite908156982e_idem(X_a,X_b),F),Z_1),G),F_1))
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),F,X_2),X_2) = ti(X_a,X_2) ) ).
fof(fact_141_folding__one__idem_Oidem,axiom,
! [X_a,X_2,F,F_1] :
( hBOOL(hAPP(fun(fun(X_a,bool),X_a),bool,hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(fun(X_a,bool),X_a),bool),finite2073411215e_idem(X_a),F),F_1))
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),F,X_2),X_2) = ti(X_a,X_2) ) ).
fof(fact_142_folding__one__idem_Oin__idem,axiom,
! [X_a,X_2,A_2,F,F_1] :
( hBOOL(hAPP(fun(fun(X_a,bool),X_a),bool,hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(fun(X_a,bool),X_a),bool),finite2073411215e_idem(X_a),F),F_1))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),A_2))
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),F,X_2),hAPP(fun(X_a,bool),X_a,F_1,A_2)) = hAPP(fun(X_a,bool),X_a,F_1,A_2) ) ) ) ).
fof(fact_143_folding__image__simple__idem_Oin__idem,axiom,
! [X_b,X_a,X_2,A_2,F,Z_1,G,F_1] :
( hBOOL(hAPP(fun(fun(X_b,bool),X_a),bool,hAPP(fun(X_b,X_a),fun(fun(fun(X_b,bool),X_a),bool),hAPP(X_a,fun(fun(X_b,X_a),fun(fun(fun(X_b,bool),X_a),bool)),hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(fun(X_b,X_a),fun(fun(fun(X_b,bool),X_a),bool))),finite908156982e_idem(X_a,X_b),F),Z_1),G),F_1))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_2))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_2),A_2))
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),F,hAPP(X_b,X_a,G,X_2)),hAPP(fun(X_b,bool),X_a,F_1,A_2)) = hAPP(fun(X_b,bool),X_a,F_1,A_2) ) ) ) ).
fof(fact_144_predicate1I,axiom,
! [X_a,Qa,Pa] :
( ! [X_1] :
( hBOOL(hAPP(X_a,bool,Pa,X_1))
=> hBOOL(hAPP(X_a,bool,Qa,X_1)) )
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),Pa),Qa)) ) ).
fof(fact_145_xt3,axiom,
! [X_b,X_a] :
( ( order(X_a)
& order(X_b) )
=> ! [Ca,F,B,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),A_1))
=> ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),Ca),hAPP(X_a,X_b,F,B)))
=> ( ! [X_1,Y_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y_1),X_1))
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(X_a,X_b,F,Y_1)),hAPP(X_a,X_b,F,X_1))) )
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),Ca),hAPP(X_a,X_b,F,A_1))) ) ) ) ) ).
fof(fact_146_xt1_I16_J,axiom,
! [X_a] :
( order(X_a)
=> ! [F,Ca,B,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),A_1))
=> ( hAPP(X_a,X_a,F,B) = ti(X_a,Ca)
=> ( ! [X_1,Y_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y_1),X_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,F,Y_1)),hAPP(X_a,X_a,F,X_1))) )
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Ca),hAPP(X_a,X_a,F,A_1))) ) ) ) ) ).
fof(fact_147_ord__le__eq__subst,axiom,
! [X_a,X_b] :
( ( ord(X_b)
& ord(X_a) )
=> ! [F,Ca,A_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B))
=> ( hAPP(X_a,X_b,F,B) = ti(X_b,Ca)
=> ( ! [X_1,Y_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_1),Y_1))
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(X_a,X_b,F,X_1)),hAPP(X_a,X_b,F,Y_1))) )
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(X_a,X_b,F,A_1)),Ca)) ) ) ) ) ).
fof(fact_148_order__subst2,axiom,
! [X_a,X_b] :
( ( order(X_b)
& order(X_a) )
=> ! [F,Ca,A_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B))
=> ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(X_a,X_b,F,B)),Ca))
=> ( ! [X_1,Y_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_1),Y_1))
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(X_a,X_b,F,X_1)),hAPP(X_a,X_b,F,Y_1))) )
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(X_a,X_b,F,A_1)),Ca)) ) ) ) ) ).
fof(fact_149_ord__eq__le__subst,axiom,
! [X_a,X_b] :
( ( ord(X_b)
& ord(X_a) )
=> ! [Ca,A_1,F,B] :
( ti(X_a,A_1) = hAPP(X_b,X_a,F,B)
=> ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),B),Ca))
=> ( ! [X_1,Y_1] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),X_1),Y_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_b,X_a,F,X_1)),hAPP(X_b,X_a,F,Y_1))) )
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),hAPP(X_b,X_a,F,Ca))) ) ) ) ) ).
fof(fact_150_xt2,axiom,
! [X_b,X_a] :
( ( order(X_a)
& ord(X_b) )
=> ! [Ca,F,B,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_b,X_a,F,B)),A_1))
=> ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),Ca),B))
=> ( ! [X_1,Y_1] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),Y_1),X_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_b,X_a,F,Y_1)),hAPP(X_b,X_a,F,X_1))) )
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_b,X_a,F,Ca)),A_1)) ) ) ) ) ).
fof(fact_151_xt1_I15_J,axiom,
! [X_a] :
( order(X_a)
=> ! [Ca,A_1,F,B] :
( ti(X_a,A_1) = hAPP(X_a,X_a,F,B)
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Ca),B))
=> ( ! [X_1,Y_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y_1),X_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,F,Y_1)),hAPP(X_a,X_a,F,X_1))) )
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,F,Ca)),A_1)) ) ) ) ) ).
fof(fact_152_order__subst1,axiom,
! [X_a,X_b] :
( ( order(X_b)
& order(X_a) )
=> ! [Ca,A_1,F,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),hAPP(X_b,X_a,F,B)))
=> ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),B),Ca))
=> ( ! [X_1,Y_1] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),X_1),Y_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_b,X_a,F,X_1)),hAPP(X_b,X_a,F,Y_1))) )
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),hAPP(X_b,X_a,F,Ca))) ) ) ) ) ).
fof(fact_153_semilattice__big_OF__eq,axiom,
! [X_a,A_2,F,F_1] :
( hBOOL(hAPP(fun(fun(X_a,bool),X_a),bool,hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(fun(X_a,bool),X_a),bool),big_semilattice_big(X_a),F),F_1))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> hAPP(fun(X_a,bool),X_a,F_1,A_2) = hAPP(fun(X_a,bool),X_a,hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,bool),X_a),finite_fold1(X_a),F),A_2) ) ) ).
fof(fact_154_folding__one_Oremove,axiom,
! [X_a,X_2,A_2,F,F_1] :
( hBOOL(hAPP(fun(fun(X_a,bool),X_a),bool,hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(fun(X_a,bool),X_a),bool),finite_folding_one(X_a),F),F_1))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),A_2))
=> ( ( hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_2),bot_bot(fun(X_a,bool)))) = bot_bot(fun(X_a,bool))
=> hAPP(fun(X_a,bool),X_a,F_1,A_2) = ti(X_a,X_2) )
& ( hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_2),bot_bot(fun(X_a,bool)))) != bot_bot(fun(X_a,bool))
=> hAPP(fun(X_a,bool),X_a,F_1,A_2) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),F,X_2),hAPP(fun(X_a,bool),X_a,F_1,hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_2),bot_bot(fun(X_a,bool)))))) ) ) ) ) ) ).
fof(fact_155_folding__one_Oinsert__remove,axiom,
! [X_a,X_2,A_2,F,F_1] :
( hBOOL(hAPP(fun(fun(X_a,bool),X_a),bool,hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(fun(X_a,bool),X_a),bool),finite_folding_one(X_a),F),F_1))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> ( ( hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_2),bot_bot(fun(X_a,bool)))) = bot_bot(fun(X_a,bool))
=> hAPP(fun(X_a,bool),X_a,F_1,hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_2),A_2)) = ti(X_a,X_2) )
& ( hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_2),bot_bot(fun(X_a,bool)))) != bot_bot(fun(X_a,bool))
=> hAPP(fun(X_a,bool),X_a,F_1,hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_2),A_2)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),F,X_2),hAPP(fun(X_a,bool),X_a,F_1,hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_2),bot_bot(fun(X_a,bool)))))) ) ) ) ) ).
fof(fact_156_DiffI,axiom,
! [X_a,B_1,Ca,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Ca),A_2))
=> ( ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Ca),B_1))
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Ca),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),B_1))) ) ) ).
fof(fact_157_DiffE,axiom,
! [X_a,Ca,A_2,B_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Ca),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),B_1)))
=> ~ ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Ca),A_2))
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Ca),B_1)) ) ) ).
fof(fact_158_finite__Diff,axiom,
! [X_a,B_1,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),B_1))) ) ).
fof(fact_159_set__diff__eq,axiom,
! [X_a,A_2,B_1] : hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),B_1) = hAPP(fun(X_a,bool),fun(X_a,bool),collect(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,fun(bool,bool)),fun(fun(X_a,bool),fun(X_a,bool)),combs(X_a,bool,bool),hAPP(fun(X_a,bool),fun(X_a,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_a,bool),fun(X_a,fun(bool,bool))),combb(bool,fun(bool,bool),X_a),fconj),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,fun(fun(X_a,bool),bool)),fun(fun(X_a,bool),fun(X_a,bool)),combc(X_a,fun(X_a,bool),bool),member(X_a)),A_2))),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(bool,bool),fun(fun(X_a,bool),fun(X_a,bool)),combb(bool,bool,X_a),fNot),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,fun(fun(X_a,bool),bool)),fun(fun(X_a,bool),fun(X_a,bool)),combc(X_a,fun(X_a,bool),bool),member(X_a)),B_1)))) ).
fof(fact_160_Diff__iff,axiom,
! [X_a,Ca,A_2,B_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Ca),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),B_1)))
<=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Ca),A_2))
& ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Ca),B_1)) ) ) ).
fof(fact_161_Diff__idemp,axiom,
! [X_a,A_2,B_1] : hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),B_1)),B_1) = hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),B_1) ).
fof(fact_162_DiffD1,axiom,
! [X_a,Ca,A_2,B_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Ca),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),B_1)))
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Ca),A_2)) ) ).
fof(fact_163_DiffD2,axiom,
! [X_a,Ca,A_2,B_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Ca),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),B_1)))
=> ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Ca),B_1)) ) ).
fof(fact_164_empty__Diff,axiom,
! [X_a,A_2] : hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),bot_bot(fun(X_a,bool))),A_2) = bot_bot(fun(X_a,bool)) ).
fof(fact_165_Diff__empty,axiom,
! [X_a,A_2] : hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),bot_bot(fun(X_a,bool))) = ti(fun(X_a,bool),A_2) ).
fof(fact_166_Diff__cancel,axiom,
! [X_a,A_2] : hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),A_2) = bot_bot(fun(X_a,bool)) ).
fof(fact_167_finite__Diff2,axiom,
! [X_a,A_2,B_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),B_1))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),B_1)))
<=> hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2)) ) ) ).
fof(fact_168_insert__Diff__if,axiom,
! [X_a,A_2,X_2,B_1] :
( ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),B_1))
=> hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_2),A_2)),B_1) = hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),B_1) )
& ( ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),B_1))
=> hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_2),A_2)),B_1) = hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_2),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),B_1)) ) ) ).
fof(fact_169_insert__Diff1,axiom,
! [X_a,A_2,X_2,B_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),B_1))
=> hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_2),A_2)),B_1) = hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),B_1) ) ).
fof(fact_170_Diff__subset,axiom,
! [X_a,A_2,B_1] : hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),B_1)),A_2)) ).
fof(fact_171_Diff__mono,axiom,
! [X_a,D_2,B_1,A_2,C] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),C))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),D_2),B_1))
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),B_1)),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),C),D_2))) ) ) ).
fof(fact_172_double__diff,axiom,
! [X_a,C,A_2,B_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),B_1),C))
=> hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),B_1),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),C),A_2)) = ti(fun(X_a,bool),A_2) ) ) ).
fof(fact_173_Diff__insert,axiom,
! [X_a,A_2,A_1,B_1] : hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),A_1),B_1)) = hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),B_1)),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),A_1),bot_bot(fun(X_a,bool)))) ).
fof(fact_174_Diff__insert2,axiom,
! [X_a,A_2,A_1,B_1] : hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),A_1),B_1)) = hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),A_1),bot_bot(fun(X_a,bool))))),B_1) ).
fof(fact_175_insert__Diff__single,axiom,
! [X_a,A_1,A_2] : hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),A_1),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),A_1),bot_bot(fun(X_a,bool))))) = hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),A_1),A_2) ).
fof(fact_176_Diff__insert__absorb,axiom,
! [X_a,X_2,A_2] :
( ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),A_2))
=> hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_2),A_2)),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_2),bot_bot(fun(X_a,bool)))) = ti(fun(X_a,bool),A_2) ) ).
fof(fact_177_insert__Diff,axiom,
! [X_a,A_1,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_1),A_2))
=> hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),A_1),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),A_1),bot_bot(fun(X_a,bool))))) = ti(fun(X_a,bool),A_2) ) ).
fof(fact_178_finite__Diff__insert,axiom,
! [X_a,A_2,A_1,B_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),A_1),B_1))))
<=> hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),B_1))) ) ).
fof(fact_179_subset__insert__iff,axiom,
! [X_a,A_2,X_2,B_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_2),B_1)))
<=> ( ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),A_2))
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_2),bot_bot(fun(X_a,bool))))),B_1)) )
& ( ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),A_2))
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1)) ) ) ) ).
fof(fact_180_diff__single__insert,axiom,
! [X_a,A_2,X_2,B_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_2),bot_bot(fun(X_a,bool))))),B_1))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),A_2))
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_2),B_1))) ) ) ).
fof(fact_181_folding__image__simple_Oinsert__remove,axiom,
! [X_b,X_a,X_2,A_2,F,Z_1,G,F_1] :
( hBOOL(hAPP(fun(fun(X_b,bool),X_a),bool,hAPP(fun(X_b,X_a),fun(fun(fun(X_b,bool),X_a),bool),hAPP(X_a,fun(fun(X_b,X_a),fun(fun(fun(X_b,bool),X_a),bool)),hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(fun(X_b,X_a),fun(fun(fun(X_b,bool),X_a),bool))),finite1357897459simple(X_a,X_b),F),Z_1),G),F_1))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_2))
=> hAPP(fun(X_b,bool),X_a,F_1,hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_2),A_2)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),F,hAPP(X_b,X_a,G,X_2)),hAPP(fun(X_b,bool),X_a,F_1,hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_2),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_2),bot_bot(fun(X_b,bool)))))) ) ) ).
fof(fact_182_folding__image__simple_Oremove,axiom,
! [X_b,X_a,X_2,A_2,F,Z_1,G,F_1] :
( hBOOL(hAPP(fun(fun(X_b,bool),X_a),bool,hAPP(fun(X_b,X_a),fun(fun(fun(X_b,bool),X_a),bool),hAPP(X_a,fun(fun(X_b,X_a),fun(fun(fun(X_b,bool),X_a),bool)),hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(fun(X_b,X_a),fun(fun(fun(X_b,bool),X_a),bool))),finite1357897459simple(X_a,X_b),F),Z_1),G),F_1))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_2))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_2),A_2))
=> hAPP(fun(X_b,bool),X_a,F_1,A_2) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),F,hAPP(X_b,X_a,G,X_2)),hAPP(fun(X_b,bool),X_a,F_1,hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_2),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_2),bot_bot(fun(X_b,bool)))))) ) ) ) ).
fof(fact_183_finite__empty__induct,axiom,
! [X_a,Pa,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,Pa,A_2))
=> ( ! [A_5,A_4] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_4))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_5),A_4))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,Pa,A_4))
=> hBOOL(hAPP(fun(X_a,bool),bool,Pa,hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_4),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),A_5),bot_bot(fun(X_a,bool)))))) ) ) )
=> hBOOL(hAPP(fun(X_a,bool),bool,Pa,bot_bot(fun(X_a,bool)))) ) ) ) ).
fof(fact_184_minus__apply,axiom,
! [X_b,X_a] :
( minus(X_a)
=> ! [A_2,B_1,X_2] : hAPP(X_b,X_a,hAPP(fun(X_b,X_a),fun(X_b,X_a),hAPP(fun(X_b,X_a),fun(fun(X_b,X_a),fun(X_b,X_a)),minus_minus(fun(X_b,X_a)),A_2),B_1),X_2) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_b,X_a,A_2,X_2)),hAPP(X_b,X_a,B_1,X_2)) ) ).
fof(fact_185_fun__diff__def,axiom,
! [X_a,X_b] :
( minus(X_b)
=> ! [A_2,B_1,X_1] : hAPP(X_a,X_b,hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_a,X_b),fun(fun(X_a,X_b),fun(X_a,X_b)),minus_minus(fun(X_a,X_b)),A_2),B_1),X_1) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),minus_minus(X_b),hAPP(X_a,X_b,A_2,X_1)),hAPP(X_a,X_b,B_1,X_1)) ) ).
fof(fact_186_diff__eq__diff__less__eq,axiom,
! [X_a] :
( ordered_ab_group_add(X_a)
=> ! [A_1,B,Ca,D] :
( hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_1),B) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),Ca),D)
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Ca),D)) ) ) ) ).
fof(fact_187_fold1__antimono,axiom,
! [X_a] :
( linorder(X_a)
=> ! [B_1,A_2] :
( ti(fun(X_a,bool),A_2) != bot_bot(fun(X_a,bool))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),B_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(fun(X_a,bool),X_a,hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,bool),X_a),finite_fold1(X_a),ord_min(X_a)),B_1)),hAPP(fun(X_a,bool),X_a,hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,bool),X_a),finite_fold1(X_a),ord_min(X_a)),A_2))) ) ) ) ) ).
fof(fact_188_comp__fun__commute_Ofold__graph__insertE__aux,axiom,
! [X_a,X_b,A_1,Z_1,A_2,Y_2,F] :
( hBOOL(hAPP(fun(X_a,fun(X_b,X_b)),bool,finite100568337ommute(X_a,X_b),F))
=> ( hBOOL(hAPP(X_b,bool,hAPP(fun(X_a,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_a,bool),fun(X_b,bool)),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(fun(X_a,bool),fun(X_b,bool))),finite_fold_graph(X_a,X_b),F),Z_1),A_2),Y_2))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_1),A_2))
=> ? [Y_3] :
( ti(X_b,Y_2) = hAPP(X_b,X_b,hAPP(X_a,fun(X_b,X_b),F,A_1),Y_3)
& hBOOL(hAPP(X_b,bool,hAPP(fun(X_a,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_a,bool),fun(X_b,bool)),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(fun(X_a,bool),fun(X_b,bool))),finite_fold_graph(X_a,X_b),F),Z_1),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),A_1),bot_bot(fun(X_a,bool))))),Y_3)) ) ) ) ) ).
fof(fact_189_min__diff__distrib__left,axiom,
! [X_a] :
( ordered_ab_group_add(X_a)
=> ! [X,Y,Z] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X),Y)),Z) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),X),Z)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),Y),Z)) ) ).
fof(fact_190_Min_Oidem,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X),X) = ti(X_a,X) ) ).
fof(fact_191_comp__fun__commute_Ofun__left__comm,axiom,
! [X_a,X_b,X_2,Y_2,Z_1,F] :
( hBOOL(hAPP(fun(X_a,fun(X_b,X_b)),bool,finite100568337ommute(X_a,X_b),F))
=> hAPP(X_b,X_b,hAPP(X_a,fun(X_b,X_b),F,X_2),hAPP(X_b,X_b,hAPP(X_a,fun(X_b,X_b),F,Y_2),Z_1)) = hAPP(X_b,X_b,hAPP(X_a,fun(X_b,X_b),F,Y_2),hAPP(X_b,X_b,hAPP(X_a,fun(X_b,X_b),F,X_2),Z_1)) ) ).
fof(fact_192_min__max_Oinf_Oidem,axiom,
! [X_a] :
( linorder(X_a)
=> ! [A_3] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),A_3),A_3) = ti(X_a,A_3) ) ).
fof(fact_193_min__max_Oinf_Ocommute,axiom,
! [X_a] :
( linorder(X_a)
=> ! [A_3,B_3] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),A_3),B_3) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),B_3),A_3) ) ).
fof(fact_194_min__max_Oinf__commute,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X,Y] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X),Y) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),Y),X) ) ).
fof(fact_195_min__max_Oinf_Oleft__idem,axiom,
! [X_a] :
( linorder(X_a)
=> ! [A_3,B_3] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),A_3),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),A_3),B_3)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),A_3),B_3) ) ).
fof(fact_196_min__max_Oinf__left__idem,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X,Y] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X),Y)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X),Y) ) ).
fof(fact_197_min__max_Oinf_Oleft__commute,axiom,
! [X_a] :
( linorder(X_a)
=> ! [B_3,A_3,C_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),B_3),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),A_3),C_1)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),A_3),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),B_3),C_1)) ) ).
fof(fact_198_min__max_Oinf__left__commute,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X,Y,Z] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),Y),Z)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),Y),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X),Z)) ) ).
fof(fact_199_min__max_Oinf_Oassoc,axiom,
! [X_a] :
( linorder(X_a)
=> ! [A_3,B_3,C_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),A_3),B_3)),C_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),A_3),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),B_3),C_1)) ) ).
fof(fact_200_min__max_Oinf__assoc,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X,Y,Z] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X),Y)),Z) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),Y),Z)) ) ).
fof(fact_201_min__max_Oinf__le1,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X,Y] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X),Y)),X)) ) ).
fof(fact_202_min__max_Oinf__le2,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X,Y] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X),Y)),Y)) ) ).
fof(fact_203_min__max_Ole__iff__inf,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X_2,Y_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_2),Y_2))
<=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X_2),Y_2) = ti(X_a,X_2) ) ) ).
fof(fact_204_min__max_Ole__inf__iff,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X_2,Y_2,Z_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_2),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),Y_2),Z_1)))
<=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_2),Y_2))
& hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_2),Z_1)) ) ) ) ).
fof(fact_205_min__max_Ole__infI1,axiom,
! [X_a] :
( linorder(X_a)
=> ! [B_3,A_3,X] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_3),X))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),A_3),B_3)),X)) ) ) ).
fof(fact_206_min__max_Ole__infI2,axiom,
! [X_a] :
( linorder(X_a)
=> ! [A_3,B_3,X] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B_3),X))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),A_3),B_3)),X)) ) ) ).
fof(fact_207_min__max_Oinf__absorb1,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X,Y] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),Y))
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X),Y) = ti(X_a,X) ) ) ).
fof(fact_208_min__max_Oinf__absorb2,axiom,
! [X_a] :
( linorder(X_a)
=> ! [Y,X] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),X))
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X),Y) = ti(X_a,Y) ) ) ).
fof(fact_209_min__max_Ole__infI,axiom,
! [X_a] :
( linorder(X_a)
=> ! [B_3,X,A_3] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),A_3))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),B_3))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),A_3),B_3))) ) ) ) ).
fof(fact_210_min__max_Oinf__greatest,axiom,
! [X_a] :
( linorder(X_a)
=> ! [Z,X,Y] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),Y))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),Z))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),Y),Z))) ) ) ) ).
fof(fact_211_min__max_Oinf__mono,axiom,
! [X_a] :
( linorder(X_a)
=> ! [B_3,D_1,A_3,C_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_3),C_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B_3),D_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),A_3),B_3)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),C_1),D_1))) ) ) ) ).
fof(fact_212_min__max_Ole__infE,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X,A_3,B_3] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),A_3),B_3)))
=> ~ ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),A_3))
=> ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),B_3)) ) ) ) ).
fof(fact_213_min__le__iff__disj,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X_2,Y_2,Z_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X_2),Y_2)),Z_1))
<=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_2),Z_1))
| hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y_2),Z_1)) ) ) ) ).
fof(fact_214_comp__fun__commute_Ofold__graph__determ,axiom,
! [X_a,X_b,Y_2,Z_1,A_2,X_2,F] :
( hBOOL(hAPP(fun(X_a,fun(X_b,X_b)),bool,finite100568337ommute(X_a,X_b),F))
=> ( hBOOL(hAPP(X_b,bool,hAPP(fun(X_a,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_a,bool),fun(X_b,bool)),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(fun(X_a,bool),fun(X_b,bool))),finite_fold_graph(X_a,X_b),F),Z_1),A_2),X_2))
=> ( hBOOL(hAPP(X_b,bool,hAPP(fun(X_a,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_a,bool),fun(X_b,bool)),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(fun(X_a,bool),fun(X_b,bool))),finite_fold_graph(X_a,X_b),F),Z_1),A_2),Y_2))
=> ti(X_b,Y_2) = ti(X_b,X_2) ) ) ) ).
fof(fact_215_min__max_Ofold1__belowI,axiom,
! [X_a] :
( linorder(X_a)
=> ! [A_1,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_1),A_2))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(fun(X_a,bool),X_a,hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,bool),X_a),finite_fold1(X_a),ord_min(X_a)),A_2)),A_1)) ) ) ) ).
fof(fact_216_diff__eq__diff__eq,axiom,
! [X_a] :
( ab_group_add(X_a)
=> ! [A_1,B,Ca,D] :
( hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_1),B) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),Ca),D)
=> ( ti(X_a,A_1) = ti(X_a,B)
<=> ti(X_a,Ca) = ti(X_a,D) ) ) ) ).
fof(fact_217_fold1__below__iff,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X_2,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> ( ti(fun(X_a,bool),A_2) != bot_bot(fun(X_a,bool))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(fun(X_a,bool),X_a,hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,bool),X_a),finite_fold1(X_a),ord_min(X_a)),A_2)),X_2))
<=> ? [X_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),A_2))
& hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_1),X_2)) ) ) ) ) ) ).
fof(fact_218_min__max_Obelow__fold1__iff,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X_2,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> ( ti(fun(X_a,bool),A_2) != bot_bot(fun(X_a,bool))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_2),hAPP(fun(X_a,bool),X_a,hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,bool),X_a),finite_fold1(X_a),ord_min(X_a)),A_2)))
<=> ! [X_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),A_2))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_2),X_1)) ) ) ) ) ) ).
fof(fact_219_comp__fun__commute_Ofold__graph__insertE,axiom,
! [X_a,X_b,Z_1,X_2,A_2,V,F] :
( hBOOL(hAPP(fun(X_a,fun(X_b,X_b)),bool,finite100568337ommute(X_a,X_b),F))
=> ( hBOOL(hAPP(X_b,bool,hAPP(fun(X_a,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_a,bool),fun(X_b,bool)),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(fun(X_a,bool),fun(X_b,bool))),finite_fold_graph(X_a,X_b),F),Z_1),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_2),A_2)),V))
=> ( ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),A_2))
=> ~ ! [Y_1] :
( ti(X_b,V) = hAPP(X_b,X_b,hAPP(X_a,fun(X_b,X_b),F,X_2),Y_1)
=> ~ hBOOL(hAPP(X_b,bool,hAPP(fun(X_a,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_a,bool),fun(X_b,bool)),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(fun(X_a,bool),fun(X_b,bool))),finite_fold_graph(X_a,X_b),F),Z_1),A_2),Y_1)) ) ) ) ) ).
fof(fact_220_min__leastR,axiom,
! [X_a] :
( order(X_a)
=> ! [X,Least] :
( ! [X_1] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Least),X_1))
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X),Least) = ti(X_a,Least) ) ) ).
fof(fact_221_min__leastL,axiom,
! [X_a] :
( ord(X_a)
=> ! [X,Least] :
( ! [X_1] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Least),X_1))
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),Least),X) = ti(X_a,Least) ) ) ).
fof(fact_222_min__ord__min,axiom,
! [X_a] :
( ord(X_a)
=> ord_min(X_a) = hAPP(fun(X_a,fun(X_a,bool)),fun(X_a,fun(X_a,X_a)),min(X_a),ord_less_eq(X_a)) ) ).
fof(fact_223_Min_Oremove,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X_2,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),A_2))
=> ( ( hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_2),bot_bot(fun(X_a,bool)))) = bot_bot(fun(X_a,bool))
=> hAPP(fun(X_a,bool),X_a,big_linorder_Min(X_a),A_2) = ti(X_a,X_2) )
& ( hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_2),bot_bot(fun(X_a,bool)))) != bot_bot(fun(X_a,bool))
=> hAPP(fun(X_a,bool),X_a,big_linorder_Min(X_a),A_2) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X_2),hAPP(fun(X_a,bool),X_a,big_linorder_Min(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_2),bot_bot(fun(X_a,bool)))))) ) ) ) ) ) ).
fof(fact_224_Min__le,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X_2,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),A_2))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(fun(X_a,bool),X_a,big_linorder_Min(X_a),A_2)),X_2)) ) ) ) ).
fof(fact_225_Min__in,axiom,
! [X_a] :
( linorder(X_a)
=> ! [A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> ( ti(fun(X_a,bool),A_2) != bot_bot(fun(X_a,bool))
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),hAPP(fun(X_a,bool),X_a,big_linorder_Min(X_a),A_2)),A_2)) ) ) ) ).
fof(fact_226_Min_Osingleton,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X_2] : hAPP(fun(X_a,bool),X_a,big_linorder_Min(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_2),bot_bot(fun(X_a,bool)))) = ti(X_a,X_2) ) ).
fof(fact_227_Min_Oin__idem,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X_2,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),A_2))
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X_2),hAPP(fun(X_a,bool),X_a,big_linorder_Min(X_a),A_2)) = hAPP(fun(X_a,bool),X_a,big_linorder_Min(X_a),A_2) ) ) ) ).
fof(fact_228_Min_OF__eq,axiom,
! [X_a] :
( linorder(X_a)
=> ! [A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> hAPP(fun(X_a,bool),X_a,big_linorder_Min(X_a),A_2) = hAPP(fun(X_a,bool),X_a,hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,bool),X_a),finite_fold1(X_a),ord_min(X_a)),A_2) ) ) ).
fof(fact_229_Min__antimono,axiom,
! [X_a] :
( linorder(X_a)
=> ! [M_1,N] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),M_1),N))
=> ( ti(fun(X_a,bool),M_1) != bot_bot(fun(X_a,bool))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),N))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(fun(X_a,bool),X_a,big_linorder_Min(X_a),N)),hAPP(fun(X_a,bool),X_a,big_linorder_Min(X_a),M_1))) ) ) ) ) ).
fof(fact_230_Min_Oinsert,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X_2,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> ( ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),A_2))
=> ( ti(fun(X_a,bool),A_2) != bot_bot(fun(X_a,bool))
=> hAPP(fun(X_a,bool),X_a,big_linorder_Min(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_2),A_2)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X_2),hAPP(fun(X_a,bool),X_a,big_linorder_Min(X_a),A_2)) ) ) ) ) ).
fof(fact_231_Min__insert,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X_2,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> ( ti(fun(X_a,bool),A_2) != bot_bot(fun(X_a,bool))
=> hAPP(fun(X_a,bool),X_a,big_linorder_Min(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_2),A_2)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X_2),hAPP(fun(X_a,bool),X_a,big_linorder_Min(X_a),A_2)) ) ) ) ).
fof(fact_232_Min_Osubset__idem,axiom,
! [X_a] :
( linorder(X_a)
=> ! [B_1,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> ( ti(fun(X_a,bool),B_1) != bot_bot(fun(X_a,bool))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),B_1),A_2))
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),hAPP(fun(X_a,bool),X_a,big_linorder_Min(X_a),B_1)),hAPP(fun(X_a,bool),X_a,big_linorder_Min(X_a),A_2)) = hAPP(fun(X_a,bool),X_a,big_linorder_Min(X_a),A_2) ) ) ) ) ).
fof(fact_233_Min_Oinsert__remove,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X_2,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> ( ( hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_2),bot_bot(fun(X_a,bool)))) = bot_bot(fun(X_a,bool))
=> hAPP(fun(X_a,bool),X_a,big_linorder_Min(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_2),A_2)) = ti(X_a,X_2) )
& ( hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_2),bot_bot(fun(X_a,bool)))) != bot_bot(fun(X_a,bool))
=> hAPP(fun(X_a,bool),X_a,big_linorder_Min(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_2),A_2)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X_2),hAPP(fun(X_a,bool),X_a,big_linorder_Min(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_2),bot_bot(fun(X_a,bool)))))) ) ) ) ) ).
fof(fact_234_Min_Oclosed,axiom,
! [X_a] :
( linorder(X_a)
=> ! [A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> ( ti(fun(X_a,bool),A_2) != bot_bot(fun(X_a,bool))
=> ( ! [X_1,Y_1] : hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X_1),Y_1)),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_1),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),Y_1),bot_bot(fun(X_a,bool))))))
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),hAPP(fun(X_a,bool),X_a,big_linorder_Min(X_a),A_2)),A_2)) ) ) ) ) ).
fof(fact_235_Min__eqI,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X_2,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> ( ! [Y_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Y_1),A_2))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_2),Y_1)) )
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),A_2))
=> hAPP(fun(X_a,bool),X_a,big_linorder_Min(X_a),A_2) = ti(X_a,X_2) ) ) ) ) ).
fof(fact_236_min__max_OInf__fin_Oinsert__remove,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X_2,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> ( ( hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_2),bot_bot(fun(X_a,bool)))) = bot_bot(fun(X_a,bool))
=> hAPP(fun(X_a,bool),X_a,hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,bool),X_a),big_Inf_fin(X_a),ord_min(X_a)),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_2),A_2)) = ti(X_a,X_2) )
& ( hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_2),bot_bot(fun(X_a,bool)))) != bot_bot(fun(X_a,bool))
=> hAPP(fun(X_a,bool),X_a,hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,bool),X_a),big_Inf_fin(X_a),ord_min(X_a)),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_2),A_2)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X_2),hAPP(fun(X_a,bool),X_a,hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,bool),X_a),big_Inf_fin(X_a),ord_min(X_a)),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_2),bot_bot(fun(X_a,bool)))))) ) ) ) ) ).
fof(fact_237_min__max_OInf__fin_Oremove,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X_2,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),A_2))
=> ( ( hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_2),bot_bot(fun(X_a,bool)))) = bot_bot(fun(X_a,bool))
=> hAPP(fun(X_a,bool),X_a,hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,bool),X_a),big_Inf_fin(X_a),ord_min(X_a)),A_2) = ti(X_a,X_2) )
& ( hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_2),bot_bot(fun(X_a,bool)))) != bot_bot(fun(X_a,bool))
=> hAPP(fun(X_a,bool),X_a,hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,bool),X_a),big_Inf_fin(X_a),ord_min(X_a)),A_2) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X_2),hAPP(fun(X_a,bool),X_a,hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,bool),X_a),big_Inf_fin(X_a),ord_min(X_a)),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_2),bot_bot(fun(X_a,bool)))))) ) ) ) ) ) ).
fof(fact_238_min__max_OInf__fin_Oin__idem,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X_2,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),A_2))
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X_2),hAPP(fun(X_a,bool),X_a,hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,bool),X_a),big_Inf_fin(X_a),ord_min(X_a)),A_2)) = hAPP(fun(X_a,bool),X_a,hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,bool),X_a),big_Inf_fin(X_a),ord_min(X_a)),A_2) ) ) ) ).
fof(fact_239_min__max_OInf__fin_Osingleton,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X_2] : hAPP(fun(X_a,bool),X_a,hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,bool),X_a),big_Inf_fin(X_a),ord_min(X_a)),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_2),bot_bot(fun(X_a,bool)))) = ti(X_a,X_2) ) ).
fof(fact_240_min__max_OInf__fin_OF__eq,axiom,
! [X_a] :
( linorder(X_a)
=> ! [A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> hAPP(fun(X_a,bool),X_a,hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,bool),X_a),big_Inf_fin(X_a),ord_min(X_a)),A_2) = hAPP(fun(X_a,bool),X_a,hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,bool),X_a),finite_fold1(X_a),ord_min(X_a)),A_2) ) ) ).
fof(fact_241_min__max_OInf__fin_Oinsert,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X_2,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> ( ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),A_2))
=> ( ti(fun(X_a,bool),A_2) != bot_bot(fun(X_a,bool))
=> hAPP(fun(X_a,bool),X_a,hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,bool),X_a),big_Inf_fin(X_a),ord_min(X_a)),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_2),A_2)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X_2),hAPP(fun(X_a,bool),X_a,hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,bool),X_a),big_Inf_fin(X_a),ord_min(X_a)),A_2)) ) ) ) ) ).
fof(fact_242_min__max_OInf__fin_Oinsert__idem,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X_2,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> ( ti(fun(X_a,bool),A_2) != bot_bot(fun(X_a,bool))
=> hAPP(fun(X_a,bool),X_a,hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,bool),X_a),big_Inf_fin(X_a),ord_min(X_a)),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_2),A_2)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X_2),hAPP(fun(X_a,bool),X_a,hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,bool),X_a),big_Inf_fin(X_a),ord_min(X_a)),A_2)) ) ) ) ).
fof(fact_243_min__max_OInf__fin_Osubset__idem,axiom,
! [X_a] :
( linorder(X_a)
=> ! [B_1,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> ( ti(fun(X_a,bool),B_1) != bot_bot(fun(X_a,bool))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),B_1),A_2))
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),hAPP(fun(X_a,bool),X_a,hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,bool),X_a),big_Inf_fin(X_a),ord_min(X_a)),B_1)),hAPP(fun(X_a,bool),X_a,hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,bool),X_a),big_Inf_fin(X_a),ord_min(X_a)),A_2)) = hAPP(fun(X_a,bool),X_a,hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,bool),X_a),big_Inf_fin(X_a),ord_min(X_a)),A_2) ) ) ) ) ).
fof(fact_244_min__max_OInf__fin_Oclosed,axiom,
! [X_a] :
( linorder(X_a)
=> ! [A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> ( ti(fun(X_a,bool),A_2) != bot_bot(fun(X_a,bool))
=> ( ! [X_1,Y_1] : hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X_1),Y_1)),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_1),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),Y_1),bot_bot(fun(X_a,bool))))))
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),hAPP(fun(X_a,bool),X_a,hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,bool),X_a),big_Inf_fin(X_a),ord_min(X_a)),A_2)),A_2)) ) ) ) ) ).
fof(fact_245_comp__fun__commute_Ofold__insert__remove,axiom,
! [X_a,X_b,Z_1,X_2,A_2,F] :
( hBOOL(hAPP(fun(X_a,fun(X_b,X_b)),bool,finite100568337ommute(X_a,X_b),F))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> hAPP(fun(X_a,bool),X_b,hAPP(X_b,fun(fun(X_a,bool),X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(fun(X_a,bool),X_b)),finite_fold(X_a,X_b),F),Z_1),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_2),A_2)) = hAPP(X_b,X_b,hAPP(X_a,fun(X_b,X_b),F,X_2),hAPP(fun(X_a,bool),X_b,hAPP(X_b,fun(fun(X_a,bool),X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(fun(X_a,bool),X_b)),finite_fold(X_a,X_b),F),Z_1),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_2),bot_bot(fun(X_a,bool)))))) ) ) ).
fof(fact_246_comp__fun__commute_Ofold__rec,axiom,
! [X_a,X_b,Z_1,X_2,A_2,F] :
( hBOOL(hAPP(fun(X_a,fun(X_b,X_b)),bool,finite100568337ommute(X_a,X_b),F))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),A_2))
=> hAPP(fun(X_a,bool),X_b,hAPP(X_b,fun(fun(X_a,bool),X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(fun(X_a,bool),X_b)),finite_fold(X_a,X_b),F),Z_1),A_2) = hAPP(X_b,X_b,hAPP(X_a,fun(X_b,X_b),F,X_2),hAPP(fun(X_a,bool),X_b,hAPP(X_b,fun(fun(X_a,bool),X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(fun(X_a,bool),X_b)),finite_fold(X_a,X_b),F),Z_1),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_2),bot_bot(fun(X_a,bool)))))) ) ) ) ).
fof(fact_247_Max__mono,axiom,
! [X_a] :
( linorder(X_a)
=> ! [M_1,N] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),M_1),N))
=> ( ti(fun(X_a,bool),M_1) != bot_bot(fun(X_a,bool))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),N))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(fun(X_a,bool),X_a,big_linorder_Max(X_a),M_1)),hAPP(fun(X_a,bool),X_a,big_linorder_Max(X_a),N))) ) ) ) ) ).
fof(fact_248_fold__empty,axiom,
! [X_b,X_a,F,Z_1] : hAPP(fun(X_b,bool),X_a,hAPP(X_a,fun(fun(X_b,bool),X_a),hAPP(fun(X_b,fun(X_a,X_a)),fun(X_a,fun(fun(X_b,bool),X_a)),finite_fold(X_b,X_a),F),Z_1),bot_bot(fun(X_b,bool))) = ti(X_a,Z_1) ).
fof(fact_249_comp__fun__commute_Ofold__fun__comm,axiom,
! [X_a,X_b,X_2,Z_1,A_2,F] :
( hBOOL(hAPP(fun(X_a,fun(X_b,X_b)),bool,finite100568337ommute(X_a,X_b),F))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> hAPP(X_b,X_b,hAPP(X_a,fun(X_b,X_b),F,X_2),hAPP(fun(X_a,bool),X_b,hAPP(X_b,fun(fun(X_a,bool),X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(fun(X_a,bool),X_b)),finite_fold(X_a,X_b),F),Z_1),A_2)) = hAPP(fun(X_a,bool),X_b,hAPP(X_b,fun(fun(X_a,bool),X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(fun(X_a,bool),X_b)),finite_fold(X_a,X_b),F),hAPP(X_b,X_b,hAPP(X_a,fun(X_b,X_b),F,X_2),Z_1)),A_2) ) ) ).
fof(fact_250_Max__ge,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X_2,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),A_2))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_2),hAPP(fun(X_a,bool),X_a,big_linorder_Max(X_a),A_2))) ) ) ) ).
fof(fact_251_Max__in,axiom,
! [X_a] :
( linorder(X_a)
=> ! [A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> ( ti(fun(X_a,bool),A_2) != bot_bot(fun(X_a,bool))
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),hAPP(fun(X_a,bool),X_a,big_linorder_Max(X_a),A_2)),A_2)) ) ) ) ).
fof(fact_252_Max_Osingleton,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X_2] : hAPP(fun(X_a,bool),X_a,big_linorder_Max(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_2),bot_bot(fun(X_a,bool)))) = ti(X_a,X_2) ) ).
fof(fact_253_comp__fun__commute_Ofold__equality,axiom,
! [X_a,X_b,Z_1,A_2,Y_2,F] :
( hBOOL(hAPP(fun(X_a,fun(X_b,X_b)),bool,finite100568337ommute(X_a,X_b),F))
=> ( hBOOL(hAPP(X_b,bool,hAPP(fun(X_a,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_a,bool),fun(X_b,bool)),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(fun(X_a,bool),fun(X_b,bool))),finite_fold_graph(X_a,X_b),F),Z_1),A_2),Y_2))
=> hAPP(fun(X_a,bool),X_b,hAPP(X_b,fun(fun(X_a,bool),X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(fun(X_a,bool),X_b)),finite_fold(X_a,X_b),F),Z_1),A_2) = ti(X_b,Y_2) ) ) ).
fof(fact_254_min__max_Ofold__inf__le__inf,axiom,
! [X_a] :
( linorder(X_a)
=> ! [B,A_1,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_1),A_2))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(fun(X_a,bool),X_a,hAPP(X_a,fun(fun(X_a,bool),X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(fun(X_a,bool),X_a)),finite_fold(X_a,X_a),ord_min(X_a)),B),A_2)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),A_1),B))) ) ) ) ).
fof(fact_255_min__max_Ofold__inf__insert,axiom,
! [X_a] :
( linorder(X_a)
=> ! [B,A_1,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> hAPP(fun(X_a,bool),X_a,hAPP(X_a,fun(fun(X_a,bool),X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(fun(X_a,bool),X_a)),finite_fold(X_a,X_a),ord_min(X_a)),B),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),A_1),A_2)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),A_1),hAPP(fun(X_a,bool),X_a,hAPP(X_a,fun(fun(X_a,bool),X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(fun(X_a,bool),X_a)),finite_fold(X_a,X_a),ord_min(X_a)),B),A_2)) ) ) ).
fof(fact_256_comp__fun__commute_Ofold__insert2,axiom,
! [X_a,X_b,Z_1,X_2,A_2,F] :
( hBOOL(hAPP(fun(X_a,fun(X_b,X_b)),bool,finite100568337ommute(X_a,X_b),F))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> ( ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),A_2))
=> hAPP(fun(X_a,bool),X_b,hAPP(X_b,fun(fun(X_a,bool),X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(fun(X_a,bool),X_b)),finite_fold(X_a,X_b),F),Z_1),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_2),A_2)) = hAPP(fun(X_a,bool),X_b,hAPP(X_b,fun(fun(X_a,bool),X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(fun(X_a,bool),X_b)),finite_fold(X_a,X_b),F),hAPP(X_b,X_b,hAPP(X_a,fun(X_b,X_b),F,X_2),Z_1)),A_2) ) ) ) ).
fof(fact_257_comp__fun__commute_Ofold__insert,axiom,
! [X_a,X_b,Z_1,X_2,A_2,F] :
( hBOOL(hAPP(fun(X_a,fun(X_b,X_b)),bool,finite100568337ommute(X_a,X_b),F))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> ( ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),A_2))
=> hAPP(fun(X_a,bool),X_b,hAPP(X_b,fun(fun(X_a,bool),X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(fun(X_a,bool),X_b)),finite_fold(X_a,X_b),F),Z_1),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_2),A_2)) = hAPP(X_b,X_b,hAPP(X_a,fun(X_b,X_b),F,X_2),hAPP(fun(X_a,bool),X_b,hAPP(X_b,fun(fun(X_a,bool),X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(fun(X_a,bool),X_b)),finite_fold(X_a,X_b),F),Z_1),A_2)) ) ) ) ).
fof(fact_258_folding__one_Oeq__fold_H,axiom,
! [X_a,X_2,A_2,F,F_1] :
( hBOOL(hAPP(fun(fun(X_a,bool),X_a),bool,hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(fun(X_a,bool),X_a),bool),finite_folding_one(X_a),F),F_1))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> ( ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),A_2))
=> hAPP(fun(X_a,bool),X_a,F_1,hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_2),A_2)) = hAPP(fun(X_a,bool),X_a,hAPP(X_a,fun(fun(X_a,bool),X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(fun(X_a,bool),X_a)),finite_fold(X_a,X_a),F),X_2),A_2) ) ) ) ).
fof(fact_259_folding__one__idem_Oeq__fold__idem_H,axiom,
! [X_a,A_1,A_2,F,F_1] :
( hBOOL(hAPP(fun(fun(X_a,bool),X_a),bool,hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(fun(X_a,bool),X_a),bool),finite2073411215e_idem(X_a),F),F_1))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> hAPP(fun(X_a,bool),X_a,F_1,hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),A_1),A_2)) = hAPP(fun(X_a,bool),X_a,hAPP(X_a,fun(fun(X_a,bool),X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(fun(X_a,bool),X_a)),finite_fold(X_a,X_a),F),A_1),A_2) ) ) ).
fof(fact_260_comp__fun__commute_Ofold__graph__fold,axiom,
! [X_a,X_b,Z_1,A_2,F] :
( hBOOL(hAPP(fun(X_a,fun(X_b,X_b)),bool,finite100568337ommute(X_a,X_b),F))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> hBOOL(hAPP(X_b,bool,hAPP(fun(X_a,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_a,bool),fun(X_b,bool)),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(fun(X_a,bool),fun(X_b,bool))),finite_fold_graph(X_a,X_b),F),Z_1),A_2),hAPP(fun(X_a,bool),X_b,hAPP(X_b,fun(fun(X_a,bool),X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(fun(X_a,bool),X_b)),finite_fold(X_a,X_b),F),Z_1),A_2))) ) ) ).
fof(fact_261_Min_Oeq__fold__idem_H,axiom,
! [X_a] :
( linorder(X_a)
=> ! [A_1,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> hAPP(fun(X_a,bool),X_a,big_linorder_Min(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),A_1),A_2)) = hAPP(fun(X_a,bool),X_a,hAPP(X_a,fun(fun(X_a,bool),X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(fun(X_a,bool),X_a)),finite_fold(X_a,X_a),ord_min(X_a)),A_1),A_2) ) ) ).
fof(fact_262_Min_Oeq__fold_H,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X_2,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> ( ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),A_2))
=> hAPP(fun(X_a,bool),X_a,big_linorder_Min(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_2),A_2)) = hAPP(fun(X_a,bool),X_a,hAPP(X_a,fun(fun(X_a,bool),X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(fun(X_a,bool),X_a)),finite_fold(X_a,X_a),ord_min(X_a)),X_2),A_2) ) ) ) ).
fof(fact_263_min__max_OInf__fin_Oeq__fold__idem_H,axiom,
! [X_a] :
( linorder(X_a)
=> ! [A_1,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> hAPP(fun(X_a,bool),X_a,hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,bool),X_a),big_Inf_fin(X_a),ord_min(X_a)),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),A_1),A_2)) = hAPP(fun(X_a,bool),X_a,hAPP(X_a,fun(fun(X_a,bool),X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(fun(X_a,bool),X_a)),finite_fold(X_a,X_a),ord_min(X_a)),A_1),A_2) ) ) ).
fof(fact_264_min__max_OInf__fin_Oeq__fold_H,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X_2,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> ( ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),A_2))
=> hAPP(fun(X_a,bool),X_a,hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,bool),X_a),big_Inf_fin(X_a),ord_min(X_a)),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_2),A_2)) = hAPP(fun(X_a,bool),X_a,hAPP(X_a,fun(fun(X_a,bool),X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(fun(X_a,bool),X_a)),finite_fold(X_a,X_a),ord_min(X_a)),X_2),A_2) ) ) ) ).
fof(fact_265_min__max_Oinf__le__fold__inf,axiom,
! [X_a] :
( linorder(X_a)
=> ! [Ca,B,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> ( ! [X_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),A_2))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),X_1)) )
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),B),Ca)),hAPP(fun(X_a,bool),X_a,hAPP(X_a,fun(fun(X_a,bool),X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(fun(X_a,bool),X_a)),finite_fold(X_a,X_a),ord_min(X_a)),Ca),A_2))) ) ) ) ).
fof(fact_266_Max__eqI,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X_2,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> ( ! [Y_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Y_1),A_2))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y_1),X_2)) )
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),A_2))
=> hAPP(fun(X_a,bool),X_a,big_linorder_Max(X_a),A_2) = ti(X_a,X_2) ) ) ) ) ).
fof(fact_267_comp__fun__idem_Ofold__insert__idem,axiom,
! [X_a,X_b,Z_1,X_2,A_2,F] :
( hBOOL(hAPP(fun(X_a,fun(X_b,X_b)),bool,finite_comp_fun_idem(X_a,X_b),F))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> hAPP(fun(X_a,bool),X_b,hAPP(X_b,fun(fun(X_a,bool),X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(fun(X_a,bool),X_b)),finite_fold(X_a,X_b),F),Z_1),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_2),A_2)) = hAPP(X_b,X_b,hAPP(X_a,fun(X_b,X_b),F,X_2),hAPP(fun(X_a,bool),X_b,hAPP(X_b,fun(fun(X_a,bool),X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(fun(X_a,bool),X_b)),finite_fold(X_a,X_b),F),Z_1),A_2)) ) ) ).
fof(fact_268_comp__fun__idem_Ofold__insert__idem2,axiom,
! [X_a,X_b,Z_1,X_2,A_2,F] :
( hBOOL(hAPP(fun(X_a,fun(X_b,X_b)),bool,finite_comp_fun_idem(X_a,X_b),F))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> hAPP(fun(X_a,bool),X_b,hAPP(X_b,fun(fun(X_a,bool),X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(fun(X_a,bool),X_b)),finite_fold(X_a,X_b),F),Z_1),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_2),A_2)) = hAPP(fun(X_a,bool),X_b,hAPP(X_b,fun(fun(X_a,bool),X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(fun(X_a,bool),X_b)),finite_fold(X_a,X_b),F),hAPP(X_b,X_b,hAPP(X_a,fun(X_b,X_b),F,X_2),Z_1)),A_2) ) ) ).
fof(fact_269_comp__fun__idem_Ofun__left__idem,axiom,
! [X_a,X_b,X_2,Z_1,F] :
( hBOOL(hAPP(fun(X_a,fun(X_b,X_b)),bool,finite_comp_fun_idem(X_a,X_b),F))
=> hAPP(X_b,X_b,hAPP(X_a,fun(X_b,X_b),F,X_2),hAPP(X_b,X_b,hAPP(X_a,fun(X_b,X_b),F,X_2),Z_1)) = hAPP(X_b,X_b,hAPP(X_a,fun(X_b,X_b),F,X_2),Z_1) ) ).
fof(fact_270_comp__fun__idem__insert,axiom,
! [X_a] : hBOOL(hAPP(fun(X_a,fun(fun(X_a,bool),fun(X_a,bool))),bool,finite_comp_fun_idem(X_a,fun(X_a,bool)),insert(X_a))) ).
fof(fact_271_min__max_Ocomp__fun__idem__inf,axiom,
! [X_a] :
( linorder(X_a)
=> hBOOL(hAPP(fun(X_a,fun(X_a,X_a)),bool,finite_comp_fun_idem(X_a,X_a),ord_min(X_a))) ) ).
fof(fact_272_Max_Oremove,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X_2,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),A_2))
=> ( ( hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_2),bot_bot(fun(X_a,bool)))) = bot_bot(fun(X_a,bool))
=> hAPP(fun(X_a,bool),X_a,big_linorder_Max(X_a),A_2) = ti(X_a,X_2) )
& ( hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_2),bot_bot(fun(X_a,bool)))) != bot_bot(fun(X_a,bool))
=> hAPP(fun(X_a,bool),X_a,big_linorder_Max(X_a),A_2) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),X_2),hAPP(fun(X_a,bool),X_a,big_linorder_Max(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_2),bot_bot(fun(X_a,bool)))))) ) ) ) ) ) ).
fof(fact_273_Max_Oinsert__remove,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X_2,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> ( ( hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_2),bot_bot(fun(X_a,bool)))) = bot_bot(fun(X_a,bool))
=> hAPP(fun(X_a,bool),X_a,big_linorder_Max(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_2),A_2)) = ti(X_a,X_2) )
& ( hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_2),bot_bot(fun(X_a,bool)))) != bot_bot(fun(X_a,bool))
=> hAPP(fun(X_a,bool),X_a,big_linorder_Max(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_2),A_2)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),X_2),hAPP(fun(X_a,bool),X_a,big_linorder_Max(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_2),bot_bot(fun(X_a,bool)))))) ) ) ) ) ).
fof(fact_274_fold__graph_H_Ointros_I2_J,axiom,
! [X_b,X_a,F,Z_1,Y_2,X_2,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),A_2))
=> ( hBOOL(hAPP(X_b,bool,hAPP(fun(X_a,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_a,bool),fun(X_b,bool)),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(fun(X_a,bool),fun(X_b,bool))),fold_graph(X_a,X_b),F),Z_1),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_2),bot_bot(fun(X_a,bool))))),Y_2))
=> hBOOL(hAPP(X_b,bool,hAPP(fun(X_a,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_a,bool),fun(X_b,bool)),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(fun(X_a,bool),fun(X_b,bool))),fold_graph(X_a,X_b),F),Z_1),A_2),hAPP(X_b,X_b,hAPP(X_a,fun(X_b,X_b),F,X_2),Y_2))) ) ) ).
fof(fact_275_setsum__diff1__ring,axiom,
! [X_a,X_b] :
( ring(X_b)
=> ! [F,A_1,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_1),A_2))
=> hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),F),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),A_1),bot_bot(fun(X_a,bool))))) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),minus_minus(X_b),hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),F),A_2)),hAPP(X_a,X_b,F,A_1)) ) ) ) ).
fof(fact_276_min__max_Ocomp__fun__idem__sup,axiom,
! [X_a] :
( linorder(X_a)
=> hBOOL(hAPP(fun(X_a,fun(X_a,X_a)),bool,finite_comp_fun_idem(X_a,X_a),ord_max(X_a))) ) ).
fof(fact_277_fold__graph_H_Ointros_I1_J,axiom,
! [X_b,X_a,F,Z_1] : hBOOL(hAPP(X_b,bool,hAPP(fun(X_a,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_a,bool),fun(X_b,bool)),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(fun(X_a,bool),fun(X_b,bool))),fold_graph(X_a,X_b),F),Z_1),bot_bot(fun(X_a,bool))),Z_1)) ).
fof(fact_278_max__diff__distrib__left,axiom,
! [X_a] :
( ordered_ab_group_add(X_a)
=> ! [X,Y,Z] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),X),Y)),Z) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),X),Z)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),Y),Z)) ) ).
fof(fact_279_min__max_Osup_Oidem,axiom,
! [X_a] :
( linorder(X_a)
=> ! [A_3] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),A_3),A_3) = ti(X_a,A_3) ) ).
fof(fact_280_min__max_Osup_Ocommute,axiom,
! [X_a] :
( linorder(X_a)
=> ! [A_3,B_3] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),A_3),B_3) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),B_3),A_3) ) ).
fof(fact_281_min__max_Osup__commute,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X,Y] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),X),Y) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),Y),X) ) ).
fof(fact_282_min__max_Osup__inf__absorb,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X,Y] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X),Y)) = ti(X_a,X) ) ).
fof(fact_283_min__max_Oinf__sup__absorb,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X,Y] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),X),Y)) = ti(X_a,X) ) ).
fof(fact_284_min__max_Osup_Oleft__idem,axiom,
! [X_a] :
( linorder(X_a)
=> ! [A_3,B_3] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),A_3),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),A_3),B_3)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),A_3),B_3) ) ).
fof(fact_285_min__max_Osup__left__idem,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X,Y] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),X),Y)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),X),Y) ) ).
fof(fact_286_min__max_Osup_Oleft__commute,axiom,
! [X_a] :
( linorder(X_a)
=> ! [B_3,A_3,C_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),B_3),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),A_3),C_1)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),A_3),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),B_3),C_1)) ) ).
fof(fact_287_min__max_Osup__left__commute,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X,Y,Z] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),Y),Z)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),Y),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),X),Z)) ) ).
fof(fact_288_min__max_Osup__inf__distrib1,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X,Y,Z] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),Y),Z)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),X),Y)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),X),Z)) ) ).
fof(fact_289_min__max_Oinf__sup__distrib1,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X,Y,Z] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),Y),Z)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X),Y)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X),Z)) ) ).
fof(fact_290_min__max_Osup_Oassoc,axiom,
! [X_a] :
( linorder(X_a)
=> ! [A_3,B_3,C_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),A_3),B_3)),C_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),A_3),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),B_3),C_1)) ) ).
fof(fact_291_min__max_Osup__assoc,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X,Y,Z] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),X),Y)),Z) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),Y),Z)) ) ).
fof(fact_292_min__max_Osup__inf__distrib2,axiom,
! [X_a] :
( linorder(X_a)
=> ! [Y,Z,X] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),Y),Z)),X) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),Y),X)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),Z),X)) ) ).
fof(fact_293_min__max_Oinf__sup__distrib2,axiom,
! [X_a] :
( linorder(X_a)
=> ! [Y,Z,X] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),Y),Z)),X) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),Y),X)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),Z),X)) ) ).
fof(fact_294_le__maxI1,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X,Y] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),X),Y))) ) ).
fof(fact_295_le__maxI2,axiom,
! [X_a] :
( linorder(X_a)
=> ! [Y,X] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),X),Y))) ) ).
fof(fact_296_min__max_Ole__iff__sup,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X_2,Y_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_2),Y_2))
<=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),X_2),Y_2) = ti(X_a,Y_2) ) ) ).
fof(fact_297_min__max_Ole__sup__iff,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X_2,Y_2,Z_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),X_2),Y_2)),Z_1))
<=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_2),Z_1))
& hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y_2),Z_1)) ) ) ) ).
fof(fact_298_min__max_Ole__supI1,axiom,
! [X_a] :
( linorder(X_a)
=> ! [B_3,X,A_3] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),A_3))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),A_3),B_3))) ) ) ).
fof(fact_299_min__max_Ole__supI2,axiom,
! [X_a] :
( linorder(X_a)
=> ! [A_3,X,B_3] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),B_3))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),A_3),B_3))) ) ) ).
fof(fact_300_min__max_Osup__absorb2,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X,Y] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),Y))
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),X),Y) = ti(X_a,Y) ) ) ).
fof(fact_301_min__max_Osup__absorb1,axiom,
! [X_a] :
( linorder(X_a)
=> ! [Y,X] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),X))
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),X),Y) = ti(X_a,X) ) ) ).
fof(fact_302_min__max_Ole__supI,axiom,
! [X_a] :
( linorder(X_a)
=> ! [B_3,A_3,X] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_3),X))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B_3),X))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),A_3),B_3)),X)) ) ) ) ).
fof(fact_303_min__max_Osup__least,axiom,
! [X_a] :
( linorder(X_a)
=> ! [Z,Y,X] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),X))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Z),X))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),Y),Z)),X)) ) ) ) ).
fof(fact_304_min__max_Osup__mono,axiom,
! [X_a] :
( linorder(X_a)
=> ! [B_3,D_1,A_3,C_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_3),C_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B_3),D_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),A_3),B_3)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),C_1),D_1))) ) ) ) ).
fof(fact_305_min__max_Ole__supE,axiom,
! [X_a] :
( linorder(X_a)
=> ! [A_3,B_3,X] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),A_3),B_3)),X))
=> ~ ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_3),X))
=> ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B_3),X)) ) ) ) ).
fof(fact_306_le__max__iff__disj,axiom,
! [X_a] :
( linorder(X_a)
=> ! [Z_1,X_2,Y_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Z_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),X_2),Y_2)))
<=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Z_1),X_2))
| hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Z_1),Y_2)) ) ) ) ).
fof(fact_307_Max_Oidem,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),X),X) = ti(X_a,X) ) ).
fof(fact_308_min__max_Odistrib__inf__le,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X,Y,Z] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X),Y)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X),Z))),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),Y),Z)))) ) ).
fof(fact_309_min__max_Odistrib__sup__le,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X,Y,Z] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),Y),Z))),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),X),Y)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),X),Z)))) ) ).
fof(fact_310_Max_Oin__idem,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X_2,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),A_2))
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),X_2),hAPP(fun(X_a,bool),X_a,big_linorder_Max(X_a),A_2)) = hAPP(fun(X_a,bool),X_a,big_linorder_Max(X_a),A_2) ) ) ) ).
fof(fact_311_setsum__diff1__nat,axiom,
! [X_a,F,A_1,A_2] :
( ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_1),A_2))
=> hAPP(fun(X_a,bool),nat,hAPP(fun(X_a,nat),fun(fun(X_a,bool),nat),big_co1399186613setsum(X_a,nat),F),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),A_1),bot_bot(fun(X_a,bool))))) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(fun(X_a,bool),nat,hAPP(fun(X_a,nat),fun(fun(X_a,bool),nat),big_co1399186613setsum(X_a,nat),F),A_2)),hAPP(X_a,nat,F,A_1)) )
& ( ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_1),A_2))
=> hAPP(fun(X_a,bool),nat,hAPP(fun(X_a,nat),fun(fun(X_a,bool),nat),big_co1399186613setsum(X_a,nat),F),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),A_1),bot_bot(fun(X_a,bool))))) = hAPP(fun(X_a,bool),nat,hAPP(fun(X_a,nat),fun(fun(X_a,bool),nat),big_co1399186613setsum(X_a,nat),F),A_2) ) ) ).
fof(fact_312_setsum__diff__nat,axiom,
! [X_a,F,A_2,B_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),B_1))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),B_1),A_2))
=> hAPP(fun(X_a,bool),nat,hAPP(fun(X_a,nat),fun(fun(X_a,bool),nat),big_co1399186613setsum(X_a,nat),F),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),B_1)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(fun(X_a,bool),nat,hAPP(fun(X_a,nat),fun(fun(X_a,bool),nat),big_co1399186613setsum(X_a,nat),F),A_2)),hAPP(fun(X_a,bool),nat,hAPP(fun(X_a,nat),fun(fun(X_a,bool),nat),big_co1399186613setsum(X_a,nat),F),B_1)) ) ) ).
fof(fact_313_min__max_Osup__le__fold__sup,axiom,
! [X_a] :
( linorder(X_a)
=> ! [B,A_1,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_1),A_2))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),A_1),B)),hAPP(fun(X_a,bool),X_a,hAPP(X_a,fun(fun(X_a,bool),X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(fun(X_a,bool),X_a)),finite_fold(X_a,X_a),ord_max(X_a)),B),A_2))) ) ) ) ).
fof(fact_314_min__max_Ofold__sup__insert,axiom,
! [X_a] :
( linorder(X_a)
=> ! [B,A_1,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> hAPP(fun(X_a,bool),X_a,hAPP(X_a,fun(fun(X_a,bool),X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(fun(X_a,bool),X_a)),finite_fold(X_a,X_a),ord_max(X_a)),B),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),A_1),A_2)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),A_1),hAPP(fun(X_a,bool),X_a,hAPP(X_a,fun(fun(X_a,bool),X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(fun(X_a,bool),X_a)),finite_fold(X_a,X_a),ord_max(X_a)),B),A_2)) ) ) ).
fof(fact_315_min__max_Osup__Inf__absorb,axiom,
! [X_a] :
( linorder(X_a)
=> ! [A_1,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_1),A_2))
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),A_1),hAPP(fun(X_a,bool),X_a,hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,bool),X_a),big_Inf_fin(X_a),ord_min(X_a)),A_2)) = ti(X_a,A_1) ) ) ) ).
fof(fact_316_Max_OF__eq,axiom,
! [X_a] :
( linorder(X_a)
=> ! [A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> hAPP(fun(X_a,bool),X_a,big_linorder_Max(X_a),A_2) = hAPP(fun(X_a,bool),X_a,hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,bool),X_a),finite_fold1(X_a),ord_max(X_a)),A_2) ) ) ).
fof(fact_317_setsum__diff,axiom,
! [X_a,X_b] :
( ab_group_add(X_b)
=> ! [F,B_1,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),B_1),A_2))
=> hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),F),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),B_1)) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),minus_minus(X_b),hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),F),A_2)),hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),F),B_1)) ) ) ) ).
fof(fact_318_Max_Oinsert,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X_2,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> ( ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),A_2))
=> ( ti(fun(X_a,bool),A_2) != bot_bot(fun(X_a,bool))
=> hAPP(fun(X_a,bool),X_a,big_linorder_Max(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_2),A_2)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),X_2),hAPP(fun(X_a,bool),X_a,big_linorder_Max(X_a),A_2)) ) ) ) ) ).
fof(fact_319_Max__insert,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X_2,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> ( ti(fun(X_a,bool),A_2) != bot_bot(fun(X_a,bool))
=> hAPP(fun(X_a,bool),X_a,big_linorder_Max(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_2),A_2)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),X_2),hAPP(fun(X_a,bool),X_a,big_linorder_Max(X_a),A_2)) ) ) ) ).
fof(fact_320_Max_Osubset__idem,axiom,
! [X_a] :
( linorder(X_a)
=> ! [B_1,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> ( ti(fun(X_a,bool),B_1) != bot_bot(fun(X_a,bool))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),B_1),A_2))
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),hAPP(fun(X_a,bool),X_a,big_linorder_Max(X_a),B_1)),hAPP(fun(X_a,bool),X_a,big_linorder_Max(X_a),A_2)) = hAPP(fun(X_a,bool),X_a,big_linorder_Max(X_a),A_2) ) ) ) ) ).
fof(fact_321_Max_Oeq__fold__idem_H,axiom,
! [X_a] :
( linorder(X_a)
=> ! [A_1,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> hAPP(fun(X_a,bool),X_a,big_linorder_Max(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),A_1),A_2)) = hAPP(fun(X_a,bool),X_a,hAPP(X_a,fun(fun(X_a,bool),X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(fun(X_a,bool),X_a)),finite_fold(X_a,X_a),ord_max(X_a)),A_1),A_2) ) ) ).
fof(fact_322_Max_Oeq__fold_H,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X_2,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> ( ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),A_2))
=> hAPP(fun(X_a,bool),X_a,big_linorder_Max(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_2),A_2)) = hAPP(fun(X_a,bool),X_a,hAPP(X_a,fun(fun(X_a,bool),X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(fun(X_a,bool),X_a)),finite_fold(X_a,X_a),ord_max(X_a)),X_2),A_2) ) ) ) ).
fof(fact_323_setsum__diff1,axiom,
! [X_a,X_b] :
( ab_group_add(X_b)
=> ! [F,A_1,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> ( ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_1),A_2))
=> hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),F),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),A_1),bot_bot(fun(X_a,bool))))) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),minus_minus(X_b),hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),F),A_2)),hAPP(X_a,X_b,F,A_1)) )
& ( ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_1),A_2))
=> hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),F),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),A_1),bot_bot(fun(X_a,bool))))) = hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),F),A_2) ) ) ) ) ).
fof(fact_324_Max_Oclosed,axiom,
! [X_a] :
( linorder(X_a)
=> ! [A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> ( ti(fun(X_a,bool),A_2) != bot_bot(fun(X_a,bool))
=> ( ! [X_1,Y_1] : hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),X_1),Y_1)),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_1),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),Y_1),bot_bot(fun(X_a,bool))))))
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),hAPP(fun(X_a,bool),X_a,big_linorder_Max(X_a),A_2)),A_2)) ) ) ) ) ).
fof(fact_325_min__max_Ofold__sup__le__sup,axiom,
! [X_a] :
( linorder(X_a)
=> ! [Ca,B,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> ( ! [X_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),A_2))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_1),B)) )
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(fun(X_a,bool),X_a,hAPP(X_a,fun(fun(X_a,bool),X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(fun(X_a,bool),X_a)),finite_fold(X_a,X_a),ord_max(X_a)),Ca),A_2)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),B),Ca))) ) ) ) ).
fof(fact_326_min__max_Odistrib__imp2,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X,Y,Z] :
( ! [X_1,Y_1,Z_2] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),X_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),Y_1),Z_2)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),X_1),Y_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),X_1),Z_2))
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),Y),Z)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X),Y)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X),Z)) ) ) ).
fof(fact_327_min__max_Odistrib__imp1,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X,Y,Z] :
( ! [X_1,Y_1,Z_2] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),Y_1),Z_2)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X_1),Y_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X_1),Z_2))
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),Y),Z)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),X),Y)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),X),Z)) ) ) ).
fof(fact_328_setsum_Ocong,axiom,
! [X_a,X_b] :
( comm_monoid_add(X_b)
=> ! [G,H,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> ( ! [X_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),A_2))
=> hAPP(X_a,X_b,G,X_1) = hAPP(X_a,X_b,H,X_1) )
=> hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),G),A_2) = hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),H),A_2) ) ) ) ).
fof(fact_329_max__leastR,axiom,
! [X_a] :
( order(X_a)
=> ! [X,Least] :
( ! [X_1] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Least),X_1))
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),X),Least) = ti(X_a,X) ) ) ).
fof(fact_330_max__leastL,axiom,
! [X_a] :
( ord(X_a)
=> ! [X,Least] :
( ! [X_1] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Least),X_1))
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),Least),X) = ti(X_a,X) ) ) ).
fof(fact_331_max__ord__max,axiom,
! [X_a] :
( ord(X_a)
=> ord_max(X_a) = hAPP(fun(X_a,fun(X_a,bool)),fun(X_a,fun(X_a,X_a)),max(X_a),ord_less_eq(X_a)) ) ).
fof(fact_332_min__max_OInf__le__Sup,axiom,
! [X_a] :
( linorder(X_a)
=> ! [A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> ( ti(fun(X_a,bool),A_2) != bot_bot(fun(X_a,bool))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(fun(X_a,bool),X_a,hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,bool),X_a),big_Inf_fin(X_a),ord_min(X_a)),A_2)),hAPP(fun(X_a,bool),X_a,hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,bool),X_a),big_Sup_fin(X_a),ord_max(X_a)),A_2))) ) ) ) ).
fof(fact_333_min__max_OSup__fin_Oinsert__remove,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X_2,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> ( ( hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_2),bot_bot(fun(X_a,bool)))) = bot_bot(fun(X_a,bool))
=> hAPP(fun(X_a,bool),X_a,hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,bool),X_a),big_Sup_fin(X_a),ord_max(X_a)),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_2),A_2)) = ti(X_a,X_2) )
& ( hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_2),bot_bot(fun(X_a,bool)))) != bot_bot(fun(X_a,bool))
=> hAPP(fun(X_a,bool),X_a,hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,bool),X_a),big_Sup_fin(X_a),ord_max(X_a)),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_2),A_2)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),X_2),hAPP(fun(X_a,bool),X_a,hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,bool),X_a),big_Sup_fin(X_a),ord_max(X_a)),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_2),bot_bot(fun(X_a,bool)))))) ) ) ) ) ).
fof(fact_334_min__max_OSup__fin_Oin__idem,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X_2,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),A_2))
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),X_2),hAPP(fun(X_a,bool),X_a,hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,bool),X_a),big_Sup_fin(X_a),ord_max(X_a)),A_2)) = hAPP(fun(X_a,bool),X_a,hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,bool),X_a),big_Sup_fin(X_a),ord_max(X_a)),A_2) ) ) ) ).
fof(fact_335_min__max_OSup__fin_Osingleton,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X_2] : hAPP(fun(X_a,bool),X_a,hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,bool),X_a),big_Sup_fin(X_a),ord_max(X_a)),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_2),bot_bot(fun(X_a,bool)))) = ti(X_a,X_2) ) ).
fof(fact_336_min__max_Oinf__Sup__absorb,axiom,
! [X_a] :
( linorder(X_a)
=> ! [A_1,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_1),A_2))
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),A_1),hAPP(fun(X_a,bool),X_a,hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,bool),X_a),big_Sup_fin(X_a),ord_max(X_a)),A_2)) = ti(X_a,A_1) ) ) ) ).
fof(fact_337_min__max_OSup__fin_OF__eq,axiom,
! [X_a] :
( linorder(X_a)
=> ! [A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> hAPP(fun(X_a,bool),X_a,hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,bool),X_a),big_Sup_fin(X_a),ord_max(X_a)),A_2) = hAPP(fun(X_a,bool),X_a,hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,bool),X_a),finite_fold1(X_a),ord_max(X_a)),A_2) ) ) ).
fof(fact_338_min__max_OSup__fin_Oinsert__idem,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X_2,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> ( ti(fun(X_a,bool),A_2) != bot_bot(fun(X_a,bool))
=> hAPP(fun(X_a,bool),X_a,hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,bool),X_a),big_Sup_fin(X_a),ord_max(X_a)),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_2),A_2)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),X_2),hAPP(fun(X_a,bool),X_a,hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,bool),X_a),big_Sup_fin(X_a),ord_max(X_a)),A_2)) ) ) ) ).
fof(fact_339_min__max_OSup__fin_Oinsert,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X_2,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> ( ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),A_2))
=> ( ti(fun(X_a,bool),A_2) != bot_bot(fun(X_a,bool))
=> hAPP(fun(X_a,bool),X_a,hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,bool),X_a),big_Sup_fin(X_a),ord_max(X_a)),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_2),A_2)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),X_2),hAPP(fun(X_a,bool),X_a,hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,bool),X_a),big_Sup_fin(X_a),ord_max(X_a)),A_2)) ) ) ) ) ).
fof(fact_340_min__max_OSup__fin_Osubset__idem,axiom,
! [X_a] :
( linorder(X_a)
=> ! [B_1,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> ( ti(fun(X_a,bool),B_1) != bot_bot(fun(X_a,bool))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),B_1),A_2))
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),hAPP(fun(X_a,bool),X_a,hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,bool),X_a),big_Sup_fin(X_a),ord_max(X_a)),B_1)),hAPP(fun(X_a,bool),X_a,hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,bool),X_a),big_Sup_fin(X_a),ord_max(X_a)),A_2)) = hAPP(fun(X_a,bool),X_a,hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,bool),X_a),big_Sup_fin(X_a),ord_max(X_a)),A_2) ) ) ) ) ).
fof(fact_341_min__max_OSup__fin_Oeq__fold__idem_H,axiom,
! [X_a] :
( linorder(X_a)
=> ! [A_1,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> hAPP(fun(X_a,bool),X_a,hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,bool),X_a),big_Sup_fin(X_a),ord_max(X_a)),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),A_1),A_2)) = hAPP(fun(X_a,bool),X_a,hAPP(X_a,fun(fun(X_a,bool),X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(fun(X_a,bool),X_a)),finite_fold(X_a,X_a),ord_max(X_a)),A_1),A_2) ) ) ).
fof(fact_342_min__max_OSup__fin_Oeq__fold_H,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X_2,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> ( ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),A_2))
=> hAPP(fun(X_a,bool),X_a,hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,bool),X_a),big_Sup_fin(X_a),ord_max(X_a)),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_2),A_2)) = hAPP(fun(X_a,bool),X_a,hAPP(X_a,fun(fun(X_a,bool),X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(fun(X_a,bool),X_a)),finite_fold(X_a,X_a),ord_max(X_a)),X_2),A_2) ) ) ) ).
fof(fact_343_min__max_OSup__fin_Oremove,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X_2,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),A_2))
=> ( ( hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_2),bot_bot(fun(X_a,bool)))) = bot_bot(fun(X_a,bool))
=> hAPP(fun(X_a,bool),X_a,hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,bool),X_a),big_Sup_fin(X_a),ord_max(X_a)),A_2) = ti(X_a,X_2) )
& ( hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_2),bot_bot(fun(X_a,bool)))) != bot_bot(fun(X_a,bool))
=> hAPP(fun(X_a,bool),X_a,hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,bool),X_a),big_Sup_fin(X_a),ord_max(X_a)),A_2) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),X_2),hAPP(fun(X_a,bool),X_a,hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,bool),X_a),big_Sup_fin(X_a),ord_max(X_a)),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_2),bot_bot(fun(X_a,bool)))))) ) ) ) ) ) ).
fof(fact_344_min__max_OSup__fin_Oclosed,axiom,
! [X_a] :
( linorder(X_a)
=> ! [A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> ( ti(fun(X_a,bool),A_2) != bot_bot(fun(X_a,bool))
=> ( ! [X_1,Y_1] : hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),X_1),Y_1)),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_1),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),Y_1),bot_bot(fun(X_a,bool))))))
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),hAPP(fun(X_a,bool),X_a,hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,bool),X_a),big_Sup_fin(X_a),ord_max(X_a)),A_2)),A_2)) ) ) ) ) ).
fof(fact_345_setsum_Oremove,axiom,
! [X_a,X_b] :
( comm_monoid_add(X_b)
=> ! [G,X_2,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),A_2))
=> hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),G),A_2) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),hAPP(X_a,X_b,G,X_2)),hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),G),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_2),bot_bot(fun(X_a,bool)))))) ) ) ) ).
fof(fact_346_setsum__diff1_H,axiom,
! [X_a,X_b] :
( comm_monoid_add(X_b)
=> ! [F,A_1,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_1),A_2))
=> hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),F),A_2) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),hAPP(X_a,X_b,F,A_1)),hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),F),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),A_1),bot_bot(fun(X_a,bool)))))) ) ) ) ).
fof(fact_347_diff__add__cancel,axiom,
! [X_a] :
( group_add(X_a)
=> ! [A_3,B_3] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_3),B_3)),B_3) = ti(X_a,A_3) ) ).
fof(fact_348_add__diff__cancel,axiom,
! [X_a] :
( group_add(X_a)
=> ! [A_3,B_3] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_3),B_3)),B_3) = ti(X_a,A_3) ) ).
fof(fact_349_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [X_a] :
( ab_semigroup_add(X_a)
=> ! [A_3,B_3,C_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_3),B_3)),C_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_3),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),B_3),C_1)) ) ).
fof(fact_350_add__left__cancel,axiom,
! [X_a] :
( cancel_semigroup_add(X_a)
=> ! [A_1,B,Ca] :
( hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),Ca)
<=> ti(X_a,B) = ti(X_a,Ca) ) ) ).
fof(fact_351_add__right__cancel,axiom,
! [X_a] :
( cancel_semigroup_add(X_a)
=> ! [B,A_1,Ca] :
( hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),B),A_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),Ca),A_1)
<=> ti(X_a,B) = ti(X_a,Ca) ) ) ).
fof(fact_352_add__left__imp__eq,axiom,
! [X_a] :
( cancel_semigroup_add(X_a)
=> ! [A_3,B_3,C_1] :
( hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_3),B_3) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_3),C_1)
=> ti(X_a,B_3) = ti(X_a,C_1) ) ) ).
fof(fact_353_add__imp__eq,axiom,
! [X_a] :
( cancel146912293up_add(X_a)
=> ! [A_3,B_3,C_1] :
( hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_3),B_3) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_3),C_1)
=> ti(X_a,B_3) = ti(X_a,C_1) ) ) ).
fof(fact_354_add__right__imp__eq,axiom,
! [X_a] :
( cancel_semigroup_add(X_a)
=> ! [B_3,A_3,C_1] :
( hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),B_3),A_3) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),C_1),A_3)
=> ti(X_a,B_3) = ti(X_a,C_1) ) ) ).
fof(fact_355_add__le__cancel__right,axiom,
! [X_a] :
( ordere236663937imp_le(X_a)
=> ! [A_1,Ca,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),Ca)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),B),Ca)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B)) ) ) ).
fof(fact_356_add__le__cancel__left,axiom,
! [X_a] :
( ordere236663937imp_le(X_a)
=> ! [Ca,A_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),Ca),A_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),Ca),B)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B)) ) ) ).
fof(fact_357_add__right__mono,axiom,
! [X_a] :
( ordere779506340up_add(X_a)
=> ! [C_1,A_3,B_3] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_3),B_3))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_3),C_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),B_3),C_1))) ) ) ).
fof(fact_358_add__left__mono,axiom,
! [X_a] :
( ordere779506340up_add(X_a)
=> ! [C_1,A_3,B_3] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_3),B_3))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),C_1),A_3)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),C_1),B_3))) ) ) ).
fof(fact_359_add__mono,axiom,
! [X_a] :
( ordere779506340up_add(X_a)
=> ! [C_1,D_1,A_3,B_3] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_3),B_3))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C_1),D_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_3),C_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),B_3),D_1))) ) ) ) ).
fof(fact_360_add__le__imp__le__right,axiom,
! [X_a] :
( ordere236663937imp_le(X_a)
=> ! [A_3,C_1,B_3] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_3),C_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),B_3),C_1)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_3),B_3)) ) ) ).
fof(fact_361_add__le__imp__le__left,axiom,
! [X_a] :
( ordere236663937imp_le(X_a)
=> ! [C_1,A_3,B_3] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),C_1),A_3)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),C_1),B_3)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_3),B_3)) ) ) ).
fof(fact_362_min__add__distrib__left,axiom,
! [X_a] :
( ordere236663937imp_le(X_a)
=> ! [X,Y,Z] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X),Y)),Z) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X),Z)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),Y),Z)) ) ).
fof(fact_363_max__add__distrib__left,axiom,
! [X_a] :
( ordere236663937imp_le(X_a)
=> ! [X,Y,Z] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),X),Y)),Z) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X),Z)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),Y),Z)) ) ).
fof(fact_364_setsum_Oinsert,axiom,
! [X_a,X_b] :
( comm_monoid_add(X_b)
=> ! [G,X_2,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> ( ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),A_2))
=> hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),G),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_2),A_2)) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),hAPP(X_a,X_b,G,X_2)),hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),G),A_2)) ) ) ) ).
fof(fact_365_setsum__insert,axiom,
! [X_a,X_b] :
( comm_monoid_add(X_b)
=> ! [F,A_1,F_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),F_1))
=> ( ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_1),F_1))
=> hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),F),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),A_1),F_1)) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),hAPP(X_a,X_b,F,A_1)),hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),F),F_1)) ) ) ) ).
fof(fact_366_setsum_Oinsert__remove,axiom,
! [X_a,X_b] :
( comm_monoid_add(X_b)
=> ! [G,X_2,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),G),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_2),A_2)) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),hAPP(X_a,X_b,G,X_2)),hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),G),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_2),bot_bot(fun(X_a,bool)))))) ) ) ).
fof(fact_367_setsum__cong2,axiom,
! [X_a,X_b] :
( comm_monoid_add(X_b)
=> ! [F,G,A_2] :
( ! [X_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),A_2))
=> hAPP(X_a,X_b,F,X_1) = hAPP(X_a,X_b,G,X_1) )
=> hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),F),A_2) = hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),G),A_2) ) ) ).
fof(fact_368_setsum__cong,axiom,
! [X_a,X_b] :
( comm_monoid_add(X_b)
=> ! [F,G,A_2,B_1] :
( ti(fun(X_a,bool),A_2) = ti(fun(X_a,bool),B_1)
=> ( ! [X_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),B_1))
=> hAPP(X_a,X_b,F,X_1) = hAPP(X_a,X_b,G,X_1) )
=> hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),F),A_2) = hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),G),B_1) ) ) ) ).
fof(fact_369_setsum_OF__cong,axiom,
! [X_a,X_b] :
( comm_monoid_add(X_b)
=> ! [H,G,A_2,B_1] :
( ti(fun(X_a,bool),A_2) = ti(fun(X_a,bool),B_1)
=> ( ! [X_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),B_1))
=> hAPP(X_a,X_b,H,X_1) = hAPP(X_a,X_b,G,X_1) )
=> hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),H),A_2) = hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),G),B_1) ) ) ) ).
fof(fact_370_fold1__eq__fold__idem,axiom,
! [X_a] :
( ab_sem1668676832m_mult(X_a)
=> ! [A_1,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> hAPP(fun(X_a,bool),X_a,hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,bool),X_a),finite_fold1(X_a),times_times(X_a)),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),A_1),A_2)) = hAPP(fun(X_a,bool),X_a,hAPP(X_a,fun(fun(X_a,bool),X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(fun(X_a,bool),X_a)),finite_fold(X_a,X_a),times_times(X_a)),A_1),A_2) ) ) ).
fof(fact_371_fold1__eq__fold,axiom,
! [X_a] :
( ab_semigroup_mult(X_a)
=> ! [A_1,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> ( ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_1),A_2))
=> hAPP(fun(X_a,bool),X_a,hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,bool),X_a),finite_fold1(X_a),times_times(X_a)),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),A_1),A_2)) = hAPP(fun(X_a,bool),X_a,hAPP(X_a,fun(fun(X_a,bool),X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(fun(X_a,bool),X_a)),finite_fold(X_a,X_a),times_times(X_a)),A_1),A_2) ) ) ) ).
fof(fact_372_times_Oidem,axiom,
! [X_a] :
( ab_sem1668676832m_mult(X_a)
=> ! [A_3] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),A_3) = ti(X_a,A_3) ) ).
fof(fact_373_mult__idem,axiom,
! [X_a] :
( ab_sem1668676832m_mult(X_a)
=> ! [X] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),X) = ti(X_a,X) ) ).
fof(fact_374_mult__left__idem,axiom,
! [X_a] :
( ab_sem1668676832m_mult(X_a)
=> ! [A_3,B_3] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),B_3)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),B_3) ) ).
fof(fact_375_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [X_a] :
( ab_semigroup_mult(X_a)
=> ! [A_3,B_3,C_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),B_3)),C_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B_3),C_1)) ) ).
fof(fact_376_comp__fun__commute,axiom,
! [X_a] :
( ab_semigroup_mult(X_a)
=> hBOOL(hAPP(fun(X_a,fun(X_a,X_a)),bool,finite100568337ommute(X_a,X_a),times_times(X_a))) ) ).
fof(fact_377_comp__fun__idem,axiom,
! [X_a] :
( ab_sem1668676832m_mult(X_a)
=> hBOOL(hAPP(fun(X_a,fun(X_a,X_a)),bool,finite_comp_fun_idem(X_a,X_a),times_times(X_a))) ) ).
fof(fact_378_fold__graph__insert__swap,axiom,
! [X_a] :
( ab_semigroup_mult(X_a)
=> ! [Z_1,B,A_2,Y_2] :
( hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(fun(X_a,bool),fun(X_a,bool))),finite_fold_graph(X_a,X_a),times_times(X_a)),B),A_2),Y_2))
=> ( ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),B),A_2))
=> hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(fun(X_a,bool),fun(X_a,bool))),finite_fold_graph(X_a,X_a),times_times(X_a)),Z_1),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),B),A_2)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Z_1),Y_2))) ) ) ) ).
fof(fact_379_fold1__insert__idem,axiom,
! [X_a] :
( ab_sem1668676832m_mult(X_a)
=> ! [X_2,A_2] :
( ti(fun(X_a,bool),A_2) != bot_bot(fun(X_a,bool))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> hAPP(fun(X_a,bool),X_a,hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,bool),X_a),finite_fold1(X_a),times_times(X_a)),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_2),A_2)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X_2),hAPP(fun(X_a,bool),X_a,hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,bool),X_a),finite_fold1(X_a),times_times(X_a)),A_2)) ) ) ) ).
fof(fact_380_fold1__insert,axiom,
! [X_a] :
( ab_semigroup_mult(X_a)
=> ! [X_2,A_2] :
( ti(fun(X_a,bool),A_2) != bot_bot(fun(X_a,bool))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> ( ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),A_2))
=> hAPP(fun(X_a,bool),X_a,hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,bool),X_a),finite_fold1(X_a),times_times(X_a)),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_2),A_2)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X_2),hAPP(fun(X_a,bool),X_a,hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,bool),X_a),finite_fold1(X_a),times_times(X_a)),A_2)) ) ) ) ) ).
fof(fact_381_fold__graph__permute__diff,axiom,
! [X_a] :
( ab_semigroup_mult(X_a)
=> ! [A_1,B,A_2,X_2] :
( hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(fun(X_a,bool),fun(X_a,bool))),finite_fold_graph(X_a,X_a),times_times(X_a)),B),A_2),X_2))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_1),A_2))
=> ( ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),B),A_2))
=> hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(fun(X_a,bool),fun(X_a,bool))),finite_fold_graph(X_a,X_a),times_times(X_a)),A_1),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),B),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),A_1),bot_bot(fun(X_a,bool)))))),X_2)) ) ) ) ) ).
fof(fact_382_fold1__in,axiom,
! [X_a] :
( ab_semigroup_mult(X_a)
=> ! [A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> ( ti(fun(X_a,bool),A_2) != bot_bot(fun(X_a,bool))
=> ( ! [X_1,Y_1] : hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X_1),Y_1)),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_1),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),Y_1),bot_bot(fun(X_a,bool))))))
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),hAPP(fun(X_a,bool),X_a,hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,bool),X_a),finite_fold1(X_a),times_times(X_a)),A_2)),A_2)) ) ) ) ) ).
fof(fact_383_le__add__iff1,axiom,
! [X_a] :
( ordered_ring(X_a)
=> ! [A_1,E,Ca,B,D] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),E)),Ca)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),E)),D)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_1),B)),E)),Ca)),D)) ) ) ).
fof(fact_384_combine__common__factor,axiom,
! [X_a] :
( semiring(X_a)
=> ! [A_3,E_1,B_3,C_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),E_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B_3),E_1)),C_1)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_3),B_3)),E_1)),C_1) ) ).
fof(fact_385_comm__semiring__class_Odistrib,axiom,
! [X_a] :
( comm_semiring(X_a)
=> ! [A_3,B_3,C_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_3),B_3)),C_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),C_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B_3),C_1)) ) ).
fof(fact_386_eq__add__iff2,axiom,
! [X_a] :
( ring(X_a)
=> ! [A_1,E,Ca,B,D] :
( hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),E)),Ca) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),E)),D)
<=> ti(X_a,Ca) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),B),A_1)),E)),D) ) ) ).
fof(fact_387_eq__add__iff1,axiom,
! [X_a] :
( ring(X_a)
=> ! [A_1,E,Ca,B,D] :
( hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),E)),Ca) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),E)),D)
<=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_1),B)),E)),Ca) = ti(X_a,D) ) ) ).
fof(fact_388_le__add__iff2,axiom,
! [X_a] :
( ordered_ring(X_a)
=> ! [A_1,E,Ca,B,D] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),E)),Ca)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),E)),D)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Ca),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),B),A_1)),E)),D))) ) ) ).
fof(fact_389_crossproduct__eq,axiom,
! [X_a] :
( semiri456707255roduct(X_a)
=> ! [W_1,Y_2,X_2,Z_1] :
( hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),W_1),Y_2)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X_2),Z_1)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),W_1),Z_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X_2),Y_2))
<=> ( ti(X_a,W_1) = ti(X_a,X_2)
| ti(X_a,Y_2) = ti(X_a,Z_1) ) ) ) ).
fof(fact_390_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_3,M,B_3] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),M)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B_3),M)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_3),B_3)),M) ) ).
fof(fact_391_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_3,B_3,C_1,D_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_3),B_3)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),C_1),D_1)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_3),C_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),B_3),D_1)) ) ).
fof(fact_392_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_3,B_3,C_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_3),B_3)),C_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_3),C_1)),B_3) ) ).
fof(fact_393_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_3,B_3,C_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_3),B_3)),C_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_3),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),B_3),C_1)) ) ).
fof(fact_394_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_3,C_1,D_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_3),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),C_1),D_1)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_3),C_1)),D_1) ) ).
fof(fact_395_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_3,C_1,D_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_3),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),C_1),D_1)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),C_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_3),D_1)) ) ).
fof(fact_396_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_3,C_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_3),C_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),C_1),A_3) ) ).
fof(fact_397_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [Lx,Ly,Rx,Ry] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Lx),Ly)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Rx),Ry)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Lx),Rx)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Ly),Ry)) ) ).
fof(fact_398_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [Lx,Ly,Rx,Ry] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Lx),Ly)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Rx),Ry)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Rx),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Lx),Ly)),Ry)) ) ).
fof(fact_399_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [Lx,Ly,Rx,Ry] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Lx),Ly)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Rx),Ry)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Lx),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Ly),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Rx),Ry))) ) ).
fof(fact_400_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [Lx,Ly,Rx] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Lx),Ly)),Rx) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Lx),Rx)),Ly) ) ).
fof(fact_401_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [Lx,Ly,Rx] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Lx),Ly)),Rx) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Lx),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Ly),Rx)) ) ).
fof(fact_402_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [Lx,Rx,Ry] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Lx),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Rx),Ry)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Lx),Rx)),Ry) ) ).
fof(fact_403_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [Lx,Rx,Ry] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Lx),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Rx),Ry)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Rx),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Lx),Ry)) ) ).
fof(fact_404_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_3,B_3] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),B_3) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B_3),A_3) ) ).
fof(fact_405_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [X,Y,Z] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),Y),Z)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),Y)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),Z)) ) ).
fof(fact_406_crossproduct__noteq,axiom,
! [X_a] :
( semiri456707255roduct(X_a)
=> ! [Ca,D,A_1,B] :
( ( ti(X_a,A_1) != ti(X_a,B)
& ti(X_a,Ca) != ti(X_a,D) )
<=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),Ca)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),D)) != hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),D)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),Ca)) ) ) ).
fof(fact_407_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_3,B_3,C_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_3),B_3)),C_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),C_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B_3),C_1)) ) ).
fof(fact_408_setprod_Oremove,axiom,
! [X_a,X_b] :
( comm_monoid_mult(X_b)
=> ! [G,X_2,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),A_2))
=> hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),G),A_2) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),hAPP(X_a,X_b,G,X_2)),hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),G),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_2),bot_bot(fun(X_a,bool)))))) ) ) ) ).
fof(fact_409_setprod_Oinsert__remove,axiom,
! [X_a,X_b] :
( comm_monoid_mult(X_b)
=> ! [G,X_2,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),G),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_2),A_2)) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),hAPP(X_a,X_b,G,X_2)),hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),G),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_2),bot_bot(fun(X_a,bool)))))) ) ) ).
fof(fact_410_setprod_Oinsert,axiom,
! [X_a,X_b] :
( comm_monoid_mult(X_b)
=> ! [G,X_2,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> ( ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),A_2))
=> hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),G),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_2),A_2)) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),hAPP(X_a,X_b,G,X_2)),hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),G),A_2)) ) ) ) ).
fof(fact_411_setprod__insert,axiom,
! [X_a,X_b] :
( comm_monoid_mult(X_b)
=> ! [F,A_1,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> ( ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_1),A_2))
=> hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),F),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),A_1),A_2)) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),hAPP(X_a,X_b,F,A_1)),hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),F),A_2)) ) ) ) ).
fof(fact_412_setprod_Ocong,axiom,
! [X_a,X_b] :
( comm_monoid_mult(X_b)
=> ! [G,H,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> ( ! [X_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),A_2))
=> hAPP(X_a,X_b,G,X_1) = hAPP(X_a,X_b,H,X_1) )
=> hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),G),A_2) = hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),H),A_2) ) ) ) ).
fof(fact_413_setprod_OF__cong,axiom,
! [X_a,X_b] :
( comm_monoid_mult(X_b)
=> ! [H,G,A_2,B_1] :
( ti(fun(X_a,bool),A_2) = ti(fun(X_a,bool),B_1)
=> ( ! [X_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),B_1))
=> hAPP(X_a,X_b,H,X_1) = hAPP(X_a,X_b,G,X_1) )
=> hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),H),A_2) = hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),G),B_1) ) ) ) ).
fof(fact_414_setprod__cong,axiom,
! [X_a,X_b] :
( comm_monoid_mult(X_b)
=> ! [F,G,A_2,B_1] :
( ti(fun(X_a,bool),A_2) = ti(fun(X_a,bool),B_1)
=> ( ! [X_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),B_1))
=> hAPP(X_a,X_b,F,X_1) = hAPP(X_a,X_b,G,X_1) )
=> hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),F),A_2) = hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),G),B_1) ) ) ) ).
fof(fact_415_strict__below__fold1__iff,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X_2,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> ( ti(fun(X_a,bool),A_2) != bot_bot(fun(X_a,bool))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_2),hAPP(fun(X_a,bool),X_a,hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,bool),X_a),finite_fold1(X_a),ord_min(X_a)),A_2)))
<=> ! [X_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),A_2))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_2),X_1)) ) ) ) ) ) ).
fof(fact_416_add__less__cancel__right,axiom,
! [X_a] :
( ordere236663937imp_le(X_a)
=> ! [A_1,Ca,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),Ca)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),B),Ca)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B)) ) ) ).
fof(fact_417_add__less__cancel__left,axiom,
! [X_a] :
( ordere236663937imp_le(X_a)
=> ! [Ca,A_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),Ca),A_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),Ca),B)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B)) ) ) ).
fof(fact_418_add__strict__right__mono,axiom,
! [X_a] :
( ordere223160158up_add(X_a)
=> ! [C_1,A_3,B_3] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_3),B_3))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_3),C_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),B_3),C_1))) ) ) ).
fof(fact_419_add__strict__left__mono,axiom,
! [X_a] :
( ordere223160158up_add(X_a)
=> ! [C_1,A_3,B_3] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_3),B_3))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),C_1),A_3)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),C_1),B_3))) ) ) ).
fof(fact_420_add__strict__mono,axiom,
! [X_a] :
( ordere223160158up_add(X_a)
=> ! [C_1,D_1,A_3,B_3] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_3),B_3))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C_1),D_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_3),C_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),B_3),D_1))) ) ) ) ).
fof(fact_421_add__less__imp__less__right,axiom,
! [X_a] :
( ordere236663937imp_le(X_a)
=> ! [A_3,C_1,B_3] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_3),C_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),B_3),C_1)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_3),B_3)) ) ) ).
fof(fact_422_add__less__imp__less__left,axiom,
! [X_a] :
( ordere236663937imp_le(X_a)
=> ! [C_1,A_3,B_3] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),C_1),A_3)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),C_1),B_3)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_3),B_3)) ) ) ).
fof(fact_423_add__less__le__mono,axiom,
! [X_a] :
( ordere223160158up_add(X_a)
=> ! [C_1,D_1,A_3,B_3] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_3),B_3))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C_1),D_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_3),C_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),B_3),D_1))) ) ) ) ).
fof(fact_424_add__le__less__mono,axiom,
! [X_a] :
( ordere223160158up_add(X_a)
=> ! [C_1,D_1,A_3,B_3] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_3),B_3))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C_1),D_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_3),C_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),B_3),D_1))) ) ) ) ).
fof(fact_425_min__max_Oless__supI1,axiom,
! [X_a] :
( linorder(X_a)
=> ! [B_3,X,A_3] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),A_3))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),A_3),B_3))) ) ) ).
fof(fact_426_min__max_Oless__supI2,axiom,
! [X_a] :
( linorder(X_a)
=> ! [A_3,X,B_3] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),B_3))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),A_3),B_3))) ) ) ).
fof(fact_427_less__max__iff__disj,axiom,
! [X_a] :
( linorder(X_a)
=> ! [Z_1,X_2,Y_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Z_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),X_2),Y_2)))
<=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Z_1),X_2))
| hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Z_1),Y_2)) ) ) ) ).
fof(fact_428_max__less__iff__conj,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X_2,Y_2,Z_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),X_2),Y_2)),Z_1))
<=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_2),Z_1))
& hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y_2),Z_1)) ) ) ) ).
fof(fact_429_not__less__bot,axiom,
! [X_a] :
( bot(X_a)
=> ! [A_3] : ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_3),bot_bot(X_a))) ) ).
fof(fact_430_bot__less,axiom,
! [X_a] :
( bot(X_a)
=> ! [A_1] :
( ti(X_a,A_1) != bot_bot(X_a)
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),bot_bot(X_a)),A_1)) ) ) ).
fof(fact_431_not__psubset__empty,axiom,
! [X_a,A_2] : ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less(fun(X_a,bool)),A_2),bot_bot(fun(X_a,bool)))) ).
fof(fact_432_psubset__eq,axiom,
! [X_a,A_2,B_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less(fun(X_a,bool)),A_2),B_1))
<=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1))
& ti(fun(X_a,bool),A_2) != ti(fun(X_a,bool),B_1) ) ) ).
fof(fact_433_subset__iff__psubset__eq,axiom,
! [X_a,A_2,B_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1))
<=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less(fun(X_a,bool)),A_2),B_1))
| ti(fun(X_a,bool),A_2) = ti(fun(X_a,bool),B_1) ) ) ).
fof(fact_434_psubset__imp__subset,axiom,
! [X_a,A_2,B_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less(fun(X_a,bool)),A_2),B_1))
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1)) ) ).
fof(fact_435_psubset__subset__trans,axiom,
! [X_a,C,A_2,B_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less(fun(X_a,bool)),A_2),B_1))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),B_1),C))
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less(fun(X_a,bool)),A_2),C)) ) ) ).
fof(fact_436_subset__psubset__trans,axiom,
! [X_a,C,A_2,B_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less(fun(X_a,bool)),B_1),C))
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less(fun(X_a,bool)),A_2),C)) ) ) ).
fof(fact_437_min__max_Oless__infI2,axiom,
! [X_a] :
( linorder(X_a)
=> ! [A_3,B_3,X] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B_3),X))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),A_3),B_3)),X)) ) ) ).
fof(fact_438_min__max_Oless__infI1,axiom,
! [X_a] :
( linorder(X_a)
=> ! [B_3,A_3,X] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_3),X))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),A_3),B_3)),X)) ) ) ).
fof(fact_439_min__less__iff__conj,axiom,
! [X_a] :
( linorder(X_a)
=> ! [Z_1,X_2,Y_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Z_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X_2),Y_2)))
<=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Z_1),X_2))
& hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Z_1),Y_2)) ) ) ) ).
fof(fact_440_min__less__iff__disj,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X_2,Y_2,Z_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X_2),Y_2)),Z_1))
<=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_2),Z_1))
| hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y_2),Z_1)) ) ) ) ).
fof(fact_441_linorder__cases,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X,Y] :
( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Y))
=> ( ti(X_a,X) != ti(X_a,Y)
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y),X)) ) ) ) ).
fof(fact_442_order__less__asym,axiom,
! [X_a] :
( preorder(X_a)
=> ! [X,Y] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Y))
=> ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y),X)) ) ) ).
fof(fact_443_xt1_I10_J,axiom,
! [X_a] :
( order(X_a)
=> ! [Z,Y,X] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y),X))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Z),Y))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Z),X)) ) ) ) ).
fof(fact_444_order__less__trans,axiom,
! [X_a] :
( preorder(X_a)
=> ! [Z,X,Y] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Y))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y),Z))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Z)) ) ) ) ).
fof(fact_445_xt1_I2_J,axiom,
! [X_a] :
( order(X_a)
=> ! [C_1,B_3,A_3] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B_3),A_3))
=> ( ti(X_a,B_3) = ti(X_a,C_1)
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C_1),A_3)) ) ) ) ).
fof(fact_446_ord__less__eq__trans,axiom,
! [X_a] :
( ord(X_a)
=> ! [C_1,A_3,B_3] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_3),B_3))
=> ( ti(X_a,B_3) = ti(X_a,C_1)
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_3),C_1)) ) ) ) ).
fof(fact_447_xt1_I1_J,axiom,
! [X_a] :
( order(X_a)
=> ! [C_1,A_3,B_3] :
( ti(X_a,A_3) = ti(X_a,B_3)
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C_1),B_3))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C_1),A_3)) ) ) ) ).
fof(fact_448_ord__eq__less__trans,axiom,
! [X_a] :
( ord(X_a)
=> ! [C_1,A_3,B_3] :
( ti(X_a,A_3) = ti(X_a,B_3)
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B_3),C_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_3),C_1)) ) ) ) ).
fof(fact_449_xt1_I9_J,axiom,
! [X_a] :
( order(X_a)
=> ! [B_3,A_3] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B_3),A_3))
=> ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_3),B_3)) ) ) ).
fof(fact_450_order__less__asym_H,axiom,
! [X_a] :
( preorder(X_a)
=> ! [A_3,B_3] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_3),B_3))
=> ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B_3),A_3)) ) ) ).
fof(fact_451_order__less__imp__triv,axiom,
! [X_a] :
( preorder(X_a)
=> ! [Pa,X_2,Y_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_2),Y_2))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y_2),X_2))
=> hBOOL(Pa) ) ) ) ).
fof(fact_452_order__less__imp__not__eq2,axiom,
! [X_a] :
( order(X_a)
=> ! [X,Y] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Y))
=> ti(X_a,Y) != ti(X_a,X) ) ) ).
fof(fact_453_order__less__imp__not__eq,axiom,
! [X_a] :
( order(X_a)
=> ! [X,Y] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Y))
=> ti(X_a,X) != ti(X_a,Y) ) ) ).
fof(fact_454_order__less__imp__not__less,axiom,
! [X_a] :
( preorder(X_a)
=> ! [X,Y] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Y))
=> ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y),X)) ) ) ).
fof(fact_455_order__less__not__sym,axiom,
! [X_a] :
( preorder(X_a)
=> ! [X,Y] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Y))
=> ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y),X)) ) ) ).
fof(fact_456_less__imp__neq,axiom,
! [X_a] :
( order(X_a)
=> ! [X,Y] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Y))
=> ti(X_a,X) != ti(X_a,Y) ) ) ).
fof(fact_457_linorder__neqE,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X,Y] :
( ti(X_a,X) != ti(X_a,Y)
=> ( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Y))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y),X)) ) ) ) ).
fof(fact_458_linorder__antisym__conv3,axiom,
! [X_a] :
( linorder(X_a)
=> ! [Y_2,X_2] :
( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y_2),X_2))
=> ( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_2),Y_2))
<=> ti(X_a,X_2) = ti(X_a,Y_2) ) ) ) ).
fof(fact_459_linorder__less__linear,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X,Y] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Y))
| ti(X_a,X) = ti(X_a,Y)
| hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y),X)) ) ) ).
fof(fact_460_not__less__iff__gr__or__eq,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X_2,Y_2] :
( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_2),Y_2))
<=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y_2),X_2))
| ti(X_a,X_2) = ti(X_a,Y_2) ) ) ) ).
fof(fact_461_linorder__neq__iff,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X_2,Y_2] :
( ti(X_a,X_2) != ti(X_a,Y_2)
<=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_2),Y_2))
| hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y_2),X_2)) ) ) ) ).
fof(fact_462_order__less__irrefl,axiom,
! [X_a] :
( preorder(X_a)
=> ! [X] : ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),X)) ) ).
fof(fact_463_linorder__not__less,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X_2,Y_2] :
( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_2),Y_2))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y_2),X_2)) ) ) ).
fof(fact_464_linorder__not__le,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X_2,Y_2] :
( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_2),Y_2))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y_2),X_2)) ) ) ).
fof(fact_465_linorder__le__less__linear,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X,Y] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),Y))
| hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y),X)) ) ) ).
fof(fact_466_order__less__le,axiom,
! [X_a] :
( order(X_a)
=> ! [X_2,Y_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_2),Y_2))
<=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_2),Y_2))
& ti(X_a,X_2) != ti(X_a,Y_2) ) ) ) ).
fof(fact_467_less__le__not__le,axiom,
! [X_a] :
( preorder(X_a)
=> ! [X_2,Y_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_2),Y_2))
<=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_2),Y_2))
& ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y_2),X_2)) ) ) ) ).
fof(fact_468_order__le__less,axiom,
! [X_a] :
( order(X_a)
=> ! [X_2,Y_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_2),Y_2))
<=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_2),Y_2))
| ti(X_a,X_2) = ti(X_a,Y_2) ) ) ) ).
fof(fact_469_leI,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X,Y] :
( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Y))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),X)) ) ) ).
fof(fact_470_not__leE,axiom,
! [X_a] :
( linorder(X_a)
=> ! [Y,X] :
( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),X))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Y)) ) ) ).
fof(fact_471_linorder__antisym__conv1,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X_2,Y_2] :
( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_2),Y_2))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_2),Y_2))
<=> ti(X_a,X_2) = ti(X_a,Y_2) ) ) ) ).
fof(fact_472_order__neq__le__trans,axiom,
! [X_a] :
( order(X_a)
=> ! [A_3,B_3] :
( ti(X_a,A_3) != ti(X_a,B_3)
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_3),B_3))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_3),B_3)) ) ) ) ).
fof(fact_473_xt1_I12_J,axiom,
! [X_a] :
( order(X_a)
=> ! [A_3,B_3] :
( ti(X_a,A_3) != ti(X_a,B_3)
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B_3),A_3))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B_3),A_3)) ) ) ) ).
fof(fact_474_leD,axiom,
! [X_a] :
( linorder(X_a)
=> ! [Y,X] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),X))
=> ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Y)) ) ) ).
fof(fact_475_order__less__imp__le,axiom,
! [X_a] :
( preorder(X_a)
=> ! [X,Y] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Y))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),Y)) ) ) ).
fof(fact_476_linorder__antisym__conv2,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X_2,Y_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_2),Y_2))
=> ( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_2),Y_2))
<=> ti(X_a,X_2) = ti(X_a,Y_2) ) ) ) ).
fof(fact_477_order__le__imp__less__or__eq,axiom,
! [X_a] :
( order(X_a)
=> ! [X,Y] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),Y))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Y))
| ti(X_a,X) = ti(X_a,Y) ) ) ) ).
fof(fact_478_order__le__neq__trans,axiom,
! [X_a] :
( order(X_a)
=> ! [A_3,B_3] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_3),B_3))
=> ( ti(X_a,A_3) != ti(X_a,B_3)
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_3),B_3)) ) ) ) ).
fof(fact_479_xt1_I11_J,axiom,
! [X_a] :
( order(X_a)
=> ! [B_3,A_3] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B_3),A_3))
=> ( ti(X_a,A_3) != ti(X_a,B_3)
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B_3),A_3)) ) ) ) ).
fof(fact_480_order__less__le__trans,axiom,
! [X_a] :
( preorder(X_a)
=> ! [Z,X,Y] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Y))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),Z))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Z)) ) ) ) ).
fof(fact_481_xt1_I7_J,axiom,
! [X_a] :
( order(X_a)
=> ! [Z,Y,X] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y),X))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Z),Y))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Z),X)) ) ) ) ).
fof(fact_482_order__le__less__trans,axiom,
! [X_a] :
( preorder(X_a)
=> ! [Z,X,Y] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),Y))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y),Z))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Z)) ) ) ) ).
fof(fact_483_xt1_I8_J,axiom,
! [X_a] :
( order(X_a)
=> ! [Z,Y,X] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),X))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Z),Y))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Z),X)) ) ) ) ).
fof(fact_484_less__fun__def,axiom,
! [X_a,X_b] :
( ord(X_b)
=> ! [F,G] :
( hBOOL(hAPP(fun(X_a,X_b),bool,hAPP(fun(X_a,X_b),fun(fun(X_a,X_b),bool),ord_less(fun(X_a,X_b)),F),G))
<=> ( hBOOL(hAPP(fun(X_a,X_b),bool,hAPP(fun(X_a,X_b),fun(fun(X_a,X_b),bool),ord_less_eq(fun(X_a,X_b)),F),G))
& ~ hBOOL(hAPP(fun(X_a,X_b),bool,hAPP(fun(X_a,X_b),fun(fun(X_a,X_b),bool),ord_less_eq(fun(X_a,X_b)),G),F)) ) ) ) ).
fof(fact_485_diff__eq__diff__less,axiom,
! [X_a] :
( ordered_ab_group_add(X_a)
=> ! [A_1,B,Ca,D] :
( hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_1),B) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),Ca),D)
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Ca),D)) ) ) ) ).
fof(fact_486_linorder__neqE__linordered__idom,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [X,Y] :
( ti(X_a,X) != ti(X_a,Y)
=> ( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Y))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y),X)) ) ) ) ).
fof(fact_487_less__add__iff1,axiom,
! [X_a] :
( ordered_ring(X_a)
=> ! [A_1,E,Ca,B,D] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),E)),Ca)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),E)),D)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_1),B)),E)),Ca)),D)) ) ) ).
fof(fact_488_less__add__iff2,axiom,
! [X_a] :
( ordered_ring(X_a)
=> ! [A_1,E,Ca,B,D] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),E)),Ca)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),E)),D)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Ca),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),B),A_1)),E)),D))) ) ) ).
fof(fact_489_psubset__insert__iff,axiom,
! [X_a,A_2,X_2,B_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less(fun(X_a,bool)),A_2),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_2),B_1)))
<=> ( ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),B_1))
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less(fun(X_a,bool)),A_2),B_1)) )
& ( ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),B_1))
=> ( ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),A_2))
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less(fun(X_a,bool)),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_2),bot_bot(fun(X_a,bool))))),B_1)) )
& ( ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),A_2))
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1)) ) ) ) ) ) ).
fof(fact_490_fold1__strict__below__iff,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X_2,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> ( ti(fun(X_a,bool),A_2) != bot_bot(fun(X_a,bool))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(fun(X_a,bool),X_a,hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,bool),X_a),finite_fold1(X_a),ord_min(X_a)),A_2)),X_2))
<=> ? [X_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),A_2))
& hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_1),X_2)) ) ) ) ) ) ).
fof(fact_491_setsum__strict__mono,axiom,
! [X_a,X_b] :
( ( comm_monoid_add(X_b)
& ordere223160158up_add(X_b) )
=> ! [F,G,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> ( ti(fun(X_a,bool),A_2) != bot_bot(fun(X_a,bool))
=> ( ! [X_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),A_2))
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),hAPP(X_a,X_b,F,X_1)),hAPP(X_a,X_b,G,X_1))) )
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),F),A_2)),hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),G),A_2))) ) ) ) ) ).
fof(fact_492_dense__le,axiom,
! [X_a] :
( dense_linorder(X_a)
=> ! [Z,Y] :
( ! [X_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_1),Y))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_1),Z)) )
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),Z)) ) ) ).
fof(fact_493_psubset__trans,axiom,
! [X_a,C,A_2,B_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less(fun(X_a,bool)),A_2),B_1))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less(fun(X_a,bool)),B_1),C))
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less(fun(X_a,bool)),A_2),C)) ) ) ).
fof(fact_494_psubsetD,axiom,
! [X_a,Ca,A_2,B_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less(fun(X_a,bool)),A_2),B_1))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Ca),A_2))
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Ca),B_1)) ) ) ).
fof(fact_495_psubset__imp__ex__mem,axiom,
! [X_a,A_2,B_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less(fun(X_a,bool)),A_2),B_1))
=> ? [B_2] : hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),B_2),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),B_1),A_2))) ) ).
fof(fact_496_xt6,axiom,
! [X_b,X_a] :
( ( order(X_a)
& ord(X_b) )
=> ! [Ca,F,B,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_b,X_a,F,B)),A_1))
=> ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),Ca),B))
=> ( ! [X_1,Y_1] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),Y_1),X_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_b,X_a,F,Y_1)),hAPP(X_b,X_a,F,X_1))) )
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_b,X_a,F,Ca)),A_1)) ) ) ) ) ).
fof(fact_497_dense__le__bounded,axiom,
! [X_a] :
( dense_linorder(X_a)
=> ! [Z,X,Y] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Y))
=> ( ! [W] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),W))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),W),Y))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),W),Z)) ) )
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),Z)) ) ) ) ).
fof(fact_498_xt5,axiom,
! [X_b,X_a] :
( ( order(X_a)
& order(X_b) )
=> ! [Ca,F,B,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),A_1))
=> ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),Ca),hAPP(X_a,X_b,F,B)))
=> ( ! [X_1,Y_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y_1),X_1))
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),hAPP(X_a,X_b,F,Y_1)),hAPP(X_a,X_b,F,X_1))) )
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),Ca),hAPP(X_a,X_b,F,A_1))) ) ) ) ) ).
fof(fact_499_xt4,axiom,
! [X_b,X_a] :
( ( order(X_a)
& order(X_b) )
=> ! [Ca,F,B,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_b,X_a,F,B)),A_1))
=> ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),Ca),B))
=> ( ! [X_1,Y_1] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),Y_1),X_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_b,X_a,F,Y_1)),hAPP(X_b,X_a,F,X_1))) )
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_b,X_a,F,Ca)),A_1)) ) ) ) ) ).
%----Arities (31)
fof(arity_fun___Orderings_Opreorder,axiom,
! [T_2,T_1] :
( preorder(T_1)
=> preorder(fun(T_2,T_1)) ) ).
fof(arity_fun___Finite__Set_Ofinite,axiom,
! [T_2,T_1] :
( ( finite_finite(T_1)
& finite_finite(T_2) )
=> finite_finite(fun(T_2,T_1)) ) ).
fof(arity_fun___Orderings_Oorder,axiom,
! [T_2,T_1] :
( order(T_1)
=> order(fun(T_2,T_1)) ) ).
fof(arity_fun___Orderings_Oord,axiom,
! [T_2,T_1] :
( ord(T_1)
=> ord(fun(T_2,T_1)) ) ).
fof(arity_fun___Orderings_Obot,axiom,
! [T_2,T_1] :
( bot(T_1)
=> bot(fun(T_2,T_1)) ) ).
fof(arity_fun___Groups_Ominus,axiom,
! [T_2,T_1] :
( minus(T_1)
=> minus(fun(T_2,T_1)) ) ).
fof(arity_Nat_Onat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduc,axiom,
semiri456707255roduct(nat) ).
fof(arity_Nat_Onat___Groups_Oordered__cancel__ab__semigroup__add,axiom,
ordere223160158up_add(nat) ).
fof(arity_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
ordere236663937imp_le(nat) ).
fof(arity_Nat_Onat___Groups_Oordered__ab__semigroup__add,axiom,
ordere779506340up_add(nat) ).
fof(arity_Nat_Onat___Groups_Ocancel__ab__semigroup__add,axiom,
cancel146912293up_add(nat) ).
fof(arity_Nat_Onat___Groups_Ocancel__semigroup__add,axiom,
cancel_semigroup_add(nat) ).
fof(arity_Nat_Onat___Groups_Oab__semigroup__mult,axiom,
ab_semigroup_mult(nat) ).
fof(arity_Nat_Onat___Groups_Ocomm__monoid__mult,axiom,
comm_monoid_mult(nat) ).
fof(arity_Nat_Onat___Groups_Oab__semigroup__add,axiom,
ab_semigroup_add(nat) ).
fof(arity_Nat_Onat___Groups_Ocomm__monoid__add,axiom,
comm_monoid_add(nat) ).
fof(arity_Nat_Onat___Rings_Ocomm__semiring__1,axiom,
comm_semiring_1(nat) ).
fof(arity_Nat_Onat___Rings_Ocomm__semiring,axiom,
comm_semiring(nat) ).
fof(arity_Nat_Onat___Orderings_Opreorder,axiom,
preorder(nat) ).
fof(arity_Nat_Onat___Orderings_Olinorder,axiom,
linorder(nat) ).
fof(arity_Nat_Onat___Orderings_Oorder,axiom,
order(nat) ).
fof(arity_Nat_Onat___Rings_Osemiring,axiom,
semiring(nat) ).
fof(arity_Nat_Onat___Orderings_Oord,axiom,
ord(nat) ).
fof(arity_Nat_Onat___Orderings_Obot,axiom,
bot(nat) ).
fof(arity_Nat_Onat___Groups_Ominus,axiom,
minus(nat) ).
fof(arity_HOL_Obool___Orderings_Opreorder,axiom,
preorder(bool) ).
fof(arity_HOL_Obool___Finite__Set_Ofinite,axiom,
finite_finite(bool) ).
fof(arity_HOL_Obool___Orderings_Oorder,axiom,
order(bool) ).
fof(arity_HOL_Obool___Orderings_Oord,axiom,
ord(bool) ).
fof(arity_HOL_Obool___Orderings_Obot,axiom,
bot(bool) ).
fof(arity_HOL_Obool___Groups_Ominus,axiom,
minus(bool) ).
%----Helper facts (20)
fof(help_ti_idem,axiom,
! [T,A] : ti(T,ti(T,A)) = ti(T,A) ).
fof(help_fNot_1_1_U,axiom,
! [P] :
( ~ hBOOL(hAPP(bool,bool,fNot,P))
| ~ hBOOL(P) ) ).
fof(help_fNot_2_1_U,axiom,
! [P] :
( hBOOL(P)
| hBOOL(hAPP(bool,bool,fNot,P)) ) ).
fof(help_COMBB_1_1_U,axiom,
! [X_a,X_c,X_b,P,Q,R] : hAPP(X_a,X_c,hAPP(fun(X_a,X_b),fun(X_a,X_c),hAPP(fun(X_b,X_c),fun(fun(X_a,X_b),fun(X_a,X_c)),combb(X_b,X_c,X_a),P),Q),R) = hAPP(X_b,X_c,P,hAPP(X_a,X_b,Q,R)) ).
fof(help_COMBC_1_1_U,axiom,
! [X_a,X_c,X_b,P,Q,R] : hAPP(X_a,X_c,hAPP(X_b,fun(X_a,X_c),hAPP(fun(X_a,fun(X_b,X_c)),fun(X_b,fun(X_a,X_c)),combc(X_a,X_b,X_c),P),Q),R) = hAPP(X_b,X_c,hAPP(X_a,fun(X_b,X_c),P,R),Q) ).
fof(help_COMBK_1_1_U,axiom,
! [X_b,X_a,P,Q] : hAPP(X_b,X_a,hAPP(X_a,fun(X_b,X_a),combk(X_a,X_b),P),Q) = ti(X_a,P) ).
fof(help_COMBS_1_1_U,axiom,
! [X_a,X_c,X_b,P,Q,R] : hAPP(X_a,X_c,hAPP(fun(X_a,X_b),fun(X_a,X_c),hAPP(fun(X_a,fun(X_b,X_c)),fun(fun(X_a,X_b),fun(X_a,X_c)),combs(X_a,X_b,X_c),P),Q),R) = hAPP(X_b,X_c,hAPP(X_a,fun(X_b,X_c),P,R),hAPP(X_a,X_b,Q,R)) ).
fof(help_fconj_1_1_U,axiom,
! [Q,P] :
( ~ hBOOL(P)
| ~ hBOOL(Q)
| hBOOL(hAPP(bool,bool,hAPP(bool,fun(bool,bool),fconj,P),Q)) ) ).
fof(help_fconj_2_1_U,axiom,
! [P,Q] :
( ~ hBOOL(hAPP(bool,bool,hAPP(bool,fun(bool,bool),fconj,P),Q))
| hBOOL(P) ) ).
fof(help_fconj_3_1_U,axiom,
! [P,Q] :
( ~ hBOOL(hAPP(bool,bool,hAPP(bool,fun(bool,bool),fconj,P),Q))
| hBOOL(Q) ) ).
fof(help_fdisj_1_1_U,axiom,
! [Q,P] :
( ~ hBOOL(P)
| hBOOL(hAPP(bool,bool,hAPP(bool,fun(bool,bool),fdisj,P),Q)) ) ).
fof(help_fdisj_2_1_U,axiom,
! [P,Q] :
( ~ hBOOL(Q)
| hBOOL(hAPP(bool,bool,hAPP(bool,fun(bool,bool),fdisj,P),Q)) ) ).
fof(help_fdisj_3_1_U,axiom,
! [P,Q] :
( ~ hBOOL(hAPP(bool,bool,hAPP(bool,fun(bool,bool),fdisj,P),Q))
| hBOOL(P)
| hBOOL(Q) ) ).
fof(help_fFalse_1_1_U,axiom,
~ hBOOL(fFalse) ).
fof(help_fFalse_1_1_T,axiom,
! [P] :
( ti(bool,P) = fTrue
| ti(bool,P) = fFalse ) ).
fof(help_fequal_1_1_T,axiom,
! [X_a,X,Y] :
( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),fequal(X_a),X),Y))
| ti(X_a,X) = ti(X_a,Y) ) ).
fof(help_fequal_2_1_T,axiom,
! [X_a,X,Y] :
( ti(X_a,X) != ti(X_a,Y)
| hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),fequal(X_a),X),Y)) ) ).
fof(help_fimplies_1_1_U,axiom,
! [Q,P] :
( hBOOL(P)
| hBOOL(hAPP(bool,bool,hAPP(bool,fun(bool,bool),fimplies,P),Q)) ) ).
fof(help_fimplies_2_1_U,axiom,
! [P,Q] :
( ~ hBOOL(Q)
| hBOOL(hAPP(bool,bool,hAPP(bool,fun(bool,bool),fimplies,P),Q)) ) ).
fof(help_fimplies_3_1_U,axiom,
! [P,Q] :
( ~ hBOOL(hAPP(bool,bool,hAPP(bool,fun(bool,bool),fimplies,P),Q))
| ~ hBOOL(P)
| hBOOL(Q) ) ).
%----Conjectures (3)
fof(conj_0,hypothesis,
hBOOL(hAPP(fun(hoare_1656922687triple(state),bool),bool,hAPP(fun(hoare_1656922687triple(state),bool),fun(fun(hoare_1656922687triple(state),bool),bool),hoare_279057269derivs(state),bot_bot(fun(hoare_1656922687triple(state),bool))),hAPP(fun(hoare_1656922687triple(state),bool),fun(hoare_1656922687triple(state),bool),hAPP(hoare_1656922687triple(state),fun(fun(hoare_1656922687triple(state),bool),fun(hoare_1656922687triple(state),bool)),insert(hoare_1656922687triple(state)),hAPP(com,hoare_1656922687triple(state),hoare_Mirabelle_MGT,c)),bot_bot(fun(hoare_1656922687triple(state),bool))))) ).
fof(conj_1,hypothesis,
hBOOL(hAPP(fun(hoare_1656922687triple(state),bool),bool,hAPP(fun(hoare_1656922687triple(state),bool),fun(fun(hoare_1656922687triple(state),bool),bool),hoare_2027193591valids(state),bot_bot(fun(hoare_1656922687triple(state),bool))),hAPP(fun(hoare_1656922687triple(state),bool),fun(hoare_1656922687triple(state),bool),hAPP(hoare_1656922687triple(state),fun(fun(hoare_1656922687triple(state),bool),fun(hoare_1656922687triple(state),bool)),insert(hoare_1656922687triple(state)),hAPP(fun(state,fun(state,bool)),hoare_1656922687triple(state),hAPP(com,fun(fun(state,fun(state,bool)),hoare_1656922687triple(state)),hAPP(fun(state,fun(state,bool)),fun(com,fun(fun(state,fun(state,bool)),hoare_1656922687triple(state))),hoare_246368825triple(state),p),c),q)),bot_bot(fun(hoare_1656922687triple(state),bool))))) ).
fof(conj_2,conjecture,
hBOOL(hAPP(fun(hoare_1656922687triple(state),bool),bool,hAPP(fun(hoare_1656922687triple(state),bool),fun(fun(hoare_1656922687triple(state),bool),bool),hoare_279057269derivs(state),bot_bot(fun(hoare_1656922687triple(state),bool))),hAPP(fun(hoare_1656922687triple(state),bool),fun(hoare_1656922687triple(state),bool),hAPP(hoare_1656922687triple(state),fun(fun(hoare_1656922687triple(state),bool),fun(hoare_1656922687triple(state),bool)),insert(hoare_1656922687triple(state)),hAPP(fun(state,fun(state,bool)),hoare_1656922687triple(state),hAPP(com,fun(fun(state,fun(state,bool)),hoare_1656922687triple(state)),hAPP(fun(state,fun(state,bool)),fun(com,fun(fun(state,fun(state,bool)),hoare_1656922687triple(state))),hoare_246368825triple(state),p),c),q)),bot_bot(fun(hoare_1656922687triple(state),bool))))) ).
%------------------------------------------------------------------------------