TPTP Problem File: SWW470+7.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SWW470+7 : TPTP v8.2.0. Released v5.3.0.
% Domain : Software Verification
% Problem : Hoare's Logic with Procedures line 200, 1000 axioms selected
% Version : Especial.
% English :
% Refs : [BN10] Boehme & Nipkow (2010), Sledgehammer: Judgement Day
% : [Bla11] Blanchette (2011), Email to Geoff Sutcliffe
% Source : [Bla11]
% Names : hoare_1000_fofpt_l200 [Bla11]
% Status : Theorem
% Rating : 0.72 v8.2.0, 0.81 v8.1.0, 0.72 v7.5.0, 0.78 v7.4.0, 0.77 v7.3.0, 0.83 v7.0.0, 0.80 v6.4.0, 0.81 v6.3.0, 0.83 v6.2.0, 0.88 v6.1.0, 0.90 v6.0.0, 0.83 v5.5.0, 0.89 v5.3.0
% Syntax : Number of formulae : 1194 ( 293 unt; 0 def)
% Number of atoms : 3158 ( 872 equ)
% Maximal formula atoms : 9 ( 2 avg)
% Number of connectives : 2218 ( 254 ~; 56 |; 98 &)
% ( 182 <=>;1628 =>; 0 <=; 0 <~>)
% Maximal formula depth : 17 ( 6 avg)
% Maximal term depth : 19 ( 2 avg)
% Number of predicates : 50 ( 49 usr; 0 prp; 1-2 aty)
% Number of functors : 85 ( 85 usr; 21 con; 0-4 aty)
% Number of variables : 3700 (3661 !; 39 ?)
% SPC : FOF_THM_RFO_SEQ
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-08-09 17:19:50
% : Encoded with polymorphic tags.
%------------------------------------------------------------------------------
%----Explicit typings (82)
fof(tsy_c_Big__Operators_Ocomm__monoid__add__class_Osetsum_res,axiom,
! [X_c,X_b] :
( comm_monoid_add(X_b)
=> ti(fun(fun(X_c,X_b),fun(fun(X_c,bool),X_b)),big_co1399186613setsum(X_c,X_b)) = big_co1399186613setsum(X_c,X_b) ) ).
fof(tsy_c_Big__Operators_Ocomm__monoid__mult__class_Osetprod_res,axiom,
! [X_b,X_c] :
( comm_monoid_mult(X_c)
=> ti(fun(fun(X_b,X_c),fun(fun(X_b,bool),X_c)),big_co302546470etprod(X_b,X_c)) = big_co302546470etprod(X_b,X_c) ) ).
fof(tsy_c_Big__Operators_Olattice_OInf__fin_res,axiom,
! [X_b] :
( linorder(X_b)
=> ti(fun(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b)),big_Inf_fin(X_b)) = big_Inf_fin(X_b) ) ).
fof(tsy_c_Big__Operators_Olattice_OSup__fin_res,axiom,
! [X_b] :
( linorder(X_b)
=> ti(fun(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b)),big_Sup_fin(X_b)) = big_Sup_fin(X_b) ) ).
fof(tsy_c_Big__Operators_Olinorder__class_OMax_res,axiom,
! [X_b] :
( linorder(X_b)
=> ti(fun(fun(X_b,bool),X_b),big_linorder_Max(X_b)) = big_linorder_Max(X_b) ) ).
fof(tsy_c_Big__Operators_Olinorder__class_OMin_res,axiom,
! [X_b] :
( linorder(X_b)
=> ti(fun(fun(X_b,bool),X_b),big_linorder_Min(X_b)) = big_linorder_Min(X_b) ) ).
fof(tsy_c_Big__Operators_Osemilattice__big_res,axiom,
! [X_b] : ti(fun(fun(X_b,fun(X_b,X_b)),fun(fun(fun(X_b,bool),X_b),bool)),big_semilattice_big(X_b)) = big_semilattice_big(X_b) ).
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_COMBI_res,axiom,
! [X_a] : ti(fun(X_a,X_a),combi(X_a)) = combi(X_a) ).
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_OAss_res,axiom,
ti(fun(vname,fun(fun(state,nat),com)),ass) = ass ).
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_Finite__Set_Ocard_res,axiom,
! [X_b] : ti(fun(fun(X_b,bool),nat),finite_card(X_b)) = finite_card(X_b) ).
fof(tsy_c_Finite__Set_Ocomp__fun__commute_res,axiom,
! [X_b,X_c] : ti(fun(fun(X_b,fun(X_c,X_c)),bool),finite100568337ommute(X_b,X_c)) = finite100568337ommute(X_b,X_c) ).
fof(tsy_c_Finite__Set_Ocomp__fun__idem_res,axiom,
! [X_b,X_c] : ti(fun(fun(X_b,fun(X_c,X_c)),bool),finite_comp_fun_idem(X_b,X_c)) = finite_comp_fun_idem(X_b,X_c) ).
fof(tsy_c_Finite__Set_Ofinite_res,axiom,
! [X_b] : ti(fun(fun(X_b,bool),bool),finite_finite_1(X_b)) = finite_finite_1(X_b) ).
fof(tsy_c_Finite__Set_Ofold_res,axiom,
! [X_b,X_c] : ti(fun(fun(X_b,fun(X_c,X_c)),fun(X_c,fun(fun(X_b,bool),X_c))),finite_fold(X_b,X_c)) = finite_fold(X_b,X_c) ).
fof(tsy_c_Finite__Set_Ofold1_res,axiom,
! [X_b] : ti(fun(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b)),finite_fold1(X_b)) = finite_fold1(X_b) ).
fof(tsy_c_Finite__Set_Ofold1Set_res,axiom,
! [X_b] : ti(fun(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),fun(X_b,bool))),finite_fold1Set(X_b)) = finite_fold1Set(X_b) ).
fof(tsy_c_Finite__Set_Ofold__graph_res,axiom,
! [X_b,X_c] : ti(fun(fun(X_b,fun(X_c,X_c)),fun(X_c,fun(fun(X_b,bool),fun(X_c,bool)))),finite_fold_graph(X_b,X_c)) = finite_fold_graph(X_b,X_c) ).
fof(tsy_c_Finite__Set_Ofolding__image__simple_res,axiom,
! [X_b,X_c] : ti(fun(fun(X_b,fun(X_b,X_b)),fun(X_b,fun(fun(X_c,X_b),fun(fun(fun(X_c,bool),X_b),bool)))),finite1357897459simple(X_b,X_c)) = finite1357897459simple(X_b,X_c) ).
fof(tsy_c_Finite__Set_Ofolding__image__simple__idem_res,axiom,
! [X_b,X_c] : ti(fun(fun(X_b,fun(X_b,X_b)),fun(X_b,fun(fun(X_c,X_b),fun(fun(fun(X_c,bool),X_b),bool)))),finite908156982e_idem(X_b,X_c)) = finite908156982e_idem(X_b,X_c) ).
fof(tsy_c_Finite__Set_Ofolding__one_res,axiom,
! [X_b] : ti(fun(fun(X_b,fun(X_b,X_b)),fun(fun(fun(X_b,bool),X_b),bool)),finite_folding_one(X_b)) = finite_folding_one(X_b) ).
fof(tsy_c_Finite__Set_Ofolding__one__idem_res,axiom,
! [X_b] : ti(fun(fun(X_b,fun(X_b,X_b)),fun(fun(fun(X_b,bool),X_b),bool)),finite2073411215e_idem(X_b)) = finite2073411215e_idem(X_b) ).
fof(tsy_c_Groups_Ominus__class_Ominus_res,axiom,
! [X_b] :
( minus(X_b)
=> ti(fun(X_b,fun(X_b,X_b)),minus_minus(X_b)) = minus_minus(X_b) ) ).
fof(tsy_c_Groups_Oone__class_Oone_res,axiom,
! [X_b] :
( one(X_b)
=> ti(X_b,one_one(X_b)) = one_one(X_b) ) ).
fof(tsy_c_Groups_Oplus__class_Oplus_0_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_Oplus__class_Oplus_1_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_Otimes__class_Otimes_0_res,axiom,
! [X_b] :
( power(X_b)
=> ti(fun(X_b,fun(X_b,X_b)),times_times(X_b)) = times_times(X_b) ) ).
fof(tsy_c_Groups_Otimes__class_Otimes_1_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_2_res,axiom,
! [X_b] :
( ab_semigroup_mult(X_b)
=> ti(fun(X_b,fun(X_b,X_b)),times_times(X_b)) = times_times(X_b) ) ).
fof(tsy_c_Groups_Ouminus__class_Ouminus_res,axiom,
! [X_c] :
( uminus(X_c)
=> ti(fun(X_c,X_c),uminus_uminus(X_c)) = uminus_uminus(X_c) ) ).
fof(tsy_c_Groups_Ozero__class_Ozero_0_res,axiom,
! [X_a] :
( ordere216010020id_add(X_a)
=> ti(X_a,zero_zero(X_a)) = zero_zero(X_a) ) ).
fof(tsy_c_Groups_Ozero__class_Ozero_1_res,axiom,
! [X_b] :
( ab_group_add(X_b)
=> ti(X_b,zero_zero(X_b)) = zero_zero(X_b) ) ).
fof(tsy_c_HOL_OThe_res,axiom,
! [X_b] : ti(fun(fun(X_b,bool),X_b),the(X_b)) = the(X_b) ).
fof(tsy_c_HOL_Oundefined_res,axiom,
! [X_a] : ti(X_a,undefined(X_a)) = undefined(X_a) ).
fof(tsy_c_Hoare__Mirabelle__lcrcocijdw_Ohoare__derivs_res,axiom,
! [X_b] : ti(fun(fun(hoare_2118899576triple(X_b),bool),fun(fun(hoare_2118899576triple(X_b),bool),bool)),hoare_1301688828derivs(X_b)) = hoare_1301688828derivs(X_b) ).
fof(tsy_c_Hoare__Mirabelle__lcrcocijdw_Otriple_Otriple_res,axiom,
! [X_b] : ti(fun(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b)))),hoare_759811442triple(X_b)) = hoare_759811442triple(X_b) ).
fof(tsy_c_Hoare__Mirabelle__lcrcocijdw_Otriple_Otriple__case_res,axiom,
! [X_c,X_b] : ti(fun(fun(fun(X_c,fun(state,bool)),fun(com,fun(fun(X_c,fun(state,bool)),X_b))),fun(hoare_2118899576triple(X_c),X_b)),hoare_225284258e_case(X_c,X_b)) = hoare_225284258e_case(X_c,X_b) ).
fof(tsy_c_Hoare__Mirabelle__lcrcocijdw_Otriple_Otriple__rec_res,axiom,
! [X_c,X_b] : ti(fun(fun(fun(X_c,fun(state,bool)),fun(com,fun(fun(X_c,fun(state,bool)),X_b))),fun(hoare_2118899576triple(X_c),X_b)),hoare_1759541758le_rec(X_c,X_b)) = hoare_1759541758le_rec(X_c,X_b) ).
fof(tsy_c_If_res,axiom,
! [X_a] : ti(fun(bool,fun(X_a,fun(X_a,X_a))),if(X_a)) = if(X_a) ).
fof(tsy_c_Nat_Osemiring__1__class_Oof__nat_res,axiom,
! [X_a] :
( semiring_1(X_a)
=> ti(fun(nat,X_a),semiring_1_of_nat(X_a)) = semiring_1_of_nat(X_a) ) ).
fof(tsy_c_Natural_Oupdate_res,axiom,
ti(fun(state,fun(vname,fun(nat,state))),update) = update ).
fof(tsy_c_Nitpick_Ofold__graph_H_res,axiom,
! [X_b,X_c] : ti(fun(fun(X_b,fun(X_c,X_c)),fun(X_c,fun(fun(X_b,bool),fun(X_c,bool)))),fold_graph(X_b,X_c)) = fold_graph(X_b,X_c) ).
fof(tsy_c_Orderings_Obot__class_Obot_res,axiom,
! [X_b] :
( bot(X_b)
=> ti(X_b,bot_bot(X_b)) = bot_bot(X_b) ) ).
fof(tsy_c_Orderings_Oord_Omax_res,axiom,
! [X_b] :
( ord(X_b)
=> ti(fun(fun(X_b,fun(X_b,bool)),fun(X_b,fun(X_b,X_b))),max(X_b)) = max(X_b) ) ).
fof(tsy_c_Orderings_Oord_Omin_res,axiom,
! [X_b] :
( ord(X_b)
=> ti(fun(fun(X_b,fun(X_b,bool)),fun(X_b,fun(X_b,X_b))),min(X_b)) = min(X_b) ) ).
fof(tsy_c_Orderings_Oord__class_Oless_res,axiom,
! [X_b] :
( ord(X_b)
=> ti(fun(X_b,fun(X_b,bool)),ord_less(X_b)) = ord_less(X_b) ) ).
fof(tsy_c_Orderings_Oord__class_Oless__eq_res,axiom,
! [X_c] :
( ord(X_c)
=> ti(fun(X_c,fun(X_c,bool)),ord_less_eq(X_c)) = ord_less_eq(X_c) ) ).
fof(tsy_c_Orderings_Oord__class_Omax_res,axiom,
! [X_b] :
( ord(X_b)
=> ti(fun(X_b,fun(X_b,X_b)),ord_max(X_b)) = ord_max(X_b) ) ).
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_Partial__Function_Oflat__lub_res,axiom,
! [X_b] : ti(fun(X_b,fun(fun(X_b,bool),X_b)),partial_flat_lub(X_b)) = partial_flat_lub(X_b) ).
fof(tsy_c_Power_Opower_Opower_res,axiom,
! [X_b] : ti(fun(X_b,fun(fun(X_b,fun(X_b,X_b)),fun(X_b,fun(nat,X_b)))),power_1(X_b)) = power_1(X_b) ).
fof(tsy_c_Power_Opower__class_Opower_res,axiom,
! [X_b] :
( power(X_b)
=> ti(fun(X_b,fun(nat,X_b)),power_power(X_b)) = power_power(X_b) ) ).
fof(tsy_c_SetInterval_Oord__class_OatLeastAtMost_res,axiom,
! [X_b] :
( order(X_b)
=> ti(fun(X_b,fun(X_b,fun(X_b,bool))),ord_atLeastAtMost(X_b)) = ord_atLeastAtMost(X_b) ) ).
fof(tsy_c_SetInterval_Oord__class_OgreaterThan_res,axiom,
! [X_b] :
( ord(X_b)
=> ti(fun(X_b,fun(X_b,bool)),ord_greaterThan(X_b)) = ord_greaterThan(X_b) ) ).
fof(tsy_c_SetInterval_Oord__class_OgreaterThanAtMost_res,axiom,
ti(fun(nat,fun(nat,fun(nat,bool))),ord_gr1471790799AtMost(nat)) = ord_gr1471790799AtMost(nat) ).
fof(tsy_c_SetInterval_Oord__class_OlessThan_res,axiom,
! [X_b] :
( ord(X_b)
=> ti(fun(X_b,fun(X_b,bool)),ord_lessThan(X_b)) = ord_lessThan(X_b) ) ).
fof(tsy_c_Set_OCollect_res,axiom,
! [X_b] : ti(fun(fun(X_b,bool),fun(X_b,bool)),collect(X_b)) = collect(X_b) ).
fof(tsy_c_Set_Oimage_res,axiom,
! [X_b,X_c] : ti(fun(fun(X_b,X_c),fun(fun(X_b,bool),fun(X_c,bool))),image(X_b,X_c)) = image(X_b,X_c) ).
fof(tsy_c_Set_Oinsert_res,axiom,
! [X_b] : ti(fun(X_b,fun(fun(X_b,bool),fun(X_b,bool))),insert(X_b)) = insert(X_b) ).
fof(tsy_c_Set_Othe__elem_res,axiom,
! [X_b] : ti(fun(fun(X_b,bool),X_b),the_elem(X_b)) = the_elem(X_b) ).
fof(tsy_c_fEx_res,axiom,
! [X_a] : ti(fun(fun(X_a,bool),bool),fEx(X_a)) = fEx(X_a) ).
fof(tsy_c_fFalse_res,hypothesis,
ti(bool,fFalse) = fFalse ).
fof(tsy_c_fNot_res,hypothesis,
ti(fun(bool,bool),fNot) = fNot ).
fof(tsy_c_fTrue_res,axiom,
ti(bool,fTrue) = fTrue ).
fof(tsy_c_fconj_res,hypothesis,
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_b] : ti(fun(X_b,fun(fun(X_b,bool),bool)),member(X_b)) = member(X_b) ).
fof(tsy_v_G_res,hypothesis,
ti(fun(hoare_2118899576triple(x_a),bool),g) = g ).
fof(tsy_v_P_res,hypothesis,
ti(fun(x_a,fun(state,bool)),p) = p ).
fof(tsy_v_b_res,hypothesis,
ti(fun(state,bool),b) = b ).
fof(tsy_v_c_res,hypothesis,
ti(com,c) = c ).
%----Relevant facts (997)
fof(fact_0_empty,axiom,
! [X_b,Ga] : hBOOL(hAPP(fun(hoare_2118899576triple(X_b),bool),bool,hAPP(fun(hoare_2118899576triple(X_b),bool),fun(fun(hoare_2118899576triple(X_b),bool),bool),hoare_1301688828derivs(X_b),Ga),bot_bot(fun(hoare_2118899576triple(X_b),bool)))) ).
fof(fact_1_triple_Oinject,axiom,
! [X_b,Fun1_1,Com_1,Fun2_1,Fun1_2,Com_2,Fun2_2] :
( hAPP(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b),hAPP(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b)),hAPP(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b))),hoare_759811442triple(X_b),Fun1_1),Com_1),Fun2_1) = hAPP(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b),hAPP(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b)),hAPP(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b))),hoare_759811442triple(X_b),Fun1_2),Com_2),Fun2_2)
<=> ( Fun1_1 = Fun1_2
& Com_1 = Com_2
& Fun2_1 = Fun2_2 ) ) ).
fof(fact_2_cut,axiom,
! [X_b,Ga,G_1,Ts] :
( hBOOL(hAPP(fun(hoare_2118899576triple(X_b),bool),bool,hAPP(fun(hoare_2118899576triple(X_b),bool),fun(fun(hoare_2118899576triple(X_b),bool),bool),hoare_1301688828derivs(X_b),G_1),Ts))
=> ( hBOOL(hAPP(fun(hoare_2118899576triple(X_b),bool),bool,hAPP(fun(hoare_2118899576triple(X_b),bool),fun(fun(hoare_2118899576triple(X_b),bool),bool),hoare_1301688828derivs(X_b),Ga),G_1))
=> hBOOL(hAPP(fun(hoare_2118899576triple(X_b),bool),bool,hAPP(fun(hoare_2118899576triple(X_b),bool),fun(fun(hoare_2118899576triple(X_b),bool),bool),hoare_1301688828derivs(X_b),Ga),Ts)) ) ) ).
fof(fact_3_hoare__derivs_Oinsert,axiom,
! [X_b,Ts,Ga,T_5] :
( hBOOL(hAPP(fun(hoare_2118899576triple(X_b),bool),bool,hAPP(fun(hoare_2118899576triple(X_b),bool),fun(fun(hoare_2118899576triple(X_b),bool),bool),hoare_1301688828derivs(X_b),Ga),hAPP(fun(hoare_2118899576triple(X_b),bool),fun(hoare_2118899576triple(X_b),bool),hAPP(hoare_2118899576triple(X_b),fun(fun(hoare_2118899576triple(X_b),bool),fun(hoare_2118899576triple(X_b),bool)),insert(hoare_2118899576triple(X_b)),T_5),bot_bot(fun(hoare_2118899576triple(X_b),bool)))))
=> ( hBOOL(hAPP(fun(hoare_2118899576triple(X_b),bool),bool,hAPP(fun(hoare_2118899576triple(X_b),bool),fun(fun(hoare_2118899576triple(X_b),bool),bool),hoare_1301688828derivs(X_b),Ga),Ts))
=> hBOOL(hAPP(fun(hoare_2118899576triple(X_b),bool),bool,hAPP(fun(hoare_2118899576triple(X_b),bool),fun(fun(hoare_2118899576triple(X_b),bool),bool),hoare_1301688828derivs(X_b),Ga),hAPP(fun(hoare_2118899576triple(X_b),bool),fun(hoare_2118899576triple(X_b),bool),hAPP(hoare_2118899576triple(X_b),fun(fun(hoare_2118899576triple(X_b),bool),fun(hoare_2118899576triple(X_b),bool)),insert(hoare_2118899576triple(X_b)),T_5),Ts))) ) ) ).
fof(fact_4_constant,axiom,
! [X_b,Ga,Pa,Ca,Q_3,C_1] :
( ( hBOOL(C_1)
=> hBOOL(hAPP(fun(hoare_2118899576triple(X_b),bool),bool,hAPP(fun(hoare_2118899576triple(X_b),bool),fun(fun(hoare_2118899576triple(X_b),bool),bool),hoare_1301688828derivs(X_b),Ga),hAPP(fun(hoare_2118899576triple(X_b),bool),fun(hoare_2118899576triple(X_b),bool),hAPP(hoare_2118899576triple(X_b),fun(fun(hoare_2118899576triple(X_b),bool),fun(hoare_2118899576triple(X_b),bool)),insert(hoare_2118899576triple(X_b)),hAPP(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b),hAPP(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b)),hAPP(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b))),hoare_759811442triple(X_b),Pa),Ca),Q_3)),bot_bot(fun(hoare_2118899576triple(X_b),bool))))) )
=> hBOOL(hAPP(fun(hoare_2118899576triple(X_b),bool),bool,hAPP(fun(hoare_2118899576triple(X_b),bool),fun(fun(hoare_2118899576triple(X_b),bool),bool),hoare_1301688828derivs(X_b),Ga),hAPP(fun(hoare_2118899576triple(X_b),bool),fun(hoare_2118899576triple(X_b),bool),hAPP(hoare_2118899576triple(X_b),fun(fun(hoare_2118899576triple(X_b),bool),fun(hoare_2118899576triple(X_b),bool)),insert(hoare_2118899576triple(X_b)),hAPP(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b),hAPP(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b)),hAPP(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b))),hoare_759811442triple(X_b),hAPP(bool,fun(X_b,fun(state,bool)),hAPP(fun(X_b,fun(bool,fun(state,bool))),fun(bool,fun(X_b,fun(state,bool))),combc(X_b,bool,fun(state,bool)),hAPP(fun(X_b,fun(state,fun(bool,bool))),fun(X_b,fun(bool,fun(state,bool))),hAPP(fun(fun(state,fun(bool,bool)),fun(bool,fun(state,bool))),fun(fun(X_b,fun(state,fun(bool,bool))),fun(X_b,fun(bool,fun(state,bool)))),combb(fun(state,fun(bool,bool)),fun(bool,fun(state,bool)),X_b),combc(state,bool,bool)),hAPP(fun(X_b,fun(state,bool)),fun(X_b,fun(state,fun(bool,bool))),hAPP(fun(fun(state,bool),fun(state,fun(bool,bool))),fun(fun(X_b,fun(state,bool)),fun(X_b,fun(state,fun(bool,bool)))),combb(fun(state,bool),fun(state,fun(bool,bool)),X_b),hAPP(fun(bool,fun(bool,bool)),fun(fun(state,bool),fun(state,fun(bool,bool))),combb(bool,fun(bool,bool),state),fconj)),Pa))),C_1)),Ca),Q_3)),bot_bot(fun(hoare_2118899576triple(X_b),bool))))) ) ).
fof(fact_5_escape,axiom,
! [X_b,Ga,Ca,Q_3,Pa] :
( ! [Z_4,S_3] :
( hBOOL(hAPP(state,bool,hAPP(X_b,fun(state,bool),Pa,Z_4),S_3))
=> hBOOL(hAPP(fun(hoare_2118899576triple(X_b),bool),bool,hAPP(fun(hoare_2118899576triple(X_b),bool),fun(fun(hoare_2118899576triple(X_b),bool),bool),hoare_1301688828derivs(X_b),Ga),hAPP(fun(hoare_2118899576triple(X_b),bool),fun(hoare_2118899576triple(X_b),bool),hAPP(hoare_2118899576triple(X_b),fun(fun(hoare_2118899576triple(X_b),bool),fun(hoare_2118899576triple(X_b),bool)),insert(hoare_2118899576triple(X_b)),hAPP(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b),hAPP(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b)),hAPP(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b))),hoare_759811442triple(X_b),hAPP(fun(state,bool),fun(X_b,fun(state,bool)),combk(fun(state,bool),X_b),hAPP(state,fun(state,bool),hAPP(fun(state,fun(state,bool)),fun(state,fun(state,bool)),combc(state,state,bool),fequal(state)),S_3))),Ca),hAPP(fun(state,bool),fun(X_b,fun(state,bool)),combk(fun(state,bool),X_b),hAPP(X_b,fun(state,bool),Q_3,Z_4)))),bot_bot(fun(hoare_2118899576triple(X_b),bool))))) )
=> hBOOL(hAPP(fun(hoare_2118899576triple(X_b),bool),bool,hAPP(fun(hoare_2118899576triple(X_b),bool),fun(fun(hoare_2118899576triple(X_b),bool),bool),hoare_1301688828derivs(X_b),Ga),hAPP(fun(hoare_2118899576triple(X_b),bool),fun(hoare_2118899576triple(X_b),bool),hAPP(hoare_2118899576triple(X_b),fun(fun(hoare_2118899576triple(X_b),bool),fun(hoare_2118899576triple(X_b),bool)),insert(hoare_2118899576triple(X_b)),hAPP(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b),hAPP(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b)),hAPP(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b))),hoare_759811442triple(X_b),Pa),Ca),Q_3)),bot_bot(fun(hoare_2118899576triple(X_b),bool))))) ) ).
fof(fact_6_conseq2,axiom,
! [X_b,Q_3,Ga,Pa,Ca,Q_5] :
( hBOOL(hAPP(fun(hoare_2118899576triple(X_b),bool),bool,hAPP(fun(hoare_2118899576triple(X_b),bool),fun(fun(hoare_2118899576triple(X_b),bool),bool),hoare_1301688828derivs(X_b),Ga),hAPP(fun(hoare_2118899576triple(X_b),bool),fun(hoare_2118899576triple(X_b),bool),hAPP(hoare_2118899576triple(X_b),fun(fun(hoare_2118899576triple(X_b),bool),fun(hoare_2118899576triple(X_b),bool)),insert(hoare_2118899576triple(X_b)),hAPP(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b),hAPP(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b)),hAPP(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b))),hoare_759811442triple(X_b),Pa),Ca),Q_5)),bot_bot(fun(hoare_2118899576triple(X_b),bool)))))
=> ( ! [Z_4,S_3] :
( hBOOL(hAPP(state,bool,hAPP(X_b,fun(state,bool),Q_5,Z_4),S_3))
=> hBOOL(hAPP(state,bool,hAPP(X_b,fun(state,bool),Q_3,Z_4),S_3)) )
=> hBOOL(hAPP(fun(hoare_2118899576triple(X_b),bool),bool,hAPP(fun(hoare_2118899576triple(X_b),bool),fun(fun(hoare_2118899576triple(X_b),bool),bool),hoare_1301688828derivs(X_b),Ga),hAPP(fun(hoare_2118899576triple(X_b),bool),fun(hoare_2118899576triple(X_b),bool),hAPP(hoare_2118899576triple(X_b),fun(fun(hoare_2118899576triple(X_b),bool),fun(hoare_2118899576triple(X_b),bool)),insert(hoare_2118899576triple(X_b)),hAPP(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b),hAPP(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b)),hAPP(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b))),hoare_759811442triple(X_b),Pa),Ca),Q_3)),bot_bot(fun(hoare_2118899576triple(X_b),bool))))) ) ) ).
fof(fact_7_conseq1,axiom,
! [X_b,Pa,Ga,P_4,Ca,Q_3] :
( hBOOL(hAPP(fun(hoare_2118899576triple(X_b),bool),bool,hAPP(fun(hoare_2118899576triple(X_b),bool),fun(fun(hoare_2118899576triple(X_b),bool),bool),hoare_1301688828derivs(X_b),Ga),hAPP(fun(hoare_2118899576triple(X_b),bool),fun(hoare_2118899576triple(X_b),bool),hAPP(hoare_2118899576triple(X_b),fun(fun(hoare_2118899576triple(X_b),bool),fun(hoare_2118899576triple(X_b),bool)),insert(hoare_2118899576triple(X_b)),hAPP(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b),hAPP(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b)),hAPP(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b))),hoare_759811442triple(X_b),P_4),Ca),Q_3)),bot_bot(fun(hoare_2118899576triple(X_b),bool)))))
=> ( ! [Z_4,S_3] :
( hBOOL(hAPP(state,bool,hAPP(X_b,fun(state,bool),Pa,Z_4),S_3))
=> hBOOL(hAPP(state,bool,hAPP(X_b,fun(state,bool),P_4,Z_4),S_3)) )
=> hBOOL(hAPP(fun(hoare_2118899576triple(X_b),bool),bool,hAPP(fun(hoare_2118899576triple(X_b),bool),fun(fun(hoare_2118899576triple(X_b),bool),bool),hoare_1301688828derivs(X_b),Ga),hAPP(fun(hoare_2118899576triple(X_b),bool),fun(hoare_2118899576triple(X_b),bool),hAPP(hoare_2118899576triple(X_b),fun(fun(hoare_2118899576triple(X_b),bool),fun(hoare_2118899576triple(X_b),bool)),insert(hoare_2118899576triple(X_b)),hAPP(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b),hAPP(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b)),hAPP(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b))),hoare_759811442triple(X_b),Pa),Ca),Q_3)),bot_bot(fun(hoare_2118899576triple(X_b),bool))))) ) ) ).
fof(fact_8_conseq12,axiom,
! [X_b,Q_3,Pa,Ga,P_4,Ca,Q_5] :
( hBOOL(hAPP(fun(hoare_2118899576triple(X_b),bool),bool,hAPP(fun(hoare_2118899576triple(X_b),bool),fun(fun(hoare_2118899576triple(X_b),bool),bool),hoare_1301688828derivs(X_b),Ga),hAPP(fun(hoare_2118899576triple(X_b),bool),fun(hoare_2118899576triple(X_b),bool),hAPP(hoare_2118899576triple(X_b),fun(fun(hoare_2118899576triple(X_b),bool),fun(hoare_2118899576triple(X_b),bool)),insert(hoare_2118899576triple(X_b)),hAPP(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b),hAPP(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b)),hAPP(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b))),hoare_759811442triple(X_b),P_4),Ca),Q_5)),bot_bot(fun(hoare_2118899576triple(X_b),bool)))))
=> ( ! [Z_4,S_3] :
( hBOOL(hAPP(state,bool,hAPP(X_b,fun(state,bool),Pa,Z_4),S_3))
=> ! [S_4] :
( ! [Z_5] :
( hBOOL(hAPP(state,bool,hAPP(X_b,fun(state,bool),P_4,Z_5),S_3))
=> hBOOL(hAPP(state,bool,hAPP(X_b,fun(state,bool),Q_5,Z_5),S_4)) )
=> hBOOL(hAPP(state,bool,hAPP(X_b,fun(state,bool),Q_3,Z_4),S_4)) ) )
=> hBOOL(hAPP(fun(hoare_2118899576triple(X_b),bool),bool,hAPP(fun(hoare_2118899576triple(X_b),bool),fun(fun(hoare_2118899576triple(X_b),bool),bool),hoare_1301688828derivs(X_b),Ga),hAPP(fun(hoare_2118899576triple(X_b),bool),fun(hoare_2118899576triple(X_b),bool),hAPP(hoare_2118899576triple(X_b),fun(fun(hoare_2118899576triple(X_b),bool),fun(hoare_2118899576triple(X_b),bool)),insert(hoare_2118899576triple(X_b)),hAPP(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b),hAPP(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b)),hAPP(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b))),hoare_759811442triple(X_b),Pa),Ca),Q_3)),bot_bot(fun(hoare_2118899576triple(X_b),bool))))) ) ) ).
fof(fact_9_insertE,axiom,
! [X_b,A_2,Ba,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),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),Ba),A_3)))
=> ( ti(X_b,A_2) != ti(X_b,Ba)
=> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),A_2),A_3)) ) ) ).
fof(fact_10_insertCI,axiom,
! [X_b,Ba,A_2,B_2] :
( ( ~ hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),A_2),B_2))
=> ti(X_b,A_2) = ti(X_b,Ba) )
=> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),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),Ba),B_2))) ) ).
fof(fact_11_emptyE,axiom,
! [X_b,A_2] : ~ hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),A_2),bot_bot(fun(X_b,bool)))) ).
fof(fact_12_singleton__conv2,axiom,
! [X_b,A_2] : hAPP(fun(X_b,bool),fun(X_b,bool),collect(X_b),hAPP(X_b,fun(X_b,bool),fequal(X_b),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),A_2),bot_bot(fun(X_b,bool))) ).
fof(fact_13_singleton__conv,axiom,
! [X_b,A_2] : hAPP(fun(X_b,bool),fun(X_b,bool),collect(X_b),hAPP(X_b,fun(X_b,bool),hAPP(fun(X_b,fun(X_b,bool)),fun(X_b,fun(X_b,bool)),combc(X_b,X_b,bool),fequal(X_b)),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),A_2),bot_bot(fun(X_b,bool))) ).
fof(fact_14_Collect__conv__if2,axiom,
! [X_b,Pa,A_2] :
( ( hBOOL(hAPP(X_b,bool,Pa,A_2))
=> hAPP(fun(X_b,bool),fun(X_b,bool),collect(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,bool)),combs(X_b,bool,bool),hAPP(fun(X_b,bool),fun(X_b,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,fun(bool,bool))),combb(bool,fun(bool,bool),X_b),fconj),hAPP(X_b,fun(X_b,bool),fequal(X_b),A_2))),Pa)) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_2),bot_bot(fun(X_b,bool))) )
& ( ~ hBOOL(hAPP(X_b,bool,Pa,A_2))
=> hAPP(fun(X_b,bool),fun(X_b,bool),collect(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,bool)),combs(X_b,bool,bool),hAPP(fun(X_b,bool),fun(X_b,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,fun(bool,bool))),combb(bool,fun(bool,bool),X_b),fconj),hAPP(X_b,fun(X_b,bool),fequal(X_b),A_2))),Pa)) = bot_bot(fun(X_b,bool)) ) ) ).
fof(fact_15_Collect__conv__if,axiom,
! [X_b,Pa,A_2] :
( ( hBOOL(hAPP(X_b,bool,Pa,A_2))
=> hAPP(fun(X_b,bool),fun(X_b,bool),collect(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,bool)),combs(X_b,bool,bool),hAPP(fun(X_b,bool),fun(X_b,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,fun(bool,bool))),combb(bool,fun(bool,bool),X_b),fconj),hAPP(X_b,fun(X_b,bool),hAPP(fun(X_b,fun(X_b,bool)),fun(X_b,fun(X_b,bool)),combc(X_b,X_b,bool),fequal(X_b)),A_2))),Pa)) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_2),bot_bot(fun(X_b,bool))) )
& ( ~ hBOOL(hAPP(X_b,bool,Pa,A_2))
=> hAPP(fun(X_b,bool),fun(X_b,bool),collect(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,bool)),combs(X_b,bool,bool),hAPP(fun(X_b,bool),fun(X_b,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,fun(bool,bool))),combb(bool,fun(bool,bool),X_b),fconj),hAPP(X_b,fun(X_b,bool),hAPP(fun(X_b,fun(X_b,bool)),fun(X_b,fun(X_b,bool)),combc(X_b,X_b,bool),fequal(X_b)),A_2))),Pa)) = bot_bot(fun(X_b,bool)) ) ) ).
fof(fact_16_triple_Orecs,axiom,
! [X_c,X_b,F1,Fun1_1,Com_1,Fun2_1] : hAPP(hoare_2118899576triple(X_c),X_b,hAPP(fun(fun(X_c,fun(state,bool)),fun(com,fun(fun(X_c,fun(state,bool)),X_b))),fun(hoare_2118899576triple(X_c),X_b),hoare_1759541758le_rec(X_c,X_b),F1),hAPP(fun(X_c,fun(state,bool)),hoare_2118899576triple(X_c),hAPP(com,fun(fun(X_c,fun(state,bool)),hoare_2118899576triple(X_c)),hAPP(fun(X_c,fun(state,bool)),fun(com,fun(fun(X_c,fun(state,bool)),hoare_2118899576triple(X_c))),hoare_759811442triple(X_c),Fun1_1),Com_1),Fun2_1)) = hAPP(fun(X_c,fun(state,bool)),X_b,hAPP(com,fun(fun(X_c,fun(state,bool)),X_b),hAPP(fun(X_c,fun(state,bool)),fun(com,fun(fun(X_c,fun(state,bool)),X_b)),F1,Fun1_1),Com_1),Fun2_1) ).
fof(fact_17_equals0D,axiom,
! [X_b,A_2,A_3] :
( ti(fun(X_b,bool),A_3) = bot_bot(fun(X_b,bool))
=> ~ hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),A_2),A_3)) ) ).
fof(fact_18_Collect__empty__eq,axiom,
! [X_b,Pa] :
( hAPP(fun(X_b,bool),fun(X_b,bool),collect(X_b),Pa) = bot_bot(fun(X_b,bool))
<=> ! [X_2] : ~ hBOOL(hAPP(X_b,bool,Pa,X_2)) ) ).
fof(fact_19_empty__iff,axiom,
! [X_b,Ca] : ~ hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),Ca),bot_bot(fun(X_b,bool)))) ).
fof(fact_20_empty__Collect__eq,axiom,
! [X_b,Pa] :
( bot_bot(fun(X_b,bool)) = hAPP(fun(X_b,bool),fun(X_b,bool),collect(X_b),Pa)
<=> ! [X_2] : ~ hBOOL(hAPP(X_b,bool,Pa,X_2)) ) ).
fof(fact_21_ex__in__conv,axiom,
! [X_b,A_3] :
( ? [X_2] : hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_2),A_3))
<=> ti(fun(X_b,bool),A_3) != bot_bot(fun(X_b,bool)) ) ).
fof(fact_22_all__not__in__conv,axiom,
! [X_b,A_3] :
( ! [X_2] : ~ hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_2),A_3))
<=> ti(fun(X_b,bool),A_3) = bot_bot(fun(X_b,bool)) ) ).
fof(fact_23_empty__def,axiom,
! [X_b] : bot_bot(fun(X_b,bool)) = hAPP(fun(X_b,bool),fun(X_b,bool),collect(X_b),hAPP(bool,fun(X_b,bool),combk(bool,X_b),fFalse)) ).
fof(fact_24_insert__absorb,axiom,
! [X_b,A_2,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),A_2),A_3))
=> hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_2),A_3) = ti(fun(X_b,bool),A_3) ) ).
fof(fact_25_insertI2,axiom,
! [X_b,Ba,A_2,B_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),A_2),B_2))
=> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),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),Ba),B_2))) ) ).
fof(fact_26_insert__ident,axiom,
! [X_b,B_2,X_1,A_3] :
( ~ hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),A_3))
=> ( ~ hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),B_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_1),A_3) = 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_1),B_2)
<=> ti(fun(X_b,bool),A_3) = ti(fun(X_b,bool),B_2) ) ) ) ).
fof(fact_27_insert__code,axiom,
! [X_b,Y_2,A_3,X_1] :
( hBOOL(hAPP(X_b,bool,hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),Y_2),A_3),X_1))
<=> ( ti(X_b,Y_2) = ti(X_b,X_1)
| hBOOL(hAPP(X_b,bool,A_3,X_1)) ) ) ).
fof(fact_28_insert__iff,axiom,
! [X_b,A_2,Ba,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),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),Ba),A_3)))
<=> ( ti(X_b,A_2) = ti(X_b,Ba)
| hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),A_2),A_3)) ) ) ).
fof(fact_29_insert__commute,axiom,
! [X_b,X_1,Y_2,A_3] : 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_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),Y_2),A_3)) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),Y_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_1),A_3)) ).
fof(fact_30_insert__absorb2,axiom,
! [X_b,X_1,A_3] : 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_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_1),A_3)) = 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_1),A_3) ).
fof(fact_31_insert__Collect,axiom,
! [X_b,A_2,Pa] : hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_2),hAPP(fun(X_b,bool),fun(X_b,bool),collect(X_b),Pa)) = hAPP(fun(X_b,bool),fun(X_b,bool),collect(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,bool)),combs(X_b,bool,bool),hAPP(fun(X_b,bool),fun(X_b,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,fun(bool,bool))),combb(bool,fun(bool,bool),X_b),fimplies),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(bool,bool),fun(fun(X_b,bool),fun(X_b,bool)),combb(bool,bool,X_b),fNot),hAPP(X_b,fun(X_b,bool),hAPP(fun(X_b,fun(X_b,bool)),fun(X_b,fun(X_b,bool)),combc(X_b,X_b,bool),fequal(X_b)),A_2)))),Pa)) ).
fof(fact_32_insert__compr,axiom,
! [X_b,A_2,B_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),A_2),B_2) = hAPP(fun(X_b,bool),fun(X_b,bool),collect(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,bool)),combs(X_b,bool,bool),hAPP(fun(X_b,bool),fun(X_b,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,fun(bool,bool))),combb(bool,fun(bool,bool),X_b),fdisj),hAPP(X_b,fun(X_b,bool),hAPP(fun(X_b,fun(X_b,bool)),fun(X_b,fun(X_b,bool)),combc(X_b,X_b,bool),fequal(X_b)),A_2))),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(fun(X_b,bool),bool)),fun(fun(X_b,bool),fun(X_b,bool)),combc(X_b,fun(X_b,bool),bool),member(X_b)),B_2))) ).
fof(fact_33_insertI1,axiom,
! [X_b,A_2,B_2] : hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),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),A_2),B_2))) ).
fof(fact_34_insert__compr__raw,axiom,
! [X_b,X_2,Xa] : 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),Xa) = hAPP(fun(X_b,bool),fun(X_b,bool),collect(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,bool)),combs(X_b,bool,bool),hAPP(fun(X_b,bool),fun(X_b,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,fun(bool,bool))),combb(bool,fun(bool,bool),X_b),fdisj),hAPP(X_b,fun(X_b,bool),hAPP(fun(X_b,fun(X_b,bool)),fun(X_b,fun(X_b,bool)),combc(X_b,X_b,bool),fequal(X_b)),X_2))),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(fun(X_b,bool),bool)),fun(fun(X_b,bool),fun(X_b,bool)),combc(X_b,fun(X_b,bool),bool),member(X_b)),Xa))) ).
fof(fact_35_singleton__inject,axiom,
! [X_b,A_2,Ba] :
( hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_2),bot_bot(fun(X_b,bool))) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),Ba),bot_bot(fun(X_b,bool)))
=> ti(X_b,A_2) = ti(X_b,Ba) ) ).
fof(fact_36_singletonE,axiom,
! [X_b,Ba,A_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),Ba),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_2),bot_bot(fun(X_b,bool)))))
=> ti(X_b,Ba) = ti(X_b,A_2) ) ).
fof(fact_37_doubleton__eq__iff,axiom,
! [X_b,A_2,Ba,Ca,D] :
( hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),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),Ba),bot_bot(fun(X_b,bool)))) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),Ca),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),D),bot_bot(fun(X_b,bool))))
<=> ( ( ti(X_b,A_2) = ti(X_b,Ca)
& ti(X_b,Ba) = ti(X_b,D) )
| ( ti(X_b,A_2) = ti(X_b,D)
& ti(X_b,Ba) = ti(X_b,Ca) ) ) ) ).
fof(fact_38_singleton__iff,axiom,
! [X_b,Ba,A_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),Ba),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_2),bot_bot(fun(X_b,bool)))))
<=> ti(X_b,Ba) = ti(X_b,A_2) ) ).
fof(fact_39_insert__not__empty,axiom,
! [X_b,A_2,A_3] : hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_2),A_3) != bot_bot(fun(X_b,bool)) ).
fof(fact_40_empty__not__insert,axiom,
! [X_b,A_2,A_3] : bot_bot(fun(X_b,bool)) != hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_2),A_3) ).
fof(fact_41_the__elem__eq,axiom,
! [X_b,X_1] : hAPP(fun(X_b,bool),X_b,the_elem(X_b),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_1),bot_bot(fun(X_b,bool)))) = ti(X_b,X_1) ).
fof(fact_42_triple_Osimps_I2_J,axiom,
! [X_c,X_b,F1,Fun1_1,Com_1,Fun2_1] : hAPP(hoare_2118899576triple(X_c),X_b,hAPP(fun(fun(X_c,fun(state,bool)),fun(com,fun(fun(X_c,fun(state,bool)),X_b))),fun(hoare_2118899576triple(X_c),X_b),hoare_225284258e_case(X_c,X_b),F1),hAPP(fun(X_c,fun(state,bool)),hoare_2118899576triple(X_c),hAPP(com,fun(fun(X_c,fun(state,bool)),hoare_2118899576triple(X_c)),hAPP(fun(X_c,fun(state,bool)),fun(com,fun(fun(X_c,fun(state,bool)),hoare_2118899576triple(X_c))),hoare_759811442triple(X_c),Fun1_1),Com_1),Fun2_1)) = hAPP(fun(X_c,fun(state,bool)),X_b,hAPP(com,fun(fun(X_c,fun(state,bool)),X_b),hAPP(fun(X_c,fun(state,bool)),fun(com,fun(fun(X_c,fun(state,bool)),X_b)),F1,Fun1_1),Com_1),Fun2_1) ).
fof(fact_43_bot__apply,axiom,
! [X_c,X_b] :
( bot(X_b)
=> ! [X_1] : hAPP(X_c,X_b,bot_bot(fun(X_c,X_b)),X_1) = bot_bot(X_b) ) ).
fof(fact_44_bot__fun__def,axiom,
! [X_b,X_c] :
( bot(X_c)
=> ! [X_2] : hAPP(X_b,X_c,bot_bot(fun(X_b,X_c)),X_2) = bot_bot(X_c) ) ).
fof(fact_45_hoare__derivs_OSkip,axiom,
! [X_b,Ga,Pa] : hBOOL(hAPP(fun(hoare_2118899576triple(X_b),bool),bool,hAPP(fun(hoare_2118899576triple(X_b),bool),fun(fun(hoare_2118899576triple(X_b),bool),bool),hoare_1301688828derivs(X_b),Ga),hAPP(fun(hoare_2118899576triple(X_b),bool),fun(hoare_2118899576triple(X_b),bool),hAPP(hoare_2118899576triple(X_b),fun(fun(hoare_2118899576triple(X_b),bool),fun(hoare_2118899576triple(X_b),bool)),insert(hoare_2118899576triple(X_b)),hAPP(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b),hAPP(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b)),hAPP(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b))),hoare_759811442triple(X_b),Pa),skip),Pa)),bot_bot(fun(hoare_2118899576triple(X_b),bool))))) ).
fof(fact_46_Comp,axiom,
! [X_b,D,R_1,Ga,Pa,Ca,Q_3] :
( hBOOL(hAPP(fun(hoare_2118899576triple(X_b),bool),bool,hAPP(fun(hoare_2118899576triple(X_b),bool),fun(fun(hoare_2118899576triple(X_b),bool),bool),hoare_1301688828derivs(X_b),Ga),hAPP(fun(hoare_2118899576triple(X_b),bool),fun(hoare_2118899576triple(X_b),bool),hAPP(hoare_2118899576triple(X_b),fun(fun(hoare_2118899576triple(X_b),bool),fun(hoare_2118899576triple(X_b),bool)),insert(hoare_2118899576triple(X_b)),hAPP(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b),hAPP(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b)),hAPP(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b))),hoare_759811442triple(X_b),Pa),Ca),Q_3)),bot_bot(fun(hoare_2118899576triple(X_b),bool)))))
=> ( hBOOL(hAPP(fun(hoare_2118899576triple(X_b),bool),bool,hAPP(fun(hoare_2118899576triple(X_b),bool),fun(fun(hoare_2118899576triple(X_b),bool),bool),hoare_1301688828derivs(X_b),Ga),hAPP(fun(hoare_2118899576triple(X_b),bool),fun(hoare_2118899576triple(X_b),bool),hAPP(hoare_2118899576triple(X_b),fun(fun(hoare_2118899576triple(X_b),bool),fun(hoare_2118899576triple(X_b),bool)),insert(hoare_2118899576triple(X_b)),hAPP(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b),hAPP(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b)),hAPP(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b))),hoare_759811442triple(X_b),Q_3),D),R_1)),bot_bot(fun(hoare_2118899576triple(X_b),bool)))))
=> hBOOL(hAPP(fun(hoare_2118899576triple(X_b),bool),bool,hAPP(fun(hoare_2118899576triple(X_b),bool),fun(fun(hoare_2118899576triple(X_b),bool),bool),hoare_1301688828derivs(X_b),Ga),hAPP(fun(hoare_2118899576triple(X_b),bool),fun(hoare_2118899576triple(X_b),bool),hAPP(hoare_2118899576triple(X_b),fun(fun(hoare_2118899576triple(X_b),bool),fun(hoare_2118899576triple(X_b),bool)),insert(hoare_2118899576triple(X_b)),hAPP(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b),hAPP(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b)),hAPP(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b))),hoare_759811442triple(X_b),Pa),hAPP(com,com,hAPP(com,fun(com,com),semi,Ca),D)),R_1)),bot_bot(fun(hoare_2118899576triple(X_b),bool))))) ) ) ).
fof(fact_47_triple_Oexhaust,axiom,
! [X_b,Y_2] :
~ ! [Fun1,Com,Fun2] : Y_2 != hAPP(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b),hAPP(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b)),hAPP(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b))),hoare_759811442triple(X_b),Fun1),Com),Fun2) ).
fof(fact_48_Set_Oset__insert,axiom,
! [X_b,X_1,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),A_3))
=> ~ ! [B_3] :
( ti(fun(X_b,bool),A_3) = 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_1),B_3)
=> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),B_3)) ) ) ).
fof(fact_49_mk__disjoint__insert,axiom,
! [X_b,A_2,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),A_2),A_3))
=> ? [B_3] :
( ti(fun(X_b,bool),A_3) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_2),B_3)
& ~ hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),A_2),B_3)) ) ) ).
fof(fact_50_equals0I,axiom,
! [X_b,A_3] :
( ! [Y_1] : ~ hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),Y_1),A_3))
=> ti(fun(X_b,bool),A_3) = bot_bot(fun(X_b,bool)) ) ).
fof(fact_51_conseq,axiom,
! [X_b,Q_3,Ga,Ca,Pa] :
( ! [Z_4,S_3] :
( hBOOL(hAPP(state,bool,hAPP(X_b,fun(state,bool),Pa,Z_4),S_3))
=> ? [P_3,Q_4] :
( hBOOL(hAPP(fun(hoare_2118899576triple(X_b),bool),bool,hAPP(fun(hoare_2118899576triple(X_b),bool),fun(fun(hoare_2118899576triple(X_b),bool),bool),hoare_1301688828derivs(X_b),Ga),hAPP(fun(hoare_2118899576triple(X_b),bool),fun(hoare_2118899576triple(X_b),bool),hAPP(hoare_2118899576triple(X_b),fun(fun(hoare_2118899576triple(X_b),bool),fun(hoare_2118899576triple(X_b),bool)),insert(hoare_2118899576triple(X_b)),hAPP(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b),hAPP(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b)),hAPP(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b))),hoare_759811442triple(X_b),P_3),Ca),Q_4)),bot_bot(fun(hoare_2118899576triple(X_b),bool)))))
& ! [S_4] :
( ! [Z_5] :
( hBOOL(hAPP(state,bool,hAPP(X_b,fun(state,bool),P_3,Z_5),S_3))
=> hBOOL(hAPP(state,bool,hAPP(X_b,fun(state,bool),Q_4,Z_5),S_4)) )
=> hBOOL(hAPP(state,bool,hAPP(X_b,fun(state,bool),Q_3,Z_4),S_4)) ) ) )
=> hBOOL(hAPP(fun(hoare_2118899576triple(X_b),bool),bool,hAPP(fun(hoare_2118899576triple(X_b),bool),fun(fun(hoare_2118899576triple(X_b),bool),bool),hoare_1301688828derivs(X_b),Ga),hAPP(fun(hoare_2118899576triple(X_b),bool),fun(hoare_2118899576triple(X_b),bool),hAPP(hoare_2118899576triple(X_b),fun(fun(hoare_2118899576triple(X_b),bool),fun(hoare_2118899576triple(X_b),bool)),insert(hoare_2118899576triple(X_b)),hAPP(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b),hAPP(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b)),hAPP(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b))),hoare_759811442triple(X_b),Pa),Ca),Q_3)),bot_bot(fun(hoare_2118899576triple(X_b),bool))))) ) ).
fof(fact_52_nonempty__iff,axiom,
! [X_b,A_3] :
( ti(fun(X_b,bool),A_3) != bot_bot(fun(X_b,bool))
<=> ? [X_2,B_3] :
( ti(fun(X_b,bool),A_3) = 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),B_3)
& ~ hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_2),B_3)) ) ) ).
fof(fact_53_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_54_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_55_the__elem__def,axiom,
! [X_b,X_3] : hAPP(fun(X_b,bool),X_b,the_elem(X_b),X_3) = hAPP(fun(X_b,bool),X_b,the(X_b),hAPP(fun(X_b,fun(X_b,bool)),fun(X_b,bool),hAPP(fun(fun(X_b,bool),bool),fun(fun(X_b,fun(X_b,bool)),fun(X_b,bool)),combb(fun(X_b,bool),bool,X_b),hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),fequal(fun(X_b,bool)),X_3)),hAPP(fun(X_b,bool),fun(X_b,fun(X_b,bool)),hAPP(fun(X_b,fun(fun(X_b,bool),fun(X_b,bool))),fun(fun(X_b,bool),fun(X_b,fun(X_b,bool))),combc(X_b,fun(X_b,bool),fun(X_b,bool)),insert(X_b)),bot_bot(fun(X_b,bool))))) ).
fof(fact_56_com_Osimps_I3_J,axiom,
! [Com1_1,Com2_1,Com1,Com2] :
( hAPP(com,com,hAPP(com,fun(com,com),semi,Com1_1),Com2_1) = hAPP(com,com,hAPP(com,fun(com,com),semi,Com1),Com2)
<=> ( Com1_1 = Com1
& Com2_1 = Com2 ) ) ).
fof(fact_57_fold1Set__sing,axiom,
! [X_b,F,A_2,Ba] :
( hBOOL(hAPP(X_b,bool,hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),fun(X_b,bool)),finite_fold1Set(X_b),F),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_2),bot_bot(fun(X_b,bool)))),Ba))
<=> ti(X_b,A_2) = ti(X_b,Ba) ) ).
fof(fact_58_folding__one_Osingleton,axiom,
! [X_b,X_1,F,F_2] :
( hBOOL(hAPP(fun(fun(X_b,bool),X_b),bool,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(fun(X_b,bool),X_b),bool),finite_folding_one(X_b),F),F_2))
=> hAPP(fun(X_b,bool),X_b,F_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_1),bot_bot(fun(X_b,bool)))) = ti(X_b,X_1) ) ).
fof(fact_59_bot__empty__eq,axiom,
! [X_b,X_2] :
( hBOOL(hAPP(X_b,bool,bot_bot(fun(X_b,bool)),X_2))
<=> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_2),bot_bot(fun(X_b,bool)))) ) ).
fof(fact_60_Ass,axiom,
! [X_b,Ga,Pa,X_3,A_2] : hBOOL(hAPP(fun(hoare_2118899576triple(X_b),bool),bool,hAPP(fun(hoare_2118899576triple(X_b),bool),fun(fun(hoare_2118899576triple(X_b),bool),bool),hoare_1301688828derivs(X_b),Ga),hAPP(fun(hoare_2118899576triple(X_b),bool),fun(hoare_2118899576triple(X_b),bool),hAPP(hoare_2118899576triple(X_b),fun(fun(hoare_2118899576triple(X_b),bool),fun(hoare_2118899576triple(X_b),bool)),insert(hoare_2118899576triple(X_b)),hAPP(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b),hAPP(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b)),hAPP(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b))),hoare_759811442triple(X_b),hAPP(fun(state,state),fun(X_b,fun(state,bool)),hAPP(fun(X_b,fun(fun(state,state),fun(state,bool))),fun(fun(state,state),fun(X_b,fun(state,bool))),combc(X_b,fun(state,state),fun(state,bool)),hAPP(fun(X_b,fun(state,bool)),fun(X_b,fun(fun(state,state),fun(state,bool))),hAPP(fun(fun(state,bool),fun(fun(state,state),fun(state,bool))),fun(fun(X_b,fun(state,bool)),fun(X_b,fun(fun(state,state),fun(state,bool)))),combb(fun(state,bool),fun(fun(state,state),fun(state,bool)),X_b),combb(state,bool,state)),Pa)),hAPP(fun(state,nat),fun(state,state),hAPP(fun(state,fun(nat,state)),fun(fun(state,nat),fun(state,state)),combs(state,nat,state),hAPP(vname,fun(state,fun(nat,state)),hAPP(fun(state,fun(vname,fun(nat,state))),fun(vname,fun(state,fun(nat,state))),combc(state,vname,fun(nat,state)),update),X_3)),A_2))),hAPP(fun(state,nat),com,hAPP(vname,fun(fun(state,nat),com),ass,X_3),A_2)),Pa)),bot_bot(fun(hoare_2118899576triple(X_b),bool))))) ).
fof(fact_61_fold1__singleton,axiom,
! [X_b,F,A_2] : hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),finite_fold1(X_b),F),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_2),bot_bot(fun(X_b,bool)))) = ti(X_b,A_2) ).
fof(fact_62_fold1__singleton__def,axiom,
! [X_b,A_2,G,F] :
( G = hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),finite_fold1(X_b),F)
=> hAPP(fun(X_b,bool),X_b,G,hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_2),bot_bot(fun(X_b,bool)))) = ti(X_b,A_2) ) ).
fof(fact_63_image__constant__conv,axiom,
! [X_c,X_b,Ca,A_3] :
( ( ti(fun(X_b,bool),A_3) = bot_bot(fun(X_b,bool))
=> hAPP(fun(X_b,bool),fun(X_c,bool),hAPP(fun(X_b,X_c),fun(fun(X_b,bool),fun(X_c,bool)),image(X_b,X_c),hAPP(X_c,fun(X_b,X_c),combk(X_c,X_b),Ca)),A_3) = bot_bot(fun(X_c,bool)) )
& ( ti(fun(X_b,bool),A_3) != bot_bot(fun(X_b,bool))
=> hAPP(fun(X_b,bool),fun(X_c,bool),hAPP(fun(X_b,X_c),fun(fun(X_b,bool),fun(X_c,bool)),image(X_b,X_c),hAPP(X_c,fun(X_b,X_c),combk(X_c,X_b),Ca)),A_3) = hAPP(fun(X_c,bool),fun(X_c,bool),hAPP(X_c,fun(fun(X_c,bool),fun(X_c,bool)),insert(X_c),Ca),bot_bot(fun(X_c,bool))) ) ) ).
fof(fact_64_image__eqI,axiom,
! [X_b,X_c,A_3,Ba,F,X_1] :
( ti(X_b,Ba) = hAPP(X_c,X_b,F,X_1)
=> ( hBOOL(hAPP(fun(X_c,bool),bool,hAPP(X_c,fun(fun(X_c,bool),bool),member(X_c),X_1),A_3))
=> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),Ba),hAPP(fun(X_c,bool),fun(X_b,bool),hAPP(fun(X_c,X_b),fun(fun(X_c,bool),fun(X_b,bool)),image(X_c,X_b),F),A_3))) ) ) ).
fof(fact_65_fold1__def,axiom,
! [X_b,F,A_3] : hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),finite_fold1(X_b),F),A_3) = hAPP(fun(X_b,bool),X_b,the(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),fun(X_b,bool)),finite_fold1Set(X_b),F),A_3)) ).
fof(fact_66_com_Osimps_I1_J,axiom,
! [Vname_1,Fun_1,Vname,Fun] :
( hAPP(fun(state,nat),com,hAPP(vname,fun(fun(state,nat),com),ass,Vname_1),Fun_1) = hAPP(fun(state,nat),com,hAPP(vname,fun(fun(state,nat),com),ass,Vname),Fun)
<=> ( ti(vname,Vname_1) = ti(vname,Vname)
& Fun_1 = Fun ) ) ).
fof(fact_67_rev__image__eqI,axiom,
! [X_c,X_b,Ba,F,X_1,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),A_3))
=> ( ti(X_c,Ba) = hAPP(X_b,X_c,F,X_1)
=> hBOOL(hAPP(fun(X_c,bool),bool,hAPP(X_c,fun(fun(X_c,bool),bool),member(X_c),Ba),hAPP(fun(X_b,bool),fun(X_c,bool),hAPP(fun(X_b,X_c),fun(fun(X_b,bool),fun(X_c,bool)),image(X_b,X_c),F),A_3))) ) ) ).
fof(fact_68_imageI,axiom,
! [X_c,X_b,F,X_1,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),A_3))
=> hBOOL(hAPP(fun(X_c,bool),bool,hAPP(X_c,fun(fun(X_c,bool),bool),member(X_c),hAPP(X_b,X_c,F,X_1)),hAPP(fun(X_b,bool),fun(X_c,bool),hAPP(fun(X_b,X_c),fun(fun(X_b,bool),fun(X_c,bool)),image(X_b,X_c),F),A_3))) ) ).
fof(fact_69_image__iff,axiom,
! [X_b,X_c,Z_2,F,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),Z_2),hAPP(fun(X_c,bool),fun(X_b,bool),hAPP(fun(X_c,X_b),fun(fun(X_c,bool),fun(X_b,bool)),image(X_c,X_b),F),A_3)))
<=> ? [X_2] :
( hBOOL(hAPP(fun(X_c,bool),bool,hAPP(X_c,fun(fun(X_c,bool),bool),member(X_c),X_2),A_3))
& ti(X_b,Z_2) = hAPP(X_c,X_b,F,X_2) ) ) ).
fof(fact_70_image__image,axiom,
! [X_c,X_b,X_d,F,G,A_3] : hAPP(fun(X_c,bool),fun(X_b,bool),hAPP(fun(X_c,X_b),fun(fun(X_c,bool),fun(X_b,bool)),image(X_c,X_b),F),hAPP(fun(X_d,bool),fun(X_c,bool),hAPP(fun(X_d,X_c),fun(fun(X_d,bool),fun(X_c,bool)),image(X_d,X_c),G),A_3)) = hAPP(fun(X_d,bool),fun(X_b,bool),hAPP(fun(X_d,X_b),fun(fun(X_d,bool),fun(X_b,bool)),image(X_d,X_b),hAPP(fun(X_d,X_c),fun(X_d,X_b),hAPP(fun(X_c,X_b),fun(fun(X_d,X_c),fun(X_d,X_b)),combb(X_c,X_b,X_d),F),G)),A_3) ).
fof(fact_71_image__ident,axiom,
! [X_b,Y_4] : hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,X_b),fun(fun(X_b,bool),fun(X_b,bool)),image(X_b,X_b),combi(X_b)),Y_4) = ti(fun(X_b,bool),Y_4) ).
fof(fact_72_com_Osimps_I24_J,axiom,
! [Vname_1,Fun_1,Com1,Com2] : hAPP(fun(state,nat),com,hAPP(vname,fun(fun(state,nat),com),ass,Vname_1),Fun_1) != hAPP(com,com,hAPP(com,fun(com,com),semi,Com1),Com2) ).
fof(fact_73_com_Osimps_I25_J,axiom,
! [Com1,Com2,Vname_1,Fun_1] : hAPP(com,com,hAPP(com,fun(com,com),semi,Com1),Com2) != hAPP(fun(state,nat),com,hAPP(vname,fun(fun(state,nat),com),ass,Vname_1),Fun_1) ).
fof(fact_74_com_Osimps_I8_J,axiom,
! [Vname,Fun] : skip != hAPP(fun(state,nat),com,hAPP(vname,fun(fun(state,nat),com),ass,Vname),Fun) ).
fof(fact_75_ext,axiom,
! [X_b,X_c,F,G] :
( ! [X_2] : hAPP(X_b,X_c,F,X_2) = hAPP(X_b,X_c,G,X_2)
=> ti(fun(X_b,X_c),F) = ti(fun(X_b,X_c),G) ) ).
fof(fact_76_mem__def,axiom,
! [X_b,X_1,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),A_3))
<=> hBOOL(hAPP(X_b,bool,A_3,X_1)) ) ).
fof(fact_77_Collect__def,axiom,
! [X_b,Pa] : hAPP(fun(X_b,bool),fun(X_b,bool),collect(X_b),Pa) = ti(fun(X_b,bool),Pa) ).
fof(fact_78_com_Osimps_I9_J,axiom,
! [Vname,Fun] : hAPP(fun(state,nat),com,hAPP(vname,fun(fun(state,nat),com),ass,Vname),Fun) != skip ).
fof(fact_79_image__is__empty,axiom,
! [X_c,X_b,F,A_3] :
( hAPP(fun(X_c,bool),fun(X_b,bool),hAPP(fun(X_c,X_b),fun(fun(X_c,bool),fun(X_b,bool)),image(X_c,X_b),F),A_3) = bot_bot(fun(X_b,bool))
<=> ti(fun(X_c,bool),A_3) = bot_bot(fun(X_c,bool)) ) ).
fof(fact_80_image__empty,axiom,
! [X_c,X_b,F] : hAPP(fun(X_c,bool),fun(X_b,bool),hAPP(fun(X_c,X_b),fun(fun(X_c,bool),fun(X_b,bool)),image(X_c,X_b),F),bot_bot(fun(X_c,bool))) = bot_bot(fun(X_b,bool)) ).
fof(fact_81_empty__is__image,axiom,
! [X_b,X_c,F,A_3] :
( bot_bot(fun(X_b,bool)) = hAPP(fun(X_c,bool),fun(X_b,bool),hAPP(fun(X_c,X_b),fun(fun(X_c,bool),fun(X_b,bool)),image(X_c,X_b),F),A_3)
<=> ti(fun(X_c,bool),A_3) = bot_bot(fun(X_c,bool)) ) ).
fof(fact_82_insert__image,axiom,
! [X_c,X_b,F,X_1,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),A_3))
=> hAPP(fun(X_c,bool),fun(X_c,bool),hAPP(X_c,fun(fun(X_c,bool),fun(X_c,bool)),insert(X_c),hAPP(X_b,X_c,F,X_1)),hAPP(fun(X_b,bool),fun(X_c,bool),hAPP(fun(X_b,X_c),fun(fun(X_b,bool),fun(X_c,bool)),image(X_b,X_c),F),A_3)) = hAPP(fun(X_b,bool),fun(X_c,bool),hAPP(fun(X_b,X_c),fun(fun(X_b,bool),fun(X_c,bool)),image(X_b,X_c),F),A_3) ) ).
fof(fact_83_image__insert,axiom,
! [X_c,X_b,F,A_2,B_2] : hAPP(fun(X_c,bool),fun(X_b,bool),hAPP(fun(X_c,X_b),fun(fun(X_c,bool),fun(X_b,bool)),image(X_c,X_b),F),hAPP(fun(X_c,bool),fun(X_c,bool),hAPP(X_c,fun(fun(X_c,bool),fun(X_c,bool)),insert(X_c),A_2),B_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),hAPP(X_c,X_b,F,A_2)),hAPP(fun(X_c,bool),fun(X_b,bool),hAPP(fun(X_c,X_b),fun(fun(X_c,bool),fun(X_b,bool)),image(X_c,X_b),F),B_2)) ).
fof(fact_84_fold1Set__nonempty,axiom,
! [X_b,F,A_3,X_1] :
( hBOOL(hAPP(X_b,bool,hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),fun(X_b,bool)),finite_fold1Set(X_b),F),A_3),X_1))
=> ti(fun(X_b,bool),A_3) != bot_bot(fun(X_b,bool)) ) ).
fof(fact_85_empty__fold1SetE,axiom,
! [X_b,F,X_1] : ~ hBOOL(hAPP(X_b,bool,hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),fun(X_b,bool)),finite_fold1Set(X_b),F),bot_bot(fun(X_b,bool))),X_1)) ).
fof(fact_86_image__constant,axiom,
! [X_c,X_b,Ca,X_1,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),A_3))
=> hAPP(fun(X_b,bool),fun(X_c,bool),hAPP(fun(X_b,X_c),fun(fun(X_b,bool),fun(X_c,bool)),image(X_b,X_c),hAPP(X_c,fun(X_b,X_c),combk(X_c,X_b),Ca)),A_3) = hAPP(fun(X_c,bool),fun(X_c,bool),hAPP(X_c,fun(fun(X_c,bool),fun(X_c,bool)),insert(X_c),Ca),bot_bot(fun(X_c,bool))) ) ).
fof(fact_87_imageE,axiom,
! [X_b,X_c,Ba,F,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),Ba),hAPP(fun(X_c,bool),fun(X_b,bool),hAPP(fun(X_c,X_b),fun(fun(X_c,bool),fun(X_b,bool)),image(X_c,X_b),F),A_3)))
=> ~ ! [X_2] :
( ti(X_b,Ba) = hAPP(X_c,X_b,F,X_2)
=> ~ hBOOL(hAPP(fun(X_c,bool),bool,hAPP(X_c,fun(fun(X_c,bool),bool),member(X_c),X_2),A_3)) ) ) ).
fof(fact_88_the__sym__eq__trivial,axiom,
! [X_b,X_1] : hAPP(fun(X_b,bool),X_b,the(X_b),hAPP(X_b,fun(X_b,bool),fequal(X_b),X_1)) = ti(X_b,X_1) ).
fof(fact_89_the__eq__trivial,axiom,
! [X_b,A_2] : hAPP(fun(X_b,bool),X_b,the(X_b),hAPP(X_b,fun(X_b,bool),hAPP(fun(X_b,fun(X_b,bool)),fun(X_b,fun(X_b,bool)),combc(X_b,X_b,bool),fequal(X_b)),A_2)) = ti(X_b,A_2) ).
fof(fact_90_If__def,axiom,
! [X_b,X_1,Y_2,Pa] :
( ( hBOOL(Pa)
=> ti(X_b,X_1) = hAPP(fun(X_b,bool),X_b,the(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,bool)),combs(X_b,bool,bool),hAPP(fun(X_b,bool),fun(X_b,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,fun(bool,bool))),combb(bool,fun(bool,bool),X_b),fconj),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(bool,bool),fun(fun(X_b,bool),fun(X_b,bool)),combb(bool,bool,X_b),hAPP(bool,fun(bool,bool),fimplies,Pa)),hAPP(X_b,fun(X_b,bool),hAPP(fun(X_b,fun(X_b,bool)),fun(X_b,fun(X_b,bool)),combc(X_b,X_b,bool),fequal(X_b)),X_1)))),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(bool,bool),fun(fun(X_b,bool),fun(X_b,bool)),combb(bool,bool,X_b),hAPP(bool,fun(bool,bool),fimplies,hAPP(bool,bool,fNot,Pa))),hAPP(X_b,fun(X_b,bool),hAPP(fun(X_b,fun(X_b,bool)),fun(X_b,fun(X_b,bool)),combc(X_b,X_b,bool),fequal(X_b)),Y_2)))) )
& ( ~ hBOOL(Pa)
=> ti(X_b,Y_2) = hAPP(fun(X_b,bool),X_b,the(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,bool)),combs(X_b,bool,bool),hAPP(fun(X_b,bool),fun(X_b,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,fun(bool,bool))),combb(bool,fun(bool,bool),X_b),fconj),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(bool,bool),fun(fun(X_b,bool),fun(X_b,bool)),combb(bool,bool,X_b),hAPP(bool,fun(bool,bool),fimplies,Pa)),hAPP(X_b,fun(X_b,bool),hAPP(fun(X_b,fun(X_b,bool)),fun(X_b,fun(X_b,bool)),combc(X_b,X_b,bool),fequal(X_b)),X_1)))),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(bool,bool),fun(fun(X_b,bool),fun(X_b,bool)),combb(bool,bool,X_b),hAPP(bool,fun(bool,bool),fimplies,hAPP(bool,bool,fNot,Pa))),hAPP(X_b,fun(X_b,bool),hAPP(fun(X_b,fun(X_b,bool)),fun(X_b,fun(X_b,bool)),combc(X_b,X_b,bool),fequal(X_b)),Y_2)))) ) ) ).
fof(fact_91_image__cong,axiom,
! [X_c,X_b,F,G,M_3,N_3] :
( ti(fun(X_b,bool),M_3) = ti(fun(X_b,bool),N_3)
=> ( ! [X_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_2),N_3))
=> hAPP(X_b,X_c,F,X_2) = hAPP(X_b,X_c,G,X_2) )
=> hAPP(fun(X_b,bool),fun(X_c,bool),hAPP(fun(X_b,X_c),fun(fun(X_b,bool),fun(X_c,bool)),image(X_b,X_c),F),M_3) = hAPP(fun(X_b,bool),fun(X_c,bool),hAPP(fun(X_b,X_c),fun(fun(X_b,bool),fun(X_c,bool)),image(X_b,X_c),G),N_3) ) ) ).
fof(fact_92_fold1Set_Ointros,axiom,
! [X_b,F,A_2,A_3,X_1] :
( hBOOL(hAPP(X_b,bool,hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),hAPP(fun(X_b,fun(X_b,X_b)),fun(X_b,fun(fun(X_b,bool),fun(X_b,bool))),finite_fold_graph(X_b,X_b),F),A_2),A_3),X_1))
=> ( ~ hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),A_2),A_3))
=> hBOOL(hAPP(X_b,bool,hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),fun(X_b,bool)),finite_fold1Set(X_b),F),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_2),A_3)),X_1)) ) ) ).
fof(fact_93_the__equality,axiom,
! [X_b,Pa,A_2] :
( hBOOL(hAPP(X_b,bool,Pa,A_2))
=> ( ! [X_2] :
( hBOOL(hAPP(X_b,bool,Pa,X_2))
=> ti(X_b,X_2) = ti(X_b,A_2) )
=> hAPP(fun(X_b,bool),X_b,the(X_b),Pa) = ti(X_b,A_2) ) ) ).
fof(fact_94_folding__one_Oinsert,axiom,
! [X_b,X_1,A_3,F,F_2] :
( hBOOL(hAPP(fun(fun(X_b,bool),X_b),bool,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(fun(X_b,bool),X_b),bool),finite_folding_one(X_b),F),F_2))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( ~ hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),A_3))
=> ( ti(fun(X_b,bool),A_3) != bot_bot(fun(X_b,bool))
=> hAPP(fun(X_b,bool),X_b,F_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_1),A_3)) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),F,X_1),hAPP(fun(X_b,bool),X_b,F_2,A_3)) ) ) ) ) ).
fof(fact_95_folding__one_Oeq__fold,axiom,
! [X_b,A_3,F,F_2] :
( hBOOL(hAPP(fun(fun(X_b,bool),X_b),bool,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(fun(X_b,bool),X_b),bool),finite_folding_one(X_b),F),F_2))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> hAPP(fun(X_b,bool),X_b,F_2,A_3) = hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),finite_fold1(X_b),F),A_3) ) ) ).
fof(fact_96_folding__one_Oclosed,axiom,
! [X_b,A_3,F,F_2] :
( hBOOL(hAPP(fun(fun(X_b,bool),X_b),bool,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(fun(X_b,bool),X_b),bool),finite_folding_one(X_b),F),F_2))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( ti(fun(X_b,bool),A_3) != bot_bot(fun(X_b,bool))
=> ( ! [X_2,Y_1] : hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),F,X_2),Y_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),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),Y_1),bot_bot(fun(X_b,bool))))))
=> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),hAPP(fun(X_b,bool),X_b,F_2,A_3)),A_3)) ) ) ) ) ).
fof(fact_97_finite__Collect__conjI,axiom,
! [X_b,Q_3,Pa] :
( ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),collect(X_b),Pa)))
| hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),collect(X_b),Q_3))) )
=> hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),collect(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,bool)),combs(X_b,bool,bool),hAPP(fun(X_b,bool),fun(X_b,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,fun(bool,bool))),combb(bool,fun(bool,bool),X_b),fconj),Pa)),Q_3)))) ) ).
fof(fact_98_finite_OemptyI,axiom,
! [X_b] : hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),bot_bot(fun(X_b,bool)))) ).
fof(fact_99_finite_OinsertI,axiom,
! [X_b,A_2,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_2),A_3))) ) ).
fof(fact_100_finite__imageI,axiom,
! [X_c,X_b,H,F_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),F_2))
=> hBOOL(hAPP(fun(X_c,bool),bool,finite_finite_1(X_c),hAPP(fun(X_b,bool),fun(X_c,bool),hAPP(fun(X_b,X_c),fun(fun(X_b,bool),fun(X_c,bool)),image(X_b,X_c),H),F_2))) ) ).
fof(fact_101_finite,axiom,
! [X_b] :
( finite_finite(X_b)
=> ! [A_3] : hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3)) ) ).
fof(fact_102_finite__code,axiom,
! [X_b] :
( finite_finite(X_b)
=> ! [A_3] : hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3)) ) ).
fof(fact_103_finite__Collect__disjI,axiom,
! [X_b,Pa,Q_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),collect(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,bool)),combs(X_b,bool,bool),hAPP(fun(X_b,bool),fun(X_b,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,fun(bool,bool))),combb(bool,fun(bool,bool),X_b),fdisj),Pa)),Q_3))))
<=> ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),collect(X_b),Pa)))
& hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),collect(X_b),Q_3))) ) ) ).
fof(fact_104_finite__insert,axiom,
! [X_b,A_2,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_2),A_3)))
<=> hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3)) ) ).
fof(fact_105_fold__graph_OemptyI,axiom,
! [X_c,X_b,F,Z_2] : hBOOL(hAPP(X_c,bool,hAPP(fun(X_b,bool),fun(X_c,bool),hAPP(X_c,fun(fun(X_b,bool),fun(X_c,bool)),hAPP(fun(X_b,fun(X_c,X_c)),fun(X_c,fun(fun(X_b,bool),fun(X_c,bool))),finite_fold_graph(X_b,X_c),F),Z_2),bot_bot(fun(X_b,bool))),Z_2)) ).
fof(fact_106_empty__fold__graphE,axiom,
! [X_c,X_b,F,Z_2,X_1] :
( hBOOL(hAPP(X_c,bool,hAPP(fun(X_b,bool),fun(X_c,bool),hAPP(X_c,fun(fun(X_b,bool),fun(X_c,bool)),hAPP(fun(X_b,fun(X_c,X_c)),fun(X_c,fun(fun(X_b,bool),fun(X_c,bool))),finite_fold_graph(X_b,X_c),F),Z_2),bot_bot(fun(X_b,bool))),X_1))
=> ti(X_c,X_1) = ti(X_c,Z_2) ) ).
fof(fact_107_fold__graph_OinsertI,axiom,
! [X_c,X_b,F,Z_2,Y_2,X_1,A_3] :
( ~ hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),A_3))
=> ( hBOOL(hAPP(X_c,bool,hAPP(fun(X_b,bool),fun(X_c,bool),hAPP(X_c,fun(fun(X_b,bool),fun(X_c,bool)),hAPP(fun(X_b,fun(X_c,X_c)),fun(X_c,fun(fun(X_b,bool),fun(X_c,bool))),finite_fold_graph(X_b,X_c),F),Z_2),A_3),Y_2))
=> hBOOL(hAPP(X_c,bool,hAPP(fun(X_b,bool),fun(X_c,bool),hAPP(X_c,fun(fun(X_b,bool),fun(X_c,bool)),hAPP(fun(X_b,fun(X_c,X_c)),fun(X_c,fun(fun(X_b,bool),fun(X_c,bool))),finite_fold_graph(X_b,X_c),F),Z_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_1),A_3)),hAPP(X_c,X_c,hAPP(X_b,fun(X_c,X_c),F,X_1),Y_2))) ) ) ).
fof(fact_108_insert__fold1SetE,axiom,
! [X_b,F,A_2,X_3,X_1] :
( hBOOL(hAPP(X_b,bool,hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),fun(X_b,bool)),finite_fold1Set(X_b),F),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_2),X_3)),X_1))
=> ~ ! [A_5,A_4] :
( hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_2),X_3) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_5),A_4)
=> ( hBOOL(hAPP(X_b,bool,hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),hAPP(fun(X_b,fun(X_b,X_b)),fun(X_b,fun(fun(X_b,bool),fun(X_b,bool))),finite_fold_graph(X_b,X_b),F),A_5),A_4),X_1))
=> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),A_5),A_4)) ) ) ) ).
fof(fact_109_finite__nonempty__imp__fold1Set,axiom,
! [X_b,F,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( ti(fun(X_b,bool),A_3) != bot_bot(fun(X_b,bool))
=> ? [X1] : hBOOL(hAPP(X_b,bool,hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),fun(X_b,bool)),finite_fold1Set(X_b),F),A_3),X1)) ) ) ).
fof(fact_110_finite_Osimps,axiom,
! [X_b,A_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_2))
<=> ( ti(fun(X_b,bool),A_2) = bot_bot(fun(X_b,bool))
| ? [A_4,A_5] :
( ti(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),A_5),A_4)
& hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_4)) ) ) ) ).
fof(fact_111_finite__induct,axiom,
! [X_b,Pa,F_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),F_2))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,Pa,bot_bot(fun(X_b,bool))))
=> ( ! [X_2,F_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),F_3))
=> ( ~ hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_2),F_3))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,Pa,F_3))
=> hBOOL(hAPP(fun(X_b,bool),bool,Pa,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),F_3))) ) ) )
=> hBOOL(hAPP(fun(X_b,bool),bool,Pa,F_2)) ) ) ) ).
fof(fact_112_pigeonhole__infinite,axiom,
! [X_c,X_b,F,A_3] :
( ~ hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( hBOOL(hAPP(fun(X_c,bool),bool,finite_finite_1(X_c),hAPP(fun(X_b,bool),fun(X_c,bool),hAPP(fun(X_b,X_c),fun(fun(X_b,bool),fun(X_c,bool)),image(X_b,X_c),F),A_3)))
=> ? [X_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_2),A_3))
& ~ hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),collect(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,bool)),combs(X_b,bool,bool),hAPP(fun(X_b,bool),fun(X_b,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,fun(bool,bool))),combb(bool,fun(bool,bool),X_b),fconj),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(fun(X_b,bool),bool)),fun(fun(X_b,bool),fun(X_b,bool)),combc(X_b,fun(X_b,bool),bool),member(X_b)),A_3))),hAPP(X_c,fun(X_b,bool),hAPP(fun(X_b,fun(X_c,bool)),fun(X_c,fun(X_b,bool)),combc(X_b,X_c,bool),hAPP(fun(X_b,X_c),fun(X_b,fun(X_c,bool)),hAPP(fun(X_c,fun(X_c,bool)),fun(fun(X_b,X_c),fun(X_b,fun(X_c,bool))),combb(X_c,fun(X_c,bool),X_b),fequal(X_c)),F)),hAPP(X_b,X_c,F,X_2)))))) ) ) ) ).
fof(fact_113_finite__imp__fold__graph,axiom,
! [X_c,X_b,F,Z_2,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ? [X1] : hBOOL(hAPP(X_c,bool,hAPP(fun(X_b,bool),fun(X_c,bool),hAPP(X_c,fun(fun(X_b,bool),fun(X_c,bool)),hAPP(fun(X_b,fun(X_c,X_c)),fun(X_c,fun(fun(X_b,bool),fun(X_c,bool))),finite_fold_graph(X_b,X_c),F),Z_2),A_3),X1)) ) ).
fof(fact_114_fold1Set_Osimps,axiom,
! [X_b,F,A1,A2] :
( hBOOL(hAPP(X_b,bool,hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),fun(X_b,bool)),finite_fold1Set(X_b),F),A1),A2))
<=> ? [A_5,A_4,X_2] :
( ti(fun(X_b,bool),A1) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_5),A_4)
& ti(X_b,A2) = ti(X_b,X_2)
& hBOOL(hAPP(X_b,bool,hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),hAPP(fun(X_b,fun(X_b,X_b)),fun(X_b,fun(fun(X_b,bool),fun(X_b,bool))),finite_fold_graph(X_b,X_b),F),A_5),A_4),X_2))
& ~ hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),A_5),A_4)) ) ) ).
fof(fact_115_theI,axiom,
! [X_b,Pa,A_2] :
( hBOOL(hAPP(X_b,bool,Pa,A_2))
=> ( ! [X_2] :
( hBOOL(hAPP(X_b,bool,Pa,X_2))
=> ti(X_b,X_2) = ti(X_b,A_2) )
=> hBOOL(hAPP(X_b,bool,Pa,hAPP(fun(X_b,bool),X_b,the(X_b),Pa))) ) ) ).
fof(fact_116_the1__equality,axiom,
! [X_b,A_2,Pa] :
( ? [X_2] :
( hBOOL(hAPP(X_b,bool,Pa,X_2))
& ! [Y_1] :
( hBOOL(hAPP(X_b,bool,Pa,Y_1))
=> ti(X_b,Y_1) = ti(X_b,X_2) ) )
=> ( hBOOL(hAPP(X_b,bool,Pa,A_2))
=> hAPP(fun(X_b,bool),X_b,the(X_b),Pa) = ti(X_b,A_2) ) ) ).
fof(fact_117_theI_H,axiom,
! [X_b,Pa] :
( ? [X_2] :
( hBOOL(hAPP(X_b,bool,Pa,X_2))
& ! [Y_1] :
( hBOOL(hAPP(X_b,bool,Pa,Y_1))
=> ti(X_b,Y_1) = ti(X_b,X_2) ) )
=> hBOOL(hAPP(X_b,bool,Pa,hAPP(fun(X_b,bool),X_b,the(X_b),Pa))) ) ).
fof(fact_118_fold__graph_Osimps,axiom,
! [X_c,X_b,F,Z_2,A1,A2] :
( hBOOL(hAPP(X_c,bool,hAPP(fun(X_b,bool),fun(X_c,bool),hAPP(X_c,fun(fun(X_b,bool),fun(X_c,bool)),hAPP(fun(X_b,fun(X_c,X_c)),fun(X_c,fun(fun(X_b,bool),fun(X_c,bool))),finite_fold_graph(X_b,X_c),F),Z_2),A1),A2))
<=> ( ( ti(fun(X_b,bool),A1) = bot_bot(fun(X_b,bool))
& ti(X_c,A2) = ti(X_c,Z_2) )
| ? [X_2,A_4,Y_1] :
( ti(fun(X_b,bool),A1) = 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_4)
& ti(X_c,A2) = hAPP(X_c,X_c,hAPP(X_b,fun(X_c,X_c),F,X_2),Y_1)
& ~ hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_2),A_4))
& hBOOL(hAPP(X_c,bool,hAPP(fun(X_b,bool),fun(X_c,bool),hAPP(X_c,fun(fun(X_b,bool),fun(X_c,bool)),hAPP(fun(X_b,fun(X_c,X_c)),fun(X_c,fun(fun(X_b,bool),fun(X_c,bool))),finite_fold_graph(X_b,X_c),F),Z_2),A_4),Y_1)) ) ) ) ).
fof(fact_119_folding__one__idem_Oinsert__idem,axiom,
! [X_b,X_1,A_3,F,F_2] :
( hBOOL(hAPP(fun(fun(X_b,bool),X_b),bool,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(fun(X_b,bool),X_b),bool),finite2073411215e_idem(X_b),F),F_2))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( ti(fun(X_b,bool),A_3) != bot_bot(fun(X_b,bool))
=> hAPP(fun(X_b,bool),X_b,F_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_1),A_3)) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),F,X_1),hAPP(fun(X_b,bool),X_b,F_2,A_3)) ) ) ) ).
fof(fact_120_finite__ne__induct,axiom,
! [X_b,Pa,F_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),F_2))
=> ( ti(fun(X_b,bool),F_2) != bot_bot(fun(X_b,bool))
=> ( ! [X_2] : hBOOL(hAPP(fun(X_b,bool),bool,Pa,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)))))
=> ( ! [X_2,F_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),F_3))
=> ( ti(fun(X_b,bool),F_3) != bot_bot(fun(X_b,bool))
=> ( ~ hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_2),F_3))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,Pa,F_3))
=> hBOOL(hAPP(fun(X_b,bool),bool,Pa,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),F_3))) ) ) ) )
=> hBOOL(hAPP(fun(X_b,bool),bool,Pa,F_2)) ) ) ) ) ).
fof(fact_121_folding__image__simple__idem_Oinsert__idem,axiom,
! [X_c,X_b,X_1,A_3,F,Z_2,G,F_2] :
( hBOOL(hAPP(fun(fun(X_c,bool),X_b),bool,hAPP(fun(X_c,X_b),fun(fun(fun(X_c,bool),X_b),bool),hAPP(X_b,fun(fun(X_c,X_b),fun(fun(fun(X_c,bool),X_b),bool)),hAPP(fun(X_b,fun(X_b,X_b)),fun(X_b,fun(fun(X_c,X_b),fun(fun(fun(X_c,bool),X_b),bool))),finite908156982e_idem(X_b,X_c),F),Z_2),G),F_2))
=> ( hBOOL(hAPP(fun(X_c,bool),bool,finite_finite_1(X_c),A_3))
=> hAPP(fun(X_c,bool),X_b,F_2,hAPP(fun(X_c,bool),fun(X_c,bool),hAPP(X_c,fun(fun(X_c,bool),fun(X_c,bool)),insert(X_c),X_1),A_3)) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),F,hAPP(X_c,X_b,G,X_1)),hAPP(fun(X_c,bool),X_b,F_2,A_3)) ) ) ).
fof(fact_122_semilattice__big_OF__eq,axiom,
! [X_b,A_3,F,F_2] :
( hBOOL(hAPP(fun(fun(X_b,bool),X_b),bool,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(fun(X_b,bool),X_b),bool),big_semilattice_big(X_b),F),F_2))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> hAPP(fun(X_b,bool),X_b,F_2,A_3) = hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),finite_fold1(X_b),F),A_3) ) ) ).
fof(fact_123_folding__one_Oremove,axiom,
! [X_b,X_1,A_3,F,F_2] :
( hBOOL(hAPP(fun(fun(X_b,bool),X_b),bool,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(fun(X_b,bool),X_b),bool),finite_folding_one(X_b),F),F_2))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),A_3))
=> ( ( 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_3),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_1),bot_bot(fun(X_b,bool)))) = bot_bot(fun(X_b,bool))
=> hAPP(fun(X_b,bool),X_b,F_2,A_3) = ti(X_b,X_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_3),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_1),bot_bot(fun(X_b,bool)))) != bot_bot(fun(X_b,bool))
=> hAPP(fun(X_b,bool),X_b,F_2,A_3) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),F,X_1),hAPP(fun(X_b,bool),X_b,F_2,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_3),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_1),bot_bot(fun(X_b,bool)))))) ) ) ) ) ) ).
fof(fact_124_folding__one_Oinsert__remove,axiom,
! [X_b,X_1,A_3,F,F_2] :
( hBOOL(hAPP(fun(fun(X_b,bool),X_b),bool,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(fun(X_b,bool),X_b),bool),finite_folding_one(X_b),F),F_2))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( ( 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_3),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_1),bot_bot(fun(X_b,bool)))) = bot_bot(fun(X_b,bool))
=> hAPP(fun(X_b,bool),X_b,F_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_1),A_3)) = ti(X_b,X_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_3),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_1),bot_bot(fun(X_b,bool)))) != bot_bot(fun(X_b,bool))
=> hAPP(fun(X_b,bool),X_b,F_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_1),A_3)) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),F,X_1),hAPP(fun(X_b,bool),X_b,F_2,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_3),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_1),bot_bot(fun(X_b,bool)))))) ) ) ) ) ).
fof(fact_125_fold1__insert,axiom,
! [X_b] :
( ab_semigroup_mult(X_b)
=> ! [X_1,A_3] :
( ti(fun(X_b,bool),A_3) != bot_bot(fun(X_b,bool))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( ~ hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),A_3))
=> hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),finite_fold1(X_b),times_times(X_b)),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_1),A_3)) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),X_1),hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),finite_fold1(X_b),times_times(X_b)),A_3)) ) ) ) ) ).
fof(fact_126_fold1__insert__idem,axiom,
! [X_b] :
( ab_sem1668676832m_mult(X_b)
=> ! [X_1,A_3] :
( ti(fun(X_b,bool),A_3) != bot_bot(fun(X_b,bool))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),finite_fold1(X_b),times_times(X_b)),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_1),A_3)) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),X_1),hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),finite_fold1(X_b),times_times(X_b)),A_3)) ) ) ) ).
fof(fact_127_DiffI,axiom,
! [X_b,B_2,Ca,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),Ca),A_3))
=> ( ~ hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),Ca),B_2))
=> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),Ca),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_3),B_2))) ) ) ).
fof(fact_128_DiffE,axiom,
! [X_b,Ca,A_3,B_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),Ca),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_3),B_2)))
=> ~ ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),Ca),A_3))
=> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),Ca),B_2)) ) ) ).
fof(fact_129_finite__Diff,axiom,
! [X_b,B_2,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),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_3),B_2))) ) ).
fof(fact_130_folding__one__idem_Oidem,axiom,
! [X_b,X_1,F,F_2] :
( hBOOL(hAPP(fun(fun(X_b,bool),X_b),bool,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(fun(X_b,bool),X_b),bool),finite2073411215e_idem(X_b),F),F_2))
=> hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),F,X_1),X_1) = ti(X_b,X_1) ) ).
fof(fact_131_set__diff__eq,axiom,
! [X_b,A_3,B_2] : 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_3),B_2) = hAPP(fun(X_b,bool),fun(X_b,bool),collect(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,bool)),combs(X_b,bool,bool),hAPP(fun(X_b,bool),fun(X_b,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,fun(bool,bool))),combb(bool,fun(bool,bool),X_b),fconj),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(fun(X_b,bool),bool)),fun(fun(X_b,bool),fun(X_b,bool)),combc(X_b,fun(X_b,bool),bool),member(X_b)),A_3))),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(bool,bool),fun(fun(X_b,bool),fun(X_b,bool)),combb(bool,bool,X_b),fNot),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(fun(X_b,bool),bool)),fun(fun(X_b,bool),fun(X_b,bool)),combc(X_b,fun(X_b,bool),bool),member(X_b)),B_2)))) ).
fof(fact_132_Diff__iff,axiom,
! [X_b,Ca,A_3,B_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),Ca),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_3),B_2)))
<=> ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),Ca),A_3))
& ~ hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),Ca),B_2)) ) ) ).
fof(fact_133_Diff__idemp,axiom,
! [X_b,A_3,B_2] : 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)),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_3),B_2)),B_2) = 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_3),B_2) ).
fof(fact_134_DiffD1,axiom,
! [X_b,Ca,A_3,B_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),Ca),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_3),B_2)))
=> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),Ca),A_3)) ) ).
fof(fact_135_DiffD2,axiom,
! [X_b,Ca,A_3,B_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),Ca),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_3),B_2)))
=> ~ hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),Ca),B_2)) ) ).
fof(fact_136_folding__image__simple__idem_Oidem,axiom,
! [X_c,X_b,X_1,F,Z_2,G,F_2] :
( hBOOL(hAPP(fun(fun(X_c,bool),X_b),bool,hAPP(fun(X_c,X_b),fun(fun(fun(X_c,bool),X_b),bool),hAPP(X_b,fun(fun(X_c,X_b),fun(fun(fun(X_c,bool),X_b),bool)),hAPP(fun(X_b,fun(X_b,X_b)),fun(X_b,fun(fun(X_c,X_b),fun(fun(fun(X_c,bool),X_b),bool))),finite908156982e_idem(X_b,X_c),F),Z_2),G),F_2))
=> hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),F,X_1),X_1) = ti(X_b,X_1) ) ).
fof(fact_137_empty__Diff,axiom,
! [X_b,A_3] : 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)),bot_bot(fun(X_b,bool))),A_3) = bot_bot(fun(X_b,bool)) ).
fof(fact_138_Diff__empty,axiom,
! [X_b,A_3] : 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_3),bot_bot(fun(X_b,bool))) = ti(fun(X_b,bool),A_3) ).
fof(fact_139_Diff__cancel,axiom,
! [X_b,A_3] : 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_3),A_3) = bot_bot(fun(X_b,bool)) ).
fof(fact_140_finite__Diff2,axiom,
! [X_b,A_3,B_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),B_2))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),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_3),B_2)))
<=> hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3)) ) ) ).
fof(fact_141_insert__Diff__if,axiom,
! [X_b,A_3,X_1,B_2] :
( ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),B_2))
=> 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)),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_1),A_3)),B_2) = 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_3),B_2) )
& ( ~ hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),B_2))
=> 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)),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_1),A_3)),B_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_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_3),B_2)) ) ) ).
fof(fact_142_insert__Diff1,axiom,
! [X_b,A_3,X_1,B_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),B_2))
=> 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)),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_1),A_3)),B_2) = 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_3),B_2) ) ).
fof(fact_143_fold__graph__permute__diff,axiom,
! [X_b] :
( ab_semigroup_mult(X_b)
=> ! [A_2,Ba,A_3,X_1] :
( hBOOL(hAPP(X_b,bool,hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),hAPP(fun(X_b,fun(X_b,X_b)),fun(X_b,fun(fun(X_b,bool),fun(X_b,bool))),finite_fold_graph(X_b,X_b),times_times(X_b)),Ba),A_3),X_1))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),A_2),A_3))
=> ( ~ hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),Ba),A_3))
=> hBOOL(hAPP(X_b,bool,hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),hAPP(fun(X_b,fun(X_b,X_b)),fun(X_b,fun(fun(X_b,bool),fun(X_b,bool))),finite_fold_graph(X_b,X_b),times_times(X_b)),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),Ba),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_3),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_2),bot_bot(fun(X_b,bool)))))),X_1)) ) ) ) ) ).
fof(fact_144_Diff__insert,axiom,
! [X_b,A_3,A_2,B_2] : 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_3),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_2),B_2)) = 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)),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_3),B_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),A_2),bot_bot(fun(X_b,bool)))) ).
fof(fact_145_Diff__insert2,axiom,
! [X_b,A_3,A_2,B_2] : 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_3),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_2),B_2)) = 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)),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_3),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_2),bot_bot(fun(X_b,bool))))),B_2) ).
fof(fact_146_insert__Diff__single,axiom,
! [X_b,A_2,A_3] : hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_2),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_3),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_2),bot_bot(fun(X_b,bool))))) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_2),A_3) ).
fof(fact_147_Diff__insert__absorb,axiom,
! [X_b,X_1,A_3] :
( ~ hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),A_3))
=> 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)),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_1),A_3)),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_1),bot_bot(fun(X_b,bool)))) = ti(fun(X_b,bool),A_3) ) ).
fof(fact_148_insert__Diff,axiom,
! [X_b,A_2,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),A_2),A_3))
=> hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_2),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_3),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_2),bot_bot(fun(X_b,bool))))) = ti(fun(X_b,bool),A_3) ) ).
fof(fact_149_finite__Diff__insert,axiom,
! [X_b,A_3,A_2,B_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),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_3),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_2),B_2))))
<=> hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),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_3),B_2))) ) ).
fof(fact_150_fold__graph__insert__swap,axiom,
! [X_b] :
( ab_semigroup_mult(X_b)
=> ! [Z_2,Ba,A_3,Y_2] :
( hBOOL(hAPP(X_b,bool,hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),hAPP(fun(X_b,fun(X_b,X_b)),fun(X_b,fun(fun(X_b,bool),fun(X_b,bool))),finite_fold_graph(X_b,X_b),times_times(X_b)),Ba),A_3),Y_2))
=> ( ~ hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),Ba),A_3))
=> hBOOL(hAPP(X_b,bool,hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),hAPP(fun(X_b,fun(X_b,X_b)),fun(X_b,fun(fun(X_b,bool),fun(X_b,bool))),finite_fold_graph(X_b,X_b),times_times(X_b)),Z_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),Ba),A_3)),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),Z_2),Y_2))) ) ) ) ).
fof(fact_151_folding__one__idem_Oin__idem,axiom,
! [X_b,X_1,A_3,F,F_2] :
( hBOOL(hAPP(fun(fun(X_b,bool),X_b),bool,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(fun(X_b,bool),X_b),bool),finite2073411215e_idem(X_b),F),F_2))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),A_3))
=> hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),F,X_1),hAPP(fun(X_b,bool),X_b,F_2,A_3)) = hAPP(fun(X_b,bool),X_b,F_2,A_3) ) ) ) ).
fof(fact_152_folding__image__simple__idem_Oin__idem,axiom,
! [X_c,X_b,X_1,A_3,F,Z_2,G,F_2] :
( hBOOL(hAPP(fun(fun(X_c,bool),X_b),bool,hAPP(fun(X_c,X_b),fun(fun(fun(X_c,bool),X_b),bool),hAPP(X_b,fun(fun(X_c,X_b),fun(fun(fun(X_c,bool),X_b),bool)),hAPP(fun(X_b,fun(X_b,X_b)),fun(X_b,fun(fun(X_c,X_b),fun(fun(fun(X_c,bool),X_b),bool))),finite908156982e_idem(X_b,X_c),F),Z_2),G),F_2))
=> ( hBOOL(hAPP(fun(X_c,bool),bool,finite_finite_1(X_c),A_3))
=> ( hBOOL(hAPP(fun(X_c,bool),bool,hAPP(X_c,fun(fun(X_c,bool),bool),member(X_c),X_1),A_3))
=> hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),F,hAPP(X_c,X_b,G,X_1)),hAPP(fun(X_c,bool),X_b,F_2,A_3)) = hAPP(fun(X_c,bool),X_b,F_2,A_3) ) ) ) ).
fof(fact_153_hom__fold1__commute,axiom,
! [X_b] :
( ab_sem1668676832m_mult(X_b)
=> ! [N_3,H] :
( ! [X_2,Y_1] : hAPP(X_b,X_b,H,hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),X_2),Y_1)) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),hAPP(X_b,X_b,H,X_2)),hAPP(X_b,X_b,H,Y_1))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),N_3))
=> ( ti(fun(X_b,bool),N_3) != bot_bot(fun(X_b,bool))
=> hAPP(X_b,X_b,H,hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),finite_fold1(X_b),times_times(X_b)),N_3)) = hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),finite_fold1(X_b),times_times(X_b)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,X_b),fun(fun(X_b,bool),fun(X_b,bool)),image(X_b,X_b),H),N_3)) ) ) ) ) ).
fof(fact_154_fold1__in,axiom,
! [X_b] :
( ab_semigroup_mult(X_b)
=> ! [A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( ti(fun(X_b,bool),A_3) != bot_bot(fun(X_b,bool))
=> ( ! [X_2,Y_1] : hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),X_2),Y_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),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),Y_1),bot_bot(fun(X_b,bool))))))
=> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),finite_fold1(X_b),times_times(X_b)),A_3)),A_3)) ) ) ) ) ).
fof(fact_155_folding__one__idem_Ohom__commute,axiom,
! [X_b,N_3,H,F,F_2] :
( hBOOL(hAPP(fun(fun(X_b,bool),X_b),bool,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(fun(X_b,bool),X_b),bool),finite2073411215e_idem(X_b),F),F_2))
=> ( ! [X_2,Y_1] : hAPP(X_b,X_b,H,hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),F,X_2),Y_1)) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),F,hAPP(X_b,X_b,H,X_2)),hAPP(X_b,X_b,H,Y_1))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),N_3))
=> ( ti(fun(X_b,bool),N_3) != bot_bot(fun(X_b,bool))
=> hAPP(X_b,X_b,H,hAPP(fun(X_b,bool),X_b,F_2,N_3)) = hAPP(fun(X_b,bool),X_b,F_2,hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,X_b),fun(fun(X_b,bool),fun(X_b,bool)),image(X_b,X_b),H),N_3)) ) ) ) ) ).
fof(fact_156_finite__empty__induct,axiom,
! [X_b,Pa,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,Pa,A_3))
=> ( ! [A_5,A_4] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_4))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),A_5),A_4))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,Pa,A_4))
=> hBOOL(hAPP(fun(X_b,bool),bool,Pa,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_4),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_5),bot_bot(fun(X_b,bool)))))) ) ) )
=> hBOOL(hAPP(fun(X_b,bool),bool,Pa,bot_bot(fun(X_b,bool)))) ) ) ) ).
fof(fact_157_folding__image__simple_Oinsert__remove,axiom,
! [X_c,X_b,X_1,A_3,F,Z_2,G,F_2] :
( hBOOL(hAPP(fun(fun(X_c,bool),X_b),bool,hAPP(fun(X_c,X_b),fun(fun(fun(X_c,bool),X_b),bool),hAPP(X_b,fun(fun(X_c,X_b),fun(fun(fun(X_c,bool),X_b),bool)),hAPP(fun(X_b,fun(X_b,X_b)),fun(X_b,fun(fun(X_c,X_b),fun(fun(fun(X_c,bool),X_b),bool))),finite1357897459simple(X_b,X_c),F),Z_2),G),F_2))
=> ( hBOOL(hAPP(fun(X_c,bool),bool,finite_finite_1(X_c),A_3))
=> hAPP(fun(X_c,bool),X_b,F_2,hAPP(fun(X_c,bool),fun(X_c,bool),hAPP(X_c,fun(fun(X_c,bool),fun(X_c,bool)),insert(X_c),X_1),A_3)) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),F,hAPP(X_c,X_b,G,X_1)),hAPP(fun(X_c,bool),X_b,F_2,hAPP(fun(X_c,bool),fun(X_c,bool),hAPP(fun(X_c,bool),fun(fun(X_c,bool),fun(X_c,bool)),minus_minus(fun(X_c,bool)),A_3),hAPP(fun(X_c,bool),fun(X_c,bool),hAPP(X_c,fun(fun(X_c,bool),fun(X_c,bool)),insert(X_c),X_1),bot_bot(fun(X_c,bool)))))) ) ) ).
fof(fact_158_folding__image__simple_Oremove,axiom,
! [X_c,X_b,X_1,A_3,F,Z_2,G,F_2] :
( hBOOL(hAPP(fun(fun(X_c,bool),X_b),bool,hAPP(fun(X_c,X_b),fun(fun(fun(X_c,bool),X_b),bool),hAPP(X_b,fun(fun(X_c,X_b),fun(fun(fun(X_c,bool),X_b),bool)),hAPP(fun(X_b,fun(X_b,X_b)),fun(X_b,fun(fun(X_c,X_b),fun(fun(fun(X_c,bool),X_b),bool))),finite1357897459simple(X_b,X_c),F),Z_2),G),F_2))
=> ( hBOOL(hAPP(fun(X_c,bool),bool,finite_finite_1(X_c),A_3))
=> ( hBOOL(hAPP(fun(X_c,bool),bool,hAPP(X_c,fun(fun(X_c,bool),bool),member(X_c),X_1),A_3))
=> hAPP(fun(X_c,bool),X_b,F_2,A_3) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),F,hAPP(X_c,X_b,G,X_1)),hAPP(fun(X_c,bool),X_b,F_2,hAPP(fun(X_c,bool),fun(X_c,bool),hAPP(fun(X_c,bool),fun(fun(X_c,bool),fun(X_c,bool)),minus_minus(fun(X_c,bool)),A_3),hAPP(fun(X_c,bool),fun(X_c,bool),hAPP(X_c,fun(fun(X_c,bool),fun(X_c,bool)),insert(X_c),X_1),bot_bot(fun(X_c,bool)))))) ) ) ) ).
fof(fact_159_minus__apply,axiom,
! [X_c,X_b] :
( minus(X_b)
=> ! [A_3,B_2,X_1] : hAPP(X_c,X_b,hAPP(fun(X_c,X_b),fun(X_c,X_b),hAPP(fun(X_c,X_b),fun(fun(X_c,X_b),fun(X_c,X_b)),minus_minus(fun(X_c,X_b)),A_3),B_2),X_1) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),minus_minus(X_b),hAPP(X_c,X_b,A_3,X_1)),hAPP(X_c,X_b,B_2,X_1)) ) ).
fof(fact_160_fun__diff__def,axiom,
! [X_b,X_c] :
( minus(X_c)
=> ! [A_3,B_2,X_2] : hAPP(X_b,X_c,hAPP(fun(X_b,X_c),fun(X_b,X_c),hAPP(fun(X_b,X_c),fun(fun(X_b,X_c),fun(X_b,X_c)),minus_minus(fun(X_b,X_c)),A_3),B_2),X_2) = hAPP(X_c,X_c,hAPP(X_c,fun(X_c,X_c),minus_minus(X_c),hAPP(X_b,X_c,A_3,X_2)),hAPP(X_b,X_c,B_2,X_2)) ) ).
fof(fact_161_comp__fun__idem__remove,axiom,
! [X_b] : hBOOL(hAPP(fun(X_b,fun(fun(X_b,bool),fun(X_b,bool))),bool,finite_comp_fun_idem(X_b,fun(X_b,bool)),hAPP(fun(X_b,fun(X_b,bool)),fun(X_b,fun(fun(X_b,bool),fun(X_b,bool))),hAPP(fun(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool))),fun(fun(X_b,fun(X_b,bool)),fun(X_b,fun(fun(X_b,bool),fun(X_b,bool)))),combb(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),X_b),hAPP(fun(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool))),fun(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool))),combc(fun(X_b,bool),fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)))),hAPP(fun(X_b,bool),fun(X_b,fun(X_b,bool)),hAPP(fun(X_b,fun(fun(X_b,bool),fun(X_b,bool))),fun(fun(X_b,bool),fun(X_b,fun(X_b,bool))),combc(X_b,fun(X_b,bool),fun(X_b,bool)),insert(X_b)),bot_bot(fun(X_b,bool)))))) ).
fof(fact_162_comp__fun__idem_Ofun__left__idem,axiom,
! [X_b,X_c,X_1,Z_2,F] :
( hBOOL(hAPP(fun(X_b,fun(X_c,X_c)),bool,finite_comp_fun_idem(X_b,X_c),F))
=> hAPP(X_c,X_c,hAPP(X_b,fun(X_c,X_c),F,X_1),hAPP(X_c,X_c,hAPP(X_b,fun(X_c,X_c),F,X_1),Z_2)) = hAPP(X_c,X_c,hAPP(X_b,fun(X_c,X_c),F,X_1),Z_2) ) ).
fof(fact_163_comp__fun__idem__insert,axiom,
! [X_b] : hBOOL(hAPP(fun(X_b,fun(fun(X_b,bool),fun(X_b,bool))),bool,finite_comp_fun_idem(X_b,fun(X_b,bool)),insert(X_b))) ).
fof(fact_164_comp__fun__idem,axiom,
! [X_b] :
( ab_sem1668676832m_mult(X_b)
=> hBOOL(hAPP(fun(X_b,fun(X_b,X_b)),bool,finite_comp_fun_idem(X_b,X_b),times_times(X_b))) ) ).
fof(fact_165_folding__image__simple_Oempty,axiom,
! [X_c,X_b,F,Z_2,G,F_2] :
( hBOOL(hAPP(fun(fun(X_c,bool),X_b),bool,hAPP(fun(X_c,X_b),fun(fun(fun(X_c,bool),X_b),bool),hAPP(X_b,fun(fun(X_c,X_b),fun(fun(fun(X_c,bool),X_b),bool)),hAPP(fun(X_b,fun(X_b,X_b)),fun(X_b,fun(fun(X_c,X_b),fun(fun(fun(X_c,bool),X_b),bool))),finite1357897459simple(X_b,X_c),F),Z_2),G),F_2))
=> hAPP(fun(X_c,bool),X_b,F_2,bot_bot(fun(X_c,bool))) = ti(X_b,Z_2) ) ).
fof(fact_166_times_Oidem,axiom,
! [X_a] :
( ab_sem1668676832m_mult(X_a)
=> ! [A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),A_1) = ti(X_a,A_1) ) ).
fof(fact_167_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_168_mult__left__idem,axiom,
! [X_a] :
( ab_sem1668676832m_mult(X_a)
=> ! [A_1,B] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B) ) ).
fof(fact_169_folding__image__simple_Oinsert,axiom,
! [X_c,X_b,X_1,A_3,F,Z_2,G,F_2] :
( hBOOL(hAPP(fun(fun(X_c,bool),X_b),bool,hAPP(fun(X_c,X_b),fun(fun(fun(X_c,bool),X_b),bool),hAPP(X_b,fun(fun(X_c,X_b),fun(fun(fun(X_c,bool),X_b),bool)),hAPP(fun(X_b,fun(X_b,X_b)),fun(X_b,fun(fun(X_c,X_b),fun(fun(fun(X_c,bool),X_b),bool))),finite1357897459simple(X_b,X_c),F),Z_2),G),F_2))
=> ( hBOOL(hAPP(fun(X_c,bool),bool,finite_finite_1(X_c),A_3))
=> ( ~ hBOOL(hAPP(fun(X_c,bool),bool,hAPP(X_c,fun(fun(X_c,bool),bool),member(X_c),X_1),A_3))
=> hAPP(fun(X_c,bool),X_b,F_2,hAPP(fun(X_c,bool),fun(X_c,bool),hAPP(X_c,fun(fun(X_c,bool),fun(X_c,bool)),insert(X_c),X_1),A_3)) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),F,hAPP(X_c,X_b,G,X_1)),hAPP(fun(X_c,bool),X_b,F_2,A_3)) ) ) ) ).
fof(fact_170_folding__image__simple_Oneutral,axiom,
! [X_c,X_b,A_3,F,Z_2,G,F_2] :
( hBOOL(hAPP(fun(fun(X_c,bool),X_b),bool,hAPP(fun(X_c,X_b),fun(fun(fun(X_c,bool),X_b),bool),hAPP(X_b,fun(fun(X_c,X_b),fun(fun(fun(X_c,bool),X_b),bool)),hAPP(fun(X_b,fun(X_b,X_b)),fun(X_b,fun(fun(X_c,X_b),fun(fun(fun(X_c,bool),X_b),bool))),finite1357897459simple(X_b,X_c),F),Z_2),G),F_2))
=> ( hBOOL(hAPP(fun(X_c,bool),bool,finite_finite_1(X_c),A_3))
=> ( ! [X_2] :
( hBOOL(hAPP(fun(X_c,bool),bool,hAPP(X_c,fun(fun(X_c,bool),bool),member(X_c),X_2),A_3))
=> hAPP(X_c,X_b,G,X_2) = ti(X_b,Z_2) )
=> hAPP(fun(X_c,bool),X_b,F_2,A_3) = ti(X_b,Z_2) ) ) ) ).
fof(fact_171_comp__fun__commute_Ofold__graph__insertE__aux,axiom,
! [X_b,X_c,A_2,Z_2,A_3,Y_2,F] :
( hBOOL(hAPP(fun(X_b,fun(X_c,X_c)),bool,finite100568337ommute(X_b,X_c),F))
=> ( hBOOL(hAPP(X_c,bool,hAPP(fun(X_b,bool),fun(X_c,bool),hAPP(X_c,fun(fun(X_b,bool),fun(X_c,bool)),hAPP(fun(X_b,fun(X_c,X_c)),fun(X_c,fun(fun(X_b,bool),fun(X_c,bool))),finite_fold_graph(X_b,X_c),F),Z_2),A_3),Y_2))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),A_2),A_3))
=> ? [Y_3] :
( ti(X_c,Y_2) = hAPP(X_c,X_c,hAPP(X_b,fun(X_c,X_c),F,A_2),Y_3)
& hBOOL(hAPP(X_c,bool,hAPP(fun(X_b,bool),fun(X_c,bool),hAPP(X_c,fun(fun(X_b,bool),fun(X_c,bool)),hAPP(fun(X_b,fun(X_c,X_c)),fun(X_c,fun(fun(X_b,bool),fun(X_c,bool))),finite_fold_graph(X_b,X_c),F),Z_2),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_3),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_2),bot_bot(fun(X_b,bool))))),Y_3)) ) ) ) ) ).
fof(fact_172_fold__graph_H_Ointros_I2_J,axiom,
! [X_c,X_b,F,Z_2,Y_2,X_1,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),A_3))
=> ( hBOOL(hAPP(X_c,bool,hAPP(fun(X_b,bool),fun(X_c,bool),hAPP(X_c,fun(fun(X_b,bool),fun(X_c,bool)),hAPP(fun(X_b,fun(X_c,X_c)),fun(X_c,fun(fun(X_b,bool),fun(X_c,bool))),fold_graph(X_b,X_c),F),Z_2),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_3),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_1),bot_bot(fun(X_b,bool))))),Y_2))
=> hBOOL(hAPP(X_c,bool,hAPP(fun(X_b,bool),fun(X_c,bool),hAPP(X_c,fun(fun(X_b,bool),fun(X_c,bool)),hAPP(fun(X_b,fun(X_c,X_c)),fun(X_c,fun(fun(X_b,bool),fun(X_c,bool))),fold_graph(X_b,X_c),F),Z_2),A_3),hAPP(X_c,X_c,hAPP(X_b,fun(X_c,X_c),F,X_1),Y_2))) ) ) ).
fof(fact_173_setsum__diff1,axiom,
! [X_b,X_c] :
( ab_group_add(X_c)
=> ! [F,A_2,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),A_2),A_3))
=> hAPP(fun(X_b,bool),X_c,hAPP(fun(X_b,X_c),fun(fun(X_b,bool),X_c),big_co1399186613setsum(X_b,X_c),F),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_3),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_2),bot_bot(fun(X_b,bool))))) = hAPP(X_c,X_c,hAPP(X_c,fun(X_c,X_c),minus_minus(X_c),hAPP(fun(X_b,bool),X_c,hAPP(fun(X_b,X_c),fun(fun(X_b,bool),X_c),big_co1399186613setsum(X_b,X_c),F),A_3)),hAPP(X_b,X_c,F,A_2)) )
& ( ~ hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),A_2),A_3))
=> hAPP(fun(X_b,bool),X_c,hAPP(fun(X_b,X_c),fun(fun(X_b,bool),X_c),big_co1399186613setsum(X_b,X_c),F),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_3),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_2),bot_bot(fun(X_b,bool))))) = hAPP(fun(X_b,bool),X_c,hAPP(fun(X_b,X_c),fun(fun(X_b,bool),X_c),big_co1399186613setsum(X_b,X_c),F),A_3) ) ) ) ) ).
fof(fact_174_setsum__diff1__ring,axiom,
! [X_b,X_c] :
( ring(X_c)
=> ! [F,A_2,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),A_2),A_3))
=> hAPP(fun(X_b,bool),X_c,hAPP(fun(X_b,X_c),fun(fun(X_b,bool),X_c),big_co1399186613setsum(X_b,X_c),F),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_3),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_2),bot_bot(fun(X_b,bool))))) = hAPP(X_c,X_c,hAPP(X_c,fun(X_c,X_c),minus_minus(X_c),hAPP(fun(X_b,bool),X_c,hAPP(fun(X_b,X_c),fun(fun(X_b,bool),X_c),big_co1399186613setsum(X_b,X_c),F),A_3)),hAPP(X_b,X_c,F,A_2)) ) ) ) ).
fof(fact_175_setprod_Oinsert__remove,axiom,
! [X_b,X_c] :
( comm_monoid_mult(X_c)
=> ! [G,X_1,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> hAPP(fun(X_b,bool),X_c,hAPP(fun(X_b,X_c),fun(fun(X_b,bool),X_c),big_co302546470etprod(X_b,X_c),G),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_1),A_3)) = hAPP(X_c,X_c,hAPP(X_c,fun(X_c,X_c),times_times(X_c),hAPP(X_b,X_c,G,X_1)),hAPP(fun(X_b,bool),X_c,hAPP(fun(X_b,X_c),fun(fun(X_b,bool),X_c),big_co302546470etprod(X_b,X_c),G),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_3),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_1),bot_bot(fun(X_b,bool)))))) ) ) ).
fof(fact_176_setprod_Oremove,axiom,
! [X_b,X_c] :
( comm_monoid_mult(X_c)
=> ! [G,X_1,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),A_3))
=> hAPP(fun(X_b,bool),X_c,hAPP(fun(X_b,X_c),fun(fun(X_b,bool),X_c),big_co302546470etprod(X_b,X_c),G),A_3) = hAPP(X_c,X_c,hAPP(X_c,fun(X_c,X_c),times_times(X_c),hAPP(X_b,X_c,G,X_1)),hAPP(fun(X_b,bool),X_c,hAPP(fun(X_b,X_c),fun(fun(X_b,bool),X_c),big_co302546470etprod(X_b,X_c),G),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_3),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_1),bot_bot(fun(X_b,bool)))))) ) ) ) ).
fof(fact_177_minus__fold__remove,axiom,
! [X_b,B_2,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> 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)),B_2),A_3) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),hAPP(fun(X_b,fun(fun(X_b,bool),fun(X_b,bool))),fun(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool))),finite_fold(X_b,fun(X_b,bool)),hAPP(fun(X_b,fun(X_b,bool)),fun(X_b,fun(fun(X_b,bool),fun(X_b,bool))),hAPP(fun(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool))),fun(fun(X_b,fun(X_b,bool)),fun(X_b,fun(fun(X_b,bool),fun(X_b,bool)))),combb(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),X_b),hAPP(fun(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool))),fun(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool))),combc(fun(X_b,bool),fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)))),hAPP(fun(X_b,bool),fun(X_b,fun(X_b,bool)),hAPP(fun(X_b,fun(fun(X_b,bool),fun(X_b,bool))),fun(fun(X_b,bool),fun(X_b,fun(X_b,bool))),combc(X_b,fun(X_b,bool),fun(X_b,bool)),insert(X_b)),bot_bot(fun(X_b,bool))))),B_2),A_3) ) ).
fof(fact_178_comp__fun__commute_Ofold__fun__comm,axiom,
! [X_b,X_c,X_1,Z_2,A_3,F] :
( hBOOL(hAPP(fun(X_b,fun(X_c,X_c)),bool,finite100568337ommute(X_b,X_c),F))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> hAPP(X_c,X_c,hAPP(X_b,fun(X_c,X_c),F,X_1),hAPP(fun(X_b,bool),X_c,hAPP(X_c,fun(fun(X_b,bool),X_c),hAPP(fun(X_b,fun(X_c,X_c)),fun(X_c,fun(fun(X_b,bool),X_c)),finite_fold(X_b,X_c),F),Z_2),A_3)) = hAPP(fun(X_b,bool),X_c,hAPP(X_c,fun(fun(X_b,bool),X_c),hAPP(fun(X_b,fun(X_c,X_c)),fun(X_c,fun(fun(X_b,bool),X_c)),finite_fold(X_b,X_c),F),hAPP(X_c,X_c,hAPP(X_b,fun(X_c,X_c),F,X_1),Z_2)),A_3) ) ) ).
fof(fact_179_comp__fun__commute_Ofun__left__comm,axiom,
! [X_b,X_c,X_1,Y_2,Z_2,F] :
( hBOOL(hAPP(fun(X_b,fun(X_c,X_c)),bool,finite100568337ommute(X_b,X_c),F))
=> hAPP(X_c,X_c,hAPP(X_b,fun(X_c,X_c),F,X_1),hAPP(X_c,X_c,hAPP(X_b,fun(X_c,X_c),F,Y_2),Z_2)) = hAPP(X_c,X_c,hAPP(X_b,fun(X_c,X_c),F,Y_2),hAPP(X_c,X_c,hAPP(X_b,fun(X_c,X_c),F,X_1),Z_2)) ) ).
fof(fact_180_setsum__commute,axiom,
! [X_c,X_d,X_b] :
( comm_monoid_add(X_b)
=> ! [F,B_2,A_3] : hAPP(fun(X_c,bool),X_b,hAPP(fun(X_c,X_b),fun(fun(X_c,bool),X_b),big_co1399186613setsum(X_c,X_b),hAPP(fun(X_d,bool),fun(X_c,X_b),hAPP(fun(X_c,fun(fun(X_d,bool),X_b)),fun(fun(X_d,bool),fun(X_c,X_b)),combc(X_c,fun(X_d,bool),X_b),hAPP(fun(X_c,fun(X_d,X_b)),fun(X_c,fun(fun(X_d,bool),X_b)),hAPP(fun(fun(X_d,X_b),fun(fun(X_d,bool),X_b)),fun(fun(X_c,fun(X_d,X_b)),fun(X_c,fun(fun(X_d,bool),X_b))),combb(fun(X_d,X_b),fun(fun(X_d,bool),X_b),X_c),big_co1399186613setsum(X_d,X_b)),F)),B_2)),A_3) = hAPP(fun(X_d,bool),X_b,hAPP(fun(X_d,X_b),fun(fun(X_d,bool),X_b),big_co1399186613setsum(X_d,X_b),hAPP(fun(X_c,bool),fun(X_d,X_b),hAPP(fun(X_d,fun(fun(X_c,bool),X_b)),fun(fun(X_c,bool),fun(X_d,X_b)),combc(X_d,fun(X_c,bool),X_b),hAPP(fun(X_d,fun(X_c,X_b)),fun(X_d,fun(fun(X_c,bool),X_b)),hAPP(fun(fun(X_c,X_b),fun(fun(X_c,bool),X_b)),fun(fun(X_d,fun(X_c,X_b)),fun(X_d,fun(fun(X_c,bool),X_b))),combb(fun(X_c,X_b),fun(fun(X_c,bool),X_b),X_d),big_co1399186613setsum(X_c,X_b)),hAPP(fun(X_c,fun(X_d,X_b)),fun(X_d,fun(X_c,X_b)),combc(X_c,X_d,X_b),F))),A_3)),B_2) ) ).
fof(fact_181_comp__fun__commute_Ofold__equality,axiom,
! [X_b,X_c,Z_2,A_3,Y_2,F] :
( hBOOL(hAPP(fun(X_b,fun(X_c,X_c)),bool,finite100568337ommute(X_b,X_c),F))
=> ( hBOOL(hAPP(X_c,bool,hAPP(fun(X_b,bool),fun(X_c,bool),hAPP(X_c,fun(fun(X_b,bool),fun(X_c,bool)),hAPP(fun(X_b,fun(X_c,X_c)),fun(X_c,fun(fun(X_b,bool),fun(X_c,bool))),finite_fold_graph(X_b,X_c),F),Z_2),A_3),Y_2))
=> hAPP(fun(X_b,bool),X_c,hAPP(X_c,fun(fun(X_b,bool),X_c),hAPP(fun(X_b,fun(X_c,X_c)),fun(X_c,fun(fun(X_b,bool),X_c)),finite_fold(X_b,X_c),F),Z_2),A_3) = ti(X_c,Y_2) ) ) ).
fof(fact_182_fold__empty,axiom,
! [X_c,X_b,F,Z_2] : hAPP(fun(X_c,bool),X_b,hAPP(X_b,fun(fun(X_c,bool),X_b),hAPP(fun(X_c,fun(X_b,X_b)),fun(X_b,fun(fun(X_c,bool),X_b)),finite_fold(X_c,X_b),F),Z_2),bot_bot(fun(X_c,bool))) = ti(X_b,Z_2) ).
fof(fact_183_comp__fun__commute_Ofold__insert2,axiom,
! [X_b,X_c,Z_2,X_1,A_3,F] :
( hBOOL(hAPP(fun(X_b,fun(X_c,X_c)),bool,finite100568337ommute(X_b,X_c),F))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( ~ hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),A_3))
=> hAPP(fun(X_b,bool),X_c,hAPP(X_c,fun(fun(X_b,bool),X_c),hAPP(fun(X_b,fun(X_c,X_c)),fun(X_c,fun(fun(X_b,bool),X_c)),finite_fold(X_b,X_c),F),Z_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_1),A_3)) = hAPP(fun(X_b,bool),X_c,hAPP(X_c,fun(fun(X_b,bool),X_c),hAPP(fun(X_b,fun(X_c,X_c)),fun(X_c,fun(fun(X_b,bool),X_c)),finite_fold(X_b,X_c),F),hAPP(X_c,X_c,hAPP(X_b,fun(X_c,X_c),F,X_1),Z_2)),A_3) ) ) ) ).
fof(fact_184_comp__fun__commute_Ofold__insert,axiom,
! [X_b,X_c,Z_2,X_1,A_3,F] :
( hBOOL(hAPP(fun(X_b,fun(X_c,X_c)),bool,finite100568337ommute(X_b,X_c),F))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( ~ hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),A_3))
=> hAPP(fun(X_b,bool),X_c,hAPP(X_c,fun(fun(X_b,bool),X_c),hAPP(fun(X_b,fun(X_c,X_c)),fun(X_c,fun(fun(X_b,bool),X_c)),finite_fold(X_b,X_c),F),Z_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_1),A_3)) = hAPP(X_c,X_c,hAPP(X_b,fun(X_c,X_c),F,X_1),hAPP(fun(X_b,bool),X_c,hAPP(X_c,fun(fun(X_b,bool),X_c),hAPP(fun(X_b,fun(X_c,X_c)),fun(X_c,fun(fun(X_b,bool),X_c)),finite_fold(X_b,X_c),F),Z_2),A_3)) ) ) ) ).
fof(fact_185_comp__fun__commute_Ofold__graph__fold,axiom,
! [X_b,X_c,Z_2,A_3,F] :
( hBOOL(hAPP(fun(X_b,fun(X_c,X_c)),bool,finite100568337ommute(X_b,X_c),F))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> hBOOL(hAPP(X_c,bool,hAPP(fun(X_b,bool),fun(X_c,bool),hAPP(X_c,fun(fun(X_b,bool),fun(X_c,bool)),hAPP(fun(X_b,fun(X_c,X_c)),fun(X_c,fun(fun(X_b,bool),fun(X_c,bool))),finite_fold_graph(X_b,X_c),F),Z_2),A_3),hAPP(fun(X_b,bool),X_c,hAPP(X_c,fun(fun(X_b,bool),X_c),hAPP(fun(X_b,fun(X_c,X_c)),fun(X_c,fun(fun(X_b,bool),X_c)),finite_fold(X_b,X_c),F),Z_2),A_3))) ) ) ).
fof(fact_186_setprod__timesf,axiom,
! [X_c,X_b] :
( comm_monoid_mult(X_b)
=> ! [F,G,A_3] : hAPP(fun(X_c,bool),X_b,hAPP(fun(X_c,X_b),fun(fun(X_c,bool),X_b),big_co302546470etprod(X_c,X_b),hAPP(fun(X_c,X_b),fun(X_c,X_b),hAPP(fun(X_c,fun(X_b,X_b)),fun(fun(X_c,X_b),fun(X_c,X_b)),combs(X_c,X_b,X_b),hAPP(fun(X_c,X_b),fun(X_c,fun(X_b,X_b)),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_c,X_b),fun(X_c,fun(X_b,X_b))),combb(X_b,fun(X_b,X_b),X_c),times_times(X_b)),F)),G)),A_3) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),hAPP(fun(X_c,bool),X_b,hAPP(fun(X_c,X_b),fun(fun(X_c,bool),X_b),big_co302546470etprod(X_c,X_b),F),A_3)),hAPP(fun(X_c,bool),X_b,hAPP(fun(X_c,X_b),fun(fun(X_c,bool),X_b),big_co302546470etprod(X_c,X_b),G),A_3)) ) ).
fof(fact_187_setsum__product,axiom,
! [X_c,X_d,X_b] :
( semiring_0(X_b)
=> ! [F,A_3,G,B_2] : hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),hAPP(fun(X_c,bool),X_b,hAPP(fun(X_c,X_b),fun(fun(X_c,bool),X_b),big_co1399186613setsum(X_c,X_b),F),A_3)),hAPP(fun(X_d,bool),X_b,hAPP(fun(X_d,X_b),fun(fun(X_d,bool),X_b),big_co1399186613setsum(X_d,X_b),G),B_2)) = hAPP(fun(X_c,bool),X_b,hAPP(fun(X_c,X_b),fun(fun(X_c,bool),X_b),big_co1399186613setsum(X_c,X_b),hAPP(fun(X_d,bool),fun(X_c,X_b),hAPP(fun(X_c,fun(fun(X_d,bool),X_b)),fun(fun(X_d,bool),fun(X_c,X_b)),combc(X_c,fun(X_d,bool),X_b),hAPP(fun(X_c,fun(X_d,X_b)),fun(X_c,fun(fun(X_d,bool),X_b)),hAPP(fun(fun(X_d,X_b),fun(fun(X_d,bool),X_b)),fun(fun(X_c,fun(X_d,X_b)),fun(X_c,fun(fun(X_d,bool),X_b))),combb(fun(X_d,X_b),fun(fun(X_d,bool),X_b),X_c),big_co1399186613setsum(X_d,X_b)),hAPP(fun(X_d,X_b),fun(X_c,fun(X_d,X_b)),hAPP(fun(X_c,fun(fun(X_d,X_b),fun(X_d,X_b))),fun(fun(X_d,X_b),fun(X_c,fun(X_d,X_b))),combc(X_c,fun(X_d,X_b),fun(X_d,X_b)),hAPP(fun(X_c,fun(X_b,X_b)),fun(X_c,fun(fun(X_d,X_b),fun(X_d,X_b))),hAPP(fun(fun(X_b,X_b),fun(fun(X_d,X_b),fun(X_d,X_b))),fun(fun(X_c,fun(X_b,X_b)),fun(X_c,fun(fun(X_d,X_b),fun(X_d,X_b)))),combb(fun(X_b,X_b),fun(fun(X_d,X_b),fun(X_d,X_b)),X_c),combb(X_b,X_b,X_d)),hAPP(fun(X_c,X_b),fun(X_c,fun(X_b,X_b)),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_c,X_b),fun(X_c,fun(X_b,X_b))),combb(X_b,fun(X_b,X_b),X_c),times_times(X_b)),F))),G))),B_2)),A_3) ) ).
fof(fact_188_setsum__left__distrib,axiom,
! [X_c,X_b] :
( semiring_0(X_b)
=> ! [F,A_3,R_2] : hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),hAPP(fun(X_c,bool),X_b,hAPP(fun(X_c,X_b),fun(fun(X_c,bool),X_b),big_co1399186613setsum(X_c,X_b),F),A_3)),R_2) = hAPP(fun(X_c,bool),X_b,hAPP(fun(X_c,X_b),fun(fun(X_c,bool),X_b),big_co1399186613setsum(X_c,X_b),hAPP(X_b,fun(X_c,X_b),hAPP(fun(X_c,fun(X_b,X_b)),fun(X_b,fun(X_c,X_b)),combc(X_c,X_b,X_b),hAPP(fun(X_c,X_b),fun(X_c,fun(X_b,X_b)),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_c,X_b),fun(X_c,fun(X_b,X_b))),combb(X_b,fun(X_b,X_b),X_c),times_times(X_b)),F)),R_2)),A_3) ) ).
fof(fact_189_setsum__right__distrib,axiom,
! [X_c,X_b] :
( semiring_0(X_b)
=> ! [R_2,F,A_3] : hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),R_2),hAPP(fun(X_c,bool),X_b,hAPP(fun(X_c,X_b),fun(fun(X_c,bool),X_b),big_co1399186613setsum(X_c,X_b),F),A_3)) = hAPP(fun(X_c,bool),X_b,hAPP(fun(X_c,X_b),fun(fun(X_c,bool),X_b),big_co1399186613setsum(X_c,X_b),hAPP(fun(X_c,X_b),fun(X_c,X_b),hAPP(fun(X_b,X_b),fun(fun(X_c,X_b),fun(X_c,X_b)),combb(X_b,X_b,X_c),hAPP(X_b,fun(X_b,X_b),times_times(X_b),R_2)),F)),A_3) ) ).
fof(fact_190_setsum__subtractf,axiom,
! [X_c,X_b] :
( ab_group_add(X_b)
=> ! [F,G,A_3] : hAPP(fun(X_c,bool),X_b,hAPP(fun(X_c,X_b),fun(fun(X_c,bool),X_b),big_co1399186613setsum(X_c,X_b),hAPP(fun(X_c,X_b),fun(X_c,X_b),hAPP(fun(X_c,fun(X_b,X_b)),fun(fun(X_c,X_b),fun(X_c,X_b)),combs(X_c,X_b,X_b),hAPP(fun(X_c,X_b),fun(X_c,fun(X_b,X_b)),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_c,X_b),fun(X_c,fun(X_b,X_b))),combb(X_b,fun(X_b,X_b),X_c),minus_minus(X_b)),F)),G)),A_3) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),minus_minus(X_b),hAPP(fun(X_c,bool),X_b,hAPP(fun(X_c,X_b),fun(fun(X_c,bool),X_b),big_co1399186613setsum(X_c,X_b),F),A_3)),hAPP(fun(X_c,bool),X_b,hAPP(fun(X_c,X_b),fun(fun(X_c,bool),X_b),big_co1399186613setsum(X_c,X_b),G),A_3)) ) ).
fof(fact_191_fold__graph_H_Ointros_I1_J,axiom,
! [X_c,X_b,F,Z_2] : hBOOL(hAPP(X_c,bool,hAPP(fun(X_b,bool),fun(X_c,bool),hAPP(X_c,fun(fun(X_b,bool),fun(X_c,bool)),hAPP(fun(X_b,fun(X_c,X_c)),fun(X_c,fun(fun(X_b,bool),fun(X_c,bool))),fold_graph(X_b,X_c),F),Z_2),bot_bot(fun(X_b,bool))),Z_2)) ).
fof(fact_192_comp__fun__commute,axiom,
! [X_b] :
( ab_semigroup_mult(X_b)
=> hBOOL(hAPP(fun(X_b,fun(X_b,X_b)),bool,finite100568337ommute(X_b,X_b),times_times(X_b))) ) ).
fof(fact_193_comp__fun__commute_Ofold__graph__determ,axiom,
! [X_b,X_c,Y_2,Z_2,A_3,X_1,F] :
( hBOOL(hAPP(fun(X_b,fun(X_c,X_c)),bool,finite100568337ommute(X_b,X_c),F))
=> ( hBOOL(hAPP(X_c,bool,hAPP(fun(X_b,bool),fun(X_c,bool),hAPP(X_c,fun(fun(X_b,bool),fun(X_c,bool)),hAPP(fun(X_b,fun(X_c,X_c)),fun(X_c,fun(fun(X_b,bool),fun(X_c,bool))),finite_fold_graph(X_b,X_c),F),Z_2),A_3),X_1))
=> ( hBOOL(hAPP(X_c,bool,hAPP(fun(X_b,bool),fun(X_c,bool),hAPP(X_c,fun(fun(X_b,bool),fun(X_c,bool)),hAPP(fun(X_b,fun(X_c,X_c)),fun(X_c,fun(fun(X_b,bool),fun(X_c,bool))),finite_fold_graph(X_b,X_c),F),Z_2),A_3),Y_2))
=> ti(X_c,Y_2) = ti(X_c,X_1) ) ) ) ).
fof(fact_194_setprod_Odistrib,axiom,
! [X_b,X_c] :
( comm_monoid_mult(X_c)
=> ! [G,H,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> hAPP(fun(X_b,bool),X_c,hAPP(fun(X_b,X_c),fun(fun(X_b,bool),X_c),big_co302546470etprod(X_b,X_c),hAPP(fun(X_b,X_c),fun(X_b,X_c),hAPP(fun(X_b,fun(X_c,X_c)),fun(fun(X_b,X_c),fun(X_b,X_c)),combs(X_b,X_c,X_c),hAPP(fun(X_b,X_c),fun(X_b,fun(X_c,X_c)),hAPP(fun(X_c,fun(X_c,X_c)),fun(fun(X_b,X_c),fun(X_b,fun(X_c,X_c))),combb(X_c,fun(X_c,X_c),X_b),times_times(X_c)),G)),H)),A_3) = hAPP(X_c,X_c,hAPP(X_c,fun(X_c,X_c),times_times(X_c),hAPP(fun(X_b,bool),X_c,hAPP(fun(X_b,X_c),fun(fun(X_b,bool),X_c),big_co302546470etprod(X_b,X_c),G),A_3)),hAPP(fun(X_b,bool),X_c,hAPP(fun(X_b,X_c),fun(fun(X_b,bool),X_c),big_co302546470etprod(X_b,X_c),H),A_3)) ) ) ).
fof(fact_195_comp__fun__commute_Ofold__rec,axiom,
! [X_b,X_c,Z_2,X_1,A_3,F] :
( hBOOL(hAPP(fun(X_b,fun(X_c,X_c)),bool,finite100568337ommute(X_b,X_c),F))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),A_3))
=> hAPP(fun(X_b,bool),X_c,hAPP(X_c,fun(fun(X_b,bool),X_c),hAPP(fun(X_b,fun(X_c,X_c)),fun(X_c,fun(fun(X_b,bool),X_c)),finite_fold(X_b,X_c),F),Z_2),A_3) = hAPP(X_c,X_c,hAPP(X_b,fun(X_c,X_c),F,X_1),hAPP(fun(X_b,bool),X_c,hAPP(X_c,fun(fun(X_b,bool),X_c),hAPP(fun(X_b,fun(X_c,X_c)),fun(X_c,fun(fun(X_b,bool),X_c)),finite_fold(X_b,X_c),F),Z_2),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_3),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_1),bot_bot(fun(X_b,bool)))))) ) ) ) ).
fof(fact_196_comp__fun__commute_Ofold__insert__remove,axiom,
! [X_b,X_c,Z_2,X_1,A_3,F] :
( hBOOL(hAPP(fun(X_b,fun(X_c,X_c)),bool,finite100568337ommute(X_b,X_c),F))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> hAPP(fun(X_b,bool),X_c,hAPP(X_c,fun(fun(X_b,bool),X_c),hAPP(fun(X_b,fun(X_c,X_c)),fun(X_c,fun(fun(X_b,bool),X_c)),finite_fold(X_b,X_c),F),Z_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_1),A_3)) = hAPP(X_c,X_c,hAPP(X_b,fun(X_c,X_c),F,X_1),hAPP(fun(X_b,bool),X_c,hAPP(X_c,fun(fun(X_b,bool),X_c),hAPP(fun(X_b,fun(X_c,X_c)),fun(X_c,fun(fun(X_b,bool),X_c)),finite_fold(X_b,X_c),F),Z_2),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_3),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_1),bot_bot(fun(X_b,bool)))))) ) ) ).
fof(fact_197_setprod__insert,axiom,
! [X_b,X_c] :
( comm_monoid_mult(X_c)
=> ! [F,A_2,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( ~ hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),A_2),A_3))
=> hAPP(fun(X_b,bool),X_c,hAPP(fun(X_b,X_c),fun(fun(X_b,bool),X_c),big_co302546470etprod(X_b,X_c),F),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_2),A_3)) = hAPP(X_c,X_c,hAPP(X_c,fun(X_c,X_c),times_times(X_c),hAPP(X_b,X_c,F,A_2)),hAPP(fun(X_b,bool),X_c,hAPP(fun(X_b,X_c),fun(fun(X_b,bool),X_c),big_co302546470etprod(X_b,X_c),F),A_3)) ) ) ) ).
fof(fact_198_setprod_Oinsert,axiom,
! [X_b,X_c] :
( comm_monoid_mult(X_c)
=> ! [G,X_1,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( ~ hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),A_3))
=> hAPP(fun(X_b,bool),X_c,hAPP(fun(X_b,X_c),fun(fun(X_b,bool),X_c),big_co302546470etprod(X_b,X_c),G),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_1),A_3)) = hAPP(X_c,X_c,hAPP(X_c,fun(X_c,X_c),times_times(X_c),hAPP(X_b,X_c,G,X_1)),hAPP(fun(X_b,bool),X_c,hAPP(fun(X_b,X_c),fun(fun(X_b,bool),X_c),big_co302546470etprod(X_b,X_c),G),A_3)) ) ) ) ).
fof(fact_199_setsum__diff1__nat,axiom,
! [X_b,F,A_2,A_3] :
( ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),A_2),A_3))
=> hAPP(fun(X_b,bool),nat,hAPP(fun(X_b,nat),fun(fun(X_b,bool),nat),big_co1399186613setsum(X_b,nat),F),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_3),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_2),bot_bot(fun(X_b,bool))))) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(fun(X_b,bool),nat,hAPP(fun(X_b,nat),fun(fun(X_b,bool),nat),big_co1399186613setsum(X_b,nat),F),A_3)),hAPP(X_b,nat,F,A_2)) )
& ( ~ hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),A_2),A_3))
=> hAPP(fun(X_b,bool),nat,hAPP(fun(X_b,nat),fun(fun(X_b,bool),nat),big_co1399186613setsum(X_b,nat),F),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_3),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_2),bot_bot(fun(X_b,bool))))) = hAPP(fun(X_b,bool),nat,hAPP(fun(X_b,nat),fun(fun(X_b,bool),nat),big_co1399186613setsum(X_b,nat),F),A_3) ) ) ).
fof(fact_200_fold__def,axiom,
! [X_c,X_b,F,Z_2,A_3] : hAPP(fun(X_c,bool),X_b,hAPP(X_b,fun(fun(X_c,bool),X_b),hAPP(fun(X_c,fun(X_b,X_b)),fun(X_b,fun(fun(X_c,bool),X_b)),finite_fold(X_c,X_b),F),Z_2),A_3) = hAPP(fun(X_b,bool),X_b,the(X_b),hAPP(fun(X_c,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_c,bool),fun(X_b,bool)),hAPP(fun(X_c,fun(X_b,X_b)),fun(X_b,fun(fun(X_c,bool),fun(X_b,bool))),finite_fold_graph(X_c,X_b),F),Z_2),A_3)) ).
fof(fact_201_comp__fun__idem_Ofold__insert__idem2,axiom,
! [X_b,X_c,Z_2,X_1,A_3,F] :
( hBOOL(hAPP(fun(X_b,fun(X_c,X_c)),bool,finite_comp_fun_idem(X_b,X_c),F))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> hAPP(fun(X_b,bool),X_c,hAPP(X_c,fun(fun(X_b,bool),X_c),hAPP(fun(X_b,fun(X_c,X_c)),fun(X_c,fun(fun(X_b,bool),X_c)),finite_fold(X_b,X_c),F),Z_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_1),A_3)) = hAPP(fun(X_b,bool),X_c,hAPP(X_c,fun(fun(X_b,bool),X_c),hAPP(fun(X_b,fun(X_c,X_c)),fun(X_c,fun(fun(X_b,bool),X_c)),finite_fold(X_b,X_c),F),hAPP(X_c,X_c,hAPP(X_b,fun(X_c,X_c),F,X_1),Z_2)),A_3) ) ) ).
fof(fact_202_comp__fun__idem_Ofold__insert__idem,axiom,
! [X_b,X_c,Z_2,X_1,A_3,F] :
( hBOOL(hAPP(fun(X_b,fun(X_c,X_c)),bool,finite_comp_fun_idem(X_b,X_c),F))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> hAPP(fun(X_b,bool),X_c,hAPP(X_c,fun(fun(X_b,bool),X_c),hAPP(fun(X_b,fun(X_c,X_c)),fun(X_c,fun(fun(X_b,bool),X_c)),finite_fold(X_b,X_c),F),Z_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_1),A_3)) = hAPP(X_c,X_c,hAPP(X_b,fun(X_c,X_c),F,X_1),hAPP(fun(X_b,bool),X_c,hAPP(X_c,fun(fun(X_b,bool),X_c),hAPP(fun(X_b,fun(X_c,X_c)),fun(X_c,fun(fun(X_b,bool),X_c)),finite_fold(X_b,X_c),F),Z_2),A_3)) ) ) ).
fof(fact_203_folding__one_Oeq__fold_H,axiom,
! [X_b,X_1,A_3,F,F_2] :
( hBOOL(hAPP(fun(fun(X_b,bool),X_b),bool,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(fun(X_b,bool),X_b),bool),finite_folding_one(X_b),F),F_2))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( ~ hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),A_3))
=> hAPP(fun(X_b,bool),X_b,F_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_1),A_3)) = hAPP(fun(X_b,bool),X_b,hAPP(X_b,fun(fun(X_b,bool),X_b),hAPP(fun(X_b,fun(X_b,X_b)),fun(X_b,fun(fun(X_b,bool),X_b)),finite_fold(X_b,X_b),F),X_1),A_3) ) ) ) ).
fof(fact_204_folding__one__idem_Oeq__fold__idem_H,axiom,
! [X_b,A_2,A_3,F,F_2] :
( hBOOL(hAPP(fun(fun(X_b,bool),X_b),bool,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(fun(X_b,bool),X_b),bool),finite2073411215e_idem(X_b),F),F_2))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> hAPP(fun(X_b,bool),X_b,F_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),A_2),A_3)) = hAPP(fun(X_b,bool),X_b,hAPP(X_b,fun(fun(X_b,bool),X_b),hAPP(fun(X_b,fun(X_b,X_b)),fun(X_b,fun(fun(X_b,bool),X_b)),finite_fold(X_b,X_b),F),A_2),A_3) ) ) ).
fof(fact_205_fold1__eq__fold,axiom,
! [X_b] :
( ab_semigroup_mult(X_b)
=> ! [A_2,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( ~ hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),A_2),A_3))
=> hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),finite_fold1(X_b),times_times(X_b)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_2),A_3)) = hAPP(fun(X_b,bool),X_b,hAPP(X_b,fun(fun(X_b,bool),X_b),hAPP(fun(X_b,fun(X_b,X_b)),fun(X_b,fun(fun(X_b,bool),X_b)),finite_fold(X_b,X_b),times_times(X_b)),A_2),A_3) ) ) ) ).
fof(fact_206_fold1__eq__fold__idem,axiom,
! [X_b] :
( ab_sem1668676832m_mult(X_b)
=> ! [A_2,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),finite_fold1(X_b),times_times(X_b)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_2),A_3)) = hAPP(fun(X_b,bool),X_b,hAPP(X_b,fun(fun(X_b,bool),X_b),hAPP(fun(X_b,fun(X_b,X_b)),fun(X_b,fun(fun(X_b,bool),X_b)),finite_fold(X_b,X_b),times_times(X_b)),A_2),A_3) ) ) ).
fof(fact_207_comp__fun__commute_Ofold__graph__insertE,axiom,
! [X_b,X_c,Z_2,X_1,A_3,V,F] :
( hBOOL(hAPP(fun(X_b,fun(X_c,X_c)),bool,finite100568337ommute(X_b,X_c),F))
=> ( hBOOL(hAPP(X_c,bool,hAPP(fun(X_b,bool),fun(X_c,bool),hAPP(X_c,fun(fun(X_b,bool),fun(X_c,bool)),hAPP(fun(X_b,fun(X_c,X_c)),fun(X_c,fun(fun(X_b,bool),fun(X_c,bool))),finite_fold_graph(X_b,X_c),F),Z_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_1),A_3)),V))
=> ( ~ hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),A_3))
=> ~ ! [Y_1] :
( ti(X_c,V) = hAPP(X_c,X_c,hAPP(X_b,fun(X_c,X_c),F,X_1),Y_1)
=> ~ hBOOL(hAPP(X_c,bool,hAPP(fun(X_b,bool),fun(X_c,bool),hAPP(X_c,fun(fun(X_b,bool),fun(X_c,bool)),hAPP(fun(X_b,fun(X_c,X_c)),fun(X_c,fun(fun(X_b,bool),fun(X_c,bool))),finite_fold_graph(X_b,X_c),F),Z_2),A_3),Y_1)) ) ) ) ) ).
fof(fact_208_setsum__image__gen,axiom,
! [X_d,X_b,X_c] :
( comm_monoid_add(X_c)
=> ! [G,F,S_1] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),S_1))
=> hAPP(fun(X_b,bool),X_c,hAPP(fun(X_b,X_c),fun(fun(X_b,bool),X_c),big_co1399186613setsum(X_b,X_c),G),S_1) = hAPP(fun(X_d,bool),X_c,hAPP(fun(X_d,X_c),fun(fun(X_d,bool),X_c),big_co1399186613setsum(X_d,X_c),hAPP(fun(X_d,fun(X_b,bool)),fun(X_d,X_c),hAPP(fun(fun(X_b,bool),X_c),fun(fun(X_d,fun(X_b,bool)),fun(X_d,X_c)),combb(fun(X_b,bool),X_c,X_d),hAPP(fun(X_b,X_c),fun(fun(X_b,bool),X_c),big_co1399186613setsum(X_b,X_c),G)),hAPP(fun(X_d,fun(X_b,bool)),fun(X_d,fun(X_b,bool)),hAPP(fun(fun(X_b,bool),fun(X_b,bool)),fun(fun(X_d,fun(X_b,bool)),fun(X_d,fun(X_b,bool))),combb(fun(X_b,bool),fun(X_b,bool),X_d),collect(X_b)),hAPP(fun(X_d,fun(X_b,bool)),fun(X_d,fun(X_b,bool)),hAPP(fun(fun(X_b,bool),fun(X_b,bool)),fun(fun(X_d,fun(X_b,bool)),fun(X_d,fun(X_b,bool))),combb(fun(X_b,bool),fun(X_b,bool),X_d),hAPP(fun(X_b,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,bool)),combs(X_b,bool,bool),hAPP(fun(X_b,bool),fun(X_b,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,fun(bool,bool))),combb(bool,fun(bool,bool),X_b),fconj),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(fun(X_b,bool),bool)),fun(fun(X_b,bool),fun(X_b,bool)),combc(X_b,fun(X_b,bool),bool),member(X_b)),S_1)))),hAPP(fun(X_b,fun(X_d,bool)),fun(X_d,fun(X_b,bool)),combc(X_b,X_d,bool),hAPP(fun(X_b,X_d),fun(X_b,fun(X_d,bool)),hAPP(fun(X_d,fun(X_d,bool)),fun(fun(X_b,X_d),fun(X_b,fun(X_d,bool))),combb(X_d,fun(X_d,bool),X_b),fequal(X_d)),F)))))),hAPP(fun(X_b,bool),fun(X_d,bool),hAPP(fun(X_b,X_d),fun(fun(X_b,bool),fun(X_d,bool)),image(X_b,X_d),F),S_1)) ) ) ).
fof(fact_209_setprod_Ocong,axiom,
! [X_b,X_c] :
( comm_monoid_mult(X_c)
=> ! [G,H,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( ! [X_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_2),A_3))
=> hAPP(X_b,X_c,G,X_2) = hAPP(X_b,X_c,H,X_2) )
=> hAPP(fun(X_b,bool),X_c,hAPP(fun(X_b,X_c),fun(fun(X_b,bool),X_c),big_co302546470etprod(X_b,X_c),G),A_3) = hAPP(fun(X_b,bool),X_c,hAPP(fun(X_b,X_c),fun(fun(X_b,bool),X_c),big_co302546470etprod(X_b,X_c),H),A_3) ) ) ) ).
fof(fact_210_setsum_Ocong,axiom,
! [X_b,X_c] :
( comm_monoid_add(X_c)
=> ! [G,H,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( ! [X_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_2),A_3))
=> hAPP(X_b,X_c,G,X_2) = hAPP(X_b,X_c,H,X_2) )
=> hAPP(fun(X_b,bool),X_c,hAPP(fun(X_b,X_c),fun(fun(X_b,bool),X_c),big_co1399186613setsum(X_b,X_c),G),A_3) = hAPP(fun(X_b,bool),X_c,hAPP(fun(X_b,X_c),fun(fun(X_b,bool),X_c),big_co1399186613setsum(X_b,X_c),H),A_3) ) ) ) ).
fof(fact_211_setsum__setsum__restrict,axiom,
! [X_c,X_b,X_d] :
( comm_monoid_add(X_d)
=> ! [F,R_1,T_4,S_1] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),S_1))
=> ( hBOOL(hAPP(fun(X_c,bool),bool,finite_finite_1(X_c),T_4))
=> hAPP(fun(X_b,bool),X_d,hAPP(fun(X_b,X_d),fun(fun(X_b,bool),X_d),big_co1399186613setsum(X_b,X_d),hAPP(fun(X_b,fun(X_c,bool)),fun(X_b,X_d),hAPP(fun(X_b,fun(fun(X_c,bool),X_d)),fun(fun(X_b,fun(X_c,bool)),fun(X_b,X_d)),combs(X_b,fun(X_c,bool),X_d),hAPP(fun(X_b,fun(X_c,X_d)),fun(X_b,fun(fun(X_c,bool),X_d)),hAPP(fun(fun(X_c,X_d),fun(fun(X_c,bool),X_d)),fun(fun(X_b,fun(X_c,X_d)),fun(X_b,fun(fun(X_c,bool),X_d))),combb(fun(X_c,X_d),fun(fun(X_c,bool),X_d),X_b),big_co1399186613setsum(X_c,X_d)),F)),hAPP(fun(X_b,fun(X_c,bool)),fun(X_b,fun(X_c,bool)),hAPP(fun(fun(X_c,bool),fun(X_c,bool)),fun(fun(X_b,fun(X_c,bool)),fun(X_b,fun(X_c,bool))),combb(fun(X_c,bool),fun(X_c,bool),X_b),collect(X_c)),hAPP(fun(X_b,fun(X_c,bool)),fun(X_b,fun(X_c,bool)),hAPP(fun(fun(X_c,bool),fun(X_c,bool)),fun(fun(X_b,fun(X_c,bool)),fun(X_b,fun(X_c,bool))),combb(fun(X_c,bool),fun(X_c,bool),X_b),hAPP(fun(X_c,fun(bool,bool)),fun(fun(X_c,bool),fun(X_c,bool)),combs(X_c,bool,bool),hAPP(fun(X_c,bool),fun(X_c,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_c,bool),fun(X_c,fun(bool,bool))),combb(bool,fun(bool,bool),X_c),fconj),hAPP(fun(X_c,bool),fun(X_c,bool),hAPP(fun(X_c,fun(fun(X_c,bool),bool)),fun(fun(X_c,bool),fun(X_c,bool)),combc(X_c,fun(X_c,bool),bool),member(X_c)),T_4)))),R_1)))),S_1) = hAPP(fun(X_c,bool),X_d,hAPP(fun(X_c,X_d),fun(fun(X_c,bool),X_d),big_co1399186613setsum(X_c,X_d),hAPP(fun(X_c,fun(X_b,bool)),fun(X_c,X_d),hAPP(fun(X_c,fun(fun(X_b,bool),X_d)),fun(fun(X_c,fun(X_b,bool)),fun(X_c,X_d)),combs(X_c,fun(X_b,bool),X_d),hAPP(fun(X_c,fun(X_b,X_d)),fun(X_c,fun(fun(X_b,bool),X_d)),hAPP(fun(fun(X_b,X_d),fun(fun(X_b,bool),X_d)),fun(fun(X_c,fun(X_b,X_d)),fun(X_c,fun(fun(X_b,bool),X_d))),combb(fun(X_b,X_d),fun(fun(X_b,bool),X_d),X_c),big_co1399186613setsum(X_b,X_d)),hAPP(fun(X_b,fun(X_c,X_d)),fun(X_c,fun(X_b,X_d)),combc(X_b,X_c,X_d),F))),hAPP(fun(X_c,fun(X_b,bool)),fun(X_c,fun(X_b,bool)),hAPP(fun(fun(X_b,bool),fun(X_b,bool)),fun(fun(X_c,fun(X_b,bool)),fun(X_c,fun(X_b,bool))),combb(fun(X_b,bool),fun(X_b,bool),X_c),collect(X_b)),hAPP(fun(X_c,fun(X_b,bool)),fun(X_c,fun(X_b,bool)),hAPP(fun(fun(X_b,bool),fun(X_b,bool)),fun(fun(X_c,fun(X_b,bool)),fun(X_c,fun(X_b,bool))),combb(fun(X_b,bool),fun(X_b,bool),X_c),hAPP(fun(X_b,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,bool)),combs(X_b,bool,bool),hAPP(fun(X_b,bool),fun(X_b,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,fun(bool,bool))),combb(bool,fun(bool,bool),X_b),fconj),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(fun(X_b,bool),bool)),fun(fun(X_b,bool),fun(X_b,bool)),combc(X_b,fun(X_b,bool),bool),member(X_b)),S_1)))),hAPP(fun(X_b,fun(X_c,bool)),fun(X_c,fun(X_b,bool)),combc(X_b,X_c,bool),R_1))))),T_4) ) ) ) ).
fof(fact_212_setsum_Oremove,axiom,
! [X_b,X_c] :
( comm_monoid_add(X_c)
=> ! [G,X_1,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),A_3))
=> hAPP(fun(X_b,bool),X_c,hAPP(fun(X_b,X_c),fun(fun(X_b,bool),X_c),big_co1399186613setsum(X_b,X_c),G),A_3) = hAPP(X_c,X_c,hAPP(X_c,fun(X_c,X_c),plus_plus(X_c),hAPP(X_b,X_c,G,X_1)),hAPP(fun(X_b,bool),X_c,hAPP(fun(X_b,X_c),fun(fun(X_b,bool),X_c),big_co1399186613setsum(X_b,X_c),G),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_3),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_1),bot_bot(fun(X_b,bool)))))) ) ) ) ).
fof(fact_213_setsum__diff1_H,axiom,
! [X_b,X_c] :
( comm_monoid_add(X_c)
=> ! [F,A_2,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),A_2),A_3))
=> hAPP(fun(X_b,bool),X_c,hAPP(fun(X_b,X_c),fun(fun(X_b,bool),X_c),big_co1399186613setsum(X_b,X_c),F),A_3) = hAPP(X_c,X_c,hAPP(X_c,fun(X_c,X_c),plus_plus(X_c),hAPP(X_b,X_c,F,A_2)),hAPP(fun(X_b,bool),X_c,hAPP(fun(X_b,X_c),fun(fun(X_b,bool),X_c),big_co1399186613setsum(X_b,X_c),F),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_3),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_2),bot_bot(fun(X_b,bool)))))) ) ) ) ).
fof(fact_214_setsum_Oinsert__remove,axiom,
! [X_b,X_c] :
( comm_monoid_add(X_c)
=> ! [G,X_1,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> hAPP(fun(X_b,bool),X_c,hAPP(fun(X_b,X_c),fun(fun(X_b,bool),X_c),big_co1399186613setsum(X_b,X_c),G),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_1),A_3)) = hAPP(X_c,X_c,hAPP(X_c,fun(X_c,X_c),plus_plus(X_c),hAPP(X_b,X_c,G,X_1)),hAPP(fun(X_b,bool),X_c,hAPP(fun(X_b,X_c),fun(fun(X_b,bool),X_c),big_co1399186613setsum(X_b,X_c),G),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_3),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_1),bot_bot(fun(X_b,bool)))))) ) ) ).
fof(fact_215_setsum__addf,axiom,
! [X_c,X_b] :
( comm_monoid_add(X_b)
=> ! [F,G,A_3] : hAPP(fun(X_c,bool),X_b,hAPP(fun(X_c,X_b),fun(fun(X_c,bool),X_b),big_co1399186613setsum(X_c,X_b),hAPP(fun(X_c,X_b),fun(X_c,X_b),hAPP(fun(X_c,fun(X_b,X_b)),fun(fun(X_c,X_b),fun(X_c,X_b)),combs(X_c,X_b,X_b),hAPP(fun(X_c,X_b),fun(X_c,fun(X_b,X_b)),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_c,X_b),fun(X_c,fun(X_b,X_b))),combb(X_b,fun(X_b,X_b),X_c),plus_plus(X_b)),F)),G)),A_3) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),hAPP(fun(X_c,bool),X_b,hAPP(fun(X_c,X_b),fun(fun(X_c,bool),X_b),big_co1399186613setsum(X_c,X_b),F),A_3)),hAPP(fun(X_c,bool),X_b,hAPP(fun(X_c,X_b),fun(fun(X_c,bool),X_b),big_co1399186613setsum(X_c,X_b),G),A_3)) ) ).
fof(fact_216_setsum_Odistrib,axiom,
! [X_b,X_c] :
( comm_monoid_add(X_c)
=> ! [G,H,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> hAPP(fun(X_b,bool),X_c,hAPP(fun(X_b,X_c),fun(fun(X_b,bool),X_c),big_co1399186613setsum(X_b,X_c),hAPP(fun(X_b,X_c),fun(X_b,X_c),hAPP(fun(X_b,fun(X_c,X_c)),fun(fun(X_b,X_c),fun(X_b,X_c)),combs(X_b,X_c,X_c),hAPP(fun(X_b,X_c),fun(X_b,fun(X_c,X_c)),hAPP(fun(X_c,fun(X_c,X_c)),fun(fun(X_b,X_c),fun(X_b,fun(X_c,X_c))),combb(X_c,fun(X_c,X_c),X_b),plus_plus(X_c)),G)),H)),A_3) = hAPP(X_c,X_c,hAPP(X_c,fun(X_c,X_c),plus_plus(X_c),hAPP(fun(X_b,bool),X_c,hAPP(fun(X_b,X_c),fun(fun(X_b,bool),X_c),big_co1399186613setsum(X_b,X_c),G),A_3)),hAPP(fun(X_b,bool),X_c,hAPP(fun(X_b,X_c),fun(fun(X_b,bool),X_c),big_co1399186613setsum(X_b,X_c),H),A_3)) ) ) ).
fof(fact_217_setsum__insert,axiom,
! [X_b,X_c] :
( comm_monoid_add(X_c)
=> ! [F,A_2,F_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),F_2))
=> ( ~ hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),A_2),F_2))
=> hAPP(fun(X_b,bool),X_c,hAPP(fun(X_b,X_c),fun(fun(X_b,bool),X_c),big_co1399186613setsum(X_b,X_c),F),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_2),F_2)) = hAPP(X_c,X_c,hAPP(X_c,fun(X_c,X_c),plus_plus(X_c),hAPP(X_b,X_c,F,A_2)),hAPP(fun(X_b,bool),X_c,hAPP(fun(X_b,X_c),fun(fun(X_b,bool),X_c),big_co1399186613setsum(X_b,X_c),F),F_2)) ) ) ) ).
fof(fact_218_setsum_Oinsert,axiom,
! [X_b,X_c] :
( comm_monoid_add(X_c)
=> ! [G,X_1,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( ~ hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),A_3))
=> hAPP(fun(X_b,bool),X_c,hAPP(fun(X_b,X_c),fun(fun(X_b,bool),X_c),big_co1399186613setsum(X_b,X_c),G),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_1),A_3)) = hAPP(X_c,X_c,hAPP(X_c,fun(X_c,X_c),plus_plus(X_c),hAPP(X_b,X_c,G,X_1)),hAPP(fun(X_b,bool),X_c,hAPP(fun(X_b,X_c),fun(fun(X_b,bool),X_c),big_co1399186613setsum(X_b,X_c),G),A_3)) ) ) ) ).
fof(fact_219_eq__add__iff2,axiom,
! [X_b] :
( ring(X_b)
=> ! [A_2,E,Ca,Ba,D] :
( hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),A_2),E)),Ca) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),Ba),E)),D)
<=> ti(X_b,Ca) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),minus_minus(X_b),Ba),A_2)),E)),D) ) ) ).
fof(fact_220_eq__add__iff1,axiom,
! [X_b] :
( ring(X_b)
=> ! [A_2,E,Ca,Ba,D] :
( hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),A_2),E)),Ca) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),Ba),E)),D)
<=> hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),minus_minus(X_b),A_2),Ba)),E)),Ca) = ti(X_b,D) ) ) ).
fof(fact_221_diff__add__cancel,axiom,
! [X_a] :
( group_add(X_a)
=> ! [A_1,B] : 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_1),B)),B) = ti(X_a,A_1) ) ).
fof(fact_222_add__diff__cancel,axiom,
! [X_a] :
( group_add(X_a)
=> ! [A_1,B] : 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_1),B)),B) = ti(X_a,A_1) ) ).
fof(fact_223_crossproduct__eq,axiom,
! [X_b] :
( semiri456707255roduct(X_b)
=> ! [W_1,Y_2,X_1,Z_2] :
( hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),W_1),Y_2)),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),X_1),Z_2)) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),W_1),Z_2)),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),X_1),Y_2))
<=> ( ti(X_b,W_1) = ti(X_b,X_1)
| ti(X_b,Y_2) = ti(X_b,Z_2) ) ) ) ).
fof(fact_224_combine__common__factor,axiom,
! [X_a] :
( semiring(X_a)
=> ! [A_1,E_1,B,C] : 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_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),E_1)),C)) = 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_1),B)),E_1)),C) ) ).
fof(fact_225_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_1,M,B] : 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),M)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),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_1),B)),M) ) ).
fof(fact_226_add__right__imp__eq,axiom,
! [X_a] :
( cancel_semigroup_add(X_a)
=> ! [B,A_1,C] :
( 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),C),A_1)
=> ti(X_a,B) = ti(X_a,C) ) ) ).
fof(fact_227_add__imp__eq,axiom,
! [X_a] :
( cancel146912293up_add(X_a)
=> ! [A_1,B,C] :
( 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),C)
=> ti(X_a,B) = ti(X_a,C) ) ) ).
fof(fact_228_add__left__imp__eq,axiom,
! [X_a] :
( cancel_semigroup_add(X_a)
=> ! [A_1,B,C] :
( 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),C)
=> ti(X_a,B) = ti(X_a,C) ) ) ).
fof(fact_229_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_1,B,C,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_1),B)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),C),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_1),C)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),B),D_1)) ) ).
fof(fact_230_add__right__cancel,axiom,
! [X_b] :
( cancel_semigroup_add(X_b)
=> ! [Ba,A_2,Ca] :
( hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),Ba),A_2) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),Ca),A_2)
<=> ti(X_b,Ba) = ti(X_b,Ca) ) ) ).
fof(fact_231_add__left__cancel,axiom,
! [X_b] :
( cancel_semigroup_add(X_b)
=> ! [A_2,Ba,Ca] :
( hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),A_2),Ba) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),A_2),Ca)
<=> ti(X_b,Ba) = ti(X_b,Ca) ) ) ).
fof(fact_232_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_1,B,C] : 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_1),B)),C) = 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_1),C)),B) ) ).
fof(fact_233_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [X_a] :
( ab_semigroup_add(X_a)
=> ! [A_1,B,C] : 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_1),B)),C) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),B),C)) ) ).
fof(fact_234_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_1,B,C] : 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_1),B)),C) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),B),C)) ) ).
fof(fact_235_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_1,C,D_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),C),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_1),C)),D_1) ) ).
fof(fact_236_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_1,C,D_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),C),D_1)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),C),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),D_1)) ) ).
fof(fact_237_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_1,C] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),C) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),C),A_1) ) ).
fof(fact_238_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_239_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_240_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_241_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_242_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_243_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [X_a] :
( ab_semigroup_mult(X_a)
=> ! [A_1,B,C] : 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_1),B)),C) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),C)) ) ).
fof(fact_244_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_245_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_246_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_1,B] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),A_1) ) ).
fof(fact_247_diff__eq__diff__eq,axiom,
! [X_b] :
( ab_group_add(X_b)
=> ! [A_2,Ba,Ca,D] :
( hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),minus_minus(X_b),A_2),Ba) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),minus_minus(X_b),Ca),D)
=> ( ti(X_b,A_2) = ti(X_b,Ba)
<=> ti(X_b,Ca) = ti(X_b,D) ) ) ) ).
fof(fact_248_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_249_crossproduct__noteq,axiom,
! [X_b] :
( semiri456707255roduct(X_b)
=> ! [Ca,D,A_2,Ba] :
( ( ti(X_b,A_2) != ti(X_b,Ba)
& ti(X_b,Ca) != ti(X_b,D) )
<=> hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),A_2),Ca)),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),Ba),D)) != hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),A_2),D)),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),Ba),Ca)) ) ) ).
fof(fact_250_comm__semiring__class_Odistrib,axiom,
! [X_a] :
( comm_semiring(X_a)
=> ! [A_1,B,C] : 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_1),B)),C) = 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),C)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),C)) ) ).
fof(fact_251_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_1,B,C] : 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_1),B)),C) = 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),C)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),C)) ) ).
fof(fact_252_setprod__cong,axiom,
! [X_b,X_c] :
( comm_monoid_mult(X_c)
=> ! [F,G,A_3,B_2] :
( ti(fun(X_b,bool),A_3) = ti(fun(X_b,bool),B_2)
=> ( ! [X_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_2),B_2))
=> hAPP(X_b,X_c,F,X_2) = hAPP(X_b,X_c,G,X_2) )
=> hAPP(fun(X_b,bool),X_c,hAPP(fun(X_b,X_c),fun(fun(X_b,bool),X_c),big_co302546470etprod(X_b,X_c),F),A_3) = hAPP(fun(X_b,bool),X_c,hAPP(fun(X_b,X_c),fun(fun(X_b,bool),X_c),big_co302546470etprod(X_b,X_c),G),B_2) ) ) ) ).
fof(fact_253_setprod_OF__cong,axiom,
! [X_b,X_c] :
( comm_monoid_mult(X_c)
=> ! [H,G,A_3,B_2] :
( ti(fun(X_b,bool),A_3) = ti(fun(X_b,bool),B_2)
=> ( ! [X_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_2),B_2))
=> hAPP(X_b,X_c,H,X_2) = hAPP(X_b,X_c,G,X_2) )
=> hAPP(fun(X_b,bool),X_c,hAPP(fun(X_b,X_c),fun(fun(X_b,bool),X_c),big_co302546470etprod(X_b,X_c),H),A_3) = hAPP(fun(X_b,bool),X_c,hAPP(fun(X_b,X_c),fun(fun(X_b,bool),X_c),big_co302546470etprod(X_b,X_c),G),B_2) ) ) ) ).
fof(fact_254_setsum__cong2,axiom,
! [X_b,X_c] :
( comm_monoid_add(X_c)
=> ! [F,G,A_3] :
( ! [X_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_2),A_3))
=> hAPP(X_b,X_c,F,X_2) = hAPP(X_b,X_c,G,X_2) )
=> hAPP(fun(X_b,bool),X_c,hAPP(fun(X_b,X_c),fun(fun(X_b,bool),X_c),big_co1399186613setsum(X_b,X_c),F),A_3) = hAPP(fun(X_b,bool),X_c,hAPP(fun(X_b,X_c),fun(fun(X_b,bool),X_c),big_co1399186613setsum(X_b,X_c),G),A_3) ) ) ).
fof(fact_255_setsum__cong,axiom,
! [X_b,X_c] :
( comm_monoid_add(X_c)
=> ! [F,G,A_3,B_2] :
( ti(fun(X_b,bool),A_3) = ti(fun(X_b,bool),B_2)
=> ( ! [X_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_2),B_2))
=> hAPP(X_b,X_c,F,X_2) = hAPP(X_b,X_c,G,X_2) )
=> hAPP(fun(X_b,bool),X_c,hAPP(fun(X_b,X_c),fun(fun(X_b,bool),X_c),big_co1399186613setsum(X_b,X_c),F),A_3) = hAPP(fun(X_b,bool),X_c,hAPP(fun(X_b,X_c),fun(fun(X_b,bool),X_c),big_co1399186613setsum(X_b,X_c),G),B_2) ) ) ) ).
fof(fact_256_setsum_OF__cong,axiom,
! [X_b,X_c] :
( comm_monoid_add(X_c)
=> ! [H,G,A_3,B_2] :
( ti(fun(X_b,bool),A_3) = ti(fun(X_b,bool),B_2)
=> ( ! [X_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_2),B_2))
=> hAPP(X_b,X_c,H,X_2) = hAPP(X_b,X_c,G,X_2) )
=> hAPP(fun(X_b,bool),X_c,hAPP(fun(X_b,X_c),fun(fun(X_b,bool),X_c),big_co1399186613setsum(X_b,X_c),H),A_3) = hAPP(fun(X_b,bool),X_c,hAPP(fun(X_b,X_c),fun(fun(X_b,bool),X_c),big_co1399186613setsum(X_b,X_c),G),B_2) ) ) ) ).
fof(fact_257_single__Diff__lessThan,axiom,
! [X_b] :
( order(X_b)
=> ! [K_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)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),K_1),bot_bot(fun(X_b,bool)))),hAPP(X_b,fun(X_b,bool),ord_lessThan(X_b),K_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),K_1),bot_bot(fun(X_b,bool))) ) ).
fof(fact_258_setsum__diff__nat,axiom,
! [X_b,F,A_3,B_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),B_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_2),A_3))
=> hAPP(fun(X_b,bool),nat,hAPP(fun(X_b,nat),fun(fun(X_b,bool),nat),big_co1399186613setsum(X_b,nat),F),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_3),B_2)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(fun(X_b,bool),nat,hAPP(fun(X_b,nat),fun(fun(X_b,bool),nat),big_co1399186613setsum(X_b,nat),F),A_3)),hAPP(fun(X_b,bool),nat,hAPP(fun(X_b,nat),fun(fun(X_b,bool),nat),big_co1399186613setsum(X_b,nat),F),B_2)) ) ) ).
fof(fact_259_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_260_equalityI,axiom,
! [X_b,A_3,B_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)),A_3),B_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_2),A_3))
=> ti(fun(X_b,bool),A_3) = ti(fun(X_b,bool),B_2) ) ) ).
fof(fact_261_subsetD,axiom,
! [X_b,Ca,A_3,B_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)),A_3),B_2))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),Ca),A_3))
=> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),Ca),B_2)) ) ) ).
fof(fact_262_empty__subsetI,axiom,
! [X_b,A_3] : 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)),bot_bot(fun(X_b,bool))),A_3)) ).
fof(fact_263_finite__Collect__subsets,axiom,
! [X_b,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> hBOOL(hAPP(fun(fun(X_b,bool),bool),bool,finite_finite_1(fun(X_b,bool)),hAPP(fun(fun(X_b,bool),bool),fun(fun(X_b,bool),bool),collect(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),hAPP(fun(fun(X_b,bool),fun(fun(X_b,bool),bool)),fun(fun(X_b,bool),fun(fun(X_b,bool),bool)),combc(fun(X_b,bool),fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool))),A_3)))) ) ).
fof(fact_264_pred__subset__eq,axiom,
! [X_b,R_1,S_1] :
( 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)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(fun(X_b,bool),bool)),fun(fun(X_b,bool),fun(X_b,bool)),combc(X_b,fun(X_b,bool),bool),member(X_b)),R_1)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(fun(X_b,bool),bool)),fun(fun(X_b,bool),fun(X_b,bool)),combc(X_b,fun(X_b,bool),bool),member(X_b)),S_1)))
<=> 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)),R_1),S_1)) ) ).
fof(fact_265_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_266_le__funE,axiom,
! [X_b,X_c] :
( ord(X_c)
=> ! [X_1,F,G] :
( hBOOL(hAPP(fun(X_b,X_c),bool,hAPP(fun(X_b,X_c),fun(fun(X_b,X_c),bool),ord_less_eq(fun(X_b,X_c)),F),G))
=> hBOOL(hAPP(X_c,bool,hAPP(X_c,fun(X_c,bool),ord_less_eq(X_c),hAPP(X_b,X_c,F,X_1)),hAPP(X_b,X_c,G,X_1))) ) ) ).
fof(fact_267_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_268_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_269_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_270_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_271_xt1_I4_J,axiom,
! [X_a] :
( order(X_a)
=> ! [C,B,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),A_1))
=> ( ti(X_a,B) = ti(X_a,C)
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C),A_1)) ) ) ) ).
fof(fact_272_ord__le__eq__trans,axiom,
! [X_a] :
( ord(X_a)
=> ! [C,A_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B))
=> ( ti(X_a,B) = ti(X_a,C)
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),C)) ) ) ) ).
fof(fact_273_xt1_I3_J,axiom,
! [X_a] :
( order(X_a)
=> ! [C,A_1,B] :
( ti(X_a,A_1) = ti(X_a,B)
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C),B))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C),A_1)) ) ) ) ).
fof(fact_274_ord__eq__le__trans,axiom,
! [X_a] :
( ord(X_a)
=> ! [C,A_1,B] :
( ti(X_a,A_1) = ti(X_a,B)
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),C))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),C)) ) ) ) ).
fof(fact_275_order__antisym__conv,axiom,
! [X_b] :
( order(X_b)
=> ! [Y_2,X_1] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),Y_2),X_1))
=> ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),X_1),Y_2))
<=> ti(X_b,X_1) = ti(X_b,Y_2) ) ) ) ).
fof(fact_276_le__funD,axiom,
! [X_b,X_c] :
( ord(X_c)
=> ! [X_1,F,G] :
( hBOOL(hAPP(fun(X_b,X_c),bool,hAPP(fun(X_b,X_c),fun(fun(X_b,X_c),bool),ord_less_eq(fun(X_b,X_c)),F),G))
=> hBOOL(hAPP(X_c,bool,hAPP(X_c,fun(X_c,bool),ord_less_eq(X_c),hAPP(X_b,X_c,F,X_1)),hAPP(X_b,X_c,G,X_1))) ) ) ).
fof(fact_277_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_278_order__eq__iff,axiom,
! [X_b] :
( order(X_b)
=> ! [X_1,Y_2] :
( ti(X_b,X_1) = ti(X_b,Y_2)
<=> ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),X_1),Y_2))
& hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),Y_2),X_1)) ) ) ) ).
fof(fact_279_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_280_le__fun__def,axiom,
! [X_b,X_c] :
( ord(X_c)
=> ! [F,G] :
( hBOOL(hAPP(fun(X_b,X_c),bool,hAPP(fun(X_b,X_c),fun(fun(X_b,X_c),bool),ord_less_eq(fun(X_b,X_c)),F),G))
<=> ! [X_2] : hBOOL(hAPP(X_c,bool,hAPP(X_c,fun(X_c,bool),ord_less_eq(X_c),hAPP(X_b,X_c,F,X_2)),hAPP(X_b,X_c,G,X_2))) ) ) ).
fof(fact_281_equalityE,axiom,
! [X_b,A_3,B_2] :
( ti(fun(X_b,bool),A_3) = ti(fun(X_b,bool),B_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)),A_3),B_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_2),A_3)) ) ) ).
fof(fact_282_subset__trans,axiom,
! [X_b,C_1,A_3,B_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)),A_3),B_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_2),C_1))
=> 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)),A_3),C_1)) ) ) ).
fof(fact_283_set__mp,axiom,
! [X_b,X_1,A_3,B_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)),A_3),B_2))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),A_3))
=> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),B_2)) ) ) ).
fof(fact_284_set__rev__mp,axiom,
! [X_b,B_2,X_1,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),A_3))
=> ( 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)),A_3),B_2))
=> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),B_2)) ) ) ).
fof(fact_285_in__mono,axiom,
! [X_b,X_1,A_3,B_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)),A_3),B_2))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),A_3))
=> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),B_2)) ) ) ).
fof(fact_286_equalityD2,axiom,
! [X_b,A_3,B_2] :
( ti(fun(X_b,bool),A_3) = ti(fun(X_b,bool),B_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_2),A_3)) ) ).
fof(fact_287_equalityD1,axiom,
! [X_b,A_3,B_2] :
( ti(fun(X_b,bool),A_3) = ti(fun(X_b,bool),B_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)),A_3),B_2)) ) ).
fof(fact_288_set__eq__subset,axiom,
! [X_b,A_3,B_2] :
( ti(fun(X_b,bool),A_3) = ti(fun(X_b,bool),B_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)),A_3),B_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_2),A_3)) ) ) ).
fof(fact_289_subset__refl,axiom,
! [X_b,A_3] : 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)),A_3),A_3)) ).
fof(fact_290_predicate1D,axiom,
! [X_b,X_1,Pa,Q_3] :
( 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)),Pa),Q_3))
=> ( hBOOL(hAPP(X_b,bool,Pa,X_1))
=> hBOOL(hAPP(X_b,bool,Q_3,X_1)) ) ) ).
fof(fact_291_rev__predicate1D,axiom,
! [X_b,Q_3,Pa,X_1] :
( hBOOL(hAPP(X_b,bool,Pa,X_1))
=> ( 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)),Pa),Q_3))
=> hBOOL(hAPP(X_b,bool,Q_3,X_1)) ) ) ).
fof(fact_292_lessThan__subset__iff,axiom,
! [X_b] :
( linorder(X_b)
=> ! [X_1,Y_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)),hAPP(X_b,fun(X_b,bool),ord_lessThan(X_b),X_1)),hAPP(X_b,fun(X_b,bool),ord_lessThan(X_b),Y_2)))
<=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),X_1),Y_2)) ) ) ).
fof(fact_293_lessThan__eq__iff,axiom,
! [X_b] :
( linorder(X_b)
=> ! [X_1,Y_2] :
( hAPP(X_b,fun(X_b,bool),ord_lessThan(X_b),X_1) = hAPP(X_b,fun(X_b,bool),ord_lessThan(X_b),Y_2)
<=> ti(X_b,X_1) = ti(X_b,Y_2) ) ) ).
fof(fact_294_add__le__imp__le__left,axiom,
! [X_a] :
( ordere236663937imp_le(X_a)
=> ! [C,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),C),A_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),C),B)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B)) ) ) ).
fof(fact_295_add__le__imp__le__right,axiom,
! [X_a] :
( ordere236663937imp_le(X_a)
=> ! [A_1,C,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),C)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),B),C)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B)) ) ) ).
fof(fact_296_add__mono,axiom,
! [X_a] :
( ordere779506340up_add(X_a)
=> ! [C,D_1,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_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C),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_1),C)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),B),D_1))) ) ) ) ).
fof(fact_297_add__left__mono,axiom,
! [X_a] :
( ordere779506340up_add(X_a)
=> ! [C,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_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),A_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),C),B))) ) ) ).
fof(fact_298_add__right__mono,axiom,
! [X_a] :
( ordere779506340up_add(X_a)
=> ! [C,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_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),C)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),B),C))) ) ) ).
fof(fact_299_add__le__cancel__left,axiom,
! [X_b] :
( ordere236663937imp_le(X_b)
=> ! [Ca,A_2,Ba] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),Ca),A_2)),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),Ca),Ba)))
<=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),A_2),Ba)) ) ) ).
fof(fact_300_add__le__cancel__right,axiom,
! [X_b] :
( ordere236663937imp_le(X_b)
=> ! [A_2,Ca,Ba] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),A_2),Ca)),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),Ba),Ca)))
<=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),A_2),Ba)) ) ) ).
fof(fact_301_le__bot,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_302_bot__unique,axiom,
! [X_b] :
( bot(X_b)
=> ! [A_2] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),A_2),bot_bot(X_b)))
<=> ti(X_b,A_2) = bot_bot(X_b) ) ) ).
fof(fact_303_bot__least,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),bot_bot(X_a)),A_1)) ) ).
fof(fact_304_diff__eq__diff__less__eq,axiom,
! [X_b] :
( ordered_ab_group_add(X_b)
=> ! [A_2,Ba,Ca,D] :
( hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),minus_minus(X_b),A_2),Ba) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),minus_minus(X_b),Ca),D)
=> ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),A_2),Ba))
<=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),Ca),D)) ) ) ) ).
fof(fact_305_subset__empty,axiom,
! [X_b,A_3] :
( 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)),A_3),bot_bot(fun(X_b,bool))))
<=> ti(fun(X_b,bool),A_3) = bot_bot(fun(X_b,bool)) ) ).
fof(fact_306_finite__subset,axiom,
! [X_b,A_3,B_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)),A_3),B_2))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),B_2))
=> hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3)) ) ) ).
fof(fact_307_rev__finite__subset,axiom,
! [X_b,A_3,B_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),B_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)),A_3),B_2))
=> hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3)) ) ) ).
fof(fact_308_insert__mono,axiom,
! [X_b,A_2,C_1,D_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)),C_1),D_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)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_2),C_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),A_2),D_2))) ) ).
fof(fact_309_subset__insertI2,axiom,
! [X_b,Ba,A_3,B_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)),A_3),B_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)),A_3),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),Ba),B_2))) ) ).
fof(fact_310_subset__insert,axiom,
! [X_b,B_2,X_1,A_3] :
( ~ hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),A_3))
=> ( 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)),A_3),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_1),B_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)),A_3),B_2)) ) ) ).
fof(fact_311_insert__subset,axiom,
! [X_b,X_1,A_3,B_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)),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_1),A_3)),B_2))
<=> ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),B_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)),A_3),B_2)) ) ) ).
fof(fact_312_subset__insertI,axiom,
! [X_b,B_2,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_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),A_2),B_2))) ).
fof(fact_313_image__mono,axiom,
! [X_c,X_b,F,A_3,B_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)),A_3),B_2))
=> hBOOL(hAPP(fun(X_c,bool),bool,hAPP(fun(X_c,bool),fun(fun(X_c,bool),bool),ord_less_eq(fun(X_c,bool)),hAPP(fun(X_b,bool),fun(X_c,bool),hAPP(fun(X_b,X_c),fun(fun(X_b,bool),fun(X_c,bool)),image(X_b,X_c),F),A_3)),hAPP(fun(X_b,bool),fun(X_c,bool),hAPP(fun(X_b,X_c),fun(fun(X_b,bool),fun(X_c,bool)),image(X_b,X_c),F),B_2))) ) ).
fof(fact_314_subset__image__iff,axiom,
! [X_b,X_c,B_2,F,A_3] :
( 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_2),hAPP(fun(X_c,bool),fun(X_b,bool),hAPP(fun(X_c,X_b),fun(fun(X_c,bool),fun(X_b,bool)),image(X_c,X_b),F),A_3)))
<=> ? [AA] :
( hBOOL(hAPP(fun(X_c,bool),bool,hAPP(fun(X_c,bool),fun(fun(X_c,bool),bool),ord_less_eq(fun(X_c,bool)),AA),A_3))
& ti(fun(X_b,bool),B_2) = hAPP(fun(X_c,bool),fun(X_b,bool),hAPP(fun(X_c,X_b),fun(fun(X_c,bool),fun(X_b,bool)),image(X_c,X_b),F),AA) ) ) ).
fof(fact_315_double__diff,axiom,
! [X_b,C_1,A_3,B_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)),A_3),B_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_2),C_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)),B_2),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)),C_1),A_3)) = ti(fun(X_b,bool),A_3) ) ) ).
fof(fact_316_Diff__mono,axiom,
! [X_b,D_2,B_2,A_3,C_1] :
( 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)),A_3),C_1))
=> ( 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)),D_2),B_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)),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_3),B_2)),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)),C_1),D_2))) ) ) ).
fof(fact_317_Diff__subset,axiom,
! [X_b,A_3,B_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)),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_3),B_2)),A_3)) ).
fof(fact_318_weaken,axiom,
! [X_b,Ts,Ga,Ts_1] :
( hBOOL(hAPP(fun(hoare_2118899576triple(X_b),bool),bool,hAPP(fun(hoare_2118899576triple(X_b),bool),fun(fun(hoare_2118899576triple(X_b),bool),bool),hoare_1301688828derivs(X_b),Ga),Ts_1))
=> ( hBOOL(hAPP(fun(hoare_2118899576triple(X_b),bool),bool,hAPP(fun(hoare_2118899576triple(X_b),bool),fun(fun(hoare_2118899576triple(X_b),bool),bool),ord_less_eq(fun(hoare_2118899576triple(X_b),bool)),Ts),Ts_1))
=> hBOOL(hAPP(fun(hoare_2118899576triple(X_b),bool),bool,hAPP(fun(hoare_2118899576triple(X_b),bool),fun(fun(hoare_2118899576triple(X_b),bool),bool),hoare_1301688828derivs(X_b),Ga),Ts)) ) ) ).
fof(fact_319_asm,axiom,
! [X_b,Ts,Ga] :
( hBOOL(hAPP(fun(hoare_2118899576triple(X_b),bool),bool,hAPP(fun(hoare_2118899576triple(X_b),bool),fun(fun(hoare_2118899576triple(X_b),bool),bool),ord_less_eq(fun(hoare_2118899576triple(X_b),bool)),Ts),Ga))
=> hBOOL(hAPP(fun(hoare_2118899576triple(X_b),bool),bool,hAPP(fun(hoare_2118899576triple(X_b),bool),fun(fun(hoare_2118899576triple(X_b),bool),bool),hoare_1301688828derivs(X_b),Ga),Ts)) ) ).
fof(fact_320_subset__singletonD,axiom,
! [X_b,A_3,X_1] :
( 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)),A_3),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_1),bot_bot(fun(X_b,bool)))))
=> ( ti(fun(X_b,bool),A_3) = bot_bot(fun(X_b,bool))
| ti(fun(X_b,bool),A_3) = 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_1),bot_bot(fun(X_b,bool))) ) ) ).
fof(fact_321_finite__surj,axiom,
! [X_c,X_b,B_2,F,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( hBOOL(hAPP(fun(X_c,bool),bool,hAPP(fun(X_c,bool),fun(fun(X_c,bool),bool),ord_less_eq(fun(X_c,bool)),B_2),hAPP(fun(X_b,bool),fun(X_c,bool),hAPP(fun(X_b,X_c),fun(fun(X_b,bool),fun(X_c,bool)),image(X_b,X_c),F),A_3)))
=> hBOOL(hAPP(fun(X_c,bool),bool,finite_finite_1(X_c),B_2)) ) ) ).
fof(fact_322_image__diff__subset,axiom,
! [X_b,X_c,F,A_3,B_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)),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)),hAPP(fun(X_c,bool),fun(X_b,bool),hAPP(fun(X_c,X_b),fun(fun(X_c,bool),fun(X_b,bool)),image(X_c,X_b),F),A_3)),hAPP(fun(X_c,bool),fun(X_b,bool),hAPP(fun(X_c,X_b),fun(fun(X_c,bool),fun(X_b,bool)),image(X_c,X_b),F),B_2))),hAPP(fun(X_c,bool),fun(X_b,bool),hAPP(fun(X_c,X_b),fun(fun(X_c,bool),fun(X_b,bool)),image(X_c,X_b),F),hAPP(fun(X_c,bool),fun(X_c,bool),hAPP(fun(X_c,bool),fun(fun(X_c,bool),fun(X_c,bool)),minus_minus(fun(X_c,bool)),A_3),B_2)))) ).
fof(fact_323_folding__image__simple__idem_Osubset__idem,axiom,
! [X_c,X_b,B_2,A_3,F,Z_2,G,F_2] :
( hBOOL(hAPP(fun(fun(X_c,bool),X_b),bool,hAPP(fun(X_c,X_b),fun(fun(fun(X_c,bool),X_b),bool),hAPP(X_b,fun(fun(X_c,X_b),fun(fun(fun(X_c,bool),X_b),bool)),hAPP(fun(X_b,fun(X_b,X_b)),fun(X_b,fun(fun(X_c,X_b),fun(fun(fun(X_c,bool),X_b),bool))),finite908156982e_idem(X_b,X_c),F),Z_2),G),F_2))
=> ( hBOOL(hAPP(fun(X_c,bool),bool,finite_finite_1(X_c),A_3))
=> ( hBOOL(hAPP(fun(X_c,bool),bool,hAPP(fun(X_c,bool),fun(fun(X_c,bool),bool),ord_less_eq(fun(X_c,bool)),B_2),A_3))
=> hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),F,hAPP(fun(X_c,bool),X_b,F_2,B_2)),hAPP(fun(X_c,bool),X_b,F_2,A_3)) = hAPP(fun(X_c,bool),X_b,F_2,A_3) ) ) ) ).
fof(fact_324_le__add__iff1,axiom,
! [X_b] :
( ordered_ring(X_b)
=> ! [A_2,E,Ca,Ba,D] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),A_2),E)),Ca)),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),Ba),E)),D)))
<=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),minus_minus(X_b),A_2),Ba)),E)),Ca)),D)) ) ) ).
fof(fact_325_le__add__iff2,axiom,
! [X_b] :
( ordered_ring(X_b)
=> ! [A_2,E,Ca,Ba,D] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),A_2),E)),Ca)),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),Ba),E)),D)))
<=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),Ca),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),minus_minus(X_b),Ba),A_2)),E)),D))) ) ) ).
fof(fact_326_diff__single__insert,axiom,
! [X_b,A_3,X_1,B_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)),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_3),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_1),bot_bot(fun(X_b,bool))))),B_2))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),A_3))
=> 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)),A_3),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_1),B_2))) ) ) ).
fof(fact_327_subset__insert__iff,axiom,
! [X_b,A_3,X_1,B_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)),A_3),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_1),B_2)))
<=> ( ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),A_3))
=> 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)),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_3),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_1),bot_bot(fun(X_b,bool))))),B_2)) )
& ( ~ hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),A_3))
=> 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)),A_3),B_2)) ) ) ) ).
fof(fact_328_folding__one__idem_Osubset__idem,axiom,
! [X_b,B_2,A_3,F,F_2] :
( hBOOL(hAPP(fun(fun(X_b,bool),X_b),bool,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(fun(X_b,bool),X_b),bool),finite2073411215e_idem(X_b),F),F_2))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( ti(fun(X_b,bool),B_2) != bot_bot(fun(X_b,bool))
=> ( 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_2),A_3))
=> hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),F,hAPP(fun(X_b,bool),X_b,F_2,B_2)),hAPP(fun(X_b,bool),X_b,F_2,A_3)) = hAPP(fun(X_b,bool),X_b,F_2,A_3) ) ) ) ) ).
fof(fact_329_setsum__diff,axiom,
! [X_b,X_c] :
( ab_group_add(X_c)
=> ! [F,B_2,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( 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_2),A_3))
=> hAPP(fun(X_b,bool),X_c,hAPP(fun(X_b,X_c),fun(fun(X_b,bool),X_c),big_co1399186613setsum(X_b,X_c),F),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_3),B_2)) = hAPP(X_c,X_c,hAPP(X_c,fun(X_c,X_c),minus_minus(X_c),hAPP(fun(X_b,bool),X_c,hAPP(fun(X_b,X_c),fun(fun(X_b,bool),X_c),big_co1399186613setsum(X_b,X_c),F),A_3)),hAPP(fun(X_b,bool),X_c,hAPP(fun(X_b,X_c),fun(fun(X_b,bool),X_c),big_co1399186613setsum(X_b,X_c),F),B_2)) ) ) ) ).
fof(fact_330_finite__subset__induct,axiom,
! [X_b,Pa,A_3,F_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),F_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)),F_2),A_3))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,Pa,bot_bot(fun(X_b,bool))))
=> ( ! [A_5,F_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),F_3))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),A_5),A_3))
=> ( ~ hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),A_5),F_3))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,Pa,F_3))
=> hBOOL(hAPP(fun(X_b,bool),bool,Pa,hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_5),F_3))) ) ) ) )
=> hBOOL(hAPP(fun(X_b,bool),bool,Pa,F_2)) ) ) ) ) ).
fof(fact_331_subsetI,axiom,
! [X_b,B_2,A_3] :
( ! [X_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_2),A_3))
=> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_2),B_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)),A_3),B_2)) ) ).
fof(fact_332_finite__subset__image,axiom,
! [X_c,X_b,F,A_3,B_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),B_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_2),hAPP(fun(X_c,bool),fun(X_b,bool),hAPP(fun(X_c,X_b),fun(fun(X_c,bool),fun(X_b,bool)),image(X_c,X_b),F),A_3)))
=> ? [C_2] :
( hBOOL(hAPP(fun(X_c,bool),bool,hAPP(fun(X_c,bool),fun(fun(X_c,bool),bool),ord_less_eq(fun(X_c,bool)),C_2),A_3))
& hBOOL(hAPP(fun(X_c,bool),bool,finite_finite_1(X_c),C_2))
& ti(fun(X_b,bool),B_2) = hAPP(fun(X_c,bool),fun(X_b,bool),hAPP(fun(X_c,X_b),fun(fun(X_c,bool),fun(X_b,bool)),image(X_c,X_b),F),C_2) ) ) ) ).
fof(fact_333_setsum__mono,axiom,
! [X_b,X_c] :
( ( comm_monoid_add(X_c)
& ordere779506340up_add(X_c) )
=> ! [F,G,K_2] :
( ! [I_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),I_2),K_2))
=> hBOOL(hAPP(X_c,bool,hAPP(X_c,fun(X_c,bool),ord_less_eq(X_c),hAPP(X_b,X_c,F,I_2)),hAPP(X_b,X_c,G,I_2))) )
=> hBOOL(hAPP(X_c,bool,hAPP(X_c,fun(X_c,bool),ord_less_eq(X_c),hAPP(fun(X_b,bool),X_c,hAPP(fun(X_b,X_c),fun(fun(X_b,bool),X_c),big_co1399186613setsum(X_b,X_c),F),K_2)),hAPP(fun(X_b,bool),X_c,hAPP(fun(X_b,X_c),fun(fun(X_b,bool),X_c),big_co1399186613setsum(X_b,X_c),G),K_2))) ) ) ).
fof(fact_334_image__subsetI,axiom,
! [X_c,X_b,F,B_2,A_3] :
( ! [X_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_2),A_3))
=> hBOOL(hAPP(fun(X_c,bool),bool,hAPP(X_c,fun(fun(X_c,bool),bool),member(X_c),hAPP(X_b,X_c,F,X_2)),B_2)) )
=> hBOOL(hAPP(fun(X_c,bool),bool,hAPP(fun(X_c,bool),fun(fun(X_c,bool),bool),ord_less_eq(fun(X_c,bool)),hAPP(fun(X_b,bool),fun(X_c,bool),hAPP(fun(X_b,X_c),fun(fun(X_b,bool),fun(X_c,bool)),image(X_b,X_c),F),A_3)),B_2)) ) ).
fof(fact_335_le__funI,axiom,
! [X_b,X_c] :
( ord(X_c)
=> ! [F,G] :
( ! [X_2] : hBOOL(hAPP(X_c,bool,hAPP(X_c,fun(X_c,bool),ord_less_eq(X_c),hAPP(X_b,X_c,F,X_2)),hAPP(X_b,X_c,G,X_2)))
=> hBOOL(hAPP(fun(X_b,X_c),bool,hAPP(fun(X_b,X_c),fun(fun(X_b,X_c),bool),ord_less_eq(fun(X_b,X_c)),F),G)) ) ) ).
fof(fact_336_flat__lub__def,axiom,
! [X_b,A_3,Ba] :
( ( 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)),A_3),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),Ba),bot_bot(fun(X_b,bool)))))
=> hAPP(fun(X_b,bool),X_b,hAPP(X_b,fun(fun(X_b,bool),X_b),partial_flat_lub(X_b),Ba),A_3) = ti(X_b,Ba) )
& ( ~ 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)),A_3),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),Ba),bot_bot(fun(X_b,bool)))))
=> hAPP(fun(X_b,bool),X_b,hAPP(X_b,fun(fun(X_b,bool),X_b),partial_flat_lub(X_b),Ba),A_3) = hAPP(fun(X_b,bool),X_b,the(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(fun(X_b,bool),bool)),fun(fun(X_b,bool),fun(X_b,bool)),combc(X_b,fun(X_b,bool),bool),member(X_b)),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_3),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),Ba),bot_bot(fun(X_b,bool)))))) ) ) ).
fof(fact_337_sum__diff__distrib,axiom,
! [N,Q_3,Pa] :
( ! [X_2] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,Q_3,X_2)),hAPP(nat,nat,Pa,X_2)))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(fun(nat,bool),nat,hAPP(fun(nat,nat),fun(fun(nat,bool),nat),big_co1399186613setsum(nat,nat),Pa),hAPP(nat,fun(nat,bool),ord_lessThan(nat),N))),hAPP(fun(nat,bool),nat,hAPP(fun(nat,nat),fun(fun(nat,bool),nat),big_co1399186613setsum(nat,nat),Q_3),hAPP(nat,fun(nat,bool),ord_lessThan(nat),N))) = hAPP(fun(nat,bool),nat,hAPP(fun(nat,nat),fun(fun(nat,bool),nat),big_co1399186613setsum(nat,nat),hAPP(fun(nat,nat),fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(fun(nat,nat),fun(nat,nat)),combs(nat,nat,nat),hAPP(fun(nat,nat),fun(nat,fun(nat,nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(nat,nat),fun(nat,fun(nat,nat))),combb(nat,fun(nat,nat),nat),minus_minus(nat)),Pa)),Q_3)),hAPP(nat,fun(nat,bool),ord_lessThan(nat),N)) ) ).
fof(fact_338_fold1__antimono,axiom,
! [X_b] :
( linorder(X_b)
=> ! [B_2,A_3] :
( ti(fun(X_b,bool),A_3) != bot_bot(fun(X_b,bool))
=> ( 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)),A_3),B_2))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),B_2))
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),finite_fold1(X_b),ord_min(X_b)),B_2)),hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),finite_fold1(X_b),ord_min(X_b)),A_3))) ) ) ) ) ).
fof(fact_339_predicate1I,axiom,
! [X_b,Q_3,Pa] :
( ! [X_2] :
( hBOOL(hAPP(X_b,bool,Pa,X_2))
=> hBOOL(hAPP(X_b,bool,Q_3,X_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)),Pa),Q_3)) ) ).
fof(fact_340_Collect__mono,axiom,
! [X_b,Q_3,Pa] :
( ! [X_2] :
( hBOOL(hAPP(X_b,bool,Pa,X_2))
=> hBOOL(hAPP(X_b,bool,Q_3,X_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)),hAPP(fun(X_b,bool),fun(X_b,bool),collect(X_b),Pa)),hAPP(fun(X_b,bool),fun(X_b,bool),collect(X_b),Q_3))) ) ).
fof(fact_341_Min__antimono,axiom,
! [X_b] :
( linorder(X_b)
=> ! [M_3,N_3] :
( 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)),M_3),N_3))
=> ( ti(fun(X_b,bool),M_3) != bot_bot(fun(X_b,bool))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),N_3))
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(fun(X_b,bool),X_b,big_linorder_Min(X_b),N_3)),hAPP(fun(X_b,bool),X_b,big_linorder_Min(X_b),M_3))) ) ) ) ) ).
fof(fact_342_Max__mono,axiom,
! [X_b] :
( linorder(X_b)
=> ! [M_3,N_3] :
( 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)),M_3),N_3))
=> ( ti(fun(X_b,bool),M_3) != bot_bot(fun(X_b,bool))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),N_3))
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(fun(X_b,bool),X_b,big_linorder_Max(X_b),M_3)),hAPP(fun(X_b,bool),X_b,big_linorder_Max(X_b),N_3))) ) ) ) ) ).
fof(fact_343_finite__lessThan,axiom,
! [K_1] : hBOOL(hAPP(fun(nat,bool),bool,finite_finite_1(nat),hAPP(nat,fun(nat,bool),ord_lessThan(nat),K_1))) ).
fof(fact_344_finite__Collect__le__nat,axiom,
! [K_1] : hBOOL(hAPP(fun(nat,bool),bool,finite_finite_1(nat),hAPP(fun(nat,bool),fun(nat,bool),collect(nat),hAPP(nat,fun(nat,bool),hAPP(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool)),combc(nat,nat,bool),ord_less_eq(nat)),K_1)))) ).
fof(fact_345_finite__nat__set__iff__bounded__le,axiom,
! [N_3] :
( hBOOL(hAPP(fun(nat,bool),bool,finite_finite_1(nat),N_3))
<=> ? [M_2] :
! [X_2] :
( hBOOL(hAPP(fun(nat,bool),bool,hAPP(nat,fun(fun(nat,bool),bool),member(nat),X_2),N_3))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),X_2),M_2)) ) ) ).
fof(fact_346_Min_Oin__idem,axiom,
! [X_b] :
( linorder(X_b)
=> ! [X_1,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),A_3))
=> hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),ord_min(X_b),X_1),hAPP(fun(X_b,bool),X_b,big_linorder_Min(X_b),A_3)) = hAPP(fun(X_b,bool),X_b,big_linorder_Min(X_b),A_3) ) ) ) ).
fof(fact_347_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_348_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_349_min__max_Oinf_Oassoc,axiom,
! [X_a] :
( linorder(X_a)
=> ! [A_1,B,C] : 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_1),B)),C) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),B),C)) ) ).
fof(fact_350_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_351_min__max_Oinf_Oleft__commute,axiom,
! [X_a] :
( linorder(X_a)
=> ! [B,A_1,C] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),B),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),A_1),C)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),B),C)) ) ).
fof(fact_352_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_353_min__max_Oinf_Oleft__idem,axiom,
! [X_a] :
( linorder(X_a)
=> ! [A_1,B] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),A_1),B)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),A_1),B) ) ).
fof(fact_354_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_355_min__max_Oinf_Ocommute,axiom,
! [X_a] :
( linorder(X_a)
=> ! [A_1,B] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),A_1),B) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),B),A_1) ) ).
fof(fact_356_min__max_Oinf_Oidem,axiom,
! [X_a] :
( linorder(X_a)
=> ! [A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),A_1),A_1) = ti(X_a,A_1) ) ).
fof(fact_357_min__le__iff__disj,axiom,
! [X_b] :
( linorder(X_b)
=> ! [X_1,Y_2,Z_2] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),ord_min(X_b),X_1),Y_2)),Z_2))
<=> ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),X_1),Z_2))
| hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),Y_2),Z_2)) ) ) ) ).
fof(fact_358_min__max_Ole__infE,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X,A_1,B] :
( 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_1),B)))
=> ~ ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),A_1))
=> ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),B)) ) ) ) ).
fof(fact_359_min__max_Oinf__mono,axiom,
! [X_a] :
( linorder(X_a)
=> ! [B,D_1,A_1,C] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),C))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),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_1),B)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),C),D_1))) ) ) ) ).
fof(fact_360_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_361_min__max_Ole__infI,axiom,
! [X_a] :
( linorder(X_a)
=> ! [B,X,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),A_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),B))
=> 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_1),B))) ) ) ) ).
fof(fact_362_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_363_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_364_min__max_Ole__infI2,axiom,
! [X_a] :
( linorder(X_a)
=> ! [A_1,B,X] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),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_1),B)),X)) ) ) ).
fof(fact_365_min__max_Ole__infI1,axiom,
! [X_a] :
( linorder(X_a)
=> ! [B,A_1,X] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),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_1),B)),X)) ) ) ).
fof(fact_366_min__max_Ole__inf__iff,axiom,
! [X_b] :
( linorder(X_b)
=> ! [X_1,Y_2,Z_2] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),X_1),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),ord_min(X_b),Y_2),Z_2)))
<=> ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),X_1),Y_2))
& hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),X_1),Z_2)) ) ) ) ).
fof(fact_367_min__max_Ole__iff__inf,axiom,
! [X_b] :
( linorder(X_b)
=> ! [X_1,Y_2] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),X_1),Y_2))
<=> hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),ord_min(X_b),X_1),Y_2) = ti(X_b,X_1) ) ) ).
fof(fact_368_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_369_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_370_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_371_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_372_Min_OF__eq,axiom,
! [X_b] :
( linorder(X_b)
=> ! [A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> hAPP(fun(X_b,bool),X_b,big_linorder_Min(X_b),A_3) = hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),finite_fold1(X_b),ord_min(X_b)),A_3) ) ) ).
fof(fact_373_min__max_Ocomp__fun__idem__inf,axiom,
! [X_b] :
( linorder(X_b)
=> hBOOL(hAPP(fun(X_b,fun(X_b,X_b)),bool,finite_comp_fun_idem(X_b,X_b),ord_min(X_b))) ) ).
fof(fact_374_Min__insert,axiom,
! [X_b] :
( linorder(X_b)
=> ! [X_1,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( ti(fun(X_b,bool),A_3) != bot_bot(fun(X_b,bool))
=> hAPP(fun(X_b,bool),X_b,big_linorder_Min(X_b),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_1),A_3)) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),ord_min(X_b),X_1),hAPP(fun(X_b,bool),X_b,big_linorder_Min(X_b),A_3)) ) ) ) ).
fof(fact_375_Min_Oinsert,axiom,
! [X_b] :
( linorder(X_b)
=> ! [X_1,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( ~ hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),A_3))
=> ( ti(fun(X_b,bool),A_3) != bot_bot(fun(X_b,bool))
=> hAPP(fun(X_b,bool),X_b,big_linorder_Min(X_b),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_1),A_3)) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),ord_min(X_b),X_1),hAPP(fun(X_b,bool),X_b,big_linorder_Min(X_b),A_3)) ) ) ) ) ).
fof(fact_376_Min_Osubset__idem,axiom,
! [X_b] :
( linorder(X_b)
=> ! [B_2,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( ti(fun(X_b,bool),B_2) != bot_bot(fun(X_b,bool))
=> ( 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_2),A_3))
=> hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),ord_min(X_b),hAPP(fun(X_b,bool),X_b,big_linorder_Min(X_b),B_2)),hAPP(fun(X_b,bool),X_b,big_linorder_Min(X_b),A_3)) = hAPP(fun(X_b,bool),X_b,big_linorder_Min(X_b),A_3) ) ) ) ) ).
fof(fact_377_Min_Oeq__fold_H,axiom,
! [X_b] :
( linorder(X_b)
=> ! [X_1,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( ~ hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),A_3))
=> hAPP(fun(X_b,bool),X_b,big_linorder_Min(X_b),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_1),A_3)) = hAPP(fun(X_b,bool),X_b,hAPP(X_b,fun(fun(X_b,bool),X_b),hAPP(fun(X_b,fun(X_b,X_b)),fun(X_b,fun(fun(X_b,bool),X_b)),finite_fold(X_b,X_b),ord_min(X_b)),X_1),A_3) ) ) ) ).
fof(fact_378_Min_Oeq__fold__idem_H,axiom,
! [X_b] :
( linorder(X_b)
=> ! [A_2,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> hAPP(fun(X_b,bool),X_b,big_linorder_Min(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_2),A_3)) = hAPP(fun(X_b,bool),X_b,hAPP(X_b,fun(fun(X_b,bool),X_b),hAPP(fun(X_b,fun(X_b,X_b)),fun(X_b,fun(fun(X_b,bool),X_b)),finite_fold(X_b,X_b),ord_min(X_b)),A_2),A_3) ) ) ).
fof(fact_379_Min_Oinsert__remove,axiom,
! [X_b] :
( linorder(X_b)
=> ! [X_1,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( ( 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_3),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_1),bot_bot(fun(X_b,bool)))) = bot_bot(fun(X_b,bool))
=> hAPP(fun(X_b,bool),X_b,big_linorder_Min(X_b),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_1),A_3)) = ti(X_b,X_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_3),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_1),bot_bot(fun(X_b,bool)))) != bot_bot(fun(X_b,bool))
=> hAPP(fun(X_b,bool),X_b,big_linorder_Min(X_b),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_1),A_3)) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),ord_min(X_b),X_1),hAPP(fun(X_b,bool),X_b,big_linorder_Min(X_b),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_3),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_1),bot_bot(fun(X_b,bool)))))) ) ) ) ) ).
fof(fact_380_Min_Oremove,axiom,
! [X_b] :
( linorder(X_b)
=> ! [X_1,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),A_3))
=> ( ( 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_3),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_1),bot_bot(fun(X_b,bool)))) = bot_bot(fun(X_b,bool))
=> hAPP(fun(X_b,bool),X_b,big_linorder_Min(X_b),A_3) = ti(X_b,X_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_3),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_1),bot_bot(fun(X_b,bool)))) != bot_bot(fun(X_b,bool))
=> hAPP(fun(X_b,bool),X_b,big_linorder_Min(X_b),A_3) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),ord_min(X_b),X_1),hAPP(fun(X_b,bool),X_b,big_linorder_Min(X_b),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_3),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_1),bot_bot(fun(X_b,bool)))))) ) ) ) ) ) ).
fof(fact_381_Max__ge,axiom,
! [X_b] :
( linorder(X_b)
=> ! [X_1,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),A_3))
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),X_1),hAPP(fun(X_b,bool),X_b,big_linorder_Max(X_b),A_3))) ) ) ) ).
fof(fact_382_Min__le,axiom,
! [X_b] :
( linorder(X_b)
=> ! [X_1,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),A_3))
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(fun(X_b,bool),X_b,big_linorder_Min(X_b),A_3)),X_1)) ) ) ) ).
fof(fact_383_Max__in,axiom,
! [X_b] :
( linorder(X_b)
=> ! [A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( ti(fun(X_b,bool),A_3) != bot_bot(fun(X_b,bool))
=> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),hAPP(fun(X_b,bool),X_b,big_linorder_Max(X_b),A_3)),A_3)) ) ) ) ).
fof(fact_384_Min__in,axiom,
! [X_b] :
( linorder(X_b)
=> ! [A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( ti(fun(X_b,bool),A_3) != bot_bot(fun(X_b,bool))
=> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),hAPP(fun(X_b,bool),X_b,big_linorder_Min(X_b),A_3)),A_3)) ) ) ) ).
fof(fact_385_Min_Osingleton,axiom,
! [X_b] :
( linorder(X_b)
=> ! [X_1] : hAPP(fun(X_b,bool),X_b,big_linorder_Min(X_b),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_1),bot_bot(fun(X_b,bool)))) = ti(X_b,X_1) ) ).
fof(fact_386_Max_Osingleton,axiom,
! [X_b] :
( linorder(X_b)
=> ! [X_1] : hAPP(fun(X_b,bool),X_b,big_linorder_Max(X_b),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_1),bot_bot(fun(X_b,bool)))) = ti(X_b,X_1) ) ).
fof(fact_387_min__max_Ofold__inf__le__inf,axiom,
! [X_b] :
( linorder(X_b)
=> ! [Ba,A_2,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),A_2),A_3))
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(fun(X_b,bool),X_b,hAPP(X_b,fun(fun(X_b,bool),X_b),hAPP(fun(X_b,fun(X_b,X_b)),fun(X_b,fun(fun(X_b,bool),X_b)),finite_fold(X_b,X_b),ord_min(X_b)),Ba),A_3)),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),ord_min(X_b),A_2),Ba))) ) ) ) ).
fof(fact_388_min__max_Ofold__inf__insert,axiom,
! [X_b] :
( linorder(X_b)
=> ! [Ba,A_2,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> hAPP(fun(X_b,bool),X_b,hAPP(X_b,fun(fun(X_b,bool),X_b),hAPP(fun(X_b,fun(X_b,X_b)),fun(X_b,fun(fun(X_b,bool),X_b)),finite_fold(X_b,X_b),ord_min(X_b)),Ba),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_2),A_3)) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),ord_min(X_b),A_2),hAPP(fun(X_b,bool),X_b,hAPP(X_b,fun(fun(X_b,bool),X_b),hAPP(fun(X_b,fun(X_b,X_b)),fun(X_b,fun(fun(X_b,bool),X_b)),finite_fold(X_b,X_b),ord_min(X_b)),Ba),A_3)) ) ) ).
fof(fact_389_min__max_Ofold1__belowI,axiom,
! [X_b] :
( linorder(X_b)
=> ! [A_2,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),A_2),A_3))
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),finite_fold1(X_b),ord_min(X_b)),A_3)),A_2)) ) ) ) ).
fof(fact_390_fold1__below__iff,axiom,
! [X_b] :
( linorder(X_b)
=> ! [X_1,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( ti(fun(X_b,bool),A_3) != bot_bot(fun(X_b,bool))
=> ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),finite_fold1(X_b),ord_min(X_b)),A_3)),X_1))
<=> ? [X_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_2),A_3))
& hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),X_2),X_1)) ) ) ) ) ) ).
fof(fact_391_min__max_Obelow__fold1__iff,axiom,
! [X_b] :
( linorder(X_b)
=> ! [X_1,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( ti(fun(X_b,bool),A_3) != bot_bot(fun(X_b,bool))
=> ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),X_1),hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),finite_fold1(X_b),ord_min(X_b)),A_3)))
<=> ! [X_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_2),A_3))
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),X_1),X_2)) ) ) ) ) ) ).
fof(fact_392_Min_Ohom__commute,axiom,
! [X_b] :
( linorder(X_b)
=> ! [N_3,H] :
( ! [X_2,Y_1] : hAPP(X_b,X_b,H,hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),ord_min(X_b),X_2),Y_1)) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),ord_min(X_b),hAPP(X_b,X_b,H,X_2)),hAPP(X_b,X_b,H,Y_1))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),N_3))
=> ( ti(fun(X_b,bool),N_3) != bot_bot(fun(X_b,bool))
=> hAPP(X_b,X_b,H,hAPP(fun(X_b,bool),X_b,big_linorder_Min(X_b),N_3)) = hAPP(fun(X_b,bool),X_b,big_linorder_Min(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,X_b),fun(fun(X_b,bool),fun(X_b,bool)),image(X_b,X_b),H),N_3)) ) ) ) ) ).
fof(fact_393_Min_Oclosed,axiom,
! [X_b] :
( linorder(X_b)
=> ! [A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( ti(fun(X_b,bool),A_3) != bot_bot(fun(X_b,bool))
=> ( ! [X_2,Y_1] : hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),ord_min(X_b),X_2),Y_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),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),Y_1),bot_bot(fun(X_b,bool))))))
=> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),hAPP(fun(X_b,bool),X_b,big_linorder_Min(X_b),A_3)),A_3)) ) ) ) ) ).
fof(fact_394_min__max_Oinf__le__fold__inf,axiom,
! [X_b] :
( linorder(X_b)
=> ! [Ca,Ba,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( ! [X_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_2),A_3))
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),Ba),X_2)) )
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),ord_min(X_b),Ba),Ca)),hAPP(fun(X_b,bool),X_b,hAPP(X_b,fun(fun(X_b,bool),X_b),hAPP(fun(X_b,fun(X_b,X_b)),fun(X_b,fun(fun(X_b,bool),X_b)),finite_fold(X_b,X_b),ord_min(X_b)),Ca),A_3))) ) ) ) ).
fof(fact_395_Min__eqI,axiom,
! [X_b] :
( linorder(X_b)
=> ! [X_1,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( ! [Y_1] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),Y_1),A_3))
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),X_1),Y_1)) )
=> ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),A_3))
=> hAPP(fun(X_b,bool),X_b,big_linorder_Min(X_b),A_3) = ti(X_b,X_1) ) ) ) ) ).
fof(fact_396_Max__eqI,axiom,
! [X_b] :
( linorder(X_b)
=> ! [X_1,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( ! [Y_1] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),Y_1),A_3))
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),Y_1),X_1)) )
=> ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),A_3))
=> hAPP(fun(X_b,bool),X_b,big_linorder_Max(X_b),A_3) = ti(X_b,X_1) ) ) ) ) ).
fof(fact_397_dual__max,axiom,
! [X_b] :
( linorder(X_b)
=> hAPP(fun(X_b,fun(X_b,bool)),fun(X_b,fun(X_b,X_b)),max(X_b),hAPP(fun(X_b,fun(X_b,bool)),fun(X_b,fun(X_b,bool)),combc(X_b,X_b,bool),ord_less_eq(X_b))) = ord_min(X_b) ) ).
fof(fact_398_finite__less__ub,axiom,
! [U,F] :
( ! [N_2] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_2),hAPP(nat,nat,F,N_2)))
=> hBOOL(hAPP(fun(nat,bool),bool,finite_finite_1(nat),hAPP(fun(nat,bool),fun(nat,bool),collect(nat),hAPP(nat,fun(nat,bool),hAPP(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool)),combc(nat,nat,bool),hAPP(fun(nat,nat),fun(nat,fun(nat,bool)),hAPP(fun(nat,fun(nat,bool)),fun(fun(nat,nat),fun(nat,fun(nat,bool))),combb(nat,fun(nat,bool),nat),ord_less_eq(nat)),F)),U)))) ) ).
fof(fact_399_min__leastR,axiom,
! [X_a] :
( order(X_a)
=> ! [X,Least] :
( ! [X_2] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Least),X_2))
=> 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_400_min__leastL,axiom,
! [X_a] :
( ord(X_a)
=> ! [X,Least] :
( ! [X_2] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Least),X_2))
=> 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_401_min__ord__min,axiom,
! [X_b] :
( ord(X_b)
=> ord_min(X_b) = hAPP(fun(X_b,fun(X_b,bool)),fun(X_b,fun(X_b,X_b)),min(X_b),ord_less_eq(X_b)) ) ).
fof(fact_402_le__diff__iff,axiom,
! [N,K_1,M_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_1),M_1))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_1),N))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M_1),K_1)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N),K_1)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_1),N)) ) ) ) ).
fof(fact_403_mult__le__mono,axiom,
! [K,L_1,I,J] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I),J))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K),L_1))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),I),K)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),J),L_1))) ) ) ).
fof(fact_404_mult__le__mono2,axiom,
! [K,I,J] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I),J))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K),I)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K),J))) ) ).
fof(fact_405_mult__le__mono1,axiom,
! [K,I,J] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I),J))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),I),K)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),J),K))) ) ).
fof(fact_406_le__cube,axiom,
! [M] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),M)))) ).
fof(fact_407_le__square,axiom,
! [M] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),M))) ).
fof(fact_408_le__refl,axiom,
! [N_1] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_1),N_1)) ).
fof(fact_409_nat__le__linear,axiom,
! [M,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N_1))
| hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_1),M)) ) ).
fof(fact_410_eq__imp__le,axiom,
! [M,N_1] :
( M = N_1
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N_1)) ) ).
fof(fact_411_le__trans,axiom,
! [K,I,J] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I),J))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),J),K))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I),K)) ) ) ).
fof(fact_412_le__antisym,axiom,
! [M,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N_1))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_1),M))
=> M = N_1 ) ) ).
fof(fact_413_add__leE,axiom,
! [M,K,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),K)),N_1))
=> ~ ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N_1))
=> ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K),N_1)) ) ) ).
fof(fact_414_add__leD1,axiom,
! [M,K,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),K)),N_1))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N_1)) ) ).
fof(fact_415_add__leD2,axiom,
! [M,K,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),K)),N_1))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K),N_1)) ) ).
fof(fact_416_add__le__mono,axiom,
! [K,L_1,I,J] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I),J))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K),L_1))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),I),K)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),J),L_1))) ) ) ).
fof(fact_417_add__le__mono1,axiom,
! [K,I,J] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I),J))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),I),K)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),J),K))) ) ).
fof(fact_418_trans__le__add2,axiom,
! [M,I,J] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I),J))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),J))) ) ).
fof(fact_419_trans__le__add1,axiom,
! [M,I,J] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I),J))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),J),M))) ) ).
fof(fact_420_nat__add__left__cancel__le,axiom,
! [K_1,M_1,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),K_1),M_1)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),K_1),N)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_1),N)) ) ).
fof(fact_421_le__iff__add,axiom,
! [M_1,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_1),N))
<=> ? [K_3] : N = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M_1),K_3) ) ).
fof(fact_422_le__add1,axiom,
! [N_1,M] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N_1),M))) ).
fof(fact_423_le__add2,axiom,
! [N_1,M] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),N_1))) ).
fof(fact_424_min__diff,axiom,
! [M,I,N_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_min(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),I)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N_1),I)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_min(nat),M),N_1)),I) ).
fof(fact_425_diff__mult__distrib2,axiom,
! [K,M,N_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),N_1)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K),M)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K),N_1)) ).
fof(fact_426_diff__add__inverse2,axiom,
! [M,N_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),N_1)),N_1) = M ).
fof(fact_427_diff__add__inverse,axiom,
! [N_1,M] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N_1),M)),N_1) = M ).
fof(fact_428_diff__diff__left,axiom,
! [I,J,K] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),I),J)),K) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),I),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),J),K)) ).
fof(fact_429_diff__commute,axiom,
! [I,J,K] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),I),J)),K) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),I),K)),J) ).
fof(fact_430_diff__mult__distrib,axiom,
! [M,N_1,K] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),N_1)),K) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),K)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),N_1),K)) ).
fof(fact_431_diff__cancel,axiom,
! [K,M,N_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),K),M)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),K),N_1)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),N_1) ).
fof(fact_432_diff__cancel2,axiom,
! [M,K,N_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),K)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N_1),K)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),N_1) ).
fof(fact_433_diff__le__self,axiom,
! [M,N_1] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),N_1)),M)) ).
fof(fact_434_diff__diff__right,axiom,
! [I,K,J] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K),J))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),I),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),J),K)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),I),K)),J) ) ).
fof(fact_435_le__diff__conv,axiom,
! [J_1,K_1,I_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),J_1),K_1)),I_1))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),J_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),I_1),K_1))) ) ).
fof(fact_436_le__add__diff,axiom,
! [M,K,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K),N_1))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N_1),M)),K))) ) ).
fof(fact_437_diff__le__mono2,axiom,
! [L_1,M,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N_1))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),L_1),N_1)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),L_1),M))) ) ).
fof(fact_438_diff__le__mono,axiom,
! [L_1,M,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N_1))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),L_1)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N_1),L_1))) ) ).
fof(fact_439_diff__diff__cancel,axiom,
! [I,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I),N_1))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N_1),I)) = I ) ).
fof(fact_440_le__add__diff__inverse,axiom,
! [N_1,M] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_1),M))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),N_1)) = M ) ).
fof(fact_441_add__diff__assoc,axiom,
! [I,K,J] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K),J))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),I),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),J),K)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),I),J)),K) ) ).
fof(fact_442_le__diff__conv2,axiom,
! [I_1,K_1,J_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_1),J_1))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),J_1),K_1)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),I_1),K_1)),J_1)) ) ) ).
fof(fact_443_le__add__diff__inverse2,axiom,
! [N_1,M] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_1),M))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),N_1)),N_1) = M ) ).
fof(fact_444_le__imp__diff__is__add,axiom,
! [K_1,I_1,J_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_1),J_1))
=> ( hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),J_1),I_1) = K_1
<=> J_1 = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),K_1),I_1) ) ) ).
fof(fact_445_diff__add__assoc,axiom,
! [I,K,J] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K),J))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),I),J)),K) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),I),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),J),K)) ) ).
fof(fact_446_add__diff__assoc2,axiom,
! [I,K,J] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K),J))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),J),K)),I) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),J),I)),K) ) ).
fof(fact_447_diff__add__assoc2,axiom,
! [I,K,J] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K),J))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),J),I)),K) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),J),K)),I) ) ).
fof(fact_448_eq__diff__iff,axiom,
! [N,K_1,M_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_1),M_1))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_1),N))
=> ( hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M_1),K_1) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N),K_1)
<=> M_1 = N ) ) ) ).
fof(fact_449_Nat_Odiff__diff__eq,axiom,
! [N_1,K,M] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K),M))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K),N_1))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),K)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N_1),K)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),N_1) ) ) ).
fof(fact_450_nat__eq__add__iff2,axiom,
! [U,M_1,N,I_1,J_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_1),J_1))
=> ( hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),I_1),U)),M_1) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),J_1),U)),N)
<=> M_1 = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),J_1),I_1)),U)),N) ) ) ).
fof(fact_451_nat__diff__add__eq2,axiom,
! [U_1,M,N_1,I,J] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I),J))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),I),U_1)),M)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),J),U_1)),N_1)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),J),I)),U_1)),N_1)) ) ).
fof(fact_452_nat__le__add__iff2,axiom,
! [U,M_1,N,I_1,J_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_1),J_1))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),I_1),U)),M_1)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),J_1),U)),N)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),J_1),I_1)),U)),N))) ) ) ).
fof(fact_453_nat__eq__add__iff1,axiom,
! [U,M_1,N,J_1,I_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),J_1),I_1))
=> ( hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),I_1),U)),M_1) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),J_1),U)),N)
<=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),I_1),J_1)),U)),M_1) = N ) ) ).
fof(fact_454_nat__diff__add__eq1,axiom,
! [U_1,M,N_1,J,I] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),J),I))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),I),U_1)),M)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),J),U_1)),N_1)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),I),J)),U_1)),M)),N_1) ) ).
fof(fact_455_nat__mult__assoc,axiom,
! [M,N_1,K] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),N_1)),K) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),N_1),K)) ).
fof(fact_456_nat__mult__commute,axiom,
! [M,N_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),N_1) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),N_1),M) ).
fof(fact_457_nat__add__right__cancel,axiom,
! [M_1,K_1,N] :
( hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M_1),K_1) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N),K_1)
<=> M_1 = N ) ).
fof(fact_458_nat__add__left__cancel,axiom,
! [K_1,M_1,N] :
( hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),K_1),M_1) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),K_1),N)
<=> M_1 = N ) ).
fof(fact_459_nat__add__assoc,axiom,
! [M,N_1,K] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),N_1)),K) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N_1),K)) ).
fof(fact_460_nat__add__left__commute,axiom,
! [X,Y,Z] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),X),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),Y),Z)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),Y),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),X),Z)) ).
fof(fact_461_nat__add__commute,axiom,
! [M,N_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),N_1) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N_1),M) ).
fof(fact_462_add__mult__distrib2,axiom,
! [K,M,N_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),N_1)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K),M)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K),N_1)) ).
fof(fact_463_add__mult__distrib,axiom,
! [M,N_1,K] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),N_1)),K) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),K)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),N_1),K)) ).
fof(fact_464_left__add__mult__distrib,axiom,
! [I,U_1,J,K] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),I),U_1)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),J),U_1)),K)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),I),J)),U_1)),K) ).
fof(fact_465_nat__le__add__iff1,axiom,
! [U,M_1,N,J_1,I_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),J_1),I_1))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),I_1),U)),M_1)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),J_1),U)),N)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),I_1),J_1)),U)),M_1)),N)) ) ) ).
fof(fact_466_termination__basic__simps_I3_J,axiom,
! [Z,X,Y] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),X),Y))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),X),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),Y),Z))) ) ).
fof(fact_467_termination__basic__simps_I4_J,axiom,
! [Y,X,Z] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),X),Z))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),X),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),Y),Z))) ) ).
fof(fact_468_min__max_OInf__fin_Oremove,axiom,
! [X_b] :
( linorder(X_b)
=> ! [X_1,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),A_3))
=> ( ( 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_3),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_1),bot_bot(fun(X_b,bool)))) = bot_bot(fun(X_b,bool))
=> hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),big_Inf_fin(X_b),ord_min(X_b)),A_3) = ti(X_b,X_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_3),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_1),bot_bot(fun(X_b,bool)))) != bot_bot(fun(X_b,bool))
=> hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),big_Inf_fin(X_b),ord_min(X_b)),A_3) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),ord_min(X_b),X_1),hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),big_Inf_fin(X_b),ord_min(X_b)),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_3),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_1),bot_bot(fun(X_b,bool)))))) ) ) ) ) ) ).
fof(fact_469_min__max_OInf__fin_Oinsert__remove,axiom,
! [X_b] :
( linorder(X_b)
=> ! [X_1,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( ( 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_3),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_1),bot_bot(fun(X_b,bool)))) = bot_bot(fun(X_b,bool))
=> hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),big_Inf_fin(X_b),ord_min(X_b)),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_1),A_3)) = ti(X_b,X_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_3),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_1),bot_bot(fun(X_b,bool)))) != bot_bot(fun(X_b,bool))
=> hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),big_Inf_fin(X_b),ord_min(X_b)),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_1),A_3)) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),ord_min(X_b),X_1),hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),big_Inf_fin(X_b),ord_min(X_b)),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_3),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_1),bot_bot(fun(X_b,bool)))))) ) ) ) ) ).
fof(fact_470_min__max_OInf__fin_Oin__idem,axiom,
! [X_b] :
( linorder(X_b)
=> ! [X_1,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),A_3))
=> hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),ord_min(X_b),X_1),hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),big_Inf_fin(X_b),ord_min(X_b)),A_3)) = hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),big_Inf_fin(X_b),ord_min(X_b)),A_3) ) ) ) ).
fof(fact_471_min__max_OInf__fin_Osingleton,axiom,
! [X_b] :
( linorder(X_b)
=> ! [X_1] : hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),big_Inf_fin(X_b),ord_min(X_b)),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_1),bot_bot(fun(X_b,bool)))) = ti(X_b,X_1) ) ).
fof(fact_472_min__max_OInf__fin_OF__eq,axiom,
! [X_b] :
( linorder(X_b)
=> ! [A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),big_Inf_fin(X_b),ord_min(X_b)),A_3) = hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),finite_fold1(X_b),ord_min(X_b)),A_3) ) ) ).
fof(fact_473_min__max_OInf__fin_Oinsert__idem,axiom,
! [X_b] :
( linorder(X_b)
=> ! [X_1,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( ti(fun(X_b,bool),A_3) != bot_bot(fun(X_b,bool))
=> hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),big_Inf_fin(X_b),ord_min(X_b)),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_1),A_3)) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),ord_min(X_b),X_1),hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),big_Inf_fin(X_b),ord_min(X_b)),A_3)) ) ) ) ).
fof(fact_474_min__max_OInf__fin_Oinsert,axiom,
! [X_b] :
( linorder(X_b)
=> ! [X_1,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( ~ hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),A_3))
=> ( ti(fun(X_b,bool),A_3) != bot_bot(fun(X_b,bool))
=> hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),big_Inf_fin(X_b),ord_min(X_b)),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_1),A_3)) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),ord_min(X_b),X_1),hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),big_Inf_fin(X_b),ord_min(X_b)),A_3)) ) ) ) ) ).
fof(fact_475_min__max_OInf__fin_Osubset__idem,axiom,
! [X_b] :
( linorder(X_b)
=> ! [B_2,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( ti(fun(X_b,bool),B_2) != bot_bot(fun(X_b,bool))
=> ( 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_2),A_3))
=> hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),ord_min(X_b),hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),big_Inf_fin(X_b),ord_min(X_b)),B_2)),hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),big_Inf_fin(X_b),ord_min(X_b)),A_3)) = hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),big_Inf_fin(X_b),ord_min(X_b)),A_3) ) ) ) ) ).
fof(fact_476_min__max_OInf__fin_Oeq__fold_H,axiom,
! [X_b] :
( linorder(X_b)
=> ! [X_1,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( ~ hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),A_3))
=> hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),big_Inf_fin(X_b),ord_min(X_b)),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_1),A_3)) = hAPP(fun(X_b,bool),X_b,hAPP(X_b,fun(fun(X_b,bool),X_b),hAPP(fun(X_b,fun(X_b,X_b)),fun(X_b,fun(fun(X_b,bool),X_b)),finite_fold(X_b,X_b),ord_min(X_b)),X_1),A_3) ) ) ) ).
fof(fact_477_min__max_OInf__fin_Oeq__fold__idem_H,axiom,
! [X_b] :
( linorder(X_b)
=> ! [A_2,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),big_Inf_fin(X_b),ord_min(X_b)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_2),A_3)) = hAPP(fun(X_b,bool),X_b,hAPP(X_b,fun(fun(X_b,bool),X_b),hAPP(fun(X_b,fun(X_b,X_b)),fun(X_b,fun(fun(X_b,bool),X_b)),finite_fold(X_b,X_b),ord_min(X_b)),A_2),A_3) ) ) ).
fof(fact_478_min__max_OInf__fin_Ohom__commute,axiom,
! [X_b] :
( linorder(X_b)
=> ! [N_3,H] :
( ! [X_2,Y_1] : hAPP(X_b,X_b,H,hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),ord_min(X_b),X_2),Y_1)) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),ord_min(X_b),hAPP(X_b,X_b,H,X_2)),hAPP(X_b,X_b,H,Y_1))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),N_3))
=> ( ti(fun(X_b,bool),N_3) != bot_bot(fun(X_b,bool))
=> hAPP(X_b,X_b,H,hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),big_Inf_fin(X_b),ord_min(X_b)),N_3)) = hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),big_Inf_fin(X_b),ord_min(X_b)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,X_b),fun(fun(X_b,bool),fun(X_b,bool)),image(X_b,X_b),H),N_3)) ) ) ) ) ).
fof(fact_479_min__max_OInf__fin_Oclosed,axiom,
! [X_b] :
( linorder(X_b)
=> ! [A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( ti(fun(X_b,bool),A_3) != bot_bot(fun(X_b,bool))
=> ( ! [X_2,Y_1] : hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),ord_min(X_b),X_2),Y_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),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),Y_1),bot_bot(fun(X_b,bool))))))
=> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),big_Inf_fin(X_b),ord_min(X_b)),A_3)),A_3)) ) ) ) ) ).
fof(fact_480_minus__Max__eq__Min,axiom,
! [X_b] :
( linord219039673up_add(X_b)
=> ! [S_1] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),S_1))
=> ( ti(fun(X_b,bool),S_1) != bot_bot(fun(X_b,bool))
=> hAPP(X_b,X_b,uminus_uminus(X_b),hAPP(fun(X_b,bool),X_b,big_linorder_Max(X_b),S_1)) = hAPP(fun(X_b,bool),X_b,big_linorder_Min(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,X_b),fun(fun(X_b,bool),fun(X_b,bool)),image(X_b,X_b),uminus_uminus(X_b)),S_1)) ) ) ) ).
fof(fact_481_minus__Min__eq__Max,axiom,
! [X_b] :
( linord219039673up_add(X_b)
=> ! [S_1] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),S_1))
=> ( ti(fun(X_b,bool),S_1) != bot_bot(fun(X_b,bool))
=> hAPP(X_b,X_b,uminus_uminus(X_b),hAPP(fun(X_b,bool),X_b,big_linorder_Min(X_b),S_1)) = hAPP(fun(X_b,bool),X_b,big_linorder_Max(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,X_b),fun(fun(X_b,bool),fun(X_b,bool)),image(X_b,X_b),uminus_uminus(X_b)),S_1)) ) ) ) ).
fof(fact_482_fun__Compl__def,axiom,
! [X_b,X_c] :
( uminus(X_c)
=> ! [A_3,X_2] : hAPP(X_b,X_c,hAPP(fun(X_b,X_c),fun(X_b,X_c),uminus_uminus(fun(X_b,X_c)),A_3),X_2) = hAPP(X_c,X_c,uminus_uminus(X_c),hAPP(X_b,X_c,A_3,X_2)) ) ).
fof(fact_483_double__compl,axiom,
! [X_a] :
( boolean_algebra(X_a)
=> ! [X] : hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),X)) = ti(X_a,X) ) ).
fof(fact_484_uminus__apply,axiom,
! [X_c,X_b] :
( uminus(X_b)
=> ! [A_3,X_1] : hAPP(X_c,X_b,hAPP(fun(X_c,X_b),fun(X_c,X_b),uminus_uminus(fun(X_c,X_b)),A_3),X_1) = hAPP(X_b,X_b,uminus_uminus(X_b),hAPP(X_c,X_b,A_3,X_1)) ) ).
fof(fact_485_compl__eq__compl__iff,axiom,
! [X_b] :
( boolean_algebra(X_b)
=> ! [X_1,Y_2] :
( hAPP(X_b,X_b,uminus_uminus(X_b),X_1) = hAPP(X_b,X_b,uminus_uminus(X_b),Y_2)
<=> ti(X_b,X_1) = ti(X_b,Y_2) ) ) ).
fof(fact_486_minus__minus,axiom,
! [X_a] :
( group_add(X_a)
=> ! [A_1] : hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),A_1)) = ti(X_a,A_1) ) ).
fof(fact_487_equation__minus__iff,axiom,
! [X_b] :
( group_add(X_b)
=> ! [A_2,Ba] :
( ti(X_b,A_2) = hAPP(X_b,X_b,uminus_uminus(X_b),Ba)
<=> ti(X_b,Ba) = hAPP(X_b,X_b,uminus_uminus(X_b),A_2) ) ) ).
fof(fact_488_minus__equation__iff,axiom,
! [X_b] :
( group_add(X_b)
=> ! [A_2,Ba] :
( hAPP(X_b,X_b,uminus_uminus(X_b),A_2) = ti(X_b,Ba)
<=> hAPP(X_b,X_b,uminus_uminus(X_b),Ba) = ti(X_b,A_2) ) ) ).
fof(fact_489_neg__equal__iff__equal,axiom,
! [X_b] :
( group_add(X_b)
=> ! [A_2,Ba] :
( hAPP(X_b,X_b,uminus_uminus(X_b),A_2) = hAPP(X_b,X_b,uminus_uminus(X_b),Ba)
<=> ti(X_b,A_2) = ti(X_b,Ba) ) ) ).
fof(fact_490_Compl__anti__mono,axiom,
! [X_b,A_3,B_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)),A_3),B_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)),hAPP(fun(X_b,bool),fun(X_b,bool),uminus_uminus(fun(X_b,bool)),B_2)),hAPP(fun(X_b,bool),fun(X_b,bool),uminus_uminus(fun(X_b,bool)),A_3))) ) ).
fof(fact_491_Compl__subset__Compl__iff,axiom,
! [X_b,A_3,B_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)),hAPP(fun(X_b,bool),fun(X_b,bool),uminus_uminus(fun(X_b,bool)),A_3)),hAPP(fun(X_b,bool),fun(X_b,bool),uminus_uminus(fun(X_b,bool)),B_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_2),A_3)) ) ).
fof(fact_492_minus__diff__eq,axiom,
! [X_a] :
( ab_group_add(X_a)
=> ! [A_1,B] : hAPP(X_a,X_a,uminus_uminus(X_a),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),B),A_1) ) ).
fof(fact_493_le__imp__neg__le,axiom,
! [X_a] :
( ordered_ab_group_add(X_a)
=> ! [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_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),B)),hAPP(X_a,X_a,uminus_uminus(X_a),A_1))) ) ) ).
fof(fact_494_neg__le__iff__le,axiom,
! [X_b] :
( ordered_ab_group_add(X_b)
=> ! [Ba,A_2] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(X_b,X_b,uminus_uminus(X_b),Ba)),hAPP(X_b,X_b,uminus_uminus(X_b),A_2)))
<=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),A_2),Ba)) ) ) ).
fof(fact_495_minus__le__iff,axiom,
! [X_b] :
( ordered_ab_group_add(X_b)
=> ! [A_2,Ba] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(X_b,X_b,uminus_uminus(X_b),A_2)),Ba))
<=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(X_b,X_b,uminus_uminus(X_b),Ba)),A_2)) ) ) ).
fof(fact_496_le__minus__iff,axiom,
! [X_b] :
( ordered_ab_group_add(X_b)
=> ! [A_2,Ba] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),A_2),hAPP(X_b,X_b,uminus_uminus(X_b),Ba)))
<=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),Ba),hAPP(X_b,X_b,uminus_uminus(X_b),A_2))) ) ) ).
fof(fact_497_compl__le__swap2,axiom,
! [X_a] :
( boolean_algebra(X_a)
=> ! [Y,X] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),Y)),X))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),X)),Y)) ) ) ).
fof(fact_498_compl__le__swap1,axiom,
! [X_a] :
( boolean_algebra(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,uminus_uminus(X_a),X)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),hAPP(X_a,X_a,uminus_uminus(X_a),Y))) ) ) ).
fof(fact_499_compl__mono,axiom,
! [X_a] :
( boolean_algebra(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),hAPP(X_a,X_a,uminus_uminus(X_a),Y)),hAPP(X_a,X_a,uminus_uminus(X_a),X))) ) ) ).
fof(fact_500_compl__le__compl__iff,axiom,
! [X_b] :
( boolean_algebra(X_b)
=> ! [X_1,Y_2] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(X_b,X_b,uminus_uminus(X_b),X_1)),hAPP(X_b,X_b,uminus_uminus(X_b),Y_2)))
<=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),Y_2),X_1)) ) ) ).
fof(fact_501_minus__add__distrib,axiom,
! [X_a] :
( ab_group_add(X_a)
=> ! [A_1,B] : hAPP(X_a,X_a,uminus_uminus(X_a),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),hAPP(X_a,X_a,uminus_uminus(X_a),A_1)),hAPP(X_a,X_a,uminus_uminus(X_a),B)) ) ).
fof(fact_502_minus__add,axiom,
! [X_a] :
( group_add(X_a)
=> ! [A_1,B] : hAPP(X_a,X_a,uminus_uminus(X_a),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),hAPP(X_a,X_a,uminus_uminus(X_a),B)),hAPP(X_a,X_a,uminus_uminus(X_a),A_1)) ) ).
fof(fact_503_add__minus__cancel,axiom,
! [X_a] :
( group_add(X_a)
=> ! [A_1,B] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),A_1)),B)) = ti(X_a,B) ) ).
fof(fact_504_minus__add__cancel,axiom,
! [X_a] :
( group_add(X_a)
=> ! [A_1,B] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),A_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B)) = ti(X_a,B) ) ).
fof(fact_505_minus__mult__right,axiom,
! [X_a] :
( ring(X_a)
=> ! [A_1,B] : hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),hAPP(X_a,X_a,uminus_uminus(X_a),B)) ) ).
fof(fact_506_minus__mult__left,axiom,
! [X_a] :
( ring(X_a)
=> ! [A_1,B] : hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),A_1)),B) ) ).
fof(fact_507_minus__mult__commute,axiom,
! [X_a] :
( ring(X_a)
=> ! [A_1,B] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),A_1)),B) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),hAPP(X_a,X_a,uminus_uminus(X_a),B)) ) ).
fof(fact_508_minus__mult__minus,axiom,
! [X_a] :
( ring(X_a)
=> ! [A_1,B] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),A_1)),hAPP(X_a,X_a,uminus_uminus(X_a),B)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B) ) ).
fof(fact_509_square__eq__iff,axiom,
! [X_b] :
( idom(X_b)
=> ! [A_2,Ba] :
( hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),A_2),A_2) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),Ba),Ba)
<=> ( ti(X_b,A_2) = ti(X_b,Ba)
| ti(X_b,A_2) = hAPP(X_b,X_b,uminus_uminus(X_b),Ba) ) ) ) ).
fof(fact_510_setsum__negf,axiom,
! [X_c,X_b] :
( ab_group_add(X_b)
=> ! [F,A_3] : hAPP(fun(X_c,bool),X_b,hAPP(fun(X_c,X_b),fun(fun(X_c,bool),X_b),big_co1399186613setsum(X_c,X_b),hAPP(fun(X_c,X_b),fun(X_c,X_b),hAPP(fun(X_b,X_b),fun(fun(X_c,X_b),fun(X_c,X_b)),combb(X_b,X_b,X_c),uminus_uminus(X_b)),F)),A_3) = hAPP(X_b,X_b,uminus_uminus(X_b),hAPP(fun(X_c,bool),X_b,hAPP(fun(X_c,X_b),fun(fun(X_c,bool),X_b),big_co1399186613setsum(X_c,X_b),F),A_3)) ) ).
fof(fact_511_diff__minus__eq__add,axiom,
! [X_a] :
( group_add(X_a)
=> ! [A_1,B] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_1),hAPP(X_a,X_a,uminus_uminus(X_a),B)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B) ) ).
fof(fact_512_comm__ring__1__class_Onormalizing__ring__rules_I2_J,axiom,
! [X_a] :
( comm_ring_1(X_a)
=> ! [X,Y] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),X),Y) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X),hAPP(X_a,X_a,uminus_uminus(X_a),Y)) ) ).
fof(fact_513_ab__diff__minus,axiom,
! [X_a] :
( ab_group_add(X_a)
=> ! [A_1,B] : 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),plus_plus(X_a),A_1),hAPP(X_a,X_a,uminus_uminus(X_a),B)) ) ).
fof(fact_514_diff__def,axiom,
! [X_a] :
( group_add(X_a)
=> ! [A_1,B] : 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),plus_plus(X_a),A_1),hAPP(X_a,X_a,uminus_uminus(X_a),B)) ) ).
fof(fact_515_subset__Compl__self__eq,axiom,
! [X_b,A_3] :
( 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)),A_3),hAPP(fun(X_b,bool),fun(X_b,bool),uminus_uminus(fun(X_b,bool)),A_3)))
<=> ti(fun(X_b,bool),A_3) = bot_bot(fun(X_b,bool)) ) ).
fof(fact_516_image__uminus__lessThan,axiom,
! [X_b] :
( ordered_ab_group_add(X_b)
=> ! [X_1] : hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,X_b),fun(fun(X_b,bool),fun(X_b,bool)),image(X_b,X_b),uminus_uminus(X_b)),hAPP(X_b,fun(X_b,bool),ord_lessThan(X_b),X_1)) = hAPP(X_b,fun(X_b,bool),ord_greaterThan(X_b),hAPP(X_b,X_b,uminus_uminus(X_b),X_1)) ) ).
fof(fact_517_image__uminus__greaterThan,axiom,
! [X_b] :
( ordered_ab_group_add(X_b)
=> ! [X_1] : hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,X_b),fun(fun(X_b,bool),fun(X_b,bool)),image(X_b,X_b),uminus_uminus(X_b)),hAPP(X_b,fun(X_b,bool),ord_greaterThan(X_b),X_1)) = hAPP(X_b,fun(X_b,bool),ord_lessThan(X_b),hAPP(X_b,X_b,uminus_uminus(X_b),X_1)) ) ).
fof(fact_518_Max_Oremove,axiom,
! [X_b] :
( linorder(X_b)
=> ! [X_1,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),A_3))
=> ( ( 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_3),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_1),bot_bot(fun(X_b,bool)))) = bot_bot(fun(X_b,bool))
=> hAPP(fun(X_b,bool),X_b,big_linorder_Max(X_b),A_3) = ti(X_b,X_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_3),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_1),bot_bot(fun(X_b,bool)))) != bot_bot(fun(X_b,bool))
=> hAPP(fun(X_b,bool),X_b,big_linorder_Max(X_b),A_3) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),ord_max(X_b),X_1),hAPP(fun(X_b,bool),X_b,big_linorder_Max(X_b),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_3),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_1),bot_bot(fun(X_b,bool)))))) ) ) ) ) ) ).
fof(fact_519_Max_Oinsert__remove,axiom,
! [X_b] :
( linorder(X_b)
=> ! [X_1,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( ( 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_3),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_1),bot_bot(fun(X_b,bool)))) = bot_bot(fun(X_b,bool))
=> hAPP(fun(X_b,bool),X_b,big_linorder_Max(X_b),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_1),A_3)) = ti(X_b,X_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_3),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_1),bot_bot(fun(X_b,bool)))) != bot_bot(fun(X_b,bool))
=> hAPP(fun(X_b,bool),X_b,big_linorder_Max(X_b),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_1),A_3)) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),ord_max(X_b),X_1),hAPP(fun(X_b,bool),X_b,big_linorder_Max(X_b),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_3),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_1),bot_bot(fun(X_b,bool)))))) ) ) ) ) ).
fof(fact_520_ComplI,axiom,
! [X_b,Ca,A_3] :
( ~ hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),Ca),A_3))
=> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),Ca),hAPP(fun(X_b,bool),fun(X_b,bool),uminus_uminus(fun(X_b,bool)),A_3))) ) ).
fof(fact_521_Collect__neg__eq,axiom,
! [X_b,Pa] : hAPP(fun(X_b,bool),fun(X_b,bool),collect(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(bool,bool),fun(fun(X_b,bool),fun(X_b,bool)),combb(bool,bool,X_b),fNot),Pa)) = hAPP(fun(X_b,bool),fun(X_b,bool),uminus_uminus(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),collect(X_b),Pa)) ).
fof(fact_522_Compl__eq,axiom,
! [X_b,A_3] : hAPP(fun(X_b,bool),fun(X_b,bool),uminus_uminus(fun(X_b,bool)),A_3) = hAPP(fun(X_b,bool),fun(X_b,bool),collect(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(bool,bool),fun(fun(X_b,bool),fun(X_b,bool)),combb(bool,bool,X_b),fNot),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(fun(X_b,bool),bool)),fun(fun(X_b,bool),fun(X_b,bool)),combc(X_b,fun(X_b,bool),bool),member(X_b)),A_3))) ).
fof(fact_523_double__complement,axiom,
! [X_b,A_3] : hAPP(fun(X_b,bool),fun(X_b,bool),uminus_uminus(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),uminus_uminus(fun(X_b,bool)),A_3)) = ti(fun(X_b,bool),A_3) ).
fof(fact_524_Compl__iff,axiom,
! [X_b,Ca,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),Ca),hAPP(fun(X_b,bool),fun(X_b,bool),uminus_uminus(fun(X_b,bool)),A_3)))
<=> ~ hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),Ca),A_3)) ) ).
fof(fact_525_Compl__eq__Compl__iff,axiom,
! [X_b,A_3,B_2] :
( hAPP(fun(X_b,bool),fun(X_b,bool),uminus_uminus(fun(X_b,bool)),A_3) = hAPP(fun(X_b,bool),fun(X_b,bool),uminus_uminus(fun(X_b,bool)),B_2)
<=> ti(fun(X_b,bool),A_3) = ti(fun(X_b,bool),B_2) ) ).
fof(fact_526_ComplD,axiom,
! [X_b,Ca,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),Ca),hAPP(fun(X_b,bool),fun(X_b,bool),uminus_uminus(fun(X_b,bool)),A_3)))
=> ~ hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),Ca),A_3)) ) ).
fof(fact_527_ComplE,axiom,
! [X_b,Ca,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),Ca),hAPP(fun(X_b,bool),fun(X_b,bool),uminus_uminus(fun(X_b,bool)),A_3)))
=> ~ hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),Ca),A_3)) ) ).
fof(fact_528_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_529_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_530_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_531_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_532_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_533_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_534_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_535_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_536_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_537_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_538_min__max_Osup_Oassoc,axiom,
! [X_a] :
( linorder(X_a)
=> ! [A_1,B,C] : 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_1),B)),C) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),B),C)) ) ).
fof(fact_539_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_540_min__max_Osup_Oleft__commute,axiom,
! [X_a] :
( linorder(X_a)
=> ! [B,A_1,C] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),B),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),A_1),C)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),B),C)) ) ).
fof(fact_541_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_542_min__max_Osup_Oleft__idem,axiom,
! [X_a] :
( linorder(X_a)
=> ! [A_1,B] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),A_1),B)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),A_1),B) ) ).
fof(fact_543_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_544_min__max_Osup_Ocommute,axiom,
! [X_a] :
( linorder(X_a)
=> ! [A_1,B] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),A_1),B) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),B),A_1) ) ).
fof(fact_545_min__max_Osup_Oidem,axiom,
! [X_a] :
( linorder(X_a)
=> ! [A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),A_1),A_1) = ti(X_a,A_1) ) ).
fof(fact_546_greaterThan__eq__iff,axiom,
! [X_b] :
( linorder(X_b)
=> ! [X_1,Y_2] :
( hAPP(X_b,fun(X_b,bool),ord_greaterThan(X_b),X_1) = hAPP(X_b,fun(X_b,bool),ord_greaterThan(X_b),Y_2)
<=> ti(X_b,X_1) = ti(X_b,Y_2) ) ) ).
fof(fact_547_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_548_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_549_min__max_Ole__iff__sup,axiom,
! [X_b] :
( linorder(X_b)
=> ! [X_1,Y_2] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),X_1),Y_2))
<=> hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),ord_max(X_b),X_1),Y_2) = ti(X_b,Y_2) ) ) ).
fof(fact_550_min__max_Ole__sup__iff,axiom,
! [X_b] :
( linorder(X_b)
=> ! [X_1,Y_2,Z_2] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),ord_max(X_b),X_1),Y_2)),Z_2))
<=> ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),X_1),Z_2))
& hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),Y_2),Z_2)) ) ) ) ).
fof(fact_551_min__max_Ole__supI1,axiom,
! [X_a] :
( linorder(X_a)
=> ! [B,X,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),A_1))
=> 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_1),B))) ) ) ).
fof(fact_552_min__max_Ole__supI2,axiom,
! [X_a] :
( linorder(X_a)
=> ! [A_1,X,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),B))
=> 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_1),B))) ) ) ).
fof(fact_553_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_554_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_555_min__max_Ole__supI,axiom,
! [X_a] :
( linorder(X_a)
=> ! [B,A_1,X] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),X))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),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_1),B)),X)) ) ) ) ).
fof(fact_556_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_557_min__max_Osup__mono,axiom,
! [X_a] :
( linorder(X_a)
=> ! [B,D_1,A_1,C] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),C))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),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_1),B)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),C),D_1))) ) ) ) ).
fof(fact_558_min__max_Ole__supE,axiom,
! [X_a] :
( linorder(X_a)
=> ! [A_1,B,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_1),B)),X))
=> ~ ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),X))
=> ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),X)) ) ) ) ).
fof(fact_559_le__max__iff__disj,axiom,
! [X_b] :
( linorder(X_b)
=> ! [Z_2,X_1,Y_2] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),Z_2),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),ord_max(X_b),X_1),Y_2)))
<=> ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),Z_2),X_1))
| hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),Z_2),Y_2)) ) ) ) ).
fof(fact_560_min__max_Ocomp__fun__idem__sup,axiom,
! [X_b] :
( linorder(X_b)
=> hBOOL(hAPP(fun(X_b,fun(X_b,X_b)),bool,finite_comp_fun_idem(X_b,X_b),ord_max(X_b))) ) ).
fof(fact_561_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_562_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_563_minus__min__eq__max,axiom,
! [X_a] :
( linord219039673up_add(X_a)
=> ! [X,Y] : hAPP(X_a,X_a,uminus_uminus(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_max(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),X)),hAPP(X_a,X_a,uminus_uminus(X_a),Y)) ) ).
fof(fact_564_minus__max__eq__min,axiom,
! [X_a] :
( linord219039673up_add(X_a)
=> ! [X,Y] : hAPP(X_a,X_a,uminus_uminus(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_min(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),X)),hAPP(X_a,X_a,uminus_uminus(X_a),Y)) ) ).
fof(fact_565_nat__minus__add__max,axiom,
! [N_1,M] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N_1),M)),M) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_max(nat),N_1),M) ).
fof(fact_566_Max_Oin__idem,axiom,
! [X_b] :
( linorder(X_b)
=> ! [X_1,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),A_3))
=> hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),ord_max(X_b),X_1),hAPP(fun(X_b,bool),X_b,big_linorder_Max(X_b),A_3)) = hAPP(fun(X_b,bool),X_b,big_linorder_Max(X_b),A_3) ) ) ) ).
fof(fact_567_min__max_Osup__le__fold__sup,axiom,
! [X_b] :
( linorder(X_b)
=> ! [Ba,A_2,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),A_2),A_3))
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),ord_max(X_b),A_2),Ba)),hAPP(fun(X_b,bool),X_b,hAPP(X_b,fun(fun(X_b,bool),X_b),hAPP(fun(X_b,fun(X_b,X_b)),fun(X_b,fun(fun(X_b,bool),X_b)),finite_fold(X_b,X_b),ord_max(X_b)),Ba),A_3))) ) ) ) ).
fof(fact_568_max__ord__max,axiom,
! [X_b] :
( ord(X_b)
=> ord_max(X_b) = hAPP(fun(X_b,fun(X_b,bool)),fun(X_b,fun(X_b,X_b)),max(X_b),ord_less_eq(X_b)) ) ).
fof(fact_569_min__max_Ofold__sup__insert,axiom,
! [X_b] :
( linorder(X_b)
=> ! [Ba,A_2,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> hAPP(fun(X_b,bool),X_b,hAPP(X_b,fun(fun(X_b,bool),X_b),hAPP(fun(X_b,fun(X_b,X_b)),fun(X_b,fun(fun(X_b,bool),X_b)),finite_fold(X_b,X_b),ord_max(X_b)),Ba),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_2),A_3)) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),ord_max(X_b),A_2),hAPP(fun(X_b,bool),X_b,hAPP(X_b,fun(fun(X_b,bool),X_b),hAPP(fun(X_b,fun(X_b,X_b)),fun(X_b,fun(fun(X_b,bool),X_b)),finite_fold(X_b,X_b),ord_max(X_b)),Ba),A_3)) ) ) ).
fof(fact_570_min__max_Osup__Inf__absorb,axiom,
! [X_b] :
( linorder(X_b)
=> ! [A_2,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),A_2),A_3))
=> hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),ord_max(X_b),A_2),hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),big_Inf_fin(X_b),ord_min(X_b)),A_3)) = ti(X_b,A_2) ) ) ) ).
fof(fact_571_greaterThan__subset__iff,axiom,
! [X_b] :
( linorder(X_b)
=> ! [X_1,Y_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)),hAPP(X_b,fun(X_b,bool),ord_greaterThan(X_b),X_1)),hAPP(X_b,fun(X_b,bool),ord_greaterThan(X_b),Y_2)))
<=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),Y_2),X_1)) ) ) ).
fof(fact_572_Max_OF__eq,axiom,
! [X_b] :
( linorder(X_b)
=> ! [A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> hAPP(fun(X_b,bool),X_b,big_linorder_Max(X_b),A_3) = hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),finite_fold1(X_b),ord_max(X_b)),A_3) ) ) ).
fof(fact_573_dual__min,axiom,
! [X_b] :
( linorder(X_b)
=> hAPP(fun(X_b,fun(X_b,bool)),fun(X_b,fun(X_b,X_b)),min(X_b),hAPP(fun(X_b,fun(X_b,bool)),fun(X_b,fun(X_b,bool)),combc(X_b,X_b,bool),ord_less_eq(X_b))) = ord_max(X_b) ) ).
fof(fact_574_Max__insert,axiom,
! [X_b] :
( linorder(X_b)
=> ! [X_1,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( ti(fun(X_b,bool),A_3) != bot_bot(fun(X_b,bool))
=> hAPP(fun(X_b,bool),X_b,big_linorder_Max(X_b),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_1),A_3)) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),ord_max(X_b),X_1),hAPP(fun(X_b,bool),X_b,big_linorder_Max(X_b),A_3)) ) ) ) ).
fof(fact_575_Max_Oinsert,axiom,
! [X_b] :
( linorder(X_b)
=> ! [X_1,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( ~ hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),A_3))
=> ( ti(fun(X_b,bool),A_3) != bot_bot(fun(X_b,bool))
=> hAPP(fun(X_b,bool),X_b,big_linorder_Max(X_b),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_1),A_3)) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),ord_max(X_b),X_1),hAPP(fun(X_b,bool),X_b,big_linorder_Max(X_b),A_3)) ) ) ) ) ).
fof(fact_576_Max_Osubset__idem,axiom,
! [X_b] :
( linorder(X_b)
=> ! [B_2,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( ti(fun(X_b,bool),B_2) != bot_bot(fun(X_b,bool))
=> ( 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_2),A_3))
=> hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),ord_max(X_b),hAPP(fun(X_b,bool),X_b,big_linorder_Max(X_b),B_2)),hAPP(fun(X_b,bool),X_b,big_linorder_Max(X_b),A_3)) = hAPP(fun(X_b,bool),X_b,big_linorder_Max(X_b),A_3) ) ) ) ) ).
fof(fact_577_Max_Oeq__fold_H,axiom,
! [X_b] :
( linorder(X_b)
=> ! [X_1,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( ~ hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),A_3))
=> hAPP(fun(X_b,bool),X_b,big_linorder_Max(X_b),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_1),A_3)) = hAPP(fun(X_b,bool),X_b,hAPP(X_b,fun(fun(X_b,bool),X_b),hAPP(fun(X_b,fun(X_b,X_b)),fun(X_b,fun(fun(X_b,bool),X_b)),finite_fold(X_b,X_b),ord_max(X_b)),X_1),A_3) ) ) ) ).
fof(fact_578_Max_Oeq__fold__idem_H,axiom,
! [X_b] :
( linorder(X_b)
=> ! [A_2,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> hAPP(fun(X_b,bool),X_b,big_linorder_Max(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_2),A_3)) = hAPP(fun(X_b,bool),X_b,hAPP(X_b,fun(fun(X_b,bool),X_b),hAPP(fun(X_b,fun(X_b,X_b)),fun(X_b,fun(fun(X_b,bool),X_b)),finite_fold(X_b,X_b),ord_max(X_b)),A_2),A_3) ) ) ).
fof(fact_579_Max_Ohom__commute,axiom,
! [X_b] :
( linorder(X_b)
=> ! [N_3,H] :
( ! [X_2,Y_1] : hAPP(X_b,X_b,H,hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),ord_max(X_b),X_2),Y_1)) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),ord_max(X_b),hAPP(X_b,X_b,H,X_2)),hAPP(X_b,X_b,H,Y_1))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),N_3))
=> ( ti(fun(X_b,bool),N_3) != bot_bot(fun(X_b,bool))
=> hAPP(X_b,X_b,H,hAPP(fun(X_b,bool),X_b,big_linorder_Max(X_b),N_3)) = hAPP(fun(X_b,bool),X_b,big_linorder_Max(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,X_b),fun(fun(X_b,bool),fun(X_b,bool)),image(X_b,X_b),H),N_3)) ) ) ) ) ).
fof(fact_580_Max_Oclosed,axiom,
! [X_b] :
( linorder(X_b)
=> ! [A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( ti(fun(X_b,bool),A_3) != bot_bot(fun(X_b,bool))
=> ( ! [X_2,Y_1] : hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),ord_max(X_b),X_2),Y_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),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),Y_1),bot_bot(fun(X_b,bool))))))
=> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),hAPP(fun(X_b,bool),X_b,big_linorder_Max(X_b),A_3)),A_3)) ) ) ) ) ).
fof(fact_581_min__max_Ofold__sup__le__sup,axiom,
! [X_b] :
( linorder(X_b)
=> ! [Ca,Ba,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( ! [X_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_2),A_3))
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),X_2),Ba)) )
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(fun(X_b,bool),X_b,hAPP(X_b,fun(fun(X_b,bool),X_b),hAPP(fun(X_b,fun(X_b,X_b)),fun(X_b,fun(fun(X_b,bool),X_b)),finite_fold(X_b,X_b),ord_max(X_b)),Ca),A_3)),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),ord_max(X_b),Ba),Ca))) ) ) ) ).
fof(fact_582_min__max_Odistrib__imp1,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X,Y,Z] :
( ! [X_2,Y_1,Z_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X_2),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),Y_1),Z_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_min(X_a),X_2),Y_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X_2),Z_1))
=> 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_583_min__max_Odistrib__imp2,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X,Y,Z] :
( ! [X_2,Y_1,Z_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),X_2),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),Y_1),Z_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_max(X_a),X_2),Y_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),X_2),Z_1))
=> 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_584_max__leastL,axiom,
! [X_a] :
( ord(X_a)
=> ! [X,Least] :
( ! [X_2] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Least),X_2))
=> 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_585_max__leastR,axiom,
! [X_a] :
( order(X_a)
=> ! [X,Least] :
( ! [X_2] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Least),X_2))
=> 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_586_min__max_OInf__le__Sup,axiom,
! [X_b] :
( linorder(X_b)
=> ! [A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( ti(fun(X_b,bool),A_3) != bot_bot(fun(X_b,bool))
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),big_Inf_fin(X_b),ord_min(X_b)),A_3)),hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),big_Sup_fin(X_b),ord_max(X_b)),A_3))) ) ) ) ).
fof(fact_587_min__max_Osup__Inf2__distrib,axiom,
! [X_b] :
( linorder(X_b)
=> ! [B_2,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( ti(fun(X_b,bool),A_3) != bot_bot(fun(X_b,bool))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),B_2))
=> ( ti(fun(X_b,bool),B_2) != bot_bot(fun(X_b,bool))
=> hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),ord_max(X_b),hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),big_Inf_fin(X_b),ord_min(X_b)),A_3)),hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),big_Inf_fin(X_b),ord_min(X_b)),B_2)) = hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),big_Inf_fin(X_b),ord_min(X_b)),hAPP(fun(X_b,bool),fun(X_b,bool),collect(X_b),hAPP(fun(X_b,fun(X_b,bool)),fun(X_b,bool),hAPP(fun(fun(X_b,bool),bool),fun(fun(X_b,fun(X_b,bool)),fun(X_b,bool)),combb(fun(X_b,bool),bool,X_b),fEx(X_b)),hAPP(fun(X_b,fun(X_b,fun(X_b,bool))),fun(X_b,fun(X_b,bool)),hAPP(fun(fun(X_b,fun(X_b,bool)),fun(X_b,bool)),fun(fun(X_b,fun(X_b,fun(X_b,bool))),fun(X_b,fun(X_b,bool))),combb(fun(X_b,fun(X_b,bool)),fun(X_b,bool),X_b),hAPP(fun(fun(X_b,bool),bool),fun(fun(X_b,fun(X_b,bool)),fun(X_b,bool)),combb(fun(X_b,bool),bool,X_b),fEx(X_b))),hAPP(fun(X_b,fun(X_b,bool)),fun(X_b,fun(X_b,fun(X_b,bool))),hAPP(fun(X_b,fun(fun(X_b,fun(X_b,bool)),fun(X_b,fun(X_b,bool)))),fun(fun(X_b,fun(X_b,bool)),fun(X_b,fun(X_b,fun(X_b,bool)))),combc(X_b,fun(X_b,fun(X_b,bool)),fun(X_b,fun(X_b,bool))),hAPP(fun(X_b,fun(X_b,fun(fun(X_b,bool),fun(X_b,bool)))),fun(X_b,fun(fun(X_b,fun(X_b,bool)),fun(X_b,fun(X_b,bool)))),hAPP(fun(fun(X_b,fun(fun(X_b,bool),fun(X_b,bool))),fun(fun(X_b,fun(X_b,bool)),fun(X_b,fun(X_b,bool)))),fun(fun(X_b,fun(X_b,fun(fun(X_b,bool),fun(X_b,bool)))),fun(X_b,fun(fun(X_b,fun(X_b,bool)),fun(X_b,fun(X_b,bool))))),combb(fun(X_b,fun(fun(X_b,bool),fun(X_b,bool))),fun(fun(X_b,fun(X_b,bool)),fun(X_b,fun(X_b,bool))),X_b),combs(X_b,fun(X_b,bool),fun(X_b,bool))),hAPP(fun(X_b,fun(X_b,fun(X_b,fun(bool,bool)))),fun(X_b,fun(X_b,fun(fun(X_b,bool),fun(X_b,bool)))),hAPP(fun(fun(X_b,fun(X_b,fun(bool,bool))),fun(X_b,fun(fun(X_b,bool),fun(X_b,bool)))),fun(fun(X_b,fun(X_b,fun(X_b,fun(bool,bool)))),fun(X_b,fun(X_b,fun(fun(X_b,bool),fun(X_b,bool))))),combb(fun(X_b,fun(X_b,fun(bool,bool))),fun(X_b,fun(fun(X_b,bool),fun(X_b,bool))),X_b),hAPP(fun(fun(X_b,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,bool))),fun(fun(X_b,fun(X_b,fun(bool,bool))),fun(X_b,fun(fun(X_b,bool),fun(X_b,bool)))),combb(fun(X_b,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,bool)),X_b),combs(X_b,bool,bool))),hAPP(fun(X_b,fun(X_b,fun(X_b,bool))),fun(X_b,fun(X_b,fun(X_b,fun(bool,bool)))),hAPP(fun(fun(X_b,fun(X_b,bool)),fun(X_b,fun(X_b,fun(bool,bool)))),fun(fun(X_b,fun(X_b,fun(X_b,bool))),fun(X_b,fun(X_b,fun(X_b,fun(bool,bool))))),combb(fun(X_b,fun(X_b,bool)),fun(X_b,fun(X_b,fun(bool,bool))),X_b),hAPP(fun(fun(X_b,bool),fun(X_b,fun(bool,bool))),fun(fun(X_b,fun(X_b,bool)),fun(X_b,fun(X_b,fun(bool,bool)))),combb(fun(X_b,bool),fun(X_b,fun(bool,bool)),X_b),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,fun(bool,bool))),combb(bool,fun(bool,bool),X_b),fconj))),hAPP(fun(X_b,fun(X_b,X_b)),fun(X_b,fun(X_b,fun(X_b,bool))),hAPP(fun(X_b,fun(fun(X_b,fun(X_b,X_b)),fun(X_b,fun(X_b,bool)))),fun(fun(X_b,fun(X_b,X_b)),fun(X_b,fun(X_b,fun(X_b,bool)))),combc(X_b,fun(X_b,fun(X_b,X_b)),fun(X_b,fun(X_b,bool))),hAPP(fun(X_b,fun(fun(X_b,X_b),fun(X_b,bool))),fun(X_b,fun(fun(X_b,fun(X_b,X_b)),fun(X_b,fun(X_b,bool)))),hAPP(fun(fun(fun(X_b,X_b),fun(X_b,bool)),fun(fun(X_b,fun(X_b,X_b)),fun(X_b,fun(X_b,bool)))),fun(fun(X_b,fun(fun(X_b,X_b),fun(X_b,bool))),fun(X_b,fun(fun(X_b,fun(X_b,X_b)),fun(X_b,fun(X_b,bool))))),combb(fun(fun(X_b,X_b),fun(X_b,bool)),fun(fun(X_b,fun(X_b,X_b)),fun(X_b,fun(X_b,bool))),X_b),combb(fun(X_b,X_b),fun(X_b,bool),X_b)),hAPP(fun(X_b,fun(X_b,bool)),fun(X_b,fun(fun(X_b,X_b),fun(X_b,bool))),hAPP(fun(fun(X_b,bool),fun(fun(X_b,X_b),fun(X_b,bool))),fun(fun(X_b,fun(X_b,bool)),fun(X_b,fun(fun(X_b,X_b),fun(X_b,bool)))),combb(fun(X_b,bool),fun(fun(X_b,X_b),fun(X_b,bool)),X_b),combb(X_b,bool,X_b)),fequal(X_b)))),ord_max(X_b)))))),hAPP(fun(X_b,bool),fun(X_b,fun(X_b,bool)),hAPP(fun(X_b,fun(fun(X_b,bool),fun(X_b,bool))),fun(fun(X_b,bool),fun(X_b,fun(X_b,bool))),combc(X_b,fun(X_b,bool),fun(X_b,bool)),hAPP(fun(X_b,fun(bool,bool)),fun(X_b,fun(fun(X_b,bool),fun(X_b,bool))),hAPP(fun(fun(bool,bool),fun(fun(X_b,bool),fun(X_b,bool))),fun(fun(X_b,fun(bool,bool)),fun(X_b,fun(fun(X_b,bool),fun(X_b,bool)))),combb(fun(bool,bool),fun(fun(X_b,bool),fun(X_b,bool)),X_b),combb(bool,bool,X_b)),hAPP(fun(X_b,bool),fun(X_b,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,fun(bool,bool))),combb(bool,fun(bool,bool),X_b),fconj),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(fun(X_b,bool),bool)),fun(fun(X_b,bool),fun(X_b,bool)),combc(X_b,fun(X_b,bool),bool),member(X_b)),A_3)))),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(fun(X_b,bool),bool)),fun(fun(X_b,bool),fun(X_b,bool)),combc(X_b,fun(X_b,bool),bool),member(X_b)),B_2))))))) ) ) ) ) ) ).
fof(fact_588_min__max_Osup__Inf1__distrib,axiom,
! [X_b] :
( linorder(X_b)
=> ! [X_1,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( ti(fun(X_b,bool),A_3) != bot_bot(fun(X_b,bool))
=> hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),ord_max(X_b),X_1),hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),big_Inf_fin(X_b),ord_min(X_b)),A_3)) = hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),big_Inf_fin(X_b),ord_min(X_b)),hAPP(fun(X_b,bool),fun(X_b,bool),collect(X_b),hAPP(fun(X_b,fun(X_b,bool)),fun(X_b,bool),hAPP(fun(fun(X_b,bool),bool),fun(fun(X_b,fun(X_b,bool)),fun(X_b,bool)),combb(fun(X_b,bool),bool,X_b),fEx(X_b)),hAPP(fun(X_b,bool),fun(X_b,fun(X_b,bool)),hAPP(fun(X_b,fun(fun(X_b,bool),fun(X_b,bool))),fun(fun(X_b,bool),fun(X_b,fun(X_b,bool))),combc(X_b,fun(X_b,bool),fun(X_b,bool)),hAPP(fun(X_b,fun(X_b,fun(bool,bool))),fun(X_b,fun(fun(X_b,bool),fun(X_b,bool))),hAPP(fun(fun(X_b,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,bool))),fun(fun(X_b,fun(X_b,fun(bool,bool))),fun(X_b,fun(fun(X_b,bool),fun(X_b,bool)))),combb(fun(X_b,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,bool)),X_b),combs(X_b,bool,bool)),hAPP(fun(X_b,fun(X_b,bool)),fun(X_b,fun(X_b,fun(bool,bool))),hAPP(fun(fun(X_b,bool),fun(X_b,fun(bool,bool))),fun(fun(X_b,fun(X_b,bool)),fun(X_b,fun(X_b,fun(bool,bool)))),combb(fun(X_b,bool),fun(X_b,fun(bool,bool)),X_b),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,fun(bool,bool))),combb(bool,fun(bool,bool),X_b),fconj)),hAPP(fun(X_b,X_b),fun(X_b,fun(X_b,bool)),hAPP(fun(X_b,fun(fun(X_b,X_b),fun(X_b,bool))),fun(fun(X_b,X_b),fun(X_b,fun(X_b,bool))),combc(X_b,fun(X_b,X_b),fun(X_b,bool)),hAPP(fun(X_b,fun(X_b,bool)),fun(X_b,fun(fun(X_b,X_b),fun(X_b,bool))),hAPP(fun(fun(X_b,bool),fun(fun(X_b,X_b),fun(X_b,bool))),fun(fun(X_b,fun(X_b,bool)),fun(X_b,fun(fun(X_b,X_b),fun(X_b,bool)))),combb(fun(X_b,bool),fun(fun(X_b,X_b),fun(X_b,bool)),X_b),combb(X_b,bool,X_b)),fequal(X_b))),hAPP(X_b,fun(X_b,X_b),ord_max(X_b),X_1))))),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(fun(X_b,bool),bool)),fun(fun(X_b,bool),fun(X_b,bool)),combc(X_b,fun(X_b,bool),bool),member(X_b)),A_3))))) ) ) ) ).
fof(fact_589_finite__Collect__bounded__ex,axiom,
! [X_c,X_b,Q_3,Pa] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),collect(X_b),Pa)))
=> ( hBOOL(hAPP(fun(X_c,bool),bool,finite_finite_1(X_c),hAPP(fun(X_c,bool),fun(X_c,bool),collect(X_c),hAPP(fun(X_c,fun(X_b,bool)),fun(X_c,bool),hAPP(fun(fun(X_b,bool),bool),fun(fun(X_c,fun(X_b,bool)),fun(X_c,bool)),combb(fun(X_b,bool),bool,X_c),fEx(X_b)),hAPP(fun(X_c,fun(X_b,bool)),fun(X_c,fun(X_b,bool)),hAPP(fun(fun(X_b,bool),fun(X_b,bool)),fun(fun(X_c,fun(X_b,bool)),fun(X_c,fun(X_b,bool))),combb(fun(X_b,bool),fun(X_b,bool),X_c),hAPP(fun(X_b,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,bool)),combs(X_b,bool,bool),hAPP(fun(X_b,bool),fun(X_b,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,fun(bool,bool))),combb(bool,fun(bool,bool),X_b),fconj),Pa))),Q_3)))))
<=> ! [Y_1] :
( hBOOL(hAPP(X_b,bool,Pa,Y_1))
=> hBOOL(hAPP(fun(X_c,bool),bool,finite_finite_1(X_c),hAPP(fun(X_c,bool),fun(X_c,bool),collect(X_c),hAPP(X_b,fun(X_c,bool),hAPP(fun(X_c,fun(X_b,bool)),fun(X_b,fun(X_c,bool)),combc(X_c,X_b,bool),Q_3),Y_1)))) ) ) ) ).
fof(fact_590_finite__image__set,axiom,
! [X_c,X_b,F,Pa] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),collect(X_b),Pa)))
=> hBOOL(hAPP(fun(X_c,bool),bool,finite_finite_1(X_c),hAPP(fun(X_c,bool),fun(X_c,bool),collect(X_c),hAPP(fun(X_c,fun(X_b,bool)),fun(X_c,bool),hAPP(fun(fun(X_b,bool),bool),fun(fun(X_c,fun(X_b,bool)),fun(X_c,bool)),combb(fun(X_b,bool),bool,X_c),fEx(X_b)),hAPP(fun(X_b,bool),fun(X_c,fun(X_b,bool)),hAPP(fun(X_c,fun(fun(X_b,bool),fun(X_b,bool))),fun(fun(X_b,bool),fun(X_c,fun(X_b,bool))),combc(X_c,fun(X_b,bool),fun(X_b,bool)),hAPP(fun(X_c,fun(X_b,fun(bool,bool))),fun(X_c,fun(fun(X_b,bool),fun(X_b,bool))),hAPP(fun(fun(X_b,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,bool))),fun(fun(X_c,fun(X_b,fun(bool,bool))),fun(X_c,fun(fun(X_b,bool),fun(X_b,bool)))),combb(fun(X_b,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,bool)),X_c),combs(X_b,bool,bool)),hAPP(fun(X_c,fun(X_b,bool)),fun(X_c,fun(X_b,fun(bool,bool))),hAPP(fun(fun(X_b,bool),fun(X_b,fun(bool,bool))),fun(fun(X_c,fun(X_b,bool)),fun(X_c,fun(X_b,fun(bool,bool)))),combb(fun(X_b,bool),fun(X_b,fun(bool,bool)),X_c),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,fun(bool,bool))),combb(bool,fun(bool,bool),X_b),fconj)),hAPP(fun(X_b,X_c),fun(X_c,fun(X_b,bool)),hAPP(fun(X_c,fun(fun(X_b,X_c),fun(X_b,bool))),fun(fun(X_b,X_c),fun(X_c,fun(X_b,bool))),combc(X_c,fun(X_b,X_c),fun(X_b,bool)),hAPP(fun(X_c,fun(X_c,bool)),fun(X_c,fun(fun(X_b,X_c),fun(X_b,bool))),hAPP(fun(fun(X_c,bool),fun(fun(X_b,X_c),fun(X_b,bool))),fun(fun(X_c,fun(X_c,bool)),fun(X_c,fun(fun(X_b,X_c),fun(X_b,bool)))),combb(fun(X_c,bool),fun(fun(X_b,X_c),fun(X_b,bool)),X_c),combb(X_c,bool,X_b)),fequal(X_c))),F)))),Pa))))) ) ).
fof(fact_591_min__max_Oinf__Sup1__distrib,axiom,
! [X_b] :
( linorder(X_b)
=> ! [X_1,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( ti(fun(X_b,bool),A_3) != bot_bot(fun(X_b,bool))
=> hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),ord_min(X_b),X_1),hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),big_Sup_fin(X_b),ord_max(X_b)),A_3)) = hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),big_Sup_fin(X_b),ord_max(X_b)),hAPP(fun(X_b,bool),fun(X_b,bool),collect(X_b),hAPP(fun(X_b,fun(X_b,bool)),fun(X_b,bool),hAPP(fun(fun(X_b,bool),bool),fun(fun(X_b,fun(X_b,bool)),fun(X_b,bool)),combb(fun(X_b,bool),bool,X_b),fEx(X_b)),hAPP(fun(X_b,bool),fun(X_b,fun(X_b,bool)),hAPP(fun(X_b,fun(fun(X_b,bool),fun(X_b,bool))),fun(fun(X_b,bool),fun(X_b,fun(X_b,bool))),combc(X_b,fun(X_b,bool),fun(X_b,bool)),hAPP(fun(X_b,fun(X_b,fun(bool,bool))),fun(X_b,fun(fun(X_b,bool),fun(X_b,bool))),hAPP(fun(fun(X_b,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,bool))),fun(fun(X_b,fun(X_b,fun(bool,bool))),fun(X_b,fun(fun(X_b,bool),fun(X_b,bool)))),combb(fun(X_b,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,bool)),X_b),combs(X_b,bool,bool)),hAPP(fun(X_b,fun(X_b,bool)),fun(X_b,fun(X_b,fun(bool,bool))),hAPP(fun(fun(X_b,bool),fun(X_b,fun(bool,bool))),fun(fun(X_b,fun(X_b,bool)),fun(X_b,fun(X_b,fun(bool,bool)))),combb(fun(X_b,bool),fun(X_b,fun(bool,bool)),X_b),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,fun(bool,bool))),combb(bool,fun(bool,bool),X_b),fconj)),hAPP(fun(X_b,X_b),fun(X_b,fun(X_b,bool)),hAPP(fun(X_b,fun(fun(X_b,X_b),fun(X_b,bool))),fun(fun(X_b,X_b),fun(X_b,fun(X_b,bool))),combc(X_b,fun(X_b,X_b),fun(X_b,bool)),hAPP(fun(X_b,fun(X_b,bool)),fun(X_b,fun(fun(X_b,X_b),fun(X_b,bool))),hAPP(fun(fun(X_b,bool),fun(fun(X_b,X_b),fun(X_b,bool))),fun(fun(X_b,fun(X_b,bool)),fun(X_b,fun(fun(X_b,X_b),fun(X_b,bool)))),combb(fun(X_b,bool),fun(fun(X_b,X_b),fun(X_b,bool)),X_b),combb(X_b,bool,X_b)),fequal(X_b))),hAPP(X_b,fun(X_b,X_b),ord_min(X_b),X_1))))),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(fun(X_b,bool),bool)),fun(fun(X_b,bool),fun(X_b,bool)),combc(X_b,fun(X_b,bool),bool),member(X_b)),A_3))))) ) ) ) ).
fof(fact_592_min__max_Oinf__Sup2__distrib,axiom,
! [X_b] :
( linorder(X_b)
=> ! [B_2,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( ti(fun(X_b,bool),A_3) != bot_bot(fun(X_b,bool))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),B_2))
=> ( ti(fun(X_b,bool),B_2) != bot_bot(fun(X_b,bool))
=> hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),ord_min(X_b),hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),big_Sup_fin(X_b),ord_max(X_b)),A_3)),hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),big_Sup_fin(X_b),ord_max(X_b)),B_2)) = hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),big_Sup_fin(X_b),ord_max(X_b)),hAPP(fun(X_b,bool),fun(X_b,bool),collect(X_b),hAPP(fun(X_b,fun(X_b,bool)),fun(X_b,bool),hAPP(fun(fun(X_b,bool),bool),fun(fun(X_b,fun(X_b,bool)),fun(X_b,bool)),combb(fun(X_b,bool),bool,X_b),fEx(X_b)),hAPP(fun(X_b,fun(X_b,fun(X_b,bool))),fun(X_b,fun(X_b,bool)),hAPP(fun(fun(X_b,fun(X_b,bool)),fun(X_b,bool)),fun(fun(X_b,fun(X_b,fun(X_b,bool))),fun(X_b,fun(X_b,bool))),combb(fun(X_b,fun(X_b,bool)),fun(X_b,bool),X_b),hAPP(fun(fun(X_b,bool),bool),fun(fun(X_b,fun(X_b,bool)),fun(X_b,bool)),combb(fun(X_b,bool),bool,X_b),fEx(X_b))),hAPP(fun(X_b,fun(X_b,bool)),fun(X_b,fun(X_b,fun(X_b,bool))),hAPP(fun(X_b,fun(fun(X_b,fun(X_b,bool)),fun(X_b,fun(X_b,bool)))),fun(fun(X_b,fun(X_b,bool)),fun(X_b,fun(X_b,fun(X_b,bool)))),combc(X_b,fun(X_b,fun(X_b,bool)),fun(X_b,fun(X_b,bool))),hAPP(fun(X_b,fun(X_b,fun(fun(X_b,bool),fun(X_b,bool)))),fun(X_b,fun(fun(X_b,fun(X_b,bool)),fun(X_b,fun(X_b,bool)))),hAPP(fun(fun(X_b,fun(fun(X_b,bool),fun(X_b,bool))),fun(fun(X_b,fun(X_b,bool)),fun(X_b,fun(X_b,bool)))),fun(fun(X_b,fun(X_b,fun(fun(X_b,bool),fun(X_b,bool)))),fun(X_b,fun(fun(X_b,fun(X_b,bool)),fun(X_b,fun(X_b,bool))))),combb(fun(X_b,fun(fun(X_b,bool),fun(X_b,bool))),fun(fun(X_b,fun(X_b,bool)),fun(X_b,fun(X_b,bool))),X_b),combs(X_b,fun(X_b,bool),fun(X_b,bool))),hAPP(fun(X_b,fun(X_b,fun(X_b,fun(bool,bool)))),fun(X_b,fun(X_b,fun(fun(X_b,bool),fun(X_b,bool)))),hAPP(fun(fun(X_b,fun(X_b,fun(bool,bool))),fun(X_b,fun(fun(X_b,bool),fun(X_b,bool)))),fun(fun(X_b,fun(X_b,fun(X_b,fun(bool,bool)))),fun(X_b,fun(X_b,fun(fun(X_b,bool),fun(X_b,bool))))),combb(fun(X_b,fun(X_b,fun(bool,bool))),fun(X_b,fun(fun(X_b,bool),fun(X_b,bool))),X_b),hAPP(fun(fun(X_b,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,bool))),fun(fun(X_b,fun(X_b,fun(bool,bool))),fun(X_b,fun(fun(X_b,bool),fun(X_b,bool)))),combb(fun(X_b,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,bool)),X_b),combs(X_b,bool,bool))),hAPP(fun(X_b,fun(X_b,fun(X_b,bool))),fun(X_b,fun(X_b,fun(X_b,fun(bool,bool)))),hAPP(fun(fun(X_b,fun(X_b,bool)),fun(X_b,fun(X_b,fun(bool,bool)))),fun(fun(X_b,fun(X_b,fun(X_b,bool))),fun(X_b,fun(X_b,fun(X_b,fun(bool,bool))))),combb(fun(X_b,fun(X_b,bool)),fun(X_b,fun(X_b,fun(bool,bool))),X_b),hAPP(fun(fun(X_b,bool),fun(X_b,fun(bool,bool))),fun(fun(X_b,fun(X_b,bool)),fun(X_b,fun(X_b,fun(bool,bool)))),combb(fun(X_b,bool),fun(X_b,fun(bool,bool)),X_b),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,fun(bool,bool))),combb(bool,fun(bool,bool),X_b),fconj))),hAPP(fun(X_b,fun(X_b,X_b)),fun(X_b,fun(X_b,fun(X_b,bool))),hAPP(fun(X_b,fun(fun(X_b,fun(X_b,X_b)),fun(X_b,fun(X_b,bool)))),fun(fun(X_b,fun(X_b,X_b)),fun(X_b,fun(X_b,fun(X_b,bool)))),combc(X_b,fun(X_b,fun(X_b,X_b)),fun(X_b,fun(X_b,bool))),hAPP(fun(X_b,fun(fun(X_b,X_b),fun(X_b,bool))),fun(X_b,fun(fun(X_b,fun(X_b,X_b)),fun(X_b,fun(X_b,bool)))),hAPP(fun(fun(fun(X_b,X_b),fun(X_b,bool)),fun(fun(X_b,fun(X_b,X_b)),fun(X_b,fun(X_b,bool)))),fun(fun(X_b,fun(fun(X_b,X_b),fun(X_b,bool))),fun(X_b,fun(fun(X_b,fun(X_b,X_b)),fun(X_b,fun(X_b,bool))))),combb(fun(fun(X_b,X_b),fun(X_b,bool)),fun(fun(X_b,fun(X_b,X_b)),fun(X_b,fun(X_b,bool))),X_b),combb(fun(X_b,X_b),fun(X_b,bool),X_b)),hAPP(fun(X_b,fun(X_b,bool)),fun(X_b,fun(fun(X_b,X_b),fun(X_b,bool))),hAPP(fun(fun(X_b,bool),fun(fun(X_b,X_b),fun(X_b,bool))),fun(fun(X_b,fun(X_b,bool)),fun(X_b,fun(fun(X_b,X_b),fun(X_b,bool)))),combb(fun(X_b,bool),fun(fun(X_b,X_b),fun(X_b,bool)),X_b),combb(X_b,bool,X_b)),fequal(X_b)))),ord_min(X_b)))))),hAPP(fun(X_b,bool),fun(X_b,fun(X_b,bool)),hAPP(fun(X_b,fun(fun(X_b,bool),fun(X_b,bool))),fun(fun(X_b,bool),fun(X_b,fun(X_b,bool))),combc(X_b,fun(X_b,bool),fun(X_b,bool)),hAPP(fun(X_b,fun(bool,bool)),fun(X_b,fun(fun(X_b,bool),fun(X_b,bool))),hAPP(fun(fun(bool,bool),fun(fun(X_b,bool),fun(X_b,bool))),fun(fun(X_b,fun(bool,bool)),fun(X_b,fun(fun(X_b,bool),fun(X_b,bool)))),combb(fun(bool,bool),fun(fun(X_b,bool),fun(X_b,bool)),X_b),combb(bool,bool,X_b)),hAPP(fun(X_b,bool),fun(X_b,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,fun(bool,bool))),combb(bool,fun(bool,bool),X_b),fconj),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(fun(X_b,bool),bool)),fun(fun(X_b,bool),fun(X_b,bool)),combc(X_b,fun(X_b,bool),bool),member(X_b)),A_3)))),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(fun(X_b,bool),bool)),fun(fun(X_b,bool),fun(X_b,bool)),combc(X_b,fun(X_b,bool),bool),member(X_b)),B_2))))))) ) ) ) ) ) ).
fof(fact_593_min__max_OSup__fin_Oin__idem,axiom,
! [X_b] :
( linorder(X_b)
=> ! [X_1,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),A_3))
=> hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),ord_max(X_b),X_1),hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),big_Sup_fin(X_b),ord_max(X_b)),A_3)) = hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),big_Sup_fin(X_b),ord_max(X_b)),A_3) ) ) ) ).
fof(fact_594_min__max_OSup__fin_Osingleton,axiom,
! [X_b] :
( linorder(X_b)
=> ! [X_1] : hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),big_Sup_fin(X_b),ord_max(X_b)),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_1),bot_bot(fun(X_b,bool)))) = ti(X_b,X_1) ) ).
fof(fact_595_min__max_Oinf__Sup__absorb,axiom,
! [X_b] :
( linorder(X_b)
=> ! [A_2,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),A_2),A_3))
=> hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),ord_min(X_b),A_2),hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),big_Sup_fin(X_b),ord_max(X_b)),A_3)) = ti(X_b,A_2) ) ) ) ).
fof(fact_596_min__max_OSup__fin_OF__eq,axiom,
! [X_b] :
( linorder(X_b)
=> ! [A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),big_Sup_fin(X_b),ord_max(X_b)),A_3) = hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),finite_fold1(X_b),ord_max(X_b)),A_3) ) ) ).
fof(fact_597_min__max_OSup__fin_Oinsert,axiom,
! [X_b] :
( linorder(X_b)
=> ! [X_1,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( ~ hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),A_3))
=> ( ti(fun(X_b,bool),A_3) != bot_bot(fun(X_b,bool))
=> hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),big_Sup_fin(X_b),ord_max(X_b)),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_1),A_3)) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),ord_max(X_b),X_1),hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),big_Sup_fin(X_b),ord_max(X_b)),A_3)) ) ) ) ) ).
fof(fact_598_min__max_OSup__fin_Oinsert__idem,axiom,
! [X_b] :
( linorder(X_b)
=> ! [X_1,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( ti(fun(X_b,bool),A_3) != bot_bot(fun(X_b,bool))
=> hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),big_Sup_fin(X_b),ord_max(X_b)),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_1),A_3)) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),ord_max(X_b),X_1),hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),big_Sup_fin(X_b),ord_max(X_b)),A_3)) ) ) ) ).
fof(fact_599_min__max_OSup__fin_Osubset__idem,axiom,
! [X_b] :
( linorder(X_b)
=> ! [B_2,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( ti(fun(X_b,bool),B_2) != bot_bot(fun(X_b,bool))
=> ( 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_2),A_3))
=> hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),ord_max(X_b),hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),big_Sup_fin(X_b),ord_max(X_b)),B_2)),hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),big_Sup_fin(X_b),ord_max(X_b)),A_3)) = hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),big_Sup_fin(X_b),ord_max(X_b)),A_3) ) ) ) ) ).
fof(fact_600_add__Max__commute,axiom,
! [X_b] :
( linord2061991079up_add(X_b)
=> ! [K_1,N_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),N_3))
=> ( ti(fun(X_b,bool),N_3) != bot_bot(fun(X_b,bool))
=> hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),K_1),hAPP(fun(X_b,bool),X_b,big_linorder_Max(X_b),N_3)) = hAPP(fun(X_b,bool),X_b,big_linorder_Max(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),collect(X_b),hAPP(fun(X_b,fun(X_b,bool)),fun(X_b,bool),hAPP(fun(fun(X_b,bool),bool),fun(fun(X_b,fun(X_b,bool)),fun(X_b,bool)),combb(fun(X_b,bool),bool,X_b),fEx(X_b)),hAPP(fun(X_b,bool),fun(X_b,fun(X_b,bool)),hAPP(fun(X_b,fun(fun(X_b,bool),fun(X_b,bool))),fun(fun(X_b,bool),fun(X_b,fun(X_b,bool))),combc(X_b,fun(X_b,bool),fun(X_b,bool)),hAPP(fun(X_b,fun(X_b,fun(bool,bool))),fun(X_b,fun(fun(X_b,bool),fun(X_b,bool))),hAPP(fun(fun(X_b,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,bool))),fun(fun(X_b,fun(X_b,fun(bool,bool))),fun(X_b,fun(fun(X_b,bool),fun(X_b,bool)))),combb(fun(X_b,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,bool)),X_b),combs(X_b,bool,bool)),hAPP(fun(X_b,fun(X_b,bool)),fun(X_b,fun(X_b,fun(bool,bool))),hAPP(fun(fun(X_b,bool),fun(X_b,fun(bool,bool))),fun(fun(X_b,fun(X_b,bool)),fun(X_b,fun(X_b,fun(bool,bool)))),combb(fun(X_b,bool),fun(X_b,fun(bool,bool)),X_b),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,fun(bool,bool))),combb(bool,fun(bool,bool),X_b),fconj)),hAPP(fun(X_b,X_b),fun(X_b,fun(X_b,bool)),hAPP(fun(X_b,fun(fun(X_b,X_b),fun(X_b,bool))),fun(fun(X_b,X_b),fun(X_b,fun(X_b,bool))),combc(X_b,fun(X_b,X_b),fun(X_b,bool)),hAPP(fun(X_b,fun(X_b,bool)),fun(X_b,fun(fun(X_b,X_b),fun(X_b,bool))),hAPP(fun(fun(X_b,bool),fun(fun(X_b,X_b),fun(X_b,bool))),fun(fun(X_b,fun(X_b,bool)),fun(X_b,fun(fun(X_b,X_b),fun(X_b,bool)))),combb(fun(X_b,bool),fun(fun(X_b,X_b),fun(X_b,bool)),X_b),combb(X_b,bool,X_b)),fequal(X_b))),hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),K_1))))),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(fun(X_b,bool),bool)),fun(fun(X_b,bool),fun(X_b,bool)),combc(X_b,fun(X_b,bool),bool),member(X_b)),N_3))))) ) ) ) ).
fof(fact_601_add__Min__commute,axiom,
! [X_b] :
( linord2061991079up_add(X_b)
=> ! [K_1,N_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),N_3))
=> ( ti(fun(X_b,bool),N_3) != bot_bot(fun(X_b,bool))
=> hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),K_1),hAPP(fun(X_b,bool),X_b,big_linorder_Min(X_b),N_3)) = hAPP(fun(X_b,bool),X_b,big_linorder_Min(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),collect(X_b),hAPP(fun(X_b,fun(X_b,bool)),fun(X_b,bool),hAPP(fun(fun(X_b,bool),bool),fun(fun(X_b,fun(X_b,bool)),fun(X_b,bool)),combb(fun(X_b,bool),bool,X_b),fEx(X_b)),hAPP(fun(X_b,bool),fun(X_b,fun(X_b,bool)),hAPP(fun(X_b,fun(fun(X_b,bool),fun(X_b,bool))),fun(fun(X_b,bool),fun(X_b,fun(X_b,bool))),combc(X_b,fun(X_b,bool),fun(X_b,bool)),hAPP(fun(X_b,fun(X_b,fun(bool,bool))),fun(X_b,fun(fun(X_b,bool),fun(X_b,bool))),hAPP(fun(fun(X_b,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,bool))),fun(fun(X_b,fun(X_b,fun(bool,bool))),fun(X_b,fun(fun(X_b,bool),fun(X_b,bool)))),combb(fun(X_b,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,bool)),X_b),combs(X_b,bool,bool)),hAPP(fun(X_b,fun(X_b,bool)),fun(X_b,fun(X_b,fun(bool,bool))),hAPP(fun(fun(X_b,bool),fun(X_b,fun(bool,bool))),fun(fun(X_b,fun(X_b,bool)),fun(X_b,fun(X_b,fun(bool,bool)))),combb(fun(X_b,bool),fun(X_b,fun(bool,bool)),X_b),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,fun(bool,bool))),combb(bool,fun(bool,bool),X_b),fconj)),hAPP(fun(X_b,X_b),fun(X_b,fun(X_b,bool)),hAPP(fun(X_b,fun(fun(X_b,X_b),fun(X_b,bool))),fun(fun(X_b,X_b),fun(X_b,fun(X_b,bool))),combc(X_b,fun(X_b,X_b),fun(X_b,bool)),hAPP(fun(X_b,fun(X_b,bool)),fun(X_b,fun(fun(X_b,X_b),fun(X_b,bool))),hAPP(fun(fun(X_b,bool),fun(fun(X_b,X_b),fun(X_b,bool))),fun(fun(X_b,fun(X_b,bool)),fun(X_b,fun(fun(X_b,X_b),fun(X_b,bool)))),combb(fun(X_b,bool),fun(fun(X_b,X_b),fun(X_b,bool)),X_b),combb(X_b,bool,X_b)),fequal(X_b))),hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),K_1))))),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(fun(X_b,bool),bool)),fun(fun(X_b,bool),fun(X_b,bool)),combc(X_b,fun(X_b,bool),bool),member(X_b)),N_3))))) ) ) ) ).
fof(fact_602_min__max_OSup__fin_Oeq__fold__idem_H,axiom,
! [X_b] :
( linorder(X_b)
=> ! [A_2,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),big_Sup_fin(X_b),ord_max(X_b)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_2),A_3)) = hAPP(fun(X_b,bool),X_b,hAPP(X_b,fun(fun(X_b,bool),X_b),hAPP(fun(X_b,fun(X_b,X_b)),fun(X_b,fun(fun(X_b,bool),X_b)),finite_fold(X_b,X_b),ord_max(X_b)),A_2),A_3) ) ) ).
fof(fact_603_min__max_OSup__fin_Oeq__fold_H,axiom,
! [X_b] :
( linorder(X_b)
=> ! [X_1,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( ~ hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),A_3))
=> hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),big_Sup_fin(X_b),ord_max(X_b)),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_1),A_3)) = hAPP(fun(X_b,bool),X_b,hAPP(X_b,fun(fun(X_b,bool),X_b),hAPP(fun(X_b,fun(X_b,X_b)),fun(X_b,fun(fun(X_b,bool),X_b)),finite_fold(X_b,X_b),ord_max(X_b)),X_1),A_3) ) ) ) ).
fof(fact_604_min__max_OSup__fin_Oremove,axiom,
! [X_b] :
( linorder(X_b)
=> ! [X_1,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),A_3))
=> ( ( 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_3),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_1),bot_bot(fun(X_b,bool)))) = bot_bot(fun(X_b,bool))
=> hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),big_Sup_fin(X_b),ord_max(X_b)),A_3) = ti(X_b,X_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_3),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_1),bot_bot(fun(X_b,bool)))) != bot_bot(fun(X_b,bool))
=> hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),big_Sup_fin(X_b),ord_max(X_b)),A_3) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),ord_max(X_b),X_1),hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),big_Sup_fin(X_b),ord_max(X_b)),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_3),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_1),bot_bot(fun(X_b,bool)))))) ) ) ) ) ) ).
fof(fact_605_min__max_OSup__fin_Oinsert__remove,axiom,
! [X_b] :
( linorder(X_b)
=> ! [X_1,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( ( 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_3),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_1),bot_bot(fun(X_b,bool)))) = bot_bot(fun(X_b,bool))
=> hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),big_Sup_fin(X_b),ord_max(X_b)),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_1),A_3)) = ti(X_b,X_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_3),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_1),bot_bot(fun(X_b,bool)))) != bot_bot(fun(X_b,bool))
=> hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),big_Sup_fin(X_b),ord_max(X_b)),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_1),A_3)) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),ord_max(X_b),X_1),hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),big_Sup_fin(X_b),ord_max(X_b)),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_3),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_1),bot_bot(fun(X_b,bool)))))) ) ) ) ) ).
fof(fact_606_min__max_OSup__fin_Ohom__commute,axiom,
! [X_b] :
( linorder(X_b)
=> ! [N_3,H] :
( ! [X_2,Y_1] : hAPP(X_b,X_b,H,hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),ord_max(X_b),X_2),Y_1)) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),ord_max(X_b),hAPP(X_b,X_b,H,X_2)),hAPP(X_b,X_b,H,Y_1))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),N_3))
=> ( ti(fun(X_b,bool),N_3) != bot_bot(fun(X_b,bool))
=> hAPP(X_b,X_b,H,hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),big_Sup_fin(X_b),ord_max(X_b)),N_3)) = hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),big_Sup_fin(X_b),ord_max(X_b)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,X_b),fun(fun(X_b,bool),fun(X_b,bool)),image(X_b,X_b),H),N_3)) ) ) ) ) ).
fof(fact_607_min__max_OSup__fin_Oclosed,axiom,
! [X_b] :
( linorder(X_b)
=> ! [A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( ti(fun(X_b,bool),A_3) != bot_bot(fun(X_b,bool))
=> ( ! [X_2,Y_1] : hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),ord_max(X_b),X_2),Y_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),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),Y_1),bot_bot(fun(X_b,bool))))))
=> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),big_Sup_fin(X_b),ord_max(X_b)),A_3)),A_3)) ) ) ) ) ).
fof(fact_608_Ex__def,axiom,
! [X_b,Pa] :
( ? [X1] : hBOOL(hAPP(X_b,bool,Pa,X1))
<=> ! [Q_2] :
( ! [X_2] :
( hBOOL(hAPP(X_b,bool,Pa,X_2))
=> hBOOL(Q_2) )
=> hBOOL(Q_2) ) ) ).
fof(fact_609_card__Diff1__le,axiom,
! [X_b,X_1,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(fun(X_b,bool),nat,finite_card(X_b),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_3),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_1),bot_bot(fun(X_b,bool)))))),hAPP(fun(X_b,bool),nat,finite_card(X_b),A_3))) ) ).
fof(fact_610_xt2,axiom,
! [X_c,X_b] :
( ( order(X_b)
& ord(X_c) )
=> ! [Ca,F,Ba,A_2] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(X_c,X_b,F,Ba)),A_2))
=> ( hBOOL(hAPP(X_c,bool,hAPP(X_c,fun(X_c,bool),ord_less_eq(X_c),Ca),Ba))
=> ( ! [X_2,Y_1] :
( hBOOL(hAPP(X_c,bool,hAPP(X_c,fun(X_c,bool),ord_less_eq(X_c),Y_1),X_2))
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(X_c,X_b,F,Y_1)),hAPP(X_c,X_b,F,X_2))) )
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(X_c,X_b,F,Ca)),A_2)) ) ) ) ) ).
fof(fact_611_xt1_I15_J,axiom,
! [X_b] :
( order(X_b)
=> ! [Ca,A_2,F,Ba] :
( ti(X_b,A_2) = hAPP(X_b,X_b,F,Ba)
=> ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),Ca),Ba))
=> ( ! [X_2,Y_1] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),Y_1),X_2))
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(X_b,X_b,F,Y_1)),hAPP(X_b,X_b,F,X_2))) )
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(X_b,X_b,F,Ca)),A_2)) ) ) ) ) ).
fof(fact_612_card__lessThan,axiom,
! [U] : hAPP(fun(nat,bool),nat,finite_card(nat),hAPP(nat,fun(nat,bool),ord_lessThan(nat),U)) = U ).
fof(fact_613_card__insert__le,axiom,
! [X_b,X_1,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(fun(X_b,bool),nat,finite_card(X_b),A_3)),hAPP(fun(X_b,bool),nat,finite_card(X_b),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_1),A_3)))) ) ).
fof(fact_614_card__seteq,axiom,
! [X_b,A_3,B_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),B_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)),A_3),B_2))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(fun(X_b,bool),nat,finite_card(X_b),B_2)),hAPP(fun(X_b,bool),nat,finite_card(X_b),A_3)))
=> ti(fun(X_b,bool),A_3) = ti(fun(X_b,bool),B_2) ) ) ) ).
fof(fact_615_card__mono,axiom,
! [X_b,A_3,B_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),B_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)),A_3),B_2))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(fun(X_b,bool),nat,finite_card(X_b),A_3)),hAPP(fun(X_b,bool),nat,finite_card(X_b),B_2))) ) ) ).
fof(fact_616_card__image__le,axiom,
! [X_c,X_b,F,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(fun(X_c,bool),nat,finite_card(X_c),hAPP(fun(X_b,bool),fun(X_c,bool),hAPP(fun(X_b,X_c),fun(fun(X_b,bool),fun(X_c,bool)),image(X_b,X_c),F),A_3))),hAPP(fun(X_b,bool),nat,finite_card(X_b),A_3))) ) ).
fof(fact_617_card__Diff__subset,axiom,
! [X_b,A_3,B_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),B_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_2),A_3))
=> hAPP(fun(X_b,bool),nat,finite_card(X_b),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_3),B_2)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(fun(X_b,bool),nat,finite_card(X_b),A_3)),hAPP(fun(X_b,bool),nat,finite_card(X_b),B_2)) ) ) ).
fof(fact_618_diff__card__le__card__Diff,axiom,
! [X_b,A_3,B_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),B_2))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(fun(X_b,bool),nat,finite_card(X_b),A_3)),hAPP(fun(X_b,bool),nat,finite_card(X_b),B_2))),hAPP(fun(X_b,bool),nat,finite_card(X_b),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_3),B_2)))) ) ).
fof(fact_619_setsum__multicount,axiom,
! [X_c,X_b,R_1,K_1,T_4,S_1] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),S_1))
=> ( hBOOL(hAPP(fun(X_c,bool),bool,finite_finite_1(X_c),T_4))
=> ( ! [X_2] :
( hBOOL(hAPP(fun(X_c,bool),bool,hAPP(X_c,fun(fun(X_c,bool),bool),member(X_c),X_2),T_4))
=> hAPP(fun(X_b,bool),nat,finite_card(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),collect(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,bool)),combs(X_b,bool,bool),hAPP(fun(X_b,bool),fun(X_b,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,fun(bool,bool))),combb(bool,fun(bool,bool),X_b),fconj),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(fun(X_b,bool),bool)),fun(fun(X_b,bool),fun(X_b,bool)),combc(X_b,fun(X_b,bool),bool),member(X_b)),S_1))),hAPP(X_c,fun(X_b,bool),hAPP(fun(X_b,fun(X_c,bool)),fun(X_c,fun(X_b,bool)),combc(X_b,X_c,bool),R_1),X_2)))) = K_1 )
=> hAPP(fun(X_b,bool),nat,hAPP(fun(X_b,nat),fun(fun(X_b,bool),nat),big_co1399186613setsum(X_b,nat),hAPP(fun(X_b,fun(X_c,bool)),fun(X_b,nat),hAPP(fun(fun(X_c,bool),nat),fun(fun(X_b,fun(X_c,bool)),fun(X_b,nat)),combb(fun(X_c,bool),nat,X_b),finite_card(X_c)),hAPP(fun(X_b,fun(X_c,bool)),fun(X_b,fun(X_c,bool)),hAPP(fun(fun(X_c,bool),fun(X_c,bool)),fun(fun(X_b,fun(X_c,bool)),fun(X_b,fun(X_c,bool))),combb(fun(X_c,bool),fun(X_c,bool),X_b),collect(X_c)),hAPP(fun(X_b,fun(X_c,bool)),fun(X_b,fun(X_c,bool)),hAPP(fun(fun(X_c,bool),fun(X_c,bool)),fun(fun(X_b,fun(X_c,bool)),fun(X_b,fun(X_c,bool))),combb(fun(X_c,bool),fun(X_c,bool),X_b),hAPP(fun(X_c,fun(bool,bool)),fun(fun(X_c,bool),fun(X_c,bool)),combs(X_c,bool,bool),hAPP(fun(X_c,bool),fun(X_c,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_c,bool),fun(X_c,fun(bool,bool))),combb(bool,fun(bool,bool),X_c),fconj),hAPP(fun(X_c,bool),fun(X_c,bool),hAPP(fun(X_c,fun(fun(X_c,bool),bool)),fun(fun(X_c,bool),fun(X_c,bool)),combc(X_c,fun(X_c,bool),bool),member(X_c)),T_4)))),R_1)))),S_1) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K_1),hAPP(fun(X_c,bool),nat,finite_card(X_c),T_4)) ) ) ) ).
fof(fact_620_setsum__multicount__gen,axiom,
! [X_c,X_b,R_1,K_1,T_5,S_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),S_2))
=> ( hBOOL(hAPP(fun(X_c,bool),bool,finite_finite_1(X_c),T_5))
=> ( ! [X_2] :
( hBOOL(hAPP(fun(X_c,bool),bool,hAPP(X_c,fun(fun(X_c,bool),bool),member(X_c),X_2),T_5))
=> hAPP(fun(X_b,bool),nat,finite_card(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),collect(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,bool)),combs(X_b,bool,bool),hAPP(fun(X_b,bool),fun(X_b,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,fun(bool,bool))),combb(bool,fun(bool,bool),X_b),fconj),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(fun(X_b,bool),bool)),fun(fun(X_b,bool),fun(X_b,bool)),combc(X_b,fun(X_b,bool),bool),member(X_b)),S_2))),hAPP(X_c,fun(X_b,bool),hAPP(fun(X_b,fun(X_c,bool)),fun(X_c,fun(X_b,bool)),combc(X_b,X_c,bool),R_1),X_2)))) = hAPP(X_c,nat,K_1,X_2) )
=> hAPP(fun(X_b,bool),nat,hAPP(fun(X_b,nat),fun(fun(X_b,bool),nat),big_co1399186613setsum(X_b,nat),hAPP(fun(X_b,fun(X_c,bool)),fun(X_b,nat),hAPP(fun(fun(X_c,bool),nat),fun(fun(X_b,fun(X_c,bool)),fun(X_b,nat)),combb(fun(X_c,bool),nat,X_b),finite_card(X_c)),hAPP(fun(X_b,fun(X_c,bool)),fun(X_b,fun(X_c,bool)),hAPP(fun(fun(X_c,bool),fun(X_c,bool)),fun(fun(X_b,fun(X_c,bool)),fun(X_b,fun(X_c,bool))),combb(fun(X_c,bool),fun(X_c,bool),X_b),collect(X_c)),hAPP(fun(X_b,fun(X_c,bool)),fun(X_b,fun(X_c,bool)),hAPP(fun(fun(X_c,bool),fun(X_c,bool)),fun(fun(X_b,fun(X_c,bool)),fun(X_b,fun(X_c,bool))),combb(fun(X_c,bool),fun(X_c,bool),X_b),hAPP(fun(X_c,fun(bool,bool)),fun(fun(X_c,bool),fun(X_c,bool)),combs(X_c,bool,bool),hAPP(fun(X_c,bool),fun(X_c,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_c,bool),fun(X_c,fun(bool,bool))),combb(bool,fun(bool,bool),X_c),fconj),hAPP(fun(X_c,bool),fun(X_c,bool),hAPP(fun(X_c,fun(fun(X_c,bool),bool)),fun(fun(X_c,bool),fun(X_c,bool)),combc(X_c,fun(X_c,bool),bool),member(X_c)),T_5)))),R_1)))),S_2) = hAPP(fun(X_c,bool),nat,hAPP(fun(X_c,nat),fun(fun(X_c,bool),nat),big_co1399186613setsum(X_c,nat),K_1),T_5) ) ) ) ).
fof(fact_621_card__Diff__singleton,axiom,
! [X_b,X_1,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),A_3))
=> hAPP(fun(X_b,bool),nat,finite_card(X_b),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_3),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_1),bot_bot(fun(X_b,bool))))) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(fun(X_b,bool),nat,finite_card(X_b),A_3)),one_one(nat)) ) ) ).
fof(fact_622_setprod__1,axiom,
! [X_c,X_b] :
( comm_monoid_mult(X_b)
=> ! [A_3] : hAPP(fun(X_c,bool),X_b,hAPP(fun(X_c,X_b),fun(fun(X_c,bool),X_b),big_co302546470etprod(X_c,X_b),hAPP(X_b,fun(X_c,X_b),combk(X_b,X_c),one_one(X_b))),A_3) = one_one(X_b) ) ).
fof(fact_623_nat__mult__eq__1__iff,axiom,
! [M_1,N] :
( hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M_1),N) = one_one(nat)
<=> ( M_1 = one_one(nat)
& N = one_one(nat) ) ) ).
fof(fact_624_nat__mult__1__right,axiom,
! [N_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),N_1),one_one(nat)) = N_1 ).
fof(fact_625_nat__1__eq__mult__iff,axiom,
! [M_1,N] :
( one_one(nat) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M_1),N)
<=> ( M_1 = one_one(nat)
& N = one_one(nat) ) ) ).
fof(fact_626_nat__mult__1,axiom,
! [N_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),one_one(nat)),N_1) = N_1 ).
fof(fact_627_mult_Ocomm__neutral,axiom,
! [X_a] :
( comm_monoid_mult(X_a)
=> ! [A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),one_one(X_a)) = ti(X_a,A_1) ) ).
fof(fact_628_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),one_one(X_a)) = ti(X_a,A_1) ) ).
fof(fact_629_mult__1__right,axiom,
! [X_a] :
( monoid_mult(X_a)
=> ! [A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),one_one(X_a)) = ti(X_a,A_1) ) ).
fof(fact_630_mult__1,axiom,
! [X_a] :
( comm_monoid_mult(X_a)
=> ! [A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),one_one(X_a)),A_1) = ti(X_a,A_1) ) ).
fof(fact_631_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),one_one(X_a)),A_1) = ti(X_a,A_1) ) ).
fof(fact_632_mult__1__left,axiom,
! [X_a] :
( monoid_mult(X_a)
=> ! [A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),one_one(X_a)),A_1) = ti(X_a,A_1) ) ).
fof(fact_633_setprod__eq__1__iff,axiom,
! [X_b,F,F_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),F_2))
=> ( hAPP(fun(X_b,bool),nat,hAPP(fun(X_b,nat),fun(fun(X_b,bool),nat),big_co302546470etprod(X_b,nat),F),F_2) = one_one(nat)
<=> ! [X_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_2),F_2))
=> hAPP(X_b,nat,F,X_2) = one_one(nat) ) ) ) ).
fof(fact_634_one__reorient,axiom,
! [X_b] :
( one(X_b)
=> ! [X_1] :
( one_one(X_b) = ti(X_b,X_1)
<=> ti(X_b,X_1) = one_one(X_b) ) ) ).
fof(fact_635_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_1,M] : 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),M)),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_1),one_one(X_a))),M) ) ).
fof(fact_636_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [M,A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),M),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),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_1),one_one(X_a))),M) ) ).
fof(fact_637_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [M] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),M),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),one_one(X_a)),one_one(X_a))),M) ) ).
fof(fact_638_square__eq__1__iff,axiom,
! [X_b] :
( ring_11004092258visors(X_b)
=> ! [X_1] :
( hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),X_1),X_1) = one_one(X_b)
<=> ( ti(X_b,X_1) = one_one(X_b)
| ti(X_b,X_1) = hAPP(X_b,X_b,uminus_uminus(X_b),one_one(X_b)) ) ) ) ).
fof(fact_639_comm__ring__1__class_Onormalizing__ring__rules_I1_J,axiom,
! [X_a] :
( comm_ring_1(X_a)
=> ! [X] : hAPP(X_a,X_a,uminus_uminus(X_a),X) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),one_one(X_a))),X) ) ).
fof(fact_640_setprod_Oempty,axiom,
! [X_c,X_b] :
( comm_monoid_mult(X_b)
=> ! [G] : hAPP(fun(X_c,bool),X_b,hAPP(fun(X_c,X_b),fun(fun(X_c,bool),X_b),big_co302546470etprod(X_c,X_b),G),bot_bot(fun(X_c,bool))) = one_one(X_b) ) ).
fof(fact_641_setprod__empty,axiom,
! [X_c,X_b] :
( comm_monoid_mult(X_b)
=> ! [F] : hAPP(fun(X_c,bool),X_b,hAPP(fun(X_c,X_b),fun(fun(X_c,bool),X_b),big_co302546470etprod(X_c,X_b),F),bot_bot(fun(X_c,bool))) = one_one(X_b) ) ).
fof(fact_642_setprod__infinite,axiom,
! [X_b,X_c] :
( comm_monoid_mult(X_c)
=> ! [F,A_3] :
( ~ hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> hAPP(fun(X_b,bool),X_c,hAPP(fun(X_b,X_c),fun(fun(X_b,bool),X_c),big_co302546470etprod(X_b,X_c),F),A_3) = one_one(X_c) ) ) ).
fof(fact_643_setprod_Oinfinite,axiom,
! [X_b,X_c] :
( comm_monoid_mult(X_c)
=> ! [G,A_3] :
( ~ hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> hAPP(fun(X_b,bool),X_c,hAPP(fun(X_b,X_c),fun(fun(X_b,bool),X_c),big_co302546470etprod(X_b,X_c),G),A_3) = one_one(X_c) ) ) ).
fof(fact_644_setprod__delta_H,axiom,
! [X_b,X_c] :
( comm_monoid_mult(X_c)
=> ! [Ba,A_2,S_1] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),S_1))
=> ( ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),A_2),S_1))
=> hAPP(fun(X_b,bool),X_c,hAPP(fun(X_b,X_c),fun(fun(X_b,bool),X_c),big_co302546470etprod(X_b,X_c),hAPP(X_c,fun(X_b,X_c),hAPP(fun(X_b,fun(X_c,X_c)),fun(X_c,fun(X_b,X_c)),combc(X_b,X_c,X_c),hAPP(fun(X_b,X_c),fun(X_b,fun(X_c,X_c)),hAPP(fun(X_b,fun(X_c,fun(X_c,X_c))),fun(fun(X_b,X_c),fun(X_b,fun(X_c,X_c))),combs(X_b,X_c,fun(X_c,X_c)),hAPP(fun(X_b,bool),fun(X_b,fun(X_c,fun(X_c,X_c))),hAPP(fun(bool,fun(X_c,fun(X_c,X_c))),fun(fun(X_b,bool),fun(X_b,fun(X_c,fun(X_c,X_c)))),combb(bool,fun(X_c,fun(X_c,X_c)),X_b),if(X_c)),hAPP(X_b,fun(X_b,bool),fequal(X_b),A_2))),Ba)),one_one(X_c))),S_1) = hAPP(X_b,X_c,Ba,A_2) )
& ( ~ hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),A_2),S_1))
=> hAPP(fun(X_b,bool),X_c,hAPP(fun(X_b,X_c),fun(fun(X_b,bool),X_c),big_co302546470etprod(X_b,X_c),hAPP(X_c,fun(X_b,X_c),hAPP(fun(X_b,fun(X_c,X_c)),fun(X_c,fun(X_b,X_c)),combc(X_b,X_c,X_c),hAPP(fun(X_b,X_c),fun(X_b,fun(X_c,X_c)),hAPP(fun(X_b,fun(X_c,fun(X_c,X_c))),fun(fun(X_b,X_c),fun(X_b,fun(X_c,X_c))),combs(X_b,X_c,fun(X_c,X_c)),hAPP(fun(X_b,bool),fun(X_b,fun(X_c,fun(X_c,X_c))),hAPP(fun(bool,fun(X_c,fun(X_c,X_c))),fun(fun(X_b,bool),fun(X_b,fun(X_c,fun(X_c,X_c)))),combb(bool,fun(X_c,fun(X_c,X_c)),X_b),if(X_c)),hAPP(X_b,fun(X_b,bool),fequal(X_b),A_2))),Ba)),one_one(X_c))),S_1) = one_one(X_c) ) ) ) ) ).
fof(fact_645_setprod__delta,axiom,
! [X_b,X_c] :
( comm_monoid_mult(X_c)
=> ! [Ba,A_2,S_1] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),S_1))
=> ( ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),A_2),S_1))
=> hAPP(fun(X_b,bool),X_c,hAPP(fun(X_b,X_c),fun(fun(X_b,bool),X_c),big_co302546470etprod(X_b,X_c),hAPP(X_c,fun(X_b,X_c),hAPP(fun(X_b,fun(X_c,X_c)),fun(X_c,fun(X_b,X_c)),combc(X_b,X_c,X_c),hAPP(fun(X_b,X_c),fun(X_b,fun(X_c,X_c)),hAPP(fun(X_b,fun(X_c,fun(X_c,X_c))),fun(fun(X_b,X_c),fun(X_b,fun(X_c,X_c))),combs(X_b,X_c,fun(X_c,X_c)),hAPP(fun(X_b,bool),fun(X_b,fun(X_c,fun(X_c,X_c))),hAPP(fun(bool,fun(X_c,fun(X_c,X_c))),fun(fun(X_b,bool),fun(X_b,fun(X_c,fun(X_c,X_c)))),combb(bool,fun(X_c,fun(X_c,X_c)),X_b),if(X_c)),hAPP(X_b,fun(X_b,bool),hAPP(fun(X_b,fun(X_b,bool)),fun(X_b,fun(X_b,bool)),combc(X_b,X_b,bool),fequal(X_b)),A_2))),Ba)),one_one(X_c))),S_1) = hAPP(X_b,X_c,Ba,A_2) )
& ( ~ hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),A_2),S_1))
=> hAPP(fun(X_b,bool),X_c,hAPP(fun(X_b,X_c),fun(fun(X_b,bool),X_c),big_co302546470etprod(X_b,X_c),hAPP(X_c,fun(X_b,X_c),hAPP(fun(X_b,fun(X_c,X_c)),fun(X_c,fun(X_b,X_c)),combc(X_b,X_c,X_c),hAPP(fun(X_b,X_c),fun(X_b,fun(X_c,X_c)),hAPP(fun(X_b,fun(X_c,fun(X_c,X_c))),fun(fun(X_b,X_c),fun(X_b,fun(X_c,X_c))),combs(X_b,X_c,fun(X_c,X_c)),hAPP(fun(X_b,bool),fun(X_b,fun(X_c,fun(X_c,X_c))),hAPP(fun(bool,fun(X_c,fun(X_c,X_c))),fun(fun(X_b,bool),fun(X_b,fun(X_c,fun(X_c,X_c)))),combb(bool,fun(X_c,fun(X_c,X_c)),X_b),if(X_c)),hAPP(X_b,fun(X_b,bool),hAPP(fun(X_b,fun(X_b,bool)),fun(X_b,fun(X_b,bool)),combc(X_b,X_b,bool),fequal(X_b)),A_2))),Ba)),one_one(X_c))),S_1) = one_one(X_c) ) ) ) ) ).
fof(fact_646_card__eq__setsum,axiom,
! [X_b,A_3] : hAPP(fun(X_b,bool),nat,finite_card(X_b),A_3) = hAPP(fun(X_b,bool),nat,hAPP(fun(X_b,nat),fun(fun(X_b,bool),nat),big_co1399186613setsum(X_b,nat),hAPP(nat,fun(X_b,nat),combk(nat,X_b),one_one(nat))),A_3) ).
fof(fact_647_card_Oinsert,axiom,
! [X_b,X_1,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( ~ hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),A_3))
=> hAPP(fun(X_b,bool),nat,finite_card(X_b),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_1),A_3)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),hAPP(fun(X_b,bool),nat,finite_card(X_b),A_3)) ) ) ).
fof(fact_648_card__Diff__insert,axiom,
! [X_b,B_2,A_2,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),A_2),A_3))
=> ( ~ hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),A_2),B_2))
=> hAPP(fun(X_b,bool),nat,finite_card(X_b),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_3),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_2),B_2))) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(fun(X_b,bool),nat,finite_card(X_b),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_3),B_2))),one_one(nat)) ) ) ) ).
fof(fact_649_card_Oinsert__remove,axiom,
! [X_b,X_1,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> hAPP(fun(X_b,bool),nat,finite_card(X_b),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_1),A_3)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),hAPP(fun(X_b,bool),nat,finite_card(X_b),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_3),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_1),bot_bot(fun(X_b,bool)))))) ) ).
fof(fact_650_card_Oremove,axiom,
! [X_b,X_1,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),A_3))
=> hAPP(fun(X_b,bool),nat,finite_card(X_b),A_3) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),hAPP(fun(X_b,bool),nat,finite_card(X_b),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_3),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_1),bot_bot(fun(X_b,bool)))))) ) ) ).
fof(fact_651_card__Diff__singleton__if,axiom,
! [X_b,X_1,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),A_3))
=> hAPP(fun(X_b,bool),nat,finite_card(X_b),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_3),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_1),bot_bot(fun(X_b,bool))))) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(fun(X_b,bool),nat,finite_card(X_b),A_3)),one_one(nat)) )
& ( ~ hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),A_3))
=> hAPP(fun(X_b,bool),nat,finite_card(X_b),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_3),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_1),bot_bot(fun(X_b,bool))))) = hAPP(fun(X_b,bool),nat,finite_card(X_b),A_3) ) ) ) ).
fof(fact_652_setprod__mono__one__right,axiom,
! [X_b,X_c] :
( comm_monoid_mult(X_c)
=> ! [F,S_1,T_4] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),T_4))
=> ( 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)),S_1),T_4))
=> ( ! [X_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_2),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)),T_4),S_1)))
=> hAPP(X_b,X_c,F,X_2) = one_one(X_c) )
=> hAPP(fun(X_b,bool),X_c,hAPP(fun(X_b,X_c),fun(fun(X_b,bool),X_c),big_co302546470etprod(X_b,X_c),F),T_4) = hAPP(fun(X_b,bool),X_c,hAPP(fun(X_b,X_c),fun(fun(X_b,bool),X_c),big_co302546470etprod(X_b,X_c),F),S_1) ) ) ) ) ).
fof(fact_653_setprod__mono__one__left,axiom,
! [X_b,X_c] :
( comm_monoid_mult(X_c)
=> ! [F,S_1,T_4] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),T_4))
=> ( 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)),S_1),T_4))
=> ( ! [X_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_2),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)),T_4),S_1)))
=> hAPP(X_b,X_c,F,X_2) = one_one(X_c) )
=> hAPP(fun(X_b,bool),X_c,hAPP(fun(X_b,X_c),fun(fun(X_b,bool),X_c),big_co302546470etprod(X_b,X_c),F),S_1) = hAPP(fun(X_b,bool),X_c,hAPP(fun(X_b,X_c),fun(fun(X_b,bool),X_c),big_co302546470etprod(X_b,X_c),F),T_4) ) ) ) ) ).
fof(fact_654_setprod_Oneutral,axiom,
! [X_b,X_c] :
( comm_monoid_mult(X_c)
=> ! [G,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( ! [X_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_2),A_3))
=> hAPP(X_b,X_c,G,X_2) = one_one(X_c) )
=> hAPP(fun(X_b,bool),X_c,hAPP(fun(X_b,X_c),fun(fun(X_b,bool),X_c),big_co302546470etprod(X_b,X_c),G),A_3) = one_one(X_c) ) ) ) ).
fof(fact_655_setprod__1_H,axiom,
! [X_b,X_c] :
( comm_monoid_mult(X_c)
=> ! [F,F_2] :
( ! [X_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_2),F_2))
=> hAPP(X_b,X_c,F,X_2) = one_one(X_c) )
=> hAPP(fun(X_b,bool),X_c,hAPP(fun(X_b,X_c),fun(fun(X_b,bool),X_c),big_co302546470etprod(X_b,X_c),F),F_2) = one_one(X_c) ) ) ).
fof(fact_656_arith__series__general,axiom,
! [X_b] :
( comm_semiring_1(X_b)
=> ! [A_2,D,N] : hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),one_one(X_b)),one_one(X_b))),hAPP(fun(nat,bool),X_b,hAPP(fun(nat,X_b),fun(fun(nat,bool),X_b),big_co1399186613setsum(nat,X_b),hAPP(fun(nat,X_b),fun(nat,X_b),hAPP(fun(X_b,X_b),fun(fun(nat,X_b),fun(nat,X_b)),combb(X_b,X_b,nat),hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),A_2)),hAPP(X_b,fun(nat,X_b),hAPP(fun(nat,fun(X_b,X_b)),fun(X_b,fun(nat,X_b)),combc(nat,X_b,X_b),hAPP(fun(nat,X_b),fun(nat,fun(X_b,X_b)),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(nat,X_b),fun(nat,fun(X_b,X_b))),combb(X_b,fun(X_b,X_b),nat),times_times(X_b)),semiring_1_of_nat(X_b))),D))),hAPP(nat,fun(nat,bool),ord_lessThan(nat),N))) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),hAPP(nat,X_b,semiring_1_of_nat(X_b),N)),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),A_2),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),A_2),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),hAPP(nat,X_b,semiring_1_of_nat(X_b),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)))),D)))) ) ).
fof(fact_657_setprod_Orelated,axiom,
! [X_c,X_b] :
( comm_monoid_mult(X_b)
=> ! [H,G,S_1,R_1] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),R_1,one_one(X_b)),one_one(X_b)))
=> ( ! [X1,Y1,X2,Y2] :
( ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),R_1,X1),X2))
& hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),R_1,Y1),Y2)) )
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),R_1,hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),X1),Y1)),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),X2),Y2))) )
=> ( hBOOL(hAPP(fun(X_c,bool),bool,finite_finite_1(X_c),S_1))
=> ( ! [X_2] :
( hBOOL(hAPP(fun(X_c,bool),bool,hAPP(X_c,fun(fun(X_c,bool),bool),member(X_c),X_2),S_1))
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),R_1,hAPP(X_c,X_b,H,X_2)),hAPP(X_c,X_b,G,X_2))) )
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),R_1,hAPP(fun(X_c,bool),X_b,hAPP(fun(X_c,X_b),fun(fun(X_c,bool),X_b),big_co302546470etprod(X_c,X_b),H),S_1)),hAPP(fun(X_c,bool),X_b,hAPP(fun(X_c,X_b),fun(fun(X_c,bool),X_b),big_co302546470etprod(X_c,X_b),G),S_1))) ) ) ) ) ) ).
fof(fact_658_of__nat__eq__iff,axiom,
! [X_b] :
( semiring_char_0(X_b)
=> ! [M_1,N] :
( hAPP(nat,X_b,semiring_1_of_nat(X_b),M_1) = hAPP(nat,X_b,semiring_1_of_nat(X_b),N)
<=> M_1 = N ) ) ).
fof(fact_659_of__nat__le__iff,axiom,
! [X_b] :
( linordered_semidom(X_b)
=> ! [M_1,N] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(nat,X_b,semiring_1_of_nat(X_b),M_1)),hAPP(nat,X_b,semiring_1_of_nat(X_b),N)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_1),N)) ) ) ).
fof(fact_660_of__nat__add,axiom,
! [X_a] :
( semiring_1(X_a)
=> ! [M,N_1] : hAPP(nat,X_a,semiring_1_of_nat(X_a),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),N_1)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),M)),hAPP(nat,X_a,semiring_1_of_nat(X_a),N_1)) ) ).
fof(fact_661_of__nat__mult,axiom,
! [X_a] :
( semiring_1(X_a)
=> ! [M,N_1] : hAPP(nat,X_a,semiring_1_of_nat(X_a),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),N_1)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),M)),hAPP(nat,X_a,semiring_1_of_nat(X_a),N_1)) ) ).
fof(fact_662_of__nat__1,axiom,
! [X_a] :
( semiring_1(X_a)
=> hAPP(nat,X_a,semiring_1_of_nat(X_a),one_one(nat)) = one_one(X_a) ) ).
fof(fact_663_of__nat__diff,axiom,
! [X_a] :
( ring_1(X_a)
=> ! [N_1,M] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_1),M))
=> hAPP(nat,X_a,semiring_1_of_nat(X_a),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),N_1)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),M)),hAPP(nat,X_a,semiring_1_of_nat(X_a),N_1)) ) ) ).
fof(fact_664_setsum__constant,axiom,
! [X_c,X_b] :
( semiring_1(X_b)
=> ! [Y_2,A_3] : hAPP(fun(X_c,bool),X_b,hAPP(fun(X_c,X_b),fun(fun(X_c,bool),X_b),big_co1399186613setsum(X_c,X_b),hAPP(X_b,fun(X_c,X_b),combk(X_b,X_c),Y_2)),A_3) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),hAPP(nat,X_b,semiring_1_of_nat(X_b),hAPP(fun(X_c,bool),nat,finite_card(X_c),A_3))),Y_2) ) ).
fof(fact_665_setsum__bounded,axiom,
! [X_b,X_c] :
( ( ordere779506340up_add(X_c)
& semiring_1(X_c) )
=> ! [F,K_2,A_3] :
( ! [I_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),I_2),A_3))
=> hBOOL(hAPP(X_c,bool,hAPP(X_c,fun(X_c,bool),ord_less_eq(X_c),hAPP(X_b,X_c,F,I_2)),K_2)) )
=> hBOOL(hAPP(X_c,bool,hAPP(X_c,fun(X_c,bool),ord_less_eq(X_c),hAPP(fun(X_b,bool),X_c,hAPP(fun(X_b,X_c),fun(fun(X_b,bool),X_c),big_co1399186613setsum(X_b,X_c),F),A_3)),hAPP(X_c,X_c,hAPP(X_c,fun(X_c,X_c),times_times(X_c),hAPP(nat,X_c,semiring_1_of_nat(X_c),hAPP(fun(X_b,bool),nat,finite_card(X_b),A_3))),K_2))) ) ) ).
fof(fact_666_of__nat__setprod,axiom,
! [X_c,X_b] :
( ( comm_monoid_mult(X_b)
& semiring_1(X_b) )
=> ! [F,A_3] : hAPP(nat,X_b,semiring_1_of_nat(X_b),hAPP(fun(X_c,bool),nat,hAPP(fun(X_c,nat),fun(fun(X_c,bool),nat),big_co302546470etprod(X_c,nat),F),A_3)) = hAPP(fun(X_c,bool),X_b,hAPP(fun(X_c,X_b),fun(fun(X_c,bool),X_b),big_co302546470etprod(X_c,X_b),hAPP(fun(X_c,nat),fun(X_c,X_b),hAPP(fun(nat,X_b),fun(fun(X_c,nat),fun(X_c,X_b)),combb(nat,X_b,X_c),semiring_1_of_nat(X_b)),F)),A_3) ) ).
fof(fact_667_of__nat__setsum,axiom,
! [X_c,X_b] :
( semiring_1(X_b)
=> ! [F,A_3] : hAPP(nat,X_b,semiring_1_of_nat(X_b),hAPP(fun(X_c,bool),nat,hAPP(fun(X_c,nat),fun(fun(X_c,bool),nat),big_co1399186613setsum(X_c,nat),F),A_3)) = hAPP(fun(X_c,bool),X_b,hAPP(fun(X_c,X_b),fun(fun(X_c,bool),X_b),big_co1399186613setsum(X_c,X_b),hAPP(fun(X_c,nat),fun(X_c,X_b),hAPP(fun(nat,X_b),fun(fun(X_c,nat),fun(X_c,X_b)),combb(nat,X_b,X_c),semiring_1_of_nat(X_b)),F)),A_3) ) ).
fof(fact_668_zadd__int__left,axiom,
! [M,N_1,Z] : hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(nat,int,semiring_1_of_nat(int),M)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(nat,int,semiring_1_of_nat(int),N_1)),Z)) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),N_1))),Z) ).
fof(fact_669_zadd__int,axiom,
! [M,N_1] : hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(nat,int,semiring_1_of_nat(int),M)),hAPP(nat,int,semiring_1_of_nat(int),N_1)) = hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),N_1)) ).
fof(fact_670_zle__int,axiom,
! [M_1,N] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(nat,int,semiring_1_of_nat(int),M_1)),hAPP(nat,int,semiring_1_of_nat(int),N)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_1),N)) ) ).
fof(fact_671_zmult__int,axiom,
! [M,N_1] : hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(nat,int,semiring_1_of_nat(int),M)),hAPP(nat,int,semiring_1_of_nat(int),N_1)) = hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),N_1)) ).
fof(fact_672_int__mult,axiom,
! [M,N_1] : hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),N_1)) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(nat,int,semiring_1_of_nat(int),M)),hAPP(nat,int,semiring_1_of_nat(int),N_1)) ).
fof(fact_673_int__1,axiom,
hAPP(nat,int,semiring_1_of_nat(int),one_one(nat)) = one_one(int) ).
fof(fact_674_int__setsum,axiom,
! [X_b,F,A_3] : hAPP(nat,int,semiring_1_of_nat(int),hAPP(fun(X_b,bool),nat,hAPP(fun(X_b,nat),fun(fun(X_b,bool),nat),big_co1399186613setsum(X_b,nat),F),A_3)) = hAPP(fun(X_b,bool),int,hAPP(fun(X_b,int),fun(fun(X_b,bool),int),big_co1399186613setsum(X_b,int),hAPP(fun(X_b,nat),fun(X_b,int),hAPP(fun(nat,int),fun(fun(X_b,nat),fun(X_b,int)),combb(nat,int,X_b),semiring_1_of_nat(int)),F)),A_3) ).
fof(fact_675_int__setprod,axiom,
! [X_b,F,A_3] : hAPP(nat,int,semiring_1_of_nat(int),hAPP(fun(X_b,bool),nat,hAPP(fun(X_b,nat),fun(fun(X_b,bool),nat),big_co302546470etprod(X_b,nat),F),A_3)) = hAPP(fun(X_b,bool),int,hAPP(fun(X_b,int),fun(fun(X_b,bool),int),big_co302546470etprod(X_b,int),hAPP(fun(X_b,nat),fun(X_b,int),hAPP(fun(nat,int),fun(fun(X_b,nat),fun(X_b,int)),combb(nat,int,X_b),semiring_1_of_nat(int)),F)),A_3) ).
fof(fact_676_zdiff__int,axiom,
! [N_1,M] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_1),M))
=> hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(nat,int,semiring_1_of_nat(int),M)),hAPP(nat,int,semiring_1_of_nat(int),N_1)) = hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),N_1)) ) ).
fof(fact_677_gauss__sum,axiom,
! [X_b] :
( comm_semiring_1(X_b)
=> ! [N] : hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),one_one(X_b)),one_one(X_b))),hAPP(fun(nat,bool),X_b,hAPP(fun(nat,X_b),fun(fun(nat,bool),X_b),big_co1399186613setsum(nat,X_b),semiring_1_of_nat(X_b)),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),one_one(nat)),N))) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),hAPP(nat,X_b,semiring_1_of_nat(X_b),N)),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),hAPP(nat,X_b,semiring_1_of_nat(X_b),N)),one_one(X_b))) ) ).
fof(fact_678_setprod__gen__delta,axiom,
! [X_b,X_c] :
( comm_monoid_mult(X_c)
=> ! [Ba,Ca,A_2,S_1] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),S_1))
=> ( ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),A_2),S_1))
=> hAPP(fun(X_b,bool),X_c,hAPP(fun(X_b,X_c),fun(fun(X_b,bool),X_c),big_co302546470etprod(X_b,X_c),hAPP(X_c,fun(X_b,X_c),hAPP(fun(X_b,fun(X_c,X_c)),fun(X_c,fun(X_b,X_c)),combc(X_b,X_c,X_c),hAPP(fun(X_b,X_c),fun(X_b,fun(X_c,X_c)),hAPP(fun(X_b,fun(X_c,fun(X_c,X_c))),fun(fun(X_b,X_c),fun(X_b,fun(X_c,X_c))),combs(X_b,X_c,fun(X_c,X_c)),hAPP(fun(X_b,bool),fun(X_b,fun(X_c,fun(X_c,X_c))),hAPP(fun(bool,fun(X_c,fun(X_c,X_c))),fun(fun(X_b,bool),fun(X_b,fun(X_c,fun(X_c,X_c)))),combb(bool,fun(X_c,fun(X_c,X_c)),X_b),if(X_c)),hAPP(X_b,fun(X_b,bool),hAPP(fun(X_b,fun(X_b,bool)),fun(X_b,fun(X_b,bool)),combc(X_b,X_b,bool),fequal(X_b)),A_2))),Ba)),Ca)),S_1) = hAPP(X_c,X_c,hAPP(X_c,fun(X_c,X_c),times_times(X_c),hAPP(X_b,X_c,Ba,A_2)),hAPP(nat,X_c,hAPP(X_c,fun(nat,X_c),power_power(X_c),Ca),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(fun(X_b,bool),nat,finite_card(X_b),S_1)),one_one(nat)))) )
& ( ~ hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),A_2),S_1))
=> hAPP(fun(X_b,bool),X_c,hAPP(fun(X_b,X_c),fun(fun(X_b,bool),X_c),big_co302546470etprod(X_b,X_c),hAPP(X_c,fun(X_b,X_c),hAPP(fun(X_b,fun(X_c,X_c)),fun(X_c,fun(X_b,X_c)),combc(X_b,X_c,X_c),hAPP(fun(X_b,X_c),fun(X_b,fun(X_c,X_c)),hAPP(fun(X_b,fun(X_c,fun(X_c,X_c))),fun(fun(X_b,X_c),fun(X_b,fun(X_c,X_c))),combs(X_b,X_c,fun(X_c,X_c)),hAPP(fun(X_b,bool),fun(X_b,fun(X_c,fun(X_c,X_c))),hAPP(fun(bool,fun(X_c,fun(X_c,X_c))),fun(fun(X_b,bool),fun(X_b,fun(X_c,fun(X_c,X_c)))),combb(bool,fun(X_c,fun(X_c,X_c)),X_b),if(X_c)),hAPP(X_b,fun(X_b,bool),hAPP(fun(X_b,fun(X_b,bool)),fun(X_b,fun(X_b,bool)),combc(X_b,X_b,bool),fequal(X_b)),A_2))),Ba)),Ca)),S_1) = hAPP(nat,X_c,hAPP(X_c,fun(nat,X_c),power_power(X_c),Ca),hAPP(fun(X_b,bool),nat,finite_card(X_b),S_1)) ) ) ) ) ).
fof(fact_679_card__Diff2__less,axiom,
! [X_b,Y_2,X_1,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),A_3))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),Y_2),A_3))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(fun(X_b,bool),nat,finite_card(X_b),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)),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_3),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_1),bot_bot(fun(X_b,bool))))),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),Y_2),bot_bot(fun(X_b,bool)))))),hAPP(fun(X_b,bool),nat,finite_card(X_b),A_3))) ) ) ) ).
fof(fact_680_negative__zle,axiom,
! [N_1,M] : hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,uminus_uminus(int),hAPP(nat,int,semiring_1_of_nat(int),N_1))),hAPP(nat,int,semiring_1_of_nat(int),M))) ).
fof(fact_681_finite__atLeastAtMost,axiom,
! [L,U] : hBOOL(hAPP(fun(nat,bool),bool,finite_finite_1(nat),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),L),U))) ).
fof(fact_682_finite__Collect__less__nat,axiom,
! [K_1] : hBOOL(hAPP(fun(nat,bool),bool,finite_finite_1(nat),hAPP(fun(nat,bool),fun(nat,bool),collect(nat),hAPP(nat,fun(nat,bool),hAPP(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool)),combc(nat,nat,bool),ord_less(nat)),K_1)))) ).
fof(fact_683_zle__iff__zadd,axiom,
! [W_1,Z_2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),W_1),Z_2))
<=> ? [N_2] : Z_2 = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),W_1),hAPP(nat,int,semiring_1_of_nat(int),N_2)) ) ).
fof(fact_684_int__power,axiom,
! [M,N_1] : hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),M),N_1)) = hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(nat,int,semiring_1_of_nat(int),M)),N_1) ).
fof(fact_685_zpower__int,axiom,
! [M,N_1] : hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(nat,int,semiring_1_of_nat(int),M)),N_1) = hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),M),N_1)) ).
fof(fact_686_int__int__eq,axiom,
! [M_1,N] :
( hAPP(nat,int,semiring_1_of_nat(int),M_1) = hAPP(nat,int,semiring_1_of_nat(int),N)
<=> M_1 = N ) ).
fof(fact_687_zmult__zminus,axiom,
! [Z,W_2] : hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,uminus_uminus(int),Z)),W_2) = hAPP(int,int,uminus_uminus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),Z),W_2)) ).
fof(fact_688_diff__int__def__symmetric,axiom,
! [Z,W_2] : hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Z),hAPP(int,int,uminus_uminus(int),W_2)) = hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),Z),W_2) ).
fof(fact_689_diff__int__def,axiom,
! [Z,W_2] : hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),Z),W_2) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Z),hAPP(int,int,uminus_uminus(int),W_2)) ).
fof(fact_690_zminus__zadd__distrib,axiom,
! [Z,W_2] : hAPP(int,int,uminus_uminus(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Z),W_2)) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,uminus_uminus(int),Z)),hAPP(int,int,uminus_uminus(int),W_2)) ).
fof(fact_691_zle__refl,axiom,
! [W_2] : hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),W_2),W_2)) ).
fof(fact_692_zle__linear,axiom,
! [Z,W_2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Z),W_2))
| hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),W_2),Z)) ) ).
fof(fact_693_zle__trans,axiom,
! [K,I,J] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),I),J))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),J),K))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),I),K)) ) ) ).
fof(fact_694_zle__antisym,axiom,
! [Z,W_2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Z),W_2))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),W_2),Z))
=> Z = W_2 ) ) ).
fof(fact_695_zmult__assoc,axiom,
! [Z1,Z2,Z3] : hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),Z1),Z2)),Z3) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),Z1),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),Z2),Z3)) ).
fof(fact_696_zpower__zpower,axiom,
! [X,Y,Z] : hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),X),Y)),Z) = hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),X),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),Y),Z)) ).
fof(fact_697_zmult__commute,axiom,
! [Z,W_2] : hAPP(int,int,hAPP(int,fun(int,int),times_times(int),Z),W_2) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),W_2),Z) ).
fof(fact_698_zdiff__zmult__distrib,axiom,
! [Z1,Z2,W_2] : hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),Z1),Z2)),W_2) = hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),Z1),W_2)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),Z2),W_2)) ).
fof(fact_699_zdiff__zmult__distrib2,axiom,
! [W_2,Z1,Z2] : hAPP(int,int,hAPP(int,fun(int,int),times_times(int),W_2),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),Z1),Z2)) = hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),W_2),Z1)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),W_2),Z2)) ).
fof(fact_700_zadd__assoc,axiom,
! [Z1,Z2,Z3] : hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Z1),Z2)),Z3) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Z1),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Z2),Z3)) ).
fof(fact_701_zadd__left__commute,axiom,
! [X,Y,Z] : hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),X),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Y),Z)) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Y),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),X),Z)) ).
fof(fact_702_zadd__commute,axiom,
! [Z,W_2] : hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Z),W_2) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),W_2),Z) ).
fof(fact_703_zpower__zadd__distrib,axiom,
! [X,Y,Z] : hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),X),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),Y),Z)) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),X),Y)),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),X),Z)) ).
fof(fact_704_zadd__zmult__distrib2,axiom,
! [W_2,Z1,Z2] : hAPP(int,int,hAPP(int,fun(int,int),times_times(int),W_2),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Z1),Z2)) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),W_2),Z1)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),W_2),Z2)) ).
fof(fact_705_zadd__zmult__distrib,axiom,
! [Z1,Z2,W_2] : hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Z1),Z2)),W_2) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),Z1),W_2)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),Z2),W_2)) ).
fof(fact_706_zadd__left__mono,axiom,
! [K,I,J] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),I),J))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),K),I)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),K),J))) ) ).
fof(fact_707_zmult__1__right,axiom,
! [Z] : hAPP(int,int,hAPP(int,fun(int,int),times_times(int),Z),one_one(int)) = Z ).
fof(fact_708_zmult__1,axiom,
! [Z] : hAPP(int,int,hAPP(int,fun(int,int),times_times(int),one_one(int)),Z) = Z ).
fof(fact_709_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [X] : hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),one_one(nat)) = ti(X_a,X) ) ).
fof(fact_710_max__less__iff__conj,axiom,
! [X_b] :
( linorder(X_b)
=> ! [X_1,Y_2,Z_2] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),ord_max(X_b),X_1),Y_2)),Z_2))
<=> ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),X_1),Z_2))
& hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),Y_2),Z_2)) ) ) ) ).
fof(fact_711_less__max__iff__disj,axiom,
! [X_b] :
( linorder(X_b)
=> ! [Z_2,X_1,Y_2] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),Z_2),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),ord_max(X_b),X_1),Y_2)))
<=> ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),Z_2),X_1))
| hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),Z_2),Y_2)) ) ) ) ).
fof(fact_712_min__max_Oless__supI2,axiom,
! [X_a] :
( linorder(X_a)
=> ! [A_1,X,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),B))
=> 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_1),B))) ) ) ).
fof(fact_713_min__max_Oless__supI1,axiom,
! [X_a] :
( linorder(X_a)
=> ! [B,X,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),A_1))
=> 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_1),B))) ) ) ).
fof(fact_714_nat__seg__image__imp__finite,axiom,
! [X_b,A_3,F,N] :
( ti(fun(X_b,bool),A_3) = hAPP(fun(nat,bool),fun(X_b,bool),hAPP(fun(nat,X_b),fun(fun(nat,bool),fun(X_b,bool)),image(nat,X_b),F),hAPP(fun(nat,bool),fun(nat,bool),collect(nat),hAPP(nat,fun(nat,bool),hAPP(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool)),combc(nat,nat,bool),ord_less(nat)),N)))
=> hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3)) ) ).
fof(fact_715_add__lessD1,axiom,
! [I,J,K] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),I),J)),K))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I),K)) ) ).
fof(fact_716_less__add__eq__less,axiom,
! [M,N_1,K,L_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),K),L_1))
=> ( hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),L_1) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),K),N_1)
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N_1)) ) ) ).
fof(fact_717_add__less__mono,axiom,
! [K,L_1,I,J] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I),J))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),K),L_1))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),I),K)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),J),L_1))) ) ) ).
fof(fact_718_add__less__mono1,axiom,
! [K,I,J] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I),J))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),I),K)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),J),K))) ) ).
fof(fact_719_trans__less__add2,axiom,
! [M,I,J] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I),J))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),J))) ) ).
fof(fact_720_trans__less__add1,axiom,
! [M,I,J] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I),J))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),J),M))) ) ).
fof(fact_721_nat__add__left__cancel__less,axiom,
! [K_1,M_1,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),K_1),M_1)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),K_1),N)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_1),N)) ) ).
fof(fact_722_not__add__less2,axiom,
! [J,I] : ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),J),I)),I)) ).
fof(fact_723_not__add__less1,axiom,
! [I,J] : ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),I),J)),I)) ).
fof(fact_724_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [X,P_2,Q_1] : hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),P_2)),Q_1) = hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),P_2),Q_1)) ) ).
fof(fact_725_add__less__imp__less__left,axiom,
! [X_a] :
( ordere236663937imp_le(X_a)
=> ! [C,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),C),A_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),C),B)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B)) ) ) ).
fof(fact_726_add__less__imp__less__right,axiom,
! [X_a] :
( ordere236663937imp_le(X_a)
=> ! [A_1,C,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),C)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),B),C)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B)) ) ) ).
fof(fact_727_add__strict__mono,axiom,
! [X_a] :
( ordere223160158up_add(X_a)
=> ! [C,D_1,A_1,B] :
( 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),C),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_1),C)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),B),D_1))) ) ) ) ).
fof(fact_728_add__strict__left__mono,axiom,
! [X_a] :
( ordere223160158up_add(X_a)
=> ! [C,A_1,B] :
( 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),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),C),A_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),C),B))) ) ) ).
fof(fact_729_add__strict__right__mono,axiom,
! [X_a] :
( ordere223160158up_add(X_a)
=> ! [C,A_1,B] :
( 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),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),C)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),B),C))) ) ) ).
fof(fact_730_add__less__cancel__left,axiom,
! [X_b] :
( ordere236663937imp_le(X_b)
=> ! [Ca,A_2,Ba] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),Ca),A_2)),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),Ca),Ba)))
<=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),A_2),Ba)) ) ) ).
fof(fact_731_add__less__cancel__right,axiom,
! [X_b] :
( ordere236663937imp_le(X_b)
=> ! [A_2,Ca,Ba] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),A_2),Ca)),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),Ba),Ca)))
<=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),A_2),Ba)) ) ) ).
fof(fact_732_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [X,Y,Q_1] : hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),Y)),Q_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),Q_1)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Y),Q_1)) ) ).
fof(fact_733_min__max_Oless__infI1,axiom,
! [X_a] :
( linorder(X_a)
=> ! [B,A_1,X] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),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_1),B)),X)) ) ) ).
fof(fact_734_min__max_Oless__infI2,axiom,
! [X_a] :
( linorder(X_a)
=> ! [A_1,B,X] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),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_1),B)),X)) ) ) ).
fof(fact_735_min__less__iff__conj,axiom,
! [X_b] :
( linorder(X_b)
=> ! [Z_2,X_1,Y_2] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),Z_2),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),ord_min(X_b),X_1),Y_2)))
<=> ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),Z_2),X_1))
& hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),Z_2),Y_2)) ) ) ) ).
fof(fact_736_min__less__iff__disj,axiom,
! [X_b] :
( linorder(X_b)
=> ! [X_1,Y_2,Z_2] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),ord_min(X_b),X_1),Y_2)),Z_2))
<=> ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),X_1),Z_2))
| hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),Y_2),Z_2)) ) ) ) ).
fof(fact_737_nat__less__le,axiom,
! [M_1,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_1),N))
<=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_1),N))
& M_1 != N ) ) ).
fof(fact_738_le__eq__less__or__eq,axiom,
! [M_1,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_1),N))
<=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_1),N))
| M_1 = N ) ) ).
fof(fact_739_less__imp__le__nat,axiom,
! [M,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N_1))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N_1)) ) ).
fof(fact_740_le__neq__implies__less,axiom,
! [M,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N_1))
=> ( M != N_1
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N_1)) ) ) ).
fof(fact_741_less__or__eq__imp__le,axiom,
! [M,N_1] :
( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N_1))
| M = N_1 )
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N_1)) ) ).
fof(fact_742_less__imp__diff__less,axiom,
! [N_1,J,K] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),J),K))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),J),N_1)),K)) ) ).
fof(fact_743_diff__less__mono2,axiom,
! [L_1,M,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N_1))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),L_1))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),L_1),N_1)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),L_1),M))) ) ) ).
fof(fact_744_lessThan__iff,axiom,
! [X_b] :
( ord(X_b)
=> ! [I_1,K_1] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),I_1),hAPP(X_b,fun(X_b,bool),ord_lessThan(X_b),K_1)))
<=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),I_1),K_1)) ) ) ).
fof(fact_745_linorder__not__less,axiom,
! [X_b] :
( linorder(X_b)
=> ! [X_1,Y_2] :
( ~ hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),X_1),Y_2))
<=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),Y_2),X_1)) ) ) ).
fof(fact_746_linorder__not__le,axiom,
! [X_b] :
( linorder(X_b)
=> ! [X_1,Y_2] :
( ~ hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),X_1),Y_2))
<=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),Y_2),X_1)) ) ) ).
fof(fact_747_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_748_order__less__le,axiom,
! [X_b] :
( order(X_b)
=> ! [X_1,Y_2] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),X_1),Y_2))
<=> ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),X_1),Y_2))
& ti(X_b,X_1) != ti(X_b,Y_2) ) ) ) ).
fof(fact_749_less__le__not__le,axiom,
! [X_b] :
( preorder(X_b)
=> ! [X_1,Y_2] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),X_1),Y_2))
<=> ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),X_1),Y_2))
& ~ hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),Y_2),X_1)) ) ) ) ).
fof(fact_750_order__le__less,axiom,
! [X_b] :
( order(X_b)
=> ! [X_1,Y_2] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),X_1),Y_2))
<=> ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),X_1),Y_2))
| ti(X_b,X_1) = ti(X_b,Y_2) ) ) ) ).
fof(fact_751_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_752_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_753_linorder__antisym__conv1,axiom,
! [X_b] :
( linorder(X_b)
=> ! [X_1,Y_2] :
( ~ hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),X_1),Y_2))
=> ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),X_1),Y_2))
<=> ti(X_b,X_1) = ti(X_b,Y_2) ) ) ) ).
fof(fact_754_order__neq__le__trans,axiom,
! [X_a] :
( order(X_a)
=> ! [A_1,B] :
( ti(X_a,A_1) != ti(X_a,B)
=> ( 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(X_a),A_1),B)) ) ) ) ).
fof(fact_755_xt1_I12_J,axiom,
! [X_a] :
( order(X_a)
=> ! [A_1,B] :
( ti(X_a,A_1) != ti(X_a,B)
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),A_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),A_1)) ) ) ) ).
fof(fact_756_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_757_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_758_linorder__antisym__conv2,axiom,
! [X_b] :
( linorder(X_b)
=> ! [X_1,Y_2] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),X_1),Y_2))
=> ( ~ hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),X_1),Y_2))
<=> ti(X_b,X_1) = ti(X_b,Y_2) ) ) ) ).
fof(fact_759_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_760_order__le__neq__trans,axiom,
! [X_a] :
( order(X_a)
=> ! [A_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B))
=> ( ti(X_a,A_1) != ti(X_a,B)
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B)) ) ) ) ).
fof(fact_761_xt1_I11_J,axiom,
! [X_a] :
( order(X_a)
=> ! [B,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),A_1))
=> ( ti(X_a,A_1) != ti(X_a,B)
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),A_1)) ) ) ) ).
fof(fact_762_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_763_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_764_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_765_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_766_not__less__bot,axiom,
! [X_a] :
( bot(X_a)
=> ! [A_1] : ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),bot_bot(X_a))) ) ).
fof(fact_767_bot__less,axiom,
! [X_b] :
( bot(X_b)
=> ! [A_2] :
( ti(X_b,A_2) != bot_bot(X_b)
<=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),bot_bot(X_b)),A_2)) ) ) ).
fof(fact_768_diff__eq__diff__less,axiom,
! [X_b] :
( ordered_ab_group_add(X_b)
=> ! [A_2,Ba,Ca,D] :
( hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),minus_minus(X_b),A_2),Ba) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),minus_minus(X_b),Ca),D)
=> ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),A_2),Ba))
<=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),Ca),D)) ) ) ) ).
fof(fact_769_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_770_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_771_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_772_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_773_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_774_xt1_I2_J,axiom,
! [X_a] :
( order(X_a)
=> ! [C,B,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),A_1))
=> ( ti(X_a,B) = ti(X_a,C)
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C),A_1)) ) ) ) ).
fof(fact_775_ord__less__eq__trans,axiom,
! [X_a] :
( ord(X_a)
=> ! [C,A_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B))
=> ( ti(X_a,B) = ti(X_a,C)
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),C)) ) ) ) ).
fof(fact_776_xt1_I1_J,axiom,
! [X_a] :
( order(X_a)
=> ! [C,A_1,B] :
( ti(X_a,A_1) = ti(X_a,B)
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C),B))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C),A_1)) ) ) ) ).
fof(fact_777_ord__eq__less__trans,axiom,
! [X_a] :
( ord(X_a)
=> ! [C,A_1,B] :
( ti(X_a,A_1) = ti(X_a,B)
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),C))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),C)) ) ) ) ).
fof(fact_778_xt1_I9_J,axiom,
! [X_a] :
( order(X_a)
=> ! [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_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B)) ) ) ).
fof(fact_779_order__less__asym_H,axiom,
! [X_a] :
( preorder(X_a)
=> ! [A_1,B] :
( 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),B),A_1)) ) ) ).
fof(fact_780_order__less__imp__triv,axiom,
! [X_b] :
( preorder(X_b)
=> ! [Pa,X_1,Y_2] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),X_1),Y_2))
=> ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),Y_2),X_1))
=> hBOOL(Pa) ) ) ) ).
fof(fact_781_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_782_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_783_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_784_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_785_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_786_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_787_linorder__antisym__conv3,axiom,
! [X_b] :
( linorder(X_b)
=> ! [Y_2,X_1] :
( ~ hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),Y_2),X_1))
=> ( ~ hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),X_1),Y_2))
<=> ti(X_b,X_1) = ti(X_b,Y_2) ) ) ) ).
fof(fact_788_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_789_not__less__iff__gr__or__eq,axiom,
! [X_b] :
( linorder(X_b)
=> ! [X_1,Y_2] :
( ~ hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),X_1),Y_2))
<=> ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),Y_2),X_1))
| ti(X_b,X_1) = ti(X_b,Y_2) ) ) ) ).
fof(fact_790_linorder__neq__iff,axiom,
! [X_b] :
( linorder(X_b)
=> ! [X_1,Y_2] :
( ti(X_b,X_1) != ti(X_b,Y_2)
<=> ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),X_1),Y_2))
| hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),Y_2),X_1)) ) ) ) ).
fof(fact_791_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_792_atLeastatMost__empty,axiom,
! [X_b] :
( order(X_b)
=> ! [Ba,A_2] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),Ba),A_2))
=> hAPP(X_b,fun(X_b,bool),hAPP(X_b,fun(X_b,fun(X_b,bool)),ord_atLeastAtMost(X_b),A_2),Ba) = bot_bot(fun(X_b,bool)) ) ) ).
fof(fact_793_lessThan__strict__subset__iff,axiom,
! [X_b] :
( linorder(X_b)
=> ! [M_1,N] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less(fun(X_b,bool)),hAPP(X_b,fun(X_b,bool),ord_lessThan(X_b),M_1)),hAPP(X_b,fun(X_b,bool),ord_lessThan(X_b),N)))
<=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),M_1),N)) ) ) ).
fof(fact_794_atLeastatMost__psubset__iff,axiom,
! [X_b] :
( order(X_b)
=> ! [A_2,Ba,Ca,D] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less(fun(X_b,bool)),hAPP(X_b,fun(X_b,bool),hAPP(X_b,fun(X_b,fun(X_b,bool)),ord_atLeastAtMost(X_b),A_2),Ba)),hAPP(X_b,fun(X_b,bool),hAPP(X_b,fun(X_b,fun(X_b,bool)),ord_atLeastAtMost(X_b),Ca),D)))
<=> ( ( ~ hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),A_2),Ba))
| ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),Ca),A_2))
& hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),Ba),D))
& ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),Ca),A_2))
| hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),Ba),D)) ) ) )
& hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),Ca),D)) ) ) ) ).
fof(fact_795_nat__less__cases,axiom,
! [Pa,M_1,N] :
( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_1),N))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),Pa,N),M_1)) )
=> ( ( M_1 = N
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),Pa,N),M_1)) )
=> ( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N),M_1))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),Pa,N),M_1)) )
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),Pa,N),M_1)) ) ) ) ).
fof(fact_796_less__not__refl3,axiom,
! [S,T_3] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),S),T_3))
=> S != T_3 ) ).
fof(fact_797_less__not__refl2,axiom,
! [N_1,M] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N_1),M))
=> M != N_1 ) ).
fof(fact_798_less__irrefl__nat,axiom,
! [N_1] : ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N_1),N_1)) ).
fof(fact_799_linorder__neqE__nat,axiom,
! [X,Y] :
( X != Y
=> ( ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),X),Y))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Y),X)) ) ) ).
fof(fact_800_nat__neq__iff,axiom,
! [M_1,N] :
( M_1 != N
<=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_1),N))
| hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N),M_1)) ) ) ).
fof(fact_801_less__not__refl,axiom,
! [N_1] : ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N_1),N_1)) ).
fof(fact_802_termination__basic__simps_I5_J,axiom,
! [X,Y] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),X),Y))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),X),Y)) ) ).
fof(fact_803_termination__basic__simps_I2_J,axiom,
! [Y,X,Z] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),X),Z))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),X),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),Y),Z))) ) ).
fof(fact_804_termination__basic__simps_I1_J,axiom,
! [Z,X,Y] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),X),Y))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),X),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),Y),Z))) ) ).
fof(fact_805_image__add__atLeastAtMost,axiom,
! [K_1,I_1,J_1] : hAPP(fun(nat,bool),fun(nat,bool),hAPP(fun(nat,nat),fun(fun(nat,bool),fun(nat,bool)),image(nat,nat),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),plus_plus(nat)),K_1)),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),I_1),J_1)) = hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),I_1),K_1)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),J_1),K_1)) ).
fof(fact_806_finite__nat__set__iff__bounded,axiom,
! [N_3] :
( hBOOL(hAPP(fun(nat,bool),bool,finite_finite_1(nat),N_3))
<=> ? [M_2] :
! [X_2] :
( hBOOL(hAPP(fun(nat,bool),bool,hAPP(nat,fun(fun(nat,bool),bool),member(nat),X_2),N_3))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),X_2),M_2)) ) ) ).
fof(fact_807_greaterThan__iff,axiom,
! [X_b] :
( ord(X_b)
=> ! [I_1,K_1] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),I_1),hAPP(X_b,fun(X_b,bool),ord_greaterThan(X_b),K_1)))
<=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),K_1),I_1)) ) ) ).
fof(fact_808_neg__less__iff__less,axiom,
! [X_b] :
( ordered_ab_group_add(X_b)
=> ! [Ba,A_2] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),hAPP(X_b,X_b,uminus_uminus(X_b),Ba)),hAPP(X_b,X_b,uminus_uminus(X_b),A_2)))
<=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),A_2),Ba)) ) ) ).
fof(fact_809_minus__less__iff,axiom,
! [X_b] :
( ordered_ab_group_add(X_b)
=> ! [A_2,Ba] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),hAPP(X_b,X_b,uminus_uminus(X_b),A_2)),Ba))
<=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),hAPP(X_b,X_b,uminus_uminus(X_b),Ba)),A_2)) ) ) ).
fof(fact_810_less__minus__iff,axiom,
! [X_b] :
( ordered_ab_group_add(X_b)
=> ! [A_2,Ba] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),A_2),hAPP(X_b,X_b,uminus_uminus(X_b),Ba)))
<=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),Ba),hAPP(X_b,X_b,uminus_uminus(X_b),A_2))) ) ) ).
fof(fact_811_compl__less__swap2,axiom,
! [X_a] :
( boolean_algebra(X_a)
=> ! [Y,X] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),Y)),X))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),X)),Y)) ) ) ).
fof(fact_812_compl__less__swap1,axiom,
! [X_a] :
( boolean_algebra(X_a)
=> ! [Y,X] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y),hAPP(X_a,X_a,uminus_uminus(X_a),X)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),hAPP(X_a,X_a,uminus_uminus(X_a),Y))) ) ) ).
fof(fact_813_compl__less__compl__iff,axiom,
! [X_b] :
( boolean_algebra(X_b)
=> ! [X_1,Y_2] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),hAPP(X_b,X_b,uminus_uminus(X_b),X_1)),hAPP(X_b,X_b,uminus_uminus(X_b),Y_2)))
<=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),Y_2),X_1)) ) ) ).
fof(fact_814_finite__M__bounded__by__nat,axiom,
! [Pa,I_1] : hBOOL(hAPP(fun(nat,bool),bool,finite_finite_1(nat),hAPP(fun(nat,bool),fun(nat,bool),collect(nat),hAPP(fun(nat,bool),fun(nat,bool),hAPP(fun(nat,fun(bool,bool)),fun(fun(nat,bool),fun(nat,bool)),combs(nat,bool,bool),hAPP(fun(nat,bool),fun(nat,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(nat,bool),fun(nat,fun(bool,bool))),combb(bool,fun(bool,bool),nat),fconj),Pa)),hAPP(nat,fun(nat,bool),hAPP(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool)),combc(nat,nat,bool),ord_less(nat)),I_1))))) ).
fof(fact_815_card__Collect__less__nat,axiom,
! [N] : hAPP(fun(nat,bool),nat,finite_card(nat),hAPP(fun(nat,bool),fun(nat,bool),collect(nat),hAPP(nat,fun(nat,bool),hAPP(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool)),combc(nat,nat,bool),ord_less(nat)),N))) = N ).
fof(fact_816_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [X,P_2,Q_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),P_2)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),Q_1)) = hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),P_2),Q_1)) ) ).
fof(fact_817_of__nat__less__iff,axiom,
! [X_b] :
( linordered_semidom(X_b)
=> ! [M_1,N] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),hAPP(nat,X_b,semiring_1_of_nat(X_b),M_1)),hAPP(nat,X_b,semiring_1_of_nat(X_b),N)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_1),N)) ) ) ).
fof(fact_818_less__imp__of__nat__less,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [M,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),M)),hAPP(nat,X_a,semiring_1_of_nat(X_a),N_1))) ) ) ).
fof(fact_819_of__nat__less__imp__less,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [M,N_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),M)),hAPP(nat,X_a,semiring_1_of_nat(X_a),N_1)))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N_1)) ) ) ).
fof(fact_820_add__less__le__mono,axiom,
! [X_a] :
( ordere223160158up_add(X_a)
=> ! [C,D_1,A_1,B] :
( 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_eq(X_a),C),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_1),C)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),B),D_1))) ) ) ) ).
fof(fact_821_add__le__less__mono,axiom,
! [X_a] :
( ordere223160158up_add(X_a)
=> ! [C,D_1,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_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C),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_1),C)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),B),D_1))) ) ) ) ).
fof(fact_822_less__add__one,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [A_1] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),one_one(X_a)))) ) ).
fof(fact_823_less__1__mult,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [N_1,M] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),M))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),N_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),M),N_1))) ) ) ) ).
fof(fact_824_add__diff__inverse,axiom,
! [M,N_1] :
( ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N_1))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),N_1)) = M ) ).
fof(fact_825_less__diff__conv,axiom,
! [I_1,J_1,K_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),J_1),K_1)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),I_1),K_1)),J_1)) ) ).
fof(fact_826_less__diff__iff,axiom,
! [N,K_1,M_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_1),M_1))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_1),N))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M_1),K_1)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N),K_1)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_1),N)) ) ) ) ).
fof(fact_827_diff__less__mono,axiom,
! [C,A_1,B] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A_1),B))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),C),A_1))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),A_1),C)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),B),C))) ) ) ).
fof(fact_828_atLeastatMost__empty__iff,axiom,
! [X_b] :
( order(X_b)
=> ! [A_2,Ba] :
( hAPP(X_b,fun(X_b,bool),hAPP(X_b,fun(X_b,fun(X_b,bool)),ord_atLeastAtMost(X_b),A_2),Ba) = bot_bot(fun(X_b,bool))
<=> ~ hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),A_2),Ba)) ) ) ).
fof(fact_829_atLeastatMost__empty__iff2,axiom,
! [X_b] :
( order(X_b)
=> ! [A_2,Ba] :
( bot_bot(fun(X_b,bool)) = hAPP(X_b,fun(X_b,bool),hAPP(X_b,fun(X_b,fun(X_b,bool)),ord_atLeastAtMost(X_b),A_2),Ba)
<=> ~ hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),A_2),Ba)) ) ) ).
fof(fact_830_atLeastatMost__subset__iff,axiom,
! [X_b] :
( order(X_b)
=> ! [A_2,Ba,Ca,D] :
( 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)),hAPP(X_b,fun(X_b,bool),hAPP(X_b,fun(X_b,fun(X_b,bool)),ord_atLeastAtMost(X_b),A_2),Ba)),hAPP(X_b,fun(X_b,bool),hAPP(X_b,fun(X_b,fun(X_b,bool)),ord_atLeastAtMost(X_b),Ca),D)))
<=> ( ~ hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),A_2),Ba))
| ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),Ca),A_2))
& hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),Ba),D)) ) ) ) ) ).
fof(fact_831_atLeastAtMost__singleton_H,axiom,
! [X_b] :
( order(X_b)
=> ! [A_2,Ba] :
( ti(X_b,A_2) = ti(X_b,Ba)
=> hAPP(X_b,fun(X_b,bool),hAPP(X_b,fun(X_b,fun(X_b,bool)),ord_atLeastAtMost(X_b),A_2),Ba) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_2),bot_bot(fun(X_b,bool))) ) ) ).
fof(fact_832_atLeastAtMost__singleton__iff,axiom,
! [X_b] :
( order(X_b)
=> ! [A_2,Ba,Ca] :
( hAPP(X_b,fun(X_b,bool),hAPP(X_b,fun(X_b,fun(X_b,bool)),ord_atLeastAtMost(X_b),A_2),Ba) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),Ca),bot_bot(fun(X_b,bool)))
<=> ( ti(X_b,A_2) = ti(X_b,Ba)
& ti(X_b,Ba) = ti(X_b,Ca) ) ) ) ).
fof(fact_833_atLeastAtMost__singleton,axiom,
! [X_b] :
( order(X_b)
=> ! [A_2] : hAPP(X_b,fun(X_b,bool),hAPP(X_b,fun(X_b,fun(X_b,bool)),ord_atLeastAtMost(X_b),A_2),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),A_2),bot_bot(fun(X_b,bool))) ) ).
fof(fact_834_image__uminus__atLeastAtMost,axiom,
! [X_b] :
( ordered_ab_group_add(X_b)
=> ! [X_1,Y_2] : hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,X_b),fun(fun(X_b,bool),fun(X_b,bool)),image(X_b,X_b),uminus_uminus(X_b)),hAPP(X_b,fun(X_b,bool),hAPP(X_b,fun(X_b,fun(X_b,bool)),ord_atLeastAtMost(X_b),X_1),Y_2)) = hAPP(X_b,fun(X_b,bool),hAPP(X_b,fun(X_b,fun(X_b,bool)),ord_atLeastAtMost(X_b),hAPP(X_b,X_b,uminus_uminus(X_b),Y_2)),hAPP(X_b,X_b,uminus_uminus(X_b),X_1)) ) ).
fof(fact_835_setsum__shift__bounds__cl__nat__ivl,axiom,
! [X_b] :
( comm_monoid_add(X_b)
=> ! [F,M_1,K_1,N] : hAPP(fun(nat,bool),X_b,hAPP(fun(nat,X_b),fun(fun(nat,bool),X_b),big_co1399186613setsum(nat,X_b),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M_1),K_1)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N),K_1))) = hAPP(fun(nat,bool),X_b,hAPP(fun(nat,X_b),fun(fun(nat,bool),X_b),big_co1399186613setsum(nat,X_b),hAPP(fun(nat,nat),fun(nat,X_b),hAPP(fun(nat,X_b),fun(fun(nat,nat),fun(nat,X_b)),combb(nat,X_b,nat),F),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),plus_plus(nat)),K_1))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),M_1),N)) ) ).
fof(fact_836_less__add__iff1,axiom,
! [X_b] :
( ordered_ring(X_b)
=> ! [A_2,E,Ca,Ba,D] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),A_2),E)),Ca)),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),Ba),E)),D)))
<=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),minus_minus(X_b),A_2),Ba)),E)),Ca)),D)) ) ) ).
fof(fact_837_less__add__iff2,axiom,
! [X_b] :
( ordered_ring(X_b)
=> ! [A_2,E,Ca,Ba,D] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),A_2),E)),Ca)),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),Ba),E)),D)))
<=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),Ca),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),minus_minus(X_b),Ba),A_2)),E)),D))) ) ) ).
fof(fact_838_setprod__constant,axiom,
! [X_b,X_c] :
( comm_monoid_mult(X_c)
=> ! [Y_2,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> hAPP(fun(X_b,bool),X_c,hAPP(fun(X_b,X_c),fun(fun(X_b,bool),X_c),big_co302546470etprod(X_b,X_c),hAPP(X_c,fun(X_b,X_c),combk(X_c,X_b),Y_2)),A_3) = hAPP(nat,X_c,hAPP(X_c,fun(nat,X_c),power_power(X_c),Y_2),hAPP(fun(X_b,bool),nat,finite_card(X_b),A_3)) ) ) ).
fof(fact_839_fold1__strict__below__iff,axiom,
! [X_b] :
( linorder(X_b)
=> ! [X_1,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( ti(fun(X_b,bool),A_3) != bot_bot(fun(X_b,bool))
=> ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),finite_fold1(X_b),ord_min(X_b)),A_3)),X_1))
<=> ? [X_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_2),A_3))
& hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),X_2),X_1)) ) ) ) ) ) ).
fof(fact_840_strict__below__fold1__iff,axiom,
! [X_b] :
( linorder(X_b)
=> ! [X_1,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( ti(fun(X_b,bool),A_3) != bot_bot(fun(X_b,bool))
=> ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),X_1),hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),finite_fold1(X_b),ord_min(X_b)),A_3)))
<=> ! [X_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_2),A_3))
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),X_1),X_2)) ) ) ) ) ) ).
fof(fact_841_nat__less__add__iff1,axiom,
! [U,M_1,N,J_1,I_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),J_1),I_1))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),I_1),U)),M_1)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),J_1),U)),N)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),I_1),J_1)),U)),M_1)),N)) ) ) ).
fof(fact_842_nat__less__add__iff2,axiom,
! [U,M_1,N,I_1,J_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_1),J_1))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),I_1),U)),M_1)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),J_1),U)),N)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),J_1),I_1)),U)),N))) ) ) ).
fof(fact_843_setsum__ub__add__nat,axiom,
! [X_b] :
( comm_monoid_add(X_b)
=> ! [F,P_1,M_1,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat))))
=> hAPP(fun(nat,bool),X_b,hAPP(fun(nat,X_b),fun(fun(nat,bool),X_b),big_co1399186613setsum(nat,X_b),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),M_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N),P_1))) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),hAPP(fun(nat,bool),X_b,hAPP(fun(nat,X_b),fun(fun(nat,bool),X_b),big_co1399186613setsum(nat,X_b),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),M_1),N))),hAPP(fun(nat,bool),X_b,hAPP(fun(nat,X_b),fun(fun(nat,bool),X_b),big_co1399186613setsum(nat,X_b),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat))),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N),P_1)))) ) ) ).
fof(fact_844_card__Diff1__less,axiom,
! [X_b,X_1,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),A_3))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(fun(X_b,bool),nat,finite_card(X_b),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_3),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_1),bot_bot(fun(X_b,bool)))))),hAPP(fun(X_b,bool),nat,finite_card(X_b),A_3))) ) ) ).
fof(fact_845_power__le__imp__le__exp,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [M,N_1,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),A_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),M)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N_1)))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N_1)) ) ) ) ).
fof(fact_846_power__increasing__iff,axiom,
! [X_b] :
( linordered_semidom(X_b)
=> ! [X_1,Y_2,Ba] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),one_one(X_b)),Ba))
=> ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(nat,X_b,hAPP(X_b,fun(nat,X_b),power_power(X_b),Ba),X_1)),hAPP(nat,X_b,hAPP(X_b,fun(nat,X_b),power_power(X_b),Ba),Y_2)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),X_1),Y_2)) ) ) ) ).
fof(fact_847_setsum__strict__mono,axiom,
! [X_b,X_c] :
( ( comm_monoid_add(X_c)
& ordere223160158up_add(X_c) )
=> ! [F,G,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( ti(fun(X_b,bool),A_3) != bot_bot(fun(X_b,bool))
=> ( ! [X_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_2),A_3))
=> hBOOL(hAPP(X_c,bool,hAPP(X_c,fun(X_c,bool),ord_less(X_c),hAPP(X_b,X_c,F,X_2)),hAPP(X_b,X_c,G,X_2))) )
=> hBOOL(hAPP(X_c,bool,hAPP(X_c,fun(X_c,bool),ord_less(X_c),hAPP(fun(X_b,bool),X_c,hAPP(fun(X_b,X_c),fun(fun(X_b,bool),X_c),big_co1399186613setsum(X_b,X_c),F),A_3)),hAPP(fun(X_b,bool),X_c,hAPP(fun(X_b,X_c),fun(fun(X_b,bool),X_c),big_co1399186613setsum(X_b,X_c),G),A_3))) ) ) ) ) ).
fof(fact_848_not__int__zless__negative,axiom,
! [N_1,M] : ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(nat,int,semiring_1_of_nat(int),N_1)),hAPP(int,int,uminus_uminus(int),hAPP(nat,int,semiring_1_of_nat(int),M)))) ).
fof(fact_849_zless__le,axiom,
! [Z_2,W_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Z_2),W_1))
<=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Z_2),W_1))
& Z_2 != W_1 ) ) ).
fof(fact_850_zadd__strict__right__mono,axiom,
! [K,I,J] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),I),J))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),I),K)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),J),K))) ) ).
fof(fact_851_less__fun__def,axiom,
! [X_b,X_c] :
( ord(X_c)
=> ! [F,G] :
( hBOOL(hAPP(fun(X_b,X_c),bool,hAPP(fun(X_b,X_c),fun(fun(X_b,X_c),bool),ord_less(fun(X_b,X_c)),F),G))
<=> ( hBOOL(hAPP(fun(X_b,X_c),bool,hAPP(fun(X_b,X_c),fun(fun(X_b,X_c),bool),ord_less_eq(fun(X_b,X_c)),F),G))
& ~ hBOOL(hAPP(fun(X_b,X_c),bool,hAPP(fun(X_b,X_c),fun(fun(X_b,X_c),bool),ord_less_eq(fun(X_b,X_c)),G),F)) ) ) ) ).
fof(fact_852_not__psubset__empty,axiom,
! [X_b,A_3] : ~ hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less(fun(X_b,bool)),A_3),bot_bot(fun(X_b,bool)))) ).
fof(fact_853_psubset__eq,axiom,
! [X_b,A_3,B_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less(fun(X_b,bool)),A_3),B_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)),A_3),B_2))
& ti(fun(X_b,bool),A_3) != ti(fun(X_b,bool),B_2) ) ) ).
fof(fact_854_subset__iff__psubset__eq,axiom,
! [X_b,A_3,B_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)),A_3),B_2))
<=> ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less(fun(X_b,bool)),A_3),B_2))
| ti(fun(X_b,bool),A_3) = ti(fun(X_b,bool),B_2) ) ) ).
fof(fact_855_psubset__imp__subset,axiom,
! [X_b,A_3,B_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less(fun(X_b,bool)),A_3),B_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)),A_3),B_2)) ) ).
fof(fact_856_psubset__subset__trans,axiom,
! [X_b,C_1,A_3,B_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less(fun(X_b,bool)),A_3),B_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_2),C_1))
=> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less(fun(X_b,bool)),A_3),C_1)) ) ) ).
fof(fact_857_subset__psubset__trans,axiom,
! [X_b,C_1,A_3,B_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)),A_3),B_2))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less(fun(X_b,bool)),B_2),C_1))
=> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less(fun(X_b,bool)),A_3),C_1)) ) ) ).
fof(fact_858_psubset__trans,axiom,
! [X_b,C_1,A_3,B_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less(fun(X_b,bool)),A_3),B_2))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less(fun(X_b,bool)),B_2),C_1))
=> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less(fun(X_b,bool)),A_3),C_1)) ) ) ).
fof(fact_859_psubsetD,axiom,
! [X_b,Ca,A_3,B_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less(fun(X_b,bool)),A_3),B_2))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),Ca),A_3))
=> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),Ca),B_2)) ) ) ).
fof(fact_860_zminus__zminus,axiom,
! [Z] : hAPP(int,int,uminus_uminus(int),hAPP(int,int,uminus_uminus(int),Z)) = Z ).
fof(fact_861_zadd__zless__mono,axiom,
! [Z_3,Z,W_3,W_2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),W_3),W_2))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Z_3),Z))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),W_3),Z_3)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),W_2),Z))) ) ) ).
fof(fact_862_zless__int,axiom,
! [M_1,N] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(nat,int,semiring_1_of_nat(int),M_1)),hAPP(nat,int,semiring_1_of_nat(int),N)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_1),N)) ) ).
fof(fact_863_zless__add1__eq,axiom,
! [W_1,Z_2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),W_1),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Z_2),one_one(int))))
<=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),W_1),Z_2))
| W_1 = Z_2 ) ) ).
fof(fact_864_atLeastAtMostPlus1__int__conv,axiom,
! [M_1,N] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),M_1),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),one_one(int)),N)))
=> hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastAtMost(int),M_1),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),one_one(int)),N)) = hAPP(fun(int,bool),fun(int,bool),hAPP(int,fun(fun(int,bool),fun(int,bool)),insert(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),one_one(int)),N)),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastAtMost(int),M_1),N)) ) ).
fof(fact_865_psubset__card__mono,axiom,
! [X_b,A_3,B_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),B_2))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less(fun(X_b,bool)),A_3),B_2))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(fun(X_b,bool),nat,finite_card(X_b),A_3)),hAPP(fun(X_b,bool),nat,finite_card(X_b),B_2))) ) ) ).
fof(fact_866_zless__imp__add1__zle,axiom,
! [W_2,Z] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),W_2),Z))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),W_2),one_one(int))),Z)) ) ).
fof(fact_867_add1__zle__eq,axiom,
! [W_1,Z_2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),W_1),one_one(int))),Z_2))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),W_1),Z_2)) ) ).
fof(fact_868_zle__add1__eq__le,axiom,
! [W_1,Z_2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),W_1),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Z_2),one_one(int))))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),W_1),Z_2)) ) ).
fof(fact_869_zle__diff1__eq,axiom,
! [W_1,Z_2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),W_1),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),Z_2),one_one(int))))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),W_1),Z_2)) ) ).
fof(fact_870_psubset__insert__iff,axiom,
! [X_b,A_3,X_1,B_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less(fun(X_b,bool)),A_3),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_1),B_2)))
<=> ( ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),B_2))
=> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less(fun(X_b,bool)),A_3),B_2)) )
& ( ~ hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),B_2))
=> ( ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),A_3))
=> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less(fun(X_b,bool)),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_3),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_1),bot_bot(fun(X_b,bool))))),B_2)) )
& ( ~ hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),A_3))
=> 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)),A_3),B_2)) ) ) ) ) ) ).
fof(fact_871_card__psubset,axiom,
! [X_b,A_3,B_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),B_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)),A_3),B_2))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(fun(X_b,bool),nat,finite_card(X_b),A_3)),hAPP(fun(X_b,bool),nat,finite_card(X_b),B_2)))
=> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less(fun(X_b,bool)),A_3),B_2)) ) ) ) ).
fof(fact_872_power__mult__distrib,axiom,
! [X_a] :
( comm_monoid_mult(X_a)
=> ! [A_1,B,N_1] : hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)),N_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N_1)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),B),N_1)) ) ).
fof(fact_873_power__commutes,axiom,
! [X_a] :
( monoid_mult(X_a)
=> ! [A_1,N_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N_1)),A_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N_1)) ) ).
fof(fact_874_power__one,axiom,
! [X_a] :
( monoid_mult(X_a)
=> ! [N_1] : hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),one_one(X_a)),N_1) = one_one(X_a) ) ).
fof(fact_875_of__nat__power,axiom,
! [X_a] :
( semiring_1(X_a)
=> ! [M,N_1] : hAPP(nat,X_a,semiring_1_of_nat(X_a),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),M),N_1)) = hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),M)),N_1) ) ).
fof(fact_876_power__mult,axiom,
! [X_a] :
( monoid_mult(X_a)
=> ! [A_1,M,N_1] : hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),N_1)) = hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),M)),N_1) ) ).
fof(fact_877_power__one__right,axiom,
! [X_a] :
( monoid_mult(X_a)
=> ! [A_1] : hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),one_one(nat)) = ti(X_a,A_1) ) ).
fof(fact_878_one__le__power,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [N_1,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),one_one(X_a)),A_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),one_one(X_a)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N_1))) ) ) ).
fof(fact_879_power__inject__exp,axiom,
! [X_b] :
( linordered_semidom(X_b)
=> ! [M_1,N,A_2] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),one_one(X_b)),A_2))
=> ( hAPP(nat,X_b,hAPP(X_b,fun(nat,X_b),power_power(X_b),A_2),M_1) = hAPP(nat,X_b,hAPP(X_b,fun(nat,X_b),power_power(X_b),A_2),N)
<=> M_1 = N ) ) ) ).
fof(fact_880_power__add,axiom,
! [X_a] :
( monoid_mult(X_a)
=> ! [A_1,M,N_1] : hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),N_1)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),M)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N_1)) ) ).
fof(fact_881_power__less__power__Suc,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [N_1,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),A_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N_1)))) ) ) ).
fof(fact_882_power__gt1__lemma,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [N_1,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),A_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N_1)))) ) ) ).
fof(fact_883_power__strict__increasing,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [A_1,N_1,N_4] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N_1),N_4))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),A_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N_1)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N_4))) ) ) ) ).
fof(fact_884_power__less__imp__less__exp,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [M,N_1,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),A_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),M)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N_1)))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N_1)) ) ) ) ).
fof(fact_885_power__strict__increasing__iff,axiom,
! [X_b] :
( linordered_semidom(X_b)
=> ! [X_1,Y_2,Ba] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),one_one(X_b)),Ba))
=> ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),hAPP(nat,X_b,hAPP(X_b,fun(nat,X_b),power_power(X_b),Ba),X_1)),hAPP(nat,X_b,hAPP(X_b,fun(nat,X_b),power_power(X_b),Ba),Y_2)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),X_1),Y_2)) ) ) ) ).
fof(fact_886_power__increasing,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [A_1,N_1,N_4] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_1),N_4))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),one_one(X_a)),A_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N_1)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N_4))) ) ) ) ).
fof(fact_887_power__minus,axiom,
! [X_a] :
( ring_1(X_a)
=> ! [A_1,N_1] : hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),A_1)),N_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),one_one(X_a))),N_1)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N_1)) ) ).
fof(fact_888_finite__conv__nat__seg__image,axiom,
! [X_b,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
<=> ? [N_2,F_1] : ti(fun(X_b,bool),A_3) = hAPP(fun(nat,bool),fun(X_b,bool),hAPP(fun(nat,X_b),fun(fun(nat,bool),fun(X_b,bool)),image(nat,X_b),F_1),hAPP(fun(nat,bool),fun(nat,bool),collect(nat),hAPP(nat,fun(nat,bool),hAPP(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool)),combc(nat,nat,bool),ord_less(nat)),N_2))) ) ).
fof(fact_889_int__le__induct,axiom,
! [Pa,I_1,K_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),I_1),K_1))
=> ( hBOOL(hAPP(int,bool,Pa,K_1))
=> ( ! [I_2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),I_2),K_1))
=> ( hBOOL(hAPP(int,bool,Pa,I_2))
=> hBOOL(hAPP(int,bool,Pa,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),I_2),one_one(int)))) ) )
=> hBOOL(hAPP(int,bool,Pa,I_1)) ) ) ) ).
fof(fact_890_finite__atLeastAtMost__int,axiom,
! [L,U] : hBOOL(hAPP(fun(int,bool),bool,finite_finite_1(int),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastAtMost(int),L),U))) ).
fof(fact_891_zless__linear,axiom,
! [X,Y] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),X),Y))
| X = Y
| hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Y),X)) ) ).
fof(fact_892_simp__from__to,axiom,
! [J_1,I_1] :
( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),J_1),I_1))
=> hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastAtMost(int),I_1),J_1) = bot_bot(fun(int,bool)) )
& ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),J_1),I_1))
=> hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastAtMost(int),I_1),J_1) = hAPP(fun(int,bool),fun(int,bool),hAPP(int,fun(fun(int,bool),fun(int,bool)),insert(int),I_1),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastAtMost(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),I_1),one_one(int))),J_1)) ) ) ).
fof(fact_893_int__less__induct,axiom,
! [Pa,I_1,K_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),I_1),K_1))
=> ( hBOOL(hAPP(int,bool,Pa,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),K_1),one_one(int))))
=> ( ! [I_2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),I_2),K_1))
=> ( hBOOL(hAPP(int,bool,Pa,I_2))
=> hBOOL(hAPP(int,bool,Pa,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),I_2),one_one(int)))) ) )
=> hBOOL(hAPP(int,bool,Pa,I_1)) ) ) ) ).
fof(fact_894_transfer__nat__int__set__relations_I1_J,axiom,
! [A_3] :
( hBOOL(hAPP(fun(nat,bool),bool,finite_finite_1(nat),A_3))
<=> hBOOL(hAPP(fun(int,bool),bool,finite_finite_1(int),hAPP(fun(nat,bool),fun(int,bool),hAPP(fun(nat,int),fun(fun(nat,bool),fun(int,bool)),image(nat,int),semiring_1_of_nat(int)),A_3))) ) ).
fof(fact_895_transfer__nat__int__set__relations_I5_J,axiom,
! [A_3,B_2] :
( hBOOL(hAPP(fun(nat,bool),bool,hAPP(fun(nat,bool),fun(fun(nat,bool),bool),ord_less_eq(fun(nat,bool)),A_3),B_2))
<=> hBOOL(hAPP(fun(int,bool),bool,hAPP(fun(int,bool),fun(fun(int,bool),bool),ord_less_eq(fun(int,bool)),hAPP(fun(nat,bool),fun(int,bool),hAPP(fun(nat,int),fun(fun(nat,bool),fun(int,bool)),image(nat,int),semiring_1_of_nat(int)),A_3)),hAPP(fun(nat,bool),fun(int,bool),hAPP(fun(nat,int),fun(fun(nat,bool),fun(int,bool)),image(nat,int),semiring_1_of_nat(int)),B_2))) ) ).
fof(fact_896_transfer__nat__int__set__relations_I2_J,axiom,
! [X_1,A_3] :
( hBOOL(hAPP(fun(nat,bool),bool,hAPP(nat,fun(fun(nat,bool),bool),member(nat),X_1),A_3))
<=> hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),hAPP(nat,int,semiring_1_of_nat(int),X_1)),hAPP(fun(nat,bool),fun(int,bool),hAPP(fun(nat,int),fun(fun(nat,bool),fun(int,bool)),image(nat,int),semiring_1_of_nat(int)),A_3))) ) ).
fof(fact_897_transfer__nat__int__set__relations_I3_J,axiom,
! [A_3,B_2] :
( A_3 = B_2
<=> hAPP(fun(nat,bool),fun(int,bool),hAPP(fun(nat,int),fun(fun(nat,bool),fun(int,bool)),image(nat,int),semiring_1_of_nat(int)),A_3) = hAPP(fun(nat,bool),fun(int,bool),hAPP(fun(nat,int),fun(fun(nat,bool),fun(int,bool)),image(nat,int),semiring_1_of_nat(int)),B_2) ) ).
fof(fact_898_int__if__cong,axiom,
! [X_1,Y_2,Pa] :
( ( hBOOL(Pa)
=> hAPP(nat,int,semiring_1_of_nat(int),X_1) = hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,hAPP(nat,fun(nat,nat),hAPP(bool,fun(nat,fun(nat,nat)),if(nat),Pa),X_1),Y_2)) )
& ( ~ hBOOL(Pa)
=> hAPP(nat,int,semiring_1_of_nat(int),Y_2) = hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,hAPP(nat,fun(nat,nat),hAPP(bool,fun(nat,fun(nat,nat)),if(nat),Pa),X_1),Y_2)) ) ) ).
fof(fact_899_transfer__int__nat__relations_I1_J,axiom,
! [X_1,Y_2] :
( hAPP(nat,int,semiring_1_of_nat(int),X_1) = hAPP(nat,int,semiring_1_of_nat(int),Y_2)
<=> X_1 = Y_2 ) ).
fof(fact_900_transfer__nat__int__set__relations_I4_J,axiom,
! [A_3,B_2] :
( hBOOL(hAPP(fun(nat,bool),bool,hAPP(fun(nat,bool),fun(fun(nat,bool),bool),ord_less(fun(nat,bool)),A_3),B_2))
<=> hBOOL(hAPP(fun(int,bool),bool,hAPP(fun(int,bool),fun(fun(int,bool),bool),ord_less(fun(int,bool)),hAPP(fun(nat,bool),fun(int,bool),hAPP(fun(nat,int),fun(fun(nat,bool),fun(int,bool)),image(nat,int),semiring_1_of_nat(int)),A_3)),hAPP(fun(nat,bool),fun(int,bool),hAPP(fun(nat,int),fun(fun(nat,bool),fun(int,bool)),image(nat,int),semiring_1_of_nat(int)),B_2))) ) ).
fof(fact_901_Nat__Transfer_Otransfer__nat__int__set__functions_I1_J,axiom,
! [A_3] : hAPP(fun(nat,bool),nat,finite_card(nat),A_3) = hAPP(fun(int,bool),nat,finite_card(int),hAPP(fun(nat,bool),fun(int,bool),hAPP(fun(nat,int),fun(fun(nat,bool),fun(int,bool)),image(nat,int),semiring_1_of_nat(int)),A_3)) ).
fof(fact_902_Nat__Transfer_Otransfer__int__nat__set__functions_I2_J,axiom,
bot_bot(fun(int,bool)) = hAPP(fun(nat,bool),fun(int,bool),hAPP(fun(nat,int),fun(fun(nat,bool),fun(int,bool)),image(nat,int),semiring_1_of_nat(int)),bot_bot(fun(nat,bool))) ).
fof(fact_903_transfer__int__nat__relations_I2_J,axiom,
! [X_1,Y_2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(nat,int,semiring_1_of_nat(int),X_1)),hAPP(nat,int,semiring_1_of_nat(int),Y_2)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),X_1),Y_2)) ) ).
fof(fact_904_transfer__int__nat__relations_I3_J,axiom,
! [X_1,Y_2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(nat,int,semiring_1_of_nat(int),X_1)),hAPP(nat,int,semiring_1_of_nat(int),Y_2)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),X_1),Y_2)) ) ).
fof(fact_905_Nat__Transfer_Otransfer__int__nat__functions_I1_J,axiom,
! [X,Y] : hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(nat,int,semiring_1_of_nat(int),X)),hAPP(nat,int,semiring_1_of_nat(int),Y)) = hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),X),Y)) ).
fof(fact_906_Nat__Transfer_Otransfer__int__nat__functions_I2_J,axiom,
! [X,Y] : hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(nat,int,semiring_1_of_nat(int),X)),hAPP(nat,int,semiring_1_of_nat(int),Y)) = hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),X),Y)) ).
fof(fact_907_transfer__int__nat__numerals_I2_J,axiom,
one_one(int) = hAPP(nat,int,semiring_1_of_nat(int),one_one(nat)) ).
fof(fact_908_Nat__Transfer_Otransfer__int__nat__functions_I4_J,axiom,
! [X,N_1] : hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(nat,int,semiring_1_of_nat(int),X)),N_1) = hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),X),N_1)) ).
fof(fact_909_int__gr__induct,axiom,
! [Pa,K_1,I_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K_1),I_1))
=> ( hBOOL(hAPP(int,bool,Pa,hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),K_1),one_one(int))))
=> ( ! [I_2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K_1),I_2))
=> ( hBOOL(hAPP(int,bool,Pa,I_2))
=> hBOOL(hAPP(int,bool,Pa,hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),I_2),one_one(int)))) ) )
=> hBOOL(hAPP(int,bool,Pa,I_1)) ) ) ) ).
fof(fact_910_int__ge__induct,axiom,
! [Pa,K_1,I_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K_1),I_1))
=> ( hBOOL(hAPP(int,bool,Pa,K_1))
=> ( ! [I_2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K_1),I_2))
=> ( hBOOL(hAPP(int,bool,Pa,I_2))
=> hBOOL(hAPP(int,bool,Pa,hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),I_2),one_one(int)))) ) )
=> hBOOL(hAPP(int,bool,Pa,I_1)) ) ) ) ).
fof(fact_911_mono__nat__linear__lb,axiom,
! [M_1,K_1,F] :
( ! [M_2,N_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_2),N_2))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,F,M_2)),hAPP(nat,nat,F,N_2))) )
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,F,M_1)),K_1)),hAPP(nat,nat,F,hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M_1),K_1)))) ) ).
fof(fact_912_bounded__nat__set__is__finite,axiom,
! [N,N_3] :
( ! [X_2] :
( hBOOL(hAPP(fun(nat,bool),bool,hAPP(nat,fun(fun(nat,bool),bool),member(nat),X_2),N_3))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),X_2),N)) )
=> hBOOL(hAPP(fun(nat,bool),bool,finite_finite_1(nat),N_3)) ) ).
fof(fact_913_psubset__imp__ex__mem,axiom,
! [X_b,A_3,B_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less(fun(X_b,bool)),A_3),B_2))
=> ? [B_1] : hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),B_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)),B_2),A_3))) ) ).
fof(fact_914_int__diff__cases,axiom,
! [Z] :
~ ! [M_2,N_2] : Z != hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(nat,int,semiring_1_of_nat(int),M_2)),hAPP(nat,int,semiring_1_of_nat(int),N_2)) ).
fof(fact_915_less__mono__imp__le__mono,axiom,
! [I_1,J_1,F] :
( ! [I_2,J_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_2),J_2))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,F,I_2)),hAPP(nat,nat,F,J_2))) )
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_1),J_1))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,F,I_1)),hAPP(nat,nat,F,J_1))) ) ) ).
fof(fact_916_power__power__power,axiom,
! [X_b] :
( power(X_b)
=> power_power(X_b) = hAPP(fun(X_b,fun(X_b,X_b)),fun(X_b,fun(nat,X_b)),hAPP(X_b,fun(fun(X_b,fun(X_b,X_b)),fun(X_b,fun(nat,X_b))),power_1(X_b),one_one(X_b)),times_times(X_b)) ) ).
fof(fact_917_dense__le,axiom,
! [X_a] :
( dense_linorder(X_a)
=> ! [Z,Y] :
( ! [X_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_2),Y))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_2),Z)) )
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),Z)) ) ) ).
fof(fact_918_minf_I8_J,axiom,
! [X_a] :
( linorder(X_a)
=> ! [T_3] :
? [Z_1] :
! [X_2] :
( 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_eq(X_a),T_3),X_2)) ) ) ).
fof(fact_919_pinf_I8_J,axiom,
! [X_a] :
( linorder(X_a)
=> ! [T_3] :
? [Z_1] :
! [X_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_eq(X_a),T_3),X_2)) ) ) ).
fof(fact_920_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_921_xt5,axiom,
! [X_c,X_b] :
( ( order(X_b)
& order(X_c) )
=> ! [Ca,F,Ba,A_2] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),Ba),A_2))
=> ( hBOOL(hAPP(X_c,bool,hAPP(X_c,fun(X_c,bool),ord_less_eq(X_c),Ca),hAPP(X_b,X_c,F,Ba)))
=> ( ! [X_2,Y_1] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),Y_1),X_2))
=> hBOOL(hAPP(X_c,bool,hAPP(X_c,fun(X_c,bool),ord_less(X_c),hAPP(X_b,X_c,F,Y_1)),hAPP(X_b,X_c,F,X_2))) )
=> hBOOL(hAPP(X_c,bool,hAPP(X_c,fun(X_c,bool),ord_less(X_c),Ca),hAPP(X_b,X_c,F,A_2))) ) ) ) ) ).
fof(fact_922_minf_I6_J,axiom,
! [X_a] :
( linorder(X_a)
=> ! [T_3] :
? [Z_1] :
! [X_2] :
( 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_eq(X_a),X_2),T_3)) ) ) ).
fof(fact_923_pinf_I6_J,axiom,
! [X_a] :
( linorder(X_a)
=> ! [T_3] :
? [Z_1] :
! [X_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_eq(X_a),X_2),T_3)) ) ) ).
fof(fact_924_xt7,axiom,
! [X_c,X_b] :
( ( order(X_b)
& order(X_c) )
=> ! [Ca,F,Ba,A_2] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),Ba),A_2))
=> ( hBOOL(hAPP(X_c,bool,hAPP(X_c,fun(X_c,bool),ord_less(X_c),Ca),hAPP(X_b,X_c,F,Ba)))
=> ( ! [X_2,Y_1] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),Y_1),X_2))
=> hBOOL(hAPP(X_c,bool,hAPP(X_c,fun(X_c,bool),ord_less_eq(X_c),hAPP(X_b,X_c,F,Y_1)),hAPP(X_b,X_c,F,X_2))) )
=> hBOOL(hAPP(X_c,bool,hAPP(X_c,fun(X_c,bool),ord_less(X_c),Ca),hAPP(X_b,X_c,F,A_2))) ) ) ) ) ).
fof(fact_925_order__less__le__subst2,axiom,
! [X_b,X_c] :
( ( order(X_c)
& order(X_b) )
=> ! [F,Ca,A_2,Ba] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),A_2),Ba))
=> ( hBOOL(hAPP(X_c,bool,hAPP(X_c,fun(X_c,bool),ord_less_eq(X_c),hAPP(X_b,X_c,F,Ba)),Ca))
=> ( ! [X_2,Y_1] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),X_2),Y_1))
=> hBOOL(hAPP(X_c,bool,hAPP(X_c,fun(X_c,bool),ord_less(X_c),hAPP(X_b,X_c,F,X_2)),hAPP(X_b,X_c,F,Y_1))) )
=> hBOOL(hAPP(X_c,bool,hAPP(X_c,fun(X_c,bool),ord_less(X_c),hAPP(X_b,X_c,F,A_2)),Ca)) ) ) ) ) ).
fof(fact_926_xt6,axiom,
! [X_c,X_b] :
( ( order(X_b)
& ord(X_c) )
=> ! [Ca,F,Ba,A_2] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(X_c,X_b,F,Ba)),A_2))
=> ( hBOOL(hAPP(X_c,bool,hAPP(X_c,fun(X_c,bool),ord_less(X_c),Ca),Ba))
=> ( ! [X_2,Y_1] :
( hBOOL(hAPP(X_c,bool,hAPP(X_c,fun(X_c,bool),ord_less(X_c),Y_1),X_2))
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),hAPP(X_c,X_b,F,Y_1)),hAPP(X_c,X_b,F,X_2))) )
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),hAPP(X_c,X_b,F,Ca)),A_2)) ) ) ) ) ).
fof(fact_927_xt4,axiom,
! [X_c,X_b] :
( ( order(X_b)
& order(X_c) )
=> ! [Ca,F,Ba,A_2] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),hAPP(X_c,X_b,F,Ba)),A_2))
=> ( hBOOL(hAPP(X_c,bool,hAPP(X_c,fun(X_c,bool),ord_less_eq(X_c),Ca),Ba))
=> ( ! [X_2,Y_1] :
( hBOOL(hAPP(X_c,bool,hAPP(X_c,fun(X_c,bool),ord_less_eq(X_c),Y_1),X_2))
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(X_c,X_b,F,Y_1)),hAPP(X_c,X_b,F,X_2))) )
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),hAPP(X_c,X_b,F,Ca)),A_2)) ) ) ) ) ).
fof(fact_928_order__le__less__subst2,axiom,
! [X_b,X_c] :
( ( order(X_c)
& order(X_b) )
=> ! [F,Ca,A_2,Ba] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),A_2),Ba))
=> ( hBOOL(hAPP(X_c,bool,hAPP(X_c,fun(X_c,bool),ord_less(X_c),hAPP(X_b,X_c,F,Ba)),Ca))
=> ( ! [X_2,Y_1] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),X_2),Y_1))
=> hBOOL(hAPP(X_c,bool,hAPP(X_c,fun(X_c,bool),ord_less_eq(X_c),hAPP(X_b,X_c,F,X_2)),hAPP(X_b,X_c,F,Y_1))) )
=> hBOOL(hAPP(X_c,bool,hAPP(X_c,fun(X_c,bool),ord_less(X_c),hAPP(X_b,X_c,F,A_2)),Ca)) ) ) ) ) ).
fof(fact_929_order__less__le__subst1,axiom,
! [X_b,X_c] :
( ( order(X_c)
& order(X_b) )
=> ! [Ca,A_2,F,Ba] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),A_2),hAPP(X_c,X_b,F,Ba)))
=> ( hBOOL(hAPP(X_c,bool,hAPP(X_c,fun(X_c,bool),ord_less_eq(X_c),Ba),Ca))
=> ( ! [X_2,Y_1] :
( hBOOL(hAPP(X_c,bool,hAPP(X_c,fun(X_c,bool),ord_less_eq(X_c),X_2),Y_1))
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(X_c,X_b,F,X_2)),hAPP(X_c,X_b,F,Y_1))) )
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),A_2),hAPP(X_c,X_b,F,Ca))) ) ) ) ) ).
fof(fact_930_order__le__less__subst1,axiom,
! [X_b,X_c] :
( ( order(X_c)
& order(X_b) )
=> ! [Ca,A_2,F,Ba] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),A_2),hAPP(X_c,X_b,F,Ba)))
=> ( hBOOL(hAPP(X_c,bool,hAPP(X_c,fun(X_c,bool),ord_less(X_c),Ba),Ca))
=> ( ! [X_2,Y_1] :
( hBOOL(hAPP(X_c,bool,hAPP(X_c,fun(X_c,bool),ord_less(X_c),X_2),Y_1))
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),hAPP(X_c,X_b,F,X_2)),hAPP(X_c,X_b,F,Y_1))) )
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),A_2),hAPP(X_c,X_b,F,Ca))) ) ) ) ) ).
fof(fact_931_int__induct,axiom,
! [I_1,Pa,K_1] :
( hBOOL(hAPP(int,bool,Pa,K_1))
=> ( ! [I_2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K_1),I_2))
=> ( hBOOL(hAPP(int,bool,Pa,I_2))
=> hBOOL(hAPP(int,bool,Pa,hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),I_2),one_one(int)))) ) )
=> ( ! [I_2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),I_2),K_1))
=> ( hBOOL(hAPP(int,bool,Pa,I_2))
=> hBOOL(hAPP(int,bool,Pa,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),I_2),one_one(int)))) ) )
=> hBOOL(hAPP(int,bool,Pa,I_1)) ) ) ) ).
fof(fact_932_finite__linorder__max__induct,axiom,
! [X_b] :
( linorder(X_b)
=> ! [Pa,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,Pa,bot_bot(fun(X_b,bool))))
=> ( ! [B_1,A_4] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_4))
=> ( ! [X_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_2),A_4))
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),X_2),B_1)) )
=> ( hBOOL(hAPP(fun(X_b,bool),bool,Pa,A_4))
=> hBOOL(hAPP(fun(X_b,bool),bool,Pa,hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),B_1),A_4))) ) ) )
=> hBOOL(hAPP(fun(X_b,bool),bool,Pa,A_3)) ) ) ) ) ).
fof(fact_933_finite__linorder__min__induct,axiom,
! [X_b] :
( linorder(X_b)
=> ! [Pa,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,Pa,bot_bot(fun(X_b,bool))))
=> ( ! [B_1,A_4] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_4))
=> ( ! [X_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_2),A_4))
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),B_1),X_2)) )
=> ( hBOOL(hAPP(fun(X_b,bool),bool,Pa,A_4))
=> hBOOL(hAPP(fun(X_b,bool),bool,Pa,hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),B_1),A_4))) ) ) )
=> hBOOL(hAPP(fun(X_b,bool),bool,Pa,A_3)) ) ) ) ) ).
fof(fact_934_setsum__natinterval__difff,axiom,
! [X_b] :
( ab_group_add(X_b)
=> ! [F,M_1,N] :
( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_1),N))
=> hAPP(fun(nat,bool),X_b,hAPP(fun(nat,X_b),fun(fun(nat,bool),X_b),big_co1399186613setsum(nat,X_b),hAPP(fun(nat,X_b),fun(nat,X_b),hAPP(fun(nat,fun(X_b,X_b)),fun(fun(nat,X_b),fun(nat,X_b)),combs(nat,X_b,X_b),hAPP(fun(nat,X_b),fun(nat,fun(X_b,X_b)),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(nat,X_b),fun(nat,fun(X_b,X_b))),combb(X_b,fun(X_b,X_b),nat),minus_minus(X_b)),F)),hAPP(fun(nat,nat),fun(nat,X_b),hAPP(fun(nat,X_b),fun(fun(nat,nat),fun(nat,X_b)),combb(nat,X_b,nat),F),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),plus_plus(nat)),one_one(nat))))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),M_1),N)) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),minus_minus(X_b),hAPP(nat,X_b,F,M_1)),hAPP(nat,X_b,F,hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)))) )
& ( ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_1),N))
=> hAPP(fun(nat,bool),X_b,hAPP(fun(nat,X_b),fun(fun(nat,bool),X_b),big_co1399186613setsum(nat,X_b),hAPP(fun(nat,X_b),fun(nat,X_b),hAPP(fun(nat,fun(X_b,X_b)),fun(fun(nat,X_b),fun(nat,X_b)),combs(nat,X_b,X_b),hAPP(fun(nat,X_b),fun(nat,fun(X_b,X_b)),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(nat,X_b),fun(nat,fun(X_b,X_b))),combb(X_b,fun(X_b,X_b),nat),minus_minus(X_b)),F)),hAPP(fun(nat,nat),fun(nat,X_b),hAPP(fun(nat,X_b),fun(fun(nat,nat),fun(nat,X_b)),combb(nat,X_b,nat),F),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),plus_plus(nat)),one_one(nat))))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),M_1),N)) = zero_zero(X_b) ) ) ) ).
fof(fact_935_setsum__head,axiom,
! [X_b] :
( comm_monoid_add(X_b)
=> ! [Pa,M_1,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_1),N))
=> hAPP(fun(nat,bool),X_b,hAPP(fun(nat,X_b),fun(fun(nat,bool),X_b),big_co1399186613setsum(nat,X_b),Pa),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),M_1),N)) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),hAPP(nat,X_b,Pa,M_1)),hAPP(fun(nat,bool),X_b,hAPP(fun(nat,X_b),fun(fun(nat,bool),X_b),big_co1399186613setsum(nat,X_b),Pa),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_gr1471790799AtMost(nat),M_1),N))) ) ) ).
fof(fact_936_less__zeroE,axiom,
! [N_1] : ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N_1),zero_zero(nat))) ).
fof(fact_937_le0,axiom,
! [N_1] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),N_1)) ).
fof(fact_938_finite__greaterThanAtMost,axiom,
! [L,U] : hBOOL(hAPP(fun(nat,bool),bool,finite_finite_1(nat),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_gr1471790799AtMost(nat),L),U))) ).
fof(fact_939_power__less__imp__less__base,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [A_1,N_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N_1)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),B),N_1)))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),B))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B)) ) ) ) ).
fof(fact_940_mult__right__le__one__le,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [Y,X] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),X))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),Y))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),one_one(X_a)))
=> 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),times_times(X_a),X),Y)),X)) ) ) ) ) ).
fof(fact_941_mult__left__le__one__le,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [Y,X] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),X))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),Y))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),one_one(X_a)))
=> 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),times_times(X_a),Y),X)),X)) ) ) ) ) ).
fof(fact_942_zero__less__two,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),one_one(X_a)),one_one(X_a)))) ) ).
fof(fact_943_power_Opower_Opower__0,axiom,
! [X_b,One,Times,A_2] : hAPP(nat,X_b,hAPP(X_b,fun(nat,X_b),hAPP(fun(X_b,fun(X_b,X_b)),fun(X_b,fun(nat,X_b)),hAPP(X_b,fun(fun(X_b,fun(X_b,X_b)),fun(X_b,fun(nat,X_b))),power_1(X_b),One),Times),A_2),zero_zero(nat)) = ti(X_b,One) ).
fof(fact_944_add__pos__nonneg,axiom,
! [X_a] :
( ordere216010020id_add(X_a)
=> ! [B,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),B))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B))) ) ) ) ).
fof(fact_945_add__nonneg__pos,axiom,
! [X_a] :
( ordere216010020id_add(X_a)
=> ! [B,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),B))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B))) ) ) ) ).
fof(fact_946_add__strict__increasing,axiom,
! [X_a] :
( ordere216010020id_add(X_a)
=> ! [B,C,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),C))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),C))) ) ) ) ).
fof(fact_947_add__strict__increasing2,axiom,
! [X_a] :
( ordere216010020id_add(X_a)
=> ! [B,C,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),C))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),C))) ) ) ) ).
fof(fact_948_add__neg__nonpos,axiom,
! [X_a] :
( ordere216010020id_add(X_a)
=> ! [B,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),zero_zero(X_a)))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),zero_zero(X_a)))
=> 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),B)),zero_zero(X_a))) ) ) ) ).
fof(fact_949_add__nonpos__neg,axiom,
! [X_a] :
( ordere216010020id_add(X_a)
=> ! [B,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),zero_zero(X_a)))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),zero_zero(X_a)))
=> 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),B)),zero_zero(X_a))) ) ) ) ).
fof(fact_950_Nat__Transfer_Otransfer__nat__int__function__closures_I4_J,axiom,
! [N_1,X] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),X),N_1))) ) ).
fof(fact_951_transfer__int__nat__quantifiers_I1_J,axiom,
! [Pa] :
( ! [X_2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X_2))
=> hBOOL(hAPP(int,bool,Pa,X_2)) )
<=> ! [X_2] : hBOOL(hAPP(int,bool,Pa,hAPP(nat,int,semiring_1_of_nat(int),X_2))) ) ).
fof(fact_952_transfer__int__nat__quantifiers_I2_J,axiom,
! [Pa] :
( ? [X_2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X_2))
& hBOOL(hAPP(int,bool,Pa,X_2)) )
<=> ? [X_2] : hBOOL(hAPP(int,bool,Pa,hAPP(nat,int,semiring_1_of_nat(int),X_2))) ) ).
fof(fact_953_Nat__Transfer_Otransfer__nat__int__function__closures_I9_J,axiom,
! [Z] : hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(nat,int,semiring_1_of_nat(int),Z))) ).
fof(fact_954_Nat__Transfer_Otransfer__nat__int__function__closures_I5_J,axiom,
hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),zero_zero(int))) ).
fof(fact_955_Nat__Transfer_Otransfer__nat__int__function__closures_I1_J,axiom,
! [Y,X] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),X),Y))) ) ) ).
fof(fact_956_Nat__Transfer_Otransfer__nat__int__function__closures_I2_J,axiom,
! [Y,X] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),X),Y))) ) ) ).
fof(fact_957_Nat__Transfer_Otransfer__nat__int__function__closures_I6_J,axiom,
hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),one_one(int))) ).
fof(fact_958_int__less__0__conv,axiom,
! [K] : ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(nat,int,semiring_1_of_nat(int),K)),zero_zero(int))) ).
fof(fact_959_zero__less__int__conv,axiom,
! [N] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),hAPP(nat,int,semiring_1_of_nat(int),N)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ).
fof(fact_960_less__bin__lemma,axiom,
! [K_1,L] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K_1),L))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),K_1),L)),zero_zero(int))) ) ).
fof(fact_961_zmult__zless__mono2,axiom,
! [K,I,J] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),I),J))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),K))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K),I)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K),J))) ) ) ).
fof(fact_962_int__0__less__1,axiom,
hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),one_one(int))) ).
fof(fact_963_nat__zero__less__power__iff,axiom,
! [X_1,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),X_1),N)))
<=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),X_1))
| N = zero_zero(nat) ) ) ).
fof(fact_964_zero__less__power,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [N_1,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N_1))) ) ) ).
fof(fact_965_nat__power__less__imp__less,axiom,
! [M,N_1,I] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),I))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),I),M)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),I),N_1)))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N_1)) ) ) ).
fof(fact_966_power__eq__imp__eq__base,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [A_1,N_1,B] :
( hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N_1) = hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),B),N_1)
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),B))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N_1))
=> ti(X_a,A_1) = ti(X_a,B) ) ) ) ) ) ).
fof(fact_967_power__strict__mono,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [N_1,A_1,B] :
( 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_eq(X_a),zero_zero(X_a)),A_1))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N_1)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),B),N_1))) ) ) ) ) ).
fof(fact_968_gr0I,axiom,
! [N_1] :
( N_1 != zero_zero(nat)
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N_1)) ) ).
fof(fact_969_gr__implies__not0,axiom,
! [M,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N_1))
=> N_1 != zero_zero(nat) ) ).
fof(fact_970_less__nat__zero__code,axiom,
! [N_1] : ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N_1),zero_zero(nat))) ).
fof(fact_971_neq0__conv,axiom,
! [N] :
( N != zero_zero(nat)
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ).
fof(fact_972_not__less0,axiom,
! [N_1] : ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N_1),zero_zero(nat))) ).
fof(fact_973_of__nat__0__less__iff,axiom,
! [X_b] :
( linordered_semidom(X_b)
=> ! [N] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),zero_zero(X_b)),hAPP(nat,X_b,semiring_1_of_nat(X_b),N)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ) ).
fof(fact_974_zero__less__double__add__iff__zero__less__single__add,axiom,
! [X_b] :
( linord219039673up_add(X_b)
=> ! [A_2] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),zero_zero(X_b)),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),A_2),A_2)))
<=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),zero_zero(X_b)),A_2)) ) ) ).
fof(fact_975_double__add__less__zero__iff__single__add__less__zero,axiom,
! [X_b] :
( linord219039673up_add(X_b)
=> ! [A_2] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),A_2),A_2)),zero_zero(X_b)))
<=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),A_2),zero_zero(X_b))) ) ) ).
fof(fact_976_add__pos__pos,axiom,
! [X_a] :
( ordere216010020id_add(X_a)
=> ! [B,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),B))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B))) ) ) ) ).
fof(fact_977_pos__add__strict,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [B,C,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),C))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),C))) ) ) ) ).
fof(fact_978_add__neg__neg,axiom,
! [X_a] :
( ordere216010020id_add(X_a)
=> ! [B,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),zero_zero(X_a)))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),zero_zero(X_a)))
=> 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),B)),zero_zero(X_a))) ) ) ) ).
fof(fact_979_even__less__0__iff,axiom,
! [X_b] :
( linordered_idom(X_b)
=> ! [A_2] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),A_2),A_2)),zero_zero(X_b)))
<=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),A_2),zero_zero(X_b))) ) ) ).
fof(fact_980_not__square__less__zero,axiom,
! [X_a] :
( linordered_ring(X_a)
=> ! [A_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),times_times(X_a),A_1),A_1)),zero_zero(X_a))) ) ).
fof(fact_981_mult__less__cancel__right__disj,axiom,
! [X_b] :
( linord581940658strict(X_b)
=> ! [A_2,Ca,Ba] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),A_2),Ca)),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),Ba),Ca)))
<=> ( ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),zero_zero(X_b)),Ca))
& hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),A_2),Ba)) )
| ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),Ca),zero_zero(X_b)))
& hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),Ba),A_2)) ) ) ) ) ).
fof(fact_982_mult__less__cancel__left__disj,axiom,
! [X_b] :
( linord581940658strict(X_b)
=> ! [Ca,A_2,Ba] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),Ca),A_2)),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),Ca),Ba)))
<=> ( ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),zero_zero(X_b)),Ca))
& hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),A_2),Ba)) )
| ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),Ca),zero_zero(X_b)))
& hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),Ba),A_2)) ) ) ) ) ).
fof(fact_983_mult__less__cancel__left__pos,axiom,
! [X_b] :
( linord581940658strict(X_b)
=> ! [A_2,Ba,Ca] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),zero_zero(X_b)),Ca))
=> ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),Ca),A_2)),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),Ca),Ba)))
<=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),A_2),Ba)) ) ) ) ).
fof(fact_984_mult__pos__pos,axiom,
! [X_a] :
( linord20386208strict(X_a)
=> ! [B,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),B))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B))) ) ) ) ).
fof(fact_985_mult__pos__neg,axiom,
! [X_a] :
( linord20386208strict(X_a)
=> ! [B,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),zero_zero(X_a)))
=> 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),times_times(X_a),A_1),B)),zero_zero(X_a))) ) ) ) ).
fof(fact_986_mult__pos__neg2,axiom,
! [X_a] :
( linord20386208strict(X_a)
=> ! [B,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),zero_zero(X_a)))
=> 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),times_times(X_a),B),A_1)),zero_zero(X_a))) ) ) ) ).
fof(fact_987_zero__less__mult__pos,axiom,
! [X_a] :
( linord20386208strict(X_a)
=> ! [A_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),B)) ) ) ) ).
fof(fact_988_zero__less__mult__pos2,axiom,
! [X_a] :
( linord20386208strict(X_a)
=> ! [B,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),A_1)))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),B)) ) ) ) ).
fof(fact_989_mult__less__cancel__left__neg,axiom,
! [X_b] :
( linord581940658strict(X_b)
=> ! [A_2,Ba,Ca] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),Ca),zero_zero(X_b)))
=> ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),Ca),A_2)),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),Ca),Ba)))
<=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),Ba),A_2)) ) ) ) ).
fof(fact_990_mult__neg__pos,axiom,
! [X_a] :
( linord20386208strict(X_a)
=> ! [B,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),zero_zero(X_a)))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),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),times_times(X_a),A_1),B)),zero_zero(X_a))) ) ) ) ).
fof(fact_991_mult__neg__neg,axiom,
! [X_a] :
( linord581940658strict(X_a)
=> ! [B,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),zero_zero(X_a)))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),zero_zero(X_a)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B))) ) ) ) ).
fof(fact_992_mult__strict__right__mono,axiom,
! [X_a] :
( linord20386208strict(X_a)
=> ! [C,A_1,B] :
( 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),zero_zero(X_a)),C))
=> 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),times_times(X_a),A_1),C)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),C))) ) ) ) ).
fof(fact_993_mult__strict__left__mono,axiom,
! [X_a] :
( linord20386208strict(X_a)
=> ! [C,A_1,B] :
( 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),zero_zero(X_a)),C))
=> 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),times_times(X_a),C),A_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C),B))) ) ) ) ).
fof(fact_994_comm__mult__strict__left__mono,axiom,
! [X_a] :
( linord893533164strict(X_a)
=> ! [C,A_1,B] :
( 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),zero_zero(X_a)),C))
=> 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),times_times(X_a),C),A_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C),B))) ) ) ) ).
fof(fact_995_mult__strict__right__mono__neg,axiom,
! [X_a] :
( linord581940658strict(X_a)
=> ! [C,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_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C),zero_zero(X_a)))
=> 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),times_times(X_a),A_1),C)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),C))) ) ) ) ).
fof(fact_996_mult__strict__left__mono__neg,axiom,
! [X_a] :
( linord581940658strict(X_a)
=> ! [C,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_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C),zero_zero(X_a)))
=> 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),times_times(X_a),C),A_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C),B))) ) ) ) ).
%----Arities (89)
fof(arity_fun___Lattices_Oboolean__algebra,axiom,
! [T_2,T_1] :
( boolean_algebra(T_1)
=> boolean_algebra(fun(T_2,T_1)) ) ).
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_Ouminus,axiom,
! [T_2,T_1] :
( uminus(T_1)
=> uminus(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_Int_Oint___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduc,axiom,
semiri456707255roduct(int) ).
fof(arity_Int_Oint___Groups_Oordered__cancel__ab__semigroup__add,axiom,
ordere223160158up_add(int) ).
fof(arity_Int_Oint___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
ordere236663937imp_le(int) ).
fof(arity_Int_Oint___Rings_Olinordered__comm__semiring__strict,axiom,
linord893533164strict(int) ).
fof(arity_Int_Oint___Groups_Olinordered__ab__semigroup__add,axiom,
linord2061991079up_add(int) ).
fof(arity_Int_Oint___Rings_Olinordered__semiring__strict,axiom,
linord20386208strict(int) ).
fof(arity_Int_Oint___Groups_Oordered__ab__semigroup__add,axiom,
ordere779506340up_add(int) ).
fof(arity_Int_Oint___Groups_Oordered__comm__monoid__add,axiom,
ordere216010020id_add(int) ).
fof(arity_Int_Oint___Groups_Olinordered__ab__group__add,axiom,
linord219039673up_add(int) ).
fof(arity_Int_Oint___Groups_Ocancel__ab__semigroup__add,axiom,
cancel146912293up_add(int) ).
fof(arity_Int_Oint___Rings_Oring__1__no__zero__divisors,axiom,
ring_11004092258visors(int) ).
fof(arity_Int_Oint___Rings_Olinordered__ring__strict,axiom,
linord581940658strict(int) ).
fof(arity_Int_Oint___Groups_Oordered__ab__group__add,axiom,
ordered_ab_group_add(int) ).
fof(arity_Int_Oint___Groups_Ocancel__semigroup__add,axiom,
cancel_semigroup_add(int) ).
fof(arity_Int_Oint___Rings_Olinordered__semidom,axiom,
linordered_semidom(int) ).
fof(arity_Int_Oint___Groups_Oab__semigroup__mult,axiom,
ab_semigroup_mult(int) ).
fof(arity_Int_Oint___Groups_Ocomm__monoid__mult,axiom,
comm_monoid_mult(int) ).
fof(arity_Int_Oint___Groups_Oab__semigroup__add,axiom,
ab_semigroup_add(int) ).
fof(arity_Int_Oint___Groups_Ocomm__monoid__add,axiom,
comm_monoid_add(int) ).
fof(arity_Int_Oint___Rings_Olinordered__ring,axiom,
linordered_ring(int) ).
fof(arity_Int_Oint___Rings_Olinordered__idom,axiom,
linordered_idom(int) ).
fof(arity_Int_Oint___Rings_Ocomm__semiring__1,axiom,
comm_semiring_1(int) ).
fof(arity_Int_Oint___Rings_Ocomm__semiring,axiom,
comm_semiring(int) ).
fof(arity_Int_Oint___Nat_Osemiring__char__0,axiom,
semiring_char_0(int) ).
fof(arity_Int_Oint___Groups_Oab__group__add,axiom,
ab_group_add(int) ).
fof(arity_Int_Oint___Rings_Oordered__ring,axiom,
ordered_ring(int) ).
fof(arity_Int_Oint___Orderings_Opreorder,axiom,
preorder(int) ).
fof(arity_Int_Oint___Orderings_Olinorder,axiom,
linorder(int) ).
fof(arity_Int_Oint___Groups_Omonoid__mult,axiom,
monoid_mult(int) ).
fof(arity_Int_Oint___Rings_Ocomm__ring__1,axiom,
comm_ring_1(int) ).
fof(arity_Int_Oint___Rings_Osemiring__1,axiom,
semiring_1(int) ).
fof(arity_Int_Oint___Rings_Osemiring__0,axiom,
semiring_0(int) ).
fof(arity_Int_Oint___Groups_Ogroup__add,axiom,
group_add(int) ).
fof(arity_Int_Oint___Orderings_Oorder,axiom,
order(int) ).
fof(arity_Int_Oint___Rings_Osemiring,axiom,
semiring(int) ).
fof(arity_Int_Oint___Orderings_Oord,axiom,
ord(int) ).
fof(arity_Int_Oint___Groups_Ouminus,axiom,
uminus(int) ).
fof(arity_Int_Oint___Rings_Oring__1,axiom,
ring_1(int) ).
fof(arity_Int_Oint___Groups_Ominus,axiom,
minus(int) ).
fof(arity_Int_Oint___Power_Opower,axiom,
power(int) ).
fof(arity_Int_Oint___Rings_Oring,axiom,
ring(int) ).
fof(arity_Int_Oint___Rings_Oidom,axiom,
idom(int) ).
fof(arity_Int_Oint___Groups_Oone,axiom,
one(int) ).
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___Rings_Olinordered__comm__semiring__strict,axiom,
linord893533164strict(nat) ).
fof(arity_Nat_Onat___Groups_Olinordered__ab__semigroup__add,axiom,
linord2061991079up_add(nat) ).
fof(arity_Nat_Onat___Rings_Olinordered__semiring__strict,axiom,
linord20386208strict(nat) ).
fof(arity_Nat_Onat___Groups_Oordered__ab__semigroup__add,axiom,
ordere779506340up_add(nat) ).
fof(arity_Nat_Onat___Groups_Oordered__comm__monoid__add,axiom,
ordere216010020id_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___Rings_Olinordered__semidom,axiom,
linordered_semidom(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___Nat_Osemiring__char__0,axiom,
semiring_char_0(nat) ).
fof(arity_Nat_Onat___Orderings_Opreorder,axiom,
preorder(nat) ).
fof(arity_Nat_Onat___Orderings_Olinorder,axiom,
linorder(nat) ).
fof(arity_Nat_Onat___Groups_Omonoid__mult,axiom,
monoid_mult(nat) ).
fof(arity_Nat_Onat___Rings_Osemiring__1,axiom,
semiring_1(nat) ).
fof(arity_Nat_Onat___Rings_Osemiring__0,axiom,
semiring_0(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_Nat_Onat___Power_Opower,axiom,
power(nat) ).
fof(arity_Nat_Onat___Groups_Oone,axiom,
one(nat) ).
fof(arity_HOL_Obool___Lattices_Oboolean__algebra,axiom,
boolean_algebra(bool) ).
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_Ouminus,axiom,
uminus(bool) ).
fof(arity_HOL_Obool___Groups_Ominus,axiom,
minus(bool) ).
%----Helper facts (25)
fof(help_ti_idem,axiom,
! [T,A] : ti(T,ti(T,A)) = ti(T,A) ).
fof(help_If_1_1_T,axiom,
! [X_a,X,Y] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),hAPP(bool,fun(X_a,fun(X_a,X_a)),if(X_a),fTrue),X),Y) = ti(X_a,X) ).
fof(help_If_2_1_T,axiom,
! [X_a,X,Y] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),hAPP(bool,fun(X_a,fun(X_a,X_a)),if(X_a),fFalse),X),Y) = ti(X_a,Y) ).
fof(help_If_3_1_T,axiom,
! [P] :
( ti(bool,P) = fTrue
| ti(bool,P) = fFalse ) ).
fof(help_fEx_1_1_U,axiom,
! [X_a,P,X] :
( ~ hBOOL(hAPP(X_a,bool,P,X))
| hBOOL(hAPP(fun(X_a,bool),bool,fEx(X_a),P)) ) ).
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_COMBI_1_1_U,axiom,
! [X_a,P] : hAPP(X_a,X_a,combi(X_a),P) = ti(X_a,P) ).
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 (1)
fof(conj_0,conjecture,
hBOOL(hAPP(fun(hoare_2118899576triple(x_a),bool),bool,hAPP(fun(hoare_2118899576triple(x_a),bool),fun(fun(hoare_2118899576triple(x_a),bool),bool),hoare_1301688828derivs(x_a),g),hAPP(fun(hoare_2118899576triple(x_a),bool),fun(hoare_2118899576triple(x_a),bool),hAPP(hoare_2118899576triple(x_a),fun(fun(hoare_2118899576triple(x_a),bool),fun(hoare_2118899576triple(x_a),bool)),insert(hoare_2118899576triple(x_a)),hAPP(fun(x_a,fun(state,bool)),hoare_2118899576triple(x_a),hAPP(com,fun(fun(x_a,fun(state,bool)),hoare_2118899576triple(x_a)),hAPP(fun(x_a,fun(state,bool)),fun(com,fun(fun(x_a,fun(state,bool)),hoare_2118899576triple(x_a))),hoare_759811442triple(x_a),hAPP(fun(state,bool),fun(x_a,fun(state,bool)),combk(fun(state,bool),x_a),hAPP(bool,fun(state,bool),combk(bool,state),fFalse))),c),hAPP(fun(state,bool),fun(x_a,fun(state,bool)),hAPP(fun(x_a,fun(fun(state,bool),fun(state,bool))),fun(fun(state,bool),fun(x_a,fun(state,bool))),combc(x_a,fun(state,bool),fun(state,bool)),hAPP(fun(x_a,fun(state,fun(bool,bool))),fun(x_a,fun(fun(state,bool),fun(state,bool))),hAPP(fun(fun(state,fun(bool,bool)),fun(fun(state,bool),fun(state,bool))),fun(fun(x_a,fun(state,fun(bool,bool))),fun(x_a,fun(fun(state,bool),fun(state,bool)))),combb(fun(state,fun(bool,bool)),fun(fun(state,bool),fun(state,bool)),x_a),combs(state,bool,bool)),hAPP(fun(x_a,fun(state,bool)),fun(x_a,fun(state,fun(bool,bool))),hAPP(fun(fun(state,bool),fun(state,fun(bool,bool))),fun(fun(x_a,fun(state,bool)),fun(x_a,fun(state,fun(bool,bool)))),combb(fun(state,bool),fun(state,fun(bool,bool)),x_a),hAPP(fun(bool,fun(bool,bool)),fun(fun(state,bool),fun(state,fun(bool,bool))),combb(bool,fun(bool,bool),state),fconj)),p))),hAPP(fun(state,bool),fun(state,bool),hAPP(fun(bool,bool),fun(fun(state,bool),fun(state,bool)),combb(bool,bool,state),fNot),b)))),bot_bot(fun(hoare_2118899576triple(x_a),bool))))) ).
%------------------------------------------------------------------------------