TPTP Problem File: SWW472+7.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SWW472+7 : TPTP v8.2.0. Released v5.3.0.
% Domain : Software Verification
% Problem : Hoare's Logic with Procedures line 327, 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_l327 [Bla11]
% Status : Theorem
% Rating : 1.00 v5.3.0
% Syntax : Number of formulae : 1204 ( 346 unt; 0 def)
% Number of atoms : 3074 ( 787 equ)
% Maximal formula atoms : 9 ( 2 avg)
% Number of connectives : 2113 ( 243 ~; 68 |; 134 &)
% ( 223 <=>;1445 =>; 0 <=; 0 <~>)
% Maximal formula depth : 17 ( 5 avg)
% Maximal term depth : 12 ( 2 avg)
% Number of predicates : 62 ( 61 usr; 0 prp; 1-2 aty)
% Number of functors : 77 ( 77 usr; 24 con; 0-4 aty)
% Number of variables : 3386 (3333 !; 53 ?)
% SPC : FOF_THM_RFO_SEQ
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-08-09 17:36:27
% : Encoded with polymorphic tags.
%------------------------------------------------------------------------------
%----Explicit typings (77)
fof(tsy_c_Big__Operators_Osemilattice__big_res,axiom,
! [X_a] : ti(fun(fun(X_a,fun(X_a,X_a)),fun(fun(fun(X_a,bool),X_a),bool)),big_semilattice_big(X_a)) = big_semilattice_big(X_a) ).
fof(tsy_c_COMBB_res,axiom,
! [X_b,X_c,X_a] : ti(fun(fun(X_b,X_c),fun(fun(X_a,X_b),fun(X_a,X_c))),combb(X_b,X_c,X_a)) = combb(X_b,X_c,X_a) ).
fof(tsy_c_COMBC_res,axiom,
! [X_a,X_b,X_c] : ti(fun(fun(X_a,fun(X_b,X_c)),fun(X_b,fun(X_a,X_c))),combc(X_a,X_b,X_c)) = combc(X_a,X_b,X_c) ).
fof(tsy_c_COMBK_res,axiom,
! [X_a,X_b] : ti(fun(X_a,fun(X_b,X_a)),combk(X_a,X_b)) = combk(X_a,X_b) ).
fof(tsy_c_COMBS_res,axiom,
! [X_a,X_b,X_c] : ti(fun(fun(X_a,fun(X_b,X_c)),fun(fun(X_a,X_b),fun(X_a,X_c))),combs(X_a,X_b,X_c)) = combs(X_a,X_b,X_c) ).
fof(tsy_c_Com_Ocom_OCond_res,axiom,
ti(fun(fun(state,bool),fun(com,fun(com,com))),cond) = cond ).
fof(tsy_c_Com_Ocom_OLocal_res,axiom,
ti(fun(loc,fun(fun(state,nat),fun(com,com))),local) = local ).
fof(tsy_c_Com_Ocom_OSKIP_res,axiom,
ti(com,skip) = skip ).
fof(tsy_c_Com_Ocom_OSemi_res,axiom,
ti(fun(com,fun(com,com)),semi) = semi ).
fof(tsy_c_Com_Ocom_OWhile_res,axiom,
ti(fun(fun(state,bool),fun(com,com)),while) = while ).
fof(tsy_c_Com_Ocom_Ocom__size_res,axiom,
ti(fun(com,nat),com_size) = com_size ).
fof(tsy_c_Finite__Set_Ocard_res,axiom,
! [X_a] : ti(fun(fun(X_a,bool),nat),finite_card(X_a)) = finite_card(X_a) ).
fof(tsy_c_Finite__Set_Ocomp__fun__commute_res,axiom,
! [X_a,X_b] : ti(fun(fun(X_a,fun(X_b,X_b)),bool),finite100568337ommute(X_a,X_b)) = finite100568337ommute(X_a,X_b) ).
fof(tsy_c_Finite__Set_Ocomp__fun__idem_res,axiom,
! [X_a,X_b] : ti(fun(fun(X_a,fun(X_b,X_b)),bool),finite_comp_fun_idem(X_a,X_b)) = finite_comp_fun_idem(X_a,X_b) ).
fof(tsy_c_Finite__Set_Ofinite_res,axiom,
! [X_a] : ti(fun(fun(X_a,bool),bool),finite_finite_1(X_a)) = finite_finite_1(X_a) ).
fof(tsy_c_Finite__Set_Ofold_res,axiom,
! [X_a,X_b] : ti(fun(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(fun(X_a,bool),X_b))),finite_fold(X_a,X_b)) = finite_fold(X_a,X_b) ).
fof(tsy_c_Finite__Set_Ofold1_res,axiom,
! [X_a] : ti(fun(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,bool),X_a)),finite_fold1(X_a)) = finite_fold1(X_a) ).
fof(tsy_c_Finite__Set_Ofold1Set_res,axiom,
! [X_a] : ti(fun(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,bool),fun(X_a,bool))),finite_fold1Set(X_a)) = finite_fold1Set(X_a) ).
fof(tsy_c_Finite__Set_Ofold__graph_res,axiom,
! [X_a,X_b] : ti(fun(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(fun(X_a,bool),fun(X_b,bool)))),finite_fold_graph(X_a,X_b)) = finite_fold_graph(X_a,X_b) ).
fof(tsy_c_Finite__Set_Ofolding__image__simple_res,axiom,
! [X_a,X_b] : ti(fun(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(fun(X_b,X_a),fun(fun(fun(X_b,bool),X_a),bool)))),finite1357897459simple(X_a,X_b)) = finite1357897459simple(X_a,X_b) ).
fof(tsy_c_Finite__Set_Ofolding__image__simple__idem_res,axiom,
! [X_a,X_b] : ti(fun(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(fun(X_b,X_a),fun(fun(fun(X_b,bool),X_a),bool)))),finite908156982e_idem(X_a,X_b)) = finite908156982e_idem(X_a,X_b) ).
fof(tsy_c_Finite__Set_Ofolding__one_res,axiom,
! [X_a] : ti(fun(fun(X_a,fun(X_a,X_a)),fun(fun(fun(X_a,bool),X_a),bool)),finite_folding_one(X_a)) = finite_folding_one(X_a) ).
fof(tsy_c_Finite__Set_Ofolding__one__idem_res,axiom,
! [X_a] : ti(fun(fun(X_a,fun(X_a,X_a)),fun(fun(fun(X_a,bool),X_a),bool)),finite2073411215e_idem(X_a)) = finite2073411215e_idem(X_a) ).
fof(tsy_c_Groups_Oabs__class_Oabs_res,axiom,
! [X_a] :
( ordere142940540dd_abs(X_a)
=> ti(fun(X_a,X_a),abs_abs(X_a)) = abs_abs(X_a) ) ).
fof(tsy_c_Groups_Ominus__class_Ominus_res,axiom,
! [X_a] :
( minus(X_a)
=> ti(fun(X_a,fun(X_a,X_a)),minus_minus(X_a)) = minus_minus(X_a) ) ).
fof(tsy_c_Groups_Oone__class_Oone_res,axiom,
! [X_a] :
( one(X_a)
=> ti(X_a,one_one(X_a)) = one_one(X_a) ) ).
fof(tsy_c_Groups_Oplus__class_Oplus_0_res,axiom,
! [X_a] :
( monoid_add(X_a)
=> ti(fun(X_a,fun(X_a,X_a)),plus_plus(X_a)) = plus_plus(X_a) ) ).
fof(tsy_c_Groups_Oplus__class_Oplus_1_res,axiom,
! [X_a] :
( ab_semigroup_add(X_a)
=> ti(fun(X_a,fun(X_a,X_a)),plus_plus(X_a)) = plus_plus(X_a) ) ).
fof(tsy_c_Groups_Oplus__class_Oplus_2_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_a] :
( mult_zero(X_a)
=> ti(fun(X_a,fun(X_a,X_a)),times_times(X_a)) = times_times(X_a) ) ).
fof(tsy_c_Groups_Otimes__class_Otimes_1_res,axiom,
! [X_a] :
( no_zero_divisors(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_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_3_res,axiom,
! [X_a] :
( monoid_mult(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_4_res,axiom,
! [X_a] :
( ab_semigroup_mult(X_a)
=> ti(fun(X_a,fun(X_a,X_a)),times_times(X_a)) = times_times(X_a) ) ).
fof(tsy_c_Groups_Ozero__class_Ozero_res,axiom,
! [X_a] :
( zero(X_a)
=> ti(X_a,zero_zero(X_a)) = zero_zero(X_a) ) ).
fof(tsy_c_HOL_Oundefined_res,axiom,
! [X_a] : ti(X_a,undefined(X_a)) = undefined(X_a) ).
fof(tsy_c_Hoare__Mirabelle__lcrcocijdw_OMGT_res,hypothesis,
ti(fun(com,hoare_2118899576triple(state)),hoare_Mirabelle_MGT) = hoare_Mirabelle_MGT ).
fof(tsy_c_Hoare__Mirabelle__lcrcocijdw_Ohoare__derivs_res,axiom,
! [X_a] : ti(fun(fun(hoare_2118899576triple(X_a),bool),fun(fun(hoare_2118899576triple(X_a),bool),bool)),hoare_1301688828derivs(X_a)) = hoare_1301688828derivs(X_a) ).
fof(tsy_c_Hoare__Mirabelle__lcrcocijdw_Ohoare__valids_res,axiom,
! [X_a] : ti(fun(fun(hoare_2118899576triple(X_a),bool),fun(fun(hoare_2118899576triple(X_a),bool),bool)),hoare_902341502valids(X_a)) = hoare_902341502valids(X_a) ).
fof(tsy_c_Hoare__Mirabelle__lcrcocijdw_Otriple_Otriple_res,axiom,
! [X_a] : ti(fun(fun(X_a,fun(state,bool)),fun(com,fun(fun(X_a,fun(state,bool)),hoare_2118899576triple(X_a)))),hoare_759811442triple(X_a)) = hoare_759811442triple(X_a) ).
fof(tsy_c_Hoare__Mirabelle__lcrcocijdw_Otriple_Otriple__case_res,axiom,
! [X_b,X_a] : ti(fun(fun(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),X_a))),fun(hoare_2118899576triple(X_b),X_a)),hoare_225284258e_case(X_b,X_a)) = hoare_225284258e_case(X_b,X_a) ).
fof(tsy_c_Hoare__Mirabelle__lcrcocijdw_Otriple_Otriple__rec_res,axiom,
! [X_b,X_a] : ti(fun(fun(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),X_a))),fun(hoare_2118899576triple(X_b),X_a)),hoare_1759541758le_rec(X_b,X_a)) = hoare_1759541758le_rec(X_b,X_a) ).
fof(tsy_c_Hoare__Mirabelle__lcrcocijdw_Otriple_Otriple__size_res,axiom,
! [X_a] : ti(fun(fun(X_a,nat),fun(hoare_2118899576triple(X_a),nat)),hoare_2043812435e_size(X_a)) = hoare_2043812435e_size(X_a) ).
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_Int_Onumber__class_Onumber__of_res,axiom,
! [X_a] :
( number(X_a)
=> ti(fun(int,X_a),number_number_of(X_a)) = number_number_of(X_a) ) ).
fof(tsy_c_Int_Oring__1__class_OInts_res,axiom,
! [X_a] :
( ring_1(X_a)
=> ti(fun(X_a,bool),ring_1_Ints(X_a)) = ring_1_Ints(X_a) ) ).
fof(tsy_c_Nat_OSuc_res,axiom,
ti(fun(nat,nat),suc) = suc ).
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_Nat_Osize__class_Osize_0_res,axiom,
ti(fun(com,nat),size_size(com)) = size_size(com) ).
fof(tsy_c_Nat_Osize__class_Osize_1_res,axiom,
! [X_a] : ti(fun(hoare_2118899576triple(X_a),nat),size_size(hoare_2118899576triple(X_a))) = size_size(hoare_2118899576triple(X_a)) ).
fof(tsy_c_Nat__Transfer_Otsub_res,axiom,
ti(fun(int,fun(int,int)),nat_tsub) = nat_tsub ).
fof(tsy_c_Nitpick_Ofold__graph_H_res,axiom,
! [X_a,X_b] : ti(fun(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(fun(X_a,bool),fun(X_b,bool)))),fold_graph(X_a,X_b)) = fold_graph(X_a,X_b) ).
fof(tsy_c_Orderings_Obot__class_Obot_res,axiom,
! [X_a] :
( bot(X_a)
=> ti(X_a,bot_bot(X_a)) = bot_bot(X_a) ) ).
fof(tsy_c_Orderings_Oord_Omin_res,axiom,
! [X_a] :
( ord(X_a)
=> ti(fun(fun(X_a,fun(X_a,bool)),fun(X_a,fun(X_a,X_a))),min(X_a)) = min(X_a) ) ).
fof(tsy_c_Orderings_Oord__class_Oless_res,axiom,
! [X_a] :
( ord(X_a)
=> ti(fun(X_a,fun(X_a,bool)),ord_less(X_a)) = ord_less(X_a) ) ).
fof(tsy_c_Orderings_Oord__class_Oless__eq_res,axiom,
! [X_b] :
( ord(X_b)
=> ti(fun(X_b,fun(X_b,bool)),ord_less_eq(X_b)) = ord_less_eq(X_b) ) ).
fof(tsy_c_Orderings_Oord__class_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_Predicate_OPowp_res,axiom,
! [X_a] : ti(fun(fun(X_a,bool),fun(fun(X_a,bool),bool)),powp(X_a)) = powp(X_a) ).
fof(tsy_c_Set_OCollect_res,axiom,
! [X_a] : ti(fun(fun(X_a,bool),fun(X_a,bool)),collect(X_a)) = collect(X_a) ).
fof(tsy_c_Set_Oinsert_res,axiom,
! [X_a] : ti(fun(X_a,fun(fun(X_a,bool),fun(X_a,bool))),insert(X_a)) = insert(X_a) ).
fof(tsy_c_Set_Othe__elem_res,axiom,
! [X_a] : ti(fun(fun(X_a,bool),X_a),the_elem(X_a)) = the_elem(X_a) ).
fof(tsy_c_Sum__Type_OPlus_res,axiom,
! [X_a,X_b] : ti(fun(fun(X_a,bool),fun(fun(X_b,bool),fun(sum_sum(X_a,X_b),bool))),sum_Plus(X_a,X_b)) = sum_Plus(X_a,X_b) ).
fof(tsy_c_fFalse_res,axiom,
ti(bool,fFalse) = fFalse ).
fof(tsy_c_fNot_res,axiom,
ti(fun(bool,bool),fNot) = fNot ).
fof(tsy_c_fTrue_res,axiom,
ti(bool,fTrue) = fTrue ).
fof(tsy_c_fconj_res,axiom,
ti(fun(bool,fun(bool,bool)),fconj) = fconj ).
fof(tsy_c_fdisj_res,axiom,
ti(fun(bool,fun(bool,bool)),fdisj) = fdisj ).
fof(tsy_c_fequal_res,axiom,
! [X_a] : ti(fun(X_a,fun(X_a,bool)),fequal(X_a)) = fequal(X_a) ).
fof(tsy_c_fimplies_res,axiom,
ti(fun(bool,fun(bool,bool)),fimplies) = fimplies ).
fof(tsy_c_hAPP_arg1,axiom,
! [X_a,X_c,B_1_1,B_2_1] : hAPP(X_a,X_c,ti(fun(X_a,X_c),B_1_1),B_2_1) = hAPP(X_a,X_c,B_1_1,B_2_1) ).
fof(tsy_c_hAPP_arg2,axiom,
! [X_a,X_c,B_1_1,B_2_1] : hAPP(X_a,X_c,B_1_1,ti(X_a,B_2_1)) = hAPP(X_a,X_c,B_1_1,B_2_1) ).
fof(tsy_c_hAPP_res,axiom,
! [X_c,X_a,B_1_1,B_2_1] : ti(X_c,hAPP(X_a,X_c,B_1_1,B_2_1)) = hAPP(X_a,X_c,B_1_1,B_2_1) ).
fof(tsy_c_hBOOL_arg1,hypothesis,
! [B_1_1] :
( hBOOL(ti(bool,B_1_1))
<=> hBOOL(B_1_1) ) ).
fof(tsy_c_member_res,axiom,
! [X_a] : ti(fun(X_a,fun(fun(X_a,bool),bool)),member(X_a)) = member(X_a) ).
fof(tsy_v_P_res,hypothesis,
ti(fun(state,fun(state,bool)),p) = p ).
fof(tsy_v_Q_res,hypothesis,
ti(fun(state,fun(state,bool)),q) = q ).
fof(tsy_v_c_res,hypothesis,
ti(com,c) = c ).
%----Relevant facts (996)
fof(fact_0_empty,axiom,
! [X_a,G_1] : 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_1),bot_bot(fun(hoare_2118899576triple(X_a),bool)))) ).
fof(fact_1_triple_Oinject,axiom,
! [X_a,Fun1,Com_1,Fun2,Fun1_2,Com,Fun2_2] :
( 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),Fun1),Com_1),Fun2) = 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),Fun1_2),Com),Fun2_2)
<=> ( Fun1 = Fun1_2
& Com_1 = Com
& Fun2 = Fun2_2 ) ) ).
fof(fact_2_hoare__sound,axiom,
! [X_a,G_1,Ts] :
( 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_1),Ts))
=> 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_902341502valids(X_a),G_1),Ts)) ) ).
fof(fact_3_cut,axiom,
! [X_a,G_1,G_2,Ts] :
( 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_2),Ts))
=> ( 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_1),G_2))
=> 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_1),Ts)) ) ) ).
fof(fact_4_hoare__derivs_Oinsert,axiom,
! [X_a,Ts,G_1,T_4] :
( 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_1),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)),T_4),bot_bot(fun(hoare_2118899576triple(X_a),bool)))))
=> ( 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_1),Ts))
=> 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_1),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)),T_4),Ts))) ) ) ).
fof(fact_5_derivs__insertD,axiom,
! [X_a,G_1,T_4,Ts] :
( 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_1),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)),T_4),Ts)))
=> ( 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_1),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)),T_4),bot_bot(fun(hoare_2118899576triple(X_a),bool)))))
& 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_1),Ts)) ) ) ).
fof(fact_6_conseq2,axiom,
! [X_a,Qa,G_1,Pa,Ca,Q_4] :
( 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_1),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),Pa),Ca),Q_4)),bot_bot(fun(hoare_2118899576triple(X_a),bool)))))
=> ( ! [Z_4,S_1] :
( hBOOL(hAPP(state,bool,hAPP(X_a,fun(state,bool),Q_4,Z_4),S_1))
=> hBOOL(hAPP(state,bool,hAPP(X_a,fun(state,bool),Qa,Z_4),S_1)) )
=> 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_1),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),Pa),Ca),Qa)),bot_bot(fun(hoare_2118899576triple(X_a),bool))))) ) ) ).
fof(fact_7_conseq1,axiom,
! [X_a,Pa,G_1,P_1,Ca,Qa] :
( 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_1),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),P_1),Ca),Qa)),bot_bot(fun(hoare_2118899576triple(X_a),bool)))))
=> ( ! [Z_4,S_1] :
( hBOOL(hAPP(state,bool,hAPP(X_a,fun(state,bool),Pa,Z_4),S_1))
=> hBOOL(hAPP(state,bool,hAPP(X_a,fun(state,bool),P_1,Z_4),S_1)) )
=> 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_1),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),Pa),Ca),Qa)),bot_bot(fun(hoare_2118899576triple(X_a),bool))))) ) ) ).
fof(fact_8_insertE,axiom,
! [X_a,A_2,B_2,A_3] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_2),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),B_2),A_3)))
=> ( ti(X_a,A_2) != ti(X_a,B_2)
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_2),A_3)) ) ) ).
fof(fact_9_insertCI,axiom,
! [X_a,B_2,A_2,B_3] :
( ( ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_2),B_3))
=> ti(X_a,A_2) = ti(X_a,B_2) )
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_2),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),B_2),B_3))) ) ).
fof(fact_10_conseq12,axiom,
! [X_a,Qa,Pa,G_1,P_1,Ca,Q_4] :
( 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_1),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),P_1),Ca),Q_4)),bot_bot(fun(hoare_2118899576triple(X_a),bool)))))
=> ( ! [Z_4,S_1] :
( hBOOL(hAPP(state,bool,hAPP(X_a,fun(state,bool),Pa,Z_4),S_1))
=> ! [S_2] :
( ! [Z_5] :
( hBOOL(hAPP(state,bool,hAPP(X_a,fun(state,bool),P_1,Z_5),S_1))
=> hBOOL(hAPP(state,bool,hAPP(X_a,fun(state,bool),Q_4,Z_5),S_2)) )
=> hBOOL(hAPP(state,bool,hAPP(X_a,fun(state,bool),Qa,Z_4),S_2)) ) )
=> 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_1),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),Pa),Ca),Qa)),bot_bot(fun(hoare_2118899576triple(X_a),bool))))) ) ) ).
fof(fact_11_emptyE,axiom,
! [X_a,A_2] : ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_2),bot_bot(fun(X_a,bool)))) ).
fof(fact_12_triple_Orecs,axiom,
! [X_b,X_a,F1,Fun1,Com_1,Fun2] : hAPP(hoare_2118899576triple(X_b),X_a,hAPP(fun(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),X_a))),fun(hoare_2118899576triple(X_b),X_a),hoare_1759541758le_rec(X_b,X_a),F1),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_1),Fun2)) = hAPP(fun(X_b,fun(state,bool)),X_a,hAPP(com,fun(fun(X_b,fun(state,bool)),X_a),hAPP(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),X_a)),F1,Fun1),Com_1),Fun2) ).
fof(fact_13_triple_Osimps_I2_J,axiom,
! [X_b,X_a,F1,Fun1,Com_1,Fun2] : hAPP(hoare_2118899576triple(X_b),X_a,hAPP(fun(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),X_a))),fun(hoare_2118899576triple(X_b),X_a),hoare_225284258e_case(X_b,X_a),F1),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_1),Fun2)) = hAPP(fun(X_b,fun(state,bool)),X_a,hAPP(com,fun(fun(X_b,fun(state,bool)),X_a),hAPP(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),X_a)),F1,Fun1),Com_1),Fun2) ).
fof(fact_14_empty__not__insert,axiom,
! [X_a,A_2,A_3] : bot_bot(fun(X_a,bool)) != hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),A_2),A_3) ).
fof(fact_15_insert__not__empty,axiom,
! [X_a,A_2,A_3] : hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),A_2),A_3) != bot_bot(fun(X_a,bool)) ).
fof(fact_16_singleton__iff,axiom,
! [X_a,B_2,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),B_2),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),A_2),bot_bot(fun(X_a,bool)))))
<=> ti(X_a,B_2) = ti(X_a,A_2) ) ).
fof(fact_17_doubleton__eq__iff,axiom,
! [X_a,A_2,B_2,Ca,D_1] :
( hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),A_2),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),B_2),bot_bot(fun(X_a,bool)))) = hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),Ca),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),D_1),bot_bot(fun(X_a,bool))))
<=> ( ( ti(X_a,A_2) = ti(X_a,Ca)
& ti(X_a,B_2) = ti(X_a,D_1) )
| ( ti(X_a,A_2) = ti(X_a,D_1)
& ti(X_a,B_2) = ti(X_a,Ca) ) ) ) ).
fof(fact_18_equals0D,axiom,
! [X_a,A_2,A_3] :
( ti(fun(X_a,bool),A_3) = bot_bot(fun(X_a,bool))
=> ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_2),A_3)) ) ).
fof(fact_19_Collect__empty__eq,axiom,
! [X_a,Pa] :
( hAPP(fun(X_a,bool),fun(X_a,bool),collect(X_a),Pa) = bot_bot(fun(X_a,bool))
<=> ! [X_2] : ~ hBOOL(hAPP(X_a,bool,Pa,X_2)) ) ).
fof(fact_20_empty__iff,axiom,
! [X_a,Ca] : ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Ca),bot_bot(fun(X_a,bool)))) ).
fof(fact_21_empty__Collect__eq,axiom,
! [X_a,Pa] :
( bot_bot(fun(X_a,bool)) = hAPP(fun(X_a,bool),fun(X_a,bool),collect(X_a),Pa)
<=> ! [X_2] : ~ hBOOL(hAPP(X_a,bool,Pa,X_2)) ) ).
fof(fact_22_ex__in__conv,axiom,
! [X_a,A_3] :
( ? [X_2] : hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),A_3))
<=> ti(fun(X_a,bool),A_3) != bot_bot(fun(X_a,bool)) ) ).
fof(fact_23_all__not__in__conv,axiom,
! [X_a,A_3] :
( ! [X_2] : ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),A_3))
<=> ti(fun(X_a,bool),A_3) = bot_bot(fun(X_a,bool)) ) ).
fof(fact_24_empty__def,axiom,
! [X_a] : bot_bot(fun(X_a,bool)) = hAPP(fun(X_a,bool),fun(X_a,bool),collect(X_a),hAPP(bool,fun(X_a,bool),combk(bool,X_a),fFalse)) ).
fof(fact_25_insert__absorb,axiom,
! [X_a,A_2,A_3] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_2),A_3))
=> hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),A_2),A_3) = ti(fun(X_a,bool),A_3) ) ).
fof(fact_26_insertI2,axiom,
! [X_a,B_2,A_2,B_3] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_2),B_3))
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_2),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),B_2),B_3))) ) ).
fof(fact_27_insert__ident,axiom,
! [X_a,B_3,X_1,A_3] :
( ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),A_3))
=> ( ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),B_3))
=> ( hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_1),A_3) = hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_1),B_3)
<=> ti(fun(X_a,bool),A_3) = ti(fun(X_a,bool),B_3) ) ) ) ).
fof(fact_28_insert__code,axiom,
! [X_a,Y_1,A_3,X_1] :
( hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),Y_1),A_3),X_1))
<=> ( ti(X_a,Y_1) = ti(X_a,X_1)
| hBOOL(hAPP(X_a,bool,A_3,X_1)) ) ) ).
fof(fact_29_insert__iff,axiom,
! [X_a,A_2,B_2,A_3] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_2),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),B_2),A_3)))
<=> ( ti(X_a,A_2) = ti(X_a,B_2)
| hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_2),A_3)) ) ) ).
fof(fact_30_insert__commute,axiom,
! [X_a,X_1,Y_1,A_3] : hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_1),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),Y_1),A_3)) = hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),Y_1),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_1),A_3)) ).
fof(fact_31_insert__absorb2,axiom,
! [X_a,X_1,A_3] : hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_1),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_1),A_3)) = hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_1),A_3) ).
fof(fact_32_insert__Collect,axiom,
! [X_a,A_2,Pa] : hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),A_2),hAPP(fun(X_a,bool),fun(X_a,bool),collect(X_a),Pa)) = hAPP(fun(X_a,bool),fun(X_a,bool),collect(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,fun(bool,bool)),fun(fun(X_a,bool),fun(X_a,bool)),combs(X_a,bool,bool),hAPP(fun(X_a,bool),fun(X_a,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_a,bool),fun(X_a,fun(bool,bool))),combb(bool,fun(bool,bool),X_a),fimplies),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(bool,bool),fun(fun(X_a,bool),fun(X_a,bool)),combb(bool,bool,X_a),fNot),hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,fun(X_a,bool)),fun(X_a,fun(X_a,bool)),combc(X_a,X_a,bool),fequal(X_a)),A_2)))),Pa)) ).
fof(fact_33_insert__compr,axiom,
! [X_a,A_2,B_3] : hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),A_2),B_3) = hAPP(fun(X_a,bool),fun(X_a,bool),collect(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,fun(bool,bool)),fun(fun(X_a,bool),fun(X_a,bool)),combs(X_a,bool,bool),hAPP(fun(X_a,bool),fun(X_a,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_a,bool),fun(X_a,fun(bool,bool))),combb(bool,fun(bool,bool),X_a),fdisj),hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,fun(X_a,bool)),fun(X_a,fun(X_a,bool)),combc(X_a,X_a,bool),fequal(X_a)),A_2))),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,fun(fun(X_a,bool),bool)),fun(fun(X_a,bool),fun(X_a,bool)),combc(X_a,fun(X_a,bool),bool),member(X_a)),B_3))) ).
fof(fact_34_insertI1,axiom,
! [X_a,A_2,B_3] : hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_2),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),A_2),B_3))) ).
fof(fact_35_singleton__inject,axiom,
! [X_a,A_2,B_2] :
( hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),A_2),bot_bot(fun(X_a,bool))) = hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),B_2),bot_bot(fun(X_a,bool)))
=> ti(X_a,A_2) = ti(X_a,B_2) ) ).
fof(fact_36_singletonE,axiom,
! [X_a,B_2,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),B_2),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),A_2),bot_bot(fun(X_a,bool)))))
=> ti(X_a,B_2) = ti(X_a,A_2) ) ).
fof(fact_37_the__elem__eq,axiom,
! [X_a,X_1] : hAPP(fun(X_a,bool),X_a,the_elem(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_1),bot_bot(fun(X_a,bool)))) = ti(X_a,X_1) ).
fof(fact_38_bot__apply,axiom,
! [X_b,X_a] :
( bot(X_a)
=> ! [X_1] : hAPP(X_b,X_a,bot_bot(fun(X_b,X_a)),X_1) = bot_bot(X_a) ) ).
fof(fact_39_bot__fun__def,axiom,
! [X_a,X_b] :
( bot(X_b)
=> ! [X_2] : hAPP(X_a,X_b,bot_bot(fun(X_a,X_b)),X_2) = bot_bot(X_b) ) ).
fof(fact_40_hoare__derivs_OSkip,axiom,
! [X_a,G_1,Pa] : 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_1),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),Pa),skip),Pa)),bot_bot(fun(hoare_2118899576triple(X_a),bool))))) ).
fof(fact_41_Comp,axiom,
! [X_a,D_1,R_3,G_1,Pa,Ca,Qa] :
( 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_1),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),Pa),Ca),Qa)),bot_bot(fun(hoare_2118899576triple(X_a),bool)))))
=> ( 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_1),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),Qa),D_1),R_3)),bot_bot(fun(hoare_2118899576triple(X_a),bool)))))
=> 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_1),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),Pa),hAPP(com,com,hAPP(com,fun(com,com),semi,Ca),D_1)),R_3)),bot_bot(fun(hoare_2118899576triple(X_a),bool))))) ) ) ).
fof(fact_42_triple_Oexhaust,axiom,
! [X_a,Y_1] :
~ ! [Fun1_1,Com_2,Fun2_1] : Y_1 != 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),Fun1_1),Com_2),Fun2_1) ).
fof(fact_43_Set_Oset__insert,axiom,
! [X_a,X_1,A_3] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),A_3))
=> ~ ! [B_4] :
( ti(fun(X_a,bool),A_3) = hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_1),B_4)
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),B_4)) ) ) ).
fof(fact_44_mk__disjoint__insert,axiom,
! [X_a,A_2,A_3] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_2),A_3))
=> ? [B_4] :
( ti(fun(X_a,bool),A_3) = hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),A_2),B_4)
& ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_2),B_4)) ) ) ).
fof(fact_45_equals0I,axiom,
! [X_a,A_3] :
( ! [Y_2] : ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Y_2),A_3))
=> ti(fun(X_a,bool),A_3) = bot_bot(fun(X_a,bool)) ) ).
fof(fact_46_conseq,axiom,
! [X_a,Qa,G_1,Ca,Pa] :
( ! [Z_4,S_1] :
( hBOOL(hAPP(state,bool,hAPP(X_a,fun(state,bool),Pa,Z_4),S_1))
=> ? [P_2,Q_3] :
( 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_1),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),P_2),Ca),Q_3)),bot_bot(fun(hoare_2118899576triple(X_a),bool)))))
& ! [S_2] :
( ! [Z_5] :
( hBOOL(hAPP(state,bool,hAPP(X_a,fun(state,bool),P_2,Z_5),S_1))
=> hBOOL(hAPP(state,bool,hAPP(X_a,fun(state,bool),Q_3,Z_5),S_2)) )
=> hBOOL(hAPP(state,bool,hAPP(X_a,fun(state,bool),Qa,Z_4),S_2)) ) ) )
=> 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_1),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),Pa),Ca),Qa)),bot_bot(fun(hoare_2118899576triple(X_a),bool))))) ) ).
fof(fact_47_nonempty__iff,axiom,
! [X_a,A_3] :
( ti(fun(X_a,bool),A_3) != bot_bot(fun(X_a,bool))
<=> ? [X_2,B_4] :
( ti(fun(X_a,bool),A_3) = hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_2),B_4)
& ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),B_4)) ) ) ).
fof(fact_48_bot__empty__eq,axiom,
! [X_a,X_2] :
( hBOOL(hAPP(X_a,bool,bot_bot(fun(X_a,bool)),X_2))
<=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),bot_bot(fun(X_a,bool)))) ) ).
fof(fact_49_com_Osimps_I13_J,axiom,
! [Com1_3,Com2_3] : hAPP(com,com,hAPP(com,fun(com,com),semi,Com1_3),Com2_3) != skip ).
fof(fact_50_com_Osimps_I12_J,axiom,
! [Com1_3,Com2_3] : skip != hAPP(com,com,hAPP(com,fun(com,com),semi,Com1_3),Com2_3) ).
fof(fact_51_com_Osimps_I3_J,axiom,
! [Com1,Com2,Com1_1,Com2_1] :
( hAPP(com,com,hAPP(com,fun(com,com),semi,Com1),Com2) = hAPP(com,com,hAPP(com,fun(com,com),semi,Com1_1),Com2_1)
<=> ( Com1 = Com1_1
& Com2 = Com2_1 ) ) ).
fof(fact_52_fold1Set__sing,axiom,
! [X_a,F,A_2,B_2] :
( hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,bool),fun(X_a,bool)),finite_fold1Set(X_a),F),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),A_2),bot_bot(fun(X_a,bool)))),B_2))
<=> ti(X_a,A_2) = ti(X_a,B_2) ) ).
fof(fact_53_folding__one_Osingleton,axiom,
! [X_a,X_1,F,F_1] :
( hBOOL(hAPP(fun(fun(X_a,bool),X_a),bool,hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(fun(X_a,bool),X_a),bool),finite_folding_one(X_a),F),F_1))
=> hAPP(fun(X_a,bool),X_a,F_1,hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_1),bot_bot(fun(X_a,bool)))) = ti(X_a,X_1) ) ).
fof(fact_54_fold1__singleton,axiom,
! [X_a,F,A_2] : hAPP(fun(X_a,bool),X_a,hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,bool),X_a),finite_fold1(X_a),F),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),A_2),bot_bot(fun(X_a,bool)))) = ti(X_a,A_2) ).
fof(fact_55_fold1__singleton__def,axiom,
! [X_a,A_2,G,F] :
( G = hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,bool),X_a),finite_fold1(X_a),F)
=> hAPP(fun(X_a,bool),X_a,G,hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),A_2),bot_bot(fun(X_a,bool)))) = ti(X_a,A_2) ) ).
fof(fact_56_empty__fold1SetE,axiom,
! [X_a,F,X_1] : ~ hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,bool),fun(X_a,bool)),finite_fold1Set(X_a),F),bot_bot(fun(X_a,bool))),X_1)) ).
fof(fact_57_fold1Set__nonempty,axiom,
! [X_a,F,A_3,X_1] :
( hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,bool),fun(X_a,bool)),finite_fold1Set(X_a),F),A_3),X_1))
=> ti(fun(X_a,bool),A_3) != bot_bot(fun(X_a,bool)) ) ).
fof(fact_58_subset__singletonD,axiom,
! [X_a,A_3,X_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_3),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_1),bot_bot(fun(X_a,bool)))))
=> ( ti(fun(X_a,bool),A_3) = bot_bot(fun(X_a,bool))
| ti(fun(X_a,bool),A_3) = hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_1),bot_bot(fun(X_a,bool))) ) ) ).
fof(fact_59_folding__image__simple_Oempty,axiom,
! [X_b,X_a,F,Z_1,G,F_1] :
( hBOOL(hAPP(fun(fun(X_b,bool),X_a),bool,hAPP(fun(X_b,X_a),fun(fun(fun(X_b,bool),X_a),bool),hAPP(X_a,fun(fun(X_b,X_a),fun(fun(fun(X_b,bool),X_a),bool)),hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(fun(X_b,X_a),fun(fun(fun(X_b,bool),X_a),bool))),finite1357897459simple(X_a,X_b),F),Z_1),G),F_1))
=> hAPP(fun(X_b,bool),X_a,F_1,bot_bot(fun(X_b,bool))) = ti(X_a,Z_1) ) ).
fof(fact_60_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_61_equalityI,axiom,
! [X_a,A_3,B_3] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_3),B_3))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),B_3),A_3))
=> ti(fun(X_a,bool),A_3) = ti(fun(X_a,bool),B_3) ) ) ).
fof(fact_62_subsetD,axiom,
! [X_a,Ca,A_3,B_3] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_3),B_3))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Ca),A_3))
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Ca),B_3)) ) ) ).
fof(fact_63_empty__subsetI,axiom,
! [X_a,A_3] : hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),bot_bot(fun(X_a,bool))),A_3)) ).
fof(fact_64_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_65_le__funE,axiom,
! [X_a,X_b] :
( ord(X_b)
=> ! [X_1,F,G] :
( hBOOL(hAPP(fun(X_a,X_b),bool,hAPP(fun(X_a,X_b),fun(fun(X_a,X_b),bool),ord_less_eq(fun(X_a,X_b)),F),G))
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(X_a,X_b,F,X_1)),hAPP(X_a,X_b,G,X_1))) ) ) ).
fof(fact_66_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_67_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_68_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_69_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_70_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_71_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_72_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_73_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_74_predicate1D,axiom,
! [X_a,X_1,Pa,Qa] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),Pa),Qa))
=> ( hBOOL(hAPP(X_a,bool,Pa,X_1))
=> hBOOL(hAPP(X_a,bool,Qa,X_1)) ) ) ).
fof(fact_75_ext,axiom,
! [X_a,X_b,F,G] :
( ! [X_2] : hAPP(X_a,X_b,F,X_2) = hAPP(X_a,X_b,G,X_2)
=> ti(fun(X_a,X_b),F) = ti(fun(X_a,X_b),G) ) ).
fof(fact_76_mem__def,axiom,
! [X_a,X_1,A_3] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),A_3))
<=> hBOOL(hAPP(X_a,bool,A_3,X_1)) ) ).
fof(fact_77_Collect__def,axiom,
! [X_a,Pa] : hAPP(fun(X_a,bool),fun(X_a,bool),collect(X_a),Pa) = ti(fun(X_a,bool),Pa) ).
fof(fact_78_order__antisym__conv,axiom,
! [X_a] :
( order(X_a)
=> ! [Y_1,X_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y_1),X_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_1),Y_1))
<=> ti(X_a,X_1) = ti(X_a,Y_1) ) ) ) ).
fof(fact_79_le__funD,axiom,
! [X_a,X_b] :
( ord(X_b)
=> ! [X_1,F,G] :
( hBOOL(hAPP(fun(X_a,X_b),bool,hAPP(fun(X_a,X_b),fun(fun(X_a,X_b),bool),ord_less_eq(fun(X_a,X_b)),F),G))
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(X_a,X_b,F,X_1)),hAPP(X_a,X_b,G,X_1))) ) ) ).
fof(fact_80_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_81_rev__predicate1D,axiom,
! [X_a,Qa,Pa,X_1] :
( hBOOL(hAPP(X_a,bool,Pa,X_1))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),Pa),Qa))
=> hBOOL(hAPP(X_a,bool,Qa,X_1)) ) ) ).
fof(fact_82_order__eq__iff,axiom,
! [X_a] :
( order(X_a)
=> ! [X_1,Y_1] :
( ti(X_a,X_1) = ti(X_a,Y_1)
<=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_1),Y_1))
& hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y_1),X_1)) ) ) ) ).
fof(fact_83_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_84_le__fun__def,axiom,
! [X_a,X_b] :
( ord(X_b)
=> ! [F,G] :
( hBOOL(hAPP(fun(X_a,X_b),bool,hAPP(fun(X_a,X_b),fun(fun(X_a,X_b),bool),ord_less_eq(fun(X_a,X_b)),F),G))
<=> ! [X_2] : hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(X_a,X_b,F,X_2)),hAPP(X_a,X_b,G,X_2))) ) ) ).
fof(fact_85_equalityE,axiom,
! [X_a,A_3,B_3] :
( ti(fun(X_a,bool),A_3) = ti(fun(X_a,bool),B_3)
=> ~ ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_3),B_3))
=> ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),B_3),A_3)) ) ) ).
fof(fact_86_subset__trans,axiom,
! [X_a,C_1,A_3,B_3] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_3),B_3))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),B_3),C_1))
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_3),C_1)) ) ) ).
fof(fact_87_set__mp,axiom,
! [X_a,X_1,A_3,B_3] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_3),B_3))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),A_3))
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),B_3)) ) ) ).
fof(fact_88_set__rev__mp,axiom,
! [X_a,B_3,X_1,A_3] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),A_3))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_3),B_3))
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),B_3)) ) ) ).
fof(fact_89_in__mono,axiom,
! [X_a,X_1,A_3,B_3] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_3),B_3))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),A_3))
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),B_3)) ) ) ).
fof(fact_90_equalityD2,axiom,
! [X_a,A_3,B_3] :
( ti(fun(X_a,bool),A_3) = ti(fun(X_a,bool),B_3)
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),B_3),A_3)) ) ).
fof(fact_91_equalityD1,axiom,
! [X_a,A_3,B_3] :
( ti(fun(X_a,bool),A_3) = ti(fun(X_a,bool),B_3)
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_3),B_3)) ) ).
fof(fact_92_set__eq__subset,axiom,
! [X_a,A_3,B_3] :
( ti(fun(X_a,bool),A_3) = ti(fun(X_a,bool),B_3)
<=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_3),B_3))
& hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),B_3),A_3)) ) ) ).
fof(fact_93_subset__refl,axiom,
! [X_a,A_3] : hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_3),A_3)) ).
fof(fact_94_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_95_bot__unique,axiom,
! [X_a] :
( bot(X_a)
=> ! [A_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_2),bot_bot(X_a)))
<=> ti(X_a,A_2) = bot_bot(X_a) ) ) ).
fof(fact_96_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_97_subset__empty,axiom,
! [X_a,A_3] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_3),bot_bot(fun(X_a,bool))))
<=> ti(fun(X_a,bool),A_3) = bot_bot(fun(X_a,bool)) ) ).
fof(fact_98_insert__mono,axiom,
! [X_a,A_2,C_1,D_3] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),C_1),D_3))
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),A_2),C_1)),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),A_2),D_3))) ) ).
fof(fact_99_subset__insertI2,axiom,
! [X_a,B_2,A_3,B_3] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_3),B_3))
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_3),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),B_2),B_3))) ) ).
fof(fact_100_subset__insert,axiom,
! [X_a,B_3,X_1,A_3] :
( ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),A_3))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_3),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_1),B_3)))
<=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_3),B_3)) ) ) ).
fof(fact_101_insert__subset,axiom,
! [X_a,X_1,A_3,B_3] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_1),A_3)),B_3))
<=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),B_3))
& hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_3),B_3)) ) ) ).
fof(fact_102_subset__insertI,axiom,
! [X_a,B_3,A_2] : hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),B_3),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),A_2),B_3))) ).
fof(fact_103_thin,axiom,
! [X_a,G_1,G_2,Ts] :
( 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_2),Ts))
=> ( hBOOL(hAPP(fun(hoare_2118899576triple(X_a),bool),bool,hAPP(fun(hoare_2118899576triple(X_a),bool),fun(fun(hoare_2118899576triple(X_a),bool),bool),ord_less_eq(fun(hoare_2118899576triple(X_a),bool)),G_2),G_1))
=> 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_1),Ts)) ) ) ).
fof(fact_104_weaken,axiom,
! [X_a,Ts,G_1,Ts_1] :
( 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_1),Ts_1))
=> ( hBOOL(hAPP(fun(hoare_2118899576triple(X_a),bool),bool,hAPP(fun(hoare_2118899576triple(X_a),bool),fun(fun(hoare_2118899576triple(X_a),bool),bool),ord_less_eq(fun(hoare_2118899576triple(X_a),bool)),Ts),Ts_1))
=> 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_1),Ts)) ) ) ).
fof(fact_105_asm,axiom,
! [X_a,Ts,G_1] :
( hBOOL(hAPP(fun(hoare_2118899576triple(X_a),bool),bool,hAPP(fun(hoare_2118899576triple(X_a),bool),fun(fun(hoare_2118899576triple(X_a),bool),bool),ord_less_eq(fun(hoare_2118899576triple(X_a),bool)),Ts),G_1))
=> 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_1),Ts)) ) ).
fof(fact_106_subsetI,axiom,
! [X_a,B_3,A_3] :
( ! [X_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),A_3))
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),B_3)) )
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_3),B_3)) ) ).
fof(fact_107_le__funI,axiom,
! [X_a,X_b] :
( ord(X_b)
=> ! [F,G] :
( ! [X_2] : hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(X_a,X_b,F,X_2)),hAPP(X_a,X_b,G,X_2)))
=> hBOOL(hAPP(fun(X_a,X_b),bool,hAPP(fun(X_a,X_b),fun(fun(X_a,X_b),bool),ord_less_eq(fun(X_a,X_b)),F),G)) ) ) ).
fof(fact_108_predicate1I,axiom,
! [X_a,Qa,Pa] :
( ! [X_2] :
( hBOOL(hAPP(X_a,bool,Pa,X_2))
=> hBOOL(hAPP(X_a,bool,Qa,X_2)) )
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),Pa),Qa)) ) ).
fof(fact_109_Collect__mono,axiom,
! [X_a,Qa,Pa] :
( ! [X_2] :
( hBOOL(hAPP(X_a,bool,Pa,X_2))
=> hBOOL(hAPP(X_a,bool,Qa,X_2)) )
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),hAPP(fun(X_a,bool),fun(X_a,bool),collect(X_a),Pa)),hAPP(fun(X_a,bool),fun(X_a,bool),collect(X_a),Qa))) ) ).
fof(fact_110_xt2,axiom,
! [X_b,X_a] :
( ( order(X_a)
& ord(X_b) )
=> ! [Ca,F,B_2,A_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_b,X_a,F,B_2)),A_2))
=> ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),Ca),B_2))
=> ( ! [X_2,Y_2] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),Y_2),X_2))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_b,X_a,F,Y_2)),hAPP(X_b,X_a,F,X_2))) )
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_b,X_a,F,Ca)),A_2)) ) ) ) ) ).
fof(fact_111_xt1_I15_J,axiom,
! [X_a] :
( order(X_a)
=> ! [Ca,A_2,F,B_2] :
( ti(X_a,A_2) = hAPP(X_a,X_a,F,B_2)
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Ca),B_2))
=> ( ! [X_2,Y_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y_2),X_2))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,F,Y_2)),hAPP(X_a,X_a,F,X_2))) )
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,F,Ca)),A_2)) ) ) ) ) ).
fof(fact_112_order__subst1,axiom,
! [X_a,X_b] :
( ( order(X_b)
& order(X_a) )
=> ! [Ca,A_2,F,B_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_2),hAPP(X_b,X_a,F,B_2)))
=> ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),B_2),Ca))
=> ( ! [X_2,Y_2] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),X_2),Y_2))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_b,X_a,F,X_2)),hAPP(X_b,X_a,F,Y_2))) )
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_2),hAPP(X_b,X_a,F,Ca))) ) ) ) ) ).
fof(fact_113_ord__eq__le__subst,axiom,
! [X_a,X_b] :
( ( ord(X_b)
& ord(X_a) )
=> ! [Ca,A_2,F,B_2] :
( ti(X_a,A_2) = hAPP(X_b,X_a,F,B_2)
=> ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),B_2),Ca))
=> ( ! [X_2,Y_2] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),X_2),Y_2))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_b,X_a,F,X_2)),hAPP(X_b,X_a,F,Y_2))) )
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_2),hAPP(X_b,X_a,F,Ca))) ) ) ) ) ).
fof(fact_114_order__subst2,axiom,
! [X_a,X_b] :
( ( order(X_b)
& order(X_a) )
=> ! [F,Ca,A_2,B_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_2),B_2))
=> ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(X_a,X_b,F,B_2)),Ca))
=> ( ! [X_2,Y_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_2),Y_2))
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(X_a,X_b,F,X_2)),hAPP(X_a,X_b,F,Y_2))) )
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(X_a,X_b,F,A_2)),Ca)) ) ) ) ) ).
fof(fact_115_ord__le__eq__subst,axiom,
! [X_a,X_b] :
( ( ord(X_b)
& ord(X_a) )
=> ! [F,Ca,A_2,B_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_2),B_2))
=> ( hAPP(X_a,X_b,F,B_2) = ti(X_b,Ca)
=> ( ! [X_2,Y_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_2),Y_2))
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(X_a,X_b,F,X_2)),hAPP(X_a,X_b,F,Y_2))) )
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(X_a,X_b,F,A_2)),Ca)) ) ) ) ) ).
fof(fact_116_xt3,axiom,
! [X_b,X_a] :
( ( order(X_a)
& order(X_b) )
=> ! [Ca,F,B_2,A_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B_2),A_2))
=> ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),Ca),hAPP(X_a,X_b,F,B_2)))
=> ( ! [X_2,Y_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y_2),X_2))
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(X_a,X_b,F,Y_2)),hAPP(X_a,X_b,F,X_2))) )
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),Ca),hAPP(X_a,X_b,F,A_2))) ) ) ) ) ).
fof(fact_117_xt1_I16_J,axiom,
! [X_a] :
( order(X_a)
=> ! [F,Ca,B_2,A_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B_2),A_2))
=> ( hAPP(X_a,X_a,F,B_2) = ti(X_a,Ca)
=> ( ! [X_2,Y_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y_2),X_2))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,F,Y_2)),hAPP(X_a,X_a,F,X_2))) )
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Ca),hAPP(X_a,X_a,F,A_2))) ) ) ) ) ).
fof(fact_118_fold1Set_Ointros,axiom,
! [X_a,F,A_2,A_3,X_1] :
( hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(fun(X_a,bool),fun(X_a,bool))),finite_fold_graph(X_a,X_a),F),A_2),A_3),X_1))
=> ( ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_2),A_3))
=> hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,bool),fun(X_a,bool)),finite_fold1Set(X_a),F),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),A_2),A_3)),X_1)) ) ) ).
fof(fact_119_folding__one_Oinsert,axiom,
! [X_a,X_1,A_3,F,F_1] :
( hBOOL(hAPP(fun(fun(X_a,bool),X_a),bool,hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(fun(X_a,bool),X_a),bool),finite_folding_one(X_a),F),F_1))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_3))
=> ( ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),A_3))
=> ( ti(fun(X_a,bool),A_3) != bot_bot(fun(X_a,bool))
=> hAPP(fun(X_a,bool),X_a,F_1,hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_1),A_3)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),F,X_1),hAPP(fun(X_a,bool),X_a,F_1,A_3)) ) ) ) ) ).
fof(fact_120_Powp__mono,axiom,
! [X_a,A_3,B_3] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_3),B_3))
=> hBOOL(hAPP(fun(fun(X_a,bool),bool),bool,hAPP(fun(fun(X_a,bool),bool),fun(fun(fun(X_a,bool),bool),bool),ord_less_eq(fun(fun(X_a,bool),bool)),hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),powp(X_a),A_3)),hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),powp(X_a),B_3))) ) ).
fof(fact_121_subset__insert__iff,axiom,
! [X_a,A_3,X_1,B_3] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_3),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_1),B_3)))
<=> ( ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),A_3))
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_3),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_1),bot_bot(fun(X_a,bool))))),B_3)) )
& ( ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),A_3))
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_3),B_3)) ) ) ) ).
fof(fact_122_diff__single__insert,axiom,
! [X_a,A_3,X_1,B_3] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_3),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_1),bot_bot(fun(X_a,bool))))),B_3))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),A_3))
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_3),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_1),B_3))) ) ) ).
fof(fact_123_folding__one_Oeq__fold,axiom,
! [X_a,A_3,F,F_1] :
( hBOOL(hAPP(fun(fun(X_a,bool),X_a),bool,hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(fun(X_a,bool),X_a),bool),finite_folding_one(X_a),F),F_1))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_3))
=> hAPP(fun(X_a,bool),X_a,F_1,A_3) = hAPP(fun(X_a,bool),X_a,hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,bool),X_a),finite_fold1(X_a),F),A_3) ) ) ).
fof(fact_124_folding__one_Oclosed,axiom,
! [X_a,A_3,F,F_1] :
( hBOOL(hAPP(fun(fun(X_a,bool),X_a),bool,hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(fun(X_a,bool),X_a),bool),finite_folding_one(X_a),F),F_1))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_3))
=> ( ti(fun(X_a,bool),A_3) != bot_bot(fun(X_a,bool))
=> ( ! [X_2,Y_2] : hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),F,X_2),Y_2)),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_2),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),Y_2),bot_bot(fun(X_a,bool))))))
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),hAPP(fun(X_a,bool),X_a,F_1,A_3)),A_3)) ) ) ) ) ).
fof(fact_125_DiffE,axiom,
! [X_a,Ca,A_3,B_3] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Ca),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_3),B_3)))
=> ~ ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Ca),A_3))
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Ca),B_3)) ) ) ).
fof(fact_126_DiffI,axiom,
! [X_a,B_3,Ca,A_3] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Ca),A_3))
=> ( ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Ca),B_3))
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Ca),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_3),B_3))) ) ) ).
fof(fact_127_finite_OemptyI,axiom,
! [X_a] : hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),bot_bot(fun(X_a,bool)))) ).
fof(fact_128_finite_OinsertI,axiom,
! [X_a,A_2,A_3] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_3))
=> hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),A_2),A_3))) ) ).
fof(fact_129_finite__Diff,axiom,
! [X_a,B_3,A_3] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_3))
=> hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_3),B_3))) ) ).
fof(fact_130_finite__Diff__insert,axiom,
! [X_a,A_3,A_2,B_3] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_3),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),A_2),B_3))))
<=> hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_3),B_3))) ) ).
fof(fact_131_finite__Diff2,axiom,
! [X_a,A_3,B_3] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),B_3))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_3),B_3)))
<=> hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_3)) ) ) ).
fof(fact_132_finite__code,axiom,
! [X_a] :
( finite_finite(X_a)
=> ! [A_3] : hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_3)) ) ).
fof(fact_133_finite,axiom,
! [X_a] :
( finite_finite(X_a)
=> ! [A_3] : hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_3)) ) ).
fof(fact_134_DiffD2,axiom,
! [X_a,Ca,A_3,B_3] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Ca),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_3),B_3)))
=> ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Ca),B_3)) ) ).
fof(fact_135_DiffD1,axiom,
! [X_a,Ca,A_3,B_3] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Ca),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_3),B_3)))
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Ca),A_3)) ) ).
fof(fact_136_Diff__idemp,axiom,
! [X_a,A_3,B_3] : hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_3),B_3)),B_3) = hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_3),B_3) ).
fof(fact_137_Diff__iff,axiom,
! [X_a,Ca,A_3,B_3] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Ca),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_3),B_3)))
<=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Ca),A_3))
& ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Ca),B_3)) ) ) ).
fof(fact_138_set__diff__eq,axiom,
! [X_a,A_3,B_3] : hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_3),B_3) = hAPP(fun(X_a,bool),fun(X_a,bool),collect(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,fun(bool,bool)),fun(fun(X_a,bool),fun(X_a,bool)),combs(X_a,bool,bool),hAPP(fun(X_a,bool),fun(X_a,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_a,bool),fun(X_a,fun(bool,bool))),combb(bool,fun(bool,bool),X_a),fconj),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,fun(fun(X_a,bool),bool)),fun(fun(X_a,bool),fun(X_a,bool)),combc(X_a,fun(X_a,bool),bool),member(X_a)),A_3))),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(bool,bool),fun(fun(X_a,bool),fun(X_a,bool)),combb(bool,bool,X_a),fNot),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,fun(fun(X_a,bool),bool)),fun(fun(X_a,bool),fun(X_a,bool)),combc(X_a,fun(X_a,bool),bool),member(X_a)),B_3)))) ).
fof(fact_139_Diff__cancel,axiom,
! [X_a,A_3] : hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_3),A_3) = bot_bot(fun(X_a,bool)) ).
fof(fact_140_Diff__empty,axiom,
! [X_a,A_3] : hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_3),bot_bot(fun(X_a,bool))) = ti(fun(X_a,bool),A_3) ).
fof(fact_141_empty__Diff,axiom,
! [X_a,A_3] : hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),bot_bot(fun(X_a,bool))),A_3) = bot_bot(fun(X_a,bool)) ).
fof(fact_142_insert__Diff1,axiom,
! [X_a,A_3,X_1,B_3] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),B_3))
=> hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_1),A_3)),B_3) = hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_3),B_3) ) ).
fof(fact_143_insert__Diff__if,axiom,
! [X_a,A_3,X_1,B_3] :
( ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),B_3))
=> hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_1),A_3)),B_3) = hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_3),B_3) )
& ( ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),B_3))
=> hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_1),A_3)),B_3) = hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_1),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_3),B_3)) ) ) ).
fof(fact_144_double__diff,axiom,
! [X_a,C_1,A_3,B_3] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_3),B_3))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),B_3),C_1))
=> hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),B_3),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),C_1),A_3)) = ti(fun(X_a,bool),A_3) ) ) ).
fof(fact_145_Diff__mono,axiom,
! [X_a,D_3,B_3,A_3,C_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_3),C_1))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),D_3),B_3))
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_3),B_3)),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),C_1),D_3))) ) ) ).
fof(fact_146_Diff__subset,axiom,
! [X_a,A_3,B_3] : hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_3),B_3)),A_3)) ).
fof(fact_147_finite__insert,axiom,
! [X_a,A_2,A_3] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),A_2),A_3)))
<=> hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_3)) ) ).
fof(fact_148_finite__subset,axiom,
! [X_a,A_3,B_3] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_3),B_3))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),B_3))
=> hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_3)) ) ) ).
fof(fact_149_rev__finite__subset,axiom,
! [X_a,A_3,B_3] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),B_3))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_3),B_3))
=> hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_3)) ) ) ).
fof(fact_150_empty__fold__graphE,axiom,
! [X_b,X_a,F,Z_1,X_1] :
( hBOOL(hAPP(X_b,bool,hAPP(fun(X_a,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_a,bool),fun(X_b,bool)),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(fun(X_a,bool),fun(X_b,bool))),finite_fold_graph(X_a,X_b),F),Z_1),bot_bot(fun(X_a,bool))),X_1))
=> ti(X_b,X_1) = ti(X_b,Z_1) ) ).
fof(fact_151_fold__graph_OemptyI,axiom,
! [X_b,X_a,F,Z_1] : hBOOL(hAPP(X_b,bool,hAPP(fun(X_a,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_a,bool),fun(X_b,bool)),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(fun(X_a,bool),fun(X_b,bool))),finite_fold_graph(X_a,X_b),F),Z_1),bot_bot(fun(X_a,bool))),Z_1)) ).
fof(fact_152_fold__graph_OinsertI,axiom,
! [X_b,X_a,F,Z_1,Y_1,X_1,A_3] :
( ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),A_3))
=> ( hBOOL(hAPP(X_b,bool,hAPP(fun(X_a,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_a,bool),fun(X_b,bool)),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(fun(X_a,bool),fun(X_b,bool))),finite_fold_graph(X_a,X_b),F),Z_1),A_3),Y_1))
=> hBOOL(hAPP(X_b,bool,hAPP(fun(X_a,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_a,bool),fun(X_b,bool)),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(fun(X_a,bool),fun(X_b,bool))),finite_fold_graph(X_a,X_b),F),Z_1),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_1),A_3)),hAPP(X_b,X_b,hAPP(X_a,fun(X_b,X_b),F,X_1),Y_1))) ) ) ).
fof(fact_153_folding__image__simple_Oremove,axiom,
! [X_b,X_a,X_1,A_3,F,Z_1,G,F_1] :
( hBOOL(hAPP(fun(fun(X_b,bool),X_a),bool,hAPP(fun(X_b,X_a),fun(fun(fun(X_b,bool),X_a),bool),hAPP(X_a,fun(fun(X_b,X_a),fun(fun(fun(X_b,bool),X_a),bool)),hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(fun(X_b,X_a),fun(fun(fun(X_b,bool),X_a),bool))),finite1357897459simple(X_a,X_b),F),Z_1),G),F_1))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_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_a,F_1,A_3) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),F,hAPP(X_b,X_a,G,X_1)),hAPP(fun(X_b,bool),X_a,F_1,hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_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_154_folding__image__simple_Oinsert__remove,axiom,
! [X_b,X_a,X_1,A_3,F,Z_1,G,F_1] :
( hBOOL(hAPP(fun(fun(X_b,bool),X_a),bool,hAPP(fun(X_b,X_a),fun(fun(fun(X_b,bool),X_a),bool),hAPP(X_a,fun(fun(X_b,X_a),fun(fun(fun(X_b,bool),X_a),bool)),hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(fun(X_b,X_a),fun(fun(fun(X_b,bool),X_a),bool))),finite1357897459simple(X_a,X_b),F),Z_1),G),F_1))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> hAPP(fun(X_b,bool),X_a,F_1,hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_1),A_3)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),F,hAPP(X_b,X_a,G,X_1)),hAPP(fun(X_b,bool),X_a,F_1,hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_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_155_folding__one_Oremove,axiom,
! [X_a,X_1,A_3,F,F_1] :
( hBOOL(hAPP(fun(fun(X_a,bool),X_a),bool,hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(fun(X_a,bool),X_a),bool),finite_folding_one(X_a),F),F_1))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_3))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),A_3))
=> ( ( hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_3),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_1),bot_bot(fun(X_a,bool)))) = bot_bot(fun(X_a,bool))
=> hAPP(fun(X_a,bool),X_a,F_1,A_3) = ti(X_a,X_1) )
& ( hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_3),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_1),bot_bot(fun(X_a,bool)))) != bot_bot(fun(X_a,bool))
=> hAPP(fun(X_a,bool),X_a,F_1,A_3) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),F,X_1),hAPP(fun(X_a,bool),X_a,F_1,hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_3),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_1),bot_bot(fun(X_a,bool)))))) ) ) ) ) ) ).
fof(fact_156_folding__one_Oinsert__remove,axiom,
! [X_a,X_1,A_3,F,F_1] :
( hBOOL(hAPP(fun(fun(X_a,bool),X_a),bool,hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(fun(X_a,bool),X_a),bool),finite_folding_one(X_a),F),F_1))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_3))
=> ( ( hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_3),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_1),bot_bot(fun(X_a,bool)))) = bot_bot(fun(X_a,bool))
=> hAPP(fun(X_a,bool),X_a,F_1,hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_1),A_3)) = ti(X_a,X_1) )
& ( hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_3),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_1),bot_bot(fun(X_a,bool)))) != bot_bot(fun(X_a,bool))
=> hAPP(fun(X_a,bool),X_a,F_1,hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_1),A_3)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),F,X_1),hAPP(fun(X_a,bool),X_a,F_1,hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_3),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_1),bot_bot(fun(X_a,bool)))))) ) ) ) ) ).
fof(fact_157_insert__Diff,axiom,
! [X_a,A_2,A_3] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_2),A_3))
=> hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),A_2),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_3),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),A_2),bot_bot(fun(X_a,bool))))) = ti(fun(X_a,bool),A_3) ) ).
fof(fact_158_Diff__insert__absorb,axiom,
! [X_a,X_1,A_3] :
( ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),A_3))
=> hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_1),A_3)),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_1),bot_bot(fun(X_a,bool)))) = ti(fun(X_a,bool),A_3) ) ).
fof(fact_159_insert__Diff__single,axiom,
! [X_a,A_2,A_3] : hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),A_2),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_3),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),A_2),bot_bot(fun(X_a,bool))))) = hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),A_2),A_3) ).
fof(fact_160_Diff__insert2,axiom,
! [X_a,A_3,A_2,B_3] : hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_3),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),A_2),B_3)) = hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_3),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),A_2),bot_bot(fun(X_a,bool))))),B_3) ).
fof(fact_161_Diff__insert,axiom,
! [X_a,A_3,A_2,B_3] : hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_3),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),A_2),B_3)) = hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_3),B_3)),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),A_2),bot_bot(fun(X_a,bool)))) ).
fof(fact_162_folding__image__simple_Oinsert,axiom,
! [X_b,X_a,X_1,A_3,F,Z_1,G,F_1] :
( hBOOL(hAPP(fun(fun(X_b,bool),X_a),bool,hAPP(fun(X_b,X_a),fun(fun(fun(X_b,bool),X_a),bool),hAPP(X_a,fun(fun(X_b,X_a),fun(fun(fun(X_b,bool),X_a),bool)),hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(fun(X_b,X_a),fun(fun(fun(X_b,bool),X_a),bool))),finite1357897459simple(X_a,X_b),F),Z_1),G),F_1))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_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_a,F_1,hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_1),A_3)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),F,hAPP(X_b,X_a,G,X_1)),hAPP(fun(X_b,bool),X_a,F_1,A_3)) ) ) ) ).
fof(fact_163_finite__empty__induct,axiom,
! [X_a,Pa,A_3] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_3))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,Pa,A_3))
=> ( ! [A_5,A_4] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_4))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_5),A_4))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,Pa,A_4))
=> hBOOL(hAPP(fun(X_a,bool),bool,Pa,hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_4),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),A_5),bot_bot(fun(X_a,bool)))))) ) ) )
=> hBOOL(hAPP(fun(X_a,bool),bool,Pa,bot_bot(fun(X_a,bool)))) ) ) ) ).
fof(fact_164_finite__subset__induct,axiom,
! [X_a,Pa,A_3,F_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),F_1))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),F_1),A_3))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,Pa,bot_bot(fun(X_a,bool))))
=> ( ! [A_5,F_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),F_2))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_5),A_3))
=> ( ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_5),F_2))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,Pa,F_2))
=> hBOOL(hAPP(fun(X_a,bool),bool,Pa,hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),A_5),F_2))) ) ) ) )
=> hBOOL(hAPP(fun(X_a,bool),bool,Pa,F_1)) ) ) ) ) ).
fof(fact_165_insert__fold1SetE,axiom,
! [X_a,F,A_2,X_3,X_1] :
( hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,bool),fun(X_a,bool)),finite_fold1Set(X_a),F),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),A_2),X_3)),X_1))
=> ~ ! [A_5,A_4] :
( hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),A_2),X_3) = hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),A_5),A_4)
=> ( hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(fun(X_a,bool),fun(X_a,bool))),finite_fold_graph(X_a,X_a),F),A_5),A_4),X_1))
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_5),A_4)) ) ) ) ).
fof(fact_166_finite__nonempty__imp__fold1Set,axiom,
! [X_a,F,A_3] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_3))
=> ( ti(fun(X_a,bool),A_3) != bot_bot(fun(X_a,bool))
=> ? [X1] : hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,bool),fun(X_a,bool)),finite_fold1Set(X_a),F),A_3),X1)) ) ) ).
fof(fact_167_finite__induct,axiom,
! [X_a,Pa,F_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),F_1))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,Pa,bot_bot(fun(X_a,bool))))
=> ( ! [X_2,F_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),F_2))
=> ( ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),F_2))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,Pa,F_2))
=> hBOOL(hAPP(fun(X_a,bool),bool,Pa,hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_2),F_2))) ) ) )
=> hBOOL(hAPP(fun(X_a,bool),bool,Pa,F_1)) ) ) ) ).
fof(fact_168_finite_Osimps,axiom,
! [X_a,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
<=> ( ti(fun(X_a,bool),A_2) = bot_bot(fun(X_a,bool))
| ? [A_4,A_5] :
( ti(fun(X_a,bool),A_2) = hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),A_5),A_4)
& hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_4)) ) ) ) ).
fof(fact_169_folding__image__simple_Oneutral,axiom,
! [X_b,X_a,A_3,F,Z_1,G,F_1] :
( hBOOL(hAPP(fun(fun(X_b,bool),X_a),bool,hAPP(fun(X_b,X_a),fun(fun(fun(X_b,bool),X_a),bool),hAPP(X_a,fun(fun(X_b,X_a),fun(fun(fun(X_b,bool),X_a),bool)),hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(fun(X_b,X_a),fun(fun(fun(X_b,bool),X_a),bool))),finite1357897459simple(X_a,X_b),F),Z_1),G),F_1))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_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_a,G,X_2) = ti(X_a,Z_1) )
=> hAPP(fun(X_b,bool),X_a,F_1,A_3) = ti(X_a,Z_1) ) ) ) ).
fof(fact_170_finite__imp__fold__graph,axiom,
! [X_b,X_a,F,Z_1,A_3] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_3))
=> ? [X1] : hBOOL(hAPP(X_b,bool,hAPP(fun(X_a,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_a,bool),fun(X_b,bool)),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(fun(X_a,bool),fun(X_b,bool))),finite_fold_graph(X_a,X_b),F),Z_1),A_3),X1)) ) ).
fof(fact_171_fold1Set_Osimps,axiom,
! [X_a,F,A1,A2] :
( hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,bool),fun(X_a,bool)),finite_fold1Set(X_a),F),A1),A2))
<=> ? [A_5,A_4,X_2] :
( ti(fun(X_a,bool),A1) = hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),A_5),A_4)
& ti(X_a,A2) = ti(X_a,X_2)
& hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(fun(X_a,bool),fun(X_a,bool))),finite_fold_graph(X_a,X_a),F),A_5),A_4),X_2))
& ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_5),A_4)) ) ) ).
fof(fact_172_minus__apply,axiom,
! [X_b,X_a] :
( minus(X_a)
=> ! [A_3,B_3,X_1] : hAPP(X_b,X_a,hAPP(fun(X_b,X_a),fun(X_b,X_a),hAPP(fun(X_b,X_a),fun(fun(X_b,X_a),fun(X_b,X_a)),minus_minus(fun(X_b,X_a)),A_3),B_3),X_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_b,X_a,A_3,X_1)),hAPP(X_b,X_a,B_3,X_1)) ) ).
fof(fact_173_fun__diff__def,axiom,
! [X_a,X_b] :
( minus(X_b)
=> ! [A_3,B_3,X_2] : hAPP(X_a,X_b,hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_a,X_b),fun(fun(X_a,X_b),fun(X_a,X_b)),minus_minus(fun(X_a,X_b)),A_3),B_3),X_2) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),minus_minus(X_b),hAPP(X_a,X_b,A_3,X_2)),hAPP(X_a,X_b,B_3,X_2)) ) ).
fof(fact_174_diff__eq__diff__less__eq,axiom,
! [X_a] :
( ordered_ab_group_add(X_a)
=> ! [A_2,B_2,Ca,D_1] :
( hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_2),B_2) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),Ca),D_1)
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_2),B_2))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Ca),D_1)) ) ) ) ).
fof(fact_175_fold__graph_Osimps,axiom,
! [X_b,X_a,F,Z_1,A1,A2] :
( hBOOL(hAPP(X_b,bool,hAPP(fun(X_a,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_a,bool),fun(X_b,bool)),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(fun(X_a,bool),fun(X_b,bool))),finite_fold_graph(X_a,X_b),F),Z_1),A1),A2))
<=> ( ( ti(fun(X_a,bool),A1) = bot_bot(fun(X_a,bool))
& ti(X_b,A2) = ti(X_b,Z_1) )
| ? [X_2,A_4,Y_2] :
( ti(fun(X_a,bool),A1) = hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_2),A_4)
& ti(X_b,A2) = hAPP(X_b,X_b,hAPP(X_a,fun(X_b,X_b),F,X_2),Y_2)
& ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),A_4))
& hBOOL(hAPP(X_b,bool,hAPP(fun(X_a,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_a,bool),fun(X_b,bool)),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(fun(X_a,bool),fun(X_b,bool))),finite_fold_graph(X_a,X_b),F),Z_1),A_4),Y_2)) ) ) ) ).
fof(fact_176_folding__one__idem_Osubset__idem,axiom,
! [X_a,B_3,A_3,F,F_1] :
( hBOOL(hAPP(fun(fun(X_a,bool),X_a),bool,hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(fun(X_a,bool),X_a),bool),finite2073411215e_idem(X_a),F),F_1))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_3))
=> ( ti(fun(X_a,bool),B_3) != bot_bot(fun(X_a,bool))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),B_3),A_3))
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),F,hAPP(fun(X_a,bool),X_a,F_1,B_3)),hAPP(fun(X_a,bool),X_a,F_1,A_3)) = hAPP(fun(X_a,bool),X_a,F_1,A_3) ) ) ) ) ).
fof(fact_177_folding__one__idem_Oinsert__idem,axiom,
! [X_a,X_1,A_3,F,F_1] :
( hBOOL(hAPP(fun(fun(X_a,bool),X_a),bool,hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(fun(X_a,bool),X_a),bool),finite2073411215e_idem(X_a),F),F_1))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_3))
=> ( ti(fun(X_a,bool),A_3) != bot_bot(fun(X_a,bool))
=> hAPP(fun(X_a,bool),X_a,F_1,hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_1),A_3)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),F,X_1),hAPP(fun(X_a,bool),X_a,F_1,A_3)) ) ) ) ).
fof(fact_178_finite__ne__induct,axiom,
! [X_a,Pa,F_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),F_1))
=> ( ti(fun(X_a,bool),F_1) != bot_bot(fun(X_a,bool))
=> ( ! [X_2] : hBOOL(hAPP(fun(X_a,bool),bool,Pa,hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_2),bot_bot(fun(X_a,bool)))))
=> ( ! [X_2,F_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),F_2))
=> ( ti(fun(X_a,bool),F_2) != bot_bot(fun(X_a,bool))
=> ( ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),F_2))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,Pa,F_2))
=> hBOOL(hAPP(fun(X_a,bool),bool,Pa,hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_2),F_2))) ) ) ) )
=> hBOOL(hAPP(fun(X_a,bool),bool,Pa,F_1)) ) ) ) ) ).
fof(fact_179_folding__image__simple__idem_Osubset__idem,axiom,
! [X_b,X_a,B_3,A_3,F,Z_1,G,F_1] :
( hBOOL(hAPP(fun(fun(X_b,bool),X_a),bool,hAPP(fun(X_b,X_a),fun(fun(fun(X_b,bool),X_a),bool),hAPP(X_a,fun(fun(X_b,X_a),fun(fun(fun(X_b,bool),X_a),bool)),hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(fun(X_b,X_a),fun(fun(fun(X_b,bool),X_a),bool))),finite908156982e_idem(X_a,X_b),F),Z_1),G),F_1))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_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_3),A_3))
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),F,hAPP(fun(X_b,bool),X_a,F_1,B_3)),hAPP(fun(X_b,bool),X_a,F_1,A_3)) = hAPP(fun(X_b,bool),X_a,F_1,A_3) ) ) ) ).
fof(fact_180_folding__image__simple__idem_Oidem,axiom,
! [X_b,X_a,X_1,F,Z_1,G,F_1] :
( hBOOL(hAPP(fun(fun(X_b,bool),X_a),bool,hAPP(fun(X_b,X_a),fun(fun(fun(X_b,bool),X_a),bool),hAPP(X_a,fun(fun(X_b,X_a),fun(fun(fun(X_b,bool),X_a),bool)),hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(fun(X_b,X_a),fun(fun(fun(X_b,bool),X_a),bool))),finite908156982e_idem(X_a,X_b),F),Z_1),G),F_1))
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),F,X_1),X_1) = ti(X_a,X_1) ) ).
fof(fact_181_folding__one__idem_Oidem,axiom,
! [X_a,X_1,F,F_1] :
( hBOOL(hAPP(fun(fun(X_a,bool),X_a),bool,hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(fun(X_a,bool),X_a),bool),finite2073411215e_idem(X_a),F),F_1))
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),F,X_1),X_1) = ti(X_a,X_1) ) ).
fof(fact_182_folding__one__idem_Oin__idem,axiom,
! [X_a,X_1,A_3,F,F_1] :
( hBOOL(hAPP(fun(fun(X_a,bool),X_a),bool,hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(fun(X_a,bool),X_a),bool),finite2073411215e_idem(X_a),F),F_1))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_3))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),A_3))
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),F,X_1),hAPP(fun(X_a,bool),X_a,F_1,A_3)) = hAPP(fun(X_a,bool),X_a,F_1,A_3) ) ) ) ).
fof(fact_183_folding__image__simple__idem_Oin__idem,axiom,
! [X_b,X_a,X_1,A_3,F,Z_1,G,F_1] :
( hBOOL(hAPP(fun(fun(X_b,bool),X_a),bool,hAPP(fun(X_b,X_a),fun(fun(fun(X_b,bool),X_a),bool),hAPP(X_a,fun(fun(X_b,X_a),fun(fun(fun(X_b,bool),X_a),bool)),hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(fun(X_b,X_a),fun(fun(fun(X_b,bool),X_a),bool))),finite908156982e_idem(X_a,X_b),F),Z_1),G),F_1))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_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_a,X_a,hAPP(X_a,fun(X_a,X_a),F,hAPP(X_b,X_a,G,X_1)),hAPP(fun(X_b,bool),X_a,F_1,A_3)) = hAPP(fun(X_b,bool),X_a,F_1,A_3) ) ) ) ).
fof(fact_184_diff__eq__diff__eq,axiom,
! [X_a] :
( ab_group_add(X_a)
=> ! [A_2,B_2,Ca,D_1] :
( hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_2),B_2) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),Ca),D_1)
=> ( ti(X_a,A_2) = ti(X_a,B_2)
<=> ti(X_a,Ca) = ti(X_a,D_1) ) ) ) ).
fof(fact_185_folding__image__simple__idem_Oinsert__idem,axiom,
! [X_b,X_a,X_1,A_3,F,Z_1,G,F_1] :
( hBOOL(hAPP(fun(fun(X_b,bool),X_a),bool,hAPP(fun(X_b,X_a),fun(fun(fun(X_b,bool),X_a),bool),hAPP(X_a,fun(fun(X_b,X_a),fun(fun(fun(X_b,bool),X_a),bool)),hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(fun(X_b,X_a),fun(fun(fun(X_b,bool),X_a),bool))),finite908156982e_idem(X_a,X_b),F),Z_1),G),F_1))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> hAPP(fun(X_b,bool),X_a,F_1,hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_1),A_3)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),F,hAPP(X_b,X_a,G,X_1)),hAPP(fun(X_b,bool),X_a,F_1,A_3)) ) ) ).
fof(fact_186_comp__fun__commute_Ofold__graph__insertE__aux,axiom,
! [X_a,X_b,A_2,Z_1,A_3,Y_1,F] :
( hBOOL(hAPP(fun(X_a,fun(X_b,X_b)),bool,finite100568337ommute(X_a,X_b),F))
=> ( hBOOL(hAPP(X_b,bool,hAPP(fun(X_a,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_a,bool),fun(X_b,bool)),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(fun(X_a,bool),fun(X_b,bool))),finite_fold_graph(X_a,X_b),F),Z_1),A_3),Y_1))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_2),A_3))
=> ? [Y_3] :
( ti(X_b,Y_1) = hAPP(X_b,X_b,hAPP(X_a,fun(X_b,X_b),F,A_2),Y_3)
& hBOOL(hAPP(X_b,bool,hAPP(fun(X_a,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_a,bool),fun(X_b,bool)),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(fun(X_a,bool),fun(X_b,bool))),finite_fold_graph(X_a,X_b),F),Z_1),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_3),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),A_2),bot_bot(fun(X_a,bool))))),Y_3)) ) ) ) ) ).
fof(fact_187_fold__graph_H_Ointros_I2_J,axiom,
! [X_b,X_a,F,Z_1,Y_1,X_1,A_3] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),A_3))
=> ( hBOOL(hAPP(X_b,bool,hAPP(fun(X_a,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_a,bool),fun(X_b,bool)),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(fun(X_a,bool),fun(X_b,bool))),fold_graph(X_a,X_b),F),Z_1),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_3),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_1),bot_bot(fun(X_a,bool))))),Y_1))
=> hBOOL(hAPP(X_b,bool,hAPP(fun(X_a,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_a,bool),fun(X_b,bool)),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(fun(X_a,bool),fun(X_b,bool))),fold_graph(X_a,X_b),F),Z_1),A_3),hAPP(X_b,X_b,hAPP(X_a,fun(X_b,X_b),F,X_1),Y_1))) ) ) ).
fof(fact_188_semilattice__big_OF__eq,axiom,
! [X_a,A_3,F,F_1] :
( hBOOL(hAPP(fun(fun(X_a,bool),X_a),bool,hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(fun(X_a,bool),X_a),bool),big_semilattice_big(X_a),F),F_1))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_3))
=> hAPP(fun(X_a,bool),X_a,F_1,A_3) = hAPP(fun(X_a,bool),X_a,hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,bool),X_a),finite_fold1(X_a),F),A_3) ) ) ).
fof(fact_189_fold__graph__permute__diff,axiom,
! [X_a] :
( ab_semigroup_mult(X_a)
=> ! [A_2,B_2,A_3,X_1] :
( hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(fun(X_a,bool),fun(X_a,bool))),finite_fold_graph(X_a,X_a),times_times(X_a)),B_2),A_3),X_1))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_2),A_3))
=> ( ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),B_2),A_3))
=> hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(fun(X_a,bool),fun(X_a,bool))),finite_fold_graph(X_a,X_a),times_times(X_a)),A_2),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),B_2),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_3),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),A_2),bot_bot(fun(X_a,bool)))))),X_1)) ) ) ) ) ).
fof(fact_190_fold1__insert,axiom,
! [X_a] :
( ab_semigroup_mult(X_a)
=> ! [X_1,A_3] :
( ti(fun(X_a,bool),A_3) != bot_bot(fun(X_a,bool))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_3))
=> ( ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),A_3))
=> hAPP(fun(X_a,bool),X_a,hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,bool),X_a),finite_fold1(X_a),times_times(X_a)),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_1),A_3)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X_1),hAPP(fun(X_a,bool),X_a,hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,bool),X_a),finite_fold1(X_a),times_times(X_a)),A_3)) ) ) ) ) ).
fof(fact_191_fold1__insert__idem,axiom,
! [X_a] :
( ab_sem1668676832m_mult(X_a)
=> ! [X_1,A_3] :
( ti(fun(X_a,bool),A_3) != bot_bot(fun(X_a,bool))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_3))
=> hAPP(fun(X_a,bool),X_a,hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,bool),X_a),finite_fold1(X_a),times_times(X_a)),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_1),A_3)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X_1),hAPP(fun(X_a,bool),X_a,hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,bool),X_a),finite_fold1(X_a),times_times(X_a)),A_3)) ) ) ) ).
fof(fact_192_psubset__insert__iff,axiom,
! [X_a,A_3,X_1,B_3] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less(fun(X_a,bool)),A_3),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_1),B_3)))
<=> ( ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),B_3))
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less(fun(X_a,bool)),A_3),B_3)) )
& ( ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),B_3))
=> ( ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),A_3))
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less(fun(X_a,bool)),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_3),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_1),bot_bot(fun(X_a,bool))))),B_3)) )
& ( ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),A_3))
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_3),B_3)) ) ) ) ) ) ).
fof(fact_193_card__Diff1__le,axiom,
! [X_a,X_1,A_3] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_3))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(fun(X_a,bool),nat,finite_card(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_3),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_1),bot_bot(fun(X_a,bool)))))),hAPP(fun(X_a,bool),nat,finite_card(X_a),A_3))) ) ).
fof(fact_194_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_195_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_196_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_197_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_198_psubsetD,axiom,
! [X_a,Ca,A_3,B_3] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less(fun(X_a,bool)),A_3),B_3))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Ca),A_3))
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Ca),B_3)) ) ) ).
fof(fact_199_psubset__trans,axiom,
! [X_a,C_1,A_3,B_3] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less(fun(X_a,bool)),A_3),B_3))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less(fun(X_a,bool)),B_3),C_1))
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less(fun(X_a,bool)),A_3),C_1)) ) ) ).
fof(fact_200_comp__fun__commute,axiom,
! [X_a] :
( ab_semigroup_mult(X_a)
=> hBOOL(hAPP(fun(X_a,fun(X_a,X_a)),bool,finite100568337ommute(X_a,X_a),times_times(X_a))) ) ).
fof(fact_201_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_202_linorder__neq__iff,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X_1,Y_1] :
( ti(X_a,X_1) != ti(X_a,Y_1)
<=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_1),Y_1))
| hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y_1),X_1)) ) ) ) ).
fof(fact_203_not__less__iff__gr__or__eq,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X_1,Y_1] :
( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_1),Y_1))
<=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y_1),X_1))
| ti(X_a,X_1) = ti(X_a,Y_1) ) ) ) ).
fof(fact_204_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_205_comp__fun__commute_Ofun__left__comm,axiom,
! [X_a,X_b,X_1,Y_1,Z_1,F] :
( hBOOL(hAPP(fun(X_a,fun(X_b,X_b)),bool,finite100568337ommute(X_a,X_b),F))
=> hAPP(X_b,X_b,hAPP(X_a,fun(X_b,X_b),F,X_1),hAPP(X_b,X_b,hAPP(X_a,fun(X_b,X_b),F,Y_1),Z_1)) = hAPP(X_b,X_b,hAPP(X_a,fun(X_b,X_b),F,Y_1),hAPP(X_b,X_b,hAPP(X_a,fun(X_b,X_b),F,X_1),Z_1)) ) ).
fof(fact_206_linorder__antisym__conv3,axiom,
! [X_a] :
( linorder(X_a)
=> ! [Y_1,X_1] :
( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y_1),X_1))
=> ( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_1),Y_1))
<=> ti(X_a,X_1) = ti(X_a,Y_1) ) ) ) ).
fof(fact_207_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_208_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_209_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_210_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_211_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_212_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_213_order__less__imp__triv,axiom,
! [X_a] :
( preorder(X_a)
=> ! [Pa,X_1,Y_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_1),Y_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y_1),X_1))
=> hBOOL(Pa) ) ) ) ).
fof(fact_214_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_215_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_216_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_217_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_218_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_219_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_220_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_221_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_222_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_223_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_224_less__fun__def,axiom,
! [X_a,X_b] :
( ord(X_b)
=> ! [F,G] :
( hBOOL(hAPP(fun(X_a,X_b),bool,hAPP(fun(X_a,X_b),fun(fun(X_a,X_b),bool),ord_less(fun(X_a,X_b)),F),G))
<=> ( hBOOL(hAPP(fun(X_a,X_b),bool,hAPP(fun(X_a,X_b),fun(fun(X_a,X_b),bool),ord_less_eq(fun(X_a,X_b)),F),G))
& ~ hBOOL(hAPP(fun(X_a,X_b),bool,hAPP(fun(X_a,X_b),fun(fun(X_a,X_b),bool),ord_less_eq(fun(X_a,X_b)),G),F)) ) ) ) ).
fof(fact_225_linorder__not__less,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X_1,Y_1] :
( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_1),Y_1))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y_1),X_1)) ) ) ).
fof(fact_226_linorder__not__le,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X_1,Y_1] :
( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_1),Y_1))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y_1),X_1)) ) ) ).
fof(fact_227_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_228_order__less__le,axiom,
! [X_a] :
( order(X_a)
=> ! [X_1,Y_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_1),Y_1))
<=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_1),Y_1))
& ti(X_a,X_1) != ti(X_a,Y_1) ) ) ) ).
fof(fact_229_less__le__not__le,axiom,
! [X_a] :
( preorder(X_a)
=> ! [X_1,Y_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_1),Y_1))
<=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_1),Y_1))
& ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y_1),X_1)) ) ) ) ).
fof(fact_230_order__le__less,axiom,
! [X_a] :
( order(X_a)
=> ! [X_1,Y_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_1),Y_1))
<=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_1),Y_1))
| ti(X_a,X_1) = ti(X_a,Y_1) ) ) ) ).
fof(fact_231_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_232_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_233_linorder__antisym__conv1,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X_1,Y_1] :
( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_1),Y_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_1),Y_1))
<=> ti(X_a,X_1) = ti(X_a,Y_1) ) ) ) ).
fof(fact_234_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_235_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_236_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_237_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_238_linorder__antisym__conv2,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X_1,Y_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_1),Y_1))
=> ( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_1),Y_1))
<=> ti(X_a,X_1) = ti(X_a,Y_1) ) ) ) ).
fof(fact_239_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_240_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_241_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_242_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_243_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_244_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_245_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_246_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_247_bot__less,axiom,
! [X_a] :
( bot(X_a)
=> ! [A_2] :
( ti(X_a,A_2) != bot_bot(X_a)
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),bot_bot(X_a)),A_2)) ) ) ).
fof(fact_248_not__psubset__empty,axiom,
! [X_a,A_3] : ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less(fun(X_a,bool)),A_3),bot_bot(fun(X_a,bool)))) ).
fof(fact_249_diff__eq__diff__less,axiom,
! [X_a] :
( ordered_ab_group_add(X_a)
=> ! [A_2,B_2,Ca,D_1] :
( hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_2),B_2) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),Ca),D_1)
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_2),B_2))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Ca),D_1)) ) ) ) ).
fof(fact_250_psubset__eq,axiom,
! [X_a,A_3,B_3] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less(fun(X_a,bool)),A_3),B_3))
<=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_3),B_3))
& ti(fun(X_a,bool),A_3) != ti(fun(X_a,bool),B_3) ) ) ).
fof(fact_251_subset__iff__psubset__eq,axiom,
! [X_a,A_3,B_3] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_3),B_3))
<=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less(fun(X_a,bool)),A_3),B_3))
| ti(fun(X_a,bool),A_3) = ti(fun(X_a,bool),B_3) ) ) ).
fof(fact_252_psubset__imp__subset,axiom,
! [X_a,A_3,B_3] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less(fun(X_a,bool)),A_3),B_3))
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_3),B_3)) ) ).
fof(fact_253_psubset__subset__trans,axiom,
! [X_a,C_1,A_3,B_3] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less(fun(X_a,bool)),A_3),B_3))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),B_3),C_1))
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less(fun(X_a,bool)),A_3),C_1)) ) ) ).
fof(fact_254_subset__psubset__trans,axiom,
! [X_a,C_1,A_3,B_3] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_3),B_3))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less(fun(X_a,bool)),B_3),C_1))
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less(fun(X_a,bool)),A_3),C_1)) ) ) ).
fof(fact_255_fold__graph_H_Ointros_I1_J,axiom,
! [X_b,X_a,F,Z_1] : hBOOL(hAPP(X_b,bool,hAPP(fun(X_a,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_a,bool),fun(X_b,bool)),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(fun(X_a,bool),fun(X_b,bool))),fold_graph(X_a,X_b),F),Z_1),bot_bot(fun(X_a,bool))),Z_1)) ).
fof(fact_256_card__insert__le,axiom,
! [X_a,X_1,A_3] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_3))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(fun(X_a,bool),nat,finite_card(X_a),A_3)),hAPP(fun(X_a,bool),nat,finite_card(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_1),A_3)))) ) ).
fof(fact_257_card__mono,axiom,
! [X_a,A_3,B_3] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),B_3))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_3),B_3))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(fun(X_a,bool),nat,finite_card(X_a),A_3)),hAPP(fun(X_a,bool),nat,finite_card(X_a),B_3))) ) ) ).
fof(fact_258_card__seteq,axiom,
! [X_a,A_3,B_3] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),B_3))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_3),B_3))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(fun(X_a,bool),nat,finite_card(X_a),B_3)),hAPP(fun(X_a,bool),nat,finite_card(X_a),A_3)))
=> ti(fun(X_a,bool),A_3) = ti(fun(X_a,bool),B_3) ) ) ) ).
fof(fact_259_diff__card__le__card__Diff,axiom,
! [X_a,A_3,B_3] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),B_3))
=> 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_a,bool),nat,finite_card(X_a),A_3)),hAPP(fun(X_a,bool),nat,finite_card(X_a),B_3))),hAPP(fun(X_a,bool),nat,finite_card(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_3),B_3)))) ) ).
fof(fact_260_comp__fun__commute_Ofold__graph__determ,axiom,
! [X_a,X_b,Y_1,Z_1,A_3,X_1,F] :
( hBOOL(hAPP(fun(X_a,fun(X_b,X_b)),bool,finite100568337ommute(X_a,X_b),F))
=> ( hBOOL(hAPP(X_b,bool,hAPP(fun(X_a,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_a,bool),fun(X_b,bool)),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(fun(X_a,bool),fun(X_b,bool))),finite_fold_graph(X_a,X_b),F),Z_1),A_3),X_1))
=> ( hBOOL(hAPP(X_b,bool,hAPP(fun(X_a,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_a,bool),fun(X_b,bool)),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(fun(X_a,bool),fun(X_b,bool))),finite_fold_graph(X_a,X_b),F),Z_1),A_3),Y_1))
=> ti(X_b,Y_1) = ti(X_b,X_1) ) ) ) ).
fof(fact_261_fold__graph__insert__swap,axiom,
! [X_a] :
( ab_semigroup_mult(X_a)
=> ! [Z_1,B_2,A_3,Y_1] :
( hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(fun(X_a,bool),fun(X_a,bool))),finite_fold_graph(X_a,X_a),times_times(X_a)),B_2),A_3),Y_1))
=> ( ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),B_2),A_3))
=> hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(fun(X_a,bool),fun(X_a,bool))),finite_fold_graph(X_a,X_a),times_times(X_a)),Z_1),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),B_2),A_3)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Z_1),Y_1))) ) ) ) ).
fof(fact_262_card__Diff__subset,axiom,
! [X_a,A_3,B_3] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),B_3))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),B_3),A_3))
=> hAPP(fun(X_a,bool),nat,finite_card(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_3),B_3)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(fun(X_a,bool),nat,finite_card(X_a),A_3)),hAPP(fun(X_a,bool),nat,finite_card(X_a),B_3)) ) ) ).
fof(fact_263_fold1__in,axiom,
! [X_a] :
( ab_semigroup_mult(X_a)
=> ! [A_3] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_3))
=> ( ti(fun(X_a,bool),A_3) != bot_bot(fun(X_a,bool))
=> ( ! [X_2,Y_2] : hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X_2),Y_2)),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_2),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),Y_2),bot_bot(fun(X_a,bool))))))
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),hAPP(fun(X_a,bool),X_a,hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,bool),X_a),finite_fold1(X_a),times_times(X_a)),A_3)),A_3)) ) ) ) ) ).
fof(fact_264_comp__fun__commute_Ofold__graph__insertE,axiom,
! [X_a,X_b,Z_1,X_1,A_3,V_1,F] :
( hBOOL(hAPP(fun(X_a,fun(X_b,X_b)),bool,finite100568337ommute(X_a,X_b),F))
=> ( hBOOL(hAPP(X_b,bool,hAPP(fun(X_a,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_a,bool),fun(X_b,bool)),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(fun(X_a,bool),fun(X_b,bool))),finite_fold_graph(X_a,X_b),F),Z_1),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_1),A_3)),V_1))
=> ( ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),A_3))
=> ~ ! [Y_2] :
( ti(X_b,V_1) = hAPP(X_b,X_b,hAPP(X_a,fun(X_b,X_b),F,X_1),Y_2)
=> ~ hBOOL(hAPP(X_b,bool,hAPP(fun(X_a,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_a,bool),fun(X_b,bool)),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(fun(X_a,bool),fun(X_b,bool))),finite_fold_graph(X_a,X_b),F),Z_1),A_3),Y_2)) ) ) ) ) ).
fof(fact_265_psubset__imp__ex__mem,axiom,
! [X_a,A_3,B_3] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less(fun(X_a,bool)),A_3),B_3))
=> ? [B_5] : hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),B_5),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),B_3),A_3))) ) ).
fof(fact_266_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_267_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_3] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),W_3))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),W_3),Y))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),W_3),Z)) ) )
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),Z)) ) ) ) ).
fof(fact_268_xt5,axiom,
! [X_b,X_a] :
( ( order(X_a)
& order(X_b) )
=> ! [Ca,F,B_2,A_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B_2),A_2))
=> ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),Ca),hAPP(X_a,X_b,F,B_2)))
=> ( ! [X_2,Y_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y_2),X_2))
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),hAPP(X_a,X_b,F,Y_2)),hAPP(X_a,X_b,F,X_2))) )
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),Ca),hAPP(X_a,X_b,F,A_2))) ) ) ) ) ).
fof(fact_269_xt7,axiom,
! [X_b,X_a] :
( ( order(X_a)
& order(X_b) )
=> ! [Ca,F,B_2,A_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B_2),A_2))
=> ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),Ca),hAPP(X_a,X_b,F,B_2)))
=> ( ! [X_2,Y_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y_2),X_2))
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(X_a,X_b,F,Y_2)),hAPP(X_a,X_b,F,X_2))) )
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),Ca),hAPP(X_a,X_b,F,A_2))) ) ) ) ) ).
fof(fact_270_order__less__le__subst2,axiom,
! [X_a,X_b] :
( ( order(X_b)
& order(X_a) )
=> ! [F,Ca,A_2,B_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_2),B_2))
=> ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(X_a,X_b,F,B_2)),Ca))
=> ( ! [X_2,Y_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_2),Y_2))
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),hAPP(X_a,X_b,F,X_2)),hAPP(X_a,X_b,F,Y_2))) )
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),hAPP(X_a,X_b,F,A_2)),Ca)) ) ) ) ) ).
fof(fact_271_psubset__card__mono,axiom,
! [X_a,A_3,B_3] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),B_3))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less(fun(X_a,bool)),A_3),B_3))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(fun(X_a,bool),nat,finite_card(X_a),A_3)),hAPP(fun(X_a,bool),nat,finite_card(X_a),B_3))) ) ) ).
fof(fact_272_card__psubset,axiom,
! [X_a,A_3,B_3] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),B_3))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_3),B_3))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(fun(X_a,bool),nat,finite_card(X_a),A_3)),hAPP(fun(X_a,bool),nat,finite_card(X_a),B_3)))
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less(fun(X_a,bool)),A_3),B_3)) ) ) ) ).
fof(fact_273_card__Diff1__less,axiom,
! [X_a,X_1,A_3] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_3))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),A_3))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(fun(X_a,bool),nat,finite_card(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_3),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_1),bot_bot(fun(X_a,bool)))))),hAPP(fun(X_a,bool),nat,finite_card(X_a),A_3))) ) ) ).
fof(fact_274_card__Diff2__less,axiom,
! [X_a,Y_1,X_1,A_3] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_3))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),A_3))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Y_1),A_3))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(fun(X_a,bool),nat,finite_card(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_3),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_1),bot_bot(fun(X_a,bool))))),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),Y_1),bot_bot(fun(X_a,bool)))))),hAPP(fun(X_a,bool),nat,finite_card(X_a),A_3))) ) ) ) ).
fof(fact_275_xt6,axiom,
! [X_b,X_a] :
( ( order(X_a)
& ord(X_b) )
=> ! [Ca,F,B_2,A_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_b,X_a,F,B_2)),A_2))
=> ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),Ca),B_2))
=> ( ! [X_2,Y_2] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),Y_2),X_2))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_b,X_a,F,Y_2)),hAPP(X_b,X_a,F,X_2))) )
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_b,X_a,F,Ca)),A_2)) ) ) ) ) ).
fof(fact_276_xt4,axiom,
! [X_b,X_a] :
( ( order(X_a)
& order(X_b) )
=> ! [Ca,F,B_2,A_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_b,X_a,F,B_2)),A_2))
=> ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),Ca),B_2))
=> ( ! [X_2,Y_2] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),Y_2),X_2))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_b,X_a,F,Y_2)),hAPP(X_b,X_a,F,X_2))) )
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_b,X_a,F,Ca)),A_2)) ) ) ) ) ).
fof(fact_277_order__le__less__subst1,axiom,
! [X_a,X_b] :
( ( order(X_b)
& order(X_a) )
=> ! [Ca,A_2,F,B_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_2),hAPP(X_b,X_a,F,B_2)))
=> ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),B_2),Ca))
=> ( ! [X_2,Y_2] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),X_2),Y_2))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_b,X_a,F,X_2)),hAPP(X_b,X_a,F,Y_2))) )
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_2),hAPP(X_b,X_a,F,Ca))) ) ) ) ) ).
fof(fact_278_order__less__le__subst1,axiom,
! [X_a,X_b] :
( ( order(X_b)
& order(X_a) )
=> ! [Ca,A_2,F,B_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_2),hAPP(X_b,X_a,F,B_2)))
=> ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),B_2),Ca))
=> ( ! [X_2,Y_2] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),X_2),Y_2))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_b,X_a,F,X_2)),hAPP(X_b,X_a,F,Y_2))) )
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_2),hAPP(X_b,X_a,F,Ca))) ) ) ) ) ).
fof(fact_279_order__le__less__subst2,axiom,
! [X_a,X_b] :
( ( order(X_b)
& order(X_a) )
=> ! [F,Ca,A_2,B_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_2),B_2))
=> ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),hAPP(X_a,X_b,F,B_2)),Ca))
=> ( ! [X_2,Y_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_2),Y_2))
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(X_a,X_b,F,X_2)),hAPP(X_a,X_b,F,Y_2))) )
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),hAPP(X_a,X_b,F,A_2)),Ca)) ) ) ) ) ).
fof(fact_280_finite__linorder__max__induct,axiom,
! [X_a] :
( linorder(X_a)
=> ! [Pa,A_3] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_3))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,Pa,bot_bot(fun(X_a,bool))))
=> ( ! [B_5,A_4] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_4))
=> ( ! [X_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),A_4))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_2),B_5)) )
=> ( hBOOL(hAPP(fun(X_a,bool),bool,Pa,A_4))
=> hBOOL(hAPP(fun(X_a,bool),bool,Pa,hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),B_5),A_4))) ) ) )
=> hBOOL(hAPP(fun(X_a,bool),bool,Pa,A_3)) ) ) ) ) ).
fof(fact_281_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_282_less__diff__iff,axiom,
! [N,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))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(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),K)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N)) ) ) ) ).
fof(fact_283_finite__linorder__min__induct,axiom,
! [X_a] :
( linorder(X_a)
=> ! [Pa,A_3] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_3))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,Pa,bot_bot(fun(X_a,bool))))
=> ( ! [B_5,A_4] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_4))
=> ( ! [X_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),A_4))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B_5),X_2)) )
=> ( hBOOL(hAPP(fun(X_a,bool),bool,Pa,A_4))
=> hBOOL(hAPP(fun(X_a,bool),bool,Pa,hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),B_5),A_4))) ) ) )
=> hBOOL(hAPP(fun(X_a,bool),bool,Pa,A_3)) ) ) ) ) ).
fof(fact_284_le__diff__iff,axiom,
! [N,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))
=> ( 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),K)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N),K)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ) ) ).
fof(fact_285_Nat_Odiff__diff__eq,axiom,
! [N_1,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_1))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(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_1),K_1)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M_1),N_1) ) ) ).
fof(fact_286_eq__diff__iff,axiom,
! [N,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))
=> ( 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),K)
<=> M = N ) ) ) ).
fof(fact_287_less__not__refl,axiom,
! [N_1] : ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N_1),N_1)) ).
fof(fact_288_nat__neq__iff,axiom,
! [M,N] :
( M != N
<=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N))
| hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N),M)) ) ) ).
fof(fact_289_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_290_less__irrefl__nat,axiom,
! [N_1] : ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N_1),N_1)) ).
fof(fact_291_less__not__refl2,axiom,
! [N_1,M_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N_1),M_1))
=> M_1 != N_1 ) ).
fof(fact_292_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_293_nat__less__cases,axiom,
! [Pa,M,N] :
( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),Pa,N),M)) )
=> ( ( M = N
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),Pa,N),M)) )
=> ( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N),M))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),Pa,N),M)) )
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),Pa,N),M)) ) ) ) ).
fof(fact_294_mult__le__mono,axiom,
! [K_1,L_1,I_1,J] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_1),J))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_1),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_1),K_1)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),J),L_1))) ) ) ).
fof(fact_295_le__antisym,axiom,
! [M_1,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_1),N_1))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_1),M_1))
=> M_1 = N_1 ) ) ).
fof(fact_296_le__trans,axiom,
! [K_1,I_1,J] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_1),J))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),J),K_1))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_1),K_1)) ) ) ).
fof(fact_297_mult__le__mono2,axiom,
! [K_1,I_1,J] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_1),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_1),I_1)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K_1),J))) ) ).
fof(fact_298_mult__le__mono1,axiom,
! [K_1,I_1,J] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_1),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_1),K_1)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),J),K_1))) ) ).
fof(fact_299_eq__imp__le,axiom,
! [M_1,N_1] :
( M_1 = N_1
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_1),N_1)) ) ).
fof(fact_300_nat__le__linear,axiom,
! [M_1,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_1),N_1))
| hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_1),M_1)) ) ).
fof(fact_301_le__cube,axiom,
! [M_1] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M_1),M_1)))) ).
fof(fact_302_le__square,axiom,
! [M_1] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M_1),M_1))) ).
fof(fact_303_le__refl,axiom,
! [N_1] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_1),N_1)) ).
fof(fact_304_diff__mult__distrib,axiom,
! [M_1,N_1,K_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M_1),N_1)),K_1) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M_1),K_1)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),N_1),K_1)) ).
fof(fact_305_diff__mult__distrib2,axiom,
! [K_1,M_1,N_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M_1),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_1),M_1)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K_1),N_1)) ).
fof(fact_306_diff__commute,axiom,
! [I_1,J,K_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),I_1),J)),K_1) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),I_1),K_1)),J) ).
fof(fact_307_nat__less__le,axiom,
! [M,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N))
<=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N))
& M != N ) ) ).
fof(fact_308_le__eq__less__or__eq,axiom,
! [M,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N))
<=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N))
| M = N ) ) ).
fof(fact_309_less__imp__le__nat,axiom,
! [M_1,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_1),N_1))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_1),N_1)) ) ).
fof(fact_310_le__neq__implies__less,axiom,
! [M_1,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_1),N_1))
=> ( M_1 != N_1
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_1),N_1)) ) ) ).
fof(fact_311_less__or__eq__imp__le,axiom,
! [M_1,N_1] :
( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_1),N_1))
| M_1 = N_1 )
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_1),N_1)) ) ).
fof(fact_312_diff__less__mono2,axiom,
! [L_1,M_1,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_1),N_1))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_1),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_1))) ) ) ).
fof(fact_313_less__imp__diff__less,axiom,
! [N_1,J,K_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),J),K_1))
=> 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_1)) ) ).
fof(fact_314_diff__le__self,axiom,
! [M_1,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_1),N_1)),M_1)) ).
fof(fact_315_diff__le__mono2,axiom,
! [L_1,M_1,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_1),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_1))) ) ).
fof(fact_316_diff__le__mono,axiom,
! [L_1,M_1,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_1),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_1),L_1)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N_1),L_1))) ) ).
fof(fact_317_diff__diff__cancel,axiom,
! [I_1,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_1),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_1)) = I_1 ) ).
fof(fact_318_less__mono__imp__le__mono,axiom,
! [I_2,J_2,F] :
( ! [I,J_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I),J_1))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,F,I)),hAPP(nat,nat,F,J_1))) )
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_2),J_2))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,F,I_2)),hAPP(nat,nat,F,J_2))) ) ) ).
fof(fact_319_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_320_card__Diff__singleton__if,axiom,
! [X_a,X_1,A_3] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_3))
=> ( ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),A_3))
=> hAPP(fun(X_a,bool),nat,finite_card(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_3),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_1),bot_bot(fun(X_a,bool))))) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(fun(X_a,bool),nat,finite_card(X_a),A_3)),one_one(nat)) )
& ( ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),A_3))
=> hAPP(fun(X_a,bool),nat,finite_card(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_3),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_1),bot_bot(fun(X_a,bool))))) = hAPP(fun(X_a,bool),nat,finite_card(X_a),A_3) ) ) ) ).
fof(fact_321_card__Diff__singleton,axiom,
! [X_a,X_1,A_3] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_3))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),A_3))
=> hAPP(fun(X_a,bool),nat,finite_card(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_3),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_1),bot_bot(fun(X_a,bool))))) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(fun(X_a,bool),nat,finite_card(X_a),A_3)),one_one(nat)) ) ) ).
fof(fact_322_card__Diff__insert,axiom,
! [X_a,B_3,A_2,A_3] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_3))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_2),A_3))
=> ( ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_2),B_3))
=> hAPP(fun(X_a,bool),nat,finite_card(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_3),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),A_2),B_3))) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(fun(X_a,bool),nat,finite_card(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_3),B_3))),one_one(nat)) ) ) ) ).
fof(fact_323_comp__fun__commute_Ofold__insert__remove,axiom,
! [X_a,X_b,Z_1,X_1,A_3,F] :
( hBOOL(hAPP(fun(X_a,fun(X_b,X_b)),bool,finite100568337ommute(X_a,X_b),F))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_3))
=> hAPP(fun(X_a,bool),X_b,hAPP(X_b,fun(fun(X_a,bool),X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(fun(X_a,bool),X_b)),finite_fold(X_a,X_b),F),Z_1),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_1),A_3)) = hAPP(X_b,X_b,hAPP(X_a,fun(X_b,X_b),F,X_1),hAPP(fun(X_a,bool),X_b,hAPP(X_b,fun(fun(X_a,bool),X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(fun(X_a,bool),X_b)),finite_fold(X_a,X_b),F),Z_1),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_3),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_1),bot_bot(fun(X_a,bool)))))) ) ) ).
fof(fact_324_nat__mult__assoc,axiom,
! [M_1,N_1,K_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M_1),N_1)),K_1) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),N_1),K_1)) ).
fof(fact_325_nat__mult__eq__1__iff,axiom,
! [M,N] :
( hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),N) = one_one(nat)
<=> ( M = one_one(nat)
& N = one_one(nat) ) ) ).
fof(fact_326_nat__mult__commute,axiom,
! [M_1,N_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M_1),N_1) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),N_1),M_1) ).
fof(fact_327_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_328_nat__1__eq__mult__iff,axiom,
! [M,N] :
( one_one(nat) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),N)
<=> ( M = one_one(nat)
& N = one_one(nat) ) ) ).
fof(fact_329_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_330_one__reorient,axiom,
! [X_a] :
( one(X_a)
=> ! [X_1] :
( one_one(X_a) = ti(X_a,X_1)
<=> ti(X_a,X_1) = one_one(X_a) ) ) ).
fof(fact_331_fold__empty,axiom,
! [X_b,X_a,F,Z_1] : hAPP(fun(X_b,bool),X_a,hAPP(X_a,fun(fun(X_b,bool),X_a),hAPP(fun(X_b,fun(X_a,X_a)),fun(X_a,fun(fun(X_b,bool),X_a)),finite_fold(X_b,X_a),F),Z_1),bot_bot(fun(X_b,bool))) = ti(X_a,Z_1) ).
fof(fact_332_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_333_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_334_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_335_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_336_comp__fun__commute_Ofold__fun__comm,axiom,
! [X_a,X_b,X_1,Z_1,A_3,F] :
( hBOOL(hAPP(fun(X_a,fun(X_b,X_b)),bool,finite100568337ommute(X_a,X_b),F))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_3))
=> hAPP(X_b,X_b,hAPP(X_a,fun(X_b,X_b),F,X_1),hAPP(fun(X_a,bool),X_b,hAPP(X_b,fun(fun(X_a,bool),X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(fun(X_a,bool),X_b)),finite_fold(X_a,X_b),F),Z_1),A_3)) = hAPP(fun(X_a,bool),X_b,hAPP(X_b,fun(fun(X_a,bool),X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(fun(X_a,bool),X_b)),finite_fold(X_a,X_b),F),hAPP(X_b,X_b,hAPP(X_a,fun(X_b,X_b),F,X_1),Z_1)),A_3) ) ) ).
fof(fact_337_comp__fun__commute_Ofold__equality,axiom,
! [X_a,X_b,Z_1,A_3,Y_1,F] :
( hBOOL(hAPP(fun(X_a,fun(X_b,X_b)),bool,finite100568337ommute(X_a,X_b),F))
=> ( hBOOL(hAPP(X_b,bool,hAPP(fun(X_a,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_a,bool),fun(X_b,bool)),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(fun(X_a,bool),fun(X_b,bool))),finite_fold_graph(X_a,X_b),F),Z_1),A_3),Y_1))
=> hAPP(fun(X_a,bool),X_b,hAPP(X_b,fun(fun(X_a,bool),X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(fun(X_a,bool),X_b)),finite_fold(X_a,X_b),F),Z_1),A_3) = ti(X_b,Y_1) ) ) ).
fof(fact_338_comp__fun__commute_Ofold__insert2,axiom,
! [X_a,X_b,Z_1,X_1,A_3,F] :
( hBOOL(hAPP(fun(X_a,fun(X_b,X_b)),bool,finite100568337ommute(X_a,X_b),F))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_3))
=> ( ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),A_3))
=> hAPP(fun(X_a,bool),X_b,hAPP(X_b,fun(fun(X_a,bool),X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(fun(X_a,bool),X_b)),finite_fold(X_a,X_b),F),Z_1),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_1),A_3)) = hAPP(fun(X_a,bool),X_b,hAPP(X_b,fun(fun(X_a,bool),X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(fun(X_a,bool),X_b)),finite_fold(X_a,X_b),F),hAPP(X_b,X_b,hAPP(X_a,fun(X_b,X_b),F,X_1),Z_1)),A_3) ) ) ) ).
fof(fact_339_comp__fun__commute_Ofold__insert,axiom,
! [X_a,X_b,Z_1,X_1,A_3,F] :
( hBOOL(hAPP(fun(X_a,fun(X_b,X_b)),bool,finite100568337ommute(X_a,X_b),F))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_3))
=> ( ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),A_3))
=> hAPP(fun(X_a,bool),X_b,hAPP(X_b,fun(fun(X_a,bool),X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(fun(X_a,bool),X_b)),finite_fold(X_a,X_b),F),Z_1),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_1),A_3)) = hAPP(X_b,X_b,hAPP(X_a,fun(X_b,X_b),F,X_1),hAPP(fun(X_a,bool),X_b,hAPP(X_b,fun(fun(X_a,bool),X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(fun(X_a,bool),X_b)),finite_fold(X_a,X_b),F),Z_1),A_3)) ) ) ) ).
fof(fact_340_folding__one_Oeq__fold_H,axiom,
! [X_a,X_1,A_3,F,F_1] :
( hBOOL(hAPP(fun(fun(X_a,bool),X_a),bool,hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(fun(X_a,bool),X_a),bool),finite_folding_one(X_a),F),F_1))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_3))
=> ( ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),A_3))
=> hAPP(fun(X_a,bool),X_a,F_1,hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_1),A_3)) = hAPP(fun(X_a,bool),X_a,hAPP(X_a,fun(fun(X_a,bool),X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(fun(X_a,bool),X_a)),finite_fold(X_a,X_a),F),X_1),A_3) ) ) ) ).
fof(fact_341_folding__one__idem_Oeq__fold__idem_H,axiom,
! [X_a,A_2,A_3,F,F_1] :
( hBOOL(hAPP(fun(fun(X_a,bool),X_a),bool,hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(fun(X_a,bool),X_a),bool),finite2073411215e_idem(X_a),F),F_1))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_3))
=> hAPP(fun(X_a,bool),X_a,F_1,hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),A_2),A_3)) = hAPP(fun(X_a,bool),X_a,hAPP(X_a,fun(fun(X_a,bool),X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(fun(X_a,bool),X_a)),finite_fold(X_a,X_a),F),A_2),A_3) ) ) ).
fof(fact_342_comp__fun__commute_Ofold__graph__fold,axiom,
! [X_a,X_b,Z_1,A_3,F] :
( hBOOL(hAPP(fun(X_a,fun(X_b,X_b)),bool,finite100568337ommute(X_a,X_b),F))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_3))
=> hBOOL(hAPP(X_b,bool,hAPP(fun(X_a,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_a,bool),fun(X_b,bool)),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(fun(X_a,bool),fun(X_b,bool))),finite_fold_graph(X_a,X_b),F),Z_1),A_3),hAPP(fun(X_a,bool),X_b,hAPP(X_b,fun(fun(X_a,bool),X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(fun(X_a,bool),X_b)),finite_fold(X_a,X_b),F),Z_1),A_3))) ) ) ).
fof(fact_343_fold1__eq__fold,axiom,
! [X_a] :
( ab_semigroup_mult(X_a)
=> ! [A_2,A_3] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_3))
=> ( ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_2),A_3))
=> hAPP(fun(X_a,bool),X_a,hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,bool),X_a),finite_fold1(X_a),times_times(X_a)),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),A_2),A_3)) = hAPP(fun(X_a,bool),X_a,hAPP(X_a,fun(fun(X_a,bool),X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(fun(X_a,bool),X_a)),finite_fold(X_a,X_a),times_times(X_a)),A_2),A_3) ) ) ) ).
fof(fact_344_fold1__eq__fold__idem,axiom,
! [X_a] :
( ab_sem1668676832m_mult(X_a)
=> ! [A_2,A_3] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_3))
=> hAPP(fun(X_a,bool),X_a,hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,bool),X_a),finite_fold1(X_a),times_times(X_a)),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),A_2),A_3)) = hAPP(fun(X_a,bool),X_a,hAPP(X_a,fun(fun(X_a,bool),X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(fun(X_a,bool),X_a)),finite_fold(X_a,X_a),times_times(X_a)),A_2),A_3) ) ) ).
fof(fact_345_comp__fun__commute_Ofold__rec,axiom,
! [X_a,X_b,Z_1,X_1,A_3,F] :
( hBOOL(hAPP(fun(X_a,fun(X_b,X_b)),bool,finite100568337ommute(X_a,X_b),F))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_3))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),A_3))
=> hAPP(fun(X_a,bool),X_b,hAPP(X_b,fun(fun(X_a,bool),X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(fun(X_a,bool),X_b)),finite_fold(X_a,X_b),F),Z_1),A_3) = hAPP(X_b,X_b,hAPP(X_a,fun(X_b,X_b),F,X_1),hAPP(fun(X_a,bool),X_b,hAPP(X_b,fun(fun(X_a,bool),X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(fun(X_a,bool),X_b)),finite_fold(X_a,X_b),F),Z_1),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_3),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_1),bot_bot(fun(X_a,bool)))))) ) ) ) ).
fof(fact_346_less__1__mult,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [N_1,M_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),M_1))
=> ( 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_1),N_1))) ) ) ) ).
fof(fact_347_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_348_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_349_card_Oremove,axiom,
! [X_a,X_1,A_3] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_3))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),A_3))
=> hAPP(fun(X_a,bool),nat,finite_card(X_a),A_3) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),hAPP(fun(X_a,bool),nat,finite_card(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_3),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_1),bot_bot(fun(X_a,bool)))))) ) ) ).
fof(fact_350_card_Oinsert__remove,axiom,
! [X_a,X_1,A_3] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_3))
=> hAPP(fun(X_a,bool),nat,finite_card(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_1),A_3)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),hAPP(fun(X_a,bool),nat,finite_card(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_3),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_1),bot_bot(fun(X_a,bool)))))) ) ).
fof(fact_351_comp__fun__idem_Ofold__insert__idem2,axiom,
! [X_a,X_b,Z_1,X_1,A_3,F] :
( hBOOL(hAPP(fun(X_a,fun(X_b,X_b)),bool,finite_comp_fun_idem(X_a,X_b),F))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_3))
=> hAPP(fun(X_a,bool),X_b,hAPP(X_b,fun(fun(X_a,bool),X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(fun(X_a,bool),X_b)),finite_fold(X_a,X_b),F),Z_1),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_1),A_3)) = hAPP(fun(X_a,bool),X_b,hAPP(X_b,fun(fun(X_a,bool),X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(fun(X_a,bool),X_b)),finite_fold(X_a,X_b),F),hAPP(X_b,X_b,hAPP(X_a,fun(X_b,X_b),F,X_1),Z_1)),A_3) ) ) ).
fof(fact_352_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_353_crossproduct__noteq,axiom,
! [X_a] :
( semiri456707255roduct(X_a)
=> ! [Ca,D_1,A_2,B_2] :
( ( ti(X_a,A_2) != ti(X_a,B_2)
& ti(X_a,Ca) != ti(X_a,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),times_times(X_a),A_2),Ca)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B_2),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),times_times(X_a),A_2),D_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B_2),Ca)) ) ) ).
fof(fact_354_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_355_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_356_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_1,M_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),times_times(X_a),A_1),M_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),M_1)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B)),M_1) ) ).
fof(fact_357_combine__common__factor,axiom,
! [X_a] :
( semiring(X_a)
=> ! [A_1,E_2,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_2)),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_2)),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_2)),C) ) ).
fof(fact_358_crossproduct__eq,axiom,
! [X_a] :
( semiri456707255roduct(X_a)
=> ! [W,Y_1,X_1,Z_1] :
( hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),W),Y_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X_1),Z_1)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),W),Z_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X_1),Y_1))
<=> ( ti(X_a,W) = ti(X_a,X_1)
| ti(X_a,Y_1) = ti(X_a,Z_1) ) ) ) ).
fof(fact_359_comp__fun__idem_Ofun__left__idem,axiom,
! [X_a,X_b,X_1,Z_1,F] :
( hBOOL(hAPP(fun(X_a,fun(X_b,X_b)),bool,finite_comp_fun_idem(X_a,X_b),F))
=> hAPP(X_b,X_b,hAPP(X_a,fun(X_b,X_b),F,X_1),hAPP(X_b,X_b,hAPP(X_a,fun(X_b,X_b),F,X_1),Z_1)) = hAPP(X_b,X_b,hAPP(X_a,fun(X_b,X_b),F,X_1),Z_1) ) ).
fof(fact_360_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_361_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_1,C,D] : 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)) = 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)) ) ).
fof(fact_362_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_1,C,D] : 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)) = 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) ) ).
fof(fact_363_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_364_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_365_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_1,B,C,D] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),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)) = 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)) ) ).
fof(fact_366_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_367_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_368_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_369_add__right__cancel,axiom,
! [X_a] :
( cancel_semigroup_add(X_a)
=> ! [B_2,A_2,Ca] :
( hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),B_2),A_2) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),Ca),A_2)
<=> ti(X_a,B_2) = ti(X_a,Ca) ) ) ).
fof(fact_370_add__left__cancel,axiom,
! [X_a] :
( cancel_semigroup_add(X_a)
=> ! [A_2,B_2,Ca] :
( hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_2),B_2) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_2),Ca)
<=> ti(X_a,B_2) = ti(X_a,Ca) ) ) ).
fof(fact_371_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_372_nat__add__right__cancel,axiom,
! [M,K,N] :
( 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),K)
<=> M = N ) ).
fof(fact_373_nat__add__left__cancel,axiom,
! [K,M,N] :
( 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)
<=> M = N ) ).
fof(fact_374_nat__add__assoc,axiom,
! [M_1,N_1,K_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M_1),N_1)),K_1) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N_1),K_1)) ).
fof(fact_375_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_376_nat__add__commute,axiom,
! [M_1,N_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M_1),N_1) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N_1),M_1) ).
fof(fact_377_add__le__cancel__right,axiom,
! [X_a] :
( ordere236663937imp_le(X_a)
=> ! [A_2,Ca,B_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_2),Ca)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),B_2),Ca)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_2),B_2)) ) ) ).
fof(fact_378_add__le__cancel__left,axiom,
! [X_a] :
( ordere236663937imp_le(X_a)
=> ! [Ca,A_2,B_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),Ca),A_2)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),Ca),B_2)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_2),B_2)) ) ) ).
fof(fact_379_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_380_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_381_add__mono,axiom,
! [X_a] :
( ordere779506340up_add(X_a)
=> ! [C,D,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))
=> 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))) ) ) ) ).
fof(fact_382_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_383_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_384_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_385_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_386_add__strict__mono,axiom,
! [X_a] :
( ordere223160158up_add(X_a)
=> ! [C,D,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))
=> 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))) ) ) ) ).
fof(fact_387_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_388_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_389_add__less__cancel__left,axiom,
! [X_a] :
( ordere236663937imp_le(X_a)
=> ! [Ca,A_2,B_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),Ca),A_2)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),Ca),B_2)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_2),B_2)) ) ) ).
fof(fact_390_add__less__cancel__right,axiom,
! [X_a] :
( ordere236663937imp_le(X_a)
=> ! [A_2,Ca,B_2] :
( 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_2),Ca)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),B_2),Ca)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_2),B_2)) ) ) ).
fof(fact_391_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_392_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_393_add__lessD1,axiom,
! [I_1,J,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),J)),K_1))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_1),K_1)) ) ).
fof(fact_394_less__add__eq__less,axiom,
! [M_1,N_1,K_1,L_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),K_1),L_1))
=> ( hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M_1),L_1) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),K_1),N_1)
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_1),N_1)) ) ) ).
fof(fact_395_add__less__mono,axiom,
! [K_1,L_1,I_1,J] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_1),J))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),K_1),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_1),K_1)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),J),L_1))) ) ) ).
fof(fact_396_add__less__mono1,axiom,
! [K_1,I_1,J] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_1),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_1),K_1)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),J),K_1))) ) ).
fof(fact_397_trans__less__add2,axiom,
! [M_1,I_1,J] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_1),J))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M_1),J))) ) ).
fof(fact_398_trans__less__add1,axiom,
! [M_1,I_1,J] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_1),J))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),J),M_1))) ) ).
fof(fact_399_nat__add__left__cancel__less,axiom,
! [K,M,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),M)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),K),N)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N)) ) ).
fof(fact_400_not__add__less2,axiom,
! [J,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),J),I_1)),I_1)) ).
fof(fact_401_not__add__less1,axiom,
! [I_1,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_1),J)),I_1)) ).
fof(fact_402_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_403_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_404_le__add2,axiom,
! [N_1,M_1] : 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_1),N_1))) ).
fof(fact_405_le__add1,axiom,
! [N_1,M_1] : 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_1))) ).
fof(fact_406_le__iff__add,axiom,
! [M,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N))
<=> ? [K_2] : N = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),K_2) ) ).
fof(fact_407_nat__add__left__cancel__le,axiom,
! [K,M,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),M)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),K),N)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ).
fof(fact_408_trans__le__add1,axiom,
! [M_1,I_1,J] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_1),J))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),J),M_1))) ) ).
fof(fact_409_trans__le__add2,axiom,
! [M_1,I_1,J] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_1),J))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M_1),J))) ) ).
fof(fact_410_add__le__mono1,axiom,
! [K_1,I_1,J] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_1),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_1),K_1)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),J),K_1))) ) ).
fof(fact_411_add__le__mono,axiom,
! [K_1,L_1,I_1,J] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_1),J))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_1),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_1),K_1)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),J),L_1))) ) ) ).
fof(fact_412_add__leD2,axiom,
! [M_1,K_1,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_1),K_1)),N_1))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_1),N_1)) ) ).
fof(fact_413_add__leD1,axiom,
! [M_1,K_1,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_1),K_1)),N_1))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_1),N_1)) ) ).
fof(fact_414_add__leE,axiom,
! [M_1,K_1,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_1),K_1)),N_1))
=> ~ ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_1),N_1))
=> ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_1),N_1)) ) ) ).
fof(fact_415_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_416_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_417_diff__cancel2,axiom,
! [M_1,K_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),M_1),K_1)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N_1),K_1)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M_1),N_1) ).
fof(fact_418_diff__cancel,axiom,
! [K_1,M_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),K_1),M_1)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),K_1),N_1)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M_1),N_1) ).
fof(fact_419_diff__diff__left,axiom,
! [I_1,J,K_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),I_1),J)),K_1) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),I_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),J),K_1)) ).
fof(fact_420_diff__add__inverse,axiom,
! [N_1,M_1] : 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_1)),N_1) = M_1 ).
fof(fact_421_diff__add__inverse2,axiom,
! [M_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),M_1),N_1)),N_1) = M_1 ).
fof(fact_422_add__mult__distrib2,axiom,
! [K_1,M_1,N_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M_1),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_1),M_1)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K_1),N_1)) ).
fof(fact_423_add__mult__distrib,axiom,
! [M_1,N_1,K_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M_1),N_1)),K_1) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M_1),K_1)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),N_1),K_1)) ).
fof(fact_424_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_425_eq__add__iff2,axiom,
! [X_a] :
( ring(X_a)
=> ! [A_2,E_1,Ca,B_2,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),times_times(X_a),A_2),E_1)),Ca) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B_2),E_1)),D_1)
<=> ti(X_a,Ca) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),B_2),A_2)),E_1)),D_1) ) ) ).
fof(fact_426_eq__add__iff1,axiom,
! [X_a] :
( ring(X_a)
=> ! [A_2,E_1,Ca,B_2,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),times_times(X_a),A_2),E_1)),Ca) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B_2),E_1)),D_1)
<=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_2),B_2)),E_1)),Ca) = ti(X_a,D_1) ) ) ).
fof(fact_427_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_1,M_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),M_1)),M_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),one_one(X_a))),M_1) ) ).
fof(fact_428_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [M_1,A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),M_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),M_1)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),one_one(X_a))),M_1) ) ).
fof(fact_429_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [M_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),M_1),M_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),one_one(X_a)),one_one(X_a))),M_1) ) ).
fof(fact_430_comp__fun__idem__insert,axiom,
! [X_a] : hBOOL(hAPP(fun(X_a,fun(fun(X_a,bool),fun(X_a,bool))),bool,finite_comp_fun_idem(X_a,fun(X_a,bool)),insert(X_a))) ).
fof(fact_431_comp__fun__idem,axiom,
! [X_a] :
( ab_sem1668676832m_mult(X_a)
=> hBOOL(hAPP(fun(X_a,fun(X_a,X_a)),bool,finite_comp_fun_idem(X_a,X_a),times_times(X_a))) ) ).
fof(fact_432_add__less__le__mono,axiom,
! [X_a] :
( ordere223160158up_add(X_a)
=> ! [C,D,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))
=> 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))) ) ) ) ).
fof(fact_433_add__le__less__mono,axiom,
! [X_a] :
( ordere223160158up_add(X_a)
=> ! [C,D,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))
=> 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))) ) ) ) ).
fof(fact_434_le__add__iff2,axiom,
! [X_a] :
( ordered_ring(X_a)
=> ! [A_2,E_1,Ca,B_2,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),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_2),E_1)),Ca)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B_2),E_1)),D_1)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Ca),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),B_2),A_2)),E_1)),D_1))) ) ) ).
fof(fact_435_le__add__iff1,axiom,
! [X_a] :
( ordered_ring(X_a)
=> ! [A_2,E_1,Ca,B_2,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),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_2),E_1)),Ca)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B_2),E_1)),D_1)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_2),B_2)),E_1)),Ca)),D_1)) ) ) ).
fof(fact_436_less__add__iff2,axiom,
! [X_a] :
( ordered_ring(X_a)
=> ! [A_2,E_1,Ca,B_2,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),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_2),E_1)),Ca)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B_2),E_1)),D_1)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Ca),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),B_2),A_2)),E_1)),D_1))) ) ) ).
fof(fact_437_less__add__iff1,axiom,
! [X_a] :
( ordered_ring(X_a)
=> ! [A_2,E_1,Ca,B_2,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),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_2),E_1)),Ca)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B_2),E_1)),D_1)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_2),B_2)),E_1)),Ca)),D_1)) ) ) ).
fof(fact_438_less__diff__conv,axiom,
! [I_2,J_2,K] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_2),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),J_2),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_2),K)),J_2)) ) ).
fof(fact_439_add__diff__inverse,axiom,
! [M_1,N_1] :
( ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_1),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_1),N_1)) = M_1 ) ).
fof(fact_440_diff__add__assoc2,axiom,
! [I_1,K_1,J] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_1),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_1)),K_1) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),J),K_1)),I_1) ) ).
fof(fact_441_add__diff__assoc2,axiom,
! [I_1,K_1,J] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_1),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_1)),I_1) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),J),I_1)),K_1) ) ).
fof(fact_442_diff__add__assoc,axiom,
! [I_1,K_1,J] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_1),J))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),I_1),J)),K_1) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),I_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),J),K_1)) ) ).
fof(fact_443_le__imp__diff__is__add,axiom,
! [K,I_2,J_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_2),J_2))
=> ( hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),J_2),I_2) = K
<=> J_2 = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),K),I_2) ) ) ).
fof(fact_444_le__add__diff__inverse2,axiom,
! [N_1,M_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_1),M_1))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M_1),N_1)),N_1) = M_1 ) ).
fof(fact_445_le__diff__conv2,axiom,
! [I_2,K,J_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K),J_2))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_2),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),J_2),K)))
<=> 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_2),K)),J_2)) ) ) ).
fof(fact_446_add__diff__assoc,axiom,
! [I_1,K_1,J] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_1),J))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),I_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),J),K_1)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),I_1),J)),K_1) ) ).
fof(fact_447_le__add__diff__inverse,axiom,
! [N_1,M_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_1),M_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_1),N_1)) = M_1 ) ).
fof(fact_448_le__add__diff,axiom,
! [M_1,K_1,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_1),N_1))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_1),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_1)),K_1))) ) ).
fof(fact_449_le__diff__conv,axiom,
! [J_2,K,I_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),J_2),K)),I_2))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),J_2),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),I_2),K))) ) ).
fof(fact_450_diff__diff__right,axiom,
! [I_1,K_1,J] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_1),J))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),I_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),J),K_1)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),I_1),K_1)),J) ) ).
fof(fact_451_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_452_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_453_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_454_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_455_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_456_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_457_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_458_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_459_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_460_card_Oinsert,axiom,
! [X_a,X_1,A_3] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_3))
=> ( ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),A_3))
=> hAPP(fun(X_a,bool),nat,finite_card(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_1),A_3)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),hAPP(fun(X_a,bool),nat,finite_card(X_a),A_3)) ) ) ).
fof(fact_461_comp__fun__idem_Ofold__insert__idem,axiom,
! [X_a,X_b,Z_1,X_1,A_3,F] :
( hBOOL(hAPP(fun(X_a,fun(X_b,X_b)),bool,finite_comp_fun_idem(X_a,X_b),F))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_3))
=> hAPP(fun(X_a,bool),X_b,hAPP(X_b,fun(fun(X_a,bool),X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(fun(X_a,bool),X_b)),finite_fold(X_a,X_b),F),Z_1),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_1),A_3)) = hAPP(X_b,X_b,hAPP(X_a,fun(X_b,X_b),F,X_1),hAPP(fun(X_a,bool),X_b,hAPP(X_b,fun(fun(X_a,bool),X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(fun(X_a,bool),X_b)),finite_fold(X_a,X_b),F),Z_1),A_3)) ) ) ).
fof(fact_462_nat__less__add__iff1,axiom,
! [U_1,M,N,J_2,I_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),J_2),I_2))
=> ( 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_2),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_2),U_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),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),I_2),J_2)),U_1)),M)),N)) ) ) ).
fof(fact_463_nat__less__add__iff2,axiom,
! [U_1,M,N,I_2,J_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_2),J_2))
=> ( 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_2),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_2),U_1)),N)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(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_2),I_2)),U_1)),N))) ) ) ).
fof(fact_464_nat__le__add__iff1,axiom,
! [U_1,M,N,J_2,I_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),J_2),I_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,hAPP(nat,fun(nat,nat),times_times(nat),I_2),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_2),U_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),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),I_2),J_2)),U_1)),M)),N)) ) ) ).
fof(fact_465_nat__diff__add__eq1,axiom,
! [U,M_1,N_1,J,I_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),J),I_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),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),U)),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_1),J)),U)),M_1)),N_1) ) ).
fof(fact_466_nat__eq__add__iff1,axiom,
! [U_1,M,N,J_2,I_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),J_2),I_2))
=> ( hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),I_2),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_2),U_1)),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_2),J_2)),U_1)),M) = N ) ) ).
fof(fact_467_left__add__mult__distrib,axiom,
! [I_1,U,J,K_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)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),J),U)),K_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),plus_plus(nat),I_1),J)),U)),K_1) ).
fof(fact_468_nat__eq__add__iff2,axiom,
! [U_1,M,N,I_2,J_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_2),J_2))
=> ( hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),I_2),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_2),U_1)),N)
<=> 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_2),I_2)),U_1)),N) ) ) ).
fof(fact_469_nat__diff__add__eq2,axiom,
! [U,M_1,N_1,I_1,J] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_1),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_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),U)),N_1)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(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),I_1)),U)),N_1)) ) ).
fof(fact_470_nat__le__add__iff2,axiom,
! [U_1,M,N,I_2,J_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_2),J_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,hAPP(nat,fun(nat,nat),times_times(nat),I_2),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_2),U_1)),N)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(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_2),I_2)),U_1)),N))) ) ) ).
fof(fact_471_mono__nat__linear__lb,axiom,
! [M,K,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)),K)),hAPP(nat,nat,F,hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),K)))) ) ).
fof(fact_472_convex__bound__lt,axiom,
! [X_a] :
( linord626643107strict(X_a)
=> ! [V,U,Y,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),Y),A_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),U))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),V))
=> ( hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),U),V) = one_one(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),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),U),X)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),V),Y))),A_1)) ) ) ) ) ) ) ).
fof(fact_473_card__Plus,axiom,
! [X_b,X_a,B_3,A_3] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_3))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),B_3))
=> hAPP(fun(sum_sum(X_a,X_b),bool),nat,finite_card(sum_sum(X_a,X_b)),hAPP(fun(X_b,bool),fun(sum_sum(X_a,X_b),bool),hAPP(fun(X_a,bool),fun(fun(X_b,bool),fun(sum_sum(X_a,X_b),bool)),sum_Plus(X_a,X_b),A_3),B_3)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(fun(X_a,bool),nat,finite_card(X_a),A_3)),hAPP(fun(X_b,bool),nat,finite_card(X_b),B_3)) ) ) ).
fof(fact_474_card__Suc__Diff1,axiom,
! [X_a,X_1,A_3] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_3))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),A_3))
=> hAPP(nat,nat,suc,hAPP(fun(X_a,bool),nat,finite_card(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_3),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_1),bot_bot(fun(X_a,bool)))))) = hAPP(fun(X_a,bool),nat,finite_card(X_a),A_3) ) ) ).
fof(fact_475_less__zeroE,axiom,
! [N_1] : ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N_1),zero_zero(nat))) ).
fof(fact_476_le0,axiom,
! [N_1] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),N_1)) ).
fof(fact_477_zero__less__Suc,axiom,
! [N_1] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(nat,nat,suc,N_1))) ).
fof(fact_478_Suc__mono,axiom,
! [M_1,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_1),N_1))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,suc,M_1)),hAPP(nat,nat,suc,N_1))) ) ).
fof(fact_479_lessI,axiom,
! [N_1] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N_1),hAPP(nat,nat,suc,N_1))) ).
fof(fact_480_add__is__1,axiom,
! [M,N] :
( hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),N) = hAPP(nat,nat,suc,zero_zero(nat))
<=> ( ( M = hAPP(nat,nat,suc,zero_zero(nat))
& N = zero_zero(nat) )
| ( M = zero_zero(nat)
& N = hAPP(nat,nat,suc,zero_zero(nat)) ) ) ) ).
fof(fact_481_one__is__add,axiom,
! [M,N] :
( hAPP(nat,nat,suc,zero_zero(nat)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),N)
<=> ( ( M = hAPP(nat,nat,suc,zero_zero(nat))
& N = zero_zero(nat) )
| ( M = zero_zero(nat)
& N = hAPP(nat,nat,suc,zero_zero(nat)) ) ) ) ).
fof(fact_482_nat__mult__eq__cancel__disj,axiom,
! [K,M,N] :
( 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)
<=> ( K = zero_zero(nat)
| M = N ) ) ).
fof(fact_483_not__less__eq,axiom,
! [M,N] :
( ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N),hAPP(nat,nat,suc,M))) ) ).
fof(fact_484_less__Suc__eq,axiom,
! [M,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),hAPP(nat,nat,suc,N)))
<=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N))
| M = N ) ) ).
fof(fact_485_Suc__less__eq,axiom,
! [M,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,suc,M)),hAPP(nat,nat,suc,N)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N)) ) ).
fof(fact_486_not__less__less__Suc__eq,axiom,
! [N,M] :
( ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N),M))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N),hAPP(nat,nat,suc,M)))
<=> N = M ) ) ).
fof(fact_487_less__antisym,axiom,
! [N_1,M_1] :
( ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N_1),M_1))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N_1),hAPP(nat,nat,suc,M_1)))
=> M_1 = N_1 ) ) ).
fof(fact_488_less__SucI,axiom,
! [M_1,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_1),N_1))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_1),hAPP(nat,nat,suc,N_1))) ) ).
fof(fact_489_Suc__lessI,axiom,
! [M_1,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_1),N_1))
=> ( hAPP(nat,nat,suc,M_1) != N_1
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,suc,M_1)),N_1)) ) ) ).
fof(fact_490_less__trans__Suc,axiom,
! [K_1,I_1,J] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_1),J))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),J),K_1))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,suc,I_1)),K_1)) ) ) ).
fof(fact_491_less__SucE,axiom,
! [M_1,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_1),hAPP(nat,nat,suc,N_1)))
=> ( ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_1),N_1))
=> M_1 = N_1 ) ) ).
fof(fact_492_Suc__lessD,axiom,
! [M_1,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,suc,M_1)),N_1))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_1),N_1)) ) ).
fof(fact_493_Suc__less__SucD,axiom,
! [M_1,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,suc,M_1)),hAPP(nat,nat,suc,N_1)))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_1),N_1)) ) ).
fof(fact_494_add__Suc__right,axiom,
! [M_1,N_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M_1),hAPP(nat,nat,suc,N_1)) = hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M_1),N_1)) ).
fof(fact_495_add__Suc,axiom,
! [M_1,N_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,suc,M_1)),N_1) = hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M_1),N_1)) ).
fof(fact_496_add__Suc__shift,axiom,
! [M_1,N_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,suc,M_1)),N_1) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M_1),hAPP(nat,nat,suc,N_1)) ).
fof(fact_497_Suc__leD,axiom,
! [M_1,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,suc,M_1)),N_1))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_1),N_1)) ) ).
fof(fact_498_le__SucE,axiom,
! [M_1,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_1),hAPP(nat,nat,suc,N_1)))
=> ( ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_1),N_1))
=> M_1 = hAPP(nat,nat,suc,N_1) ) ) ).
fof(fact_499_le__SucI,axiom,
! [M_1,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_1),N_1))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_1),hAPP(nat,nat,suc,N_1))) ) ).
fof(fact_500_Suc__le__mono,axiom,
! [N,M] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,suc,N)),hAPP(nat,nat,suc,M)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N),M)) ) ).
fof(fact_501_le__Suc__eq,axiom,
! [M,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),hAPP(nat,nat,suc,N)))
<=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N))
| M = hAPP(nat,nat,suc,N) ) ) ).
fof(fact_502_not__less__eq__eq,axiom,
! [M,N] :
( ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,suc,N)),M)) ) ).
fof(fact_503_Suc__n__not__le__n,axiom,
! [N_1] : ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,suc,N_1)),N_1)) ).
fof(fact_504_diff__Suc__Suc,axiom,
! [M_1,N_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,suc,M_1)),hAPP(nat,nat,suc,N_1)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M_1),N_1) ).
fof(fact_505_Suc__diff__diff,axiom,
! [M_1,N_1,K_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,suc,M_1)),N_1)),hAPP(nat,nat,suc,K_1)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M_1),N_1)),K_1) ).
fof(fact_506_Suc__mult__cancel1,axiom,
! [K,M,N] :
( hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,suc,K)),M) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,suc,K)),N)
<=> M = N ) ).
fof(fact_507_add__0__left,axiom,
! [X_a] :
( monoid_add(X_a)
=> ! [A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),zero_zero(X_a)),A_1) = ti(X_a,A_1) ) ).
fof(fact_508_add__0,axiom,
! [X_a] :
( comm_monoid_add(X_a)
=> ! [A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),zero_zero(X_a)),A_1) = ti(X_a,A_1) ) ).
fof(fact_509_double__zero__sym,axiom,
! [X_a] :
( linord219039673up_add(X_a)
=> ! [A_2] :
( zero_zero(X_a) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_2),A_2)
<=> ti(X_a,A_2) = zero_zero(X_a) ) ) ).
fof(fact_510_add__0__right,axiom,
! [X_a] :
( monoid_add(X_a)
=> ! [A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),zero_zero(X_a)) = ti(X_a,A_1) ) ).
fof(fact_511_add_Ocomm__neutral,axiom,
! [X_a] :
( comm_monoid_add(X_a)
=> ! [A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),zero_zero(X_a)) = ti(X_a,A_1) ) ).
fof(fact_512_add__0__iff,axiom,
! [X_a] :
( semiri456707255roduct(X_a)
=> ! [B_2,A_2] :
( ti(X_a,B_2) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),B_2),A_2)
<=> ti(X_a,A_2) = zero_zero(X_a) ) ) ).
fof(fact_513_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),zero_zero(X_a)) = ti(X_a,A_1) ) ).
fof(fact_514_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),zero_zero(X_a)),A_1) = ti(X_a,A_1) ) ).
fof(fact_515_divisors__zero,axiom,
! [X_a] :
( no_zero_divisors(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) = zero_zero(X_a)
=> ( ti(X_a,A_1) = zero_zero(X_a)
| ti(X_a,B) = zero_zero(X_a) ) ) ) ).
fof(fact_516_no__zero__divisors,axiom,
! [X_a] :
( no_zero_divisors(X_a)
=> ! [B,A_1] :
( ti(X_a,A_1) != zero_zero(X_a)
=> ( ti(X_a,B) != zero_zero(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_517_mult__eq__0__iff,axiom,
! [X_a] :
( ring_n68954251visors(X_a)
=> ! [A_2,B_2] :
( hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_2),B_2) = zero_zero(X_a)
<=> ( ti(X_a,A_2) = zero_zero(X_a)
| ti(X_a,B_2) = zero_zero(X_a) ) ) ) ).
fof(fact_518_comm__semiring__1__class_Onormalizing__semiring__rules_I10_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),zero_zero(X_a)) = zero_zero(X_a) ) ).
fof(fact_519_mult__zero__right,axiom,
! [X_a] :
( mult_zero(X_a)
=> ! [A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),zero_zero(X_a)) = zero_zero(X_a) ) ).
fof(fact_520_comm__semiring__1__class_Onormalizing__semiring__rules_I9_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),zero_zero(X_a)),A_1) = zero_zero(X_a) ) ).
fof(fact_521_mult__zero__left,axiom,
! [X_a] :
( mult_zero(X_a)
=> ! [A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),zero_zero(X_a)),A_1) = zero_zero(X_a) ) ).
fof(fact_522_diff__0__right,axiom,
! [X_a] :
( group_add(X_a)
=> ! [A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_1),zero_zero(X_a)) = ti(X_a,A_1) ) ).
fof(fact_523_diff__self,axiom,
! [X_a] :
( group_add(X_a)
=> ! [A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_1),A_1) = zero_zero(X_a) ) ).
fof(fact_524_eq__iff__diff__eq__0,axiom,
! [X_a] :
( ab_group_add(X_a)
=> ! [A_2,B_2] :
( ti(X_a,A_2) = ti(X_a,B_2)
<=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_2),B_2) = zero_zero(X_a) ) ) ).
fof(fact_525_right__minus__eq,axiom,
! [X_a] :
( group_add(X_a)
=> ! [A_2,B_2] :
( hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_2),B_2) = zero_zero(X_a)
<=> ti(X_a,A_2) = ti(X_a,B_2) ) ) ).
fof(fact_526_one__neq__zero,axiom,
! [X_a] :
( zero_neq_one(X_a)
=> one_one(X_a) != zero_zero(X_a) ) ).
fof(fact_527_zero__neq__one,axiom,
! [X_a] :
( zero_neq_one(X_a)
=> zero_zero(X_a) != one_one(X_a) ) ).
fof(fact_528_not__less0,axiom,
! [N_1] : ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N_1),zero_zero(nat))) ).
fof(fact_529_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_530_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_531_gr__implies__not0,axiom,
! [M_1,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_1),N_1))
=> N_1 != zero_zero(nat) ) ).
fof(fact_532_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_533_plus__nat_Oadd__0,axiom,
! [N_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),zero_zero(nat)),N_1) = N_1 ).
fof(fact_534_Nat_Oadd__0__right,axiom,
! [M_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M_1),zero_zero(nat)) = M_1 ).
fof(fact_535_add__is__0,axiom,
! [M,N] :
( hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),N) = zero_zero(nat)
<=> ( M = zero_zero(nat)
& N = zero_zero(nat) ) ) ).
fof(fact_536_add__eq__self__zero,axiom,
! [M_1,N_1] :
( hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M_1),N_1) = M_1
=> N_1 = zero_zero(nat) ) ).
fof(fact_537_le__0__eq,axiom,
! [N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N),zero_zero(nat)))
<=> N = zero_zero(nat) ) ).
fof(fact_538_less__eq__nat_Osimps_I1_J,axiom,
! [N_1] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),N_1)) ).
fof(fact_539_diff__0__eq__0,axiom,
! [N_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),zero_zero(nat)),N_1) = zero_zero(nat) ).
fof(fact_540_minus__nat_Odiff__0,axiom,
! [M_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M_1),zero_zero(nat)) = M_1 ).
fof(fact_541_diff__self__eq__0,axiom,
! [M_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M_1),M_1) = zero_zero(nat) ).
fof(fact_542_diffs0__imp__equal,axiom,
! [M_1,N_1] :
( hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M_1),N_1) = zero_zero(nat)
=> ( hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N_1),M_1) = zero_zero(nat)
=> M_1 = N_1 ) ) ).
fof(fact_543_mult__cancel2,axiom,
! [M,K,N] :
( 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),K)
<=> ( M = N
| K = zero_zero(nat) ) ) ).
fof(fact_544_mult__cancel1,axiom,
! [K,M,N] :
( 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)
<=> ( M = N
| K = zero_zero(nat) ) ) ).
fof(fact_545_mult__is__0,axiom,
! [M,N] :
( hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),N) = zero_zero(nat)
<=> ( M = zero_zero(nat)
| N = zero_zero(nat) ) ) ).
fof(fact_546_mult__0__right,axiom,
! [M_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M_1),zero_zero(nat)) = zero_zero(nat) ).
fof(fact_547_mult__0,axiom,
! [N_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),zero_zero(nat)),N_1) = zero_zero(nat) ).
fof(fact_548_mult__eq__1__iff,axiom,
! [M,N] :
( hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),N) = hAPP(nat,nat,suc,zero_zero(nat))
<=> ( M = hAPP(nat,nat,suc,zero_zero(nat))
& N = hAPP(nat,nat,suc,zero_zero(nat)) ) ) ).
fof(fact_549_zero__reorient,axiom,
! [X_a] :
( zero(X_a)
=> ! [X_1] :
( zero_zero(X_a) = ti(X_a,X_1)
<=> ti(X_a,X_1) = zero_zero(X_a) ) ) ).
fof(fact_550_Suc__inject,axiom,
! [X,Y] :
( hAPP(nat,nat,suc,X) = hAPP(nat,nat,suc,Y)
=> X = Y ) ).
fof(fact_551_Suc__neq__Zero,axiom,
! [M_1] : hAPP(nat,nat,suc,M_1) != zero_zero(nat) ).
fof(fact_552_Zero__neq__Suc,axiom,
! [M_1] : zero_zero(nat) != hAPP(nat,nat,suc,M_1) ).
fof(fact_553_nat_Oinject,axiom,
! [Nat_4,Nat_3] :
( hAPP(nat,nat,suc,Nat_4) = hAPP(nat,nat,suc,Nat_3)
<=> Nat_4 = Nat_3 ) ).
fof(fact_554_Suc__n__not__n,axiom,
! [N_1] : hAPP(nat,nat,suc,N_1) != N_1 ).
fof(fact_555_nat_Osimps_I3_J,axiom,
! [Nat_2] : hAPP(nat,nat,suc,Nat_2) != zero_zero(nat) ).
fof(fact_556_Suc__not__Zero,axiom,
! [M_1] : hAPP(nat,nat,suc,M_1) != zero_zero(nat) ).
fof(fact_557_n__not__Suc__n,axiom,
! [N_1] : N_1 != hAPP(nat,nat,suc,N_1) ).
fof(fact_558_nat_Osimps_I2_J,axiom,
! [Nat_1] : zero_zero(nat) != hAPP(nat,nat,suc,Nat_1) ).
fof(fact_559_Zero__not__Suc,axiom,
! [M_1] : zero_zero(nat) != hAPP(nat,nat,suc,M_1) ).
fof(fact_560_gr0__conv__Suc,axiom,
! [N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
<=> ? [M_2] : N = hAPP(nat,nat,suc,M_2) ) ).
fof(fact_561_less__Suc0,axiom,
! [N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N),hAPP(nat,nat,suc,zero_zero(nat))))
<=> N = zero_zero(nat) ) ).
fof(fact_562_less__Suc__eq__0__disj,axiom,
! [M,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),hAPP(nat,nat,suc,N)))
<=> ( M = zero_zero(nat)
| ? [J_1] :
( M = hAPP(nat,nat,suc,J_1)
& hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),J_1),N)) ) ) ) ).
fof(fact_563_one__le__mult__iff,axiom,
! [M,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,suc,zero_zero(nat))),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),N)))
<=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,suc,zero_zero(nat))),M))
& hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,suc,zero_zero(nat))),N)) ) ) ).
fof(fact_564_one__less__mult,axiom,
! [M_1,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,suc,zero_zero(nat))),N_1))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,suc,zero_zero(nat))),M_1))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,suc,zero_zero(nat))),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M_1),N_1))) ) ) ).
fof(fact_565_n__less__n__mult__m,axiom,
! [M_1,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,suc,zero_zero(nat))),N_1))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,suc,zero_zero(nat))),M_1))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),N_1),M_1))) ) ) ).
fof(fact_566_n__less__m__mult__n,axiom,
! [M_1,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,suc,zero_zero(nat))),N_1))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,suc,zero_zero(nat))),M_1))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M_1),N_1))) ) ) ).
fof(fact_567_Suc__pred,axiom,
! [N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N_1))
=> hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N_1),hAPP(nat,nat,suc,zero_zero(nat)))) = N_1 ) ).
fof(fact_568_diff__Suc__less,axiom,
! [I_1,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N_1))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N_1),hAPP(nat,nat,suc,I_1))),N_1)) ) ).
fof(fact_569_One__nat__def,axiom,
one_one(nat) = hAPP(nat,nat,suc,zero_zero(nat)) ).
fof(fact_570_Suc__diff__1,axiom,
! [N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N_1))
=> hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N_1),one_one(nat))) = N_1 ) ).
fof(fact_571_finite__PlusD_I2_J,axiom,
! [X_a,X_b,A_3,B_3] :
( hBOOL(hAPP(fun(sum_sum(X_a,X_b),bool),bool,finite_finite_1(sum_sum(X_a,X_b)),hAPP(fun(X_b,bool),fun(sum_sum(X_a,X_b),bool),hAPP(fun(X_a,bool),fun(fun(X_b,bool),fun(sum_sum(X_a,X_b),bool)),sum_Plus(X_a,X_b),A_3),B_3)))
=> hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),B_3)) ) ).
fof(fact_572_finite__PlusD_I1_J,axiom,
! [X_a,X_b,A_3,B_3] :
( hBOOL(hAPP(fun(sum_sum(X_a,X_b),bool),bool,finite_finite_1(sum_sum(X_a,X_b)),hAPP(fun(X_b,bool),fun(sum_sum(X_a,X_b),bool),hAPP(fun(X_a,bool),fun(fun(X_b,bool),fun(sum_sum(X_a,X_b),bool)),sum_Plus(X_a,X_b),A_3),B_3)))
=> hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_3)) ) ).
fof(fact_573_finite__Plus,axiom,
! [X_b,X_a,B_3,A_3] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_3))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),B_3))
=> hBOOL(hAPP(fun(sum_sum(X_a,X_b),bool),bool,finite_finite_1(sum_sum(X_a,X_b)),hAPP(fun(X_b,bool),fun(sum_sum(X_a,X_b),bool),hAPP(fun(X_a,bool),fun(fun(X_b,bool),fun(sum_sum(X_a,X_b),bool)),sum_Plus(X_a,X_b),A_3),B_3))) ) ) ).
fof(fact_574_finite__Plus__iff,axiom,
! [X_a,X_b,A_3,B_3] :
( hBOOL(hAPP(fun(sum_sum(X_a,X_b),bool),bool,finite_finite_1(sum_sum(X_a,X_b)),hAPP(fun(X_b,bool),fun(sum_sum(X_a,X_b),bool),hAPP(fun(X_a,bool),fun(fun(X_b,bool),fun(sum_sum(X_a,X_b),bool)),sum_Plus(X_a,X_b),A_3),B_3)))
<=> ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_3))
& hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),B_3)) ) ) ).
fof(fact_575_card__Plus__conv__if,axiom,
! [X_b,X_a,B_3,A_3] :
( ( ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_3))
& hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),B_3)) )
=> hAPP(fun(sum_sum(X_a,X_b),bool),nat,finite_card(sum_sum(X_a,X_b)),hAPP(fun(X_b,bool),fun(sum_sum(X_a,X_b),bool),hAPP(fun(X_a,bool),fun(fun(X_b,bool),fun(sum_sum(X_a,X_b),bool)),sum_Plus(X_a,X_b),A_3),B_3)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(fun(X_a,bool),nat,finite_card(X_a),A_3)),hAPP(fun(X_b,bool),nat,finite_card(X_b),B_3)) )
& ( ~ ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_3))
& hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),B_3)) )
=> hAPP(fun(sum_sum(X_a,X_b),bool),nat,finite_card(sum_sum(X_a,X_b)),hAPP(fun(X_b,bool),fun(sum_sum(X_a,X_b),bool),hAPP(fun(X_a,bool),fun(fun(X_b,bool),fun(sum_sum(X_a,X_b),bool)),sum_Plus(X_a,X_b),A_3),B_3)) = zero_zero(nat) ) ) ).
fof(fact_576_less__add__Suc1,axiom,
! [I_1,M_1] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_1),hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),I_1),M_1)))) ).
fof(fact_577_less__add__Suc2,axiom,
! [I_1,M_1] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_1),hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M_1),I_1)))) ).
fof(fact_578_less__iff__Suc__add,axiom,
! [M,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N))
<=> ? [K_2] : N = hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),K_2)) ) ).
fof(fact_579_Suc__le__lessD,axiom,
! [M_1,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,suc,M_1)),N_1))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_1),N_1)) ) ).
fof(fact_580_le__less__Suc__eq,axiom,
! [M,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N),hAPP(nat,nat,suc,M)))
<=> N = M ) ) ).
fof(fact_581_Suc__leI,axiom,
! [M_1,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_1),N_1))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,suc,M_1)),N_1)) ) ).
fof(fact_582_le__imp__less__Suc,axiom,
! [M_1,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_1),N_1))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_1),hAPP(nat,nat,suc,N_1))) ) ).
fof(fact_583_Suc__le__eq,axiom,
! [M,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,suc,M)),N))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N)) ) ).
fof(fact_584_less__Suc__eq__le,axiom,
! [M,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),hAPP(nat,nat,suc,N)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ).
fof(fact_585_less__eq__Suc__le,axiom,
! [N,M] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N),M))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,suc,N)),M)) ) ).
fof(fact_586_diff__less__Suc,axiom,
! [M_1,N_1] : 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),N_1)),hAPP(nat,nat,suc,M_1))) ).
fof(fact_587_Suc__mult__less__cancel1,axiom,
! [K,M,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,suc,K)),M)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,suc,K)),N)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N)) ) ).
fof(fact_588_mult__Suc,axiom,
! [M_1,N_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,suc,M_1)),N_1) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M_1),N_1)) ).
fof(fact_589_mult__Suc__right,axiom,
! [M_1,N_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M_1),hAPP(nat,nat,suc,N_1)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M_1),N_1)) ).
fof(fact_590_Suc__diff__le,axiom,
! [N_1,M_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_1),M_1))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,suc,M_1)),N_1) = hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M_1),N_1)) ) ).
fof(fact_591_Suc__mult__le__cancel1,axiom,
! [K,M,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,suc,K)),M)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,suc,K)),N)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ).
fof(fact_592_add__nonpos__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_eq(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_eq(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_593_add__increasing2,axiom,
! [X_a] :
( ordere216010020id_add(X_a)
=> ! [B,A_1,C] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),C))
=> ( 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_eq(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_594_add__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_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),B),C))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(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_595_add__nonneg__eq__0__iff,axiom,
! [X_a] :
( ordere216010020id_add(X_a)
=> ! [Y_1,X_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),X_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),Y_1))
=> ( hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X_1),Y_1) = zero_zero(X_a)
<=> ( ti(X_a,X_1) = zero_zero(X_a)
& ti(X_a,Y_1) = zero_zero(X_a) ) ) ) ) ) ).
fof(fact_596_add__nonneg__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_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(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(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_597_double__add__le__zero__iff__single__add__le__zero,axiom,
! [X_a] :
( linord219039673up_add(X_a)
=> ! [A_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_2),A_2)),zero_zero(X_a)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_2),zero_zero(X_a))) ) ) ).
fof(fact_598_zero__le__double__add__iff__zero__le__single__add,axiom,
! [X_a] :
( linord219039673up_add(X_a)
=> ! [A_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_2),A_2)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_2)) ) ) ).
fof(fact_599_split__mult__neg__le,axiom,
! [X_a] :
( ordere453448008miring(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_eq(X_a),B),zero_zero(X_a))) )
| ( 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_eq(X_a),zero_zero(X_a)),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),times_times(X_a),A_1),B)),zero_zero(X_a))) ) ) ).
fof(fact_600_split__mult__pos__le,axiom,
! [X_a] :
( ordered_ring(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_eq(X_a),zero_zero(X_a)),B)) )
| ( 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_eq(X_a),B),zero_zero(X_a))) ) )
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(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_601_mult__mono,axiom,
! [X_a] :
( ordered_semiring(X_a)
=> ! [C,D,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))
=> ( 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_eq(X_a),zero_zero(X_a)),C))
=> 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),A_1),C)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),D))) ) ) ) ) ) ).
fof(fact_602_mult__mono_H,axiom,
! [X_a] :
( ordered_semiring(X_a)
=> ! [C,D,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))
=> ( 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)),C))
=> 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),A_1),C)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),D))) ) ) ) ) ) ).
fof(fact_603_mult__left__mono__neg,axiom,
! [X_a] :
( ordered_ring(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))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C),zero_zero(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),C),A_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C),B))) ) ) ) ).
fof(fact_604_mult__right__mono__neg,axiom,
! [X_a] :
( ordered_ring(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))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C),zero_zero(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),A_1),C)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),C))) ) ) ) ).
fof(fact_605_comm__mult__left__mono,axiom,
! [X_a] :
( ordere1490568538miring(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),zero_zero(X_a)),C))
=> 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),C),A_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C),B))) ) ) ) ).
fof(fact_606_mult__left__mono,axiom,
! [X_a] :
( ordered_semiring(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),zero_zero(X_a)),C))
=> 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),C),A_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C),B))) ) ) ) ).
fof(fact_607_mult__right__mono,axiom,
! [X_a] :
( ordered_semiring(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),zero_zero(X_a)),C))
=> 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),A_1),C)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),C))) ) ) ) ).
fof(fact_608_mult__nonpos__nonpos,axiom,
! [X_a] :
( ordered_ring(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_eq(X_a),B),zero_zero(X_a)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(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_609_mult__nonpos__nonneg,axiom,
! [X_a] :
( ordere453448008miring(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_eq(X_a),zero_zero(X_a)),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),times_times(X_a),A_1),B)),zero_zero(X_a))) ) ) ) ).
fof(fact_610_mult__nonneg__nonpos2,axiom,
! [X_a] :
( ordere453448008miring(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_eq(X_a),B),zero_zero(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),B),A_1)),zero_zero(X_a))) ) ) ) ).
fof(fact_611_mult__nonneg__nonpos,axiom,
! [X_a] :
( ordere453448008miring(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_eq(X_a),B),zero_zero(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),A_1),B)),zero_zero(X_a))) ) ) ) ).
fof(fact_612_mult__nonneg__nonneg,axiom,
! [X_a] :
( ordere453448008miring(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_eq(X_a),zero_zero(X_a)),B))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(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_613_mult__le__0__iff,axiom,
! [X_a] :
( linord581940658strict(X_a)
=> ! [A_2,B_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_2),B_2)),zero_zero(X_a)))
<=> ( ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_2))
& hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B_2),zero_zero(X_a))) )
| ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_2),zero_zero(X_a)))
& hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),B_2)) ) ) ) ) ).
fof(fact_614_zero__le__mult__iff,axiom,
! [X_a] :
( linord581940658strict(X_a)
=> ! [A_2,B_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_2),B_2)))
<=> ( ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_2))
& hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),B_2)) )
| ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_2),zero_zero(X_a)))
& hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B_2),zero_zero(X_a))) ) ) ) ) ).
fof(fact_615_zero__le__square,axiom,
! [X_a] :
( linordered_ring(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)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),A_1))) ) ).
fof(fact_616_zero__less__double__add__iff__zero__less__single__add,axiom,
! [X_a] :
( linord219039673up_add(X_a)
=> ! [A_2] :
( 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_2),A_2)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_2)) ) ) ).
fof(fact_617_double__add__less__zero__iff__single__add__less__zero,axiom,
! [X_a] :
( linord219039673up_add(X_a)
=> ! [A_2] :
( 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_2),A_2)),zero_zero(X_a)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_2),zero_zero(X_a))) ) ) ).
fof(fact_618_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_619_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_620_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_621_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_622_mult__less__cancel__right__disj,axiom,
! [X_a] :
( linord581940658strict(X_a)
=> ! [A_2,Ca,B_2] :
( 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_2),Ca)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B_2),Ca)))
<=> ( ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),Ca))
& hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_2),B_2)) )
| ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Ca),zero_zero(X_a)))
& hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B_2),A_2)) ) ) ) ) ).
fof(fact_623_mult__less__cancel__left__disj,axiom,
! [X_a] :
( linord581940658strict(X_a)
=> ! [Ca,A_2,B_2] :
( 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),Ca),A_2)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Ca),B_2)))
<=> ( ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),Ca))
& hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_2),B_2)) )
| ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Ca),zero_zero(X_a)))
& hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B_2),A_2)) ) ) ) ) ).
fof(fact_624_mult__less__cancel__left__pos,axiom,
! [X_a] :
( linord581940658strict(X_a)
=> ! [A_2,B_2,Ca] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),Ca))
=> ( 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),Ca),A_2)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Ca),B_2)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_2),B_2)) ) ) ) ).
fof(fact_625_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_626_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_627_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_628_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_629_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_630_mult__less__cancel__left__neg,axiom,
! [X_a] :
( linord581940658strict(X_a)
=> ! [A_2,B_2,Ca] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Ca),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),Ca),A_2)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Ca),B_2)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B_2),A_2)) ) ) ) ).
fof(fact_631_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_632_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_633_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_634_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_635_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_636_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_637_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))) ) ) ) ).
fof(fact_638_le__iff__diff__le__0,axiom,
! [X_a] :
( ordered_ab_group_add(X_a)
=> ! [A_2,B_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_2),B_2))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_2),B_2)),zero_zero(X_a))) ) ) ).
fof(fact_639_less__iff__diff__less__0,axiom,
! [X_a] :
( ordered_ab_group_add(X_a)
=> ! [A_2,B_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_2),B_2))
<=> 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),minus_minus(X_a),A_2),B_2)),zero_zero(X_a))) ) ) ).
fof(fact_640_not__one__le__zero,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),one_one(X_a)),zero_zero(X_a))) ) ).
fof(fact_641_zero__le__one,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),one_one(X_a))) ) ).
fof(fact_642_add__scale__eq__noteq,axiom,
! [X_a] :
( semiri456707255roduct(X_a)
=> ! [C,D,A_1,B,R_2] :
( ti(X_a,R_2) != zero_zero(X_a)
=> ( ( ti(X_a,A_1) = ti(X_a,B)
& ti(X_a,C) != ti(X_a,D) )
=> 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),times_times(X_a),R_2),C)) != hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),B),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),R_2),D)) ) ) ) ).
fof(fact_643_zero__less__one,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)),one_one(X_a))) ) ).
fof(fact_644_not__one__less__zero,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),zero_zero(X_a))) ) ).
fof(fact_645_diff__Suc__1,axiom,
! [N_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,suc,N_1)),one_one(nat)) = N_1 ).
fof(fact_646_add__gr__0,axiom,
! [M,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),N)))
<=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M))
| hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ) ).
fof(fact_647_card_Oempty,axiom,
! [X_a] : hAPP(fun(X_a,bool),nat,finite_card(X_a),bot_bot(fun(X_a,bool))) = zero_zero(nat) ).
fof(fact_648_card__infinite,axiom,
! [X_a,A_3] :
( ~ hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_3))
=> hAPP(fun(X_a,bool),nat,finite_card(X_a),A_3) = zero_zero(nat) ) ).
fof(fact_649_zero__less__diff,axiom,
! [N,M] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N),M)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N)) ) ).
fof(fact_650_diff__less,axiom,
! [M_1,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N_1))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M_1))
=> 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),N_1)),M_1)) ) ) ).
fof(fact_651_mult__less__mono2,axiom,
! [K_1,I_1,J] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_1),J))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K_1))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K_1),I_1)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K_1),J))) ) ) ).
fof(fact_652_mult__less__mono1,axiom,
! [K_1,I_1,J] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_1),J))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K_1))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),I_1),K_1)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),J),K_1))) ) ) ).
fof(fact_653_mult__less__cancel2,axiom,
! [M,K,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(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),K)))
<=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
& hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N)) ) ) ).
fof(fact_654_mult__less__cancel1,axiom,
! [K,M,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(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)))
<=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
& hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N)) ) ) ).
fof(fact_655_nat__0__less__mult__iff,axiom,
! [M,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),N)))
<=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M))
& hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ) ).
fof(fact_656_nat__mult__less__cancel1,axiom,
! [M,N,K] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(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)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N)) ) ) ).
fof(fact_657_nat__mult__eq__cancel1,axiom,
! [M,N,K] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
=> ( 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)
<=> M = N ) ) ).
fof(fact_658_diff__add__0,axiom,
! [N_1,M_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N_1),M_1)) = zero_zero(nat) ).
fof(fact_659_diff__is__0__eq_H,axiom,
! [M_1,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_1),N_1))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M_1),N_1) = zero_zero(nat) ) ).
fof(fact_660_diff__is__0__eq,axiom,
! [M,N] :
( hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),N) = zero_zero(nat)
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ).
fof(fact_661_mult__eq__self__implies__10,axiom,
! [M_1,N_1] :
( M_1 = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M_1),N_1)
=> ( N_1 = one_one(nat)
| M_1 = zero_zero(nat) ) ) ).
fof(fact_662_card__insert__if,axiom,
! [X_a,X_1,A_3] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_3))
=> ( ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),A_3))
=> hAPP(fun(X_a,bool),nat,finite_card(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_1),A_3)) = hAPP(fun(X_a,bool),nat,finite_card(X_a),A_3) )
& ( ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),A_3))
=> hAPP(fun(X_a,bool),nat,finite_card(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_1),A_3)) = hAPP(nat,nat,suc,hAPP(fun(X_a,bool),nat,finite_card(X_a),A_3)) ) ) ) ).
fof(fact_663_card__insert__disjoint,axiom,
! [X_a,X_1,A_3] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_3))
=> ( ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),A_3))
=> hAPP(fun(X_a,bool),nat,finite_card(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_1),A_3)) = hAPP(nat,nat,suc,hAPP(fun(X_a,bool),nat,finite_card(X_a),A_3)) ) ) ).
fof(fact_664_diff__Suc__diff__eq1,axiom,
! [M_1,K_1,J] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_1),J))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M_1),hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),J),K_1))) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M_1),K_1)),hAPP(nat,nat,suc,J)) ) ).
fof(fact_665_diff__Suc__diff__eq2,axiom,
! [M_1,K_1,J] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_1),J))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),J),K_1))),M_1) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,suc,J)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),K_1),M_1)) ) ).
fof(fact_666_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_667_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_668_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_669_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_670_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_671_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_672_mult__left__le__imp__le,axiom,
! [X_a] :
( linord20386208strict(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),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)))
=> ( 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_eq(X_a),A_1),B)) ) ) ) ).
fof(fact_673_mult__right__le__imp__le,axiom,
! [X_a] :
( linord20386208strict(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),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)))
=> ( 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_eq(X_a),A_1),B)) ) ) ) ).
fof(fact_674_mult__less__imp__less__left,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),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)))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),C))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B)) ) ) ) ).
fof(fact_675_mult__left__less__imp__less,axiom,
! [X_a] :
( linordered_semiring(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),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)))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),C))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B)) ) ) ) ).
fof(fact_676_mult__less__imp__less__right,axiom,
! [X_a] :
( linord20386208strict(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),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)))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),C))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B)) ) ) ) ).
fof(fact_677_mult__right__less__imp__less,axiom,
! [X_a] :
( linordered_semiring(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),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)))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),C))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B)) ) ) ) ).
fof(fact_678_mult__le__less__imp__less,axiom,
! [X_a] :
( linord20386208strict(X_a)
=> ! [C,D,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))
=> ( 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)),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),D))) ) ) ) ) ) ).
fof(fact_679_mult__less__le__imp__less,axiom,
! [X_a] :
( linord20386208strict(X_a)
=> ! [C,D,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))
=> ( 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)),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),D))) ) ) ) ) ) ).
fof(fact_680_mult__strict__mono_H,axiom,
! [X_a] :
( linord20386208strict(X_a)
=> ! [C,D,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))
=> ( 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)),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),D))) ) ) ) ) ) ).
fof(fact_681_mult__strict__mono,axiom,
! [X_a] :
( linord20386208strict(X_a)
=> ! [C,D,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))
=> ( 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_eq(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),D))) ) ) ) ) ) ).
fof(fact_682_mult__le__cancel__left__neg,axiom,
! [X_a] :
( linord581940658strict(X_a)
=> ! [A_2,B_2,Ca] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Ca),zero_zero(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),Ca),A_2)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Ca),B_2)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B_2),A_2)) ) ) ) ).
fof(fact_683_mult__le__cancel__left__pos,axiom,
! [X_a] :
( linord581940658strict(X_a)
=> ! [A_2,B_2,Ca] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),Ca))
=> ( 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),Ca),A_2)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Ca),B_2)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_2),B_2)) ) ) ) ).
fof(fact_684_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_685_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_686_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_687_card__eq__0__iff,axiom,
! [X_a,A_3] :
( hAPP(fun(X_a,bool),nat,finite_card(X_a),A_3) = zero_zero(nat)
<=> ( ti(fun(X_a,bool),A_3) = bot_bot(fun(X_a,bool))
| ~ hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_3)) ) ) ).
fof(fact_688_card__ge__0__finite,axiom,
! [X_a,A_3] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(fun(X_a,bool),nat,finite_card(X_a),A_3)))
=> hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_3)) ) ).
fof(fact_689_nat__diff__split,axiom,
! [Pa,A_2,B_2] :
( hBOOL(hAPP(nat,bool,Pa,hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),A_2),B_2)))
<=> ( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A_2),B_2))
=> hBOOL(hAPP(nat,bool,Pa,zero_zero(nat))) )
& ! [D_2] :
( A_2 = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),B_2),D_2)
=> hBOOL(hAPP(nat,bool,Pa,D_2)) ) ) ) ).
fof(fact_690_nat__diff__split__asm,axiom,
! [Pa,A_2,B_2] :
( hBOOL(hAPP(nat,bool,Pa,hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),A_2),B_2)))
<=> ~ ( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A_2),B_2))
& ~ hBOOL(hAPP(nat,bool,Pa,zero_zero(nat))) )
| ? [D_2] :
( A_2 = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),B_2),D_2)
& ~ hBOOL(hAPP(nat,bool,Pa,D_2)) ) ) ) ).
fof(fact_691_mult__le__cancel2,axiom,
! [M,K,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(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),K)))
<=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ) ).
fof(fact_692_mult__le__cancel1,axiom,
! [K,M,N] :
( 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),M)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K),N)))
<=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ) ).
fof(fact_693_nat__mult__le__cancel1,axiom,
! [M,N,K] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
=> ( 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),M)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K),N)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ) ).
fof(fact_694_convex__bound__le,axiom,
! [X_a] :
( linord1278240602ring_1(X_a)
=> ! [V,U,Y,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),Y),A_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),U))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),V))
=> ( hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),U),V) = 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),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),U),X)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),V),Y))),A_1)) ) ) ) ) ) ) ).
fof(fact_695_card__gt__0__iff,axiom,
! [X_a,A_3] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(fun(X_a,bool),nat,finite_card(X_a),A_3)))
<=> ( ti(fun(X_a,bool),A_3) != bot_bot(fun(X_a,bool))
& hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_3)) ) ) ).
fof(fact_696_card__insert,axiom,
! [X_a,X_1,A_3] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_3))
=> hAPP(fun(X_a,bool),nat,finite_card(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_1),A_3)) = hAPP(nat,nat,suc,hAPP(fun(X_a,bool),nat,finite_card(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_3),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),X_1),bot_bot(fun(X_a,bool)))))) ) ).
fof(fact_697_ex__least__nat__less,axiom,
! [N,Pa] :
( ~ hBOOL(hAPP(nat,bool,Pa,zero_zero(nat)))
=> ( hBOOL(hAPP(nat,bool,Pa,N))
=> ? [K_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),K_2),N))
& ! [I] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I),K_2))
=> ~ hBOOL(hAPP(nat,bool,Pa,I)) )
& hBOOL(hAPP(nat,bool,Pa,hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),K_2),one_one(nat)))) ) ) ) ).
fof(fact_698_card__Suc__eq,axiom,
! [X_a,A_3,K] :
( hAPP(fun(X_a,bool),nat,finite_card(X_a),A_3) = hAPP(nat,nat,suc,K)
<=> ? [B_5,B_4] :
( ti(fun(X_a,bool),A_3) = hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),B_5),B_4)
& ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),B_5),B_4))
& hAPP(fun(X_a,bool),nat,finite_card(X_a),B_4) = K
& ( K = zero_zero(nat)
=> ti(fun(X_a,bool),B_4) = bot_bot(fun(X_a,bool)) ) ) ) ).
fof(fact_699_mult__eq__if,axiom,
! [N_1,M_1] :
( ( M_1 = zero_zero(nat)
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M_1),N_1) = zero_zero(nat) )
& ( M_1 != zero_zero(nat)
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M_1),N_1) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M_1),one_one(nat))),N_1)) ) ) ).
fof(fact_700_add__eq__if,axiom,
! [N_1,M_1] :
( ( M_1 = zero_zero(nat)
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M_1),N_1) = N_1 )
& ( M_1 != zero_zero(nat)
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M_1),N_1) = hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M_1),one_one(nat))),N_1)) ) ) ).
fof(fact_701_bot__nat__def,axiom,
bot_bot(nat) = zero_zero(nat) ).
fof(fact_702_sum__squares__eq__zero__iff,axiom,
! [X_a] :
( linord581940658strict(X_a)
=> ! [X_1,Y_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),X_1),X_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Y_1),Y_1)) = zero_zero(X_a)
<=> ( ti(X_a,X_1) = zero_zero(X_a)
& ti(X_a,Y_1) = zero_zero(X_a) ) ) ) ).
fof(fact_703_Suc__eq__plus1__left,axiom,
! [N_1] : hAPP(nat,nat,suc,N_1) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),N_1) ).
fof(fact_704_Suc__eq__plus1,axiom,
! [N_1] : hAPP(nat,nat,suc,N_1) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N_1),one_one(nat)) ).
fof(fact_705_diff__Suc__eq__diff__pred,axiom,
! [M_1,N_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M_1),hAPP(nat,nat,suc,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_1),one_one(nat))),N_1) ).
fof(fact_706_sum__squares__ge__zero,axiom,
! [X_a] :
( linordered_ring(X_a)
=> ! [X,Y] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),X)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Y),Y)))) ) ).
fof(fact_707_sum__squares__le__zero__iff,axiom,
! [X_a] :
( linord581940658strict(X_a)
=> ! [X_1,Y_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),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X_1),X_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Y_1),Y_1))),zero_zero(X_a)))
<=> ( ti(X_a,X_1) = zero_zero(X_a)
& ti(X_a,Y_1) = zero_zero(X_a) ) ) ) ).
fof(fact_708_sum__squares__gt__zero__iff,axiom,
! [X_a] :
( linord581940658strict(X_a)
=> ! [X_1,Y_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),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X_1),X_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Y_1),Y_1))))
<=> ( ti(X_a,X_1) != zero_zero(X_a)
| ti(X_a,Y_1) != zero_zero(X_a) ) ) ) ).
fof(fact_709_not__sum__squares__lt__zero,axiom,
! [X_a] :
( linordered_ring(X_a)
=> ! [X,Y] : ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),X)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Y),Y))),zero_zero(X_a))) ) ).
fof(fact_710_Suc__pred_H,axiom,
! [N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N_1))
=> N_1 = hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N_1),one_one(nat))) ) ).
fof(fact_711_card_Oneutral,axiom,
! [X_a,A_3] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_3))
=> ( ( ? [X_2] : hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),A_3))
=> one_one(nat) = zero_zero(nat) )
=> hAPP(fun(X_a,bool),nat,finite_card(X_a),A_3) = zero_zero(nat) ) ) ).
fof(fact_712_triple_Osize_I1_J,axiom,
! [X_a,Fa,Fun1,Com_1,Fun2] : hAPP(hoare_2118899576triple(X_a),nat,hAPP(fun(X_a,nat),fun(hoare_2118899576triple(X_a),nat),hoare_2043812435e_size(X_a),Fa),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),Fun1),Com_1),Fun2)) = zero_zero(nat) ).
fof(fact_713_inc__induct,axiom,
! [Pa,I_2,J_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_2),J_2))
=> ( hBOOL(hAPP(nat,bool,Pa,J_2))
=> ( ! [I] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I),J_2))
=> ( hBOOL(hAPP(nat,bool,Pa,hAPP(nat,nat,suc,I)))
=> hBOOL(hAPP(nat,bool,Pa,I)) ) )
=> hBOOL(hAPP(nat,bool,Pa,I_2)) ) ) ) ).
fof(fact_714_ex__least__nat__le,axiom,
! [N,Pa] :
( ~ hBOOL(hAPP(nat,bool,Pa,zero_zero(nat)))
=> ( hBOOL(hAPP(nat,bool,Pa,N))
=> ? [K_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_2),N))
& ! [I] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I),K_2))
=> ~ hBOOL(hAPP(nat,bool,Pa,I)) )
& hBOOL(hAPP(nat,bool,Pa,K_2)) ) ) ) ).
fof(fact_715_less__imp__Suc__add,axiom,
! [M_1,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_1),N_1))
=> ? [K_2] : N_1 = hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M_1),K_2)) ) ).
fof(fact_716_less__imp__add__positive,axiom,
! [I_1,J] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_1),J))
=> ? [K_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K_2))
& hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),I_1),K_2) = J ) ) ).
fof(fact_717_gr0__implies__Suc,axiom,
! [N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N_1))
=> ? [M_2] : N_1 = hAPP(nat,nat,suc,M_2) ) ).
fof(fact_718_lift__Suc__mono__le,axiom,
! [X_a] :
( order(X_a)
=> ! [N,N_3,F] :
( ! [N_2] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(nat,X_a,F,N_2)),hAPP(nat,X_a,F,hAPP(nat,nat,suc,N_2))))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N),N_3))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(nat,X_a,F,N)),hAPP(nat,X_a,F,N_3))) ) ) ) ).
fof(fact_719_lift__Suc__mono__less__iff,axiom,
! [X_a] :
( order(X_a)
=> ! [N,M,F] :
( ! [N_2] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,F,N_2)),hAPP(nat,X_a,F,hAPP(nat,nat,suc,N_2))))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,F,N)),hAPP(nat,X_a,F,M)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N),M)) ) ) ) ).
fof(fact_720_lift__Suc__mono__less,axiom,
! [X_a] :
( order(X_a)
=> ! [N,N_3,F] :
( ! [N_2] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,F,N_2)),hAPP(nat,X_a,F,hAPP(nat,nat,suc,N_2))))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N),N_3))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,F,N)),hAPP(nat,X_a,F,N_3))) ) ) ) ).
fof(fact_721_even__less__0__iff,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [A_2] :
( 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_2),A_2)),zero_zero(X_a)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_2),zero_zero(X_a))) ) ) ).
fof(fact_722_card__eq__SucD,axiom,
! [X_a,A_3,K] :
( hAPP(fun(X_a,bool),nat,finite_card(X_a),A_3) = hAPP(nat,nat,suc,K)
=> ? [B_5,B_4] :
( ti(fun(X_a,bool),A_3) = hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),B_5),B_4)
& ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),B_5),B_4))
& hAPP(fun(X_a,bool),nat,finite_card(X_a),B_4) = K
& ( K = zero_zero(nat)
=> ti(fun(X_a,bool),B_4) = bot_bot(fun(X_a,bool)) ) ) ) ).
fof(fact_723_double__eq__0__iff,axiom,
! [X_a] :
( linord219039673up_add(X_a)
=> ! [A_2] :
( hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_2),A_2) = zero_zero(X_a)
<=> ti(X_a,A_2) = zero_zero(X_a) ) ) ).
fof(fact_724_triple_Osize_I2_J,axiom,
! [X_a,Fun1,Com_1,Fun2] : hAPP(hoare_2118899576triple(X_a),nat,size_size(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),Fun1),Com_1),Fun2)) = zero_zero(nat) ).
fof(fact_725_field__le__mult__one__interval,axiom,
! [X_a] :
( linord1117847801e_zero(X_a)
=> ! [X,Y] :
( ! [Z_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),Z_2))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Z_2),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),Z_2),X)),Y)) ) )
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),Y)) ) ) ).
fof(fact_726_field__le__epsilon,axiom,
! [X_a] :
( linordered_field(X_a)
=> ! [X,Y] :
( ! [E] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),E))
=> 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),plus_plus(X_a),Y),E))) )
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),Y)) ) ) ).
fof(fact_727_zero__induct__lemma,axiom,
! [I_2,Pa,K] :
( hBOOL(hAPP(nat,bool,Pa,K))
=> ( ! [N_2] :
( hBOOL(hAPP(nat,bool,Pa,hAPP(nat,nat,suc,N_2)))
=> hBOOL(hAPP(nat,bool,Pa,N_2)) )
=> hBOOL(hAPP(nat,bool,Pa,hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),K),I_2))) ) ) ).
fof(fact_728_Suc__le__D,axiom,
! [N_1,M_3] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,suc,N_1)),M_3))
=> ? [M_2] : M_3 = hAPP(nat,nat,suc,M_2) ) ).
fof(fact_729_lessE,axiom,
! [I_1,K_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_1),K_1))
=> ( K_1 != hAPP(nat,nat,suc,I_1)
=> ~ ! [J_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_1),J_1))
=> K_1 != hAPP(nat,nat,suc,J_1) ) ) ) ).
fof(fact_730_Suc__lessE,axiom,
! [I_1,K_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,suc,I_1)),K_1))
=> ~ ! [J_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_1),J_1))
=> K_1 != hAPP(nat,nat,suc,J_1) ) ) ).
fof(fact_731_not0__implies__Suc,axiom,
! [N_1] :
( N_1 != zero_zero(nat)
=> ? [M_2] : N_1 = hAPP(nat,nat,suc,M_2) ) ).
fof(fact_732_nat_Oexhaust,axiom,
! [Y] :
( Y != zero_zero(nat)
=> ~ ! [Nat] : Y != hAPP(nat,nat,suc,Nat) ) ).
fof(fact_733_zero__induct,axiom,
! [Pa,K] :
( hBOOL(hAPP(nat,bool,Pa,K))
=> ( ! [N_2] :
( hBOOL(hAPP(nat,bool,Pa,hAPP(nat,nat,suc,N_2)))
=> hBOOL(hAPP(nat,bool,Pa,N_2)) )
=> hBOOL(hAPP(nat,bool,Pa,zero_zero(nat))) ) ) ).
fof(fact_734_nat__induct,axiom,
! [N,Pa] :
( hBOOL(hAPP(nat,bool,Pa,zero_zero(nat)))
=> ( ! [N_2] :
( hBOOL(hAPP(nat,bool,Pa,N_2))
=> hBOOL(hAPP(nat,bool,Pa,hAPP(nat,nat,suc,N_2))) )
=> hBOOL(hAPP(nat,bool,Pa,N)) ) ) ).
fof(fact_735_Plus__eq__empty__conv,axiom,
! [X_b,X_a,A_3,B_3] :
( hAPP(fun(X_b,bool),fun(sum_sum(X_a,X_b),bool),hAPP(fun(X_a,bool),fun(fun(X_b,bool),fun(sum_sum(X_a,X_b),bool)),sum_Plus(X_a,X_b),A_3),B_3) = bot_bot(fun(sum_sum(X_a,X_b),bool))
<=> ( ti(fun(X_a,bool),A_3) = bot_bot(fun(X_a,bool))
& ti(fun(X_b,bool),B_3) = bot_bot(fun(X_b,bool)) ) ) ).
fof(fact_736_com_Osize_I4_J,axiom,
! [Com1_2,Com2_2] : hAPP(com,nat,com_size,hAPP(com,com,hAPP(com,fun(com,com),semi,Com1_2),Com2_2)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(com,nat,com_size,Com1_2)),hAPP(com,nat,com_size,Com2_2))),hAPP(nat,nat,suc,zero_zero(nat))) ).
fof(fact_737_com_Osize_I12_J,axiom,
! [Com1_2,Com2_2] : hAPP(com,nat,size_size(com),hAPP(com,com,hAPP(com,fun(com,com),semi,Com1_2),Com2_2)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(com,nat,size_size(com),Com1_2)),hAPP(com,nat,size_size(com),Com2_2))),hAPP(nat,nat,suc,zero_zero(nat))) ).
fof(fact_738_com_Osize_I9_J,axiom,
hAPP(com,nat,size_size(com),skip) = zero_zero(nat) ).
fof(fact_739_com_Osize_I1_J,axiom,
hAPP(com,nat,com_size,skip) = zero_zero(nat) ).
fof(fact_740_Ints__odd__less__0,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_2),ring_1_Ints(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),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),one_one(X_a)),A_2)),A_2)),zero_zero(X_a)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_2),zero_zero(X_a))) ) ) ) ).
fof(fact_741_com_Osize_I5_J,axiom,
! [Fun_1,Com1,Com2] : hAPP(com,nat,com_size,hAPP(com,com,hAPP(com,fun(com,com),hAPP(fun(state,bool),fun(com,fun(com,com)),cond,Fun_1),Com1),Com2)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(com,nat,com_size,Com1)),hAPP(com,nat,com_size,Com2))),hAPP(nat,nat,suc,zero_zero(nat))) ).
fof(fact_742_com_Osimps_I45_J,axiom,
! [Fun,Com1_1,Com2_1,Com1,Com2] : hAPP(com,com,hAPP(com,fun(com,com),hAPP(fun(state,bool),fun(com,fun(com,com)),cond,Fun),Com1_1),Com2_1) != hAPP(com,com,hAPP(com,fun(com,com),semi,Com1),Com2) ).
fof(fact_743_com_Osimps_I44_J,axiom,
! [Com1,Com2,Fun,Com1_1,Com2_1] : hAPP(com,com,hAPP(com,fun(com,com),semi,Com1),Com2) != hAPP(com,com,hAPP(com,fun(com,com),hAPP(fun(state,bool),fun(com,fun(com,com)),cond,Fun),Com1_1),Com2_1) ).
fof(fact_744_com_Osimps_I15_J,axiom,
! [Fun,Com1_1,Com2_1] : hAPP(com,com,hAPP(com,fun(com,com),hAPP(fun(state,bool),fun(com,fun(com,com)),cond,Fun),Com1_1),Com2_1) != skip ).
fof(fact_745_com_Osimps_I14_J,axiom,
! [Fun,Com1_1,Com2_1] : skip != hAPP(com,com,hAPP(com,fun(com,com),hAPP(fun(state,bool),fun(com,fun(com,com)),cond,Fun),Com1_1),Com2_1) ).
fof(fact_746_com_Osimps_I4_J,axiom,
! [Fun_1,Com1,Com2,Fun,Com1_1,Com2_1] :
( hAPP(com,com,hAPP(com,fun(com,com),hAPP(fun(state,bool),fun(com,fun(com,com)),cond,Fun_1),Com1),Com2) = hAPP(com,com,hAPP(com,fun(com,com),hAPP(fun(state,bool),fun(com,fun(com,com)),cond,Fun),Com1_1),Com2_1)
<=> ( Fun_1 = Fun
& Com1 = Com1_1
& Com2 = Com2_1 ) ) ).
fof(fact_747_Ints__1,axiom,
! [X_a] :
( ring_1(X_a)
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),one_one(X_a)),ring_1_Ints(X_a))) ) ).
fof(fact_748_Ints__diff,axiom,
! [X_a] :
( ring_1(X_a)
=> ! [B_2,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_2),ring_1_Ints(X_a)))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),B_2),ring_1_Ints(X_a)))
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_2),B_2)),ring_1_Ints(X_a))) ) ) ) ).
fof(fact_749_Ints__mult,axiom,
! [X_a] :
( ring_1(X_a)
=> ! [B_2,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_2),ring_1_Ints(X_a)))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),B_2),ring_1_Ints(X_a)))
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_2),B_2)),ring_1_Ints(X_a))) ) ) ) ).
fof(fact_750_Ints__add,axiom,
! [X_a] :
( ring_1(X_a)
=> ! [B_2,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_2),ring_1_Ints(X_a)))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),B_2),ring_1_Ints(X_a)))
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_2),B_2)),ring_1_Ints(X_a))) ) ) ) ).
fof(fact_751_Ints__0,axiom,
! [X_a] :
( ring_1(X_a)
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),zero_zero(X_a)),ring_1_Ints(X_a))) ) ).
fof(fact_752_Ints__double__eq__0__iff,axiom,
! [X_a] :
( ring_char_0(X_a)
=> ! [A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_2),ring_1_Ints(X_a)))
=> ( hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_2),A_2) = zero_zero(X_a)
<=> ti(X_a,A_2) = zero_zero(X_a) ) ) ) ).
fof(fact_753_Ints__odd__nonzero,axiom,
! [X_a] :
( ring_char_0(X_a)
=> ! [A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_2),ring_1_Ints(X_a)))
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),one_one(X_a)),A_2)),A_2) != zero_zero(X_a) ) ) ).
fof(fact_754_com_Osize_I13_J,axiom,
! [Fun_1,Com1,Com2] : hAPP(com,nat,size_size(com),hAPP(com,com,hAPP(com,fun(com,com),hAPP(fun(state,bool),fun(com,fun(com,com)),cond,Fun_1),Com1),Com2)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(com,nat,size_size(com),Com1)),hAPP(com,nat,size_size(com),Com2))),hAPP(nat,nat,suc,zero_zero(nat))) ).
fof(fact_755_com_Osize_I11_J,axiom,
! [Loc,Fun_1,Com_1] : hAPP(com,nat,size_size(com),hAPP(com,com,hAPP(fun(state,nat),fun(com,com),hAPP(loc,fun(fun(state,nat),fun(com,com)),local,Loc),Fun_1),Com_1)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(com,nat,size_size(com),Com_1)),hAPP(nat,nat,suc,zero_zero(nat))) ).
fof(fact_756_com_Osize_I3_J,axiom,
! [Loc,Fun_1,Com_1] : hAPP(com,nat,com_size,hAPP(com,com,hAPP(fun(state,nat),fun(com,com),hAPP(loc,fun(fun(state,nat),fun(com,com)),local,Loc),Fun_1),Com_1)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(com,nat,com_size,Com_1)),hAPP(nat,nat,suc,zero_zero(nat))) ).
fof(fact_757_com_Osimps_I36_J,axiom,
! [Loc,Fun_1,Com_1,Fun,Com1_1,Com2_1] : hAPP(com,com,hAPP(fun(state,nat),fun(com,com),hAPP(loc,fun(fun(state,nat),fun(com,com)),local,Loc),Fun_1),Com_1) != hAPP(com,com,hAPP(com,fun(com,com),hAPP(fun(state,bool),fun(com,fun(com,com)),cond,Fun),Com1_1),Com2_1) ).
fof(fact_758_com_Osimps_I37_J,axiom,
! [Fun,Com1_1,Com2_1,Loc,Fun_1,Com_1] : hAPP(com,com,hAPP(com,fun(com,com),hAPP(fun(state,bool),fun(com,fun(com,com)),cond,Fun),Com1_1),Com2_1) != hAPP(com,com,hAPP(fun(state,nat),fun(com,com),hAPP(loc,fun(fun(state,nat),fun(com,com)),local,Loc),Fun_1),Com_1) ).
fof(fact_759_com_Osimps_I2_J,axiom,
! [Loc,Fun_1,Com_1,Loc_1,Fun,Com] :
( hAPP(com,com,hAPP(fun(state,nat),fun(com,com),hAPP(loc,fun(fun(state,nat),fun(com,com)),local,Loc),Fun_1),Com_1) = hAPP(com,com,hAPP(fun(state,nat),fun(com,com),hAPP(loc,fun(fun(state,nat),fun(com,com)),local,Loc_1),Fun),Com)
<=> ( ti(loc,Loc) = ti(loc,Loc_1)
& Fun_1 = Fun
& Com_1 = Com ) ) ).
fof(fact_760_com_Osimps_I10_J,axiom,
! [Loc_1,Fun,Com] : skip != hAPP(com,com,hAPP(fun(state,nat),fun(com,com),hAPP(loc,fun(fun(state,nat),fun(com,com)),local,Loc_1),Fun),Com) ).
fof(fact_761_com_Osimps_I11_J,axiom,
! [Loc_1,Fun,Com] : hAPP(com,com,hAPP(fun(state,nat),fun(com,com),hAPP(loc,fun(fun(state,nat),fun(com,com)),local,Loc_1),Fun),Com) != skip ).
fof(fact_762_com_Osimps_I35_J,axiom,
! [Com1_1,Com2_1,Loc,Fun_1,Com_1] : hAPP(com,com,hAPP(com,fun(com,com),semi,Com1_1),Com2_1) != hAPP(com,com,hAPP(fun(state,nat),fun(com,com),hAPP(loc,fun(fun(state,nat),fun(com,com)),local,Loc),Fun_1),Com_1) ).
fof(fact_763_com_Osimps_I34_J,axiom,
! [Loc,Fun_1,Com_1,Com1_1,Com2_1] : hAPP(com,com,hAPP(fun(state,nat),fun(com,com),hAPP(loc,fun(fun(state,nat),fun(com,com)),local,Loc),Fun_1),Com_1) != hAPP(com,com,hAPP(com,fun(com,com),semi,Com1_1),Com2_1) ).
fof(fact_764_com_Osize_I6_J,axiom,
! [Fun_1,Com_1] : hAPP(com,nat,com_size,hAPP(com,com,hAPP(fun(state,bool),fun(com,com),while,Fun_1),Com_1)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(com,nat,com_size,Com_1)),hAPP(nat,nat,suc,zero_zero(nat))) ).
fof(fact_765_com_Osize_I14_J,axiom,
! [Fun_1,Com_1] : hAPP(com,nat,size_size(com),hAPP(com,com,hAPP(fun(state,bool),fun(com,com),while,Fun_1),Com_1)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(com,nat,size_size(com),Com_1)),hAPP(nat,nat,suc,zero_zero(nat))) ).
fof(fact_766_com_Osimps_I39_J,axiom,
! [Fun,Com,Loc,Fun_1,Com_1] : hAPP(com,com,hAPP(fun(state,bool),fun(com,com),while,Fun),Com) != hAPP(com,com,hAPP(fun(state,nat),fun(com,com),hAPP(loc,fun(fun(state,nat),fun(com,com)),local,Loc),Fun_1),Com_1) ).
fof(fact_767_com_Osimps_I38_J,axiom,
! [Loc,Fun_1,Com_1,Fun,Com] : hAPP(com,com,hAPP(fun(state,nat),fun(com,com),hAPP(loc,fun(fun(state,nat),fun(com,com)),local,Loc),Fun_1),Com_1) != hAPP(com,com,hAPP(fun(state,bool),fun(com,com),while,Fun),Com) ).
fof(fact_768_com_Osimps_I47_J,axiom,
! [Fun,Com,Com1,Com2] : hAPP(com,com,hAPP(fun(state,bool),fun(com,com),while,Fun),Com) != hAPP(com,com,hAPP(com,fun(com,com),semi,Com1),Com2) ).
fof(fact_769_com_Osimps_I46_J,axiom,
! [Com1,Com2,Fun,Com] : hAPP(com,com,hAPP(com,fun(com,com),semi,Com1),Com2) != hAPP(com,com,hAPP(fun(state,bool),fun(com,com),while,Fun),Com) ).
fof(fact_770_com_Osimps_I17_J,axiom,
! [Fun,Com] : hAPP(com,com,hAPP(fun(state,bool),fun(com,com),while,Fun),Com) != skip ).
fof(fact_771_com_Osimps_I16_J,axiom,
! [Fun,Com] : skip != hAPP(com,com,hAPP(fun(state,bool),fun(com,com),while,Fun),Com) ).
fof(fact_772_com_Osimps_I5_J,axiom,
! [Fun_1,Com_1,Fun,Com] :
( hAPP(com,com,hAPP(fun(state,bool),fun(com,com),while,Fun_1),Com_1) = hAPP(com,com,hAPP(fun(state,bool),fun(com,com),while,Fun),Com)
<=> ( Fun_1 = Fun
& Com_1 = Com ) ) ).
fof(fact_773_com_Osimps_I53_J,axiom,
! [Fun,Com,Fun_1,Com1,Com2] : hAPP(com,com,hAPP(fun(state,bool),fun(com,com),while,Fun),Com) != hAPP(com,com,hAPP(com,fun(com,com),hAPP(fun(state,bool),fun(com,fun(com,com)),cond,Fun_1),Com1),Com2) ).
fof(fact_774_com_Osimps_I52_J,axiom,
! [Fun_1,Com1,Com2,Fun,Com] : hAPP(com,com,hAPP(com,fun(com,com),hAPP(fun(state,bool),fun(com,fun(com,com)),cond,Fun_1),Com1),Com2) != hAPP(com,com,hAPP(fun(state,bool),fun(com,com),while,Fun),Com) ).
fof(fact_775_expand__Suc,axiom,
! [V] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(int,nat,number_number_of(nat),V)))
=> hAPP(int,nat,number_number_of(nat),V) = hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(int,nat,number_number_of(nat),V)),one_one(nat))) ) ).
fof(fact_776_fold1__antimono,axiom,
! [X_a] :
( linorder(X_a)
=> ! [B_3,A_3] :
( ti(fun(X_a,bool),A_3) != bot_bot(fun(X_a,bool))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_3),B_3))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),B_3))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(fun(X_a,bool),X_a,hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,bool),X_a),finite_fold1(X_a),ord_min(X_a)),B_3)),hAPP(fun(X_a,bool),X_a,hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,bool),X_a),finite_fold1(X_a),ord_min(X_a)),A_3))) ) ) ) ) ).
fof(fact_777_Ints__number__of,axiom,
! [X_a] :
( number_ring(X_a)
=> ! [W] : hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),hAPP(int,X_a,number_number_of(X_a),W)),ring_1_Ints(X_a))) ) ).
fof(fact_778_number__of__add,axiom,
! [X_a] :
( number_ring(X_a)
=> ! [V,W_1] : hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),V),W_1)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(int,X_a,number_number_of(X_a),V)),hAPP(int,X_a,number_number_of(X_a),W_1)) ) ).
fof(fact_779_add__number__of__eq,axiom,
! [X_a] :
( number_ring(X_a)
=> ! [V,W_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(int,X_a,number_number_of(X_a),V)),hAPP(int,X_a,number_number_of(X_a),W_1)) = hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),V),W_1)) ) ).
fof(fact_780_add__number__of__left,axiom,
! [X_a] :
( number_ring(X_a)
=> ! [V,W_1,Z] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(int,X_a,number_number_of(X_a),V)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(int,X_a,number_number_of(X_a),W_1)),Z)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),V),W_1))),Z) ) ).
fof(fact_781_le__number__of__eq__not__less,axiom,
! [X_a] :
( ( number(X_a)
& linorder(X_a) )
=> ! [V_1,W] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(int,X_a,number_number_of(X_a),V_1)),hAPP(int,X_a,number_number_of(X_a),W)))
<=> ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(int,X_a,number_number_of(X_a),W)),hAPP(int,X_a,number_number_of(X_a),V_1))) ) ) ).
fof(fact_782_left__distrib__number__of,axiom,
! [X_b] :
( ( number(X_b)
& semiring(X_b) )
=> ! [A_1,B,V] : 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),A_1),B)),hAPP(int,X_b,number_number_of(X_b),V)) = 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_1),hAPP(int,X_b,number_number_of(X_b),V))),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),B),hAPP(int,X_b,number_number_of(X_b),V))) ) ).
fof(fact_783_right__distrib__number__of,axiom,
! [X_b] :
( ( number(X_b)
& semiring(X_b) )
=> ! [V,B,C] : hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),hAPP(int,X_b,number_number_of(X_b),V)),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),B),C)) = 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(int,X_b,number_number_of(X_b),V)),B)),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),hAPP(int,X_b,number_number_of(X_b),V)),C)) ) ).
fof(fact_784_add__number__of__diff1,axiom,
! [X_a] :
( number_ring(X_a)
=> ! [V,W_1,C] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(int,X_a,number_number_of(X_a),V)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(int,X_a,number_number_of(X_a),W_1)),C)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),V),W_1))),C) ) ).
fof(fact_785_left__diff__distrib__number__of,axiom,
! [X_b] :
( ( number(X_b)
& ring(X_b) )
=> ! [A_1,B,V] : 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_1),B)),hAPP(int,X_b,number_number_of(X_b),V)) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),minus_minus(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),A_1),hAPP(int,X_b,number_number_of(X_b),V))),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),B),hAPP(int,X_b,number_number_of(X_b),V))) ) ).
fof(fact_786_right__diff__distrib__number__of,axiom,
! [X_b] :
( ( number(X_b)
& ring(X_b) )
=> ! [V,B,C] : hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),hAPP(int,X_b,number_number_of(X_b),V)),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),minus_minus(X_b),B),C)) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),minus_minus(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),hAPP(int,X_b,number_number_of(X_b),V)),B)),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),hAPP(int,X_b,number_number_of(X_b),V)),C)) ) ).
fof(fact_787_number__of__mult,axiom,
! [X_a] :
( number_ring(X_a)
=> ! [V,W_1] : hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),V),W_1)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(int,X_a,number_number_of(X_a),V)),hAPP(int,X_a,number_number_of(X_a),W_1)) ) ).
fof(fact_788_arith__simps_I32_J,axiom,
! [X_a] :
( number_ring(X_a)
=> ! [V,W_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(int,X_a,number_number_of(X_a),V)),hAPP(int,X_a,number_number_of(X_a),W_1)) = hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),V),W_1)) ) ).
fof(fact_789_mult__number__of__left,axiom,
! [X_a] :
( number_ring(X_a)
=> ! [V,W_1,Z] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(int,X_a,number_number_of(X_a),V)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(int,X_a,number_number_of(X_a),W_1)),Z)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),V),W_1))),Z) ) ).
fof(fact_790_less__number__of,axiom,
! [X_a] :
( ( number_ring(X_a)
& linordered_idom(X_a) )
=> ! [X_1,Y_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(int,X_a,number_number_of(X_a),X_1)),hAPP(int,X_a,number_number_of(X_a),Y_1)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),X_1),Y_1)) ) ) ).
fof(fact_791_min__number__of,axiom,
! [X_b] :
( ( number(X_b)
& ord(X_b) )
=> ! [U,V] :
( ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(int,X_b,number_number_of(X_b),U)),hAPP(int,X_b,number_number_of(X_b),V)))
=> hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),ord_min(X_b),hAPP(int,X_b,number_number_of(X_b),U)),hAPP(int,X_b,number_number_of(X_b),V)) = hAPP(int,X_b,number_number_of(X_b),U) )
& ( ~ hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(int,X_b,number_number_of(X_b),U)),hAPP(int,X_b,number_number_of(X_b),V)))
=> hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),ord_min(X_b),hAPP(int,X_b,number_number_of(X_b),U)),hAPP(int,X_b,number_number_of(X_b),V)) = hAPP(int,X_b,number_number_of(X_b),V) ) ) ) ).
fof(fact_792_le__number__of,axiom,
! [X_a] :
( ( number_ring(X_a)
& linordered_idom(X_a) )
=> ! [X_1,Y_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(int,X_a,number_number_of(X_a),X_1)),hAPP(int,X_a,number_number_of(X_a),Y_1)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),X_1),Y_1)) ) ) ).
fof(fact_793_number__of__reorient,axiom,
! [X_a] :
( number(X_a)
=> ! [W,X_1] :
( hAPP(int,X_a,number_number_of(X_a),W) = ti(X_a,X_1)
<=> ti(X_a,X_1) = hAPP(int,X_a,number_number_of(X_a),W) ) ) ).
fof(fact_794_eq__number__of,axiom,
! [X_a] :
( ( number_ring(X_a)
& ring_char_0(X_a) )
=> ! [X_1,Y_1] :
( hAPP(int,X_a,number_number_of(X_a),X_1) = hAPP(int,X_a,number_number_of(X_a),Y_1)
<=> X_1 = Y_1 ) ) ).
fof(fact_795_number__of__diff,axiom,
! [X_a] :
( number_ring(X_a)
=> ! [V,W_1] : hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),V),W_1)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(int,X_a,number_number_of(X_a),V)),hAPP(int,X_a,number_number_of(X_a),W_1)) ) ).
fof(fact_796_min__Suc__Suc,axiom,
! [M_1,N_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_min(nat),hAPP(nat,nat,suc,M_1)),hAPP(nat,nat,suc,N_1)) = hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_min(nat),M_1),N_1)) ).
fof(fact_797_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_798_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_799_min__max_Ole__iff__inf,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X_1,Y_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_1),Y_1))
<=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X_1),Y_1) = ti(X_a,X_1) ) ) ).
fof(fact_800_min__max_Ole__inf__iff,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X_1,Y_1,Z_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),Y_1),Z_1)))
<=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_1),Y_1))
& hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_1),Z_1)) ) ) ) ).
fof(fact_801_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_802_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_803_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_804_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_805_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_806_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_807_min__max_Oinf__mono,axiom,
! [X_a] :
( linorder(X_a)
=> ! [B,D,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))
=> 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))) ) ) ) ).
fof(fact_808_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_809_min__le__iff__disj,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X_1,Y_1,Z_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X_1),Y_1)),Z_1))
<=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_1),Z_1))
| hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y_1),Z_1)) ) ) ) ).
fof(fact_810_min__less__iff__disj,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X_1,Y_1,Z_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X_1),Y_1)),Z_1))
<=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_1),Z_1))
| hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y_1),Z_1)) ) ) ) ).
fof(fact_811_min__less__iff__conj,axiom,
! [X_a] :
( linorder(X_a)
=> ! [Z_1,X_1,Y_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Z_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X_1),Y_1)))
<=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Z_1),X_1))
& hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Z_1),Y_1)) ) ) ) ).
fof(fact_812_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_813_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_814_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_815_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_816_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_817_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_818_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_819_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_820_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_821_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_822_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_823_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_824_min__max_Ocomp__fun__idem__inf,axiom,
! [X_a] :
( linorder(X_a)
=> hBOOL(hAPP(fun(X_a,fun(X_a,X_a)),bool,finite_comp_fun_idem(X_a,X_a),ord_min(X_a))) ) ).
fof(fact_825_min__diff,axiom,
! [M_1,I_1,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_1),I_1)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N_1),I_1)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_min(nat),M_1),N_1)),I_1) ).
fof(fact_826_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_827_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_828_min__0L,axiom,
! [N_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_min(nat),zero_zero(nat)),N_1) = zero_zero(nat) ).
fof(fact_829_min__0R,axiom,
! [N_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_min(nat),N_1),zero_zero(nat)) = zero_zero(nat) ).
fof(fact_830_min__max_Ofold__inf__le__inf,axiom,
! [X_a] :
( linorder(X_a)
=> ! [B_2,A_2,A_3] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_3))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_2),A_3))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(fun(X_a,bool),X_a,hAPP(X_a,fun(fun(X_a,bool),X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(fun(X_a,bool),X_a)),finite_fold(X_a,X_a),ord_min(X_a)),B_2),A_3)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),A_2),B_2))) ) ) ) ).
fof(fact_831_min__max_Ofold__inf__insert,axiom,
! [X_a] :
( linorder(X_a)
=> ! [B_2,A_2,A_3] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_3))
=> hAPP(fun(X_a,bool),X_a,hAPP(X_a,fun(fun(X_a,bool),X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(fun(X_a,bool),X_a)),finite_fold(X_a,X_a),ord_min(X_a)),B_2),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(X_a,fun(fun(X_a,bool),fun(X_a,bool)),insert(X_a),A_2),A_3)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),A_2),hAPP(fun(X_a,bool),X_a,hAPP(X_a,fun(fun(X_a,bool),X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(fun(X_a,bool),X_a)),finite_fold(X_a,X_a),ord_min(X_a)),B_2),A_3)) ) ) ).
fof(fact_832_min__max_Ofold1__belowI,axiom,
! [X_a] :
( linorder(X_a)
=> ! [A_2,A_3] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_3))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_2),A_3))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(fun(X_a,bool),X_a,hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,bool),X_a),finite_fold1(X_a),ord_min(X_a)),A_3)),A_2)) ) ) ) ).
fof(fact_833_fold1__below__iff,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X_1,A_3] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_3))
=> ( ti(fun(X_a,bool),A_3) != bot_bot(fun(X_a,bool))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(fun(X_a,bool),X_a,hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,bool),X_a),finite_fold1(X_a),ord_min(X_a)),A_3)),X_1))
<=> ? [X_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),A_3))
& hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_2),X_1)) ) ) ) ) ) ).
fof(fact_834_min__max_Obelow__fold1__iff,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X_1,A_3] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_3))
=> ( ti(fun(X_a,bool),A_3) != bot_bot(fun(X_a,bool))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_1),hAPP(fun(X_a,bool),X_a,hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,bool),X_a),finite_fold1(X_a),ord_min(X_a)),A_3)))
<=> ! [X_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),A_3))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_1),X_2)) ) ) ) ) ) ).
fof(fact_835_strict__below__fold1__iff,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X_1,A_3] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_3))
=> ( ti(fun(X_a,bool),A_3) != bot_bot(fun(X_a,bool))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_1),hAPP(fun(X_a,bool),X_a,hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,bool),X_a),finite_fold1(X_a),ord_min(X_a)),A_3)))
<=> ! [X_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),A_3))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_1),X_2)) ) ) ) ) ) ).
fof(fact_836_fold1__strict__below__iff,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X_1,A_3] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_3))
=> ( ti(fun(X_a,bool),A_3) != bot_bot(fun(X_a,bool))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(fun(X_a,bool),X_a,hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,bool),X_a),finite_fold1(X_a),ord_min(X_a)),A_3)),X_1))
<=> ? [X_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),A_3))
& hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_2),X_1)) ) ) ) ) ) ).
fof(fact_837_min__max_Oinf__le__fold__inf,axiom,
! [X_a] :
( linorder(X_a)
=> ! [Ca,B_2,A_3] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_3))
=> ( ! [X_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),A_3))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B_2),X_2)) )
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),B_2),Ca)),hAPP(fun(X_a,bool),X_a,hAPP(X_a,fun(fun(X_a,bool),X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(fun(X_a,bool),X_a)),finite_fold(X_a,X_a),ord_min(X_a)),Ca),A_3))) ) ) ) ).
fof(fact_838_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_839_plus__numeral__code_I9_J,axiom,
! [V,W_1] : hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,number_number_of(int),V)),hAPP(int,int,number_number_of(int),W_1)) = hAPP(int,int,number_number_of(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),V),W_1)) ).
fof(fact_840_times__numeral__code_I5_J,axiom,
! [V,W_1] : hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),V)),hAPP(int,int,number_number_of(int),W_1)) = hAPP(int,int,number_number_of(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),V),W_1)) ).
fof(fact_841_less__number__of__int__code,axiom,
! [K,L] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),K)),hAPP(int,int,number_number_of(int),L)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),L)) ) ).
fof(fact_842_less__eq__number__of__int__code,axiom,
! [K,L] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,number_number_of(int),K)),hAPP(int,int,number_number_of(int),L)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K),L)) ) ).
fof(fact_843_zmult__1,axiom,
! [Z] : hAPP(int,int,hAPP(int,fun(int,int),times_times(int),one_one(int)),Z) = Z ).
fof(fact_844_zmult__1__right,axiom,
! [Z] : hAPP(int,int,hAPP(int,fun(int,int),times_times(int),Z),one_one(int)) = Z ).
fof(fact_845_zle__diff1__eq,axiom,
! [W,Z_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),W),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),Z_1),one_one(int))))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),W),Z_1)) ) ).
fof(fact_846_zle__antisym,axiom,
! [Z,W_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Z),W_1))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),W_1),Z))
=> Z = W_1 ) ) ).
fof(fact_847_zle__trans,axiom,
! [K_1,I_1,J] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),I_1),J))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),J),K_1))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),I_1),K_1)) ) ) ).
fof(fact_848_zle__linear,axiom,
! [Z,W_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Z),W_1))
| hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),W_1),Z)) ) ).
fof(fact_849_zle__refl,axiom,
! [W_1] : hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),W_1),W_1)) ).
fof(fact_850_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_851_zless__le,axiom,
! [Z_1,W] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Z_1),W))
<=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Z_1),W))
& Z_1 != W ) ) ).
fof(fact_852_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_853_zmult__commute,axiom,
! [Z,W_1] : hAPP(int,int,hAPP(int,fun(int,int),times_times(int),Z),W_1) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),W_1),Z) ).
fof(fact_854_zdiff__zmult__distrib2,axiom,
! [W_1,Z1,Z2] : hAPP(int,int,hAPP(int,fun(int,int),times_times(int),W_1),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_1),Z1)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),W_1),Z2)) ).
fof(fact_855_zdiff__zmult__distrib,axiom,
! [Z1,Z2,W_1] : 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_1) = hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),Z1),W_1)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),Z2),W_1)) ).
fof(fact_856_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_857_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_858_zadd__commute,axiom,
! [Z,W_1] : hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Z),W_1) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),W_1),Z) ).
fof(fact_859_zadd__left__mono,axiom,
! [K_1,I_1,J] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),I_1),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_1),I_1)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),K_1),J))) ) ).
fof(fact_860_zadd__strict__right__mono,axiom,
! [K_1,I_1,J] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),I_1),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_1),K_1)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),J),K_1))) ) ).
fof(fact_861_zadd__zless__mono,axiom,
! [Z_3,Z,W_2,W_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),W_2),W_1))
=> ( 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_2),Z_3)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),W_1),Z))) ) ) ).
fof(fact_862_zle__add1__eq__le,axiom,
! [W,Z_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),W),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Z_1),one_one(int))))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),W),Z_1)) ) ).
fof(fact_863_add1__zle__eq,axiom,
! [W,Z_1] :
( 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),one_one(int))),Z_1))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),W),Z_1)) ) ).
fof(fact_864_zless__imp__add1__zle,axiom,
! [W_1,Z] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),W_1),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_1),one_one(int))),Z)) ) ).
fof(fact_865_zless__add1__eq,axiom,
! [W,Z_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),W),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Z_1),one_one(int))))
<=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),W),Z_1))
| W = Z_1 ) ) ).
fof(fact_866_zadd__zmult__distrib,axiom,
! [Z1,Z2,W_1] : 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_1) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),Z1),W_1)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),Z2),W_1)) ).
fof(fact_867_zadd__zmult__distrib2,axiom,
! [W_1,Z1,Z2] : hAPP(int,int,hAPP(int,fun(int,int),times_times(int),W_1),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_1),Z1)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),W_1),Z2)) ).
fof(fact_868_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_869_pos__zmult__eq__1__iff,axiom,
! [N,M] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),M))
=> ( hAPP(int,int,hAPP(int,fun(int,int),times_times(int),M),N) = one_one(int)
<=> ( M = one_one(int)
& N = one_one(int) ) ) ) ).
fof(fact_870_int__one__le__iff__zero__less,axiom,
! [Z_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),one_one(int)),Z_1))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),Z_1)) ) ).
fof(fact_871_less__bin__lemma,axiom,
! [K,L] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),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),L)),zero_zero(int))) ) ).
fof(fact_872_zmult__zless__mono2,axiom,
! [K_1,I_1,J] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),I_1),J))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),K_1))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K_1),I_1)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K_1),J))) ) ) ).
fof(fact_873_zadd__0__right,axiom,
! [Z] : hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Z),zero_zero(int)) = Z ).
fof(fact_874_zadd__0,axiom,
! [Z] : hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),zero_zero(int)),Z) = Z ).
fof(fact_875_le__imp__0__less,axiom,
! [Z] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),one_one(int)),Z))) ) ).
fof(fact_876_odd__less__0,axiom,
! [Z_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),one_one(int)),Z_1)),Z_1)),zero_zero(int)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Z_1),zero_zero(int))) ) ).
fof(fact_877_odd__nonzero,axiom,
! [Z] : hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),one_one(int)),Z)),Z) != zero_zero(int) ).
fof(fact_878_zdiv__mono2__neg__lemma,axiom,
! [B,Q_2,R_2,B_1,Q_1,R_1] :
( hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B),Q_2)),R_2) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B_1),Q_1)),R_1)
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B_1),Q_1)),R_1)),zero_zero(int)))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),R_2),B))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),R_1))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B_1))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B_1),B))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Q_1),Q_2)) ) ) ) ) ) ) ).
fof(fact_879_unique__quotient__lemma__neg,axiom,
! [B,Q_1,R_1,Q_2,R_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),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B),Q_1)),R_1)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B),Q_2)),R_2)))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),R_2),zero_zero(int)))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),R_2))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),R_1))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Q_2),Q_1)) ) ) ) ) ).
fof(fact_880_number__of__is__id,axiom,
! [K_1] : hAPP(int,int,number_number_of(int),K_1) = K_1 ).
fof(fact_881_int__0__neq__1,axiom,
zero_zero(int) != one_one(int) ).
fof(fact_882_self__quotient__aux2,axiom,
! [R_2,Q_2,A_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A_1))
=> ( A_1 = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),R_2),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),Q_2))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),R_2))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Q_2),one_one(int))) ) ) ) ).
fof(fact_883_self__quotient__aux1,axiom,
! [R_2,Q_2,A_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A_1))
=> ( A_1 = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),R_2),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),Q_2))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),R_2),A_1))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),one_one(int)),Q_2)) ) ) ) ).
fof(fact_884_q__pos__lemma,axiom,
! [B_1,Q_1,R_1] :
( 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),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B_1),Q_1)),R_1)))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),R_1),B_1))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B_1))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Q_1)) ) ) ) ).
fof(fact_885_q__neg__lemma,axiom,
! [B_1,Q_1,R_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B_1),Q_1)),R_1)),zero_zero(int)))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),R_1))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B_1))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Q_1),zero_zero(int))) ) ) ) ).
fof(fact_886_unique__quotient__lemma,axiom,
! [B,Q_1,R_1,Q_2,R_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),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B),Q_1)),R_1)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B),Q_2)),R_2)))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),R_1))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),R_1),B))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),R_2),B))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Q_1),Q_2)) ) ) ) ) ).
fof(fact_887_zdiv__mono2__lemma,axiom,
! [B,Q_2,R_2,B_1,Q_1,R_1] :
( hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B),Q_2)),R_2) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B_1),Q_1)),R_1)
=> ( 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),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B_1),Q_1)),R_1)))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),R_1),B_1))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),R_2))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B_1))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B_1),B))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Q_2),Q_1)) ) ) ) ) ) ) ).
fof(fact_888_int__less__induct,axiom,
! [Pa,I_2,K] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),I_2),K))
=> ( hBOOL(hAPP(int,bool,Pa,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),K),one_one(int))))
=> ( ! [I] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),I),K))
=> ( hBOOL(hAPP(int,bool,Pa,I))
=> hBOOL(hAPP(int,bool,Pa,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),I),one_one(int)))) ) )
=> hBOOL(hAPP(int,bool,Pa,I_2)) ) ) ) ).
fof(fact_889_int__le__induct,axiom,
! [Pa,I_2,K] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),I_2),K))
=> ( hBOOL(hAPP(int,bool,Pa,K))
=> ( ! [I] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),I),K))
=> ( hBOOL(hAPP(int,bool,Pa,I))
=> hBOOL(hAPP(int,bool,Pa,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),I),one_one(int)))) ) )
=> hBOOL(hAPP(int,bool,Pa,I_2)) ) ) ) ).
fof(fact_890_int__gr__induct,axiom,
! [Pa,K,I_2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),I_2))
=> ( hBOOL(hAPP(int,bool,Pa,hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),K),one_one(int))))
=> ( ! [I] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),I))
=> ( hBOOL(hAPP(int,bool,Pa,I))
=> hBOOL(hAPP(int,bool,Pa,hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),I),one_one(int)))) ) )
=> hBOOL(hAPP(int,bool,Pa,I_2)) ) ) ) ).
fof(fact_891_int__ge__induct,axiom,
! [Pa,K,I_2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K),I_2))
=> ( hBOOL(hAPP(int,bool,Pa,K))
=> ( ! [I] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K),I))
=> ( hBOOL(hAPP(int,bool,Pa,I))
=> hBOOL(hAPP(int,bool,Pa,hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),I),one_one(int)))) ) )
=> hBOOL(hAPP(int,bool,Pa,I_2)) ) ) ) ).
fof(fact_892_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_893_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_894_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_895_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_896_decr__mult__lemma,axiom,
! [K,Pa,D_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),D_1))
=> ( ! [X_2] :
( hBOOL(hAPP(int,bool,Pa,X_2))
=> hBOOL(hAPP(int,bool,Pa,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),X_2),D_1))) )
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K))
=> ! [X_2] :
( hBOOL(hAPP(int,bool,Pa,X_2))
=> hBOOL(hAPP(int,bool,Pa,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),X_2),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K),D_1)))) ) ) ) ) ).
fof(fact_897_incr__mult__lemma,axiom,
! [K,Pa,D_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),D_1))
=> ( ! [X_2] :
( hBOOL(hAPP(int,bool,Pa,X_2))
=> hBOOL(hAPP(int,bool,Pa,hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),X_2),D_1))) )
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K))
=> ! [X_2] :
( hBOOL(hAPP(int,bool,Pa,X_2))
=> hBOOL(hAPP(int,bool,Pa,hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),X_2),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K),D_1)))) ) ) ) ) ).
fof(fact_898_imp__le__cong,axiom,
! [P_1,Pa,X_1] :
( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X_1))
=> ( hBOOL(Pa)
<=> hBOOL(P_1) ) )
=> ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X_1))
=> hBOOL(Pa) )
<=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X_1))
=> hBOOL(P_1) ) ) ) ).
fof(fact_899_conj__le__cong,axiom,
! [P_1,Pa,X_1] :
( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X_1))
=> ( hBOOL(Pa)
<=> hBOOL(P_1) ) )
=> ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X_1))
& hBOOL(Pa) )
<=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X_1))
& hBOOL(P_1) ) ) ) ).
fof(fact_900_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_901_min__ord__min,axiom,
! [X_a] :
( ord(X_a)
=> ord_min(X_a) = hAPP(fun(X_a,fun(X_a,bool)),fun(X_a,fun(X_a,X_a)),min(X_a),ord_less_eq(X_a)) ) ).
fof(fact_902_pinf_I6_J,axiom,
! [X_a] :
( linorder(X_a)
=> ! [T_3] :
? [Z_2] :
! [X_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Z_2),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_903_minf_I6_J,axiom,
! [X_a] :
( linorder(X_a)
=> ! [T_3] :
? [Z_2] :
! [X_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_2),Z_2))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_2),T_3)) ) ) ).
fof(fact_904_minf_I8_J,axiom,
! [X_a] :
( linorder(X_a)
=> ! [T_3] :
? [Z_2] :
! [X_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_2),Z_2))
=> ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),T_3),X_2)) ) ) ).
fof(fact_905_pinf_I8_J,axiom,
! [X_a] :
( linorder(X_a)
=> ! [T_3] :
? [Z_2] :
! [X_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Z_2),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_906_int__induct,axiom,
! [I_2,Pa,K] :
( hBOOL(hAPP(int,bool,Pa,K))
=> ( ! [I] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K),I))
=> ( hBOOL(hAPP(int,bool,Pa,I))
=> hBOOL(hAPP(int,bool,Pa,hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),I),one_one(int)))) ) )
=> ( ! [I] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),I),K))
=> ( hBOOL(hAPP(int,bool,Pa,I))
=> hBOOL(hAPP(int,bool,Pa,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),I),one_one(int)))) ) )
=> hBOOL(hAPP(int,bool,Pa,I_2)) ) ) ) ).
fof(fact_907_plusinfinity,axiom,
! [Pa,P_1,D_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),D_1))
=> ( ! [X_2,K_2] :
( hBOOL(hAPP(int,bool,P_1,X_2))
<=> hBOOL(hAPP(int,bool,P_1,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),X_2),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K_2),D_1)))) )
=> ( ? [Z_2] :
! [X_2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Z_2),X_2))
=> ( hBOOL(hAPP(int,bool,Pa,X_2))
<=> hBOOL(hAPP(int,bool,P_1,X_2)) ) )
=> ( ? [X1] : hBOOL(hAPP(int,bool,P_1,X1))
=> ? [X1] : hBOOL(hAPP(int,bool,Pa,X1)) ) ) ) ) ).
fof(fact_908_minusinfinity,axiom,
! [Pa,P1,D_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),D_1))
=> ( ! [X_2,K_2] :
( hBOOL(hAPP(int,bool,P1,X_2))
<=> hBOOL(hAPP(int,bool,P1,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),X_2),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K_2),D_1)))) )
=> ( ? [Z_2] :
! [X_2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),X_2),Z_2))
=> ( hBOOL(hAPP(int,bool,Pa,X_2))
<=> hBOOL(hAPP(int,bool,P1,X_2)) ) )
=> ( ? [X1] : hBOOL(hAPP(int,bool,P1,X1))
=> ? [X1] : hBOOL(hAPP(int,bool,Pa,X1)) ) ) ) ) ).
fof(fact_909_tsub__def,axiom,
! [Y,X] :
( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Y),X))
=> hAPP(int,int,hAPP(int,fun(int,int),nat_tsub,X),Y) = hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),X),Y) )
& ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Y),X))
=> hAPP(int,int,hAPP(int,fun(int,int),nat_tsub,X),Y) = zero_zero(int) ) ) ).
fof(fact_910_Nat__Transfer_Otransfer__nat__int__function__closures_I3_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),nat_tsub,X),Y))) ) ) ).
fof(fact_911_tsub__eq,axiom,
! [Y,X] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Y),X))
=> hAPP(int,int,hAPP(int,fun(int,int),nat_tsub,X),Y) = hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),X),Y) ) ).
fof(fact_912_of__nat__number__of__lemma,axiom,
! [X_a] :
( number_ring(X_a)
=> ! [V] :
( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,number_number_of(int),V)))
=> hAPP(nat,X_a,semiring_1_of_nat(X_a),hAPP(int,nat,number_number_of(nat),V)) = hAPP(int,X_a,number_number_of(X_a),V) )
& ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,number_number_of(int),V)))
=> hAPP(nat,X_a,semiring_1_of_nat(X_a),hAPP(int,nat,number_number_of(nat),V)) = zero_zero(X_a) ) ) ) ).
fof(fact_913_incr__lemma,axiom,
! [Z,X,D] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),D))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Z),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),X),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,abs_abs(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),X),Z))),one_one(int))),D)))) ) ).
fof(fact_914_Nat__Transfer_Otransfer__int__nat__functions_I3_J,axiom,
! [X,Y] : hAPP(int,int,hAPP(int,fun(int,int),nat_tsub,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),minus_minus(nat),X),Y)) ).
fof(fact_915_of__nat__Suc,axiom,
! [X_a] :
( semiring_1(X_a)
=> ! [M_1] : hAPP(nat,X_a,semiring_1_of_nat(X_a),hAPP(nat,nat,suc,M_1)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),one_one(X_a)),hAPP(nat,X_a,semiring_1_of_nat(X_a),M_1)) ) ).
fof(fact_916_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_917_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_918_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_919_transfer__int__nat__numerals_I2_J,axiom,
one_one(int) = hAPP(nat,int,semiring_1_of_nat(int),one_one(nat)) ).
fof(fact_920_transfer__int__nat__numerals_I1_J,axiom,
zero_zero(int) = hAPP(nat,int,semiring_1_of_nat(int),zero_zero(nat)) ).
fof(fact_921_transfer__int__nat__relations_I3_J,axiom,
! [X_1,Y_1] :
( 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_1)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),X_1),Y_1)) ) ).
fof(fact_922_transfer__int__nat__relations_I2_J,axiom,
! [X_1,Y_1] :
( 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_1)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),X_1),Y_1)) ) ).
fof(fact_923_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_924_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_925_int__Suc,axiom,
! [M_1] : hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,suc,M_1)) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),one_one(int)),hAPP(nat,int,semiring_1_of_nat(int),M_1)) ).
fof(fact_926_int__Suc0__eq__1,axiom,
hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,suc,zero_zero(nat))) = one_one(int) ).
fof(fact_927_zless__iff__Suc__zadd,axiom,
! [W,Z_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),W),Z_1))
<=> ? [N_2] : Z_1 = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),W),hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,suc,N_2))) ) ).
fof(fact_928_abs__triangle__ineq,axiom,
! [X_a] :
( ordere142940540dd_abs(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,abs_abs(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,abs_abs(X_a),A_1)),hAPP(X_a,X_a,abs_abs(X_a),B)))) ) ).
fof(fact_929_abs__mult__less,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [B,D,A_1,C] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,abs_abs(X_a),A_1)),C))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,abs_abs(X_a),B)),D))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,abs_abs(X_a),A_1)),hAPP(X_a,X_a,abs_abs(X_a),B))),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C),D))) ) ) ) ).
fof(fact_930_abs__triangle__ineq2__sym,axiom,
! [X_a] :
( ordere142940540dd_abs(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),minus_minus(X_a),hAPP(X_a,X_a,abs_abs(X_a),A_1)),hAPP(X_a,X_a,abs_abs(X_a),B))),hAPP(X_a,X_a,abs_abs(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),B),A_1)))) ) ).
fof(fact_931_abs__triangle__ineq2,axiom,
! [X_a] :
( ordere142940540dd_abs(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),minus_minus(X_a),hAPP(X_a,X_a,abs_abs(X_a),A_1)),hAPP(X_a,X_a,abs_abs(X_a),B))),hAPP(X_a,X_a,abs_abs(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_1),B)))) ) ).
fof(fact_932_abs__triangle__ineq3,axiom,
! [X_a] :
( ordere142940540dd_abs(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,abs_abs(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,abs_abs(X_a),A_1)),hAPP(X_a,X_a,abs_abs(X_a),B)))),hAPP(X_a,X_a,abs_abs(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_1),B)))) ) ).
fof(fact_933_of__nat__less__imp__less,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [M_1,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_1)),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_1),N_1)) ) ) ).
fof(fact_934_less__imp__of__nat__less,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [M_1,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_1),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_1)),hAPP(nat,X_a,semiring_1_of_nat(X_a),N_1))) ) ) ).
fof(fact_935_of__nat__less__iff,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [M,N] :
( 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)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N)) ) ) ).
fof(fact_936_of__nat__le__iff,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [M,N] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),M)),hAPP(nat,X_a,semiring_1_of_nat(X_a),N)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ) ).
fof(fact_937_of__nat__add,axiom,
! [X_a] :
( semiring_1(X_a)
=> ! [M_1,N_1] : hAPP(nat,X_a,semiring_1_of_nat(X_a),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M_1),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_1)),hAPP(nat,X_a,semiring_1_of_nat(X_a),N_1)) ) ).
fof(fact_938_abs__add__abs,axiom,
! [X_a] :
( ordere142940540dd_abs(X_a)
=> ! [A_1,B] : hAPP(X_a,X_a,abs_abs(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,abs_abs(X_a),A_1)),hAPP(X_a,X_a,abs_abs(X_a),B))) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,abs_abs(X_a),A_1)),hAPP(X_a,X_a,abs_abs(X_a),B)) ) ).
fof(fact_939_abs__minus__commute,axiom,
! [X_a] :
( ordere142940540dd_abs(X_a)
=> ! [A_1,B] : hAPP(X_a,X_a,abs_abs(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,abs_abs(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),B),A_1)) ) ).
fof(fact_940_abs__le__D1,axiom,
! [X_a] :
( ordere142940540dd_abs(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,abs_abs(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)) ) ) ).
fof(fact_941_abs__ge__self,axiom,
! [X_a] :
( ordere142940540dd_abs(X_a)
=> ! [A_1] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),hAPP(X_a,X_a,abs_abs(X_a),A_1))) ) ).
fof(fact_942_abs__mult__self,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,abs_abs(X_a),A_1)),hAPP(X_a,X_a,abs_abs(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) ) ).
fof(fact_943_abs__mult,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [A_1,B] : hAPP(X_a,X_a,abs_abs(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,abs_abs(X_a),A_1)),hAPP(X_a,X_a,abs_abs(X_a),B)) ) ).
fof(fact_944_of__nat__mult,axiom,
! [X_a] :
( semiring_1(X_a)
=> ! [M_1,N_1] : hAPP(nat,X_a,semiring_1_of_nat(X_a),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M_1),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_1)),hAPP(nat,X_a,semiring_1_of_nat(X_a),N_1)) ) ).
fof(fact_945_zle__int,axiom,
! [M,N] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(nat,int,semiring_1_of_nat(int),M)),hAPP(nat,int,semiring_1_of_nat(int),N)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ).
fof(fact_946_zless__int,axiom,
! [M,N] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(nat,int,semiring_1_of_nat(int),M)),hAPP(nat,int,semiring_1_of_nat(int),N)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N)) ) ).
fof(fact_947_zmult__int,axiom,
! [M_1,N_1] : hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(nat,int,semiring_1_of_nat(int),M_1)),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_1),N_1)) ).
fof(fact_948_int__mult,axiom,
! [M_1,N_1] : hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M_1),N_1)) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(nat,int,semiring_1_of_nat(int),M_1)),hAPP(nat,int,semiring_1_of_nat(int),N_1)) ).
fof(fact_949_zadd__int__left,axiom,
! [M_1,N_1,Z] : hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(nat,int,semiring_1_of_nat(int),M_1)),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_1),N_1))),Z) ).
fof(fact_950_zadd__int,axiom,
! [M_1,N_1] : hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(nat,int,semiring_1_of_nat(int),M_1)),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_1),N_1)) ).
fof(fact_951_zle__iff__zadd,axiom,
! [W,Z_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),W),Z_1))
<=> ? [N_2] : Z_1 = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),W),hAPP(nat,int,semiring_1_of_nat(int),N_2)) ) ).
fof(fact_952_abs__triangle__ineq4,axiom,
! [X_a] :
( ordere142940540dd_abs(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,abs_abs(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),plus_plus(X_a),hAPP(X_a,X_a,abs_abs(X_a),A_1)),hAPP(X_a,X_a,abs_abs(X_a),B)))) ) ).
fof(fact_953_abs__diff__triangle__ineq,axiom,
! [X_a] :
( ordere142940540dd_abs(X_a)
=> ! [A_1,B,C,D] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,abs_abs(X_a),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)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),C),D)))),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,abs_abs(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_1),C))),hAPP(X_a,X_a,abs_abs(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),B),D))))) ) ).
fof(fact_954_Ints__of__nat,axiom,
! [X_a] :
( ring_1(X_a)
=> ! [N] : hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),N)),ring_1_Ints(X_a))) ) ).
fof(fact_955_abs__int__eq,axiom,
! [M_1] : hAPP(int,int,abs_abs(int),hAPP(nat,int,semiring_1_of_nat(int),M_1)) = hAPP(nat,int,semiring_1_of_nat(int),M_1) ).
fof(fact_956_abs__of__nat,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [N_1] : hAPP(X_a,X_a,abs_abs(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),N_1)) = hAPP(nat,X_a,semiring_1_of_nat(X_a),N_1) ) ).
fof(fact_957_of__nat__eq__iff,axiom,
! [X_a] :
( semiring_char_0(X_a)
=> ! [M,N] :
( hAPP(nat,X_a,semiring_1_of_nat(X_a),M) = hAPP(nat,X_a,semiring_1_of_nat(X_a),N)
<=> M = N ) ) ).
fof(fact_958_abs__idempotent,axiom,
! [X_a] :
( ordere142940540dd_abs(X_a)
=> ! [A_1] : hAPP(X_a,X_a,abs_abs(X_a),hAPP(X_a,X_a,abs_abs(X_a),A_1)) = hAPP(X_a,X_a,abs_abs(X_a),A_1) ) ).
fof(fact_959_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_960_abs__eq__0,axiom,
! [X_a] :
( ordere142940540dd_abs(X_a)
=> ! [A_2] :
( hAPP(X_a,X_a,abs_abs(X_a),A_2) = zero_zero(X_a)
<=> ti(X_a,A_2) = zero_zero(X_a) ) ) ).
fof(fact_961_abs__zero,axiom,
! [X_a] :
( ordere142940540dd_abs(X_a)
=> hAPP(X_a,X_a,abs_abs(X_a),zero_zero(X_a)) = zero_zero(X_a) ) ).
fof(fact_962_of__nat__0,axiom,
! [X_a] :
( semiring_1(X_a)
=> hAPP(nat,X_a,semiring_1_of_nat(X_a),zero_zero(nat)) = zero_zero(X_a) ) ).
fof(fact_963_abs__one,axiom,
! [X_a] :
( linordered_idom(X_a)
=> hAPP(X_a,X_a,abs_abs(X_a),one_one(X_a)) = one_one(X_a) ) ).
fof(fact_964_int__1,axiom,
hAPP(nat,int,semiring_1_of_nat(int),one_one(nat)) = one_one(int) ).
fof(fact_965_int__eq__0__conv,axiom,
! [N] :
( hAPP(nat,int,semiring_1_of_nat(int),N) = zero_zero(int)
<=> N = zero_zero(nat) ) ).
fof(fact_966_int__0,axiom,
hAPP(nat,int,semiring_1_of_nat(int),zero_zero(nat)) = zero_zero(int) ).
fof(fact_967_abs__eq__mult,axiom,
! [X_a] :
( ordered_ring_abs(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_eq(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),zero_zero(X_a)),B))
| hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),zero_zero(X_a))) ) )
=> hAPP(X_a,X_a,abs_abs(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,abs_abs(X_a),A_1)),hAPP(X_a,X_a,abs_abs(X_a),B)) ) ) ).
fof(fact_968_abs__mult__pos,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))
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,abs_abs(X_a),Y)),X) = hAPP(X_a,X_a,abs_abs(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Y),X)) ) ) ).
fof(fact_969_int__less__0__conv,axiom,
! [K_1] : ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(nat,int,semiring_1_of_nat(int),K_1)),zero_zero(int))) ).
fof(fact_970_zero__zle__int,axiom,
! [N_1] : hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(nat,int,semiring_1_of_nat(int),N_1))) ).
fof(fact_971_int__le__0__conv,axiom,
! [N] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(nat,int,semiring_1_of_nat(int),N)),zero_zero(int)))
<=> N = zero_zero(nat) ) ).
fof(fact_972_zabs__less__one__iff,axiom,
! [Z_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,abs_abs(int),Z_1)),one_one(int)))
<=> Z_1 = zero_zero(int) ) ).
fof(fact_973_abs__zmult__eq__1,axiom,
! [M_1,N_1] :
( hAPP(int,int,abs_abs(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),M_1),N_1)) = one_one(int)
=> hAPP(int,int,abs_abs(int),M_1) = one_one(int) ) ).
fof(fact_974_of__nat__less__0__iff,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [M_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_1)),zero_zero(X_a))) ) ).
fof(fact_975_of__nat__0__le__iff,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [N_1] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(nat,X_a,semiring_1_of_nat(X_a),N_1))) ) ).
fof(fact_976_zero__le__imp__of__nat,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [M_1] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(nat,X_a,semiring_1_of_nat(X_a),M_1))) ) ).
fof(fact_977_abs__not__less__zero,axiom,
! [X_a] :
( ordere142940540dd_abs(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,abs_abs(X_a),A_1)),zero_zero(X_a))) ) ).
fof(fact_978_zero__less__abs__iff,axiom,
! [X_a] :
( ordere142940540dd_abs(X_a)
=> ! [A_2] :
( 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,abs_abs(X_a),A_2)))
<=> ti(X_a,A_2) != zero_zero(X_a) ) ) ).
fof(fact_979_abs__of__pos,axiom,
! [X_a] :
( ordere142940540dd_abs(X_a)
=> ! [A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1))
=> hAPP(X_a,X_a,abs_abs(X_a),A_1) = ti(X_a,A_1) ) ) ).
fof(fact_980_abs__of__nonneg,axiom,
! [X_a] :
( ordere142940540dd_abs(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)),A_1))
=> hAPP(X_a,X_a,abs_abs(X_a),A_1) = ti(X_a,A_1) ) ) ).
fof(fact_981_abs__le__zero__iff,axiom,
! [X_a] :
( ordere142940540dd_abs(X_a)
=> ! [A_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,abs_abs(X_a),A_2)),zero_zero(X_a)))
<=> ti(X_a,A_2) = zero_zero(X_a) ) ) ).
fof(fact_982_abs__ge__zero,axiom,
! [X_a] :
( ordere142940540dd_abs(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)),hAPP(X_a,X_a,abs_abs(X_a),A_1))) ) ).
fof(fact_983_number__of__int,axiom,
! [X_a] :
( number_semiring(X_a)
=> ! [N_1] : hAPP(int,X_a,number_number_of(X_a),hAPP(nat,int,semiring_1_of_nat(int),N_1)) = hAPP(nat,X_a,semiring_1_of_nat(X_a),N_1) ) ).
fof(fact_984_zdiff__int,axiom,
! [N_1,M_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_1),M_1))
=> hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(nat,int,semiring_1_of_nat(int),M_1)),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_1),N_1)) ) ).
fof(fact_985_of__nat__diff,axiom,
! [X_a] :
( ring_1(X_a)
=> ! [N_1,M_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_1),M_1))
=> hAPP(nat,X_a,semiring_1_of_nat(X_a),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M_1),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_1)),hAPP(nat,X_a,semiring_1_of_nat(X_a),N_1)) ) ) ).
fof(fact_986_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_987_zmult__zless__mono2__lemma,axiom,
! [K_1,I_1,J] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),I_1),J))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K_1))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(nat,int,semiring_1_of_nat(int),K_1)),I_1)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(nat,int,semiring_1_of_nat(int),K_1)),J))) ) ) ).
fof(fact_988_of__nat__0__less__iff,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [N] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(nat,X_a,semiring_1_of_nat(X_a),N)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ) ).
fof(fact_989_zdiff__int__split,axiom,
! [Pa,X_1,Y_1] :
( hBOOL(hAPP(int,bool,Pa,hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),X_1),Y_1))))
<=> ( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Y_1),X_1))
=> hBOOL(hAPP(int,bool,Pa,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(nat,int,semiring_1_of_nat(int),X_1)),hAPP(nat,int,semiring_1_of_nat(int),Y_1)))) )
& ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),X_1),Y_1))
=> hBOOL(hAPP(int,bool,Pa,zero_zero(int))) ) ) ) ).
fof(fact_990_decr__lemma,axiom,
! [X,Z,D] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),D))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),X),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,abs_abs(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),X),Z))),one_one(int))),D))),Z)) ) ).
fof(fact_991_int__val__lemma,axiom,
! [K,F,N] :
( ! [I] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I),N))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,abs_abs(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(nat,int,F,hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),I),one_one(nat)))),hAPP(nat,int,F,I)))),one_one(int))) )
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(nat,int,F,zero_zero(nat))),K))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K),hAPP(nat,int,F,N)))
=> ? [I] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I),N))
& hAPP(nat,int,F,I) = K ) ) ) ) ).
fof(fact_992_nat0__intermed__int__val,axiom,
! [K,F,N] :
( ! [I] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I),N))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,abs_abs(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(nat,int,F,hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),I),one_one(nat)))),hAPP(nat,int,F,I)))),one_one(int))) )
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(nat,int,F,zero_zero(nat))),K))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K),hAPP(nat,int,F,N)))
=> ? [I] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I),N))
& hAPP(nat,int,F,I) = K ) ) ) ) ).
fof(fact_993_transfer__int__nat__relations_I1_J,axiom,
! [X_1,Y_1] :
( hAPP(nat,int,semiring_1_of_nat(int),X_1) = hAPP(nat,int,semiring_1_of_nat(int),Y_1)
<=> X_1 = Y_1 ) ).
fof(fact_994_int__if__cong,axiom,
! [X_1,Y_1,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_1)) )
& ( ~ hBOOL(Pa)
=> hAPP(nat,int,semiring_1_of_nat(int),Y_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_1)) ) ) ).
fof(fact_995_int__int__eq,axiom,
! [M,N] :
( hAPP(nat,int,semiring_1_of_nat(int),M) = hAPP(nat,int,semiring_1_of_nat(int),N)
<=> M = N ) ).
%----Arities (105)
fof(arity_fun___Orderings_Opreorder,axiom,
! [T_2,T_1] :
( preorder(T_1)
=> preorder(fun(T_2,T_1)) ) ).
fof(arity_fun___Finite__Set_Ofinite,axiom,
! [T_2,T_1] :
( ( finite_finite(T_1)
& finite_finite(T_2) )
=> finite_finite(fun(T_2,T_1)) ) ).
fof(arity_fun___Orderings_Oorder,axiom,
! [T_2,T_1] :
( order(T_1)
=> order(fun(T_2,T_1)) ) ).
fof(arity_fun___Orderings_Oord,axiom,
! [T_2,T_1] :
( ord(T_1)
=> ord(fun(T_2,T_1)) ) ).
fof(arity_fun___Orderings_Obot,axiom,
! [T_2,T_1] :
( bot(T_1)
=> bot(fun(T_2,T_1)) ) ).
fof(arity_fun___Groups_Ominus,axiom,
! [T_2,T_1] :
( minus(T_1)
=> minus(fun(T_2,T_1)) ) ).
fof(arity_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___Rings_Olinordered__semiring__1__strict,axiom,
linord626643107strict(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__ab__group__add__abs,axiom,
ordere142940540dd_abs(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_Oordered__cancel__semiring,axiom,
ordere453448008miring(int) ).
fof(arity_Int_Oint___Rings_Olinordered__ring__strict,axiom,
linord581940658strict(int) ).
fof(arity_Int_Oint___Rings_Oring__no__zero__divisors,axiom,
ring_n68954251visors(int) ).
fof(arity_Int_Oint___Rings_Oordered__comm__semiring,axiom,
ordere1490568538miring(int) ).
fof(arity_Int_Oint___Rings_Olinordered__semiring__1,axiom,
linord1278240602ring_1(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__semiring,axiom,
linordered_semiring(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___Rings_Oordered__semiring,axiom,
ordered_semiring(int) ).
fof(arity_Int_Oint___Rings_Oordered__ring__abs,axiom,
ordered_ring_abs(int) ).
fof(arity_Int_Oint___Rings_Ono__zero__divisors,axiom,
no_zero_divisors(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___Int_Onumber__semiring,axiom,
number_semiring(int) ).
fof(arity_Int_Oint___Groups_Oab__group__add,axiom,
ab_group_add(int) ).
fof(arity_Int_Oint___Rings_Ozero__neq__one,axiom,
zero_neq_one(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___Groups_Omonoid__add,axiom,
monoid_add(int) ).
fof(arity_Int_Oint___Rings_Osemiring__1,axiom,
semiring_1(int) ).
fof(arity_Int_Oint___Groups_Ogroup__add,axiom,
group_add(int) ).
fof(arity_Int_Oint___Rings_Omult__zero,axiom,
mult_zero(int) ).
fof(arity_Int_Oint___Orderings_Oorder,axiom,
order(int) ).
fof(arity_Int_Oint___Int_Oring__char__0,axiom,
ring_char_0(int) ).
fof(arity_Int_Oint___Int_Onumber__ring,axiom,
number_ring(int) ).
fof(arity_Int_Oint___Rings_Osemiring,axiom,
semiring(int) ).
fof(arity_Int_Oint___Orderings_Oord,axiom,
ord(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___Groups_Ozero,axiom,
zero(int) ).
fof(arity_Int_Oint___Rings_Oring,axiom,
ring(int) ).
fof(arity_Int_Oint___Int_Onumber,axiom,
number(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___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___Rings_Oordered__cancel__semiring,axiom,
ordere453448008miring(nat) ).
fof(arity_Nat_Onat___Rings_Oordered__comm__semiring,axiom,
ordere1490568538miring(nat) ).
fof(arity_Nat_Onat___Groups_Ocancel__semigroup__add,axiom,
cancel_semigroup_add(nat) ).
fof(arity_Nat_Onat___Rings_Olinordered__semiring,axiom,
linordered_semiring(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___Rings_Oordered__semiring,axiom,
ordered_semiring(nat) ).
fof(arity_Nat_Onat___Rings_Ono__zero__divisors,axiom,
no_zero_divisors(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___Int_Onumber__semiring,axiom,
number_semiring(nat) ).
fof(arity_Nat_Onat___Rings_Ozero__neq__one,axiom,
zero_neq_one(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___Groups_Omonoid__add,axiom,
monoid_add(nat) ).
fof(arity_Nat_Onat___Rings_Osemiring__1,axiom,
semiring_1(nat) ).
fof(arity_Nat_Onat___Rings_Omult__zero,axiom,
mult_zero(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___Groups_Ozero,axiom,
zero(nat) ).
fof(arity_Nat_Onat___Int_Onumber,axiom,
number(nat) ).
fof(arity_Nat_Onat___Groups_Oone,axiom,
one(nat) ).
fof(arity_HOL_Obool___Orderings_Opreorder,axiom,
preorder(bool) ).
fof(arity_HOL_Obool___Finite__Set_Ofinite,axiom,
finite_finite(bool) ).
fof(arity_HOL_Obool___Orderings_Oorder,axiom,
order(bool) ).
fof(arity_HOL_Obool___Orderings_Oord,axiom,
ord(bool) ).
fof(arity_HOL_Obool___Orderings_Obot,axiom,
bot(bool) ).
fof(arity_HOL_Obool___Groups_Ominus,axiom,
minus(bool) ).
fof(arity_sum___Finite__Set_Ofinite,axiom,
! [T_2,T_1] :
( ( finite_finite(T_1)
& finite_finite(T_2) )
=> finite_finite(sum_sum(T_2,T_1)) ) ).
%----Helper facts (23)
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_fNot_1_1_U,axiom,
! [P] :
( ~ hBOOL(hAPP(bool,bool,fNot,P))
| ~ hBOOL(P) ) ).
fof(help_fNot_2_1_U,axiom,
! [P] :
( hBOOL(P)
| hBOOL(hAPP(bool,bool,fNot,P)) ) ).
fof(help_COMBB_1_1_U,axiom,
! [X_a,X_c,X_b,P,Q,R] : hAPP(X_a,X_c,hAPP(fun(X_a,X_b),fun(X_a,X_c),hAPP(fun(X_b,X_c),fun(fun(X_a,X_b),fun(X_a,X_c)),combb(X_b,X_c,X_a),P),Q),R) = hAPP(X_b,X_c,P,hAPP(X_a,X_b,Q,R)) ).
fof(help_COMBC_1_1_U,axiom,
! [X_a,X_c,X_b,P,Q,R] : hAPP(X_a,X_c,hAPP(X_b,fun(X_a,X_c),hAPP(fun(X_a,fun(X_b,X_c)),fun(X_b,fun(X_a,X_c)),combc(X_a,X_b,X_c),P),Q),R) = hAPP(X_b,X_c,hAPP(X_a,fun(X_b,X_c),P,R),Q) ).
fof(help_COMBK_1_1_U,axiom,
! [X_b,X_a,P,Q] : hAPP(X_b,X_a,hAPP(X_a,fun(X_b,X_a),combk(X_a,X_b),P),Q) = ti(X_a,P) ).
fof(help_COMBS_1_1_U,axiom,
! [X_a,X_c,X_b,P,Q,R] : hAPP(X_a,X_c,hAPP(fun(X_a,X_b),fun(X_a,X_c),hAPP(fun(X_a,fun(X_b,X_c)),fun(fun(X_a,X_b),fun(X_a,X_c)),combs(X_a,X_b,X_c),P),Q),R) = hAPP(X_b,X_c,hAPP(X_a,fun(X_b,X_c),P,R),hAPP(X_a,X_b,Q,R)) ).
fof(help_fconj_1_1_U,axiom,
! [Q,P] :
( ~ hBOOL(P)
| ~ hBOOL(Q)
| hBOOL(hAPP(bool,bool,hAPP(bool,fun(bool,bool),fconj,P),Q)) ) ).
fof(help_fconj_2_1_U,axiom,
! [P,Q] :
( ~ hBOOL(hAPP(bool,bool,hAPP(bool,fun(bool,bool),fconj,P),Q))
| hBOOL(P) ) ).
fof(help_fconj_3_1_U,axiom,
! [P,Q] :
( ~ hBOOL(hAPP(bool,bool,hAPP(bool,fun(bool,bool),fconj,P),Q))
| hBOOL(Q) ) ).
fof(help_fdisj_1_1_U,axiom,
! [Q,P] :
( ~ hBOOL(P)
| hBOOL(hAPP(bool,bool,hAPP(bool,fun(bool,bool),fdisj,P),Q)) ) ).
fof(help_fdisj_2_1_U,axiom,
! [P,Q] :
( ~ hBOOL(Q)
| hBOOL(hAPP(bool,bool,hAPP(bool,fun(bool,bool),fdisj,P),Q)) ) ).
fof(help_fdisj_3_1_U,axiom,
! [P,Q] :
( ~ hBOOL(hAPP(bool,bool,hAPP(bool,fun(bool,bool),fdisj,P),Q))
| hBOOL(P)
| hBOOL(Q) ) ).
fof(help_fFalse_1_1_U,axiom,
~ hBOOL(fFalse) ).
fof(help_fFalse_1_1_T,axiom,
! [P] :
( ti(bool,P) = fTrue
| ti(bool,P) = fFalse ) ).
fof(help_fequal_1_1_T,axiom,
! [X_a,X,Y] :
( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),fequal(X_a),X),Y))
| ti(X_a,X) = ti(X_a,Y) ) ).
fof(help_fequal_2_1_T,axiom,
! [X_a,X,Y] :
( ti(X_a,X) != ti(X_a,Y)
| hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),fequal(X_a),X),Y)) ) ).
fof(help_fimplies_1_1_U,axiom,
! [Q,P] :
( hBOOL(P)
| hBOOL(hAPP(bool,bool,hAPP(bool,fun(bool,bool),fimplies,P),Q)) ) ).
fof(help_fimplies_2_1_U,axiom,
! [P,Q] :
( ~ hBOOL(Q)
| hBOOL(hAPP(bool,bool,hAPP(bool,fun(bool,bool),fimplies,P),Q)) ) ).
fof(help_fimplies_3_1_U,axiom,
! [P,Q] :
( ~ hBOOL(hAPP(bool,bool,hAPP(bool,fun(bool,bool),fimplies,P),Q))
| ~ hBOOL(P)
| hBOOL(Q) ) ).
%----Conjectures (3)
fof(conj_0,hypothesis,
hBOOL(hAPP(fun(hoare_2118899576triple(state),bool),bool,hAPP(fun(hoare_2118899576triple(state),bool),fun(fun(hoare_2118899576triple(state),bool),bool),hoare_1301688828derivs(state),bot_bot(fun(hoare_2118899576triple(state),bool))),hAPP(fun(hoare_2118899576triple(state),bool),fun(hoare_2118899576triple(state),bool),hAPP(hoare_2118899576triple(state),fun(fun(hoare_2118899576triple(state),bool),fun(hoare_2118899576triple(state),bool)),insert(hoare_2118899576triple(state)),hAPP(com,hoare_2118899576triple(state),hoare_Mirabelle_MGT,c)),bot_bot(fun(hoare_2118899576triple(state),bool))))) ).
fof(conj_1,hypothesis,
hBOOL(hAPP(fun(hoare_2118899576triple(state),bool),bool,hAPP(fun(hoare_2118899576triple(state),bool),fun(fun(hoare_2118899576triple(state),bool),bool),hoare_902341502valids(state),bot_bot(fun(hoare_2118899576triple(state),bool))),hAPP(fun(hoare_2118899576triple(state),bool),fun(hoare_2118899576triple(state),bool),hAPP(hoare_2118899576triple(state),fun(fun(hoare_2118899576triple(state),bool),fun(hoare_2118899576triple(state),bool)),insert(hoare_2118899576triple(state)),hAPP(fun(state,fun(state,bool)),hoare_2118899576triple(state),hAPP(com,fun(fun(state,fun(state,bool)),hoare_2118899576triple(state)),hAPP(fun(state,fun(state,bool)),fun(com,fun(fun(state,fun(state,bool)),hoare_2118899576triple(state))),hoare_759811442triple(state),p),c),q)),bot_bot(fun(hoare_2118899576triple(state),bool))))) ).
fof(conj_2,conjecture,
hBOOL(hAPP(fun(hoare_2118899576triple(state),bool),bool,hAPP(fun(hoare_2118899576triple(state),bool),fun(fun(hoare_2118899576triple(state),bool),bool),hoare_1301688828derivs(state),bot_bot(fun(hoare_2118899576triple(state),bool))),hAPP(fun(hoare_2118899576triple(state),bool),fun(hoare_2118899576triple(state),bool),hAPP(hoare_2118899576triple(state),fun(fun(hoare_2118899576triple(state),bool),fun(hoare_2118899576triple(state),bool)),insert(hoare_2118899576triple(state)),hAPP(fun(state,fun(state,bool)),hoare_2118899576triple(state),hAPP(com,fun(fun(state,fun(state,bool)),hoare_2118899576triple(state)),hAPP(fun(state,fun(state,bool)),fun(com,fun(fun(state,fun(state,bool)),hoare_2118899576triple(state))),hoare_759811442triple(state),p),c),q)),bot_bot(fun(hoare_2118899576triple(state),bool))))) ).
%------------------------------------------------------------------------------