TPTP Problem File: SWW471+7.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SWW471+7 : TPTP v8.2.0. Released v5.3.0.
% Domain : Software Verification
% Problem : Hoare's Logic with Procedures line 269, 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_l269 [Bla11]
% Status : Theorem
% Rating : 1.00 v5.3.0
% Syntax : Number of formulae : 1168 ( 331 unt; 0 def)
% Number of atoms : 2885 ( 911 equ)
% Maximal formula atoms : 10 ( 2 avg)
% Number of connectives : 1994 ( 277 ~; 60 |; 107 &)
% ( 179 <=>;1371 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 6 avg)
% Maximal term depth : 18 ( 2 avg)
% Number of predicates : 54 ( 53 usr; 0 prp; 1-2 aty)
% Number of functors : 85 ( 85 usr; 28 con; 0-4 aty)
% Number of variables : 3557 (3524 !; 33 ?)
% SPC : FOF_THM_RFO_SEQ
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-08-09 17:28:13
% : Encoded with polymorphic tags.
%------------------------------------------------------------------------------
%----Explicit typings (84)
fof(tsy_c_Big__Operators_Ocomm__monoid__big_res,axiom,
! [X_b,X_c] : ti(fun(fun(X_b,fun(X_b,X_b)),fun(X_b,fun(fun(fun(X_c,X_b),fun(fun(X_c,bool),X_b)),bool))),big_comm_monoid_big(X_b,X_c)) = big_comm_monoid_big(X_b,X_c) ).
fof(tsy_c_Big__Operators_Olattice__class_OInf__fin_res,axiom,
! [X_b] :
( lattice(X_b)
=> ti(fun(fun(X_b,bool),X_b),big_lattice_Inf_fin(X_b)) = big_lattice_Inf_fin(X_b) ) ).
fof(tsy_c_Big__Operators_Olattice__class_OSup__fin_res,axiom,
! [X_b] :
( lattice(X_b)
=> ti(fun(fun(X_b,bool),X_b),big_lattice_Sup_fin(X_b)) = big_lattice_Sup_fin(X_b) ) ).
fof(tsy_c_Big__Operators_Osemilattice__big_res,axiom,
! [X_b] : ti(fun(fun(X_b,fun(X_b,X_b)),fun(fun(fun(X_b,bool),X_b),bool)),big_semilattice_big(X_b)) = big_semilattice_big(X_b) ).
fof(tsy_c_COMBB_res,axiom,
! [X_b,X_c,X_a] : ti(fun(fun(X_b,X_c),fun(fun(X_a,X_b),fun(X_a,X_c))),combb(X_b,X_c,X_a)) = combb(X_b,X_c,X_a) ).
fof(tsy_c_COMBC_res,axiom,
! [X_a,X_b,X_c] : ti(fun(fun(X_a,fun(X_b,X_c)),fun(X_b,fun(X_a,X_c))),combc(X_a,X_b,X_c)) = combc(X_a,X_b,X_c) ).
fof(tsy_c_COMBI_res,axiom,
! [X_a] : ti(fun(X_a,X_a),combi(X_a)) = combi(X_a) ).
fof(tsy_c_COMBK_res,axiom,
! [X_a,X_b] : ti(fun(X_a,fun(X_b,X_a)),combk(X_a,X_b)) = combk(X_a,X_b) ).
fof(tsy_c_COMBS_res,axiom,
! [X_a,X_b,X_c] : ti(fun(fun(X_a,fun(X_b,X_c)),fun(fun(X_a,X_b),fun(X_a,X_c))),combs(X_a,X_b,X_c)) = combs(X_a,X_b,X_c) ).
fof(tsy_c_Com_Obody_res,hypothesis,
ti(fun(pname,option(com)),body_1) = body_1 ).
fof(tsy_c_Com_Ocom_OBODY_res,hypothesis,
ti(fun(pname,com),body) = body ).
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_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_b] : ti(fun(fun(X_b,bool),nat),finite_card(X_b)) = finite_card(X_b) ).
fof(tsy_c_Finite__Set_Ocomp__fun__idem_res,axiom,
! [X_b,X_c] : ti(fun(fun(X_b,fun(X_c,X_c)),bool),finite_comp_fun_idem(X_b,X_c)) = finite_comp_fun_idem(X_b,X_c) ).
fof(tsy_c_Finite__Set_Ofinite_res,axiom,
! [X_b] : ti(fun(fun(X_b,bool),bool),finite_finite_1(X_b)) = finite_finite_1(X_b) ).
fof(tsy_c_Finite__Set_Ofold1_res,axiom,
! [X_b] : ti(fun(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b)),finite_fold1(X_b)) = finite_fold1(X_b) ).
fof(tsy_c_Finite__Set_Ofold1Set_res,axiom,
! [X_b] : ti(fun(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),fun(X_b,bool))),finite_fold1Set(X_b)) = finite_fold1Set(X_b) ).
fof(tsy_c_Finite__Set_Ofold__image_res,axiom,
! [X_b,X_c] : ti(fun(fun(X_b,fun(X_b,X_b)),fun(fun(X_c,X_b),fun(X_b,fun(fun(X_c,bool),X_b)))),finite_fold_image(X_b,X_c)) = finite_fold_image(X_b,X_c) ).
fof(tsy_c_Finite__Set_Ofolding__image__simple_res,axiom,
! [X_b,X_c] : ti(fun(fun(X_b,fun(X_b,X_b)),fun(X_b,fun(fun(X_c,X_b),fun(fun(fun(X_c,bool),X_b),bool)))),finite1357897459simple(X_b,X_c)) = finite1357897459simple(X_b,X_c) ).
fof(tsy_c_Finite__Set_Ofolding__image__simple__idem_res,axiom,
! [X_b,X_c] : ti(fun(fun(X_b,fun(X_b,X_b)),fun(X_b,fun(fun(X_c,X_b),fun(fun(fun(X_c,bool),X_b),bool)))),finite908156982e_idem(X_b,X_c)) = finite908156982e_idem(X_b,X_c) ).
fof(tsy_c_Finite__Set_Ofolding__one_res,axiom,
! [X_b] : ti(fun(fun(X_b,fun(X_b,X_b)),fun(fun(fun(X_b,bool),X_b),bool)),finite_folding_one(X_b)) = finite_folding_one(X_b) ).
fof(tsy_c_Finite__Set_Ofolding__one__idem_res,axiom,
! [X_b] : ti(fun(fun(X_b,fun(X_b,X_b)),fun(fun(fun(X_b,bool),X_b),bool)),finite2073411215e_idem(X_b)) = finite2073411215e_idem(X_b) ).
fof(tsy_c_Groups_Ominus__class_Ominus_res,axiom,
! [X_c] :
( minus(X_c)
=> ti(fun(X_c,fun(X_c,X_c)),minus_minus(X_c)) = minus_minus(X_c) ) ).
fof(tsy_c_Groups_Oone__class_Oone_res,axiom,
! [X_b] :
( one(X_b)
=> ti(X_b,one_one(X_b)) = one_one(X_b) ) ).
fof(tsy_c_Groups_Oplus__class_Oplus_0_res,axiom,
! [X_a] :
( 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_b] :
( cancel_semigroup_add(X_b)
=> ti(fun(X_b,fun(X_b,X_b)),plus_plus(X_b)) = plus_plus(X_b) ) ).
fof(tsy_c_Groups_Oplus__class_Oplus_2_res,axiom,
! [X_a] :
( ab_semigroup_add(X_a)
=> ti(fun(X_a,fun(X_a,X_a)),plus_plus(X_a)) = plus_plus(X_a) ) ).
fof(tsy_c_Groups_Otimes__class_Otimes_0_res,axiom,
! [X_a] :
( 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_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] :
( 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_3_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_Otimes__class_Otimes_4_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_Ozero__class_Ozero_res,axiom,
! [X_b] :
( zero(X_b)
=> ti(X_b,zero_zero(X_b)) = zero_zero(X_b) ) ).
fof(tsy_c_HOL_OThe_res,axiom,
! [X_b] : ti(fun(fun(X_b,bool),X_b),the_1(X_b)) = the_1(X_b) ).
fof(tsy_c_HOL_Oundefined_res,axiom,
! [X_a] : ti(X_a,undefined(X_a)) = undefined(X_a) ).
fof(tsy_c_Hoare__Mirabelle__lcrcocijdw_OMGT_res,axiom,
ti(fun(com,hoare_2118899576triple(state)),hoare_Mirabelle_MGT) = hoare_Mirabelle_MGT ).
fof(tsy_c_Hoare__Mirabelle__lcrcocijdw_Ohoare__derivs_res,axiom,
! [X_b] : ti(fun(fun(hoare_2118899576triple(X_b),bool),fun(fun(hoare_2118899576triple(X_b),bool),bool)),hoare_1301688828derivs(X_b)) = hoare_1301688828derivs(X_b) ).
fof(tsy_c_Hoare__Mirabelle__lcrcocijdw_Ohoare__valids_res,axiom,
! [X_b] : ti(fun(fun(hoare_2118899576triple(X_b),bool),fun(fun(hoare_2118899576triple(X_b),bool),bool)),hoare_902341502valids(X_b)) = hoare_902341502valids(X_b) ).
fof(tsy_c_Hoare__Mirabelle__lcrcocijdw_Otriple_Otriple_res,axiom,
! [X_b] : ti(fun(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b)))),hoare_759811442triple(X_b)) = hoare_759811442triple(X_b) ).
fof(tsy_c_Hoare__Mirabelle__lcrcocijdw_Otriple_Otriple__case_res,axiom,
! [X_c,X_b] : ti(fun(fun(fun(X_c,fun(state,bool)),fun(com,fun(fun(X_c,fun(state,bool)),X_b))),fun(hoare_2118899576triple(X_c),X_b)),hoare_225284258e_case(X_c,X_b)) = hoare_225284258e_case(X_c,X_b) ).
fof(tsy_c_Hoare__Mirabelle__lcrcocijdw_Otriple_Otriple__rec_res,axiom,
! [X_c,X_b] : ti(fun(fun(fun(X_c,fun(state,bool)),fun(com,fun(fun(X_c,fun(state,bool)),X_b))),fun(hoare_2118899576triple(X_c),X_b)),hoare_1759541758le_rec(X_c,X_b)) = hoare_1759541758le_rec(X_c,X_b) ).
fof(tsy_c_Hoare__Mirabelle__lcrcocijdw_Otriple_Otriple__size_res,axiom,
! [X_b] : ti(fun(fun(X_b,nat),fun(hoare_2118899576triple(X_b),nat)),hoare_2043812435e_size(X_b)) = hoare_2043812435e_size(X_b) ).
fof(tsy_c_Hoare__Mirabelle__lcrcocijdw_Otriple__valid_res,axiom,
! [X_b] : ti(fun(nat,fun(hoare_2118899576triple(X_b),bool)),hoare_1942962616_valid(X_b)) = hoare_1942962616_valid(X_b) ).
fof(tsy_c_If_res,axiom,
! [X_a] : ti(fun(bool,fun(X_a,fun(X_a,X_a))),if(X_a)) = if(X_a) ).
fof(tsy_c_Lattices_Osemilattice__inf__class_Oinf_res,axiom,
! [X_a] :
( semilattice_inf(X_a)
=> ti(fun(X_a,fun(X_a,X_a)),semilattice_inf_inf(X_a)) = semilattice_inf_inf(X_a) ) ).
fof(tsy_c_Lattices_Osemilattice__sup__class_Osup_res,axiom,
! [X_a] :
( semilattice_sup(X_a)
=> ti(fun(X_a,fun(X_a,X_a)),semilattice_sup_sup(X_a)) = semilattice_sup_sup(X_a) ) ).
fof(tsy_c_Nat_OSuc_res,axiom,
ti(fun(nat,nat),suc) = suc ).
fof(tsy_c_Nat_Onat_Onat__case_res,axiom,
! [X_b] : ti(fun(X_b,fun(fun(nat,X_b),fun(nat,X_b))),nat_case(X_b)) = nat_case(X_b) ).
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_b] : ti(fun(hoare_2118899576triple(X_b),nat),size_size(hoare_2118899576triple(X_b))) = size_size(hoare_2118899576triple(X_b)) ).
fof(tsy_c_Natural_Oevalc_res,axiom,
ti(fun(com,fun(state,fun(state,bool))),evalc) = evalc ).
fof(tsy_c_Natural_Oevaln_res,axiom,
ti(fun(com,fun(state,fun(nat,fun(state,bool)))),evaln) = evaln ).
fof(tsy_c_Option_Othe_res,hypothesis,
ti(fun(option(com),com),the(com)) = the(com) ).
fof(tsy_c_Orderings_Obot__class_Obot_res,axiom,
! [X_c] :
( bot(X_c)
=> ti(X_c,bot_bot(X_c)) = bot_bot(X_c) ) ).
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_c] :
( ord(X_c)
=> ti(fun(X_c,fun(X_c,bool)),ord_less_eq(X_c)) = ord_less_eq(X_c) ) ).
fof(tsy_c_Partial__Function_Oflat__lub_res,axiom,
! [X_b] : ti(fun(X_b,fun(fun(X_b,bool),X_b)),partial_flat_lub(X_b)) = partial_flat_lub(X_b) ).
fof(tsy_c_Predicate_OPowp_res,axiom,
! [X_b] : ti(fun(fun(X_b,bool),fun(fun(X_b,bool),bool)),powp(X_b)) = powp(X_b) ).
fof(tsy_c_Set_OCollect_res,axiom,
! [X_b] : ti(fun(fun(X_b,bool),fun(X_b,bool)),collect(X_b)) = collect(X_b) ).
fof(tsy_c_Set_Oimage_res,axiom,
! [X_c,X_b] : ti(fun(fun(X_c,X_b),fun(fun(X_c,bool),fun(X_b,bool))),image(X_c,X_b)) = image(X_c,X_b) ).
fof(tsy_c_Set_Oinsert_res,axiom,
! [X_b] : ti(fun(X_b,fun(fun(X_b,bool),fun(X_b,bool))),insert(X_b)) = insert(X_b) ).
fof(tsy_c_Set_Othe__elem_res,axiom,
! [X_b] : ti(fun(fun(X_b,bool),X_b),the_elem(X_b)) = the_elem(X_b) ).
fof(tsy_c_Sum__Type_OPlus_res,axiom,
! [X_b,X_c] : ti(fun(fun(X_b,bool),fun(fun(X_c,bool),fun(sum_sum(X_b,X_c),bool))),sum_Plus(X_b,X_c)) = sum_Plus(X_b,X_c) ).
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_b] : ti(fun(X_b,fun(fun(X_b,bool),bool)),member(X_b)) = member(X_b) ).
fof(tsy_v_G_res,hypothesis,
ti(fun(hoare_2118899576triple(x_a),bool),g) = g ).
fof(tsy_v_P_res,hypothesis,
ti(fun(pname,fun(x_a,fun(state,bool))),p) = p ).
fof(tsy_v_Procs_res,hypothesis,
ti(fun(pname,bool),procs) = procs ).
fof(tsy_v_Q_res,hypothesis,
ti(fun(pname,fun(x_a,fun(state,bool))),q) = q ).
fof(tsy_v_n_res,hypothesis,
ti(nat,n) = n ).
%----Relevant facts (996)
fof(fact_0_triple_Oinject,axiom,
! [X_b,Fun1,Com_1,Fun2,Fun1_2,Com,Fun2_2] :
( hAPP(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b),hAPP(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b)),hAPP(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b))),hoare_759811442triple(X_b),Fun1),Com_1),Fun2) = hAPP(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b),hAPP(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b)),hAPP(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b))),hoare_759811442triple(X_b),Fun1_2),Com),Fun2_2)
<=> ( Fun1 = Fun1_2
& Com_1 = Com
& Fun2 = Fun2_2 ) ) ).
fof(fact_1_hoare__valids__def,axiom,
! [X_b,Ga,Ts] :
( hBOOL(hAPP(fun(hoare_2118899576triple(X_b),bool),bool,hAPP(fun(hoare_2118899576triple(X_b),bool),fun(fun(hoare_2118899576triple(X_b),bool),bool),hoare_902341502valids(X_b),Ga),Ts))
<=> ! [N] :
( ! [X] :
( hBOOL(hAPP(fun(hoare_2118899576triple(X_b),bool),bool,hAPP(hoare_2118899576triple(X_b),fun(fun(hoare_2118899576triple(X_b),bool),bool),member(hoare_2118899576triple(X_b)),X),Ga))
=> hBOOL(hAPP(hoare_2118899576triple(X_b),bool,hAPP(nat,fun(hoare_2118899576triple(X_b),bool),hoare_1942962616_valid(X_b),N),X)) )
=> ! [X] :
( hBOOL(hAPP(fun(hoare_2118899576triple(X_b),bool),bool,hAPP(hoare_2118899576triple(X_b),fun(fun(hoare_2118899576triple(X_b),bool),bool),member(hoare_2118899576triple(X_b)),X),Ts))
=> hBOOL(hAPP(hoare_2118899576triple(X_b),bool,hAPP(nat,fun(hoare_2118899576triple(X_b),bool),hoare_1942962616_valid(X_b),N),X)) ) ) ) ).
fof(fact_2_hoare__derivs_OBody,axiom,
! [X_b,Ga,Pa,Qa,Procsa] :
( hBOOL(hAPP(fun(hoare_2118899576triple(X_b),bool),bool,hAPP(fun(hoare_2118899576triple(X_b),bool),fun(fun(hoare_2118899576triple(X_b),bool),bool),hoare_1301688828derivs(X_b),hAPP(fun(hoare_2118899576triple(X_b),bool),fun(hoare_2118899576triple(X_b),bool),hAPP(fun(hoare_2118899576triple(X_b),bool),fun(fun(hoare_2118899576triple(X_b),bool),fun(hoare_2118899576triple(X_b),bool)),semilattice_sup_sup(fun(hoare_2118899576triple(X_b),bool)),Ga),hAPP(fun(pname,bool),fun(hoare_2118899576triple(X_b),bool),hAPP(fun(pname,hoare_2118899576triple(X_b)),fun(fun(pname,bool),fun(hoare_2118899576triple(X_b),bool)),image(pname,hoare_2118899576triple(X_b)),hAPP(fun(pname,fun(X_b,fun(state,bool))),fun(pname,hoare_2118899576triple(X_b)),hAPP(fun(pname,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b))),fun(fun(pname,fun(X_b,fun(state,bool))),fun(pname,hoare_2118899576triple(X_b))),combs(pname,fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b)),hAPP(fun(pname,com),fun(pname,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b))),hAPP(fun(pname,fun(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b)))),fun(fun(pname,com),fun(pname,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b)))),combs(pname,com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b))),hAPP(fun(pname,fun(X_b,fun(state,bool))),fun(pname,fun(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b)))),hAPP(fun(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b)))),fun(fun(pname,fun(X_b,fun(state,bool))),fun(pname,fun(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b))))),combb(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b))),pname),hoare_759811442triple(X_b)),Pa)),body)),Qa)),Procsa))),hAPP(fun(pname,bool),fun(hoare_2118899576triple(X_b),bool),hAPP(fun(pname,hoare_2118899576triple(X_b)),fun(fun(pname,bool),fun(hoare_2118899576triple(X_b),bool)),image(pname,hoare_2118899576triple(X_b)),hAPP(fun(pname,fun(X_b,fun(state,bool))),fun(pname,hoare_2118899576triple(X_b)),hAPP(fun(pname,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b))),fun(fun(pname,fun(X_b,fun(state,bool))),fun(pname,hoare_2118899576triple(X_b))),combs(pname,fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b)),hAPP(fun(pname,com),fun(pname,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b))),hAPP(fun(pname,fun(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b)))),fun(fun(pname,com),fun(pname,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b)))),combs(pname,com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b))),hAPP(fun(pname,fun(X_b,fun(state,bool))),fun(pname,fun(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b)))),hAPP(fun(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b)))),fun(fun(pname,fun(X_b,fun(state,bool))),fun(pname,fun(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b))))),combb(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b))),pname),hoare_759811442triple(X_b)),Pa)),hAPP(fun(pname,option(com)),fun(pname,com),hAPP(fun(option(com),com),fun(fun(pname,option(com)),fun(pname,com)),combb(option(com),com,pname),the(com)),body_1))),Qa)),Procsa)))
=> hBOOL(hAPP(fun(hoare_2118899576triple(X_b),bool),bool,hAPP(fun(hoare_2118899576triple(X_b),bool),fun(fun(hoare_2118899576triple(X_b),bool),bool),hoare_1301688828derivs(X_b),Ga),hAPP(fun(pname,bool),fun(hoare_2118899576triple(X_b),bool),hAPP(fun(pname,hoare_2118899576triple(X_b)),fun(fun(pname,bool),fun(hoare_2118899576triple(X_b),bool)),image(pname,hoare_2118899576triple(X_b)),hAPP(fun(pname,fun(X_b,fun(state,bool))),fun(pname,hoare_2118899576triple(X_b)),hAPP(fun(pname,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b))),fun(fun(pname,fun(X_b,fun(state,bool))),fun(pname,hoare_2118899576triple(X_b))),combs(pname,fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b)),hAPP(fun(pname,com),fun(pname,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b))),hAPP(fun(pname,fun(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b)))),fun(fun(pname,com),fun(pname,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b)))),combs(pname,com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b))),hAPP(fun(pname,fun(X_b,fun(state,bool))),fun(pname,fun(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b)))),hAPP(fun(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b)))),fun(fun(pname,fun(X_b,fun(state,bool))),fun(pname,fun(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b))))),combb(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b))),pname),hoare_759811442triple(X_b)),Pa)),body)),Qa)),Procsa))) ) ).
fof(fact_3_UnE,axiom,
! [X_b,C_1,A_3,B_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),C_1),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),A_3),B_2)))
=> ( ~ hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),C_1),A_3))
=> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),C_1),B_2)) ) ) ).
fof(fact_4_sup1E,axiom,
! [X_b,A_3,B_2,X_2] :
( hBOOL(hAPP(X_b,bool,hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),A_3),B_2),X_2))
=> ( ~ hBOOL(hAPP(X_b,bool,A_3,X_2))
=> hBOOL(hAPP(X_b,bool,B_2,X_2)) ) ) ).
fof(fact_5_sup1CI,axiom,
! [X_b,A_3,B_2,X_2] :
( ( ~ hBOOL(hAPP(X_b,bool,B_2,X_2))
=> hBOOL(hAPP(X_b,bool,A_3,X_2)) )
=> hBOOL(hAPP(X_b,bool,hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),A_3),B_2),X_2)) ) ).
fof(fact_6_UnCI,axiom,
! [X_b,A_3,C_1,B_2] :
( ( ~ hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),C_1),B_2))
=> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),C_1),A_3)) )
=> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),C_1),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),A_3),B_2))) ) ).
fof(fact_7_image__eqI,axiom,
! [X_b,X_c,A_3,B_1,F,X_2] :
( ti(X_b,B_1) = hAPP(X_c,X_b,F,X_2)
=> ( hBOOL(hAPP(fun(X_c,bool),bool,hAPP(X_c,fun(fun(X_c,bool),bool),member(X_c),X_2),A_3))
=> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),B_1),hAPP(fun(X_c,bool),fun(X_b,bool),hAPP(fun(X_c,X_b),fun(fun(X_c,bool),fun(X_b,bool)),image(X_c,X_b),F),A_3))) ) ) ).
fof(fact_8_triple_Orecs,axiom,
! [X_c,X_b,F1,Fun1,Com_1,Fun2] : hAPP(hoare_2118899576triple(X_c),X_b,hAPP(fun(fun(X_c,fun(state,bool)),fun(com,fun(fun(X_c,fun(state,bool)),X_b))),fun(hoare_2118899576triple(X_c),X_b),hoare_1759541758le_rec(X_c,X_b),F1),hAPP(fun(X_c,fun(state,bool)),hoare_2118899576triple(X_c),hAPP(com,fun(fun(X_c,fun(state,bool)),hoare_2118899576triple(X_c)),hAPP(fun(X_c,fun(state,bool)),fun(com,fun(fun(X_c,fun(state,bool)),hoare_2118899576triple(X_c))),hoare_759811442triple(X_c),Fun1),Com_1),Fun2)) = hAPP(fun(X_c,fun(state,bool)),X_b,hAPP(com,fun(fun(X_c,fun(state,bool)),X_b),hAPP(fun(X_c,fun(state,bool)),fun(com,fun(fun(X_c,fun(state,bool)),X_b)),F1,Fun1),Com_1),Fun2) ).
fof(fact_9_triple_Osimps_I2_J,axiom,
! [X_c,X_b,F1,Fun1,Com_1,Fun2] : hAPP(hoare_2118899576triple(X_c),X_b,hAPP(fun(fun(X_c,fun(state,bool)),fun(com,fun(fun(X_c,fun(state,bool)),X_b))),fun(hoare_2118899576triple(X_c),X_b),hoare_225284258e_case(X_c,X_b),F1),hAPP(fun(X_c,fun(state,bool)),hoare_2118899576triple(X_c),hAPP(com,fun(fun(X_c,fun(state,bool)),hoare_2118899576triple(X_c)),hAPP(fun(X_c,fun(state,bool)),fun(com,fun(fun(X_c,fun(state,bool)),hoare_2118899576triple(X_c))),hoare_759811442triple(X_c),Fun1),Com_1),Fun2)) = hAPP(fun(X_c,fun(state,bool)),X_b,hAPP(com,fun(fun(X_c,fun(state,bool)),X_b),hAPP(fun(X_c,fun(state,bool)),fun(com,fun(fun(X_c,fun(state,bool)),X_b)),F1,Fun1),Com_1),Fun2) ).
fof(fact_10_image__Un,axiom,
! [X_c,X_b,F,A_3,B_2] : hAPP(fun(X_c,bool),fun(X_b,bool),hAPP(fun(X_c,X_b),fun(fun(X_c,bool),fun(X_b,bool)),image(X_c,X_b),F),hAPP(fun(X_c,bool),fun(X_c,bool),hAPP(fun(X_c,bool),fun(fun(X_c,bool),fun(X_c,bool)),semilattice_sup_sup(fun(X_c,bool)),A_3),B_2)) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),hAPP(fun(X_c,bool),fun(X_b,bool),hAPP(fun(X_c,X_b),fun(fun(X_c,bool),fun(X_b,bool)),image(X_c,X_b),F),A_3)),hAPP(fun(X_c,bool),fun(X_b,bool),hAPP(fun(X_c,X_b),fun(fun(X_c,bool),fun(X_b,bool)),image(X_c,X_b),F),B_2)) ).
fof(fact_11_sup__fun__def,axiom,
! [X_b,X_c] :
( lattice(X_c)
=> ! [F,G,X] : hAPP(X_b,X_c,hAPP(fun(X_b,X_c),fun(X_b,X_c),hAPP(fun(X_b,X_c),fun(fun(X_b,X_c),fun(X_b,X_c)),semilattice_sup_sup(fun(X_b,X_c)),F),G),X) = hAPP(X_c,X_c,hAPP(X_c,fun(X_c,X_c),semilattice_sup_sup(X_c),hAPP(X_b,X_c,F,X)),hAPP(X_b,X_c,G,X)) ) ).
fof(fact_12_sup__apply,axiom,
! [X_c,X_b] :
( lattice(X_b)
=> ! [F,G,X_2] : hAPP(X_c,X_b,hAPP(fun(X_c,X_b),fun(X_c,X_b),hAPP(fun(X_c,X_b),fun(fun(X_c,X_b),fun(X_c,X_b)),semilattice_sup_sup(fun(X_c,X_b)),F),G),X_2) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),semilattice_sup_sup(X_b),hAPP(X_c,X_b,F,X_2)),hAPP(X_c,X_b,G,X_2)) ) ).
fof(fact_13_cut,axiom,
! [X_b,Ga,G_1,Ts] :
( hBOOL(hAPP(fun(hoare_2118899576triple(X_b),bool),bool,hAPP(fun(hoare_2118899576triple(X_b),bool),fun(fun(hoare_2118899576triple(X_b),bool),bool),hoare_1301688828derivs(X_b),G_1),Ts))
=> ( hBOOL(hAPP(fun(hoare_2118899576triple(X_b),bool),bool,hAPP(fun(hoare_2118899576triple(X_b),bool),fun(fun(hoare_2118899576triple(X_b),bool),bool),hoare_1301688828derivs(X_b),Ga),G_1))
=> hBOOL(hAPP(fun(hoare_2118899576triple(X_b),bool),bool,hAPP(fun(hoare_2118899576triple(X_b),bool),fun(fun(hoare_2118899576triple(X_b),bool),bool),hoare_1301688828derivs(X_b),Ga),Ts)) ) ) ).
fof(fact_14_sup__assoc,axiom,
! [X_a] :
( semilattice_sup(X_a)
=> ! [X_1,Y,Z] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),X_1),Y)),Z) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),X_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),Y),Z)) ) ).
fof(fact_15_inf__sup__aci_I6_J,axiom,
! [X_a] :
( lattice(X_a)
=> ! [X_1,Y,Z] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),X_1),Y)),Z) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),X_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),Y),Z)) ) ).
fof(fact_16_sup_Oassoc,axiom,
! [X_a] :
( semilattice_sup(X_a)
=> ! [A_1,B,C] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),A_1),B)),C) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),B),C)) ) ).
fof(fact_17_sup__left__commute,axiom,
! [X_a] :
( semilattice_sup(X_a)
=> ! [X_1,Y,Z] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),X_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),Y),Z)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),Y),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),X_1),Z)) ) ).
fof(fact_18_inf__sup__aci_I7_J,axiom,
! [X_a] :
( lattice(X_a)
=> ! [X_1,Y,Z] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),X_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),Y),Z)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),Y),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),X_1),Z)) ) ).
fof(fact_19_sup_Oleft__commute,axiom,
! [X_a] :
( semilattice_sup(X_a)
=> ! [B,A_1,C] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),B),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),A_1),C)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),B),C)) ) ).
fof(fact_20_sup__left__idem,axiom,
! [X_a] :
( semilattice_sup(X_a)
=> ! [X_1,Y] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),X_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),X_1),Y)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),X_1),Y) ) ).
fof(fact_21_inf__sup__aci_I8_J,axiom,
! [X_a] :
( lattice(X_a)
=> ! [X_1,Y] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),X_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),X_1),Y)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),X_1),Y) ) ).
fof(fact_22_sup_Oleft__idem,axiom,
! [X_a] :
( semilattice_sup(X_a)
=> ! [A_1,B] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),A_1),B)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),A_1),B) ) ).
fof(fact_23_sup__commute,axiom,
! [X_a] :
( semilattice_sup(X_a)
=> ! [X_1,Y] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),X_1),Y) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),Y),X_1) ) ).
fof(fact_24_inf__sup__aci_I5_J,axiom,
! [X_a] :
( lattice(X_a)
=> ! [X_1,Y] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),X_1),Y) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),Y),X_1) ) ).
fof(fact_25_sup_Ocommute,axiom,
! [X_a] :
( semilattice_sup(X_a)
=> ! [A_1,B] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),A_1),B) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),B),A_1) ) ).
fof(fact_26_sup__idem,axiom,
! [X_a] :
( semilattice_sup(X_a)
=> ! [X_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),X_1),X_1) = ti(X_a,X_1) ) ).
fof(fact_27_sup_Oidem,axiom,
! [X_a] :
( semilattice_sup(X_a)
=> ! [A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),A_1),A_1) = ti(X_a,A_1) ) ).
fof(fact_28_rev__image__eqI,axiom,
! [X_c,X_b,B_1,F,X_2,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_2),A_3))
=> ( ti(X_c,B_1) = hAPP(X_b,X_c,F,X_2)
=> hBOOL(hAPP(fun(X_c,bool),bool,hAPP(X_c,fun(fun(X_c,bool),bool),member(X_c),B_1),hAPP(fun(X_b,bool),fun(X_c,bool),hAPP(fun(X_b,X_c),fun(fun(X_b,bool),fun(X_c,bool)),image(X_b,X_c),F),A_3))) ) ) ).
fof(fact_29_imageI,axiom,
! [X_c,X_b,F,X_2,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_2),A_3))
=> hBOOL(hAPP(fun(X_c,bool),bool,hAPP(X_c,fun(fun(X_c,bool),bool),member(X_c),hAPP(X_b,X_c,F,X_2)),hAPP(fun(X_b,bool),fun(X_c,bool),hAPP(fun(X_b,X_c),fun(fun(X_b,bool),fun(X_c,bool)),image(X_b,X_c),F),A_3))) ) ).
fof(fact_30_image__iff,axiom,
! [X_b,X_c,Z_1,F,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),Z_1),hAPP(fun(X_c,bool),fun(X_b,bool),hAPP(fun(X_c,X_b),fun(fun(X_c,bool),fun(X_b,bool)),image(X_c,X_b),F),A_3)))
<=> ? [X] :
( hBOOL(hAPP(fun(X_c,bool),bool,hAPP(X_c,fun(fun(X_c,bool),bool),member(X_c),X),A_3))
& ti(X_b,Z_1) = hAPP(X_c,X_b,F,X) ) ) ).
fof(fact_31_UnI2,axiom,
! [X_b,A_3,C_1,B_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),C_1),B_2))
=> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),C_1),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),A_3),B_2))) ) ).
fof(fact_32_UnI1,axiom,
! [X_b,B_2,C_1,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),C_1),A_3))
=> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),C_1),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),A_3),B_2))) ) ).
fof(fact_33_sup1I2,axiom,
! [X_b,A_3,B_2,X_2] :
( hBOOL(hAPP(X_b,bool,B_2,X_2))
=> hBOOL(hAPP(X_b,bool,hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),A_3),B_2),X_2)) ) ).
fof(fact_34_sup1I1,axiom,
! [X_b,B_2,A_3,X_2] :
( hBOOL(hAPP(X_b,bool,A_3,X_2))
=> hBOOL(hAPP(X_b,bool,hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),A_3),B_2),X_2)) ) ).
fof(fact_35_ball__Un,axiom,
! [X_b,Pa,A_3,B_2] :
( ! [X] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),A_3),B_2)))
=> hBOOL(hAPP(X_b,bool,Pa,X)) )
<=> ( ! [X] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X),A_3))
=> hBOOL(hAPP(X_b,bool,Pa,X)) )
& ! [X] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X),B_2))
=> hBOOL(hAPP(X_b,bool,Pa,X)) ) ) ) ).
fof(fact_36_bex__Un,axiom,
! [X_b,Pa,A_3,B_2] :
( ? [X] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),A_3),B_2)))
& hBOOL(hAPP(X_b,bool,Pa,X)) )
<=> ( ? [X] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X),A_3))
& hBOOL(hAPP(X_b,bool,Pa,X)) )
| ? [X] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X),B_2))
& hBOOL(hAPP(X_b,bool,Pa,X)) ) ) ) ).
fof(fact_37_Un__assoc,axiom,
! [X_b,A_3,B_2,C_3] : hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),A_3),B_2)),C_3) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),A_3),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),B_2),C_3)) ).
fof(fact_38_Un__iff,axiom,
! [X_b,C_1,A_3,B_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),C_1),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),A_3),B_2)))
<=> ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),C_1),A_3))
| hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),C_1),B_2)) ) ) ).
fof(fact_39_Un__left__commute,axiom,
! [X_b,A_3,B_2,C_3] : hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),A_3),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),B_2),C_3)) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),B_2),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),A_3),C_3)) ).
fof(fact_40_Un__left__absorb,axiom,
! [X_b,A_3,B_2] : hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),A_3),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),A_3),B_2)) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),A_3),B_2) ).
fof(fact_41_Un__commute,axiom,
! [X_b,A_3,B_2] : hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),A_3),B_2) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),B_2),A_3) ).
fof(fact_42_Un__def,axiom,
! [X_b,A_3,B_2] : hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),A_3),B_2) = hAPP(fun(X_b,bool),fun(X_b,bool),collect(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,bool)),combs(X_b,bool,bool),hAPP(fun(X_b,bool),fun(X_b,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,fun(bool,bool))),combb(bool,fun(bool,bool),X_b),fdisj),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(fun(X_b,bool),bool)),fun(fun(X_b,bool),fun(X_b,bool)),combc(X_b,fun(X_b,bool),bool),member(X_b)),A_3))),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(fun(X_b,bool),bool)),fun(fun(X_b,bool),fun(X_b,bool)),combc(X_b,fun(X_b,bool),bool),member(X_b)),B_2))) ).
fof(fact_43_Un__absorb,axiom,
! [X_b,A_3] : hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),A_3),A_3) = ti(fun(X_b,bool),A_3) ).
fof(fact_44_image__ident,axiom,
! [X_b,Y_3] : hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,X_b),fun(fun(X_b,bool),fun(X_b,bool)),image(X_b,X_b),combi(X_b)),Y_3) = ti(fun(X_b,bool),Y_3) ).
fof(fact_45_image__image,axiom,
! [X_c,X_b,X_d,F,G,A_3] : hAPP(fun(X_c,bool),fun(X_b,bool),hAPP(fun(X_c,X_b),fun(fun(X_c,bool),fun(X_b,bool)),image(X_c,X_b),F),hAPP(fun(X_d,bool),fun(X_c,bool),hAPP(fun(X_d,X_c),fun(fun(X_d,bool),fun(X_c,bool)),image(X_d,X_c),G),A_3)) = hAPP(fun(X_d,bool),fun(X_b,bool),hAPP(fun(X_d,X_b),fun(fun(X_d,bool),fun(X_b,bool)),image(X_d,X_b),hAPP(fun(X_d,X_c),fun(X_d,X_b),hAPP(fun(X_c,X_b),fun(fun(X_d,X_c),fun(X_d,X_b)),combb(X_c,X_b,X_d),F),G)),A_3) ).
fof(fact_46_sup__Un__eq,axiom,
! [X_b,R_1,S_1,X] :
( hBOOL(hAPP(X_b,bool,hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(fun(X_b,bool),bool)),fun(fun(X_b,bool),fun(X_b,bool)),combc(X_b,fun(X_b,bool),bool),member(X_b)),R_1)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(fun(X_b,bool),bool)),fun(fun(X_b,bool),fun(X_b,bool)),combc(X_b,fun(X_b,bool),bool),member(X_b)),S_1)),X))
<=> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),R_1),S_1))) ) ).
fof(fact_47_Collect__disj__eq,axiom,
! [X_b,Pa,Qa] : hAPP(fun(X_b,bool),fun(X_b,bool),collect(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,bool)),combs(X_b,bool,bool),hAPP(fun(X_b,bool),fun(X_b,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,fun(bool,bool))),combb(bool,fun(bool,bool),X_b),fdisj),Pa)),Qa)) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),collect(X_b),Pa)),hAPP(fun(X_b,bool),fun(X_b,bool),collect(X_b),Qa)) ).
fof(fact_48_imageE,axiom,
! [X_b,X_c,B_1,F,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),B_1),hAPP(fun(X_c,bool),fun(X_b,bool),hAPP(fun(X_c,X_b),fun(fun(X_c,bool),fun(X_b,bool)),image(X_c,X_b),F),A_3)))
=> ~ ! [X] :
( ti(X_b,B_1) = hAPP(X_c,X_b,F,X)
=> ~ hBOOL(hAPP(fun(X_c,bool),bool,hAPP(X_c,fun(fun(X_c,bool),bool),member(X_c),X),A_3)) ) ) ).
fof(fact_49_Body__triple__valid__Suc,axiom,
! [X_b,N_1,Pa,Pn_1,Qa] :
( hBOOL(hAPP(hoare_2118899576triple(X_b),bool,hAPP(nat,fun(hoare_2118899576triple(X_b),bool),hoare_1942962616_valid(X_b),N_1),hAPP(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b),hAPP(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b)),hAPP(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b))),hoare_759811442triple(X_b),Pa),hAPP(option(com),com,the(com),hAPP(pname,option(com),body_1,Pn_1))),Qa)))
<=> hBOOL(hAPP(hoare_2118899576triple(X_b),bool,hAPP(nat,fun(hoare_2118899576triple(X_b),bool),hoare_1942962616_valid(X_b),hAPP(nat,nat,suc,N_1)),hAPP(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b),hAPP(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b)),hAPP(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b))),hoare_759811442triple(X_b),Pa),hAPP(pname,com,body,Pn_1)),Qa))) ) ).
fof(fact_50_triple_Oexhaust,axiom,
! [X_b,Y_1] :
~ ! [Fun1_1,Com_2,Fun2_1] : Y_1 != hAPP(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b),hAPP(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b)),hAPP(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b))),hoare_759811442triple(X_b),Fun1_1),Com_2),Fun2_1) ).
fof(fact_51_Body1,axiom,
! [X_b,Pn_1,Ga,Pa,Qa,Procsa] :
( hBOOL(hAPP(fun(hoare_2118899576triple(X_b),bool),bool,hAPP(fun(hoare_2118899576triple(X_b),bool),fun(fun(hoare_2118899576triple(X_b),bool),bool),hoare_1301688828derivs(X_b),hAPP(fun(hoare_2118899576triple(X_b),bool),fun(hoare_2118899576triple(X_b),bool),hAPP(fun(hoare_2118899576triple(X_b),bool),fun(fun(hoare_2118899576triple(X_b),bool),fun(hoare_2118899576triple(X_b),bool)),semilattice_sup_sup(fun(hoare_2118899576triple(X_b),bool)),Ga),hAPP(fun(pname,bool),fun(hoare_2118899576triple(X_b),bool),hAPP(fun(pname,hoare_2118899576triple(X_b)),fun(fun(pname,bool),fun(hoare_2118899576triple(X_b),bool)),image(pname,hoare_2118899576triple(X_b)),hAPP(fun(pname,fun(X_b,fun(state,bool))),fun(pname,hoare_2118899576triple(X_b)),hAPP(fun(pname,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b))),fun(fun(pname,fun(X_b,fun(state,bool))),fun(pname,hoare_2118899576triple(X_b))),combs(pname,fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b)),hAPP(fun(pname,com),fun(pname,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b))),hAPP(fun(pname,fun(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b)))),fun(fun(pname,com),fun(pname,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b)))),combs(pname,com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b))),hAPP(fun(pname,fun(X_b,fun(state,bool))),fun(pname,fun(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b)))),hAPP(fun(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b)))),fun(fun(pname,fun(X_b,fun(state,bool))),fun(pname,fun(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b))))),combb(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b))),pname),hoare_759811442triple(X_b)),Pa)),body)),Qa)),Procsa))),hAPP(fun(pname,bool),fun(hoare_2118899576triple(X_b),bool),hAPP(fun(pname,hoare_2118899576triple(X_b)),fun(fun(pname,bool),fun(hoare_2118899576triple(X_b),bool)),image(pname,hoare_2118899576triple(X_b)),hAPP(fun(pname,fun(X_b,fun(state,bool))),fun(pname,hoare_2118899576triple(X_b)),hAPP(fun(pname,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b))),fun(fun(pname,fun(X_b,fun(state,bool))),fun(pname,hoare_2118899576triple(X_b))),combs(pname,fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b)),hAPP(fun(pname,com),fun(pname,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b))),hAPP(fun(pname,fun(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b)))),fun(fun(pname,com),fun(pname,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b)))),combs(pname,com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b))),hAPP(fun(pname,fun(X_b,fun(state,bool))),fun(pname,fun(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b)))),hAPP(fun(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b)))),fun(fun(pname,fun(X_b,fun(state,bool))),fun(pname,fun(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b))))),combb(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b))),pname),hoare_759811442triple(X_b)),Pa)),hAPP(fun(pname,option(com)),fun(pname,com),hAPP(fun(option(com),com),fun(fun(pname,option(com)),fun(pname,com)),combb(option(com),com,pname),the(com)),body_1))),Qa)),Procsa)))
=> ( hBOOL(hAPP(fun(pname,bool),bool,hAPP(pname,fun(fun(pname,bool),bool),member(pname),Pn_1),Procsa))
=> hBOOL(hAPP(fun(hoare_2118899576triple(X_b),bool),bool,hAPP(fun(hoare_2118899576triple(X_b),bool),fun(fun(hoare_2118899576triple(X_b),bool),bool),hoare_1301688828derivs(X_b),Ga),hAPP(fun(hoare_2118899576triple(X_b),bool),fun(hoare_2118899576triple(X_b),bool),hAPP(hoare_2118899576triple(X_b),fun(fun(hoare_2118899576triple(X_b),bool),fun(hoare_2118899576triple(X_b),bool)),insert(hoare_2118899576triple(X_b)),hAPP(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b),hAPP(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b)),hAPP(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b))),hoare_759811442triple(X_b),hAPP(pname,fun(X_b,fun(state,bool)),Pa,Pn_1)),hAPP(pname,com,body,Pn_1)),hAPP(pname,fun(X_b,fun(state,bool)),Qa,Pn_1))),bot_bot(fun(hoare_2118899576triple(X_b),bool))))) ) ) ).
fof(fact_52_image__cong,axiom,
! [X_c,X_b,F,G,M_5,N_3] :
( ti(fun(X_b,bool),M_5) = ti(fun(X_b,bool),N_3)
=> ( ! [X] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X),N_3))
=> hAPP(X_b,X_c,F,X) = hAPP(X_b,X_c,G,X) )
=> hAPP(fun(X_b,bool),fun(X_c,bool),hAPP(fun(X_b,X_c),fun(fun(X_b,bool),fun(X_c,bool)),image(X_b,X_c),F),M_5) = hAPP(fun(X_b,bool),fun(X_c,bool),hAPP(fun(X_b,X_c),fun(fun(X_b,bool),fun(X_c,bool)),image(X_b,X_c),G),N_3) ) ) ).
fof(fact_53_Powp__def,axiom,
! [X_b,A_3,X] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),powp(X_b),A_3),X))
<=> ! [Xa] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),Xa),X))
=> hBOOL(hAPP(X_b,bool,A_3,Xa)) ) ) ).
fof(fact_54_Body__triple__valid__0,axiom,
! [X_b,Pa,Pn_1,Qa] : hBOOL(hAPP(hoare_2118899576triple(X_b),bool,hAPP(nat,fun(hoare_2118899576triple(X_b),bool),hoare_1942962616_valid(X_b),zero_zero(nat)),hAPP(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b),hAPP(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b)),hAPP(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b))),hoare_759811442triple(X_b),Pa),hAPP(pname,com,body,Pn_1)),Qa))) ).
fof(fact_55_com_Osimps_I6_J,axiom,
! [Pname_3,Pname] :
( hAPP(pname,com,body,Pname_3) = hAPP(pname,com,body,Pname)
<=> ti(pname,Pname_3) = ti(pname,Pname) ) ).
fof(fact_56_evalc_OBody,axiom,
! [Pn,S0,S1_2] :
( hBOOL(hAPP(state,bool,hAPP(state,fun(state,bool),hAPP(com,fun(state,fun(state,bool)),evalc,hAPP(option(com),com,the(com),hAPP(pname,option(com),body_1,Pn))),S0),S1_2))
=> hBOOL(hAPP(state,bool,hAPP(state,fun(state,bool),hAPP(com,fun(state,fun(state,bool)),evalc,hAPP(pname,com,body,Pn)),S0),S1_2)) ) ).
fof(fact_57_evalc__elim__cases_I6_J,axiom,
! [P,S,S1_2] :
( hBOOL(hAPP(state,bool,hAPP(state,fun(state,bool),hAPP(com,fun(state,fun(state,bool)),evalc,hAPP(pname,com,body,P)),S),S1_2))
=> hBOOL(hAPP(state,bool,hAPP(state,fun(state,bool),hAPP(com,fun(state,fun(state,bool)),evalc,hAPP(option(com),com,the(com),hAPP(pname,option(com),body_1,P))),S),S1_2)) ) ).
fof(fact_58_Sup__fin_Oidem,axiom,
! [X_a] :
( lattice(X_a)
=> ! [X_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),X_1),X_1) = ti(X_a,X_1) ) ).
fof(fact_59_triples__valid__Suc,axiom,
! [X_b,N_1,Ts] :
( ! [X] :
( hBOOL(hAPP(fun(hoare_2118899576triple(X_b),bool),bool,hAPP(hoare_2118899576triple(X_b),fun(fun(hoare_2118899576triple(X_b),bool),bool),member(hoare_2118899576triple(X_b)),X),Ts))
=> hBOOL(hAPP(hoare_2118899576triple(X_b),bool,hAPP(nat,fun(hoare_2118899576triple(X_b),bool),hoare_1942962616_valid(X_b),hAPP(nat,nat,suc,N_1)),X)) )
=> ! [X] :
( hBOOL(hAPP(fun(hoare_2118899576triple(X_b),bool),bool,hAPP(hoare_2118899576triple(X_b),fun(fun(hoare_2118899576triple(X_b),bool),bool),member(hoare_2118899576triple(X_b)),X),Ts))
=> hBOOL(hAPP(hoare_2118899576triple(X_b),bool,hAPP(nat,fun(hoare_2118899576triple(X_b),bool),hoare_1942962616_valid(X_b),N_1),X)) ) ) ).
fof(fact_60_emptyE,axiom,
! [X_b,A_2] : ~ hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),A_2),bot_bot(fun(X_b,bool)))) ).
fof(fact_61_insertE,axiom,
! [X_b,A_2,B_1,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),A_2),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),B_1),A_3)))
=> ( ti(X_b,A_2) != ti(X_b,B_1)
=> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),A_2),A_3)) ) ) ).
fof(fact_62_insertCI,axiom,
! [X_b,B_1,A_2,B_2] :
( ( ~ hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),A_2),B_2))
=> ti(X_b,A_2) = ti(X_b,B_1) )
=> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),A_2),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),B_1),B_2))) ) ).
fof(fact_63_empty__not__insert,axiom,
! [X_b,A_2,A_3] : bot_bot(fun(X_b,bool)) != hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_2),A_3) ).
fof(fact_64_insert__not__empty,axiom,
! [X_b,A_2,A_3] : hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_2),A_3) != bot_bot(fun(X_b,bool)) ).
fof(fact_65_bot__empty__eq,axiom,
! [X_b,X] :
( hBOOL(hAPP(X_b,bool,bot_bot(fun(X_b,bool)),X))
<=> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X),bot_bot(fun(X_b,bool)))) ) ).
fof(fact_66_empty__def,axiom,
! [X_b] : bot_bot(fun(X_b,bool)) = hAPP(fun(X_b,bool),fun(X_b,bool),collect(X_b),hAPP(bool,fun(X_b,bool),combk(bool,X_b),fFalse)) ).
fof(fact_67_insertI1,axiom,
! [X_b,A_2,B_2] : hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),A_2),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_2),B_2))) ).
fof(fact_68_all__not__in__conv,axiom,
! [X_b,A_3] :
( ! [X] : ~ hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X),A_3))
<=> ti(fun(X_b,bool),A_3) = bot_bot(fun(X_b,bool)) ) ).
fof(fact_69_singleton__conv2,axiom,
! [X_b,A_2] : hAPP(fun(X_b,bool),fun(X_b,bool),collect(X_b),hAPP(X_b,fun(X_b,bool),fequal(X_b),A_2)) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_2),bot_bot(fun(X_b,bool))) ).
fof(fact_70_ex__in__conv,axiom,
! [X_b,A_3] :
( ? [X] : hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X),A_3))
<=> ti(fun(X_b,bool),A_3) != bot_bot(fun(X_b,bool)) ) ).
fof(fact_71_singleton__conv,axiom,
! [X_b,A_2] : hAPP(fun(X_b,bool),fun(X_b,bool),collect(X_b),hAPP(X_b,fun(X_b,bool),hAPP(fun(X_b,fun(X_b,bool)),fun(X_b,fun(X_b,bool)),combc(X_b,X_b,bool),fequal(X_b)),A_2)) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_2),bot_bot(fun(X_b,bool))) ).
fof(fact_72_Collect__conv__if2,axiom,
! [X_b,Pa,A_2] :
( ( hBOOL(hAPP(X_b,bool,Pa,A_2))
=> hAPP(fun(X_b,bool),fun(X_b,bool),collect(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,bool)),combs(X_b,bool,bool),hAPP(fun(X_b,bool),fun(X_b,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,fun(bool,bool))),combb(bool,fun(bool,bool),X_b),fconj),hAPP(X_b,fun(X_b,bool),fequal(X_b),A_2))),Pa)) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_2),bot_bot(fun(X_b,bool))) )
& ( ~ hBOOL(hAPP(X_b,bool,Pa,A_2))
=> hAPP(fun(X_b,bool),fun(X_b,bool),collect(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,bool)),combs(X_b,bool,bool),hAPP(fun(X_b,bool),fun(X_b,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,fun(bool,bool))),combb(bool,fun(bool,bool),X_b),fconj),hAPP(X_b,fun(X_b,bool),fequal(X_b),A_2))),Pa)) = bot_bot(fun(X_b,bool)) ) ) ).
fof(fact_73_Collect__conv__if,axiom,
! [X_b,Pa,A_2] :
( ( hBOOL(hAPP(X_b,bool,Pa,A_2))
=> hAPP(fun(X_b,bool),fun(X_b,bool),collect(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,bool)),combs(X_b,bool,bool),hAPP(fun(X_b,bool),fun(X_b,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,fun(bool,bool))),combb(bool,fun(bool,bool),X_b),fconj),hAPP(X_b,fun(X_b,bool),hAPP(fun(X_b,fun(X_b,bool)),fun(X_b,fun(X_b,bool)),combc(X_b,X_b,bool),fequal(X_b)),A_2))),Pa)) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_2),bot_bot(fun(X_b,bool))) )
& ( ~ hBOOL(hAPP(X_b,bool,Pa,A_2))
=> hAPP(fun(X_b,bool),fun(X_b,bool),collect(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,bool)),combs(X_b,bool,bool),hAPP(fun(X_b,bool),fun(X_b,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,fun(bool,bool))),combb(bool,fun(bool,bool),X_b),fconj),hAPP(X_b,fun(X_b,bool),hAPP(fun(X_b,fun(X_b,bool)),fun(X_b,fun(X_b,bool)),combc(X_b,X_b,bool),fequal(X_b)),A_2))),Pa)) = bot_bot(fun(X_b,bool)) ) ) ).
fof(fact_74_ext,axiom,
! [X_b,X_c,F,G] :
( ! [X] : hAPP(X_b,X_c,F,X) = hAPP(X_b,X_c,G,X)
=> ti(fun(X_b,X_c),F) = ti(fun(X_b,X_c),G) ) ).
fof(fact_75_mem__def,axiom,
! [X_b,X_2,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_2),A_3))
<=> hBOOL(hAPP(X_b,bool,A_3,X_2)) ) ).
fof(fact_76_Collect__def,axiom,
! [X_b,Pa] : hAPP(fun(X_b,bool),fun(X_b,bool),collect(X_b),Pa) = ti(fun(X_b,bool),Pa) ).
fof(fact_77_empty__Collect__eq,axiom,
! [X_b,Pa] :
( bot_bot(fun(X_b,bool)) = hAPP(fun(X_b,bool),fun(X_b,bool),collect(X_b),Pa)
<=> ! [X] : ~ hBOOL(hAPP(X_b,bool,Pa,X)) ) ).
fof(fact_78_empty__iff,axiom,
! [X_b,C_1] : ~ hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),C_1),bot_bot(fun(X_b,bool)))) ).
fof(fact_79_insert__compr,axiom,
! [X_b,A_2,B_2] : hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_2),B_2) = hAPP(fun(X_b,bool),fun(X_b,bool),collect(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,bool)),combs(X_b,bool,bool),hAPP(fun(X_b,bool),fun(X_b,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,fun(bool,bool))),combb(bool,fun(bool,bool),X_b),fdisj),hAPP(X_b,fun(X_b,bool),hAPP(fun(X_b,fun(X_b,bool)),fun(X_b,fun(X_b,bool)),combc(X_b,X_b,bool),fequal(X_b)),A_2))),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(fun(X_b,bool),bool)),fun(fun(X_b,bool),fun(X_b,bool)),combc(X_b,fun(X_b,bool),bool),member(X_b)),B_2))) ).
fof(fact_80_insert__Collect,axiom,
! [X_b,A_2,Pa] : hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_2),hAPP(fun(X_b,bool),fun(X_b,bool),collect(X_b),Pa)) = hAPP(fun(X_b,bool),fun(X_b,bool),collect(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,bool)),combs(X_b,bool,bool),hAPP(fun(X_b,bool),fun(X_b,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,fun(bool,bool))),combb(bool,fun(bool,bool),X_b),fimplies),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(bool,bool),fun(fun(X_b,bool),fun(X_b,bool)),combb(bool,bool,X_b),fNot),hAPP(X_b,fun(X_b,bool),hAPP(fun(X_b,fun(X_b,bool)),fun(X_b,fun(X_b,bool)),combc(X_b,X_b,bool),fequal(X_b)),A_2)))),Pa)) ).
fof(fact_81_singleton__iff,axiom,
! [X_b,B_1,A_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),B_1),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_2),bot_bot(fun(X_b,bool)))))
<=> ti(X_b,B_1) = ti(X_b,A_2) ) ).
fof(fact_82_insert__absorb2,axiom,
! [X_b,X_2,A_3] : hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_2),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_2),A_3)) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_2),A_3) ).
fof(fact_83_insert__commute,axiom,
! [X_b,X_2,Y_1,A_3] : hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_2),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),Y_1),A_3)) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),Y_1),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_2),A_3)) ).
fof(fact_84_insert__iff,axiom,
! [X_b,A_2,B_1,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),A_2),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),B_1),A_3)))
<=> ( ti(X_b,A_2) = ti(X_b,B_1)
| hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),A_2),A_3)) ) ) ).
fof(fact_85_Collect__empty__eq,axiom,
! [X_b,Pa] :
( hAPP(fun(X_b,bool),fun(X_b,bool),collect(X_b),Pa) = bot_bot(fun(X_b,bool))
<=> ! [X] : ~ hBOOL(hAPP(X_b,bool,Pa,X)) ) ).
fof(fact_86_doubleton__eq__iff,axiom,
! [X_b,A_2,B_1,C_1,D] :
( hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_2),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),B_1),bot_bot(fun(X_b,bool)))) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),C_1),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),D),bot_bot(fun(X_b,bool))))
<=> ( ( ti(X_b,A_2) = ti(X_b,C_1)
& ti(X_b,B_1) = ti(X_b,D) )
| ( ti(X_b,A_2) = ti(X_b,D)
& ti(X_b,B_1) = ti(X_b,C_1) ) ) ) ).
fof(fact_87_insert__code,axiom,
! [X_b,Y_1,A_3,X_2] :
( hBOOL(hAPP(X_b,bool,hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),Y_1),A_3),X_2))
<=> ( ti(X_b,Y_1) = ti(X_b,X_2)
| hBOOL(hAPP(X_b,bool,A_3,X_2)) ) ) ).
fof(fact_88_insert__ident,axiom,
! [X_b,B_2,X_2,A_3] :
( ~ hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_2),A_3))
=> ( ~ hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_2),B_2))
=> ( hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_2),A_3) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_2),B_2)
<=> ti(fun(X_b,bool),A_3) = ti(fun(X_b,bool),B_2) ) ) ) ).
fof(fact_89_equals0D,axiom,
! [X_b,A_2,A_3] :
( ti(fun(X_b,bool),A_3) = bot_bot(fun(X_b,bool))
=> ~ hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),A_2),A_3)) ) ).
fof(fact_90_insertI2,axiom,
! [X_b,B_1,A_2,B_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),A_2),B_2))
=> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),A_2),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),B_1),B_2))) ) ).
fof(fact_91_insert__absorb,axiom,
! [X_b,A_2,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),A_2),A_3))
=> hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_2),A_3) = ti(fun(X_b,bool),A_3) ) ).
fof(fact_92_singletonE,axiom,
! [X_b,B_1,A_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),B_1),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_2),bot_bot(fun(X_b,bool)))))
=> ti(X_b,B_1) = ti(X_b,A_2) ) ).
fof(fact_93_singleton__inject,axiom,
! [X_b,A_2,B_1] :
( hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_2),bot_bot(fun(X_b,bool))) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),B_1),bot_bot(fun(X_b,bool)))
=> ti(X_b,A_2) = ti(X_b,B_1) ) ).
fof(fact_94_com__det,axiom,
! [U_1,C,S,T_3] :
( hBOOL(hAPP(state,bool,hAPP(state,fun(state,bool),hAPP(com,fun(state,fun(state,bool)),evalc,C),S),T_3))
=> ( hBOOL(hAPP(state,bool,hAPP(state,fun(state,bool),hAPP(com,fun(state,fun(state,bool)),evalc,C),S),U_1))
=> U_1 = T_3 ) ) ).
fof(fact_95_insert__is__Un,axiom,
! [X_b,A_2,A_3] : hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_2),A_3) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_2),bot_bot(fun(X_b,bool)))),A_3) ).
fof(fact_96_insert__compr__raw,axiom,
! [X_b,X,Xa] : hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X),Xa) = hAPP(fun(X_b,bool),fun(X_b,bool),collect(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,bool)),combs(X_b,bool,bool),hAPP(fun(X_b,bool),fun(X_b,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,fun(bool,bool))),combb(bool,fun(bool,bool),X_b),fdisj),hAPP(X_b,fun(X_b,bool),hAPP(fun(X_b,fun(X_b,bool)),fun(X_b,fun(X_b,bool)),combc(X_b,X_b,bool),fequal(X_b)),X))),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(fun(X_b,bool),bool)),fun(fun(X_b,bool),fun(X_b,bool)),combc(X_b,fun(X_b,bool),bool),member(X_b)),Xa))) ).
fof(fact_97_derivs__insertD,axiom,
! [X_b,Ga,T_4,Ts] :
( hBOOL(hAPP(fun(hoare_2118899576triple(X_b),bool),bool,hAPP(fun(hoare_2118899576triple(X_b),bool),fun(fun(hoare_2118899576triple(X_b),bool),bool),hoare_1301688828derivs(X_b),Ga),hAPP(fun(hoare_2118899576triple(X_b),bool),fun(hoare_2118899576triple(X_b),bool),hAPP(hoare_2118899576triple(X_b),fun(fun(hoare_2118899576triple(X_b),bool),fun(hoare_2118899576triple(X_b),bool)),insert(hoare_2118899576triple(X_b)),T_4),Ts)))
=> ( hBOOL(hAPP(fun(hoare_2118899576triple(X_b),bool),bool,hAPP(fun(hoare_2118899576triple(X_b),bool),fun(fun(hoare_2118899576triple(X_b),bool),bool),hoare_1301688828derivs(X_b),Ga),hAPP(fun(hoare_2118899576triple(X_b),bool),fun(hoare_2118899576triple(X_b),bool),hAPP(hoare_2118899576triple(X_b),fun(fun(hoare_2118899576triple(X_b),bool),fun(hoare_2118899576triple(X_b),bool)),insert(hoare_2118899576triple(X_b)),T_4),bot_bot(fun(hoare_2118899576triple(X_b),bool)))))
& hBOOL(hAPP(fun(hoare_2118899576triple(X_b),bool),bool,hAPP(fun(hoare_2118899576triple(X_b),bool),fun(fun(hoare_2118899576triple(X_b),bool),bool),hoare_1301688828derivs(X_b),Ga),Ts)) ) ) ).
fof(fact_98_hoare__derivs_Oinsert,axiom,
! [X_b,Ts,Ga,T_4] :
( hBOOL(hAPP(fun(hoare_2118899576triple(X_b),bool),bool,hAPP(fun(hoare_2118899576triple(X_b),bool),fun(fun(hoare_2118899576triple(X_b),bool),bool),hoare_1301688828derivs(X_b),Ga),hAPP(fun(hoare_2118899576triple(X_b),bool),fun(hoare_2118899576triple(X_b),bool),hAPP(hoare_2118899576triple(X_b),fun(fun(hoare_2118899576triple(X_b),bool),fun(hoare_2118899576triple(X_b),bool)),insert(hoare_2118899576triple(X_b)),T_4),bot_bot(fun(hoare_2118899576triple(X_b),bool)))))
=> ( hBOOL(hAPP(fun(hoare_2118899576triple(X_b),bool),bool,hAPP(fun(hoare_2118899576triple(X_b),bool),fun(fun(hoare_2118899576triple(X_b),bool),bool),hoare_1301688828derivs(X_b),Ga),Ts))
=> hBOOL(hAPP(fun(hoare_2118899576triple(X_b),bool),bool,hAPP(fun(hoare_2118899576triple(X_b),bool),fun(fun(hoare_2118899576triple(X_b),bool),bool),hoare_1301688828derivs(X_b),Ga),hAPP(fun(hoare_2118899576triple(X_b),bool),fun(hoare_2118899576triple(X_b),bool),hAPP(hoare_2118899576triple(X_b),fun(fun(hoare_2118899576triple(X_b),bool),fun(hoare_2118899576triple(X_b),bool)),insert(hoare_2118899576triple(X_b)),T_4),Ts))) ) ) ).
fof(fact_99_image__constant__conv,axiom,
! [X_c,X_b,C_1,A_3] :
( ( ti(fun(X_b,bool),A_3) = bot_bot(fun(X_b,bool))
=> hAPP(fun(X_b,bool),fun(X_c,bool),hAPP(fun(X_b,X_c),fun(fun(X_b,bool),fun(X_c,bool)),image(X_b,X_c),hAPP(X_c,fun(X_b,X_c),combk(X_c,X_b),C_1)),A_3) = bot_bot(fun(X_c,bool)) )
& ( ti(fun(X_b,bool),A_3) != bot_bot(fun(X_b,bool))
=> hAPP(fun(X_b,bool),fun(X_c,bool),hAPP(fun(X_b,X_c),fun(fun(X_b,bool),fun(X_c,bool)),image(X_b,X_c),hAPP(X_c,fun(X_b,X_c),combk(X_c,X_b),C_1)),A_3) = hAPP(fun(X_c,bool),fun(X_c,bool),hAPP(X_c,fun(fun(X_c,bool),fun(X_c,bool)),insert(X_c),C_1),bot_bot(fun(X_c,bool))) ) ) ).
fof(fact_100_image__constant,axiom,
! [X_c,X_b,C_1,X_2,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_2),A_3))
=> hAPP(fun(X_b,bool),fun(X_c,bool),hAPP(fun(X_b,X_c),fun(fun(X_b,bool),fun(X_c,bool)),image(X_b,X_c),hAPP(X_c,fun(X_b,X_c),combk(X_c,X_b),C_1)),A_3) = hAPP(fun(X_c,bool),fun(X_c,bool),hAPP(X_c,fun(fun(X_c,bool),fun(X_c,bool)),insert(X_c),C_1),bot_bot(fun(X_c,bool))) ) ).
fof(fact_101_image__insert,axiom,
! [X_c,X_b,F,A_2,B_2] : hAPP(fun(X_c,bool),fun(X_b,bool),hAPP(fun(X_c,X_b),fun(fun(X_c,bool),fun(X_b,bool)),image(X_c,X_b),F),hAPP(fun(X_c,bool),fun(X_c,bool),hAPP(X_c,fun(fun(X_c,bool),fun(X_c,bool)),insert(X_c),A_2),B_2)) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),hAPP(X_c,X_b,F,A_2)),hAPP(fun(X_c,bool),fun(X_b,bool),hAPP(fun(X_c,X_b),fun(fun(X_c,bool),fun(X_b,bool)),image(X_c,X_b),F),B_2)) ).
fof(fact_102_insert__image,axiom,
! [X_c,X_b,F,X_2,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_2),A_3))
=> hAPP(fun(X_c,bool),fun(X_c,bool),hAPP(X_c,fun(fun(X_c,bool),fun(X_c,bool)),insert(X_c),hAPP(X_b,X_c,F,X_2)),hAPP(fun(X_b,bool),fun(X_c,bool),hAPP(fun(X_b,X_c),fun(fun(X_b,bool),fun(X_c,bool)),image(X_b,X_c),F),A_3)) = hAPP(fun(X_b,bool),fun(X_c,bool),hAPP(fun(X_b,X_c),fun(fun(X_b,bool),fun(X_c,bool)),image(X_b,X_c),F),A_3) ) ).
fof(fact_103_Un__insert__right,axiom,
! [X_b,A_3,A_2,B_2] : hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),A_3),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_2),B_2)) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_2),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),A_3),B_2)) ).
fof(fact_104_Un__insert__left,axiom,
! [X_b,A_2,B_2,C_3] : hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_2),B_2)),C_3) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_2),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),B_2),C_3)) ).
fof(fact_105_empty__is__image,axiom,
! [X_b,X_c,F,A_3] :
( bot_bot(fun(X_b,bool)) = hAPP(fun(X_c,bool),fun(X_b,bool),hAPP(fun(X_c,X_b),fun(fun(X_c,bool),fun(X_b,bool)),image(X_c,X_b),F),A_3)
<=> ti(fun(X_c,bool),A_3) = bot_bot(fun(X_c,bool)) ) ).
fof(fact_106_image__empty,axiom,
! [X_c,X_b,F] : hAPP(fun(X_c,bool),fun(X_b,bool),hAPP(fun(X_c,X_b),fun(fun(X_c,bool),fun(X_b,bool)),image(X_c,X_b),F),bot_bot(fun(X_c,bool))) = bot_bot(fun(X_b,bool)) ).
fof(fact_107_image__is__empty,axiom,
! [X_c,X_b,F,A_3] :
( hAPP(fun(X_c,bool),fun(X_b,bool),hAPP(fun(X_c,X_b),fun(fun(X_c,bool),fun(X_b,bool)),image(X_c,X_b),F),A_3) = bot_bot(fun(X_b,bool))
<=> ti(fun(X_c,bool),A_3) = bot_bot(fun(X_c,bool)) ) ).
fof(fact_108_ball__empty,axiom,
! [X_b,Pa,X] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X),bot_bot(fun(X_b,bool))))
=> hBOOL(hAPP(X_b,bool,Pa,X)) ) ).
fof(fact_109_Un__empty__left,axiom,
! [X_b,B_2] : hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),bot_bot(fun(X_b,bool))),B_2) = ti(fun(X_b,bool),B_2) ).
fof(fact_110_Un__empty__right,axiom,
! [X_b,A_3] : hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),A_3),bot_bot(fun(X_b,bool))) = ti(fun(X_b,bool),A_3) ).
fof(fact_111_Un__empty,axiom,
! [X_b,A_3,B_2] :
( hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),A_3),B_2) = bot_bot(fun(X_b,bool))
<=> ( ti(fun(X_b,bool),A_3) = bot_bot(fun(X_b,bool))
& ti(fun(X_b,bool),B_2) = bot_bot(fun(X_b,bool)) ) ) ).
fof(fact_112_constant,axiom,
! [X_b,Ga,Pa,C_1,Qa,C_3] :
( ( hBOOL(C_3)
=> hBOOL(hAPP(fun(hoare_2118899576triple(X_b),bool),bool,hAPP(fun(hoare_2118899576triple(X_b),bool),fun(fun(hoare_2118899576triple(X_b),bool),bool),hoare_1301688828derivs(X_b),Ga),hAPP(fun(hoare_2118899576triple(X_b),bool),fun(hoare_2118899576triple(X_b),bool),hAPP(hoare_2118899576triple(X_b),fun(fun(hoare_2118899576triple(X_b),bool),fun(hoare_2118899576triple(X_b),bool)),insert(hoare_2118899576triple(X_b)),hAPP(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b),hAPP(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b)),hAPP(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b))),hoare_759811442triple(X_b),Pa),C_1),Qa)),bot_bot(fun(hoare_2118899576triple(X_b),bool))))) )
=> hBOOL(hAPP(fun(hoare_2118899576triple(X_b),bool),bool,hAPP(fun(hoare_2118899576triple(X_b),bool),fun(fun(hoare_2118899576triple(X_b),bool),bool),hoare_1301688828derivs(X_b),Ga),hAPP(fun(hoare_2118899576triple(X_b),bool),fun(hoare_2118899576triple(X_b),bool),hAPP(hoare_2118899576triple(X_b),fun(fun(hoare_2118899576triple(X_b),bool),fun(hoare_2118899576triple(X_b),bool)),insert(hoare_2118899576triple(X_b)),hAPP(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b),hAPP(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b)),hAPP(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b))),hoare_759811442triple(X_b),hAPP(bool,fun(X_b,fun(state,bool)),hAPP(fun(X_b,fun(bool,fun(state,bool))),fun(bool,fun(X_b,fun(state,bool))),combc(X_b,bool,fun(state,bool)),hAPP(fun(X_b,fun(state,fun(bool,bool))),fun(X_b,fun(bool,fun(state,bool))),hAPP(fun(fun(state,fun(bool,bool)),fun(bool,fun(state,bool))),fun(fun(X_b,fun(state,fun(bool,bool))),fun(X_b,fun(bool,fun(state,bool)))),combb(fun(state,fun(bool,bool)),fun(bool,fun(state,bool)),X_b),combc(state,bool,bool)),hAPP(fun(X_b,fun(state,bool)),fun(X_b,fun(state,fun(bool,bool))),hAPP(fun(fun(state,bool),fun(state,fun(bool,bool))),fun(fun(X_b,fun(state,bool)),fun(X_b,fun(state,fun(bool,bool)))),combb(fun(state,bool),fun(state,fun(bool,bool)),X_b),hAPP(fun(bool,fun(bool,bool)),fun(fun(state,bool),fun(state,fun(bool,bool))),combb(bool,fun(bool,bool),state),fconj)),Pa))),C_3)),C_1),Qa)),bot_bot(fun(hoare_2118899576triple(X_b),bool))))) ) ).
fof(fact_113_empty,axiom,
! [X_b,Ga] : hBOOL(hAPP(fun(hoare_2118899576triple(X_b),bool),bool,hAPP(fun(hoare_2118899576triple(X_b),bool),fun(fun(hoare_2118899576triple(X_b),bool),bool),hoare_1301688828derivs(X_b),Ga),bot_bot(fun(hoare_2118899576triple(X_b),bool)))) ).
fof(fact_114_sup__bot__left,axiom,
! [X_a] :
( bounded_lattice_bot(X_a)
=> ! [X_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),bot_bot(X_a)),X_1) = ti(X_a,X_1) ) ).
fof(fact_115_sup__bot__right,axiom,
! [X_a] :
( bounded_lattice_bot(X_a)
=> ! [X_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),X_1),bot_bot(X_a)) = ti(X_a,X_1) ) ).
fof(fact_116_sup__eq__bot__iff,axiom,
! [X_b] :
( bounded_lattice_bot(X_b)
=> ! [X_2,Y_1] :
( hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),semilattice_sup_sup(X_b),X_2),Y_1) = bot_bot(X_b)
<=> ( ti(X_b,X_2) = bot_bot(X_b)
& ti(X_b,Y_1) = bot_bot(X_b) ) ) ) ).
fof(fact_117_triple__valid__Suc,axiom,
! [X_a,N_2,T_3] :
( hBOOL(hAPP(hoare_2118899576triple(X_a),bool,hAPP(nat,fun(hoare_2118899576triple(X_a),bool),hoare_1942962616_valid(X_a),hAPP(nat,nat,suc,N_2)),T_3))
=> hBOOL(hAPP(hoare_2118899576triple(X_a),bool,hAPP(nat,fun(hoare_2118899576triple(X_a),bool),hoare_1942962616_valid(X_a),N_2),T_3)) ) ).
fof(fact_118_insert__def,axiom,
! [X_b,A_2,B_2] : hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_2),B_2) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),collect(X_b),hAPP(X_b,fun(X_b,bool),hAPP(fun(X_b,fun(X_b,bool)),fun(X_b,fun(X_b,bool)),combc(X_b,X_b,bool),fequal(X_b)),A_2))),B_2) ).
fof(fact_119_weak__Body,axiom,
! [X_b,Ga,Pa,Pn_1,Qa] :
( hBOOL(hAPP(fun(hoare_2118899576triple(X_b),bool),bool,hAPP(fun(hoare_2118899576triple(X_b),bool),fun(fun(hoare_2118899576triple(X_b),bool),bool),hoare_1301688828derivs(X_b),Ga),hAPP(fun(hoare_2118899576triple(X_b),bool),fun(hoare_2118899576triple(X_b),bool),hAPP(hoare_2118899576triple(X_b),fun(fun(hoare_2118899576triple(X_b),bool),fun(hoare_2118899576triple(X_b),bool)),insert(hoare_2118899576triple(X_b)),hAPP(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b),hAPP(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b)),hAPP(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b))),hoare_759811442triple(X_b),Pa),hAPP(option(com),com,the(com),hAPP(pname,option(com),body_1,Pn_1))),Qa)),bot_bot(fun(hoare_2118899576triple(X_b),bool)))))
=> hBOOL(hAPP(fun(hoare_2118899576triple(X_b),bool),bool,hAPP(fun(hoare_2118899576triple(X_b),bool),fun(fun(hoare_2118899576triple(X_b),bool),bool),hoare_1301688828derivs(X_b),Ga),hAPP(fun(hoare_2118899576triple(X_b),bool),fun(hoare_2118899576triple(X_b),bool),hAPP(hoare_2118899576triple(X_b),fun(fun(hoare_2118899576triple(X_b),bool),fun(hoare_2118899576triple(X_b),bool)),insert(hoare_2118899576triple(X_b)),hAPP(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b),hAPP(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b)),hAPP(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b))),hoare_759811442triple(X_b),Pa),hAPP(pname,com,body,Pn_1)),Qa)),bot_bot(fun(hoare_2118899576triple(X_b),bool))))) ) ).
fof(fact_120_BodyN,axiom,
! [X_b,Pa,Pn_1,Qa,Ga] :
( hBOOL(hAPP(fun(hoare_2118899576triple(X_b),bool),bool,hAPP(fun(hoare_2118899576triple(X_b),bool),fun(fun(hoare_2118899576triple(X_b),bool),bool),hoare_1301688828derivs(X_b),hAPP(fun(hoare_2118899576triple(X_b),bool),fun(hoare_2118899576triple(X_b),bool),hAPP(hoare_2118899576triple(X_b),fun(fun(hoare_2118899576triple(X_b),bool),fun(hoare_2118899576triple(X_b),bool)),insert(hoare_2118899576triple(X_b)),hAPP(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b),hAPP(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b)),hAPP(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b))),hoare_759811442triple(X_b),Pa),hAPP(pname,com,body,Pn_1)),Qa)),Ga)),hAPP(fun(hoare_2118899576triple(X_b),bool),fun(hoare_2118899576triple(X_b),bool),hAPP(hoare_2118899576triple(X_b),fun(fun(hoare_2118899576triple(X_b),bool),fun(hoare_2118899576triple(X_b),bool)),insert(hoare_2118899576triple(X_b)),hAPP(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b),hAPP(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b)),hAPP(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b))),hoare_759811442triple(X_b),Pa),hAPP(option(com),com,the(com),hAPP(pname,option(com),body_1,Pn_1))),Qa)),bot_bot(fun(hoare_2118899576triple(X_b),bool)))))
=> hBOOL(hAPP(fun(hoare_2118899576triple(X_b),bool),bool,hAPP(fun(hoare_2118899576triple(X_b),bool),fun(fun(hoare_2118899576triple(X_b),bool),bool),hoare_1301688828derivs(X_b),Ga),hAPP(fun(hoare_2118899576triple(X_b),bool),fun(hoare_2118899576triple(X_b),bool),hAPP(hoare_2118899576triple(X_b),fun(fun(hoare_2118899576triple(X_b),bool),fun(hoare_2118899576triple(X_b),bool)),insert(hoare_2118899576triple(X_b)),hAPP(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b),hAPP(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b)),hAPP(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b))),hoare_759811442triple(X_b),Pa),hAPP(pname,com,body,Pn_1)),Qa)),bot_bot(fun(hoare_2118899576triple(X_b),bool))))) ) ).
fof(fact_121_escape,axiom,
! [X_b,Ga,C_1,Qa,Pa] :
( ! [Z_3,S_3] :
( hBOOL(hAPP(state,bool,hAPP(X_b,fun(state,bool),Pa,Z_3),S_3))
=> hBOOL(hAPP(fun(hoare_2118899576triple(X_b),bool),bool,hAPP(fun(hoare_2118899576triple(X_b),bool),fun(fun(hoare_2118899576triple(X_b),bool),bool),hoare_1301688828derivs(X_b),Ga),hAPP(fun(hoare_2118899576triple(X_b),bool),fun(hoare_2118899576triple(X_b),bool),hAPP(hoare_2118899576triple(X_b),fun(fun(hoare_2118899576triple(X_b),bool),fun(hoare_2118899576triple(X_b),bool)),insert(hoare_2118899576triple(X_b)),hAPP(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b),hAPP(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b)),hAPP(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b))),hoare_759811442triple(X_b),hAPP(fun(state,bool),fun(X_b,fun(state,bool)),combk(fun(state,bool),X_b),hAPP(state,fun(state,bool),hAPP(fun(state,fun(state,bool)),fun(state,fun(state,bool)),combc(state,state,bool),fequal(state)),S_3))),C_1),hAPP(fun(state,bool),fun(X_b,fun(state,bool)),combk(fun(state,bool),X_b),hAPP(X_b,fun(state,bool),Qa,Z_3)))),bot_bot(fun(hoare_2118899576triple(X_b),bool))))) )
=> hBOOL(hAPP(fun(hoare_2118899576triple(X_b),bool),bool,hAPP(fun(hoare_2118899576triple(X_b),bool),fun(fun(hoare_2118899576triple(X_b),bool),bool),hoare_1301688828derivs(X_b),Ga),hAPP(fun(hoare_2118899576triple(X_b),bool),fun(hoare_2118899576triple(X_b),bool),hAPP(hoare_2118899576triple(X_b),fun(fun(hoare_2118899576triple(X_b),bool),fun(hoare_2118899576triple(X_b),bool)),insert(hoare_2118899576triple(X_b)),hAPP(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b),hAPP(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b)),hAPP(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b))),hoare_759811442triple(X_b),Pa),C_1),Qa)),bot_bot(fun(hoare_2118899576triple(X_b),bool))))) ) ).
fof(fact_122_conseq1,axiom,
! [X_b,Pa,Ga,P_1,C_1,Qa] :
( hBOOL(hAPP(fun(hoare_2118899576triple(X_b),bool),bool,hAPP(fun(hoare_2118899576triple(X_b),bool),fun(fun(hoare_2118899576triple(X_b),bool),bool),hoare_1301688828derivs(X_b),Ga),hAPP(fun(hoare_2118899576triple(X_b),bool),fun(hoare_2118899576triple(X_b),bool),hAPP(hoare_2118899576triple(X_b),fun(fun(hoare_2118899576triple(X_b),bool),fun(hoare_2118899576triple(X_b),bool)),insert(hoare_2118899576triple(X_b)),hAPP(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b),hAPP(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b)),hAPP(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b))),hoare_759811442triple(X_b),P_1),C_1),Qa)),bot_bot(fun(hoare_2118899576triple(X_b),bool)))))
=> ( ! [Z_3,S_3] :
( hBOOL(hAPP(state,bool,hAPP(X_b,fun(state,bool),Pa,Z_3),S_3))
=> hBOOL(hAPP(state,bool,hAPP(X_b,fun(state,bool),P_1,Z_3),S_3)) )
=> hBOOL(hAPP(fun(hoare_2118899576triple(X_b),bool),bool,hAPP(fun(hoare_2118899576triple(X_b),bool),fun(fun(hoare_2118899576triple(X_b),bool),bool),hoare_1301688828derivs(X_b),Ga),hAPP(fun(hoare_2118899576triple(X_b),bool),fun(hoare_2118899576triple(X_b),bool),hAPP(hoare_2118899576triple(X_b),fun(fun(hoare_2118899576triple(X_b),bool),fun(hoare_2118899576triple(X_b),bool)),insert(hoare_2118899576triple(X_b)),hAPP(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b),hAPP(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b)),hAPP(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b))),hoare_759811442triple(X_b),Pa),C_1),Qa)),bot_bot(fun(hoare_2118899576triple(X_b),bool))))) ) ) ).
fof(fact_123_conseq2,axiom,
! [X_b,Qa,Ga,Pa,C_1,Q_1] :
( hBOOL(hAPP(fun(hoare_2118899576triple(X_b),bool),bool,hAPP(fun(hoare_2118899576triple(X_b),bool),fun(fun(hoare_2118899576triple(X_b),bool),bool),hoare_1301688828derivs(X_b),Ga),hAPP(fun(hoare_2118899576triple(X_b),bool),fun(hoare_2118899576triple(X_b),bool),hAPP(hoare_2118899576triple(X_b),fun(fun(hoare_2118899576triple(X_b),bool),fun(hoare_2118899576triple(X_b),bool)),insert(hoare_2118899576triple(X_b)),hAPP(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b),hAPP(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b)),hAPP(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b))),hoare_759811442triple(X_b),Pa),C_1),Q_1)),bot_bot(fun(hoare_2118899576triple(X_b),bool)))))
=> ( ! [Z_3,S_3] :
( hBOOL(hAPP(state,bool,hAPP(X_b,fun(state,bool),Q_1,Z_3),S_3))
=> hBOOL(hAPP(state,bool,hAPP(X_b,fun(state,bool),Qa,Z_3),S_3)) )
=> hBOOL(hAPP(fun(hoare_2118899576triple(X_b),bool),bool,hAPP(fun(hoare_2118899576triple(X_b),bool),fun(fun(hoare_2118899576triple(X_b),bool),bool),hoare_1301688828derivs(X_b),Ga),hAPP(fun(hoare_2118899576triple(X_b),bool),fun(hoare_2118899576triple(X_b),bool),hAPP(hoare_2118899576triple(X_b),fun(fun(hoare_2118899576triple(X_b),bool),fun(hoare_2118899576triple(X_b),bool)),insert(hoare_2118899576triple(X_b)),hAPP(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b),hAPP(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b)),hAPP(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b))),hoare_759811442triple(X_b),Pa),C_1),Qa)),bot_bot(fun(hoare_2118899576triple(X_b),bool))))) ) ) ).
fof(fact_124_triple_Osize_I1_J,axiom,
! [X_b,Fa,Fun1,Com_1,Fun2] : hAPP(hoare_2118899576triple(X_b),nat,hAPP(fun(X_b,nat),fun(hoare_2118899576triple(X_b),nat),hoare_2043812435e_size(X_b),Fa),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)) = zero_zero(nat) ).
fof(fact_125_MGT__def,axiom,
! [C_1] : hAPP(com,hoare_2118899576triple(state),hoare_Mirabelle_MGT,C_1) = 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),fequal(state)),C_1),hAPP(com,fun(state,fun(state,bool)),evalc,C_1)) ).
fof(fact_126_triple_Osize_I2_J,axiom,
! [X_b,Fun1,Com_1,Fun2] : hAPP(hoare_2118899576triple(X_b),nat,size_size(hoare_2118899576triple(X_b)),hAPP(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b),hAPP(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b)),hAPP(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b))),hoare_759811442triple(X_b),Fun1),Com_1),Fun2)) = zero_zero(nat) ).
fof(fact_127_conseq12,axiom,
! [X_b,Qa,Pa,Ga,P_1,C_1,Q_1] :
( hBOOL(hAPP(fun(hoare_2118899576triple(X_b),bool),bool,hAPP(fun(hoare_2118899576triple(X_b),bool),fun(fun(hoare_2118899576triple(X_b),bool),bool),hoare_1301688828derivs(X_b),Ga),hAPP(fun(hoare_2118899576triple(X_b),bool),fun(hoare_2118899576triple(X_b),bool),hAPP(hoare_2118899576triple(X_b),fun(fun(hoare_2118899576triple(X_b),bool),fun(hoare_2118899576triple(X_b),bool)),insert(hoare_2118899576triple(X_b)),hAPP(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b),hAPP(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b)),hAPP(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b))),hoare_759811442triple(X_b),P_1),C_1),Q_1)),bot_bot(fun(hoare_2118899576triple(X_b),bool)))))
=> ( ! [Z_3,S_3] :
( hBOOL(hAPP(state,bool,hAPP(X_b,fun(state,bool),Pa,Z_3),S_3))
=> ! [S_4] :
( ! [Z_4] :
( hBOOL(hAPP(state,bool,hAPP(X_b,fun(state,bool),P_1,Z_4),S_3))
=> hBOOL(hAPP(state,bool,hAPP(X_b,fun(state,bool),Q_1,Z_4),S_4)) )
=> hBOOL(hAPP(state,bool,hAPP(X_b,fun(state,bool),Qa,Z_3),S_4)) ) )
=> hBOOL(hAPP(fun(hoare_2118899576triple(X_b),bool),bool,hAPP(fun(hoare_2118899576triple(X_b),bool),fun(fun(hoare_2118899576triple(X_b),bool),bool),hoare_1301688828derivs(X_b),Ga),hAPP(fun(hoare_2118899576triple(X_b),bool),fun(hoare_2118899576triple(X_b),bool),hAPP(hoare_2118899576triple(X_b),fun(fun(hoare_2118899576triple(X_b),bool),fun(hoare_2118899576triple(X_b),bool)),insert(hoare_2118899576triple(X_b)),hAPP(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b),hAPP(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b)),hAPP(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b))),hoare_759811442triple(X_b),Pa),C_1),Qa)),bot_bot(fun(hoare_2118899576triple(X_b),bool))))) ) ) ).
fof(fact_128_the__elem__eq,axiom,
! [X_b,X_2] : hAPP(fun(X_b,bool),X_b,the_elem(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_2),bot_bot(fun(X_b,bool)))) = ti(X_b,X_2) ).
fof(fact_129_Suc__neq__Zero,axiom,
! [M] : hAPP(nat,nat,suc,M) != zero_zero(nat) ).
fof(fact_130_bot__nat__def,axiom,
bot_bot(nat) = zero_zero(nat) ).
fof(fact_131_n__not__Suc__n,axiom,
! [N_2] : N_2 != hAPP(nat,nat,suc,N_2) ).
fof(fact_132_Suc__n__not__n,axiom,
! [N_2] : hAPP(nat,nat,suc,N_2) != N_2 ).
fof(fact_133_nat_Oinject,axiom,
! [Nat,Nat_4] :
( hAPP(nat,nat,suc,Nat) = hAPP(nat,nat,suc,Nat_4)
<=> Nat = Nat_4 ) ).
fof(fact_134_Suc__inject,axiom,
! [X_1,Y] :
( hAPP(nat,nat,suc,X_1) = hAPP(nat,nat,suc,Y)
=> X_1 = Y ) ).
fof(fact_135_Zero__not__Suc,axiom,
! [M] : zero_zero(nat) != hAPP(nat,nat,suc,M) ).
fof(fact_136_nat_Osimps_I2_J,axiom,
! [Nat_3] : zero_zero(nat) != hAPP(nat,nat,suc,Nat_3) ).
fof(fact_137_Suc__not__Zero,axiom,
! [M] : hAPP(nat,nat,suc,M) != zero_zero(nat) ).
fof(fact_138_nat_Osimps_I3_J,axiom,
! [Nat_2] : hAPP(nat,nat,suc,Nat_2) != zero_zero(nat) ).
fof(fact_139_Zero__neq__Suc,axiom,
! [M] : zero_zero(nat) != hAPP(nat,nat,suc,M) ).
fof(fact_140_not0__implies__Suc,axiom,
! [N_2] :
( N_2 != zero_zero(nat)
=> ? [M_2] : N_2 = hAPP(nat,nat,suc,M_2) ) ).
fof(fact_141_nat__induct,axiom,
! [N_1,Pa] :
( hBOOL(hAPP(nat,bool,Pa,zero_zero(nat)))
=> ( ! [N] :
( hBOOL(hAPP(nat,bool,Pa,N))
=> hBOOL(hAPP(nat,bool,Pa,hAPP(nat,nat,suc,N))) )
=> hBOOL(hAPP(nat,bool,Pa,N_1)) ) ) ).
fof(fact_142_zero__induct,axiom,
! [Pa,K_1] :
( hBOOL(hAPP(nat,bool,Pa,K_1))
=> ( ! [N] :
( hBOOL(hAPP(nat,bool,Pa,hAPP(nat,nat,suc,N)))
=> hBOOL(hAPP(nat,bool,Pa,N)) )
=> hBOOL(hAPP(nat,bool,Pa,zero_zero(nat))) ) ) ).
fof(fact_143_nat_Oexhaust,axiom,
! [Y] :
( Y != zero_zero(nat)
=> ~ ! [Nat_1] : Y != hAPP(nat,nat,suc,Nat_1) ) ).
fof(fact_144_bot__fun__def,axiom,
! [X_b,X_c] :
( bot(X_c)
=> ! [X] : hAPP(X_b,X_c,bot_bot(fun(X_b,X_c)),X) = bot_bot(X_c) ) ).
fof(fact_145_bot__apply,axiom,
! [X_c,X_b] :
( bot(X_b)
=> ! [X_2] : hAPP(X_c,X_b,bot_bot(fun(X_c,X_b)),X_2) = bot_bot(X_b) ) ).
fof(fact_146_evaln_OBody,axiom,
! [Pn,S0,N_2,S1_2] :
( hBOOL(hAPP(state,bool,hAPP(nat,fun(state,bool),hAPP(state,fun(nat,fun(state,bool)),hAPP(com,fun(state,fun(nat,fun(state,bool))),evaln,hAPP(option(com),com,the(com),hAPP(pname,option(com),body_1,Pn))),S0),N_2),S1_2))
=> hBOOL(hAPP(state,bool,hAPP(nat,fun(state,bool),hAPP(state,fun(nat,fun(state,bool)),hAPP(com,fun(state,fun(nat,fun(state,bool))),evaln,hAPP(pname,com,body,Pn)),S0),hAPP(nat,nat,suc,N_2)),S1_2)) ) ).
fof(fact_147_hoare__derivs_OSkip,axiom,
! [X_b,Ga,Pa] : hBOOL(hAPP(fun(hoare_2118899576triple(X_b),bool),bool,hAPP(fun(hoare_2118899576triple(X_b),bool),fun(fun(hoare_2118899576triple(X_b),bool),bool),hoare_1301688828derivs(X_b),Ga),hAPP(fun(hoare_2118899576triple(X_b),bool),fun(hoare_2118899576triple(X_b),bool),hAPP(hoare_2118899576triple(X_b),fun(fun(hoare_2118899576triple(X_b),bool),fun(hoare_2118899576triple(X_b),bool)),insert(hoare_2118899576triple(X_b)),hAPP(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b),hAPP(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b)),hAPP(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b))),hoare_759811442triple(X_b),Pa),skip),Pa)),bot_bot(fun(hoare_2118899576triple(X_b),bool))))) ).
fof(fact_148_LoopF,axiom,
! [X_b,Ga,Pa,B_1,C_1] : hBOOL(hAPP(fun(hoare_2118899576triple(X_b),bool),bool,hAPP(fun(hoare_2118899576triple(X_b),bool),fun(fun(hoare_2118899576triple(X_b),bool),bool),hoare_1301688828derivs(X_b),Ga),hAPP(fun(hoare_2118899576triple(X_b),bool),fun(hoare_2118899576triple(X_b),bool),hAPP(hoare_2118899576triple(X_b),fun(fun(hoare_2118899576triple(X_b),bool),fun(hoare_2118899576triple(X_b),bool)),insert(hoare_2118899576triple(X_b)),hAPP(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b),hAPP(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b)),hAPP(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b))),hoare_759811442triple(X_b),hAPP(fun(state,bool),fun(X_b,fun(state,bool)),hAPP(fun(X_b,fun(fun(state,bool),fun(state,bool))),fun(fun(state,bool),fun(X_b,fun(state,bool))),combc(X_b,fun(state,bool),fun(state,bool)),hAPP(fun(X_b,fun(state,fun(bool,bool))),fun(X_b,fun(fun(state,bool),fun(state,bool))),hAPP(fun(fun(state,fun(bool,bool)),fun(fun(state,bool),fun(state,bool))),fun(fun(X_b,fun(state,fun(bool,bool))),fun(X_b,fun(fun(state,bool),fun(state,bool)))),combb(fun(state,fun(bool,bool)),fun(fun(state,bool),fun(state,bool)),X_b),combs(state,bool,bool)),hAPP(fun(X_b,fun(state,bool)),fun(X_b,fun(state,fun(bool,bool))),hAPP(fun(fun(state,bool),fun(state,fun(bool,bool))),fun(fun(X_b,fun(state,bool)),fun(X_b,fun(state,fun(bool,bool)))),combb(fun(state,bool),fun(state,fun(bool,bool)),X_b),hAPP(fun(bool,fun(bool,bool)),fun(fun(state,bool),fun(state,fun(bool,bool))),combb(bool,fun(bool,bool),state),fconj)),Pa))),hAPP(fun(state,bool),fun(state,bool),hAPP(fun(bool,bool),fun(fun(state,bool),fun(state,bool)),combb(bool,bool,state),fNot),B_1))),hAPP(com,com,hAPP(fun(state,bool),fun(com,com),while,B_1),C_1)),Pa)),bot_bot(fun(hoare_2118899576triple(X_b),bool))))) ).
fof(fact_149_evaln_OWhileFalse,axiom,
! [C_1,N_1,B_1,S_2] :
( ~ hBOOL(hAPP(state,bool,B_1,S_2))
=> hBOOL(hAPP(state,bool,hAPP(nat,fun(state,bool),hAPP(state,fun(nat,fun(state,bool)),hAPP(com,fun(state,fun(nat,fun(state,bool))),evaln,hAPP(com,com,hAPP(fun(state,bool),fun(com,com),while,B_1),C_1)),S_2),N_1),S_2)) ) ).
fof(fact_150_evaln_OWhileTrue,axiom,
! [S2_1,C_1,N_1,S1,B_1,S0_1] :
( hBOOL(hAPP(state,bool,B_1,S0_1))
=> ( hBOOL(hAPP(state,bool,hAPP(nat,fun(state,bool),hAPP(state,fun(nat,fun(state,bool)),hAPP(com,fun(state,fun(nat,fun(state,bool))),evaln,C_1),S0_1),N_1),S1))
=> ( hBOOL(hAPP(state,bool,hAPP(nat,fun(state,bool),hAPP(state,fun(nat,fun(state,bool)),hAPP(com,fun(state,fun(nat,fun(state,bool))),evaln,hAPP(com,com,hAPP(fun(state,bool),fun(com,com),while,B_1),C_1)),S1),N_1),S2_1))
=> hBOOL(hAPP(state,bool,hAPP(nat,fun(state,bool),hAPP(state,fun(nat,fun(state,bool)),hAPP(com,fun(state,fun(nat,fun(state,bool))),evaln,hAPP(com,com,hAPP(fun(state,bool),fun(com,com),while,B_1),C_1)),S0_1),N_1),S2_1)) ) ) ) ).
fof(fact_151_evalc_OWhileTrue,axiom,
! [S2_1,C_1,S1,B_1,S0_1] :
( hBOOL(hAPP(state,bool,B_1,S0_1))
=> ( hBOOL(hAPP(state,bool,hAPP(state,fun(state,bool),hAPP(com,fun(state,fun(state,bool)),evalc,C_1),S0_1),S1))
=> ( hBOOL(hAPP(state,bool,hAPP(state,fun(state,bool),hAPP(com,fun(state,fun(state,bool)),evalc,hAPP(com,com,hAPP(fun(state,bool),fun(com,com),while,B_1),C_1)),S1),S2_1))
=> hBOOL(hAPP(state,bool,hAPP(state,fun(state,bool),hAPP(com,fun(state,fun(state,bool)),evalc,hAPP(com,com,hAPP(fun(state,bool),fun(com,com),while,B_1),C_1)),S0_1),S2_1)) ) ) ) ).
fof(fact_152_evalc_OWhileFalse,axiom,
! [C_1,B_1,S_2] :
( ~ hBOOL(hAPP(state,bool,B_1,S_2))
=> hBOOL(hAPP(state,bool,hAPP(state,fun(state,bool),hAPP(com,fun(state,fun(state,bool)),evalc,hAPP(com,com,hAPP(fun(state,bool),fun(com,com),while,B_1),C_1)),S_2),S_2)) ) ).
fof(fact_153_evaln_OSkip,axiom,
! [S,N_2] : hBOOL(hAPP(state,bool,hAPP(nat,fun(state,bool),hAPP(state,fun(nat,fun(state,bool)),hAPP(com,fun(state,fun(nat,fun(state,bool))),evaln,skip),S),N_2),S)) ).
fof(fact_154_evaln__elim__cases_I1_J,axiom,
! [S,N_2,T_3] :
( hBOOL(hAPP(state,bool,hAPP(nat,fun(state,bool),hAPP(state,fun(nat,fun(state,bool)),hAPP(com,fun(state,fun(nat,fun(state,bool))),evaln,skip),S),N_2),T_3))
=> T_3 = S ) ).
fof(fact_155_evalc__elim__cases_I1_J,axiom,
! [S,T_3] :
( hBOOL(hAPP(state,bool,hAPP(state,fun(state,bool),hAPP(com,fun(state,fun(state,bool)),evalc,skip),S),T_3))
=> T_3 = S ) ).
fof(fact_156_evalc_OSkip,axiom,
! [S] : hBOOL(hAPP(state,bool,hAPP(state,fun(state,bool),hAPP(com,fun(state,fun(state,bool)),evalc,skip),S),S)) ).
fof(fact_157_com_Osimps_I16_J,axiom,
! [Fun,Com] : skip != hAPP(com,com,hAPP(fun(state,bool),fun(com,com),while,Fun),Com) ).
fof(fact_158_com_Osimps_I17_J,axiom,
! [Fun,Com] : hAPP(com,com,hAPP(fun(state,bool),fun(com,com),while,Fun),Com) != skip ).
fof(fact_159_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_160_evaln__Suc,axiom,
! [C,S,N_2,S_5] :
( hBOOL(hAPP(state,bool,hAPP(nat,fun(state,bool),hAPP(state,fun(nat,fun(state,bool)),hAPP(com,fun(state,fun(nat,fun(state,bool))),evaln,C),S),N_2),S_5))
=> hBOOL(hAPP(state,bool,hAPP(nat,fun(state,bool),hAPP(state,fun(nat,fun(state,bool)),hAPP(com,fun(state,fun(nat,fun(state,bool))),evaln,C),S),hAPP(nat,nat,suc,N_2)),S_5)) ) ).
fof(fact_161_evaln__evalc,axiom,
! [C,S,N_2,T_3] :
( hBOOL(hAPP(state,bool,hAPP(nat,fun(state,bool),hAPP(state,fun(nat,fun(state,bool)),hAPP(com,fun(state,fun(nat,fun(state,bool))),evaln,C),S),N_2),T_3))
=> hBOOL(hAPP(state,bool,hAPP(state,fun(state,bool),hAPP(com,fun(state,fun(state,bool)),evalc,C),S),T_3)) ) ).
fof(fact_162_eval__eq,axiom,
! [C_1,S_2,T_4] :
( hBOOL(hAPP(state,bool,hAPP(state,fun(state,bool),hAPP(com,fun(state,fun(state,bool)),evalc,C_1),S_2),T_4))
<=> ? [N] : hBOOL(hAPP(state,bool,hAPP(nat,fun(state,bool),hAPP(state,fun(nat,fun(state,bool)),hAPP(com,fun(state,fun(nat,fun(state,bool))),evaln,C_1),S_2),N),T_4)) ) ).
fof(fact_163_com_Osimps_I59_J,axiom,
! [Pname,Fun_1,Com_1] : hAPP(pname,com,body,Pname) != hAPP(com,com,hAPP(fun(state,bool),fun(com,com),while,Fun_1),Com_1) ).
fof(fact_164_com_Osimps_I58_J,axiom,
! [Fun_1,Com_1,Pname] : hAPP(com,com,hAPP(fun(state,bool),fun(com,com),while,Fun_1),Com_1) != hAPP(pname,com,body,Pname) ).
fof(fact_165_com_Osimps_I18_J,axiom,
! [Pname_2] : skip != hAPP(pname,com,body,Pname_2) ).
fof(fact_166_com_Osimps_I19_J,axiom,
! [Pname_2] : hAPP(pname,com,body,Pname_2) != skip ).
fof(fact_167_triple__valid__def2,axiom,
! [X_b,N_1,Pa,C_1,Qa] :
( hBOOL(hAPP(hoare_2118899576triple(X_b),bool,hAPP(nat,fun(hoare_2118899576triple(X_b),bool),hoare_1942962616_valid(X_b),N_1),hAPP(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b),hAPP(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b)),hAPP(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b))),hoare_759811442triple(X_b),Pa),C_1),Qa)))
<=> ! [Z_3,S_3] :
( hBOOL(hAPP(state,bool,hAPP(X_b,fun(state,bool),Pa,Z_3),S_3))
=> ! [S_4] :
( hBOOL(hAPP(state,bool,hAPP(nat,fun(state,bool),hAPP(state,fun(nat,fun(state,bool)),hAPP(com,fun(state,fun(nat,fun(state,bool))),evaln,C_1),S_3),N_1),S_4))
=> hBOOL(hAPP(state,bool,hAPP(X_b,fun(state,bool),Qa,Z_3),S_4)) ) ) ) ).
fof(fact_168_evaln__elim__cases_I6_J,axiom,
! [P,S,N_2,S1_2] :
( hBOOL(hAPP(state,bool,hAPP(nat,fun(state,bool),hAPP(state,fun(nat,fun(state,bool)),hAPP(com,fun(state,fun(nat,fun(state,bool))),evaln,hAPP(pname,com,body,P)),S),N_2),S1_2))
=> ~ ! [N] :
( N_2 = hAPP(nat,nat,suc,N)
=> ~ hBOOL(hAPP(state,bool,hAPP(nat,fun(state,bool),hAPP(state,fun(nat,fun(state,bool)),hAPP(com,fun(state,fun(nat,fun(state,bool))),evaln,hAPP(option(com),com,the(com),hAPP(pname,option(com),body_1,P))),S),N),S1_2)) ) ) ).
fof(fact_169_evalc__WHILE__case,axiom,
! [B_1,C_1,S_2,T_4] :
( hBOOL(hAPP(state,bool,hAPP(state,fun(state,bool),hAPP(com,fun(state,fun(state,bool)),evalc,hAPP(com,com,hAPP(fun(state,bool),fun(com,com),while,B_1),C_1)),S_2),T_4))
=> ( ( T_4 = S_2
=> hBOOL(hAPP(state,bool,B_1,S_2)) )
=> ~ ( hBOOL(hAPP(state,bool,B_1,S_2))
=> ! [S1_1] :
( hBOOL(hAPP(state,bool,hAPP(state,fun(state,bool),hAPP(com,fun(state,fun(state,bool)),evalc,C_1),S_2),S1_1))
=> ~ hBOOL(hAPP(state,bool,hAPP(state,fun(state,bool),hAPP(com,fun(state,fun(state,bool)),evalc,hAPP(com,com,hAPP(fun(state,bool),fun(com,com),while,B_1),C_1)),S1_1),T_4)) ) ) ) ) ).
fof(fact_170_evaln__WHILE__case,axiom,
! [B_1,C_1,S_2,N_1,T_4] :
( hBOOL(hAPP(state,bool,hAPP(nat,fun(state,bool),hAPP(state,fun(nat,fun(state,bool)),hAPP(com,fun(state,fun(nat,fun(state,bool))),evaln,hAPP(com,com,hAPP(fun(state,bool),fun(com,com),while,B_1),C_1)),S_2),N_1),T_4))
=> ( ( T_4 = S_2
=> hBOOL(hAPP(state,bool,B_1,S_2)) )
=> ~ ( hBOOL(hAPP(state,bool,B_1,S_2))
=> ! [S1_1] :
( hBOOL(hAPP(state,bool,hAPP(nat,fun(state,bool),hAPP(state,fun(nat,fun(state,bool)),hAPP(com,fun(state,fun(nat,fun(state,bool))),evaln,C_1),S_2),N_1),S1_1))
=> ~ hBOOL(hAPP(state,bool,hAPP(nat,fun(state,bool),hAPP(state,fun(nat,fun(state,bool)),hAPP(com,fun(state,fun(nat,fun(state,bool))),evaln,hAPP(com,com,hAPP(fun(state,bool),fun(com,com),while,B_1),C_1)),S1_1),N_1),T_4)) ) ) ) ) ).
fof(fact_171_evalc__evaln,axiom,
! [C,S,T_3] :
( hBOOL(hAPP(state,bool,hAPP(state,fun(state,bool),hAPP(com,fun(state,fun(state,bool)),evalc,C),S),T_3))
=> ? [N] : hBOOL(hAPP(state,bool,hAPP(nat,fun(state,bool),hAPP(state,fun(nat,fun(state,bool)),hAPP(com,fun(state,fun(nat,fun(state,bool))),evaln,C),S),N),T_3)) ) ).
fof(fact_172_Comp,axiom,
! [X_b,D,R_1,Ga,Pa,C_1,Qa] :
( hBOOL(hAPP(fun(hoare_2118899576triple(X_b),bool),bool,hAPP(fun(hoare_2118899576triple(X_b),bool),fun(fun(hoare_2118899576triple(X_b),bool),bool),hoare_1301688828derivs(X_b),Ga),hAPP(fun(hoare_2118899576triple(X_b),bool),fun(hoare_2118899576triple(X_b),bool),hAPP(hoare_2118899576triple(X_b),fun(fun(hoare_2118899576triple(X_b),bool),fun(hoare_2118899576triple(X_b),bool)),insert(hoare_2118899576triple(X_b)),hAPP(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b),hAPP(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b)),hAPP(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b))),hoare_759811442triple(X_b),Pa),C_1),Qa)),bot_bot(fun(hoare_2118899576triple(X_b),bool)))))
=> ( hBOOL(hAPP(fun(hoare_2118899576triple(X_b),bool),bool,hAPP(fun(hoare_2118899576triple(X_b),bool),fun(fun(hoare_2118899576triple(X_b),bool),bool),hoare_1301688828derivs(X_b),Ga),hAPP(fun(hoare_2118899576triple(X_b),bool),fun(hoare_2118899576triple(X_b),bool),hAPP(hoare_2118899576triple(X_b),fun(fun(hoare_2118899576triple(X_b),bool),fun(hoare_2118899576triple(X_b),bool)),insert(hoare_2118899576triple(X_b)),hAPP(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b),hAPP(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b)),hAPP(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b))),hoare_759811442triple(X_b),Qa),D),R_1)),bot_bot(fun(hoare_2118899576triple(X_b),bool)))))
=> hBOOL(hAPP(fun(hoare_2118899576triple(X_b),bool),bool,hAPP(fun(hoare_2118899576triple(X_b),bool),fun(fun(hoare_2118899576triple(X_b),bool),bool),hoare_1301688828derivs(X_b),Ga),hAPP(fun(hoare_2118899576triple(X_b),bool),fun(hoare_2118899576triple(X_b),bool),hAPP(hoare_2118899576triple(X_b),fun(fun(hoare_2118899576triple(X_b),bool),fun(hoare_2118899576triple(X_b),bool)),insert(hoare_2118899576triple(X_b)),hAPP(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b),hAPP(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b)),hAPP(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_2118899576triple(X_b))),hoare_759811442triple(X_b),Pa),hAPP(com,com,hAPP(com,fun(com,com),semi,C_1),D)),R_1)),bot_bot(fun(hoare_2118899576triple(X_b),bool))))) ) ) ).
fof(fact_173_the__elem__def,axiom,
! [X_b,X_3] : hAPP(fun(X_b,bool),X_b,the_elem(X_b),X_3) = hAPP(fun(X_b,bool),X_b,the_1(X_b),hAPP(fun(X_b,fun(X_b,bool)),fun(X_b,bool),hAPP(fun(fun(X_b,bool),bool),fun(fun(X_b,fun(X_b,bool)),fun(X_b,bool)),combb(fun(X_b,bool),bool,X_b),hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),fequal(fun(X_b,bool)),X_3)),hAPP(fun(X_b,bool),fun(X_b,fun(X_b,bool)),hAPP(fun(X_b,fun(fun(X_b,bool),fun(X_b,bool))),fun(fun(X_b,bool),fun(X_b,fun(X_b,bool))),combc(X_b,fun(X_b,bool),fun(X_b,bool)),insert(X_b)),bot_bot(fun(X_b,bool))))) ).
fof(fact_174_finite__pointwise,axiom,
! [X_c,X_b,Pa,Qa,Ga,P_1,C0,Q_1,U_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),U_2))
=> ( ! [P_2] :
( hBOOL(hAPP(fun(hoare_2118899576triple(X_c),bool),bool,hAPP(fun(hoare_2118899576triple(X_c),bool),fun(fun(hoare_2118899576triple(X_c),bool),bool),hoare_1301688828derivs(X_c),Ga),hAPP(fun(hoare_2118899576triple(X_c),bool),fun(hoare_2118899576triple(X_c),bool),hAPP(hoare_2118899576triple(X_c),fun(fun(hoare_2118899576triple(X_c),bool),fun(hoare_2118899576triple(X_c),bool)),insert(hoare_2118899576triple(X_c)),hAPP(fun(X_c,fun(state,bool)),hoare_2118899576triple(X_c),hAPP(com,fun(fun(X_c,fun(state,bool)),hoare_2118899576triple(X_c)),hAPP(fun(X_c,fun(state,bool)),fun(com,fun(fun(X_c,fun(state,bool)),hoare_2118899576triple(X_c))),hoare_759811442triple(X_c),hAPP(X_b,fun(X_c,fun(state,bool)),P_1,P_2)),hAPP(X_b,com,C0,P_2)),hAPP(X_b,fun(X_c,fun(state,bool)),Q_1,P_2))),bot_bot(fun(hoare_2118899576triple(X_c),bool)))))
=> hBOOL(hAPP(fun(hoare_2118899576triple(X_c),bool),bool,hAPP(fun(hoare_2118899576triple(X_c),bool),fun(fun(hoare_2118899576triple(X_c),bool),bool),hoare_1301688828derivs(X_c),Ga),hAPP(fun(hoare_2118899576triple(X_c),bool),fun(hoare_2118899576triple(X_c),bool),hAPP(hoare_2118899576triple(X_c),fun(fun(hoare_2118899576triple(X_c),bool),fun(hoare_2118899576triple(X_c),bool)),insert(hoare_2118899576triple(X_c)),hAPP(fun(X_c,fun(state,bool)),hoare_2118899576triple(X_c),hAPP(com,fun(fun(X_c,fun(state,bool)),hoare_2118899576triple(X_c)),hAPP(fun(X_c,fun(state,bool)),fun(com,fun(fun(X_c,fun(state,bool)),hoare_2118899576triple(X_c))),hoare_759811442triple(X_c),hAPP(X_b,fun(X_c,fun(state,bool)),Pa,P_2)),hAPP(X_b,com,C0,P_2)),hAPP(X_b,fun(X_c,fun(state,bool)),Qa,P_2))),bot_bot(fun(hoare_2118899576triple(X_c),bool))))) )
=> ( hBOOL(hAPP(fun(hoare_2118899576triple(X_c),bool),bool,hAPP(fun(hoare_2118899576triple(X_c),bool),fun(fun(hoare_2118899576triple(X_c),bool),bool),hoare_1301688828derivs(X_c),Ga),hAPP(fun(X_b,bool),fun(hoare_2118899576triple(X_c),bool),hAPP(fun(X_b,hoare_2118899576triple(X_c)),fun(fun(X_b,bool),fun(hoare_2118899576triple(X_c),bool)),image(X_b,hoare_2118899576triple(X_c)),hAPP(fun(X_b,fun(X_c,fun(state,bool))),fun(X_b,hoare_2118899576triple(X_c)),hAPP(fun(X_b,fun(fun(X_c,fun(state,bool)),hoare_2118899576triple(X_c))),fun(fun(X_b,fun(X_c,fun(state,bool))),fun(X_b,hoare_2118899576triple(X_c))),combs(X_b,fun(X_c,fun(state,bool)),hoare_2118899576triple(X_c)),hAPP(fun(X_b,com),fun(X_b,fun(fun(X_c,fun(state,bool)),hoare_2118899576triple(X_c))),hAPP(fun(X_b,fun(com,fun(fun(X_c,fun(state,bool)),hoare_2118899576triple(X_c)))),fun(fun(X_b,com),fun(X_b,fun(fun(X_c,fun(state,bool)),hoare_2118899576triple(X_c)))),combs(X_b,com,fun(fun(X_c,fun(state,bool)),hoare_2118899576triple(X_c))),hAPP(fun(X_b,fun(X_c,fun(state,bool))),fun(X_b,fun(com,fun(fun(X_c,fun(state,bool)),hoare_2118899576triple(X_c)))),hAPP(fun(fun(X_c,fun(state,bool)),fun(com,fun(fun(X_c,fun(state,bool)),hoare_2118899576triple(X_c)))),fun(fun(X_b,fun(X_c,fun(state,bool))),fun(X_b,fun(com,fun(fun(X_c,fun(state,bool)),hoare_2118899576triple(X_c))))),combb(fun(X_c,fun(state,bool)),fun(com,fun(fun(X_c,fun(state,bool)),hoare_2118899576triple(X_c))),X_b),hoare_759811442triple(X_c)),P_1)),C0)),Q_1)),U_2)))
=> hBOOL(hAPP(fun(hoare_2118899576triple(X_c),bool),bool,hAPP(fun(hoare_2118899576triple(X_c),bool),fun(fun(hoare_2118899576triple(X_c),bool),bool),hoare_1301688828derivs(X_c),Ga),hAPP(fun(X_b,bool),fun(hoare_2118899576triple(X_c),bool),hAPP(fun(X_b,hoare_2118899576triple(X_c)),fun(fun(X_b,bool),fun(hoare_2118899576triple(X_c),bool)),image(X_b,hoare_2118899576triple(X_c)),hAPP(fun(X_b,fun(X_c,fun(state,bool))),fun(X_b,hoare_2118899576triple(X_c)),hAPP(fun(X_b,fun(fun(X_c,fun(state,bool)),hoare_2118899576triple(X_c))),fun(fun(X_b,fun(X_c,fun(state,bool))),fun(X_b,hoare_2118899576triple(X_c))),combs(X_b,fun(X_c,fun(state,bool)),hoare_2118899576triple(X_c)),hAPP(fun(X_b,com),fun(X_b,fun(fun(X_c,fun(state,bool)),hoare_2118899576triple(X_c))),hAPP(fun(X_b,fun(com,fun(fun(X_c,fun(state,bool)),hoare_2118899576triple(X_c)))),fun(fun(X_b,com),fun(X_b,fun(fun(X_c,fun(state,bool)),hoare_2118899576triple(X_c)))),combs(X_b,com,fun(fun(X_c,fun(state,bool)),hoare_2118899576triple(X_c))),hAPP(fun(X_b,fun(X_c,fun(state,bool))),fun(X_b,fun(com,fun(fun(X_c,fun(state,bool)),hoare_2118899576triple(X_c)))),hAPP(fun(fun(X_c,fun(state,bool)),fun(com,fun(fun(X_c,fun(state,bool)),hoare_2118899576triple(X_c)))),fun(fun(X_b,fun(X_c,fun(state,bool))),fun(X_b,fun(com,fun(fun(X_c,fun(state,bool)),hoare_2118899576triple(X_c))))),combb(fun(X_c,fun(state,bool)),fun(com,fun(fun(X_c,fun(state,bool)),hoare_2118899576triple(X_c))),X_b),hoare_759811442triple(X_c)),Pa)),C0)),Qa)),U_2))) ) ) ) ).
fof(fact_175_evaln__max2,axiom,
! [C2_1,S2,N2,T2,C1_1,S1_2,N1,T1] :
( hBOOL(hAPP(state,bool,hAPP(nat,fun(state,bool),hAPP(state,fun(nat,fun(state,bool)),hAPP(com,fun(state,fun(nat,fun(state,bool))),evaln,C1_1),S1_2),N1),T1))
=> ( hBOOL(hAPP(state,bool,hAPP(nat,fun(state,bool),hAPP(state,fun(nat,fun(state,bool)),hAPP(com,fun(state,fun(nat,fun(state,bool))),evaln,C2_1),S2),N2),T2))
=> ? [N] :
( hBOOL(hAPP(state,bool,hAPP(nat,fun(state,bool),hAPP(state,fun(nat,fun(state,bool)),hAPP(com,fun(state,fun(nat,fun(state,bool))),evaln,C1_1),S1_2),N),T1))
& hBOOL(hAPP(state,bool,hAPP(nat,fun(state,bool),hAPP(state,fun(nat,fun(state,bool)),hAPP(com,fun(state,fun(nat,fun(state,bool))),evaln,C2_1),S2),N),T2)) ) ) ) ).
fof(fact_176_mk__disjoint__insert,axiom,
! [X_b,A_2,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),A_2),A_3))
=> ? [B_3] :
( ti(fun(X_b,bool),A_3) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_2),B_3)
& ~ hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),A_2),B_3)) ) ) ).
fof(fact_177_evaln_OSemi,axiom,
! [C1_1,S2,C0_1,S0,N_2,S1_2] :
( hBOOL(hAPP(state,bool,hAPP(nat,fun(state,bool),hAPP(state,fun(nat,fun(state,bool)),hAPP(com,fun(state,fun(nat,fun(state,bool))),evaln,C0_1),S0),N_2),S1_2))
=> ( hBOOL(hAPP(state,bool,hAPP(nat,fun(state,bool),hAPP(state,fun(nat,fun(state,bool)),hAPP(com,fun(state,fun(nat,fun(state,bool))),evaln,C1_1),S1_2),N_2),S2))
=> hBOOL(hAPP(state,bool,hAPP(nat,fun(state,bool),hAPP(state,fun(nat,fun(state,bool)),hAPP(com,fun(state,fun(nat,fun(state,bool))),evaln,hAPP(com,com,hAPP(com,fun(com,com),semi,C0_1),C1_1)),S0),N_2),S2)) ) ) ).
fof(fact_178_evalc_OSemi,axiom,
! [C1_1,S2,C0_1,S0,S1_2] :
( hBOOL(hAPP(state,bool,hAPP(state,fun(state,bool),hAPP(com,fun(state,fun(state,bool)),evalc,C0_1),S0),S1_2))
=> ( hBOOL(hAPP(state,bool,hAPP(state,fun(state,bool),hAPP(com,fun(state,fun(state,bool)),evalc,C1_1),S1_2),S2))
=> hBOOL(hAPP(state,bool,hAPP(state,fun(state,bool),hAPP(com,fun(state,fun(state,bool)),evalc,hAPP(com,com,hAPP(com,fun(com,com),semi,C0_1),C1_1)),S0),S2)) ) ) ).
fof(fact_179_com_Osimps_I3_J,axiom,
! [Com1_1,Com2_1,Com1,Com2] :
( hAPP(com,com,hAPP(com,fun(com,com),semi,Com1_1),Com2_1) = hAPP(com,com,hAPP(com,fun(com,com),semi,Com1),Com2)
<=> ( Com1_1 = Com1
& Com2_1 = Com2 ) ) ).
fof(fact_180_com_Osimps_I48_J,axiom,
! [Com1_2,Com2_2,Pname_2] : hAPP(com,com,hAPP(com,fun(com,com),semi,Com1_2),Com2_2) != hAPP(pname,com,body,Pname_2) ).
fof(fact_181_com_Osimps_I49_J,axiom,
! [Pname_2,Com1_2,Com2_2] : hAPP(pname,com,body,Pname_2) != hAPP(com,com,hAPP(com,fun(com,com),semi,Com1_2),Com2_2) ).
fof(fact_182_com_Osimps_I47_J,axiom,
! [Fun,Com,Com1_1,Com2_1] : hAPP(com,com,hAPP(fun(state,bool),fun(com,com),while,Fun),Com) != hAPP(com,com,hAPP(com,fun(com,com),semi,Com1_1),Com2_1) ).
fof(fact_183_com_Osimps_I46_J,axiom,
! [Com1_1,Com2_1,Fun,Com] : hAPP(com,com,hAPP(com,fun(com,com),semi,Com1_1),Com2_1) != hAPP(com,com,hAPP(fun(state,bool),fun(com,com),while,Fun),Com) ).
fof(fact_184_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_185_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_186_evalc__elim__cases_I4_J,axiom,
! [C1_1,C2_1,S,T_3] :
( hBOOL(hAPP(state,bool,hAPP(state,fun(state,bool),hAPP(com,fun(state,fun(state,bool)),evalc,hAPP(com,com,hAPP(com,fun(com,com),semi,C1_1),C2_1)),S),T_3))
=> ~ ! [S1_1] :
( hBOOL(hAPP(state,bool,hAPP(state,fun(state,bool),hAPP(com,fun(state,fun(state,bool)),evalc,C1_1),S),S1_1))
=> ~ hBOOL(hAPP(state,bool,hAPP(state,fun(state,bool),hAPP(com,fun(state,fun(state,bool)),evalc,C2_1),S1_1),T_3)) ) ) ).
fof(fact_187_evaln__elim__cases_I4_J,axiom,
! [C1_1,C2_1,S,N_2,T_3] :
( hBOOL(hAPP(state,bool,hAPP(nat,fun(state,bool),hAPP(state,fun(nat,fun(state,bool)),hAPP(com,fun(state,fun(nat,fun(state,bool))),evaln,hAPP(com,com,hAPP(com,fun(com,com),semi,C1_1),C2_1)),S),N_2),T_3))
=> ~ ! [S1_1] :
( hBOOL(hAPP(state,bool,hAPP(nat,fun(state,bool),hAPP(state,fun(nat,fun(state,bool)),hAPP(com,fun(state,fun(nat,fun(state,bool))),evaln,C1_1),S),N_2),S1_1))
=> ~ hBOOL(hAPP(state,bool,hAPP(nat,fun(state,bool),hAPP(state,fun(nat,fun(state,bool)),hAPP(com,fun(state,fun(nat,fun(state,bool))),evaln,C2_1),S1_1),N_2),T_3)) ) ) ).
fof(fact_188_finite__imageI,axiom,
! [X_c,X_b,H,F_1] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),F_1))
=> hBOOL(hAPP(fun(X_c,bool),bool,finite_finite_1(X_c),hAPP(fun(X_b,bool),fun(X_c,bool),hAPP(fun(X_b,X_c),fun(fun(X_b,bool),fun(X_c,bool)),image(X_b,X_c),H),F_1))) ) ).
fof(fact_189_finite_OinsertI,axiom,
! [X_b,A_2,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_2),A_3))) ) ).
fof(fact_190_finite_OemptyI,axiom,
! [X_b] : hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),bot_bot(fun(X_b,bool)))) ).
fof(fact_191_finite__Collect__conjI,axiom,
! [X_b,Qa,Pa] :
( ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),collect(X_b),Pa)))
| hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),collect(X_b),Qa))) )
=> hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),collect(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,bool)),combs(X_b,bool,bool),hAPP(fun(X_b,bool),fun(X_b,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,fun(bool,bool))),combb(bool,fun(bool,bool),X_b),fconj),Pa)),Qa)))) ) ).
fof(fact_192_finite__Un,axiom,
! [X_b,F_1,Ga] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),F_1),Ga)))
<=> ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),F_1))
& hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),Ga)) ) ) ).
fof(fact_193_finite__UnI,axiom,
! [X_b,Ga,F_1] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),F_1))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),Ga))
=> hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),F_1),Ga))) ) ) ).
fof(fact_194_finite__code,axiom,
! [X_b] :
( finite_finite(X_b)
=> ! [A_3] : hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3)) ) ).
fof(fact_195_finite,axiom,
! [X_b] :
( finite_finite(X_b)
=> ! [A_3] : hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3)) ) ).
fof(fact_196_finite__Collect__disjI,axiom,
! [X_b,Pa,Qa] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),collect(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,bool)),combs(X_b,bool,bool),hAPP(fun(X_b,bool),fun(X_b,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,fun(bool,bool))),combb(bool,fun(bool,bool),X_b),fdisj),Pa)),Qa))))
<=> ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),collect(X_b),Pa)))
& hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),collect(X_b),Qa))) ) ) ).
fof(fact_197_finite__insert,axiom,
! [X_b,A_2,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_2),A_3)))
<=> hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3)) ) ).
fof(fact_198_finite__induct,axiom,
! [X_b,Pa,F_1] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),F_1))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,Pa,bot_bot(fun(X_b,bool))))
=> ( ! [X,F_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),F_2))
=> ( ~ hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X),F_2))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,Pa,F_2))
=> hBOOL(hAPP(fun(X_b,bool),bool,Pa,hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X),F_2))) ) ) )
=> hBOOL(hAPP(fun(X_b,bool),bool,Pa,F_1)) ) ) ) ).
fof(fact_199_finite_Osimps,axiom,
! [X_b,A_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_2))
<=> ( ti(fun(X_b,bool),A_2) = bot_bot(fun(X_b,bool))
| ? [A_5,A_4] :
( ti(fun(X_b,bool),A_2) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_4),A_5)
& hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_5)) ) ) ) ).
fof(fact_200_pigeonhole__infinite,axiom,
! [X_c,X_b,F,A_3] :
( ~ hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( hBOOL(hAPP(fun(X_c,bool),bool,finite_finite_1(X_c),hAPP(fun(X_b,bool),fun(X_c,bool),hAPP(fun(X_b,X_c),fun(fun(X_b,bool),fun(X_c,bool)),image(X_b,X_c),F),A_3)))
=> ? [X] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X),A_3))
& ~ hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),collect(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,bool)),combs(X_b,bool,bool),hAPP(fun(X_b,bool),fun(X_b,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,fun(bool,bool))),combb(bool,fun(bool,bool),X_b),fconj),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(fun(X_b,bool),bool)),fun(fun(X_b,bool),fun(X_b,bool)),combc(X_b,fun(X_b,bool),bool),member(X_b)),A_3))),hAPP(X_c,fun(X_b,bool),hAPP(fun(X_b,fun(X_c,bool)),fun(X_c,fun(X_b,bool)),combc(X_b,X_c,bool),hAPP(fun(X_b,X_c),fun(X_b,fun(X_c,bool)),hAPP(fun(X_c,fun(X_c,bool)),fun(fun(X_b,X_c),fun(X_b,fun(X_c,bool))),combb(X_c,fun(X_c,bool),X_b),fequal(X_c)),F)),hAPP(X_b,X_c,F,X)))))) ) ) ) ).
fof(fact_201_nonempty__iff,axiom,
! [X_b,A_3] :
( ti(fun(X_b,bool),A_3) != bot_bot(fun(X_b,bool))
<=> ? [X,B_3] :
( ti(fun(X_b,bool),A_3) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X),B_3)
& ~ hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X),B_3)) ) ) ).
fof(fact_202_folding__one__idem_Ounion__idem,axiom,
! [X_b,B_2,A_3,F,F_1] :
( hBOOL(hAPP(fun(fun(X_b,bool),X_b),bool,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(fun(X_b,bool),X_b),bool),finite2073411215e_idem(X_b),F),F_1))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( ti(fun(X_b,bool),A_3) != bot_bot(fun(X_b,bool))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),B_2))
=> ( ti(fun(X_b,bool),B_2) != bot_bot(fun(X_b,bool))
=> hAPP(fun(X_b,bool),X_b,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)),semilattice_sup_sup(fun(X_b,bool)),A_3),B_2)) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),F,hAPP(fun(X_b,bool),X_b,F_1,A_3)),hAPP(fun(X_b,bool),X_b,F_1,B_2)) ) ) ) ) ) ).
fof(fact_203_folding__one__idem_Oinsert__idem,axiom,
! [X_b,X_2,A_3,F,F_1] :
( hBOOL(hAPP(fun(fun(X_b,bool),X_b),bool,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(fun(X_b,bool),X_b),bool),finite2073411215e_idem(X_b),F),F_1))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( ti(fun(X_b,bool),A_3) != bot_bot(fun(X_b,bool))
=> hAPP(fun(X_b,bool),X_b,F_1,hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_2),A_3)) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),F,X_2),hAPP(fun(X_b,bool),X_b,F_1,A_3)) ) ) ) ).
fof(fact_204_image__eq__fold__image,axiom,
! [X_c,X_b,F,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> hAPP(fun(X_b,bool),fun(X_c,bool),hAPP(fun(X_b,X_c),fun(fun(X_b,bool),fun(X_c,bool)),image(X_b,X_c),F),A_3) = hAPP(fun(X_b,bool),fun(X_c,bool),hAPP(fun(X_c,bool),fun(fun(X_b,bool),fun(X_c,bool)),hAPP(fun(X_b,fun(X_c,bool)),fun(fun(X_c,bool),fun(fun(X_b,bool),fun(X_c,bool))),hAPP(fun(fun(X_c,bool),fun(fun(X_c,bool),fun(X_c,bool))),fun(fun(X_b,fun(X_c,bool)),fun(fun(X_c,bool),fun(fun(X_b,bool),fun(X_c,bool)))),finite_fold_image(fun(X_c,bool),X_b),semilattice_sup_sup(fun(X_c,bool))),hAPP(fun(X_c,bool),fun(X_b,fun(X_c,bool)),hAPP(fun(X_b,fun(fun(X_c,bool),fun(X_c,bool))),fun(fun(X_c,bool),fun(X_b,fun(X_c,bool))),combc(X_b,fun(X_c,bool),fun(X_c,bool)),hAPP(fun(X_b,X_c),fun(X_b,fun(fun(X_c,bool),fun(X_c,bool))),hAPP(fun(X_c,fun(fun(X_c,bool),fun(X_c,bool))),fun(fun(X_b,X_c),fun(X_b,fun(fun(X_c,bool),fun(X_c,bool)))),combb(X_c,fun(fun(X_c,bool),fun(X_c,bool)),X_b),insert(X_c)),F)),bot_bot(fun(X_c,bool)))),bot_bot(fun(X_c,bool))),A_3) ) ).
fof(fact_205_finite__ne__induct,axiom,
! [X_b,Pa,F_1] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),F_1))
=> ( ti(fun(X_b,bool),F_1) != bot_bot(fun(X_b,bool))
=> ( ! [X] : hBOOL(hAPP(fun(X_b,bool),bool,Pa,hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X),bot_bot(fun(X_b,bool)))))
=> ( ! [X,F_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),F_2))
=> ( ti(fun(X_b,bool),F_2) != bot_bot(fun(X_b,bool))
=> ( ~ hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X),F_2))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,Pa,F_2))
=> hBOOL(hAPP(fun(X_b,bool),bool,Pa,hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X),F_2))) ) ) ) )
=> hBOOL(hAPP(fun(X_b,bool),bool,Pa,F_1)) ) ) ) ) ).
fof(fact_206_folding__one__idem_Oidem,axiom,
! [X_b,X_2,F,F_1] :
( hBOOL(hAPP(fun(fun(X_b,bool),X_b),bool,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(fun(X_b,bool),X_b),bool),finite2073411215e_idem(X_b),F),F_1))
=> hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),F,X_2),X_2) = ti(X_b,X_2) ) ).
fof(fact_207_fold__image__empty,axiom,
! [X_c,X_b,F,G,Z_1] : hAPP(fun(X_c,bool),X_b,hAPP(X_b,fun(fun(X_c,bool),X_b),hAPP(fun(X_c,X_b),fun(X_b,fun(fun(X_c,bool),X_b)),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_c,X_b),fun(X_b,fun(fun(X_c,bool),X_b))),finite_fold_image(X_b,X_c),F),G),Z_1),bot_bot(fun(X_c,bool))) = ti(X_b,Z_1) ).
fof(fact_208_folding__one__idem_Oin__idem,axiom,
! [X_b,X_2,A_3,F,F_1] :
( hBOOL(hAPP(fun(fun(X_b,bool),X_b),bool,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(fun(X_b,bool),X_b),bool),finite2073411215e_idem(X_b),F),F_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_2),A_3))
=> hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),F,X_2),hAPP(fun(X_b,bool),X_b,F_1,A_3)) = hAPP(fun(X_b,bool),X_b,F_1,A_3) ) ) ) ).
fof(fact_209_folding__one__idem_Ohom__commute,axiom,
! [X_b,N_3,H,F,F_1] :
( hBOOL(hAPP(fun(fun(X_b,bool),X_b),bool,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(fun(X_b,bool),X_b),bool),finite2073411215e_idem(X_b),F),F_1))
=> ( ! [X,Y_2] : hAPP(X_b,X_b,H,hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),F,X),Y_2)) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),F,hAPP(X_b,X_b,H,X)),hAPP(X_b,X_b,H,Y_2))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),N_3))
=> ( ti(fun(X_b,bool),N_3) != bot_bot(fun(X_b,bool))
=> hAPP(X_b,X_b,H,hAPP(fun(X_b,bool),X_b,F_1,N_3)) = hAPP(fun(X_b,bool),X_b,F_1,hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,X_b),fun(fun(X_b,bool),fun(X_b,bool)),image(X_b,X_b),H),N_3)) ) ) ) ) ).
fof(fact_210_comm__monoid__big_OF__eq,axiom,
! [X_c,X_b,G,A_3,F,Z_1,F_1] :
( hBOOL(hAPP(fun(fun(X_c,X_b),fun(fun(X_c,bool),X_b)),bool,hAPP(X_b,fun(fun(fun(X_c,X_b),fun(fun(X_c,bool),X_b)),bool),hAPP(fun(X_b,fun(X_b,X_b)),fun(X_b,fun(fun(fun(X_c,X_b),fun(fun(X_c,bool),X_b)),bool)),big_comm_monoid_big(X_b,X_c),F),Z_1),F_1))
=> ( ( hBOOL(hAPP(fun(X_c,bool),bool,finite_finite_1(X_c),A_3))
=> hAPP(fun(X_c,bool),X_b,hAPP(fun(X_c,X_b),fun(fun(X_c,bool),X_b),F_1,G),A_3) = hAPP(fun(X_c,bool),X_b,hAPP(X_b,fun(fun(X_c,bool),X_b),hAPP(fun(X_c,X_b),fun(X_b,fun(fun(X_c,bool),X_b)),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_c,X_b),fun(X_b,fun(fun(X_c,bool),X_b))),finite_fold_image(X_b,X_c),F),G),Z_1),A_3) )
& ( ~ hBOOL(hAPP(fun(X_c,bool),bool,finite_finite_1(X_c),A_3))
=> hAPP(fun(X_c,bool),X_b,hAPP(fun(X_c,X_b),fun(fun(X_c,bool),X_b),F_1,G),A_3) = ti(X_b,Z_1) ) ) ) ).
fof(fact_211_folding__one_Oinsert,axiom,
! [X_b,X_2,A_3,F,F_1] :
( hBOOL(hAPP(fun(fun(X_b,bool),X_b),bool,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(fun(X_b,bool),X_b),bool),finite_folding_one(X_b),F),F_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_2),A_3))
=> ( ti(fun(X_b,bool),A_3) != bot_bot(fun(X_b,bool))
=> hAPP(fun(X_b,bool),X_b,F_1,hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_2),A_3)) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),F,X_2),hAPP(fun(X_b,bool),X_b,F_1,A_3)) ) ) ) ) ).
fof(fact_212_the__sym__eq__trivial,axiom,
! [X_b,X_2] : hAPP(fun(X_b,bool),X_b,the_1(X_b),hAPP(X_b,fun(X_b,bool),fequal(X_b),X_2)) = ti(X_b,X_2) ).
fof(fact_213_the__eq__trivial,axiom,
! [X_b,A_2] : hAPP(fun(X_b,bool),X_b,the_1(X_b),hAPP(X_b,fun(X_b,bool),hAPP(fun(X_b,fun(X_b,bool)),fun(X_b,fun(X_b,bool)),combc(X_b,X_b,bool),fequal(X_b)),A_2)) = ti(X_b,A_2) ).
fof(fact_214_If__def,axiom,
! [X_b,X_2,Y_1,Pa] :
( ( hBOOL(Pa)
=> ti(X_b,X_2) = hAPP(fun(X_b,bool),X_b,the_1(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,bool)),combs(X_b,bool,bool),hAPP(fun(X_b,bool),fun(X_b,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,fun(bool,bool))),combb(bool,fun(bool,bool),X_b),fconj),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(bool,bool),fun(fun(X_b,bool),fun(X_b,bool)),combb(bool,bool,X_b),hAPP(bool,fun(bool,bool),fimplies,Pa)),hAPP(X_b,fun(X_b,bool),hAPP(fun(X_b,fun(X_b,bool)),fun(X_b,fun(X_b,bool)),combc(X_b,X_b,bool),fequal(X_b)),X_2)))),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(bool,bool),fun(fun(X_b,bool),fun(X_b,bool)),combb(bool,bool,X_b),hAPP(bool,fun(bool,bool),fimplies,hAPP(bool,bool,fNot,Pa))),hAPP(X_b,fun(X_b,bool),hAPP(fun(X_b,fun(X_b,bool)),fun(X_b,fun(X_b,bool)),combc(X_b,X_b,bool),fequal(X_b)),Y_1)))) )
& ( ~ hBOOL(Pa)
=> ti(X_b,Y_1) = hAPP(fun(X_b,bool),X_b,the_1(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,bool)),combs(X_b,bool,bool),hAPP(fun(X_b,bool),fun(X_b,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,fun(bool,bool))),combb(bool,fun(bool,bool),X_b),fconj),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(bool,bool),fun(fun(X_b,bool),fun(X_b,bool)),combb(bool,bool,X_b),hAPP(bool,fun(bool,bool),fimplies,Pa)),hAPP(X_b,fun(X_b,bool),hAPP(fun(X_b,fun(X_b,bool)),fun(X_b,fun(X_b,bool)),combc(X_b,X_b,bool),fequal(X_b)),X_2)))),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(bool,bool),fun(fun(X_b,bool),fun(X_b,bool)),combb(bool,bool,X_b),hAPP(bool,fun(bool,bool),fimplies,hAPP(bool,bool,fNot,Pa))),hAPP(X_b,fun(X_b,bool),hAPP(fun(X_b,fun(X_b,bool)),fun(X_b,fun(X_b,bool)),combc(X_b,X_b,bool),fequal(X_b)),Y_1)))) ) ) ).
fof(fact_215_folding__image__simple__idem_Ounion__idem,axiom,
! [X_c,X_b,B_2,A_3,F,Z_1,G,F_1] :
( hBOOL(hAPP(fun(fun(X_c,bool),X_b),bool,hAPP(fun(X_c,X_b),fun(fun(fun(X_c,bool),X_b),bool),hAPP(X_b,fun(fun(X_c,X_b),fun(fun(fun(X_c,bool),X_b),bool)),hAPP(fun(X_b,fun(X_b,X_b)),fun(X_b,fun(fun(X_c,X_b),fun(fun(fun(X_c,bool),X_b),bool))),finite908156982e_idem(X_b,X_c),F),Z_1),G),F_1))
=> ( hBOOL(hAPP(fun(X_c,bool),bool,finite_finite_1(X_c),A_3))
=> ( hBOOL(hAPP(fun(X_c,bool),bool,finite_finite_1(X_c),B_2))
=> hAPP(fun(X_c,bool),X_b,F_1,hAPP(fun(X_c,bool),fun(X_c,bool),hAPP(fun(X_c,bool),fun(fun(X_c,bool),fun(X_c,bool)),semilattice_sup_sup(fun(X_c,bool)),A_3),B_2)) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),F,hAPP(fun(X_c,bool),X_b,F_1,A_3)),hAPP(fun(X_c,bool),X_b,F_1,B_2)) ) ) ) ).
fof(fact_216_folding__image__simple__idem_Oidem,axiom,
! [X_c,X_b,X_2,F,Z_1,G,F_1] :
( hBOOL(hAPP(fun(fun(X_c,bool),X_b),bool,hAPP(fun(X_c,X_b),fun(fun(fun(X_c,bool),X_b),bool),hAPP(X_b,fun(fun(X_c,X_b),fun(fun(fun(X_c,bool),X_b),bool)),hAPP(fun(X_b,fun(X_b,X_b)),fun(X_b,fun(fun(X_c,X_b),fun(fun(fun(X_c,bool),X_b),bool))),finite908156982e_idem(X_b,X_c),F),Z_1),G),F_1))
=> hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),F,X_2),X_2) = ti(X_b,X_2) ) ).
fof(fact_217_folding__image__simple__idem_Oin__idem,axiom,
! [X_c,X_b,X_2,A_3,F,Z_1,G,F_1] :
( hBOOL(hAPP(fun(fun(X_c,bool),X_b),bool,hAPP(fun(X_c,X_b),fun(fun(fun(X_c,bool),X_b),bool),hAPP(X_b,fun(fun(X_c,X_b),fun(fun(fun(X_c,bool),X_b),bool)),hAPP(fun(X_b,fun(X_b,X_b)),fun(X_b,fun(fun(X_c,X_b),fun(fun(fun(X_c,bool),X_b),bool))),finite908156982e_idem(X_b,X_c),F),Z_1),G),F_1))
=> ( hBOOL(hAPP(fun(X_c,bool),bool,finite_finite_1(X_c),A_3))
=> ( hBOOL(hAPP(fun(X_c,bool),bool,hAPP(X_c,fun(fun(X_c,bool),bool),member(X_c),X_2),A_3))
=> hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),F,hAPP(X_c,X_b,G,X_2)),hAPP(fun(X_c,bool),X_b,F_1,A_3)) = hAPP(fun(X_c,bool),X_b,F_1,A_3) ) ) ) ).
fof(fact_218_comm__monoid__big_Oinfinite,axiom,
! [X_c,X_b,G,A_3,F,Z_1,F_1] :
( hBOOL(hAPP(fun(fun(X_c,X_b),fun(fun(X_c,bool),X_b)),bool,hAPP(X_b,fun(fun(fun(X_c,X_b),fun(fun(X_c,bool),X_b)),bool),hAPP(fun(X_b,fun(X_b,X_b)),fun(X_b,fun(fun(fun(X_c,X_b),fun(fun(X_c,bool),X_b)),bool)),big_comm_monoid_big(X_b,X_c),F),Z_1),F_1))
=> ( ~ hBOOL(hAPP(fun(X_c,bool),bool,finite_finite_1(X_c),A_3))
=> hAPP(fun(X_c,bool),X_b,hAPP(fun(X_c,X_b),fun(fun(X_c,bool),X_b),F_1,G),A_3) = ti(X_b,Z_1) ) ) ).
fof(fact_219_folding__one_Osingleton,axiom,
! [X_b,X_2,F,F_1] :
( hBOOL(hAPP(fun(fun(X_b,bool),X_b),bool,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(fun(X_b,bool),X_b),bool),finite_folding_one(X_b),F),F_1))
=> hAPP(fun(X_b,bool),X_b,F_1,hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_2),bot_bot(fun(X_b,bool)))) = ti(X_b,X_2) ) ).
fof(fact_220_folding__image__simple__idem_Oinsert__idem,axiom,
! [X_c,X_b,X_2,A_3,F,Z_1,G,F_1] :
( hBOOL(hAPP(fun(fun(X_c,bool),X_b),bool,hAPP(fun(X_c,X_b),fun(fun(fun(X_c,bool),X_b),bool),hAPP(X_b,fun(fun(X_c,X_b),fun(fun(fun(X_c,bool),X_b),bool)),hAPP(fun(X_b,fun(X_b,X_b)),fun(X_b,fun(fun(X_c,X_b),fun(fun(fun(X_c,bool),X_b),bool))),finite908156982e_idem(X_b,X_c),F),Z_1),G),F_1))
=> ( hBOOL(hAPP(fun(X_c,bool),bool,finite_finite_1(X_c),A_3))
=> hAPP(fun(X_c,bool),X_b,F_1,hAPP(fun(X_c,bool),fun(X_c,bool),hAPP(X_c,fun(fun(X_c,bool),fun(X_c,bool)),insert(X_c),X_2),A_3)) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),F,hAPP(X_c,X_b,G,X_2)),hAPP(fun(X_c,bool),X_b,F_1,A_3)) ) ) ).
fof(fact_221_the__equality,axiom,
! [X_b,Pa,A_2] :
( hBOOL(hAPP(X_b,bool,Pa,A_2))
=> ( ! [X] :
( hBOOL(hAPP(X_b,bool,Pa,X))
=> ti(X_b,X) = ti(X_b,A_2) )
=> hAPP(fun(X_b,bool),X_b,the_1(X_b),Pa) = ti(X_b,A_2) ) ) ).
fof(fact_222_folding__one_Oclosed,axiom,
! [X_b,A_3,F,F_1] :
( hBOOL(hAPP(fun(fun(X_b,bool),X_b),bool,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(fun(X_b,bool),X_b),bool),finite_folding_one(X_b),F),F_1))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( ti(fun(X_b,bool),A_3) != bot_bot(fun(X_b,bool))
=> ( ! [X,Y_2] : hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),F,X),Y_2)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),Y_2),bot_bot(fun(X_b,bool))))))
=> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),hAPP(fun(X_b,bool),X_b,F_1,A_3)),A_3)) ) ) ) ) ).
fof(fact_223_comm__monoid__big_OF__cong,axiom,
! [X_c,X_b,H,G,A_3,B_2,F,Z_1,F_1] :
( hBOOL(hAPP(fun(fun(X_c,X_b),fun(fun(X_c,bool),X_b)),bool,hAPP(X_b,fun(fun(fun(X_c,X_b),fun(fun(X_c,bool),X_b)),bool),hAPP(fun(X_b,fun(X_b,X_b)),fun(X_b,fun(fun(fun(X_c,X_b),fun(fun(X_c,bool),X_b)),bool)),big_comm_monoid_big(X_b,X_c),F),Z_1),F_1))
=> ( ti(fun(X_c,bool),A_3) = ti(fun(X_c,bool),B_2)
=> ( ! [X] :
( hBOOL(hAPP(fun(X_c,bool),bool,hAPP(X_c,fun(fun(X_c,bool),bool),member(X_c),X),B_2))
=> hAPP(X_c,X_b,H,X) = hAPP(X_c,X_b,G,X) )
=> hAPP(fun(X_c,bool),X_b,hAPP(fun(X_c,X_b),fun(fun(X_c,bool),X_b),F_1,H),A_3) = hAPP(fun(X_c,bool),X_b,hAPP(fun(X_c,X_b),fun(fun(X_c,bool),X_b),F_1,G),B_2) ) ) ) ).
fof(fact_224_theI,axiom,
! [X_b,Pa,A_2] :
( hBOOL(hAPP(X_b,bool,Pa,A_2))
=> ( ! [X] :
( hBOOL(hAPP(X_b,bool,Pa,X))
=> ti(X_b,X) = ti(X_b,A_2) )
=> hBOOL(hAPP(X_b,bool,Pa,hAPP(fun(X_b,bool),X_b,the_1(X_b),Pa))) ) ) ).
fof(fact_225_the1__equality,axiom,
! [X_b,A_2,Pa] :
( ? [X] :
( hBOOL(hAPP(X_b,bool,Pa,X))
& ! [Y_2] :
( hBOOL(hAPP(X_b,bool,Pa,Y_2))
=> ti(X_b,Y_2) = ti(X_b,X) ) )
=> ( hBOOL(hAPP(X_b,bool,Pa,A_2))
=> hAPP(fun(X_b,bool),X_b,the_1(X_b),Pa) = ti(X_b,A_2) ) ) ).
fof(fact_226_theI_H,axiom,
! [X_b,Pa] :
( ? [X] :
( hBOOL(hAPP(X_b,bool,Pa,X))
& ! [Y_2] :
( hBOOL(hAPP(X_b,bool,Pa,Y_2))
=> ti(X_b,Y_2) = ti(X_b,X) ) )
=> hBOOL(hAPP(X_b,bool,Pa,hAPP(fun(X_b,bool),X_b,the_1(X_b),Pa))) ) ).
fof(fact_227_Set_Oset__insert,axiom,
! [X_b,X_2,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_2),A_3))
=> ~ ! [B_3] :
( ti(fun(X_b,bool),A_3) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_2),B_3)
=> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_2),B_3)) ) ) ).
fof(fact_228_equals0I,axiom,
! [X_b,A_3] :
( ! [Y_2] : ~ hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),Y_2),A_3))
=> ti(fun(X_b,bool),A_3) = bot_bot(fun(X_b,bool)) ) ).
fof(fact_229_Sup__fin_Ounion__idem,axiom,
! [X_b] :
( lattice(X_b)
=> ! [B_2,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( ti(fun(X_b,bool),A_3) != bot_bot(fun(X_b,bool))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),B_2))
=> ( ti(fun(X_b,bool),B_2) != bot_bot(fun(X_b,bool))
=> hAPP(fun(X_b,bool),X_b,big_lattice_Sup_fin(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),A_3),B_2)) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),semilattice_sup_sup(X_b),hAPP(fun(X_b,bool),X_b,big_lattice_Sup_fin(X_b),A_3)),hAPP(fun(X_b,bool),X_b,big_lattice_Sup_fin(X_b),B_2)) ) ) ) ) ) ).
fof(fact_230_Sup__fin_Oinsert,axiom,
! [X_b] :
( lattice(X_b)
=> ! [X_2,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( ~ hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_2),A_3))
=> ( ti(fun(X_b,bool),A_3) != bot_bot(fun(X_b,bool))
=> hAPP(fun(X_b,bool),X_b,big_lattice_Sup_fin(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_2),A_3)) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),semilattice_sup_sup(X_b),X_2),hAPP(fun(X_b,bool),X_b,big_lattice_Sup_fin(X_b),A_3)) ) ) ) ) ).
fof(fact_231_Sup__fin_Oinsert__idem,axiom,
! [X_b] :
( lattice(X_b)
=> ! [X_2,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( ti(fun(X_b,bool),A_3) != bot_bot(fun(X_b,bool))
=> hAPP(fun(X_b,bool),X_b,big_lattice_Sup_fin(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_2),A_3)) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),semilattice_sup_sup(X_b),X_2),hAPP(fun(X_b,bool),X_b,big_lattice_Sup_fin(X_b),A_3)) ) ) ) ).
fof(fact_232_folding__one_Ounion__disjoint,axiom,
! [X_b,B_2,A_3,F,F_1] :
( hBOOL(hAPP(fun(fun(X_b,bool),X_b),bool,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(fun(X_b,bool),X_b),bool),finite_folding_one(X_b),F),F_1))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( ti(fun(X_b,bool),A_3) != bot_bot(fun(X_b,bool))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),B_2))
=> ( ti(fun(X_b,bool),B_2) != bot_bot(fun(X_b,bool))
=> ( hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_3),B_2) = bot_bot(fun(X_b,bool))
=> hAPP(fun(X_b,bool),X_b,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)),semilattice_sup_sup(fun(X_b,bool)),A_3),B_2)) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),F,hAPP(fun(X_b,bool),X_b,F_1,A_3)),hAPP(fun(X_b,bool),X_b,F_1,B_2)) ) ) ) ) ) ) ).
fof(fact_233_folding__one_Ounion__inter,axiom,
! [X_b,B_2,A_3,F,F_1] :
( hBOOL(hAPP(fun(fun(X_b,bool),X_b),bool,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(fun(X_b,bool),X_b),bool),finite_folding_one(X_b),F),F_1))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),B_2))
=> ( hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_3),B_2) != bot_bot(fun(X_b,bool))
=> hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),F,hAPP(fun(X_b,bool),X_b,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)),semilattice_sup_sup(fun(X_b,bool)),A_3),B_2))),hAPP(fun(X_b,bool),X_b,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)),semilattice_inf_inf(fun(X_b,bool)),A_3),B_2))) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),F,hAPP(fun(X_b,bool),X_b,F_1,A_3)),hAPP(fun(X_b,bool),X_b,F_1,B_2)) ) ) ) ) ).
fof(fact_234_folding__one_Oinsert__remove,axiom,
! [X_b,X_2,A_3,F,F_1] :
( hBOOL(hAPP(fun(fun(X_b,bool),X_b),bool,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(fun(X_b,bool),X_b),bool),finite_folding_one(X_b),F),F_1))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( ( hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_3),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_2),bot_bot(fun(X_b,bool)))) = bot_bot(fun(X_b,bool))
=> hAPP(fun(X_b,bool),X_b,F_1,hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_2),A_3)) = ti(X_b,X_2) )
& ( hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_3),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_2),bot_bot(fun(X_b,bool)))) != bot_bot(fun(X_b,bool))
=> hAPP(fun(X_b,bool),X_b,F_1,hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_2),A_3)) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),F,X_2),hAPP(fun(X_b,bool),X_b,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_2),bot_bot(fun(X_b,bool)))))) ) ) ) ) ).
fof(fact_235_inf1I,axiom,
! [X_b,B_2,A_3,X_2] :
( hBOOL(hAPP(X_b,bool,A_3,X_2))
=> ( hBOOL(hAPP(X_b,bool,B_2,X_2))
=> hBOOL(hAPP(X_b,bool,hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_3),B_2),X_2)) ) ) ).
fof(fact_236_IntI,axiom,
! [X_b,B_2,C_1,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),C_1),A_3))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),C_1),B_2))
=> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),C_1),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_3),B_2))) ) ) ).
fof(fact_237_IntE,axiom,
! [X_b,C_1,A_3,B_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),C_1),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_3),B_2)))
=> ~ ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),C_1),A_3))
=> ~ hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),C_1),B_2)) ) ) ).
fof(fact_238_inf1E,axiom,
! [X_b,A_3,B_2,X_2] :
( hBOOL(hAPP(X_b,bool,hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_3),B_2),X_2))
=> ~ ( hBOOL(hAPP(X_b,bool,A_3,X_2))
=> ~ hBOOL(hAPP(X_b,bool,B_2,X_2)) ) ) ).
fof(fact_239_DiffI,axiom,
! [X_b,B_2,C_1,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),C_1),A_3))
=> ( ~ hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),C_1),B_2))
=> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),C_1),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_3),B_2))) ) ) ).
fof(fact_240_DiffE,axiom,
! [X_b,C_1,A_3,B_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),C_1),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_3),B_2)))
=> ~ ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),C_1),A_3))
=> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),C_1),B_2)) ) ) ).
fof(fact_241_finite__Int,axiom,
! [X_b,Ga,F_1] :
( ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),F_1))
| hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),Ga)) )
=> hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),F_1),Ga))) ) ).
fof(fact_242_finite__Diff,axiom,
! [X_b,B_2,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_3),B_2))) ) ).
fof(fact_243_inf__Sup__absorb,axiom,
! [X_b] :
( lattice(X_b)
=> ! [A_2,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),A_2),A_3))
=> hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),semilattice_inf_inf(X_b),A_2),hAPP(fun(X_b,bool),X_b,big_lattice_Sup_fin(X_b),A_3)) = ti(X_b,A_2) ) ) ) ).
fof(fact_244_Diff__Int,axiom,
! [X_b,A_3,B_2,C_3] : hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_3),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),B_2),C_3)) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_3),B_2)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_3),C_3)) ).
fof(fact_245_Diff__Un,axiom,
! [X_b,A_3,B_2,C_3] : hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_3),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),B_2),C_3)) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_3),B_2)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_3),C_3)) ).
fof(fact_246_Un__Diff__Int,axiom,
! [X_b,A_3,B_2] : hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_3),B_2)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_3),B_2)) = ti(fun(X_b,bool),A_3) ).
fof(fact_247_Collect__conj__eq,axiom,
! [X_b,Pa,Qa] : hAPP(fun(X_b,bool),fun(X_b,bool),collect(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,bool)),combs(X_b,bool,bool),hAPP(fun(X_b,bool),fun(X_b,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,fun(bool,bool))),combb(bool,fun(bool,bool),X_b),fconj),Pa)),Qa)) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),collect(X_b),Pa)),hAPP(fun(X_b,bool),fun(X_b,bool),collect(X_b),Qa)) ).
fof(fact_248_Int__Collect,axiom,
! [X_b,X_2,A_3,Pa] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_2),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_3),hAPP(fun(X_b,bool),fun(X_b,bool),collect(X_b),Pa))))
<=> ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_2),A_3))
& hBOOL(hAPP(X_b,bool,Pa,X_2)) ) ) ).
fof(fact_249_inf__Int__eq,axiom,
! [X_b,R_1,S_1,X] :
( hBOOL(hAPP(X_b,bool,hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(fun(X_b,bool),bool)),fun(fun(X_b,bool),fun(X_b,bool)),combc(X_b,fun(X_b,bool),bool),member(X_b)),R_1)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(fun(X_b,bool),bool)),fun(fun(X_b,bool),fun(X_b,bool)),combc(X_b,fun(X_b,bool),bool),member(X_b)),S_1)),X))
<=> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),R_1),S_1))) ) ).
fof(fact_250_Int__absorb,axiom,
! [X_b,A_3] : hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_3),A_3) = ti(fun(X_b,bool),A_3) ).
fof(fact_251_inf_Oidem,axiom,
! [X_a] :
( semilattice_inf(X_a)
=> ! [A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),A_1),A_1) = ti(X_a,A_1) ) ).
fof(fact_252_inf__idem,axiom,
! [X_a] :
( semilattice_inf(X_a)
=> ! [X_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),X_1),X_1) = ti(X_a,X_1) ) ).
fof(fact_253_fun__diff__def,axiom,
! [X_b,X_c] :
( minus(X_c)
=> ! [A_3,B_2,X] : hAPP(X_b,X_c,hAPP(fun(X_b,X_c),fun(X_b,X_c),hAPP(fun(X_b,X_c),fun(fun(X_b,X_c),fun(X_b,X_c)),minus_minus(fun(X_b,X_c)),A_3),B_2),X) = hAPP(X_c,X_c,hAPP(X_c,fun(X_c,X_c),minus_minus(X_c),hAPP(X_b,X_c,A_3,X)),hAPP(X_b,X_c,B_2,X)) ) ).
fof(fact_254_inf__fun__def,axiom,
! [X_b,X_c] :
( lattice(X_c)
=> ! [F,G,X] : hAPP(X_b,X_c,hAPP(fun(X_b,X_c),fun(X_b,X_c),hAPP(fun(X_b,X_c),fun(fun(X_b,X_c),fun(X_b,X_c)),semilattice_inf_inf(fun(X_b,X_c)),F),G),X) = hAPP(X_c,X_c,hAPP(X_c,fun(X_c,X_c),semilattice_inf_inf(X_c),hAPP(X_b,X_c,F,X)),hAPP(X_b,X_c,G,X)) ) ).
fof(fact_255_set__diff__eq,axiom,
! [X_b,A_3,B_2] : hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_3),B_2) = hAPP(fun(X_b,bool),fun(X_b,bool),collect(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,bool)),combs(X_b,bool,bool),hAPP(fun(X_b,bool),fun(X_b,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,fun(bool,bool))),combb(bool,fun(bool,bool),X_b),fconj),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(fun(X_b,bool),bool)),fun(fun(X_b,bool),fun(X_b,bool)),combc(X_b,fun(X_b,bool),bool),member(X_b)),A_3))),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(bool,bool),fun(fun(X_b,bool),fun(X_b,bool)),combb(bool,bool,X_b),fNot),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(fun(X_b,bool),bool)),fun(fun(X_b,bool),fun(X_b,bool)),combc(X_b,fun(X_b,bool),bool),member(X_b)),B_2)))) ).
fof(fact_256_Int__def,axiom,
! [X_b,A_3,B_2] : hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_3),B_2) = hAPP(fun(X_b,bool),fun(X_b,bool),collect(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,bool)),combs(X_b,bool,bool),hAPP(fun(X_b,bool),fun(X_b,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,fun(bool,bool))),combb(bool,fun(bool,bool),X_b),fconj),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(fun(X_b,bool),bool)),fun(fun(X_b,bool),fun(X_b,bool)),combc(X_b,fun(X_b,bool),bool),member(X_b)),A_3))),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(fun(X_b,bool),bool)),fun(fun(X_b,bool),fun(X_b,bool)),combc(X_b,fun(X_b,bool),bool),member(X_b)),B_2))) ).
fof(fact_257_Int__commute,axiom,
! [X_b,A_3,B_2] : hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_3),B_2) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),B_2),A_3) ).
fof(fact_258_inf_Ocommute,axiom,
! [X_a] :
( semilattice_inf(X_a)
=> ! [A_1,B] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),A_1),B) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),B),A_1) ) ).
fof(fact_259_inf__sup__aci_I1_J,axiom,
! [X_a] :
( lattice(X_a)
=> ! [X_1,Y] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),X_1),Y) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),Y),X_1) ) ).
fof(fact_260_inf__commute,axiom,
! [X_a] :
( semilattice_inf(X_a)
=> ! [X_1,Y] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),X_1),Y) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),Y),X_1) ) ).
fof(fact_261_Int__left__absorb,axiom,
! [X_b,A_3,B_2] : hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_3),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_3),B_2)) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_3),B_2) ).
fof(fact_262_inf_Oleft__idem,axiom,
! [X_a] :
( semilattice_inf(X_a)
=> ! [A_1,B] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),A_1),B)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),A_1),B) ) ).
fof(fact_263_inf__sup__aci_I4_J,axiom,
! [X_a] :
( lattice(X_a)
=> ! [X_1,Y] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),X_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),X_1),Y)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),X_1),Y) ) ).
fof(fact_264_inf__left__idem,axiom,
! [X_a] :
( semilattice_inf(X_a)
=> ! [X_1,Y] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),X_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),X_1),Y)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),X_1),Y) ) ).
fof(fact_265_Int__left__commute,axiom,
! [X_b,A_3,B_2,C_3] : hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_3),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),B_2),C_3)) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),B_2),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_3),C_3)) ).
fof(fact_266_inf_Oleft__commute,axiom,
! [X_a] :
( semilattice_inf(X_a)
=> ! [B,A_1,C] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),B),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),A_1),C)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),B),C)) ) ).
fof(fact_267_inf__sup__aci_I3_J,axiom,
! [X_a] :
( lattice(X_a)
=> ! [X_1,Y,Z] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),X_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),Y),Z)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),Y),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),X_1),Z)) ) ).
fof(fact_268_inf__left__commute,axiom,
! [X_a] :
( semilattice_inf(X_a)
=> ! [X_1,Y,Z] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),X_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),Y),Z)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),Y),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),X_1),Z)) ) ).
fof(fact_269_Diff__iff,axiom,
! [X_b,C_1,A_3,B_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),C_1),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_3),B_2)))
<=> ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),C_1),A_3))
& ~ hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),C_1),B_2)) ) ) ).
fof(fact_270_Int__iff,axiom,
! [X_b,C_1,A_3,B_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),C_1),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_3),B_2)))
<=> ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),C_1),A_3))
& hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),C_1),B_2)) ) ) ).
fof(fact_271_Diff__Int__distrib,axiom,
! [X_b,C_3,A_3,B_2] : hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),C_3),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_3),B_2)) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),C_3),A_3)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),C_3),B_2)) ).
fof(fact_272_Diff__idemp,axiom,
! [X_b,A_3,B_2] : hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_3),B_2)),B_2) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_3),B_2) ).
fof(fact_273_Int__Diff,axiom,
! [X_b,A_3,B_2,C_3] : hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_3),B_2)),C_3) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_3),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),B_2),C_3)) ).
fof(fact_274_Int__assoc,axiom,
! [X_b,A_3,B_2,C_3] : hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_3),B_2)),C_3) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_3),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),B_2),C_3)) ).
fof(fact_275_inf_Oassoc,axiom,
! [X_a] :
( semilattice_inf(X_a)
=> ! [A_1,B,C] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),A_1),B)),C) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),B),C)) ) ).
fof(fact_276_inf__sup__aci_I2_J,axiom,
! [X_a] :
( lattice(X_a)
=> ! [X_1,Y,Z] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),X_1),Y)),Z) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),X_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),Y),Z)) ) ).
fof(fact_277_inf__assoc,axiom,
! [X_a] :
( semilattice_inf(X_a)
=> ! [X_1,Y,Z] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),X_1),Y)),Z) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),X_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),Y),Z)) ) ).
fof(fact_278_Diff__Int__distrib2,axiom,
! [X_b,A_3,B_2,C_3] : hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_3),B_2)),C_3) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_3),C_3)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),B_2),C_3)) ).
fof(fact_279_Diff__Int2,axiom,
! [X_b,A_3,C_3,B_2] : hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_3),C_3)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),B_2),C_3)) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_3),C_3)),B_2) ).
fof(fact_280_minus__apply,axiom,
! [X_c,X_b] :
( minus(X_b)
=> ! [A_3,B_2,X_2] : hAPP(X_c,X_b,hAPP(fun(X_c,X_b),fun(X_c,X_b),hAPP(fun(X_c,X_b),fun(fun(X_c,X_b),fun(X_c,X_b)),minus_minus(fun(X_c,X_b)),A_3),B_2),X_2) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),minus_minus(X_b),hAPP(X_c,X_b,A_3,X_2)),hAPP(X_c,X_b,B_2,X_2)) ) ).
fof(fact_281_inf__apply,axiom,
! [X_c,X_b] :
( lattice(X_b)
=> ! [F,G,X_2] : hAPP(X_c,X_b,hAPP(fun(X_c,X_b),fun(X_c,X_b),hAPP(fun(X_c,X_b),fun(fun(X_c,X_b),fun(X_c,X_b)),semilattice_inf_inf(fun(X_c,X_b)),F),G),X_2) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),semilattice_inf_inf(X_b),hAPP(X_c,X_b,F,X_2)),hAPP(X_c,X_b,G,X_2)) ) ).
fof(fact_282_DiffD1,axiom,
! [X_b,C_1,A_3,B_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),C_1),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_3),B_2)))
=> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),C_1),A_3)) ) ).
fof(fact_283_IntD1,axiom,
! [X_b,C_1,A_3,B_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),C_1),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_3),B_2)))
=> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),C_1),A_3)) ) ).
fof(fact_284_IntD2,axiom,
! [X_b,C_1,A_3,B_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),C_1),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_3),B_2)))
=> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),C_1),B_2)) ) ).
fof(fact_285_DiffD2,axiom,
! [X_b,C_1,A_3,B_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),C_1),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_3),B_2)))
=> ~ hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),C_1),B_2)) ) ).
fof(fact_286_inf1D1,axiom,
! [X_b,A_3,B_2,X_2] :
( hBOOL(hAPP(X_b,bool,hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_3),B_2),X_2))
=> hBOOL(hAPP(X_b,bool,A_3,X_2)) ) ).
fof(fact_287_inf1D2,axiom,
! [X_b,A_3,B_2,X_2] :
( hBOOL(hAPP(X_b,bool,hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_3),B_2),X_2))
=> hBOOL(hAPP(X_b,bool,B_2,X_2)) ) ).
fof(fact_288_Inf__fin_Oidem,axiom,
! [X_a] :
( lattice(X_a)
=> ! [X_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),X_1),X_1) = ti(X_a,X_1) ) ).
fof(fact_289_Diff__triv,axiom,
! [X_b,A_3,B_2] :
( hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_3),B_2) = bot_bot(fun(X_b,bool))
=> hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_3),B_2) = ti(fun(X_b,bool),A_3) ) ).
fof(fact_290_Diff__disjoint,axiom,
! [X_b,A_3,B_2] : hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_3),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),B_2),A_3)) = bot_bot(fun(X_b,bool)) ).
fof(fact_291_inf__bot__left,axiom,
! [X_a] :
( bounded_lattice_bot(X_a)
=> ! [X_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),bot_bot(X_a)),X_1) = bot_bot(X_a) ) ).
fof(fact_292_inf__bot__right,axiom,
! [X_a] :
( bounded_lattice_bot(X_a)
=> ! [X_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),X_1),bot_bot(X_a)) = bot_bot(X_a) ) ).
fof(fact_293_inf__sup__absorb,axiom,
! [X_a] :
( lattice(X_a)
=> ! [X_1,Y] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),X_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),X_1),Y)) = ti(X_a,X_1) ) ).
fof(fact_294_sup__inf__absorb,axiom,
! [X_a] :
( lattice(X_a)
=> ! [X_1,Y] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),X_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),X_1),Y)) = ti(X_a,X_1) ) ).
fof(fact_295_inf__sup__distrib1,axiom,
! [X_a] :
( distrib_lattice(X_a)
=> ! [X_1,Y,Z] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),X_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),Y),Z)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),X_1),Y)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),X_1),Z)) ) ).
fof(fact_296_sup__inf__distrib1,axiom,
! [X_a] :
( distrib_lattice(X_a)
=> ! [X_1,Y,Z] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),X_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),Y),Z)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),X_1),Y)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),X_1),Z)) ) ).
fof(fact_297_inf__sup__distrib2,axiom,
! [X_a] :
( distrib_lattice(X_a)
=> ! [Y,Z,X_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),Y),Z)),X_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),Y),X_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),Z),X_1)) ) ).
fof(fact_298_sup__inf__distrib2,axiom,
! [X_a] :
( distrib_lattice(X_a)
=> ! [Y,Z,X_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),Y),Z)),X_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),Y),X_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),Z),X_1)) ) ).
fof(fact_299_empty__Diff,axiom,
! [X_b,A_3] : hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),bot_bot(fun(X_b,bool))),A_3) = bot_bot(fun(X_b,bool)) ).
fof(fact_300_Diff__empty,axiom,
! [X_b,A_3] : hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_3),bot_bot(fun(X_b,bool))) = ti(fun(X_b,bool),A_3) ).
fof(fact_301_Diff__cancel,axiom,
! [X_b,A_3] : hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_3),A_3) = bot_bot(fun(X_b,bool)) ).
fof(fact_302_finite__Diff2,axiom,
! [X_b,A_3,B_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),B_2))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_3),B_2)))
<=> hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3)) ) ) ).
fof(fact_303_insert__Diff__if,axiom,
! [X_b,A_3,X_2,B_2] :
( ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_2),B_2))
=> hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_2),A_3)),B_2) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_3),B_2) )
& ( ~ hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_2),B_2))
=> hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_2),A_3)),B_2) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_2),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_3),B_2)) ) ) ).
fof(fact_304_insert__Diff1,axiom,
! [X_b,A_3,X_2,B_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_2),B_2))
=> hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_2),A_3)),B_2) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_3),B_2) ) ).
fof(fact_305_Int__empty__left,axiom,
! [X_b,B_2] : hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),bot_bot(fun(X_b,bool))),B_2) = bot_bot(fun(X_b,bool)) ).
fof(fact_306_Int__empty__right,axiom,
! [X_b,A_3] : hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_3),bot_bot(fun(X_b,bool))) = bot_bot(fun(X_b,bool)) ).
fof(fact_307_disjoint__iff__not__equal,axiom,
! [X_b,A_3,B_2] :
( hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_3),B_2) = bot_bot(fun(X_b,bool))
<=> ! [X] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X),A_3))
=> ! [Xa] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),Xa),B_2))
=> ti(X_b,X) != ti(X_b,Xa) ) ) ) ).
fof(fact_308_Un__Diff__cancel,axiom,
! [X_b,A_3,B_2] : hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),A_3),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),B_2),A_3)) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),A_3),B_2) ).
fof(fact_309_Un__Diff__cancel2,axiom,
! [X_b,B_2,A_3] : hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),B_2),A_3)),A_3) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),B_2),A_3) ).
fof(fact_310_Un__Diff,axiom,
! [X_b,A_3,B_2,C_3] : hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),A_3),B_2)),C_3) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_3),C_3)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),B_2),C_3)) ).
fof(fact_311_Int__insert__right,axiom,
! [X_b,B_2,A_2,A_3] :
( ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),A_2),A_3))
=> hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_3),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_2),B_2)) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_2),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_3),B_2)) )
& ( ~ hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),A_2),A_3))
=> hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_3),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_2),B_2)) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_3),B_2) ) ) ).
fof(fact_312_Int__insert__left,axiom,
! [X_b,B_2,A_2,C_3] :
( ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),A_2),C_3))
=> hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_2),B_2)),C_3) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_2),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),B_2),C_3)) )
& ( ~ hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),A_2),C_3))
=> hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_2),B_2)),C_3) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),B_2),C_3) ) ) ).
fof(fact_313_insert__inter__insert,axiom,
! [X_b,A_2,A_3,B_2] : hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_2),A_3)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_2),B_2)) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_2),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_3),B_2)) ).
fof(fact_314_Int__insert__right__if0,axiom,
! [X_b,B_2,A_2,A_3] :
( ~ hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),A_2),A_3))
=> hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_3),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_2),B_2)) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_3),B_2) ) ).
fof(fact_315_Int__insert__left__if0,axiom,
! [X_b,B_2,A_2,C_3] :
( ~ hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),A_2),C_3))
=> hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_2),B_2)),C_3) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),B_2),C_3) ) ).
fof(fact_316_Int__insert__right__if1,axiom,
! [X_b,B_2,A_2,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),A_2),A_3))
=> hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_3),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_2),B_2)) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_2),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_3),B_2)) ) ).
fof(fact_317_Int__insert__left__if1,axiom,
! [X_b,B_2,A_2,C_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),A_2),C_3))
=> hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_2),B_2)),C_3) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_2),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),B_2),C_3)) ) ).
fof(fact_318_Int__Un__distrib,axiom,
! [X_b,A_3,B_2,C_3] : hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_3),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),B_2),C_3)) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_3),B_2)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_3),C_3)) ).
fof(fact_319_Un__Int__distrib,axiom,
! [X_b,A_3,B_2,C_3] : hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),A_3),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),B_2),C_3)) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),A_3),B_2)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),A_3),C_3)) ).
fof(fact_320_Int__Un__distrib2,axiom,
! [X_b,B_2,C_3,A_3] : hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),B_2),C_3)),A_3) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),B_2),A_3)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),C_3),A_3)) ).
fof(fact_321_Un__Int__distrib2,axiom,
! [X_b,B_2,C_3,A_3] : hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),B_2),C_3)),A_3) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),B_2),A_3)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),C_3),A_3)) ).
fof(fact_322_Un__Int__crazy,axiom,
! [X_b,A_3,B_2,C_3] : hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_3),B_2)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),B_2),C_3))),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),C_3),A_3)) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),A_3),B_2)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),B_2),C_3))),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),C_3),A_3)) ).
fof(fact_323_Sup__fin_Oinsert__remove,axiom,
! [X_b] :
( lattice(X_b)
=> ! [X_2,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( ( hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_3),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_2),bot_bot(fun(X_b,bool)))) = bot_bot(fun(X_b,bool))
=> hAPP(fun(X_b,bool),X_b,big_lattice_Sup_fin(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_2),A_3)) = ti(X_b,X_2) )
& ( hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_3),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_2),bot_bot(fun(X_b,bool)))) != bot_bot(fun(X_b,bool))
=> hAPP(fun(X_b,bool),X_b,big_lattice_Sup_fin(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_2),A_3)) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),semilattice_sup_sup(X_b),X_2),hAPP(fun(X_b,bool),X_b,big_lattice_Sup_fin(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_3),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_2),bot_bot(fun(X_b,bool)))))) ) ) ) ) ).
fof(fact_324_Sup__fin_Oremove,axiom,
! [X_b] :
( lattice(X_b)
=> ! [X_2,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_2),A_3))
=> ( ( hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),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_2),bot_bot(fun(X_b,bool)))) = bot_bot(fun(X_b,bool))
=> hAPP(fun(X_b,bool),X_b,big_lattice_Sup_fin(X_b),A_3) = ti(X_b,X_2) )
& ( hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_3),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_2),bot_bot(fun(X_b,bool)))) != bot_bot(fun(X_b,bool))
=> hAPP(fun(X_b,bool),X_b,big_lattice_Sup_fin(X_b),A_3) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),semilattice_sup_sup(X_b),X_2),hAPP(fun(X_b,bool),X_b,big_lattice_Sup_fin(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_3),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_2),bot_bot(fun(X_b,bool)))))) ) ) ) ) ) ).
fof(fact_325_Diff__insert,axiom,
! [X_b,A_3,A_2,B_2] : hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_3),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_2),B_2)) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_3),B_2)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_2),bot_bot(fun(X_b,bool)))) ).
fof(fact_326_Diff__insert2,axiom,
! [X_b,A_3,A_2,B_2] : hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_3),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_2),B_2)) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_3),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_2),bot_bot(fun(X_b,bool))))),B_2) ).
fof(fact_327_insert__Diff__single,axiom,
! [X_b,A_2,A_3] : hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_2),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_3),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_2),bot_bot(fun(X_b,bool))))) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_2),A_3) ).
fof(fact_328_Diff__insert__absorb,axiom,
! [X_b,X_2,A_3] :
( ~ hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_2),A_3))
=> hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_2),A_3)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_2),bot_bot(fun(X_b,bool)))) = ti(fun(X_b,bool),A_3) ) ).
fof(fact_329_insert__Diff,axiom,
! [X_b,A_2,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),A_2),A_3))
=> hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_2),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_3),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_2),bot_bot(fun(X_b,bool))))) = ti(fun(X_b,bool),A_3) ) ).
fof(fact_330_finite__Diff__insert,axiom,
! [X_b,A_3,A_2,B_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_3),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_2),B_2))))
<=> hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_3),B_2))) ) ).
fof(fact_331_Sup__fin_Ounion__inter,axiom,
! [X_b] :
( lattice(X_b)
=> ! [B_2,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),B_2))
=> ( hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_3),B_2) != bot_bot(fun(X_b,bool))
=> hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),semilattice_sup_sup(X_b),hAPP(fun(X_b,bool),X_b,big_lattice_Sup_fin(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),A_3),B_2))),hAPP(fun(X_b,bool),X_b,big_lattice_Sup_fin(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_3),B_2))) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),semilattice_sup_sup(X_b),hAPP(fun(X_b,bool),X_b,big_lattice_Sup_fin(X_b),A_3)),hAPP(fun(X_b,bool),X_b,big_lattice_Sup_fin(X_b),B_2)) ) ) ) ) ).
fof(fact_332_Sup__fin_Ounion__disjoint,axiom,
! [X_b] :
( lattice(X_b)
=> ! [B_2,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( ti(fun(X_b,bool),A_3) != bot_bot(fun(X_b,bool))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),B_2))
=> ( ti(fun(X_b,bool),B_2) != bot_bot(fun(X_b,bool))
=> ( hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_3),B_2) = bot_bot(fun(X_b,bool))
=> hAPP(fun(X_b,bool),X_b,big_lattice_Sup_fin(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),A_3),B_2)) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),semilattice_sup_sup(X_b),hAPP(fun(X_b,bool),X_b,big_lattice_Sup_fin(X_b),A_3)),hAPP(fun(X_b,bool),X_b,big_lattice_Sup_fin(X_b),B_2)) ) ) ) ) ) ) ).
fof(fact_333_Sup__fin_Osingleton,axiom,
! [X_b] :
( lattice(X_b)
=> ! [X_2] : hAPP(fun(X_b,bool),X_b,big_lattice_Sup_fin(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_2),bot_bot(fun(X_b,bool)))) = ti(X_b,X_2) ) ).
fof(fact_334_Sup__fin_Oin__idem,axiom,
! [X_b] :
( lattice(X_b)
=> ! [X_2,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_2),A_3))
=> hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),semilattice_sup_sup(X_b),X_2),hAPP(fun(X_b,bool),X_b,big_lattice_Sup_fin(X_b),A_3)) = hAPP(fun(X_b,bool),X_b,big_lattice_Sup_fin(X_b),A_3) ) ) ) ).
fof(fact_335_if__image__distrib,axiom,
! [X_c,X_b,Pa,F,G,S_1] : hAPP(fun(X_c,bool),fun(X_b,bool),hAPP(fun(X_c,X_b),fun(fun(X_c,bool),fun(X_b,bool)),image(X_c,X_b),hAPP(fun(X_c,X_b),fun(X_c,X_b),hAPP(fun(X_c,fun(X_b,X_b)),fun(fun(X_c,X_b),fun(X_c,X_b)),combs(X_c,X_b,X_b),hAPP(fun(X_c,X_b),fun(X_c,fun(X_b,X_b)),hAPP(fun(X_c,fun(X_b,fun(X_b,X_b))),fun(fun(X_c,X_b),fun(X_c,fun(X_b,X_b))),combs(X_c,X_b,fun(X_b,X_b)),hAPP(fun(X_c,bool),fun(X_c,fun(X_b,fun(X_b,X_b))),hAPP(fun(bool,fun(X_b,fun(X_b,X_b))),fun(fun(X_c,bool),fun(X_c,fun(X_b,fun(X_b,X_b)))),combb(bool,fun(X_b,fun(X_b,X_b)),X_c),if(X_b)),Pa)),F)),G)),S_1) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),hAPP(fun(X_c,bool),fun(X_b,bool),hAPP(fun(X_c,X_b),fun(fun(X_c,bool),fun(X_b,bool)),image(X_c,X_b),F),hAPP(fun(X_c,bool),fun(X_c,bool),hAPP(fun(X_c,bool),fun(fun(X_c,bool),fun(X_c,bool)),semilattice_inf_inf(fun(X_c,bool)),S_1),hAPP(fun(X_c,bool),fun(X_c,bool),collect(X_c),Pa)))),hAPP(fun(X_c,bool),fun(X_b,bool),hAPP(fun(X_c,X_b),fun(fun(X_c,bool),fun(X_b,bool)),image(X_c,X_b),G),hAPP(fun(X_c,bool),fun(X_c,bool),hAPP(fun(X_c,bool),fun(fun(X_c,bool),fun(X_c,bool)),semilattice_inf_inf(fun(X_c,bool)),S_1),hAPP(fun(X_c,bool),fun(X_c,bool),collect(X_c),hAPP(fun(X_c,bool),fun(X_c,bool),hAPP(fun(bool,bool),fun(fun(X_c,bool),fun(X_c,bool)),combb(bool,bool,X_c),fNot),Pa))))) ).
fof(fact_336_folding__one_Oremove,axiom,
! [X_b,X_2,A_3,F,F_1] :
( hBOOL(hAPP(fun(fun(X_b,bool),X_b),bool,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(fun(X_b,bool),X_b),bool),finite_folding_one(X_b),F),F_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_2),A_3))
=> ( ( hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),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_2),bot_bot(fun(X_b,bool)))) = bot_bot(fun(X_b,bool))
=> hAPP(fun(X_b,bool),X_b,F_1,A_3) = ti(X_b,X_2) )
& ( hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_3),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_2),bot_bot(fun(X_b,bool)))) != bot_bot(fun(X_b,bool))
=> hAPP(fun(X_b,bool),X_b,F_1,A_3) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),F,X_2),hAPP(fun(X_b,bool),X_b,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_2),bot_bot(fun(X_b,bool)))))) ) ) ) ) ) ).
fof(fact_337_Sup__fin_Ohom__commute,axiom,
! [X_b] :
( lattice(X_b)
=> ! [N_3,H] :
( ! [X,Y_2] : hAPP(X_b,X_b,H,hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),semilattice_sup_sup(X_b),X),Y_2)) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),semilattice_sup_sup(X_b),hAPP(X_b,X_b,H,X)),hAPP(X_b,X_b,H,Y_2))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),N_3))
=> ( ti(fun(X_b,bool),N_3) != bot_bot(fun(X_b,bool))
=> hAPP(X_b,X_b,H,hAPP(fun(X_b,bool),X_b,big_lattice_Sup_fin(X_b),N_3)) = hAPP(fun(X_b,bool),X_b,big_lattice_Sup_fin(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,X_b),fun(fun(X_b,bool),fun(X_b,bool)),image(X_b,X_b),H),N_3)) ) ) ) ) ).
fof(fact_338_Sup__fin_Oclosed,axiom,
! [X_b] :
( lattice(X_b)
=> ! [A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( ti(fun(X_b,bool),A_3) != bot_bot(fun(X_b,bool))
=> ( ! [X,Y_2] : hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),semilattice_sup_sup(X_b),X),Y_2)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),Y_2),bot_bot(fun(X_b,bool))))))
=> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),hAPP(fun(X_b,bool),X_b,big_lattice_Sup_fin(X_b),A_3)),A_3)) ) ) ) ) ).
fof(fact_339_finite__empty__induct,axiom,
! [X_b,Pa,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,Pa,A_3))
=> ( ! [A_4,A_5] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_5))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),A_4),A_5))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,Pa,A_5))
=> hBOOL(hAPP(fun(X_b,bool),bool,Pa,hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_5),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_4),bot_bot(fun(X_b,bool)))))) ) ) )
=> hBOOL(hAPP(fun(X_b,bool),bool,Pa,bot_bot(fun(X_b,bool)))) ) ) ) ).
fof(fact_340_distrib__imp1,axiom,
! [X_a] :
( lattice(X_a)
=> ! [X_1,Y,Z] :
( ! [X,Y_2,Z_2] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),Y_2),Z_2)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),X),Y_2)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),X),Z_2))
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),X_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),Y),Z)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),X_1),Y)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),X_1),Z)) ) ) ).
fof(fact_341_distrib__imp2,axiom,
! [X_a] :
( lattice(X_a)
=> ! [X_1,Y,Z] :
( ! [X,Y_2,Z_2] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),Y_2),Z_2)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),X),Y_2)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),X),Z_2))
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),X_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),Y),Z)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),X_1),Y)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),X_1),Z)) ) ) ).
fof(fact_342_folding__image__simple_Ounion__disjoint,axiom,
! [X_c,X_b,B_2,A_3,F,Z_1,G,F_1] :
( hBOOL(hAPP(fun(fun(X_c,bool),X_b),bool,hAPP(fun(X_c,X_b),fun(fun(fun(X_c,bool),X_b),bool),hAPP(X_b,fun(fun(X_c,X_b),fun(fun(fun(X_c,bool),X_b),bool)),hAPP(fun(X_b,fun(X_b,X_b)),fun(X_b,fun(fun(X_c,X_b),fun(fun(fun(X_c,bool),X_b),bool))),finite1357897459simple(X_b,X_c),F),Z_1),G),F_1))
=> ( hBOOL(hAPP(fun(X_c,bool),bool,finite_finite_1(X_c),A_3))
=> ( hBOOL(hAPP(fun(X_c,bool),bool,finite_finite_1(X_c),B_2))
=> ( hAPP(fun(X_c,bool),fun(X_c,bool),hAPP(fun(X_c,bool),fun(fun(X_c,bool),fun(X_c,bool)),semilattice_inf_inf(fun(X_c,bool)),A_3),B_2) = bot_bot(fun(X_c,bool))
=> hAPP(fun(X_c,bool),X_b,F_1,hAPP(fun(X_c,bool),fun(X_c,bool),hAPP(fun(X_c,bool),fun(fun(X_c,bool),fun(X_c,bool)),semilattice_sup_sup(fun(X_c,bool)),A_3),B_2)) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),F,hAPP(fun(X_c,bool),X_b,F_1,A_3)),hAPP(fun(X_c,bool),X_b,F_1,B_2)) ) ) ) ) ).
fof(fact_343_folding__image__simple_Oremove,axiom,
! [X_c,X_b,X_2,A_3,F,Z_1,G,F_1] :
( hBOOL(hAPP(fun(fun(X_c,bool),X_b),bool,hAPP(fun(X_c,X_b),fun(fun(fun(X_c,bool),X_b),bool),hAPP(X_b,fun(fun(X_c,X_b),fun(fun(fun(X_c,bool),X_b),bool)),hAPP(fun(X_b,fun(X_b,X_b)),fun(X_b,fun(fun(X_c,X_b),fun(fun(fun(X_c,bool),X_b),bool))),finite1357897459simple(X_b,X_c),F),Z_1),G),F_1))
=> ( hBOOL(hAPP(fun(X_c,bool),bool,finite_finite_1(X_c),A_3))
=> ( hBOOL(hAPP(fun(X_c,bool),bool,hAPP(X_c,fun(fun(X_c,bool),bool),member(X_c),X_2),A_3))
=> hAPP(fun(X_c,bool),X_b,F_1,A_3) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),F,hAPP(X_c,X_b,G,X_2)),hAPP(fun(X_c,bool),X_b,F_1,hAPP(fun(X_c,bool),fun(X_c,bool),hAPP(fun(X_c,bool),fun(fun(X_c,bool),fun(X_c,bool)),minus_minus(fun(X_c,bool)),A_3),hAPP(fun(X_c,bool),fun(X_c,bool),hAPP(X_c,fun(fun(X_c,bool),fun(X_c,bool)),insert(X_c),X_2),bot_bot(fun(X_c,bool)))))) ) ) ) ).
fof(fact_344_diff__0__eq__0,axiom,
! [N_2] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),zero_zero(nat)),N_2) = zero_zero(nat) ).
fof(fact_345_minus__nat_Odiff__0,axiom,
! [M] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),zero_zero(nat)) = M ).
fof(fact_346_diff__self__eq__0,axiom,
! [M] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),M) = zero_zero(nat) ).
fof(fact_347_diffs0__imp__equal,axiom,
! [M,N_2] :
( hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),N_2) = zero_zero(nat)
=> ( hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N_2),M) = zero_zero(nat)
=> M = N_2 ) ) ).
fof(fact_348_diff__Suc__Suc,axiom,
! [M,N_2] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,suc,M)),hAPP(nat,nat,suc,N_2)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),N_2) ).
fof(fact_349_Suc__diff__diff,axiom,
! [M,N_2,K_2] : 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)),N_2)),hAPP(nat,nat,suc,K_2)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),N_2)),K_2) ).
fof(fact_350_folding__image__simple_Oempty,axiom,
! [X_c,X_b,F,Z_1,G,F_1] :
( hBOOL(hAPP(fun(fun(X_c,bool),X_b),bool,hAPP(fun(X_c,X_b),fun(fun(fun(X_c,bool),X_b),bool),hAPP(X_b,fun(fun(X_c,X_b),fun(fun(fun(X_c,bool),X_b),bool)),hAPP(fun(X_b,fun(X_b,X_b)),fun(X_b,fun(fun(X_c,X_b),fun(fun(fun(X_c,bool),X_b),bool))),finite1357897459simple(X_b,X_c),F),Z_1),G),F_1))
=> hAPP(fun(X_c,bool),X_b,F_1,bot_bot(fun(X_c,bool))) = ti(X_b,Z_1) ) ).
fof(fact_351_folding__image__simple_Oinsert,axiom,
! [X_c,X_b,X_2,A_3,F,Z_1,G,F_1] :
( hBOOL(hAPP(fun(fun(X_c,bool),X_b),bool,hAPP(fun(X_c,X_b),fun(fun(fun(X_c,bool),X_b),bool),hAPP(X_b,fun(fun(X_c,X_b),fun(fun(fun(X_c,bool),X_b),bool)),hAPP(fun(X_b,fun(X_b,X_b)),fun(X_b,fun(fun(X_c,X_b),fun(fun(fun(X_c,bool),X_b),bool))),finite1357897459simple(X_b,X_c),F),Z_1),G),F_1))
=> ( hBOOL(hAPP(fun(X_c,bool),bool,finite_finite_1(X_c),A_3))
=> ( ~ hBOOL(hAPP(fun(X_c,bool),bool,hAPP(X_c,fun(fun(X_c,bool),bool),member(X_c),X_2),A_3))
=> hAPP(fun(X_c,bool),X_b,F_1,hAPP(fun(X_c,bool),fun(X_c,bool),hAPP(X_c,fun(fun(X_c,bool),fun(X_c,bool)),insert(X_c),X_2),A_3)) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),F,hAPP(X_c,X_b,G,X_2)),hAPP(fun(X_c,bool),X_b,F_1,A_3)) ) ) ) ).
fof(fact_352_folding__image__simple_Oeq__fold__g,axiom,
! [X_c,X_b,A_3,F,Z_1,G,F_1] :
( hBOOL(hAPP(fun(fun(X_c,bool),X_b),bool,hAPP(fun(X_c,X_b),fun(fun(fun(X_c,bool),X_b),bool),hAPP(X_b,fun(fun(X_c,X_b),fun(fun(fun(X_c,bool),X_b),bool)),hAPP(fun(X_b,fun(X_b,X_b)),fun(X_b,fun(fun(X_c,X_b),fun(fun(fun(X_c,bool),X_b),bool))),finite1357897459simple(X_b,X_c),F),Z_1),G),F_1))
=> ( hBOOL(hAPP(fun(X_c,bool),bool,finite_finite_1(X_c),A_3))
=> hAPP(fun(X_c,bool),X_b,F_1,A_3) = hAPP(fun(X_c,bool),X_b,hAPP(X_b,fun(fun(X_c,bool),X_b),hAPP(fun(X_c,X_b),fun(X_b,fun(fun(X_c,bool),X_b)),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_c,X_b),fun(X_b,fun(fun(X_c,bool),X_b))),finite_fold_image(X_b,X_c),F),G),Z_1),A_3) ) ) ).
fof(fact_353_folding__image__simple_Ounion__inter,axiom,
! [X_c,X_b,B_2,A_3,F,Z_1,G,F_1] :
( hBOOL(hAPP(fun(fun(X_c,bool),X_b),bool,hAPP(fun(X_c,X_b),fun(fun(fun(X_c,bool),X_b),bool),hAPP(X_b,fun(fun(X_c,X_b),fun(fun(fun(X_c,bool),X_b),bool)),hAPP(fun(X_b,fun(X_b,X_b)),fun(X_b,fun(fun(X_c,X_b),fun(fun(fun(X_c,bool),X_b),bool))),finite1357897459simple(X_b,X_c),F),Z_1),G),F_1))
=> ( hBOOL(hAPP(fun(X_c,bool),bool,finite_finite_1(X_c),A_3))
=> ( hBOOL(hAPP(fun(X_c,bool),bool,finite_finite_1(X_c),B_2))
=> hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),F,hAPP(fun(X_c,bool),X_b,F_1,hAPP(fun(X_c,bool),fun(X_c,bool),hAPP(fun(X_c,bool),fun(fun(X_c,bool),fun(X_c,bool)),semilattice_sup_sup(fun(X_c,bool)),A_3),B_2))),hAPP(fun(X_c,bool),X_b,F_1,hAPP(fun(X_c,bool),fun(X_c,bool),hAPP(fun(X_c,bool),fun(fun(X_c,bool),fun(X_c,bool)),semilattice_inf_inf(fun(X_c,bool)),A_3),B_2))) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),F,hAPP(fun(X_c,bool),X_b,F_1,A_3)),hAPP(fun(X_c,bool),X_b,F_1,B_2)) ) ) ) ).
fof(fact_354_folding__image__simple_Oinsert__remove,axiom,
! [X_c,X_b,X_2,A_3,F,Z_1,G,F_1] :
( hBOOL(hAPP(fun(fun(X_c,bool),X_b),bool,hAPP(fun(X_c,X_b),fun(fun(fun(X_c,bool),X_b),bool),hAPP(X_b,fun(fun(X_c,X_b),fun(fun(fun(X_c,bool),X_b),bool)),hAPP(fun(X_b,fun(X_b,X_b)),fun(X_b,fun(fun(X_c,X_b),fun(fun(fun(X_c,bool),X_b),bool))),finite1357897459simple(X_b,X_c),F),Z_1),G),F_1))
=> ( hBOOL(hAPP(fun(X_c,bool),bool,finite_finite_1(X_c),A_3))
=> hAPP(fun(X_c,bool),X_b,F_1,hAPP(fun(X_c,bool),fun(X_c,bool),hAPP(X_c,fun(fun(X_c,bool),fun(X_c,bool)),insert(X_c),X_2),A_3)) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),F,hAPP(X_c,X_b,G,X_2)),hAPP(fun(X_c,bool),X_b,F_1,hAPP(fun(X_c,bool),fun(X_c,bool),hAPP(fun(X_c,bool),fun(fun(X_c,bool),fun(X_c,bool)),minus_minus(fun(X_c,bool)),A_3),hAPP(fun(X_c,bool),fun(X_c,bool),hAPP(X_c,fun(fun(X_c,bool),fun(X_c,bool)),insert(X_c),X_2),bot_bot(fun(X_c,bool)))))) ) ) ).
fof(fact_355_folding__image__simple_Ounion__inter__neutral,axiom,
! [X_c,X_b,B_2,A_3,F,Z_1,G,F_1] :
( hBOOL(hAPP(fun(fun(X_c,bool),X_b),bool,hAPP(fun(X_c,X_b),fun(fun(fun(X_c,bool),X_b),bool),hAPP(X_b,fun(fun(X_c,X_b),fun(fun(fun(X_c,bool),X_b),bool)),hAPP(fun(X_b,fun(X_b,X_b)),fun(X_b,fun(fun(X_c,X_b),fun(fun(fun(X_c,bool),X_b),bool))),finite1357897459simple(X_b,X_c),F),Z_1),G),F_1))
=> ( hBOOL(hAPP(fun(X_c,bool),bool,finite_finite_1(X_c),A_3))
=> ( hBOOL(hAPP(fun(X_c,bool),bool,finite_finite_1(X_c),B_2))
=> ( ! [X] :
( hBOOL(hAPP(fun(X_c,bool),bool,hAPP(X_c,fun(fun(X_c,bool),bool),member(X_c),X),hAPP(fun(X_c,bool),fun(X_c,bool),hAPP(fun(X_c,bool),fun(fun(X_c,bool),fun(X_c,bool)),semilattice_inf_inf(fun(X_c,bool)),A_3),B_2)))
=> hAPP(X_c,X_b,G,X) = ti(X_b,Z_1) )
=> hAPP(fun(X_c,bool),X_b,F_1,hAPP(fun(X_c,bool),fun(X_c,bool),hAPP(fun(X_c,bool),fun(fun(X_c,bool),fun(X_c,bool)),semilattice_sup_sup(fun(X_c,bool)),A_3),B_2)) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),F,hAPP(fun(X_c,bool),X_b,F_1,A_3)),hAPP(fun(X_c,bool),X_b,F_1,B_2)) ) ) ) ) ).
fof(fact_356_folding__image__simple_Oneutral,axiom,
! [X_c,X_b,A_3,F,Z_1,G,F_1] :
( hBOOL(hAPP(fun(fun(X_c,bool),X_b),bool,hAPP(fun(X_c,X_b),fun(fun(fun(X_c,bool),X_b),bool),hAPP(X_b,fun(fun(X_c,X_b),fun(fun(fun(X_c,bool),X_b),bool)),hAPP(fun(X_b,fun(X_b,X_b)),fun(X_b,fun(fun(X_c,X_b),fun(fun(fun(X_c,bool),X_b),bool))),finite1357897459simple(X_b,X_c),F),Z_1),G),F_1))
=> ( hBOOL(hAPP(fun(X_c,bool),bool,finite_finite_1(X_c),A_3))
=> ( ! [X] :
( hBOOL(hAPP(fun(X_c,bool),bool,hAPP(X_c,fun(fun(X_c,bool),bool),member(X_c),X),A_3))
=> hAPP(X_c,X_b,G,X) = ti(X_b,Z_1) )
=> hAPP(fun(X_c,bool),X_b,F_1,A_3) = ti(X_b,Z_1) ) ) ) ).
fof(fact_357_right__minus__eq,axiom,
! [X_b] :
( group_add(X_b)
=> ! [A_2,B_1] :
( hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),minus_minus(X_b),A_2),B_1) = zero_zero(X_b)
<=> ti(X_b,A_2) = ti(X_b,B_1) ) ) ).
fof(fact_358_eq__iff__diff__eq__0,axiom,
! [X_b] :
( ab_group_add(X_b)
=> ! [A_2,B_1] :
( ti(X_b,A_2) = ti(X_b,B_1)
<=> hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),minus_minus(X_b),A_2),B_1) = zero_zero(X_b) ) ) ).
fof(fact_359_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_360_diff__commute,axiom,
! [I,J_1,K_2] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),I),J_1)),K_2) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),I),K_2)),J_1) ).
fof(fact_361_zero__reorient,axiom,
! [X_b] :
( zero(X_b)
=> ! [X_2] :
( zero_zero(X_b) = ti(X_b,X_2)
<=> ti(X_b,X_2) = zero_zero(X_b) ) ) ).
fof(fact_362_diff__eq__diff__eq,axiom,
! [X_b] :
( ab_group_add(X_b)
=> ! [A_2,B_1,C_1,D] :
( hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),minus_minus(X_b),A_2),B_1) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),minus_minus(X_b),C_1),D)
=> ( ti(X_b,A_2) = ti(X_b,B_1)
<=> ti(X_b,C_1) = ti(X_b,D) ) ) ) ).
fof(fact_363_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_364_zero__induct__lemma,axiom,
! [I_1,Pa,K_1] :
( hBOOL(hAPP(nat,bool,Pa,K_1))
=> ( ! [N] :
( hBOOL(hAPP(nat,bool,Pa,hAPP(nat,nat,suc,N)))
=> hBOOL(hAPP(nat,bool,Pa,N)) )
=> hBOOL(hAPP(nat,bool,Pa,hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),K_1),I_1))) ) ) ).
fof(fact_365_Inf__fin_Ounion__disjoint,axiom,
! [X_b] :
( lattice(X_b)
=> ! [B_2,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( ti(fun(X_b,bool),A_3) != bot_bot(fun(X_b,bool))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),B_2))
=> ( ti(fun(X_b,bool),B_2) != bot_bot(fun(X_b,bool))
=> ( hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_3),B_2) = bot_bot(fun(X_b,bool))
=> hAPP(fun(X_b,bool),X_b,big_lattice_Inf_fin(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),A_3),B_2)) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),semilattice_inf_inf(X_b),hAPP(fun(X_b,bool),X_b,big_lattice_Inf_fin(X_b),A_3)),hAPP(fun(X_b,bool),X_b,big_lattice_Inf_fin(X_b),B_2)) ) ) ) ) ) ) ).
fof(fact_366_Inf__fin_Ounion__inter,axiom,
! [X_b] :
( lattice(X_b)
=> ! [B_2,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),B_2))
=> ( hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_3),B_2) != bot_bot(fun(X_b,bool))
=> hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),semilattice_inf_inf(X_b),hAPP(fun(X_b,bool),X_b,big_lattice_Inf_fin(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),A_3),B_2))),hAPP(fun(X_b,bool),X_b,big_lattice_Inf_fin(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_3),B_2))) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),semilattice_inf_inf(X_b),hAPP(fun(X_b,bool),X_b,big_lattice_Inf_fin(X_b),A_3)),hAPP(fun(X_b,bool),X_b,big_lattice_Inf_fin(X_b),B_2)) ) ) ) ) ).
fof(fact_367_Inf__fin_Oinsert__remove,axiom,
! [X_b] :
( lattice(X_b)
=> ! [X_2,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( ( hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_3),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_2),bot_bot(fun(X_b,bool)))) = bot_bot(fun(X_b,bool))
=> hAPP(fun(X_b,bool),X_b,big_lattice_Inf_fin(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_2),A_3)) = ti(X_b,X_2) )
& ( hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_3),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_2),bot_bot(fun(X_b,bool)))) != bot_bot(fun(X_b,bool))
=> hAPP(fun(X_b,bool),X_b,big_lattice_Inf_fin(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_2),A_3)) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),semilattice_inf_inf(X_b),X_2),hAPP(fun(X_b,bool),X_b,big_lattice_Inf_fin(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_3),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_2),bot_bot(fun(X_b,bool)))))) ) ) ) ) ).
fof(fact_368_Inf__fin_Oremove,axiom,
! [X_b] :
( lattice(X_b)
=> ! [X_2,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_2),A_3))
=> ( ( hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),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_2),bot_bot(fun(X_b,bool)))) = bot_bot(fun(X_b,bool))
=> hAPP(fun(X_b,bool),X_b,big_lattice_Inf_fin(X_b),A_3) = ti(X_b,X_2) )
& ( hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_3),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_2),bot_bot(fun(X_b,bool)))) != bot_bot(fun(X_b,bool))
=> hAPP(fun(X_b,bool),X_b,big_lattice_Inf_fin(X_b),A_3) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),semilattice_inf_inf(X_b),X_2),hAPP(fun(X_b,bool),X_b,big_lattice_Inf_fin(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_3),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_2),bot_bot(fun(X_b,bool)))))) ) ) ) ) ) ).
fof(fact_369_Inf__fin_Osingleton,axiom,
! [X_b] :
( lattice(X_b)
=> ! [X_2] : hAPP(fun(X_b,bool),X_b,big_lattice_Inf_fin(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_2),bot_bot(fun(X_b,bool)))) = ti(X_b,X_2) ) ).
fof(fact_370_sup__Inf__absorb,axiom,
! [X_b] :
( lattice(X_b)
=> ! [A_2,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),A_2),A_3))
=> hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),semilattice_sup_sup(X_b),A_2),hAPP(fun(X_b,bool),X_b,big_lattice_Inf_fin(X_b),A_3)) = ti(X_b,A_2) ) ) ) ).
fof(fact_371_Inf__fin_Oin__idem,axiom,
! [X_b] :
( lattice(X_b)
=> ! [X_2,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_2),A_3))
=> hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),semilattice_inf_inf(X_b),X_2),hAPP(fun(X_b,bool),X_b,big_lattice_Inf_fin(X_b),A_3)) = hAPP(fun(X_b,bool),X_b,big_lattice_Inf_fin(X_b),A_3) ) ) ) ).
fof(fact_372_Inf__fin_Oinsert,axiom,
! [X_b] :
( lattice(X_b)
=> ! [X_2,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( ~ hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_2),A_3))
=> ( ti(fun(X_b,bool),A_3) != bot_bot(fun(X_b,bool))
=> hAPP(fun(X_b,bool),X_b,big_lattice_Inf_fin(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_2),A_3)) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),semilattice_inf_inf(X_b),X_2),hAPP(fun(X_b,bool),X_b,big_lattice_Inf_fin(X_b),A_3)) ) ) ) ) ).
fof(fact_373_Inf__fin_Oinsert__idem,axiom,
! [X_b] :
( lattice(X_b)
=> ! [X_2,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( ti(fun(X_b,bool),A_3) != bot_bot(fun(X_b,bool))
=> hAPP(fun(X_b,bool),X_b,big_lattice_Inf_fin(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_2),A_3)) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),semilattice_inf_inf(X_b),X_2),hAPP(fun(X_b,bool),X_b,big_lattice_Inf_fin(X_b),A_3)) ) ) ) ).
fof(fact_374_Inf__fin_Ounion__idem,axiom,
! [X_b] :
( lattice(X_b)
=> ! [B_2,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( ti(fun(X_b,bool),A_3) != bot_bot(fun(X_b,bool))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),B_2))
=> ( ti(fun(X_b,bool),B_2) != bot_bot(fun(X_b,bool))
=> hAPP(fun(X_b,bool),X_b,big_lattice_Inf_fin(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),A_3),B_2)) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),semilattice_inf_inf(X_b),hAPP(fun(X_b,bool),X_b,big_lattice_Inf_fin(X_b),A_3)),hAPP(fun(X_b,bool),X_b,big_lattice_Inf_fin(X_b),B_2)) ) ) ) ) ) ).
fof(fact_375_Inf__fin_Ohom__commute,axiom,
! [X_b] :
( lattice(X_b)
=> ! [N_3,H] :
( ! [X,Y_2] : hAPP(X_b,X_b,H,hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),semilattice_inf_inf(X_b),X),Y_2)) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),semilattice_inf_inf(X_b),hAPP(X_b,X_b,H,X)),hAPP(X_b,X_b,H,Y_2))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),N_3))
=> ( ti(fun(X_b,bool),N_3) != bot_bot(fun(X_b,bool))
=> hAPP(X_b,X_b,H,hAPP(fun(X_b,bool),X_b,big_lattice_Inf_fin(X_b),N_3)) = hAPP(fun(X_b,bool),X_b,big_lattice_Inf_fin(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,X_b),fun(fun(X_b,bool),fun(X_b,bool)),image(X_b,X_b),H),N_3)) ) ) ) ) ).
fof(fact_376_Inf__fin_Oclosed,axiom,
! [X_b] :
( lattice(X_b)
=> ! [A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( ti(fun(X_b,bool),A_3) != bot_bot(fun(X_b,bool))
=> ( ! [X,Y_2] : hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),semilattice_inf_inf(X_b),X),Y_2)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),Y_2),bot_bot(fun(X_b,bool))))))
=> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),hAPP(fun(X_b,bool),X_b,big_lattice_Inf_fin(X_b),A_3)),A_3)) ) ) ) ) ).
fof(fact_377_diff__Suc,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(fun(nat,nat),fun(nat,nat),hAPP(nat,fun(fun(nat,nat),fun(nat,nat)),nat_case(nat),zero_zero(nat)),combi(nat)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M_1),N_1)) ).
fof(fact_378_comp__fun__idem__remove,axiom,
! [X_b] : hBOOL(hAPP(fun(X_b,fun(fun(X_b,bool),fun(X_b,bool))),bool,finite_comp_fun_idem(X_b,fun(X_b,bool)),hAPP(fun(X_b,fun(X_b,bool)),fun(X_b,fun(fun(X_b,bool),fun(X_b,bool))),hAPP(fun(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool))),fun(fun(X_b,fun(X_b,bool)),fun(X_b,fun(fun(X_b,bool),fun(X_b,bool)))),combb(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),X_b),hAPP(fun(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool))),fun(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool))),combc(fun(X_b,bool),fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)))),hAPP(fun(X_b,bool),fun(X_b,fun(X_b,bool)),hAPP(fun(X_b,fun(fun(X_b,bool),fun(X_b,bool))),fun(fun(X_b,bool),fun(X_b,fun(X_b,bool))),combc(X_b,fun(X_b,bool),fun(X_b,bool)),insert(X_b)),bot_bot(fun(X_b,bool)))))) ).
fof(fact_379_card__insert,axiom,
! [X_b,X_2,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> hAPP(fun(X_b,bool),nat,finite_card(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_2),A_3)) = hAPP(nat,nat,suc,hAPP(fun(X_b,bool),nat,finite_card(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_3),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_2),bot_bot(fun(X_b,bool)))))) ) ).
fof(fact_380_comp__fun__idem_Ofun__left__idem,axiom,
! [X_b,X_c,X_2,Z_1,F] :
( hBOOL(hAPP(fun(X_b,fun(X_c,X_c)),bool,finite_comp_fun_idem(X_b,X_c),F))
=> hAPP(X_c,X_c,hAPP(X_b,fun(X_c,X_c),F,X_2),hAPP(X_c,X_c,hAPP(X_b,fun(X_c,X_c),F,X_2),Z_1)) = hAPP(X_c,X_c,hAPP(X_b,fun(X_c,X_c),F,X_2),Z_1) ) ).
fof(fact_381_comp__fun__idem__insert,axiom,
! [X_b] : hBOOL(hAPP(fun(X_b,fun(fun(X_b,bool),fun(X_b,bool))),bool,finite_comp_fun_idem(X_b,fun(X_b,bool)),insert(X_b))) ).
fof(fact_382_card_Oempty,axiom,
! [X_b] : hAPP(fun(X_b,bool),nat,finite_card(X_b),bot_bot(fun(X_b,bool))) = zero_zero(nat) ).
fof(fact_383_card__infinite,axiom,
! [X_b,A_3] :
( ~ hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> hAPP(fun(X_b,bool),nat,finite_card(X_b),A_3) = zero_zero(nat) ) ).
fof(fact_384_comp__fun__idem__sup,axiom,
! [X_b] :
( semilattice_sup(X_b)
=> hBOOL(hAPP(fun(X_b,fun(X_b,X_b)),bool,finite_comp_fun_idem(X_b,X_b),semilattice_sup_sup(X_b))) ) ).
fof(fact_385_comp__fun__idem__inf,axiom,
! [X_b] :
( semilattice_inf(X_b)
=> hBOOL(hAPP(fun(X_b,fun(X_b,X_b)),bool,finite_comp_fun_idem(X_b,X_b),semilattice_inf_inf(X_b))) ) ).
fof(fact_386_nat__case__0,axiom,
! [X_b,F1,F2] : hAPP(nat,X_b,hAPP(fun(nat,X_b),fun(nat,X_b),hAPP(X_b,fun(fun(nat,X_b),fun(nat,X_b)),nat_case(X_b),F1),F2),zero_zero(nat)) = ti(X_b,F1) ).
fof(fact_387_nat__case__Suc,axiom,
! [X_b,F1,F2,Nat] : hAPP(nat,X_b,hAPP(fun(nat,X_b),fun(nat,X_b),hAPP(X_b,fun(fun(nat,X_b),fun(nat,X_b)),nat_case(X_b),F1),F2),hAPP(nat,nat,suc,Nat)) = hAPP(nat,X_b,F2,Nat) ).
fof(fact_388_card__eq__0__iff,axiom,
! [X_b,A_3] :
( hAPP(fun(X_b,bool),nat,finite_card(X_b),A_3) = zero_zero(nat)
<=> ( ti(fun(X_b,bool),A_3) = bot_bot(fun(X_b,bool))
| ~ hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3)) ) ) ).
fof(fact_389_card__insert__disjoint,axiom,
! [X_b,X_2,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( ~ hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_2),A_3))
=> hAPP(fun(X_b,bool),nat,finite_card(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_2),A_3)) = hAPP(nat,nat,suc,hAPP(fun(X_b,bool),nat,finite_card(X_b),A_3)) ) ) ).
fof(fact_390_card__insert__if,axiom,
! [X_b,X_2,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_2),A_3))
=> hAPP(fun(X_b,bool),nat,finite_card(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_2),A_3)) = hAPP(fun(X_b,bool),nat,finite_card(X_b),A_3) )
& ( ~ hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_2),A_3))
=> hAPP(fun(X_b,bool),nat,finite_card(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_2),A_3)) = hAPP(nat,nat,suc,hAPP(fun(X_b,bool),nat,finite_card(X_b),A_3)) ) ) ) ).
fof(fact_391_card__Diff__subset__Int,axiom,
! [X_b,A_3,B_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_3),B_2)))
=> hAPP(fun(X_b,bool),nat,finite_card(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_3),B_2)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(fun(X_b,bool),nat,finite_card(X_b),A_3)),hAPP(fun(X_b,bool),nat,finite_card(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_3),B_2))) ) ).
fof(fact_392_card__Suc__Diff1,axiom,
! [X_b,X_2,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_2),A_3))
=> hAPP(nat,nat,suc,hAPP(fun(X_b,bool),nat,finite_card(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_3),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_2),bot_bot(fun(X_b,bool)))))) = hAPP(fun(X_b,bool),nat,finite_card(X_b),A_3) ) ) ).
fof(fact_393_card__Suc__eq,axiom,
! [X_b,A_3,K_1] :
( hAPP(fun(X_b,bool),nat,finite_card(X_b),A_3) = hAPP(nat,nat,suc,K_1)
<=> ? [B_4,B_3] :
( ti(fun(X_b,bool),A_3) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),B_4),B_3)
& ~ hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),B_4),B_3))
& hAPP(fun(X_b,bool),nat,finite_card(X_b),B_3) = K_1
& ( K_1 = zero_zero(nat)
=> ti(fun(X_b,bool),B_3) = bot_bot(fun(X_b,bool)) ) ) ) ).
fof(fact_394_card__eq__SucD,axiom,
! [X_b,A_3,K_1] :
( hAPP(fun(X_b,bool),nat,finite_card(X_b),A_3) = hAPP(nat,nat,suc,K_1)
=> ? [B_4,B_3] :
( ti(fun(X_b,bool),A_3) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),B_4),B_3)
& ~ hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),B_4),B_3))
& hAPP(fun(X_b,bool),nat,finite_card(X_b),B_3) = K_1
& ( K_1 = zero_zero(nat)
=> ti(fun(X_b,bool),B_3) = bot_bot(fun(X_b,bool)) ) ) ) ).
fof(fact_395_card__Diff__singleton__if,axiom,
! [X_b,X_2,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_2),A_3))
=> hAPP(fun(X_b,bool),nat,finite_card(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_3),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_2),bot_bot(fun(X_b,bool))))) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(fun(X_b,bool),nat,finite_card(X_b),A_3)),one_one(nat)) )
& ( ~ hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_2),A_3))
=> hAPP(fun(X_b,bool),nat,finite_card(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_3),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_2),bot_bot(fun(X_b,bool))))) = hAPP(fun(X_b,bool),nat,finite_card(X_b),A_3) ) ) ) ).
fof(fact_396_card__Diff__singleton,axiom,
! [X_b,X_2,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_2),A_3))
=> hAPP(fun(X_b,bool),nat,finite_card(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_3),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_2),bot_bot(fun(X_b,bool))))) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(fun(X_b,bool),nat,finite_card(X_b),A_3)),one_one(nat)) ) ) ).
fof(fact_397_card__Diff__insert,axiom,
! [X_b,B_2,A_2,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),A_2),A_3))
=> ( ~ hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),A_2),B_2))
=> hAPP(fun(X_b,bool),nat,finite_card(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_3),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_2),B_2))) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(fun(X_b,bool),nat,finite_card(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_3),B_2))),one_one(nat)) ) ) ) ).
fof(fact_398_one__reorient,axiom,
! [X_b] :
( one(X_b)
=> ! [X_2] :
( one_one(X_b) = ti(X_b,X_2)
<=> ti(X_b,X_2) = one_one(X_b) ) ) ).
fof(fact_399_One__nat__def,axiom,
one_one(nat) = hAPP(nat,nat,suc,zero_zero(nat)) ).
fof(fact_400_diff__Suc__1,axiom,
! [N_2] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,suc,N_2)),one_one(nat)) = N_2 ).
fof(fact_401_diff__Suc__eq__diff__pred,axiom,
! [M,N_2] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),hAPP(nat,nat,suc,N_2)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),one_one(nat))),N_2) ).
fof(fact_402_card_Oneutral,axiom,
! [X_b,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( ( ? [X] : hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X),A_3))
=> one_one(nat) = zero_zero(nat) )
=> hAPP(fun(X_b,bool),nat,finite_card(X_b),A_3) = zero_zero(nat) ) ) ).
fof(fact_403_one__neq__zero,axiom,
! [X_a] :
( zero_neq_one(X_a)
=> one_one(X_a) != zero_zero(X_a) ) ).
fof(fact_404_zero__neq__one,axiom,
! [X_a] :
( zero_neq_one(X_a)
=> zero_zero(X_a) != one_one(X_a) ) ).
fof(fact_405_card_Oremove,axiom,
! [X_b,X_2,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_2),A_3))
=> hAPP(fun(X_b,bool),nat,finite_card(X_b),A_3) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),hAPP(fun(X_b,bool),nat,finite_card(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_3),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_2),bot_bot(fun(X_b,bool)))))) ) ) ).
fof(fact_406_card_Oinsert__remove,axiom,
! [X_b,X_2,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> hAPP(fun(X_b,bool),nat,finite_card(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_2),A_3)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),hAPP(fun(X_b,bool),nat,finite_card(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_3),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_2),bot_bot(fun(X_b,bool)))))) ) ).
fof(fact_407_plus__nat_Oadd__0,axiom,
! [N_2] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),zero_zero(nat)),N_2) = N_2 ).
fof(fact_408_Nat_Oadd__0__right,axiom,
! [M] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),zero_zero(nat)) = M ).
fof(fact_409_add__is__0,axiom,
! [M_1,N_1] :
( hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M_1),N_1) = zero_zero(nat)
<=> ( M_1 = zero_zero(nat)
& N_1 = zero_zero(nat) ) ) ).
fof(fact_410_add__eq__self__zero,axiom,
! [M,N_2] :
( hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),N_2) = M
=> N_2 = zero_zero(nat) ) ).
fof(fact_411_add__Suc__right,axiom,
! [M,N_2] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),hAPP(nat,nat,suc,N_2)) = hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),N_2)) ).
fof(fact_412_add__Suc,axiom,
! [M,N_2] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,suc,M)),N_2) = hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),N_2)) ).
fof(fact_413_add__Suc__shift,axiom,
! [M,N_2] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,suc,M)),N_2) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),hAPP(nat,nat,suc,N_2)) ).
fof(fact_414_nat__add__commute,axiom,
! [M,N_2] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),N_2) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N_2),M) ).
fof(fact_415_nat__add__left__commute,axiom,
! [X_1,Y,Z] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),X_1),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_1),Z)) ).
fof(fact_416_nat__add__assoc,axiom,
! [M,N_2,K_2] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),N_2)),K_2) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N_2),K_2)) ).
fof(fact_417_nat__add__left__cancel,axiom,
! [K_1,M_1,N_1] :
( 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)
<=> M_1 = N_1 ) ).
fof(fact_418_nat__add__right__cancel,axiom,
! [M_1,K_1,N_1] :
( 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)
<=> M_1 = N_1 ) ).
fof(fact_419_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_420_add__left__cancel,axiom,
! [X_b] :
( cancel_semigroup_add(X_b)
=> ! [A_2,B_1,C_1] :
( hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),A_2),B_1) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),A_2),C_1)
<=> ti(X_b,B_1) = ti(X_b,C_1) ) ) ).
fof(fact_421_add__right__cancel,axiom,
! [X_b] :
( cancel_semigroup_add(X_b)
=> ! [B_1,A_2,C_1] :
( hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),B_1),A_2) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),C_1),A_2)
<=> ti(X_b,B_1) = ti(X_b,C_1) ) ) ).
fof(fact_422_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_423_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_424_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_425_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_426_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_427_double__zero__sym,axiom,
! [X_b] :
( linord219039673up_add(X_b)
=> ! [A_2] :
( zero_zero(X_b) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),A_2),A_2)
<=> ti(X_b,A_2) = zero_zero(X_b) ) ) ).
fof(fact_428_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_429_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_430_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_431_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_432_diff__add__inverse2,axiom,
! [M,N_2] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),N_2)),N_2) = M ).
fof(fact_433_diff__add__inverse,axiom,
! [N_2,M] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N_2),M)),N_2) = M ).
fof(fact_434_diff__diff__left,axiom,
! [I,J_1,K_2] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),I),J_1)),K_2) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),I),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),J_1),K_2)) ).
fof(fact_435_diff__cancel,axiom,
! [K_2,M,N_2] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),K_2),M)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),K_2),N_2)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),N_2) ).
fof(fact_436_diff__cancel2,axiom,
! [M,K_2,N_2] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),K_2)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N_2),K_2)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),N_2) ).
fof(fact_437_add__is__1,axiom,
! [M_1,N_1] :
( hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M_1),N_1) = hAPP(nat,nat,suc,zero_zero(nat))
<=> ( ( M_1 = hAPP(nat,nat,suc,zero_zero(nat))
& N_1 = zero_zero(nat) )
| ( M_1 = zero_zero(nat)
& N_1 = hAPP(nat,nat,suc,zero_zero(nat)) ) ) ) ).
fof(fact_438_one__is__add,axiom,
! [M_1,N_1] :
( hAPP(nat,nat,suc,zero_zero(nat)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M_1),N_1)
<=> ( ( M_1 = hAPP(nat,nat,suc,zero_zero(nat))
& N_1 = zero_zero(nat) )
| ( M_1 = zero_zero(nat)
& N_1 = hAPP(nat,nat,suc,zero_zero(nat)) ) ) ) ).
fof(fact_439_diff__add__0,axiom,
! [N_2,M] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N_2),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N_2),M)) = zero_zero(nat) ).
fof(fact_440_Suc__eq__plus1,axiom,
! [N_2] : hAPP(nat,nat,suc,N_2) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N_2),one_one(nat)) ).
fof(fact_441_Suc__eq__plus1__left,axiom,
! [N_2] : hAPP(nat,nat,suc,N_2) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),N_2) ).
fof(fact_442_card__Un__Int,axiom,
! [X_b,B_2,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),B_2))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(fun(X_b,bool),nat,finite_card(X_b),A_3)),hAPP(fun(X_b,bool),nat,finite_card(X_b),B_2)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(fun(X_b,bool),nat,finite_card(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),A_3),B_2))),hAPP(fun(X_b,bool),nat,finite_card(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_3),B_2))) ) ) ).
fof(fact_443_card_Ounion__inter,axiom,
! [X_b,B_2,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),B_2))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(fun(X_b,bool),nat,finite_card(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),A_3),B_2))),hAPP(fun(X_b,bool),nat,finite_card(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_3),B_2))) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(fun(X_b,bool),nat,finite_card(X_b),A_3)),hAPP(fun(X_b,bool),nat,finite_card(X_b),B_2)) ) ) ).
fof(fact_444_add__eq__if,axiom,
! [N_2,M] :
( ( M = zero_zero(nat)
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),N_2) = N_2 )
& ( M != zero_zero(nat)
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),N_2) = 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),one_one(nat))),N_2)) ) ) ).
fof(fact_445_card_Oinsert,axiom,
! [X_b,X_2,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( ~ hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_2),A_3))
=> hAPP(fun(X_b,bool),nat,finite_card(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_2),A_3)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),hAPP(fun(X_b,bool),nat,finite_card(X_b),A_3)) ) ) ).
fof(fact_446_card_Oeq__fold__g,axiom,
! [X_b,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> hAPP(fun(X_b,bool),nat,finite_card(X_b),A_3) = hAPP(fun(X_b,bool),nat,hAPP(nat,fun(fun(X_b,bool),nat),hAPP(fun(X_b,nat),fun(nat,fun(fun(X_b,bool),nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(X_b,nat),fun(nat,fun(fun(X_b,bool),nat))),finite_fold_image(nat,X_b),plus_plus(nat)),hAPP(nat,fun(X_b,nat),combk(nat,X_b),one_one(nat))),zero_zero(nat)),A_3) ) ).
fof(fact_447_card__def,axiom,
! [X_b,A_3] :
( ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> hAPP(fun(X_b,bool),nat,finite_card(X_b),A_3) = hAPP(fun(X_b,bool),nat,hAPP(nat,fun(fun(X_b,bool),nat),hAPP(fun(X_b,nat),fun(nat,fun(fun(X_b,bool),nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(X_b,nat),fun(nat,fun(fun(X_b,bool),nat))),finite_fold_image(nat,X_b),plus_plus(nat)),hAPP(nat,fun(X_b,nat),combk(nat,X_b),one_one(nat))),zero_zero(nat)),A_3) )
& ( ~ hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> hAPP(fun(X_b,bool),nat,finite_card(X_b),A_3) = zero_zero(nat) ) ) ).
fof(fact_448_card__Un__disjoint,axiom,
! [X_b,B_2,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),B_2))
=> ( hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_3),B_2) = bot_bot(fun(X_b,bool))
=> hAPP(fun(X_b,bool),nat,finite_card(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),A_3),B_2)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(fun(X_b,bool),nat,finite_card(X_b),A_3)),hAPP(fun(X_b,bool),nat,finite_card(X_b),B_2)) ) ) ) ).
fof(fact_449_card_Ounion__inter__neutral,axiom,
! [X_b,B_2,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),B_2))
=> ( ( ? [X] : hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_3),B_2)))
=> one_one(nat) = zero_zero(nat) )
=> hAPP(fun(X_b,bool),nat,finite_card(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),A_3),B_2)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(fun(X_b,bool),nat,finite_card(X_b),A_3)),hAPP(fun(X_b,bool),nat,finite_card(X_b),B_2)) ) ) ) ).
fof(fact_450_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_451_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_452_double__eq__0__iff,axiom,
! [X_b] :
( linord219039673up_add(X_b)
=> ! [A_2] :
( hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),A_2),A_2) = zero_zero(X_b)
<=> ti(X_b,A_2) = zero_zero(X_b) ) ) ).
fof(fact_453_add__0__iff,axiom,
! [X_b] :
( semiri456707255roduct(X_b)
=> ! [B_1,A_2] :
( ti(X_b,B_1) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),B_1),A_2)
<=> ti(X_b,A_2) = zero_zero(X_b) ) ) ).
fof(fact_454_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_455_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_1,C,D_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),C),D_1)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),C),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),D_1)) ) ).
fof(fact_456_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_1,C,D_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),C),D_1)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),C)),D_1) ) ).
fof(fact_457_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_458_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_459_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_1,B,C,D_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),C),D_1)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),C)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),B),D_1)) ) ).
fof(fact_460_com_Osize_I7_J,axiom,
! [Pname_1] : hAPP(com,nat,com_size,hAPP(pname,com,body,Pname_1)) = zero_zero(nat) ).
fof(fact_461_com_Osize_I1_J,axiom,
hAPP(com,nat,com_size,skip) = zero_zero(nat) ).
fof(fact_462_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_463_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_464_card__Plus__conv__if,axiom,
! [X_c,X_b,B_2,A_3] :
( ( ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
& hBOOL(hAPP(fun(X_c,bool),bool,finite_finite_1(X_c),B_2)) )
=> hAPP(fun(sum_sum(X_b,X_c),bool),nat,finite_card(sum_sum(X_b,X_c)),hAPP(fun(X_c,bool),fun(sum_sum(X_b,X_c),bool),hAPP(fun(X_b,bool),fun(fun(X_c,bool),fun(sum_sum(X_b,X_c),bool)),sum_Plus(X_b,X_c),A_3),B_2)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(fun(X_b,bool),nat,finite_card(X_b),A_3)),hAPP(fun(X_c,bool),nat,finite_card(X_c),B_2)) )
& ( ~ ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
& hBOOL(hAPP(fun(X_c,bool),bool,finite_finite_1(X_c),B_2)) )
=> hAPP(fun(sum_sum(X_b,X_c),bool),nat,finite_card(sum_sum(X_b,X_c)),hAPP(fun(X_c,bool),fun(sum_sum(X_b,X_c),bool),hAPP(fun(X_b,bool),fun(fun(X_c,bool),fun(sum_sum(X_b,X_c),bool)),sum_Plus(X_b,X_c),A_3),B_2)) = zero_zero(nat) ) ) ).
fof(fact_465_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_466_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_467_fold__Un__disjoint,axiom,
! [X_b,X_c] :
( comm_monoid_mult(X_c)
=> ! [G,B_2,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),B_2))
=> ( hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_3),B_2) = bot_bot(fun(X_b,bool))
=> hAPP(fun(X_b,bool),X_c,hAPP(X_c,fun(fun(X_b,bool),X_c),hAPP(fun(X_b,X_c),fun(X_c,fun(fun(X_b,bool),X_c)),hAPP(fun(X_c,fun(X_c,X_c)),fun(fun(X_b,X_c),fun(X_c,fun(fun(X_b,bool),X_c))),finite_fold_image(X_c,X_b),times_times(X_c)),G),one_one(X_c)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),A_3),B_2)) = hAPP(X_c,X_c,hAPP(X_c,fun(X_c,X_c),times_times(X_c),hAPP(fun(X_b,bool),X_c,hAPP(X_c,fun(fun(X_b,bool),X_c),hAPP(fun(X_b,X_c),fun(X_c,fun(fun(X_b,bool),X_c)),hAPP(fun(X_c,fun(X_c,X_c)),fun(fun(X_b,X_c),fun(X_c,fun(fun(X_b,bool),X_c))),finite_fold_image(X_c,X_b),times_times(X_c)),G),one_one(X_c)),A_3)),hAPP(fun(X_b,bool),X_c,hAPP(X_c,fun(fun(X_b,bool),X_c),hAPP(fun(X_b,X_c),fun(X_c,fun(fun(X_b,bool),X_c)),hAPP(fun(X_c,fun(X_c,X_c)),fun(fun(X_b,X_c),fun(X_c,fun(fun(X_b,bool),X_c))),finite_fold_image(X_c,X_b),times_times(X_c)),G),one_one(X_c)),B_2)) ) ) ) ) ).
fof(fact_468_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_469_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_470_add__mult__distrib2,axiom,
! [K_2,M,N_2] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K_2),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),N_2)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K_2),M)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K_2),N_2)) ).
fof(fact_471_add__mult__distrib,axiom,
! [M,N_2,K_2] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),N_2)),K_2) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),K_2)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),N_2),K_2)) ).
fof(fact_472_crossproduct__eq,axiom,
! [X_b] :
( semiri456707255roduct(X_b)
=> ! [W,Y_1,X_2,Z_1] :
( hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),W),Y_1)),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),X_2),Z_1)) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),W),Z_1)),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),X_2),Y_1))
<=> ( ti(X_b,W) = ti(X_b,X_2)
| ti(X_b,Y_1) = ti(X_b,Z_1) ) ) ) ).
fof(fact_473_combine__common__factor,axiom,
! [X_a] :
( semiring(X_a)
=> ! [A_1,E_1,B,C] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),E_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),E_1)),C)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B)),E_1)),C) ) ).
fof(fact_474_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_1,M,B] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),M)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),M)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B)),M) ) ).
fof(fact_475_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_476_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_477_crossproduct__noteq,axiom,
! [X_b] :
( semiri456707255roduct(X_b)
=> ! [C_1,D,A_2,B_1] :
( ( ti(X_b,A_2) != ti(X_b,B_1)
& ti(X_b,C_1) != ti(X_b,D) )
<=> hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),A_2),C_1)),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),B_1),D)) != hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),A_2),D)),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),B_1),C_1)) ) ) ).
fof(fact_478_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [X_1,Y,Z] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X_1),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_1),Y)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X_1),Z)) ) ).
fof(fact_479_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_480_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_481_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_482_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_483_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_484_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_485_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_486_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_487_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_488_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_489_nat__mult__eq__1__iff,axiom,
! [M_1,N_1] :
( hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M_1),N_1) = one_one(nat)
<=> ( M_1 = one_one(nat)
& N_1 = one_one(nat) ) ) ).
fof(fact_490_nat__mult__1__right,axiom,
! [N_2] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),N_2),one_one(nat)) = N_2 ).
fof(fact_491_nat__1__eq__mult__iff,axiom,
! [M_1,N_1] :
( one_one(nat) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M_1),N_1)
<=> ( M_1 = one_one(nat)
& N_1 = one_one(nat) ) ) ).
fof(fact_492_nat__mult__1,axiom,
! [N_2] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),one_one(nat)),N_2) = N_2 ).
fof(fact_493_mult__0,axiom,
! [N_2] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),zero_zero(nat)),N_2) = zero_zero(nat) ).
fof(fact_494_mult__0__right,axiom,
! [M] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),zero_zero(nat)) = zero_zero(nat) ).
fof(fact_495_mult__is__0,axiom,
! [M_1,N_1] :
( hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M_1),N_1) = zero_zero(nat)
<=> ( M_1 = zero_zero(nat)
| N_1 = zero_zero(nat) ) ) ).
fof(fact_496_mult__cancel1,axiom,
! [K_1,M_1,N_1] :
( 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)
<=> ( M_1 = N_1
| K_1 = zero_zero(nat) ) ) ).
fof(fact_497_mult__cancel2,axiom,
! [M_1,K_1,N_1] :
( 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)
<=> ( M_1 = N_1
| K_1 = zero_zero(nat) ) ) ).
fof(fact_498_Suc__mult__cancel1,axiom,
! [K_1,M_1,N_1] :
( hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,suc,K_1)),M_1) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,suc,K_1)),N_1)
<=> M_1 = N_1 ) ).
fof(fact_499_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_500_mult__idem,axiom,
! [X_a] :
( ab_sem1668676832m_mult(X_a)
=> ! [X_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X_1),X_1) = ti(X_a,X_1) ) ).
fof(fact_501_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_502_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_503_diff__mult__distrib,axiom,
! [M,N_2,K_2] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),N_2)),K_2) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),K_2)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),N_2),K_2)) ).
fof(fact_504_diff__mult__distrib2,axiom,
! [K_2,M,N_2] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K_2),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),N_2)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K_2),M)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K_2),N_2)) ).
fof(fact_505_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_506_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_507_mult__eq__0__iff,axiom,
! [X_b] :
( ring_n68954251visors(X_b)
=> ! [A_2,B_1] :
( hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),A_2),B_1) = zero_zero(X_b)
<=> ( ti(X_b,A_2) = zero_zero(X_b)
| ti(X_b,B_1) = zero_zero(X_b) ) ) ) ).
fof(fact_508_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_509_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_510_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_511_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_512_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_513_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_514_comp__fun__idem,axiom,
! [X_b] :
( ab_sem1668676832m_mult(X_b)
=> hBOOL(hAPP(fun(X_b,fun(X_b,X_b)),bool,finite_comp_fun_idem(X_b,X_b),times_times(X_b))) ) ).
fof(fact_515_finite__PlusD_I2_J,axiom,
! [X_b,X_c,A_3,B_2] :
( hBOOL(hAPP(fun(sum_sum(X_b,X_c),bool),bool,finite_finite_1(sum_sum(X_b,X_c)),hAPP(fun(X_c,bool),fun(sum_sum(X_b,X_c),bool),hAPP(fun(X_b,bool),fun(fun(X_c,bool),fun(sum_sum(X_b,X_c),bool)),sum_Plus(X_b,X_c),A_3),B_2)))
=> hBOOL(hAPP(fun(X_c,bool),bool,finite_finite_1(X_c),B_2)) ) ).
fof(fact_516_finite__PlusD_I1_J,axiom,
! [X_b,X_c,A_3,B_2] :
( hBOOL(hAPP(fun(sum_sum(X_b,X_c),bool),bool,finite_finite_1(sum_sum(X_b,X_c)),hAPP(fun(X_c,bool),fun(sum_sum(X_b,X_c),bool),hAPP(fun(X_b,bool),fun(fun(X_c,bool),fun(sum_sum(X_b,X_c),bool)),sum_Plus(X_b,X_c),A_3),B_2)))
=> hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3)) ) ).
fof(fact_517_finite__Plus,axiom,
! [X_c,X_b,B_2,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( hBOOL(hAPP(fun(X_c,bool),bool,finite_finite_1(X_c),B_2))
=> hBOOL(hAPP(fun(sum_sum(X_b,X_c),bool),bool,finite_finite_1(sum_sum(X_b,X_c)),hAPP(fun(X_c,bool),fun(sum_sum(X_b,X_c),bool),hAPP(fun(X_b,bool),fun(fun(X_c,bool),fun(sum_sum(X_b,X_c),bool)),sum_Plus(X_b,X_c),A_3),B_2))) ) ) ).
fof(fact_518_finite__Plus__iff,axiom,
! [X_b,X_c,A_3,B_2] :
( hBOOL(hAPP(fun(sum_sum(X_b,X_c),bool),bool,finite_finite_1(sum_sum(X_b,X_c)),hAPP(fun(X_c,bool),fun(sum_sum(X_b,X_c),bool),hAPP(fun(X_b,bool),fun(fun(X_c,bool),fun(sum_sum(X_b,X_c),bool)),sum_Plus(X_b,X_c),A_3),B_2)))
<=> ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
& hBOOL(hAPP(fun(X_c,bool),bool,finite_finite_1(X_c),B_2)) ) ) ).
fof(fact_519_sum__squares__eq__zero__iff,axiom,
! [X_b] :
( linord581940658strict(X_b)
=> ! [X_2,Y_1] :
( 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),X_2),X_2)),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),Y_1),Y_1)) = zero_zero(X_b)
<=> ( ti(X_b,X_2) = zero_zero(X_b)
& ti(X_b,Y_1) = zero_zero(X_b) ) ) ) ).
fof(fact_520_add__scale__eq__noteq,axiom,
! [X_a] :
( semiri456707255roduct(X_a)
=> ! [C,D_1,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_1) )
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),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_1)) ) ) ) ).
fof(fact_521_eq__add__iff2,axiom,
! [X_b] :
( ring(X_b)
=> ! [A_2,E,C_1,B_1,D] :
( hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),A_2),E)),C_1) = 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),B_1),E)),D)
<=> ti(X_b,C_1) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),minus_minus(X_b),B_1),A_2)),E)),D) ) ) ).
fof(fact_522_eq__add__iff1,axiom,
! [X_b] :
( ring(X_b)
=> ! [A_2,E,C_1,B_1,D] :
( hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),A_2),E)),C_1) = 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),B_1),E)),D)
<=> hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),minus_minus(X_b),A_2),B_1)),E)),C_1) = ti(X_b,D) ) ) ).
fof(fact_523_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [M] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),M),M) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),one_one(X_a)),one_one(X_a))),M) ) ).
fof(fact_524_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [M,A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),M),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),M)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),one_one(X_a))),M) ) ).
fof(fact_525_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_1,M] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),M)),M) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),one_one(X_a))),M) ) ).
fof(fact_526_mult__eq__1__iff,axiom,
! [M_1,N_1] :
( hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M_1),N_1) = hAPP(nat,nat,suc,zero_zero(nat))
<=> ( M_1 = hAPP(nat,nat,suc,zero_zero(nat))
& N_1 = hAPP(nat,nat,suc,zero_zero(nat)) ) ) ).
fof(fact_527_mult__Suc,axiom,
! [M,N_2] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,suc,M)),N_2) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N_2),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),N_2)) ).
fof(fact_528_mult__Suc__right,axiom,
! [M,N_2] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),hAPP(nat,nat,suc,N_2)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),N_2)) ).
fof(fact_529_mult__eq__self__implies__10,axiom,
! [M,N_2] :
( M = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),N_2)
=> ( N_2 = one_one(nat)
| M = zero_zero(nat) ) ) ).
fof(fact_530_fold__image__insert,axiom,
! [X_b,X_c] :
( ab_semigroup_mult(X_c)
=> ! [G,Z_1,A_2,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( ~ hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),A_2),A_3))
=> hAPP(fun(X_b,bool),X_c,hAPP(X_c,fun(fun(X_b,bool),X_c),hAPP(fun(X_b,X_c),fun(X_c,fun(fun(X_b,bool),X_c)),hAPP(fun(X_c,fun(X_c,X_c)),fun(fun(X_b,X_c),fun(X_c,fun(fun(X_b,bool),X_c))),finite_fold_image(X_c,X_b),times_times(X_c)),G),Z_1),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_2),A_3)) = hAPP(X_c,X_c,hAPP(X_c,fun(X_c,X_c),times_times(X_c),hAPP(X_b,X_c,G,A_2)),hAPP(fun(X_b,bool),X_c,hAPP(X_c,fun(fun(X_b,bool),X_c),hAPP(fun(X_b,X_c),fun(X_c,fun(fun(X_b,bool),X_c)),hAPP(fun(X_c,fun(X_c,X_c)),fun(fun(X_b,X_c),fun(X_c,fun(fun(X_b,bool),X_c))),finite_fold_image(X_c,X_b),times_times(X_c)),G),Z_1),A_3)) ) ) ) ).
fof(fact_531_com_Osize_I15_J,axiom,
! [Pname_1] : hAPP(com,nat,size_size(com),hAPP(pname,com,body,Pname_1)) = zero_zero(nat) ).
fof(fact_532_fold__image__distrib,axiom,
! [X_b,X_c] :
( comm_monoid_mult(X_c)
=> ! [G,H,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> hAPP(fun(X_b,bool),X_c,hAPP(X_c,fun(fun(X_b,bool),X_c),hAPP(fun(X_b,X_c),fun(X_c,fun(fun(X_b,bool),X_c)),hAPP(fun(X_c,fun(X_c,X_c)),fun(fun(X_b,X_c),fun(X_c,fun(fun(X_b,bool),X_c))),finite_fold_image(X_c,X_b),times_times(X_c)),hAPP(fun(X_b,X_c),fun(X_b,X_c),hAPP(fun(X_b,fun(X_c,X_c)),fun(fun(X_b,X_c),fun(X_b,X_c)),combs(X_b,X_c,X_c),hAPP(fun(X_b,X_c),fun(X_b,fun(X_c,X_c)),hAPP(fun(X_c,fun(X_c,X_c)),fun(fun(X_b,X_c),fun(X_b,fun(X_c,X_c))),combb(X_c,fun(X_c,X_c),X_b),times_times(X_c)),G)),H)),one_one(X_c)),A_3) = hAPP(X_c,X_c,hAPP(X_c,fun(X_c,X_c),times_times(X_c),hAPP(fun(X_b,bool),X_c,hAPP(X_c,fun(fun(X_b,bool),X_c),hAPP(fun(X_b,X_c),fun(X_c,fun(fun(X_b,bool),X_c)),hAPP(fun(X_c,fun(X_c,X_c)),fun(fun(X_b,X_c),fun(X_c,fun(fun(X_b,bool),X_c))),finite_fold_image(X_c,X_b),times_times(X_c)),G),one_one(X_c)),A_3)),hAPP(fun(X_b,bool),X_c,hAPP(X_c,fun(fun(X_b,bool),X_c),hAPP(fun(X_b,X_c),fun(X_c,fun(fun(X_b,bool),X_c)),hAPP(fun(X_c,fun(X_c,X_c)),fun(fun(X_b,X_c),fun(X_c,fun(fun(X_b,bool),X_c))),finite_fold_image(X_c,X_b),times_times(X_c)),H),one_one(X_c)),A_3)) ) ) ).
fof(fact_533_com_Osize_I9_J,axiom,
hAPP(com,nat,size_size(com),skip) = zero_zero(nat) ).
fof(fact_534_mult__eq__if,axiom,
! [N_2,M] :
( ( M = zero_zero(nat)
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),N_2) = zero_zero(nat) )
& ( M != zero_zero(nat)
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),N_2) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N_2),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),one_one(nat))),N_2)) ) ) ).
fof(fact_535_card__Plus,axiom,
! [X_c,X_b,B_2,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( hBOOL(hAPP(fun(X_c,bool),bool,finite_finite_1(X_c),B_2))
=> hAPP(fun(sum_sum(X_b,X_c),bool),nat,finite_card(sum_sum(X_b,X_c)),hAPP(fun(X_c,bool),fun(sum_sum(X_b,X_c),bool),hAPP(fun(X_b,bool),fun(fun(X_c,bool),fun(sum_sum(X_b,X_c),bool)),sum_Plus(X_b,X_c),A_3),B_2)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(fun(X_b,bool),nat,finite_card(X_b),A_3)),hAPP(fun(X_c,bool),nat,finite_card(X_c),B_2)) ) ) ).
fof(fact_536_fold__image__Un__Int,axiom,
! [X_b,X_c] :
( comm_monoid_mult(X_c)
=> ! [G,B_2,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),B_2))
=> hAPP(X_c,X_c,hAPP(X_c,fun(X_c,X_c),times_times(X_c),hAPP(fun(X_b,bool),X_c,hAPP(X_c,fun(fun(X_b,bool),X_c),hAPP(fun(X_b,X_c),fun(X_c,fun(fun(X_b,bool),X_c)),hAPP(fun(X_c,fun(X_c,X_c)),fun(fun(X_b,X_c),fun(X_c,fun(fun(X_b,bool),X_c))),finite_fold_image(X_c,X_b),times_times(X_c)),G),one_one(X_c)),A_3)),hAPP(fun(X_b,bool),X_c,hAPP(X_c,fun(fun(X_b,bool),X_c),hAPP(fun(X_b,X_c),fun(X_c,fun(fun(X_b,bool),X_c)),hAPP(fun(X_c,fun(X_c,X_c)),fun(fun(X_b,X_c),fun(X_c,fun(fun(X_b,bool),X_c))),finite_fold_image(X_c,X_b),times_times(X_c)),G),one_one(X_c)),B_2)) = hAPP(X_c,X_c,hAPP(X_c,fun(X_c,X_c),times_times(X_c),hAPP(fun(X_b,bool),X_c,hAPP(X_c,fun(fun(X_b,bool),X_c),hAPP(fun(X_b,X_c),fun(X_c,fun(fun(X_b,bool),X_c)),hAPP(fun(X_c,fun(X_c,X_c)),fun(fun(X_b,X_c),fun(X_c,fun(fun(X_b,bool),X_c))),finite_fold_image(X_c,X_b),times_times(X_c)),G),one_one(X_c)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),A_3),B_2))),hAPP(fun(X_b,bool),X_c,hAPP(X_c,fun(fun(X_b,bool),X_c),hAPP(fun(X_b,X_c),fun(X_c,fun(fun(X_b,bool),X_c)),hAPP(fun(X_c,fun(X_c,X_c)),fun(fun(X_b,X_c),fun(X_c,fun(fun(X_b,bool),X_c))),finite_fold_image(X_c,X_b),times_times(X_c)),G),one_one(X_c)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_3),B_2))) ) ) ) ).
fof(fact_537_fold__image__Un__one,axiom,
! [X_b,X_c] :
( comm_monoid_mult(X_c)
=> ! [F,T_5,S_1] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),S_1))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),T_5))
=> ( ! [X] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),S_1),T_5)))
=> hAPP(X_b,X_c,F,X) = one_one(X_c) )
=> hAPP(fun(X_b,bool),X_c,hAPP(X_c,fun(fun(X_b,bool),X_c),hAPP(fun(X_b,X_c),fun(X_c,fun(fun(X_b,bool),X_c)),hAPP(fun(X_c,fun(X_c,X_c)),fun(fun(X_b,X_c),fun(X_c,fun(fun(X_b,bool),X_c))),finite_fold_image(X_c,X_b),times_times(X_c)),F),one_one(X_c)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),S_1),T_5)) = hAPP(X_c,X_c,hAPP(X_c,fun(X_c,X_c),times_times(X_c),hAPP(fun(X_b,bool),X_c,hAPP(X_c,fun(fun(X_b,bool),X_c),hAPP(fun(X_b,X_c),fun(X_c,fun(fun(X_b,bool),X_c)),hAPP(fun(X_c,fun(X_c,X_c)),fun(fun(X_b,X_c),fun(X_c,fun(fun(X_b,bool),X_c))),finite_fold_image(X_c,X_b),times_times(X_c)),F),one_one(X_c)),S_1)),hAPP(fun(X_b,bool),X_c,hAPP(X_c,fun(fun(X_b,bool),X_c),hAPP(fun(X_b,X_c),fun(X_c,fun(fun(X_b,bool),X_c)),hAPP(fun(X_c,fun(X_c,X_c)),fun(fun(X_b,X_c),fun(X_c,fun(fun(X_b,bool),X_c))),finite_fold_image(X_c,X_b),times_times(X_c)),F),one_one(X_c)),T_5)) ) ) ) ) ).
fof(fact_538_fold__image__1,axiom,
! [X_b,X_c] :
( comm_monoid_mult(X_c)
=> ! [F,S_1] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),S_1))
=> ( ! [X] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X),S_1))
=> hAPP(X_b,X_c,F,X) = one_one(X_c) )
=> hAPP(fun(X_b,bool),X_c,hAPP(X_c,fun(fun(X_b,bool),X_c),hAPP(fun(X_b,X_c),fun(X_c,fun(fun(X_b,bool),X_c)),hAPP(fun(X_c,fun(X_c,X_c)),fun(fun(X_b,X_c),fun(X_c,fun(fun(X_b,bool),X_c))),finite_fold_image(X_c,X_b),times_times(X_c)),F),one_one(X_c)),S_1) = one_one(X_c) ) ) ) ).
fof(fact_539_fold__image__cong,axiom,
! [X_b,X_c] :
( ab_semigroup_mult(X_c)
=> ! [Z_1,G,H,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( ! [X] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X),A_3))
=> hAPP(X_b,X_c,G,X) = hAPP(X_b,X_c,H,X) )
=> hAPP(fun(X_b,bool),X_c,hAPP(X_c,fun(fun(X_b,bool),X_c),hAPP(fun(X_b,X_c),fun(X_c,fun(fun(X_b,bool),X_c)),hAPP(fun(X_c,fun(X_c,X_c)),fun(fun(X_b,X_c),fun(X_c,fun(fun(X_b,bool),X_c))),finite_fold_image(X_c,X_b),times_times(X_c)),G),Z_1),A_3) = hAPP(fun(X_b,bool),X_c,hAPP(X_c,fun(fun(X_b,bool),X_c),hAPP(fun(X_b,X_c),fun(X_c,fun(fun(X_b,bool),X_c)),hAPP(fun(X_c,fun(X_c,X_c)),fun(fun(X_b,X_c),fun(X_c,fun(fun(X_b,bool),X_c))),finite_fold_image(X_c,X_b),times_times(X_c)),H),Z_1),A_3) ) ) ) ).
fof(fact_540_Plus__eq__empty__conv,axiom,
! [X_c,X_b,A_3,B_2] :
( hAPP(fun(X_c,bool),fun(sum_sum(X_b,X_c),bool),hAPP(fun(X_b,bool),fun(fun(X_c,bool),fun(sum_sum(X_b,X_c),bool)),sum_Plus(X_b,X_c),A_3),B_2) = bot_bot(fun(sum_sum(X_b,X_c),bool))
<=> ( ti(fun(X_b,bool),A_3) = bot_bot(fun(X_b,bool))
& ti(fun(X_c,bool),B_2) = bot_bot(fun(X_c,bool)) ) ) ).
fof(fact_541_nat__mult__commute,axiom,
! [M,N_2] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),N_2) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),N_2),M) ).
fof(fact_542_nat__mult__assoc,axiom,
! [M,N_2,K_2] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),N_2)),K_2) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),N_2),K_2)) ).
fof(fact_543_left__add__mult__distrib,axiom,
! [I,U_1,J_1,K_2] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),I),U_1)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),J_1),U_1)),K_2)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),I),J_1)),U_1)),K_2) ).
fof(fact_544_nat__mult__eq__cancel__disj,axiom,
! [K_1,M_1,N_1] :
( 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)
<=> ( K_1 = zero_zero(nat)
| M_1 = N_1 ) ) ).
fof(fact_545_fold__image__eq__general__inverses,axiom,
! [X_c,X_b,X_d] :
( comm_monoid_mult(X_d)
=> ! [E,G,F,H,K_1,T_5,S_1] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),S_1))
=> ( ! [Y_2] :
( hBOOL(hAPP(fun(X_c,bool),bool,hAPP(X_c,fun(fun(X_c,bool),bool),member(X_c),Y_2),T_5))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),hAPP(X_c,X_b,K_1,Y_2)),S_1))
& hAPP(X_b,X_c,H,hAPP(X_c,X_b,K_1,Y_2)) = ti(X_c,Y_2) ) )
=> ( ! [X] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X),S_1))
=> ( hBOOL(hAPP(fun(X_c,bool),bool,hAPP(X_c,fun(fun(X_c,bool),bool),member(X_c),hAPP(X_b,X_c,H,X)),T_5))
& hAPP(X_c,X_b,K_1,hAPP(X_b,X_c,H,X)) = ti(X_b,X)
& hAPP(X_c,X_d,G,hAPP(X_b,X_c,H,X)) = hAPP(X_b,X_d,F,X) ) )
=> hAPP(fun(X_b,bool),X_d,hAPP(X_d,fun(fun(X_b,bool),X_d),hAPP(fun(X_b,X_d),fun(X_d,fun(fun(X_b,bool),X_d)),hAPP(fun(X_d,fun(X_d,X_d)),fun(fun(X_b,X_d),fun(X_d,fun(fun(X_b,bool),X_d))),finite_fold_image(X_d,X_b),times_times(X_d)),F),E),S_1) = hAPP(fun(X_c,bool),X_d,hAPP(X_d,fun(fun(X_c,bool),X_d),hAPP(fun(X_c,X_d),fun(X_d,fun(fun(X_c,bool),X_d)),hAPP(fun(X_d,fun(X_d,X_d)),fun(fun(X_c,X_d),fun(X_d,fun(fun(X_c,bool),X_d))),finite_fold_image(X_d,X_c),times_times(X_d)),G),E),T_5) ) ) ) ) ).
fof(fact_546_fold__image__related,axiom,
! [X_c,X_b] :
( comm_monoid_mult(X_b)
=> ! [H,G,S_1,R_1,E] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),R_1,E),E))
=> ( ! [X1,Y1,X2,Y2] :
( ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),R_1,X1),X2))
& hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),R_1,Y1),Y2)) )
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),R_1,hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),X1),Y1)),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),X2),Y2))) )
=> ( hBOOL(hAPP(fun(X_c,bool),bool,finite_finite_1(X_c),S_1))
=> ( ! [X] :
( hBOOL(hAPP(fun(X_c,bool),bool,hAPP(X_c,fun(fun(X_c,bool),bool),member(X_c),X),S_1))
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),R_1,hAPP(X_c,X_b,H,X)),hAPP(X_c,X_b,G,X))) )
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),R_1,hAPP(fun(X_c,bool),X_b,hAPP(X_b,fun(fun(X_c,bool),X_b),hAPP(fun(X_c,X_b),fun(X_b,fun(fun(X_c,bool),X_b)),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_c,X_b),fun(X_b,fun(fun(X_c,bool),X_b))),finite_fold_image(X_b,X_c),times_times(X_b)),H),E),S_1)),hAPP(fun(X_c,bool),X_b,hAPP(X_b,fun(fun(X_c,bool),X_b),hAPP(fun(X_c,X_b),fun(X_b,fun(fun(X_c,bool),X_b)),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_c,X_b),fun(X_b,fun(fun(X_c,bool),X_b))),finite_fold_image(X_b,X_c),times_times(X_b)),G),E),S_1))) ) ) ) ) ) ).
fof(fact_547_fold1__Un,axiom,
! [X_b] :
( ab_semigroup_mult(X_b)
=> ! [B_2,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( ti(fun(X_b,bool),A_3) != bot_bot(fun(X_b,bool))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),B_2))
=> ( ti(fun(X_b,bool),B_2) != bot_bot(fun(X_b,bool))
=> ( hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_3),B_2) = bot_bot(fun(X_b,bool))
=> hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),finite_fold1(X_b),times_times(X_b)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),A_3),B_2)) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),finite_fold1(X_b),times_times(X_b)),A_3)),hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),finite_fold1(X_b),times_times(X_b)),B_2)) ) ) ) ) ) ) ).
fof(fact_548_com_Osize_I13_J,axiom,
! [Fun_1,Com1_1,Com2_1] : 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_1),Com2_1)) = 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_1)),hAPP(com,nat,size_size(com),Com2_1))),hAPP(nat,nat,suc,zero_zero(nat))) ).
fof(fact_549_com_Osize_I5_J,axiom,
! [Fun_1,Com1_1,Com2_1] : 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_1),Com2_1)) = 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_1)),hAPP(com,nat,com_size,Com2_1))),hAPP(nat,nat,suc,zero_zero(nat))) ).
fof(fact_550_card__Diff1__le,axiom,
! [X_b,X_2,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(fun(X_b,bool),nat,finite_card(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_3),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_2),bot_bot(fun(X_b,bool)))))),hAPP(fun(X_b,bool),nat,finite_card(X_b),A_3))) ) ).
fof(fact_551_order__refl,axiom,
! [X_a] :
( preorder(X_a)
=> ! [X_1] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_1),X_1)) ) ).
fof(fact_552_finite__Collect__le__nat,axiom,
! [K_1] : hBOOL(hAPP(fun(nat,bool),bool,finite_finite_1(nat),hAPP(fun(nat,bool),fun(nat,bool),collect(nat),hAPP(nat,fun(nat,bool),hAPP(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool)),combc(nat,nat,bool),ord_less_eq(nat)),K_1)))) ).
fof(fact_553_le0,axiom,
! [N_2] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),N_2)) ).
fof(fact_554_evaln__elim__cases_I5_J,axiom,
! [B_1,C1,C2,S_2,N_1,T_4] :
( hBOOL(hAPP(state,bool,hAPP(nat,fun(state,bool),hAPP(state,fun(nat,fun(state,bool)),hAPP(com,fun(state,fun(nat,fun(state,bool))),evaln,hAPP(com,com,hAPP(com,fun(com,com),hAPP(fun(state,bool),fun(com,fun(com,com)),cond,B_1),C1),C2)),S_2),N_1),T_4))
=> ( ( hBOOL(hAPP(state,bool,B_1,S_2))
=> ~ hBOOL(hAPP(state,bool,hAPP(nat,fun(state,bool),hAPP(state,fun(nat,fun(state,bool)),hAPP(com,fun(state,fun(nat,fun(state,bool))),evaln,C1),S_2),N_1),T_4)) )
=> ~ ( ~ hBOOL(hAPP(state,bool,B_1,S_2))
=> ~ hBOOL(hAPP(state,bool,hAPP(nat,fun(state,bool),hAPP(state,fun(nat,fun(state,bool)),hAPP(com,fun(state,fun(nat,fun(state,bool))),evaln,C2),S_2),N_1),T_4)) ) ) ) ).
fof(fact_555_evaln_OIfTrue,axiom,
! [C1,C0,N_1,S1,B_1,S_2] :
( hBOOL(hAPP(state,bool,B_1,S_2))
=> ( hBOOL(hAPP(state,bool,hAPP(nat,fun(state,bool),hAPP(state,fun(nat,fun(state,bool)),hAPP(com,fun(state,fun(nat,fun(state,bool))),evaln,C0),S_2),N_1),S1))
=> hBOOL(hAPP(state,bool,hAPP(nat,fun(state,bool),hAPP(state,fun(nat,fun(state,bool)),hAPP(com,fun(state,fun(nat,fun(state,bool))),evaln,hAPP(com,com,hAPP(com,fun(com,com),hAPP(fun(state,bool),fun(com,fun(com,com)),cond,B_1),C0),C1)),S_2),N_1),S1)) ) ) ).
fof(fact_556_evaln_OIfFalse,axiom,
! [C0,C1,N_1,S1,B_1,S_2] :
( ~ hBOOL(hAPP(state,bool,B_1,S_2))
=> ( hBOOL(hAPP(state,bool,hAPP(nat,fun(state,bool),hAPP(state,fun(nat,fun(state,bool)),hAPP(com,fun(state,fun(nat,fun(state,bool))),evaln,C1),S_2),N_1),S1))
=> hBOOL(hAPP(state,bool,hAPP(nat,fun(state,bool),hAPP(state,fun(nat,fun(state,bool)),hAPP(com,fun(state,fun(nat,fun(state,bool))),evaln,hAPP(com,com,hAPP(com,fun(com,com),hAPP(fun(state,bool),fun(com,fun(com,com)),cond,B_1),C0),C1)),S_2),N_1),S1)) ) ) ).
fof(fact_557_evalc_OIfFalse,axiom,
! [C0,C1,S1,B_1,S_2] :
( ~ hBOOL(hAPP(state,bool,B_1,S_2))
=> ( hBOOL(hAPP(state,bool,hAPP(state,fun(state,bool),hAPP(com,fun(state,fun(state,bool)),evalc,C1),S_2),S1))
=> hBOOL(hAPP(state,bool,hAPP(state,fun(state,bool),hAPP(com,fun(state,fun(state,bool)),evalc,hAPP(com,com,hAPP(com,fun(com,com),hAPP(fun(state,bool),fun(com,fun(com,com)),cond,B_1),C0),C1)),S_2),S1)) ) ) ).
fof(fact_558_evalc_OIfTrue,axiom,
! [C1,C0,S1,B_1,S_2] :
( hBOOL(hAPP(state,bool,B_1,S_2))
=> ( hBOOL(hAPP(state,bool,hAPP(state,fun(state,bool),hAPP(com,fun(state,fun(state,bool)),evalc,C0),S_2),S1))
=> hBOOL(hAPP(state,bool,hAPP(state,fun(state,bool),hAPP(com,fun(state,fun(state,bool)),evalc,hAPP(com,com,hAPP(com,fun(com,com),hAPP(fun(state,bool),fun(com,fun(com,com)),cond,B_1),C0),C1)),S_2),S1)) ) ) ).
fof(fact_559_evalc__elim__cases_I5_J,axiom,
! [B_1,C1,C2,S_2,T_4] :
( hBOOL(hAPP(state,bool,hAPP(state,fun(state,bool),hAPP(com,fun(state,fun(state,bool)),evalc,hAPP(com,com,hAPP(com,fun(com,com),hAPP(fun(state,bool),fun(com,fun(com,com)),cond,B_1),C1),C2)),S_2),T_4))
=> ( ( hBOOL(hAPP(state,bool,B_1,S_2))
=> ~ hBOOL(hAPP(state,bool,hAPP(state,fun(state,bool),hAPP(com,fun(state,fun(state,bool)),evalc,C1),S_2),T_4)) )
=> ~ ( ~ hBOOL(hAPP(state,bool,B_1,S_2))
=> ~ hBOOL(hAPP(state,bool,hAPP(state,fun(state,bool),hAPP(com,fun(state,fun(state,bool)),evalc,C2),S_2),T_4)) ) ) ) ).
fof(fact_560_com_Osimps_I55_J,axiom,
! [Pname,Fun_1,Com1_1,Com2_1] : hAPP(pname,com,body,Pname) != hAPP(com,com,hAPP(com,fun(com,com),hAPP(fun(state,bool),fun(com,fun(com,com)),cond,Fun_1),Com1_1),Com2_1) ).
fof(fact_561_com_Osimps_I54_J,axiom,
! [Fun_1,Com1_1,Com2_1,Pname] : hAPP(com,com,hAPP(com,fun(com,com),hAPP(fun(state,bool),fun(com,fun(com,com)),cond,Fun_1),Com1_1),Com2_1) != hAPP(pname,com,body,Pname) ).
fof(fact_562_linorder__le__cases,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X_1,Y] :
( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_1),Y))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),X_1)) ) ) ).
fof(fact_563_le__funE,axiom,
! [X_b,X_c] :
( ord(X_c)
=> ! [X_2,F,G] :
( hBOOL(hAPP(fun(X_b,X_c),bool,hAPP(fun(X_b,X_c),fun(fun(X_b,X_c),bool),ord_less_eq(fun(X_b,X_c)),F),G))
=> hBOOL(hAPP(X_c,bool,hAPP(X_c,fun(X_c,bool),ord_less_eq(X_c),hAPP(X_b,X_c,F,X_2)),hAPP(X_b,X_c,G,X_2))) ) ) ).
fof(fact_564_xt1_I6_J,axiom,
! [X_a] :
( order(X_a)
=> ! [Z,Y,X_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),X_1))
=> ( 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_1)) ) ) ) ).
fof(fact_565_xt1_I5_J,axiom,
! [X_a] :
( order(X_a)
=> ! [Y,X_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),X_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_1),Y))
=> ti(X_a,X_1) = ti(X_a,Y) ) ) ) ).
fof(fact_566_order__trans,axiom,
! [X_a] :
( preorder(X_a)
=> ! [Z,X_1,Y] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_1),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_1),Z)) ) ) ) ).
fof(fact_567_order__antisym,axiom,
! [X_a] :
( order(X_a)
=> ! [X_1,Y] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_1),Y))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),X_1))
=> ti(X_a,X_1) = ti(X_a,Y) ) ) ) ).
fof(fact_568_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_569_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))
=> ( B = C
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),C)) ) ) ) ).
fof(fact_570_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_571_ord__eq__le__trans,axiom,
! [X_a] :
( ord(X_a)
=> ! [C,A_1,B] :
( A_1 = 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_572_order__antisym__conv,axiom,
! [X_b] :
( order(X_b)
=> ! [Y_1,X_2] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),Y_1),X_2))
=> ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),X_2),Y_1))
<=> ti(X_b,X_2) = ti(X_b,Y_1) ) ) ) ).
fof(fact_573_le__funD,axiom,
! [X_b,X_c] :
( ord(X_c)
=> ! [X_2,F,G] :
( hBOOL(hAPP(fun(X_b,X_c),bool,hAPP(fun(X_b,X_c),fun(fun(X_b,X_c),bool),ord_less_eq(fun(X_b,X_c)),F),G))
=> hBOOL(hAPP(X_c,bool,hAPP(X_c,fun(X_c,bool),ord_less_eq(X_c),hAPP(X_b,X_c,F,X_2)),hAPP(X_b,X_c,G,X_2))) ) ) ).
fof(fact_574_order__eq__refl,axiom,
! [X_a] :
( preorder(X_a)
=> ! [X_1,Y] :
( X_1 = Y
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_1),Y)) ) ) ).
fof(fact_575_order__eq__iff,axiom,
! [X_b] :
( order(X_b)
=> ! [X_2,Y_1] :
( ti(X_b,X_2) = ti(X_b,Y_1)
<=> ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),X_2),Y_1))
& hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),Y_1),X_2)) ) ) ) ).
fof(fact_576_linorder__linear,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X_1,Y] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_1),Y))
| hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),X_1)) ) ) ).
fof(fact_577_le__fun__def,axiom,
! [X_b,X_c] :
( ord(X_c)
=> ! [F,G] :
( hBOOL(hAPP(fun(X_b,X_c),bool,hAPP(fun(X_b,X_c),fun(fun(X_b,X_c),bool),ord_less_eq(fun(X_b,X_c)),F),G))
<=> ! [X] : hBOOL(hAPP(X_c,bool,hAPP(X_c,fun(X_c,bool),ord_less_eq(X_c),hAPP(X_b,X_c,F,X)),hAPP(X_b,X_c,G,X))) ) ) ).
fof(fact_578_com_Osimps_I4_J,axiom,
! [Fun_1,Com1_1,Com2_1,Fun,Com1,Com2] :
( hAPP(com,com,hAPP(com,fun(com,com),hAPP(fun(state,bool),fun(com,fun(com,com)),cond,Fun_1),Com1_1),Com2_1) = hAPP(com,com,hAPP(com,fun(com,com),hAPP(fun(state,bool),fun(com,fun(com,com)),cond,Fun),Com1),Com2)
<=> ( Fun_1 = Fun
& Com1_1 = Com1
& Com2_1 = Com2 ) ) ).
fof(fact_579_le__antisym,axiom,
! [M,N_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N_2))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_2),M))
=> M = N_2 ) ) ).
fof(fact_580_le__trans,axiom,
! [K_2,I,J_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I),J_1))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),J_1),K_2))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I),K_2)) ) ) ).
fof(fact_581_eq__imp__le,axiom,
! [M,N_2] :
( M = N_2
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N_2)) ) ).
fof(fact_582_nat__le__linear,axiom,
! [M,N_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N_2))
| hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_2),M)) ) ).
fof(fact_583_le__refl,axiom,
! [N_2] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_2),N_2)) ).
fof(fact_584_Suc__leD,axiom,
! [M,N_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,suc,M)),N_2))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N_2)) ) ).
fof(fact_585_le__SucE,axiom,
! [M,N_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),hAPP(nat,nat,suc,N_2)))
=> ( ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N_2))
=> M = hAPP(nat,nat,suc,N_2) ) ) ).
fof(fact_586_le__SucI,axiom,
! [M,N_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N_2))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),hAPP(nat,nat,suc,N_2))) ) ).
fof(fact_587_Suc__le__mono,axiom,
! [N_1,M_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,suc,N_1)),hAPP(nat,nat,suc,M_1)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_1),M_1)) ) ).
fof(fact_588_le__Suc__eq,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_589_not__less__eq__eq,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),hAPP(nat,nat,suc,N_1)),M_1)) ) ).
fof(fact_590_Suc__n__not__le__n,axiom,
! [N_2] : ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,suc,N_2)),N_2)) ).
fof(fact_591_le__0__eq,axiom,
! [N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_1),zero_zero(nat)))
<=> N_1 = zero_zero(nat) ) ).
fof(fact_592_less__eq__nat_Osimps_I1_J,axiom,
! [N_2] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),N_2)) ).
fof(fact_593_inf__sup__ord_I1_J,axiom,
! [X_a] :
( lattice(X_a)
=> ! [X_1,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),semilattice_inf_inf(X_a),X_1),Y)),X_1)) ) ).
fof(fact_594_inf__le1,axiom,
! [X_a] :
( semilattice_inf(X_a)
=> ! [X_1,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),semilattice_inf_inf(X_a),X_1),Y)),X_1)) ) ).
fof(fact_595_inf__sup__ord_I2_J,axiom,
! [X_a] :
( lattice(X_a)
=> ! [X_1,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),semilattice_inf_inf(X_a),X_1),Y)),Y)) ) ).
fof(fact_596_inf__le2,axiom,
! [X_a] :
( semilattice_inf(X_a)
=> ! [X_1,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),semilattice_inf_inf(X_a),X_1),Y)),Y)) ) ).
fof(fact_597_le__iff__inf,axiom,
! [X_b] :
( semilattice_inf(X_b)
=> ! [X_2,Y_1] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),X_2),Y_1))
<=> hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),semilattice_inf_inf(X_b),X_2),Y_1) = ti(X_b,X_2) ) ) ).
fof(fact_598_le__inf__iff,axiom,
! [X_b] :
( semilattice_inf(X_b)
=> ! [X_2,Y_1,Z_1] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),X_2),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),semilattice_inf_inf(X_b),Y_1),Z_1)))
<=> ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),X_2),Y_1))
& hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),X_2),Z_1)) ) ) ) ).
fof(fact_599_le__infI1,axiom,
! [X_a] :
( semilattice_inf(X_a)
=> ! [B,A_1,X_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),X_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),A_1),B)),X_1)) ) ) ).
fof(fact_600_le__infI2,axiom,
! [X_a] :
( semilattice_inf(X_a)
=> ! [A_1,B,X_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),X_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),A_1),B)),X_1)) ) ) ).
fof(fact_601_inf__absorb1,axiom,
! [X_a] :
( semilattice_inf(X_a)
=> ! [X_1,Y] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_1),Y))
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),X_1),Y) = ti(X_a,X_1) ) ) ).
fof(fact_602_inf__absorb2,axiom,
! [X_a] :
( semilattice_inf(X_a)
=> ! [Y,X_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),X_1))
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),X_1),Y) = ti(X_a,Y) ) ) ).
fof(fact_603_le__infI,axiom,
! [X_a] :
( semilattice_inf(X_a)
=> ! [B,X_1,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_1),A_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_1),B))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),A_1),B))) ) ) ) ).
fof(fact_604_inf__greatest,axiom,
! [X_a] :
( semilattice_inf(X_a)
=> ! [Z,X_1,Y] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_1),Y))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_1),Z))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),Y),Z))) ) ) ) ).
fof(fact_605_inf__mono,axiom,
! [X_a] :
( semilattice_inf(X_a)
=> ! [B,D_1,A_1,C] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),C))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),D_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),A_1),B)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),C),D_1))) ) ) ) ).
fof(fact_606_le__infE,axiom,
! [X_a] :
( semilattice_inf(X_a)
=> ! [X_1,A_1,B] :
( 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),semilattice_inf_inf(X_a),A_1),B)))
=> ~ ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_1),A_1))
=> ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_1),B)) ) ) ) ).
fof(fact_607_inf__sup__ord_I3_J,axiom,
! [X_a] :
( lattice(X_a)
=> ! [X_1,Y] : 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),semilattice_sup_sup(X_a),X_1),Y))) ) ).
fof(fact_608_sup__ge1,axiom,
! [X_a] :
( semilattice_sup(X_a)
=> ! [X_1,Y] : 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),semilattice_sup_sup(X_a),X_1),Y))) ) ).
fof(fact_609_inf__sup__ord_I4_J,axiom,
! [X_a] :
( lattice(X_a)
=> ! [Y,X_1] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),X_1),Y))) ) ).
fof(fact_610_sup__ge2,axiom,
! [X_a] :
( semilattice_sup(X_a)
=> ! [Y,X_1] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),X_1),Y))) ) ).
fof(fact_611_le__iff__sup,axiom,
! [X_b] :
( semilattice_sup(X_b)
=> ! [X_2,Y_1] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),X_2),Y_1))
<=> hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),semilattice_sup_sup(X_b),X_2),Y_1) = ti(X_b,Y_1) ) ) ).
fof(fact_612_le__sup__iff,axiom,
! [X_b] :
( semilattice_sup(X_b)
=> ! [X_2,Y_1,Z_1] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),semilattice_sup_sup(X_b),X_2),Y_1)),Z_1))
<=> ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),X_2),Z_1))
& hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),Y_1),Z_1)) ) ) ) ).
fof(fact_613_le__supI1,axiom,
! [X_a] :
( semilattice_sup(X_a)
=> ! [B,X_1,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_1),A_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),semilattice_sup_sup(X_a),A_1),B))) ) ) ).
fof(fact_614_le__supI2,axiom,
! [X_a] :
( semilattice_sup(X_a)
=> ! [A_1,X_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_1),B))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),A_1),B))) ) ) ).
fof(fact_615_sup__absorb2,axiom,
! [X_a] :
( semilattice_sup(X_a)
=> ! [X_1,Y] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_1),Y))
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),X_1),Y) = ti(X_a,Y) ) ) ).
fof(fact_616_sup__absorb1,axiom,
! [X_a] :
( semilattice_sup(X_a)
=> ! [Y,X_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),X_1))
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),X_1),Y) = ti(X_a,X_1) ) ) ).
fof(fact_617_le__supI,axiom,
! [X_a] :
( semilattice_sup(X_a)
=> ! [B,A_1,X_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),X_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),X_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),A_1),B)),X_1)) ) ) ) ).
fof(fact_618_sup__least,axiom,
! [X_a] :
( semilattice_sup(X_a)
=> ! [Z,Y,X_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),X_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Z),X_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),Y),Z)),X_1)) ) ) ) ).
fof(fact_619_sup__mono,axiom,
! [X_a] :
( semilattice_sup(X_a)
=> ! [B,D_1,A_1,C] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),C))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),D_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),A_1),B)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),C),D_1))) ) ) ) ).
fof(fact_620_le__supE,axiom,
! [X_a] :
( semilattice_sup(X_a)
=> ! [A_1,B,X_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),A_1),B)),X_1))
=> ~ ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),X_1))
=> ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),X_1)) ) ) ) ).
fof(fact_621_diff__eq__diff__less__eq,axiom,
! [X_b] :
( ordered_ab_group_add(X_b)
=> ! [A_2,B_1,C_1,D] :
( hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),minus_minus(X_b),A_2),B_1) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),minus_minus(X_b),C_1),D)
=> ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),A_2),B_1))
<=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),C_1),D)) ) ) ) ).
fof(fact_622_evaln__nonstrict,axiom,
! [M,C,S,N_2,T_3] :
( hBOOL(hAPP(state,bool,hAPP(nat,fun(state,bool),hAPP(state,fun(nat,fun(state,bool)),hAPP(com,fun(state,fun(nat,fun(state,bool))),evaln,C),S),N_2),T_3))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_2),M))
=> hBOOL(hAPP(state,bool,hAPP(nat,fun(state,bool),hAPP(state,fun(nat,fun(state,bool)),hAPP(com,fun(state,fun(nat,fun(state,bool))),evaln,C),S),M),T_3)) ) ) ).
fof(fact_623_diff__le__self,axiom,
! [M,N_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),M),N_2)),M)) ).
fof(fact_624_diff__le__mono2,axiom,
! [L,M,N_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N_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),L),N_2)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),L),M))) ) ).
fof(fact_625_diff__le__mono,axiom,
! [L,M,N_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N_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),M),L)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N_2),L))) ) ).
fof(fact_626_diff__diff__cancel,axiom,
! [I,N_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I),N_2))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N_2),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N_2),I)) = I ) ).
fof(fact_627_eq__diff__iff,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),M_1),K_1) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N_1),K_1)
<=> M_1 = N_1 ) ) ) ).
fof(fact_628_Nat_Odiff__diff__eq,axiom,
! [N_2,K_2,M] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_2),M))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_2),N_2))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),K_2)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N_2),K_2)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),N_2) ) ) ).
fof(fact_629_le__diff__iff,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))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M_1),K_1)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N_1),K_1)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_1),N_1)) ) ) ) ).
fof(fact_630_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_631_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_632_add__mono,axiom,
! [X_a] :
( ordere779506340up_add(X_a)
=> ! [C,D_1,A_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C),D_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),C)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),B),D_1))) ) ) ) ).
fof(fact_633_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_634_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_635_add__le__cancel__left,axiom,
! [X_b] :
( ordere236663937imp_le(X_b)
=> ! [C_1,A_2,B_1] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),C_1),A_2)),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),C_1),B_1)))
<=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),A_2),B_1)) ) ) ).
fof(fact_636_add__le__cancel__right,axiom,
! [X_b] :
( ordere236663937imp_le(X_b)
=> ! [A_2,C_1,B_1] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),A_2),C_1)),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),B_1),C_1)))
<=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),A_2),B_1)) ) ) ).
fof(fact_637_add__leE,axiom,
! [M,K_2,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),M),K_2)),N_2))
=> ~ ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N_2))
=> ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_2),N_2)) ) ) ).
fof(fact_638_add__leD1,axiom,
! [M,K_2,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),M),K_2)),N_2))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N_2)) ) ).
fof(fact_639_add__leD2,axiom,
! [M,K_2,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),M),K_2)),N_2))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_2),N_2)) ) ).
fof(fact_640_add__le__mono,axiom,
! [K_2,L,I,J_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I),J_1))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_2),L))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),I),K_2)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),J_1),L))) ) ) ).
fof(fact_641_add__le__mono1,axiom,
! [K_2,I,J_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I),J_1))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),I),K_2)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),J_1),K_2))) ) ).
fof(fact_642_trans__le__add2,axiom,
! [M,I,J_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I),J_1))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),J_1))) ) ).
fof(fact_643_trans__le__add1,axiom,
! [M,I,J_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I),J_1))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),J_1),M))) ) ).
fof(fact_644_nat__add__left__cancel__le,axiom,
! [K_1,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),plus_plus(nat),K_1),M_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_eq(nat),M_1),N_1)) ) ).
fof(fact_645_le__iff__add,axiom,
! [M_1,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_1),N_1))
<=> ? [K] : N_1 = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M_1),K) ) ).
fof(fact_646_le__add1,axiom,
! [N_2,M] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_2),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N_2),M))) ).
fof(fact_647_le__add2,axiom,
! [N_2,M] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_2),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),N_2))) ).
fof(fact_648_card__Collect__le__nat,axiom,
! [N_1] : hAPP(fun(nat,bool),nat,finite_card(nat),hAPP(fun(nat,bool),fun(nat,bool),collect(nat),hAPP(nat,fun(nat,bool),hAPP(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool)),combc(nat,nat,bool),ord_less_eq(nat)),N_1))) = hAPP(nat,nat,suc,N_1) ).
fof(fact_649_less__eq__nat_Osimps_I2_J,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(fun(nat,bool),fun(nat,bool),hAPP(bool,fun(fun(nat,bool),fun(nat,bool)),nat_case(bool),fFalse),hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_1)),N_1)) ) ).
fof(fact_650_below__fold1__iff,axiom,
! [X_b] :
( semilattice_inf(X_b)
=> ! [X_2,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( ti(fun(X_b,bool),A_3) != bot_bot(fun(X_b,bool))
=> ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),X_2),hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),finite_fold1(X_b),semilattice_inf_inf(X_b)),A_3)))
<=> ! [X] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X),A_3))
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),X_2),X)) ) ) ) ) ) ).
fof(fact_651_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_652_bot__unique,axiom,
! [X_b] :
( bot(X_b)
=> ! [A_2] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),A_2),bot_bot(X_b)))
<=> ti(X_b,A_2) = bot_bot(X_b) ) ) ).
fof(fact_653_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_654_fold1__belowI,axiom,
! [X_b] :
( semilattice_inf(X_b)
=> ! [A_2,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),A_2),A_3))
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),finite_fold1(X_b),semilattice_inf_inf(X_b)),A_3)),A_2)) ) ) ) ).
fof(fact_655_mult__le__mono,axiom,
! [K_2,L,I,J_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I),J_1))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_2),L))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),I),K_2)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),J_1),L))) ) ) ).
fof(fact_656_mult__le__mono2,axiom,
! [K_2,I,J_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I),J_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),K_2),I)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K_2),J_1))) ) ).
fof(fact_657_mult__le__mono1,axiom,
! [K_2,I,J_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I),J_1))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),I),K_2)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),J_1),K_2))) ) ).
fof(fact_658_le__cube,axiom,
! [M] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),M)))) ).
fof(fact_659_le__square,axiom,
! [M] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),M))) ).
fof(fact_660_com_Osimps_I52_J,axiom,
! [Fun_1,Com1_1,Com2_1,Fun,Com] : hAPP(com,com,hAPP(com,fun(com,com),hAPP(fun(state,bool),fun(com,fun(com,com)),cond,Fun_1),Com1_1),Com2_1) != hAPP(com,com,hAPP(fun(state,bool),fun(com,com),while,Fun),Com) ).
fof(fact_661_com_Osimps_I53_J,axiom,
! [Fun,Com,Fun_1,Com1_1,Com2_1] : 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_1),Com2_1) ).
fof(fact_662_com_Osimps_I45_J,axiom,
! [Fun,Com1,Com2,Com1_1,Com2_1] : hAPP(com,com,hAPP(com,fun(com,com),hAPP(fun(state,bool),fun(com,fun(com,com)),cond,Fun),Com1),Com2) != hAPP(com,com,hAPP(com,fun(com,com),semi,Com1_1),Com2_1) ).
fof(fact_663_com_Osimps_I44_J,axiom,
! [Com1_1,Com2_1,Fun,Com1,Com2] : hAPP(com,com,hAPP(com,fun(com,com),semi,Com1_1),Com2_1) != hAPP(com,com,hAPP(com,fun(com,com),hAPP(fun(state,bool),fun(com,fun(com,com)),cond,Fun),Com1),Com2) ).
fof(fact_664_com_Osimps_I15_J,axiom,
! [Fun,Com1,Com2] : hAPP(com,com,hAPP(com,fun(com,com),hAPP(fun(state,bool),fun(com,fun(com,com)),cond,Fun),Com1),Com2) != skip ).
fof(fact_665_com_Osimps_I14_J,axiom,
! [Fun,Com1,Com2] : skip != hAPP(com,com,hAPP(com,fun(com,com),hAPP(fun(state,bool),fun(com,fun(com,com)),cond,Fun),Com1),Com2) ).
fof(fact_666_zero__le__double__add__iff__zero__le__single__add,axiom,
! [X_b] :
( linord219039673up_add(X_b)
=> ! [A_2] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),zero_zero(X_b)),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),A_2),A_2)))
<=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),zero_zero(X_b)),A_2)) ) ) ).
fof(fact_667_double__add__le__zero__iff__single__add__le__zero,axiom,
! [X_b] :
( linord219039673up_add(X_b)
=> ! [A_2] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),A_2),A_2)),zero_zero(X_b)))
<=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),A_2),zero_zero(X_b))) ) ) ).
fof(fact_668_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_669_add__nonneg__eq__0__iff,axiom,
! [X_b] :
( ordere216010020id_add(X_b)
=> ! [Y_1,X_2] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),zero_zero(X_b)),X_2))
=> ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),zero_zero(X_b)),Y_1))
=> ( hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),X_2),Y_1) = zero_zero(X_b)
<=> ( ti(X_b,X_2) = zero_zero(X_b)
& ti(X_b,Y_1) = zero_zero(X_b) ) ) ) ) ) ).
fof(fact_670_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_671_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_672_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_673_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_674_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_675_mult__mono,axiom,
! [X_a] :
( ordered_semiring(X_a)
=> ! [C,D_1,A_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C),D_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),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_1))) ) ) ) ) ) ).
fof(fact_676_mult__mono_H,axiom,
! [X_a] :
( ordered_semiring(X_a)
=> ! [C,D_1,A_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C),D_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),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_1))) ) ) ) ) ) ).
fof(fact_677_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_678_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_679_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_680_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_681_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_682_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_683_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_684_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_685_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_686_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_687_mult__le__0__iff,axiom,
! [X_b] :
( linord581940658strict(X_b)
=> ! [A_2,B_1] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),A_2),B_1)),zero_zero(X_b)))
<=> ( ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),zero_zero(X_b)),A_2))
& hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),B_1),zero_zero(X_b))) )
| ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),A_2),zero_zero(X_b)))
& hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),zero_zero(X_b)),B_1)) ) ) ) ) ).
fof(fact_688_zero__le__mult__iff,axiom,
! [X_b] :
( linord581940658strict(X_b)
=> ! [A_2,B_1] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),zero_zero(X_b)),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),A_2),B_1)))
<=> ( ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),zero_zero(X_b)),A_2))
& hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),zero_zero(X_b)),B_1)) )
| ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),A_2),zero_zero(X_b)))
& hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),B_1),zero_zero(X_b))) ) ) ) ) ).
fof(fact_689_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_690_le__iff__diff__le__0,axiom,
! [X_b] :
( ordered_ab_group_add(X_b)
=> ! [A_2,B_1] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),A_2),B_1))
<=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),minus_minus(X_b),A_2),B_1)),zero_zero(X_b))) ) ) ).
fof(fact_691_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_692_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_693_distrib__inf__le,axiom,
! [X_a] :
( lattice(X_a)
=> ! [X_1,Y,Z] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),X_1),Y)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),X_1),Z))),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),X_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),Y),Z)))) ) ).
fof(fact_694_distrib__sup__le,axiom,
! [X_a] :
( lattice(X_a)
=> ! [X_1,Y,Z] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),X_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),Y),Z))),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_inf_inf(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),X_1),Y)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),X_1),Z)))) ) ).
fof(fact_695_diff__is__0__eq_H,axiom,
! [M,N_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N_2))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),N_2) = zero_zero(nat) ) ).
fof(fact_696_diff__is__0__eq,axiom,
! [M_1,N_1] :
( hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M_1),N_1) = zero_zero(nat)
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_1),N_1)) ) ).
fof(fact_697_Suc__diff__le,axiom,
! [N_2,M] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_2),M))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,suc,M)),N_2) = hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),N_2)) ) ).
fof(fact_698_Suc__mult__le__cancel1,axiom,
! [K_1,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),times_times(nat),hAPP(nat,nat,suc,K_1)),M_1)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,suc,K_1)),N_1)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_1),N_1)) ) ).
fof(fact_699_diff__diff__right,axiom,
! [I,K_2,J_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_2),J_1))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),I),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),J_1),K_2)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),I),K_2)),J_1) ) ).
fof(fact_700_le__diff__conv,axiom,
! [J_2,K_1,I_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),J_2),K_1)),I_1))
<=> 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_1),K_1))) ) ).
fof(fact_701_le__add__diff,axiom,
! [M,K_2,N_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_2),N_2))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N_2),M)),K_2))) ) ).
fof(fact_702_le__add__diff__inverse,axiom,
! [N_2,M] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_2),M))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N_2),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),N_2)) = M ) ).
fof(fact_703_add__diff__assoc,axiom,
! [I,K_2,J_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_2),J_1))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),I),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),J_1),K_2)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),I),J_1)),K_2) ) ).
fof(fact_704_le__diff__conv2,axiom,
! [I_1,K_1,J_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_1),J_2))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),J_2),K_1)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),I_1),K_1)),J_2)) ) ) ).
fof(fact_705_le__add__diff__inverse2,axiom,
! [N_2,M] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_2),M))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),N_2)),N_2) = M ) ).
fof(fact_706_le__imp__diff__is__add,axiom,
! [K_1,I_1,J_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_1),J_2))
=> ( hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),J_2),I_1) = K_1
<=> J_2 = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),K_1),I_1) ) ) ).
fof(fact_707_diff__add__assoc,axiom,
! [I,K_2,J_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_2),J_1))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),I),J_1)),K_2) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),I),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),J_1),K_2)) ) ).
fof(fact_708_add__diff__assoc2,axiom,
! [I,K_2,J_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_2),J_1))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),J_1),K_2)),I) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),J_1),I)),K_2) ) ).
fof(fact_709_diff__add__assoc2,axiom,
! [I,K_2,J_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_2),J_1))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),J_1),I)),K_2) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),J_1),K_2)),I) ) ).
fof(fact_710_fold1__singleton,axiom,
! [X_b,F,A_2] : hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),finite_fold1(X_b),F),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_2),bot_bot(fun(X_b,bool)))) = ti(X_b,A_2) ).
fof(fact_711_fold1__singleton__def,axiom,
! [X_b,A_2,G,F] :
( G = hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),finite_fold1(X_b),F)
=> hAPP(fun(X_b,bool),X_b,G,hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_2),bot_bot(fun(X_b,bool)))) = ti(X_b,A_2) ) ).
fof(fact_712_folding__one_Oeq__fold,axiom,
! [X_b,A_3,F,F_1] :
( hBOOL(hAPP(fun(fun(X_b,bool),X_b),bool,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(fun(X_b,bool),X_b),bool),finite_folding_one(X_b),F),F_1))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> hAPP(fun(X_b,bool),X_b,F_1,A_3) = hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),finite_fold1(X_b),F),A_3) ) ) ).
fof(fact_713_sum__squares__le__zero__iff,axiom,
! [X_b] :
( linord581940658strict(X_b)
=> ! [X_2,Y_1] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),X_2),X_2)),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),Y_1),Y_1))),zero_zero(X_b)))
<=> ( ti(X_b,X_2) = zero_zero(X_b)
& ti(X_b,Y_1) = zero_zero(X_b) ) ) ) ).
fof(fact_714_sum__squares__ge__zero,axiom,
! [X_a] :
( linordered_ring(X_a)
=> ! [X_1,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_1),X_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Y),Y)))) ) ).
fof(fact_715_mult__right__le__one__le,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [Y,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))
=> ( 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_1),Y)),X_1)) ) ) ) ) ).
fof(fact_716_mult__left__le__one__le,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [Y,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))
=> ( 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_1)),X_1)) ) ) ) ) ).
fof(fact_717_le__add__iff1,axiom,
! [X_b] :
( ordered_ring(X_b)
=> ! [A_2,E,C_1,B_1,D] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),A_2),E)),C_1)),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),B_1),E)),D)))
<=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),minus_minus(X_b),A_2),B_1)),E)),C_1)),D)) ) ) ).
fof(fact_718_le__add__iff2,axiom,
! [X_b] :
( ordered_ring(X_b)
=> ! [A_2,E,C_1,B_1,D] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),A_2),E)),C_1)),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),B_1),E)),D)))
<=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),C_1),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),minus_minus(X_b),B_1),A_2)),E)),D))) ) ) ).
fof(fact_719_card__insert__le,axiom,
! [X_b,X_2,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(fun(X_b,bool),nat,finite_card(X_b),A_3)),hAPP(fun(X_b,bool),nat,finite_card(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_2),A_3)))) ) ).
fof(fact_720_one__le__mult__iff,axiom,
! [M_1,N_1] :
( 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_1),N_1)))
<=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,suc,zero_zero(nat))),M_1))
& hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,suc,zero_zero(nat))),N_1)) ) ) ).
fof(fact_721_card__image__le,axiom,
! [X_c,X_b,F,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(fun(X_c,bool),nat,finite_card(X_c),hAPP(fun(X_b,bool),fun(X_c,bool),hAPP(fun(X_b,X_c),fun(fun(X_b,bool),fun(X_c,bool)),image(X_b,X_c),F),A_3))),hAPP(fun(X_b,bool),nat,finite_card(X_b),A_3))) ) ).
fof(fact_722_diff__Suc__diff__eq2,axiom,
! [M,K_2,J_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_2),J_1))
=> 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_1),K_2))),M) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,suc,J_1)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),K_2),M)) ) ).
fof(fact_723_diff__Suc__diff__eq1,axiom,
! [M,K_2,J_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_2),J_1))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),J_1),K_2))) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),K_2)),hAPP(nat,nat,suc,J_1)) ) ).
fof(fact_724_nat__le__add__iff1,axiom,
! [U,M_1,N_1,J_2,I_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),J_2),I_1))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),I_1),U)),M_1)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),J_2),U)),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),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_2)),U)),M_1)),N_1)) ) ) ).
fof(fact_725_nat__diff__add__eq1,axiom,
! [U_1,M,N_2,J_1,I] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),J_1),I))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),I),U_1)),M)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),J_1),U_1)),N_2)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),I),J_1)),U_1)),M)),N_2) ) ).
fof(fact_726_nat__eq__add__iff1,axiom,
! [U,M_1,N_1,J_2,I_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),J_2),I_1))
=> ( hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),I_1),U)),M_1) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),J_2),U)),N_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),I_1),J_2)),U)),M_1) = N_1 ) ) ).
fof(fact_727_nat__le__add__iff2,axiom,
! [U,M_1,N_1,I_1,J_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_1),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_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_2),U)),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),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_1)),U)),N_1))) ) ) ).
fof(fact_728_nat__diff__add__eq2,axiom,
! [U_1,M,N_2,I,J_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I),J_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),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_1),U_1)),N_2)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),J_1),I)),U_1)),N_2)) ) ).
fof(fact_729_nat__eq__add__iff2,axiom,
! [U,M_1,N_1,I_1,J_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_1),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_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_2),U)),N_1)
<=> 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_2),I_1)),U)),N_1) ) ) ).
fof(fact_730_Sup__fin_OF__eq,axiom,
! [X_b] :
( lattice(X_b)
=> ! [A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> hAPP(fun(X_b,bool),X_b,big_lattice_Sup_fin(X_b),A_3) = hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),finite_fold1(X_b),semilattice_sup_sup(X_b)),A_3) ) ) ).
fof(fact_731_Inf__fin_OF__eq,axiom,
! [X_b] :
( lattice(X_b)
=> ! [A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> hAPP(fun(X_b,bool),X_b,big_lattice_Inf_fin(X_b),A_3) = hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),finite_fold1(X_b),semilattice_inf_inf(X_b)),A_3) ) ) ).
fof(fact_732_convex__bound__le,axiom,
! [X_a] :
( linord1278240602ring_1(X_a)
=> ! [V,U_1,Y,X_1,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_1),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_1))
=> ( 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_1),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_1),X_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),V),Y))),A_1)) ) ) ) ) ) ) ).
fof(fact_733_diff__card__le__card__Diff,axiom,
! [X_b,A_3,B_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),B_2))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(fun(X_b,bool),nat,finite_card(X_b),A_3)),hAPP(fun(X_b,bool),nat,finite_card(X_b),B_2))),hAPP(fun(X_b,bool),nat,finite_card(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_3),B_2)))) ) ).
fof(fact_734_Inf__le__Sup,axiom,
! [X_b] :
( lattice(X_b)
=> ! [A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( ti(fun(X_b,bool),A_3) != bot_bot(fun(X_b,bool))
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(fun(X_b,bool),X_b,big_lattice_Inf_fin(X_b),A_3)),hAPP(fun(X_b,bool),X_b,big_lattice_Sup_fin(X_b),A_3))) ) ) ) ).
fof(fact_735_fold1__insert,axiom,
! [X_b] :
( ab_semigroup_mult(X_b)
=> ! [X_2,A_3] :
( ti(fun(X_b,bool),A_3) != bot_bot(fun(X_b,bool))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( ~ hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_2),A_3))
=> hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),finite_fold1(X_b),times_times(X_b)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_2),A_3)) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),X_2),hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),finite_fold1(X_b),times_times(X_b)),A_3)) ) ) ) ) ).
fof(fact_736_fold1__insert__idem,axiom,
! [X_b] :
( ab_sem1668676832m_mult(X_b)
=> ! [X_2,A_3] :
( ti(fun(X_b,bool),A_3) != bot_bot(fun(X_b,bool))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),finite_fold1(X_b),times_times(X_b)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_2),A_3)) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),X_2),hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),finite_fold1(X_b),times_times(X_b)),A_3)) ) ) ) ).
fof(fact_737_fold1__Un2,axiom,
! [X_b] :
( ab_sem1668676832m_mult(X_b)
=> ! [B_2,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( ti(fun(X_b,bool),A_3) != bot_bot(fun(X_b,bool))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),B_2))
=> ( ti(fun(X_b,bool),B_2) != bot_bot(fun(X_b,bool))
=> hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),finite_fold1(X_b),times_times(X_b)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),A_3),B_2)) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),finite_fold1(X_b),times_times(X_b)),A_3)),hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),finite_fold1(X_b),times_times(X_b)),B_2)) ) ) ) ) ) ).
fof(fact_738_hom__fold1__commute,axiom,
! [X_b] :
( ab_sem1668676832m_mult(X_b)
=> ! [N_3,H] :
( ! [X,Y_2] : hAPP(X_b,X_b,H,hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),X),Y_2)) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),hAPP(X_b,X_b,H,X)),hAPP(X_b,X_b,H,Y_2))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),N_3))
=> ( ti(fun(X_b,bool),N_3) != bot_bot(fun(X_b,bool))
=> hAPP(X_b,X_b,H,hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),finite_fold1(X_b),times_times(X_b)),N_3)) = hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),finite_fold1(X_b),times_times(X_b)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,X_b),fun(fun(X_b,bool),fun(X_b,bool)),image(X_b,X_b),H),N_3)) ) ) ) ) ).
fof(fact_739_fold1__in,axiom,
! [X_b] :
( ab_semigroup_mult(X_b)
=> ! [A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( ti(fun(X_b,bool),A_3) != bot_bot(fun(X_b,bool))
=> ( ! [X,Y_2] : hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),X),Y_2)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),Y_2),bot_bot(fun(X_b,bool))))))
=> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),finite_fold1(X_b),times_times(X_b)),A_3)),A_3)) ) ) ) ) ).
fof(fact_740_semilattice__big_OF__eq,axiom,
! [X_b,A_3,F,F_1] :
( hBOOL(hAPP(fun(fun(X_b,bool),X_b),bool,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(fun(X_b,bool),X_b),bool),big_semilattice_big(X_b),F),F_1))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> hAPP(fun(X_b,bool),X_b,F_1,A_3) = hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),finite_fold1(X_b),F),A_3) ) ) ).
fof(fact_741_empty__subsetI,axiom,
! [X_b,A_3] : hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),bot_bot(fun(X_b,bool))),A_3)) ).
fof(fact_742_finite__Collect__subsets,axiom,
! [X_b,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> hBOOL(hAPP(fun(fun(X_b,bool),bool),bool,finite_finite_1(fun(X_b,bool)),hAPP(fun(fun(X_b,bool),bool),fun(fun(X_b,bool),bool),collect(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),hAPP(fun(fun(X_b,bool),fun(fun(X_b,bool),bool)),fun(fun(X_b,bool),fun(fun(X_b,bool),bool)),combc(fun(X_b,bool),fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool))),A_3)))) ) ).
fof(fact_743_subset__image__iff,axiom,
! [X_b,X_c,B_2,F,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),B_2),hAPP(fun(X_c,bool),fun(X_b,bool),hAPP(fun(X_c,X_b),fun(fun(X_c,bool),fun(X_b,bool)),image(X_c,X_b),F),A_3)))
<=> ? [AA] :
( hBOOL(hAPP(fun(X_c,bool),bool,hAPP(fun(X_c,bool),fun(fun(X_c,bool),bool),ord_less_eq(fun(X_c,bool)),AA),A_3))
& ti(fun(X_b,bool),B_2) = hAPP(fun(X_c,bool),fun(X_b,bool),hAPP(fun(X_c,X_b),fun(fun(X_c,bool),fun(X_b,bool)),image(X_c,X_b),F),AA) ) ) ).
fof(fact_744_image__mono,axiom,
! [X_c,X_b,F,A_3,B_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),A_3),B_2))
=> hBOOL(hAPP(fun(X_c,bool),bool,hAPP(fun(X_c,bool),fun(fun(X_c,bool),bool),ord_less_eq(fun(X_c,bool)),hAPP(fun(X_b,bool),fun(X_c,bool),hAPP(fun(X_b,X_c),fun(fun(X_b,bool),fun(X_c,bool)),image(X_b,X_c),F),A_3)),hAPP(fun(X_b,bool),fun(X_c,bool),hAPP(fun(X_b,X_c),fun(fun(X_b,bool),fun(X_c,bool)),image(X_b,X_c),F),B_2))) ) ).
fof(fact_745_Int__lower1,axiom,
! [X_b,A_3,B_2] : hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_3),B_2)),A_3)) ).
fof(fact_746_Int__lower2,axiom,
! [X_b,A_3,B_2] : hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_3),B_2)),B_2)) ).
fof(fact_747_Int__absorb2,axiom,
! [X_b,A_3,B_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),A_3),B_2))
=> hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_3),B_2) = ti(fun(X_b,bool),A_3) ) ).
fof(fact_748_Int__absorb1,axiom,
! [X_b,B_2,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),B_2),A_3))
=> hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_3),B_2) = ti(fun(X_b,bool),B_2) ) ).
fof(fact_749_Int__greatest,axiom,
! [X_b,B_2,C_3,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)),C_3),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)),C_3),B_2))
=> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),C_3),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_3),B_2))) ) ) ).
fof(fact_750_Int__mono,axiom,
! [X_b,B_2,D_2,A_3,C_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),A_3),C_3))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),B_2),D_2))
=> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_3),B_2)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),C_3),D_2))) ) ) ).
fof(fact_751_Un__upper1,axiom,
! [X_b,A_3,B_2] : hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),A_3),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),A_3),B_2))) ).
fof(fact_752_Un__upper2,axiom,
! [X_b,B_2,A_3] : hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),B_2),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),A_3),B_2))) ).
fof(fact_753_subset__Un__eq,axiom,
! [X_b,A_3,B_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),A_3),B_2))
<=> hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),A_3),B_2) = ti(fun(X_b,bool),B_2) ) ).
fof(fact_754_Un__absorb1,axiom,
! [X_b,A_3,B_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),A_3),B_2))
=> hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),A_3),B_2) = ti(fun(X_b,bool),B_2) ) ).
fof(fact_755_Un__absorb2,axiom,
! [X_b,B_2,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),B_2),A_3))
=> hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),A_3),B_2) = ti(fun(X_b,bool),A_3) ) ).
fof(fact_756_Un__least,axiom,
! [X_b,B_2,A_3,C_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),A_3),C_3))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),B_2),C_3))
=> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),A_3),B_2)),C_3)) ) ) ).
fof(fact_757_Un__mono,axiom,
! [X_b,B_2,D_2,A_3,C_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),A_3),C_3))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),B_2),D_2))
=> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),A_3),B_2)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),C_3),D_2))) ) ) ).
fof(fact_758_Powp__mono,axiom,
! [X_b,A_3,B_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),A_3),B_2))
=> hBOOL(hAPP(fun(fun(X_b,bool),bool),bool,hAPP(fun(fun(X_b,bool),bool),fun(fun(fun(X_b,bool),bool),bool),ord_less_eq(fun(fun(X_b,bool),bool)),hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),powp(X_b),A_3)),hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),powp(X_b),B_2))) ) ).
fof(fact_759_subset__insertI,axiom,
! [X_b,B_2,A_2] : hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),B_2),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_2),B_2))) ).
fof(fact_760_insert__subset,axiom,
! [X_b,X_2,A_3,B_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_2),A_3)),B_2))
<=> ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_2),B_2))
& hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),A_3),B_2)) ) ) ).
fof(fact_761_subset__insert,axiom,
! [X_b,B_2,X_2,A_3] :
( ~ hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_2),A_3))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),A_3),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_2),B_2)))
<=> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),A_3),B_2)) ) ) ).
fof(fact_762_subset__insertI2,axiom,
! [X_b,B_1,A_3,B_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),A_3),B_2))
=> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),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),B_1),B_2))) ) ).
fof(fact_763_insert__mono,axiom,
! [X_b,A_2,C_3,D_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),C_3),D_2))
=> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_2),C_3)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_2),D_2))) ) ).
fof(fact_764_thin,axiom,
! [X_b,Ga,G_1,Ts] :
( hBOOL(hAPP(fun(hoare_2118899576triple(X_b),bool),bool,hAPP(fun(hoare_2118899576triple(X_b),bool),fun(fun(hoare_2118899576triple(X_b),bool),bool),hoare_1301688828derivs(X_b),G_1),Ts))
=> ( hBOOL(hAPP(fun(hoare_2118899576triple(X_b),bool),bool,hAPP(fun(hoare_2118899576triple(X_b),bool),fun(fun(hoare_2118899576triple(X_b),bool),bool),ord_less_eq(fun(hoare_2118899576triple(X_b),bool)),G_1),Ga))
=> hBOOL(hAPP(fun(hoare_2118899576triple(X_b),bool),bool,hAPP(fun(hoare_2118899576triple(X_b),bool),fun(fun(hoare_2118899576triple(X_b),bool),bool),hoare_1301688828derivs(X_b),Ga),Ts)) ) ) ).
fof(fact_765_weaken,axiom,
! [X_b,Ts,Ga,Ts_1] :
( hBOOL(hAPP(fun(hoare_2118899576triple(X_b),bool),bool,hAPP(fun(hoare_2118899576triple(X_b),bool),fun(fun(hoare_2118899576triple(X_b),bool),bool),hoare_1301688828derivs(X_b),Ga),Ts_1))
=> ( hBOOL(hAPP(fun(hoare_2118899576triple(X_b),bool),bool,hAPP(fun(hoare_2118899576triple(X_b),bool),fun(fun(hoare_2118899576triple(X_b),bool),bool),ord_less_eq(fun(hoare_2118899576triple(X_b),bool)),Ts),Ts_1))
=> hBOOL(hAPP(fun(hoare_2118899576triple(X_b),bool),bool,hAPP(fun(hoare_2118899576triple(X_b),bool),fun(fun(hoare_2118899576triple(X_b),bool),bool),hoare_1301688828derivs(X_b),Ga),Ts)) ) ) ).
fof(fact_766_asm,axiom,
! [X_b,Ts,Ga] :
( hBOOL(hAPP(fun(hoare_2118899576triple(X_b),bool),bool,hAPP(fun(hoare_2118899576triple(X_b),bool),fun(fun(hoare_2118899576triple(X_b),bool),bool),ord_less_eq(fun(hoare_2118899576triple(X_b),bool)),Ts),Ga))
=> hBOOL(hAPP(fun(hoare_2118899576triple(X_b),bool),bool,hAPP(fun(hoare_2118899576triple(X_b),bool),fun(fun(hoare_2118899576triple(X_b),bool),bool),hoare_1301688828derivs(X_b),Ga),Ts)) ) ).
fof(fact_767_Diff__subset,axiom,
! [X_b,A_3,B_2] : hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_3),B_2)),A_3)) ).
fof(fact_768_Diff__mono,axiom,
! [X_b,D_2,B_2,A_3,C_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),A_3),C_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)),D_2),B_2))
=> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_3),B_2)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),C_3),D_2))) ) ) ).
fof(fact_769_double__diff,axiom,
! [X_b,C_3,A_3,B_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),A_3),B_2))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),B_2),C_3))
=> hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),B_2),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),C_3),A_3)) = ti(fun(X_b,bool),A_3) ) ) ).
fof(fact_770_subset__empty,axiom,
! [X_b,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),A_3),bot_bot(fun(X_b,bool))))
<=> ti(fun(X_b,bool),A_3) = bot_bot(fun(X_b,bool)) ) ).
fof(fact_771_finite__subset,axiom,
! [X_b,A_3,B_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),A_3),B_2))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),B_2))
=> hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3)) ) ) ).
fof(fact_772_rev__finite__subset,axiom,
! [X_b,A_3,B_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),B_2))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),A_3),B_2))
=> hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3)) ) ) ).
fof(fact_773_subset__singletonD,axiom,
! [X_b,A_3,X_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),A_3),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_2),bot_bot(fun(X_b,bool)))))
=> ( ti(fun(X_b,bool),A_3) = bot_bot(fun(X_b,bool))
| ti(fun(X_b,bool),A_3) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_2),bot_bot(fun(X_b,bool))) ) ) ).
fof(fact_774_finite__surj,axiom,
! [X_c,X_b,B_2,F,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( hBOOL(hAPP(fun(X_c,bool),bool,hAPP(fun(X_c,bool),fun(fun(X_c,bool),bool),ord_less_eq(fun(X_c,bool)),B_2),hAPP(fun(X_b,bool),fun(X_c,bool),hAPP(fun(X_b,X_c),fun(fun(X_b,bool),fun(X_c,bool)),image(X_b,X_c),F),A_3)))
=> hBOOL(hAPP(fun(X_c,bool),bool,finite_finite_1(X_c),B_2)) ) ) ).
fof(fact_775_image__Int__subset,axiom,
! [X_b,X_c,F,A_3,B_2] : hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),hAPP(fun(X_c,bool),fun(X_b,bool),hAPP(fun(X_c,X_b),fun(fun(X_c,bool),fun(X_b,bool)),image(X_c,X_b),F),hAPP(fun(X_c,bool),fun(X_c,bool),hAPP(fun(X_c,bool),fun(fun(X_c,bool),fun(X_c,bool)),semilattice_inf_inf(fun(X_c,bool)),A_3),B_2))),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),hAPP(fun(X_c,bool),fun(X_b,bool),hAPP(fun(X_c,X_b),fun(fun(X_c,bool),fun(X_b,bool)),image(X_c,X_b),F),A_3)),hAPP(fun(X_c,bool),fun(X_b,bool),hAPP(fun(X_c,X_b),fun(fun(X_c,bool),fun(X_b,bool)),image(X_c,X_b),F),B_2)))) ).
fof(fact_776_Un__Int__assoc__eq,axiom,
! [X_b,A_3,B_2,C_3] :
( hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_3),B_2)),C_3) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_3),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),B_2),C_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)),C_3),A_3)) ) ).
fof(fact_777_image__diff__subset,axiom,
! [X_b,X_c,F,A_3,B_2] : hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),hAPP(fun(X_c,bool),fun(X_b,bool),hAPP(fun(X_c,X_b),fun(fun(X_c,bool),fun(X_b,bool)),image(X_c,X_b),F),A_3)),hAPP(fun(X_c,bool),fun(X_b,bool),hAPP(fun(X_c,X_b),fun(fun(X_c,bool),fun(X_b,bool)),image(X_c,X_b),F),B_2))),hAPP(fun(X_c,bool),fun(X_b,bool),hAPP(fun(X_c,X_b),fun(fun(X_c,bool),fun(X_b,bool)),image(X_c,X_b),F),hAPP(fun(X_c,bool),fun(X_c,bool),hAPP(fun(X_c,bool),fun(fun(X_c,bool),fun(X_c,bool)),minus_minus(fun(X_c,bool)),A_3),B_2)))) ).
fof(fact_778_Diff__subset__conv,axiom,
! [X_b,A_3,B_2,C_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_3),B_2)),C_3))
<=> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),A_3),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),B_2),C_3))) ) ).
fof(fact_779_Diff__partition,axiom,
! [X_b,A_3,B_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),A_3),B_2))
=> hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),A_3),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),B_2),A_3)) = ti(fun(X_b,bool),B_2) ) ).
fof(fact_780_folding__image__simple__idem_Osubset__idem,axiom,
! [X_c,X_b,B_2,A_3,F,Z_1,G,F_1] :
( hBOOL(hAPP(fun(fun(X_c,bool),X_b),bool,hAPP(fun(X_c,X_b),fun(fun(fun(X_c,bool),X_b),bool),hAPP(X_b,fun(fun(X_c,X_b),fun(fun(fun(X_c,bool),X_b),bool)),hAPP(fun(X_b,fun(X_b,X_b)),fun(X_b,fun(fun(X_c,X_b),fun(fun(fun(X_c,bool),X_b),bool))),finite908156982e_idem(X_b,X_c),F),Z_1),G),F_1))
=> ( hBOOL(hAPP(fun(X_c,bool),bool,finite_finite_1(X_c),A_3))
=> ( hBOOL(hAPP(fun(X_c,bool),bool,hAPP(fun(X_c,bool),fun(fun(X_c,bool),bool),ord_less_eq(fun(X_c,bool)),B_2),A_3))
=> hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),F,hAPP(fun(X_c,bool),X_b,F_1,B_2)),hAPP(fun(X_c,bool),X_b,F_1,A_3)) = hAPP(fun(X_c,bool),X_b,F_1,A_3) ) ) ) ).
fof(fact_781_subset__insert__iff,axiom,
! [X_b,A_3,X_2,B_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),A_3),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_2),B_2)))
<=> ( ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_2),A_3))
=> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_3),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_2),bot_bot(fun(X_b,bool))))),B_2)) )
& ( ~ hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_2),A_3))
=> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),A_3),B_2)) ) ) ) ).
fof(fact_782_diff__single__insert,axiom,
! [X_b,A_3,X_2,B_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),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_2),bot_bot(fun(X_b,bool))))),B_2))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_2),A_3))
=> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),A_3),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_2),B_2))) ) ) ).
fof(fact_783_card__mono,axiom,
! [X_b,A_3,B_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),B_2))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),A_3),B_2))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(fun(X_b,bool),nat,finite_card(X_b),A_3)),hAPP(fun(X_b,bool),nat,finite_card(X_b),B_2))) ) ) ).
fof(fact_784_card__seteq,axiom,
! [X_b,A_3,B_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),B_2))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),A_3),B_2))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(fun(X_b,bool),nat,finite_card(X_b),B_2)),hAPP(fun(X_b,bool),nat,finite_card(X_b),A_3)))
=> ti(fun(X_b,bool),A_3) = ti(fun(X_b,bool),B_2) ) ) ) ).
fof(fact_785_folding__one__idem_Osubset__idem,axiom,
! [X_b,B_2,A_3,F,F_1] :
( hBOOL(hAPP(fun(fun(X_b,bool),X_b),bool,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(fun(X_b,bool),X_b),bool),finite2073411215e_idem(X_b),F),F_1))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( ti(fun(X_b,bool),B_2) != bot_bot(fun(X_b,bool))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),B_2),A_3))
=> hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),F,hAPP(fun(X_b,bool),X_b,F_1,B_2)),hAPP(fun(X_b,bool),X_b,F_1,A_3)) = hAPP(fun(X_b,bool),X_b,F_1,A_3) ) ) ) ) ).
fof(fact_786_card__Diff__subset,axiom,
! [X_b,A_3,B_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),B_2))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),B_2),A_3))
=> hAPP(fun(X_b,bool),nat,finite_card(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_3),B_2)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(fun(X_b,bool),nat,finite_card(X_b),A_3)),hAPP(fun(X_b,bool),nat,finite_card(X_b),B_2)) ) ) ).
fof(fact_787_Sup__fin_Osubset__idem,axiom,
! [X_b] :
( lattice(X_b)
=> ! [B_2,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( ti(fun(X_b,bool),B_2) != bot_bot(fun(X_b,bool))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),B_2),A_3))
=> hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),semilattice_sup_sup(X_b),hAPP(fun(X_b,bool),X_b,big_lattice_Sup_fin(X_b),B_2)),hAPP(fun(X_b,bool),X_b,big_lattice_Sup_fin(X_b),A_3)) = hAPP(fun(X_b,bool),X_b,big_lattice_Sup_fin(X_b),A_3) ) ) ) ) ).
fof(fact_788_Inf__fin_Osubset__idem,axiom,
! [X_b] :
( lattice(X_b)
=> ! [B_2,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( ti(fun(X_b,bool),B_2) != bot_bot(fun(X_b,bool))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),B_2),A_3))
=> hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),semilattice_inf_inf(X_b),hAPP(fun(X_b,bool),X_b,big_lattice_Inf_fin(X_b),B_2)),hAPP(fun(X_b,bool),X_b,big_lattice_Inf_fin(X_b),A_3)) = hAPP(fun(X_b,bool),X_b,big_lattice_Inf_fin(X_b),A_3) ) ) ) ) ).
fof(fact_789_lift__Suc__mono__le,axiom,
! [X_b] :
( order(X_b)
=> ! [N_1,N_4,F] :
( ! [N] : hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(nat,X_b,F,N)),hAPP(nat,X_b,F,hAPP(nat,nat,suc,N))))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_1),N_4))
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(nat,X_b,F,N_1)),hAPP(nat,X_b,F,N_4))) ) ) ) ).
fof(fact_790_finite__subset__induct,axiom,
! [X_b,Pa,A_3,F_1] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),F_1))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),F_1),A_3))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,Pa,bot_bot(fun(X_b,bool))))
=> ( ! [A_4,F_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),F_2))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),A_4),A_3))
=> ( ~ hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),A_4),F_2))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,Pa,F_2))
=> hBOOL(hAPP(fun(X_b,bool),bool,Pa,hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_4),F_2))) ) ) ) )
=> hBOOL(hAPP(fun(X_b,bool),bool,Pa,F_1)) ) ) ) ) ).
fof(fact_791_equalityI,axiom,
! [X_b,A_3,B_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),A_3),B_2))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),B_2),A_3))
=> ti(fun(X_b,bool),A_3) = ti(fun(X_b,bool),B_2) ) ) ).
fof(fact_792_subsetD,axiom,
! [X_b,C_1,A_3,B_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),A_3),B_2))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),C_1),A_3))
=> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),C_1),B_2)) ) ) ).
fof(fact_793_equalityE,axiom,
! [X_b,A_3,B_2] :
( ti(fun(X_b,bool),A_3) = ti(fun(X_b,bool),B_2)
=> ~ ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),A_3),B_2))
=> ~ hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),B_2),A_3)) ) ) ).
fof(fact_794_subset__trans,axiom,
! [X_b,C_3,A_3,B_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),A_3),B_2))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),B_2),C_3))
=> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),A_3),C_3)) ) ) ).
fof(fact_795_set__mp,axiom,
! [X_b,X_2,A_3,B_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),A_3),B_2))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_2),A_3))
=> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_2),B_2)) ) ) ).
fof(fact_796_set__rev__mp,axiom,
! [X_b,B_2,X_2,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_2),A_3))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),A_3),B_2))
=> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_2),B_2)) ) ) ).
fof(fact_797_predicate1D,axiom,
! [X_b,X_2,Pa,Qa] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),Pa),Qa))
=> ( hBOOL(hAPP(X_b,bool,Pa,X_2))
=> hBOOL(hAPP(X_b,bool,Qa,X_2)) ) ) ).
fof(fact_798_in__mono,axiom,
! [X_b,X_2,A_3,B_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),A_3),B_2))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_2),A_3))
=> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_2),B_2)) ) ) ).
fof(fact_799_equalityD2,axiom,
! [X_b,A_3,B_2] :
( ti(fun(X_b,bool),A_3) = ti(fun(X_b,bool),B_2)
=> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),B_2),A_3)) ) ).
fof(fact_800_equalityD1,axiom,
! [X_b,A_3,B_2] :
( ti(fun(X_b,bool),A_3) = ti(fun(X_b,bool),B_2)
=> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),A_3),B_2)) ) ).
fof(fact_801_rev__predicate1D,axiom,
! [X_b,Qa,Pa,X_2] :
( hBOOL(hAPP(X_b,bool,Pa,X_2))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),Pa),Qa))
=> hBOOL(hAPP(X_b,bool,Qa,X_2)) ) ) ).
fof(fact_802_set__eq__subset,axiom,
! [X_b,A_3,B_2] :
( ti(fun(X_b,bool),A_3) = ti(fun(X_b,bool),B_2)
<=> ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),A_3),B_2))
& hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),B_2),A_3)) ) ) ).
fof(fact_803_subset__refl,axiom,
! [X_b,A_3] : hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),A_3),A_3)) ).
fof(fact_804_pred__subset__eq,axiom,
! [X_b,R_1,S_1] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(fun(X_b,bool),bool)),fun(fun(X_b,bool),fun(X_b,bool)),combc(X_b,fun(X_b,bool),bool),member(X_b)),R_1)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(fun(X_b,bool),bool)),fun(fun(X_b,bool),fun(X_b,bool)),combc(X_b,fun(X_b,bool),bool),member(X_b)),S_1)))
<=> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),R_1),S_1)) ) ).
fof(fact_805_subsetI,axiom,
! [X_b,B_2,A_3] :
( ! [X] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X),A_3))
=> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X),B_2)) )
=> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),A_3),B_2)) ) ).
fof(fact_806_finite__subset__image,axiom,
! [X_c,X_b,F,A_3,B_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),B_2))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),B_2),hAPP(fun(X_c,bool),fun(X_b,bool),hAPP(fun(X_c,X_b),fun(fun(X_c,bool),fun(X_b,bool)),image(X_c,X_b),F),A_3)))
=> ? [C_2] :
( hBOOL(hAPP(fun(X_c,bool),bool,hAPP(fun(X_c,bool),fun(fun(X_c,bool),bool),ord_less_eq(fun(X_c,bool)),C_2),A_3))
& hBOOL(hAPP(fun(X_c,bool),bool,finite_finite_1(X_c),C_2))
& ti(fun(X_b,bool),B_2) = hAPP(fun(X_c,bool),fun(X_b,bool),hAPP(fun(X_c,X_b),fun(fun(X_c,bool),fun(X_b,bool)),image(X_c,X_b),F),C_2) ) ) ) ).
fof(fact_807_Int__Collect__mono,axiom,
! [X_b,Qa,Pa,A_3,B_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),A_3),B_2))
=> ( ! [X] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X),A_3))
=> ( hBOOL(hAPP(X_b,bool,Pa,X))
=> hBOOL(hAPP(X_b,bool,Qa,X)) ) )
=> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),A_3),hAPP(fun(X_b,bool),fun(X_b,bool),collect(X_b),Pa))),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),B_2),hAPP(fun(X_b,bool),fun(X_b,bool),collect(X_b),Qa)))) ) ) ).
fof(fact_808_Suc__le__D,axiom,
! [N_2,M_4] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,suc,N_2)),M_4))
=> ? [M_2] : M_4 = hAPP(nat,nat,suc,M_2) ) ).
fof(fact_809_Suc__le__D__lemma,axiom,
! [Pa,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] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_1),M_2))
=> hBOOL(hAPP(nat,bool,Pa,hAPP(nat,nat,suc,M_2))) )
=> hBOOL(hAPP(nat,bool,Pa,M_3)) ) ) ).
fof(fact_810_image__subsetI,axiom,
! [X_c,X_b,F,B_2,A_3] :
( ! [X] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X),A_3))
=> hBOOL(hAPP(fun(X_c,bool),bool,hAPP(X_c,fun(fun(X_c,bool),bool),member(X_c),hAPP(X_b,X_c,F,X)),B_2)) )
=> hBOOL(hAPP(fun(X_c,bool),bool,hAPP(fun(X_c,bool),fun(fun(X_c,bool),bool),ord_less_eq(fun(X_c,bool)),hAPP(fun(X_b,bool),fun(X_c,bool),hAPP(fun(X_b,X_c),fun(fun(X_b,bool),fun(X_c,bool)),image(X_b,X_c),F),A_3)),B_2)) ) ).
fof(fact_811_le__funI,axiom,
! [X_b,X_c] :
( ord(X_c)
=> ! [F,G] :
( ! [X] : hBOOL(hAPP(X_c,bool,hAPP(X_c,fun(X_c,bool),ord_less_eq(X_c),hAPP(X_b,X_c,F,X)),hAPP(X_b,X_c,G,X)))
=> hBOOL(hAPP(fun(X_b,X_c),bool,hAPP(fun(X_b,X_c),fun(fun(X_b,X_c),bool),ord_less_eq(fun(X_b,X_c)),F),G)) ) ) ).
fof(fact_812_finite__nat__set__iff__bounded__le,axiom,
! [N_3] :
( hBOOL(hAPP(fun(nat,bool),bool,finite_finite_1(nat),N_3))
<=> ? [M_2] :
! [X] :
( hBOOL(hAPP(fun(nat,bool),bool,hAPP(nat,fun(fun(nat,bool),bool),member(nat),X),N_3))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),X),M_2)) ) ) ).
fof(fact_813_finite__less__ub,axiom,
! [U,F] :
( ! [N] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N),hAPP(nat,nat,F,N)))
=> hBOOL(hAPP(fun(nat,bool),bool,finite_finite_1(nat),hAPP(fun(nat,bool),fun(nat,bool),collect(nat),hAPP(nat,fun(nat,bool),hAPP(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool)),combc(nat,nat,bool),hAPP(fun(nat,nat),fun(nat,fun(nat,bool)),hAPP(fun(nat,fun(nat,bool)),fun(fun(nat,nat),fun(nat,fun(nat,bool))),combb(nat,fun(nat,bool),nat),ord_less_eq(nat)),F)),U)))) ) ).
fof(fact_814_termination__basic__simps_I3_J,axiom,
! [Z,X_1,Y] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),X_1),Y))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),X_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),Y),Z))) ) ).
fof(fact_815_termination__basic__simps_I4_J,axiom,
! [Y,X_1,Z] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),X_1),Z))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),X_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),Y),Z))) ) ).
fof(fact_816_flat__lub__def,axiom,
! [X_b,A_3,B_1] :
( ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),A_3),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),B_1),bot_bot(fun(X_b,bool)))))
=> hAPP(fun(X_b,bool),X_b,hAPP(X_b,fun(fun(X_b,bool),X_b),partial_flat_lub(X_b),B_1),A_3) = ti(X_b,B_1) )
& ( ~ hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),A_3),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),B_1),bot_bot(fun(X_b,bool)))))
=> hAPP(fun(X_b,bool),X_b,hAPP(X_b,fun(fun(X_b,bool),X_b),partial_flat_lub(X_b),B_1),A_3) = hAPP(fun(X_b,bool),X_b,the_1(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(fun(X_b,bool),bool)),fun(fun(X_b,bool),fun(X_b,bool)),combc(X_b,fun(X_b,bool),bool),member(X_b)),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_3),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),B_1),bot_bot(fun(X_b,bool)))))) ) ) ).
fof(fact_817_fold1__def,axiom,
! [X_b,F,A_3] : hAPP(fun(X_b,bool),X_b,hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),X_b),finite_fold1(X_b),F),A_3) = hAPP(fun(X_b,bool),X_b,the_1(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),fun(X_b,bool)),finite_fold1Set(X_b),F),A_3)) ).
fof(fact_818_fold1Set__nonempty,axiom,
! [X_b,F,A_3,X_2] :
( hBOOL(hAPP(X_b,bool,hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),fun(X_b,bool)),finite_fold1Set(X_b),F),A_3),X_2))
=> ti(fun(X_b,bool),A_3) != bot_bot(fun(X_b,bool)) ) ).
fof(fact_819_empty__fold1SetE,axiom,
! [X_b,F,X_2] : ~ hBOOL(hAPP(X_b,bool,hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),fun(X_b,bool)),finite_fold1Set(X_b),F),bot_bot(fun(X_b,bool))),X_2)) ).
fof(fact_820_fold1Set__sing,axiom,
! [X_b,F,A_2,B_1] :
( hBOOL(hAPP(X_b,bool,hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),fun(X_b,bool)),finite_fold1Set(X_b),F),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_2),bot_bot(fun(X_b,bool)))),B_1))
<=> ti(X_b,A_2) = ti(X_b,B_1) ) ).
fof(fact_821_finite__nonempty__imp__fold1Set,axiom,
! [X_b,F,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( ti(fun(X_b,bool),A_3) != bot_bot(fun(X_b,bool))
=> ? [X1] : hBOOL(hAPP(X_b,bool,hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_b,bool),fun(X_b,bool)),finite_fold1Set(X_b),F),A_3),X1)) ) ) ).
fof(fact_822_card__Diff1__less,axiom,
! [X_b,X_2,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_2),A_3))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(fun(X_b,bool),nat,finite_card(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_3),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),X_2),bot_bot(fun(X_b,bool)))))),hAPP(fun(X_b,bool),nat,finite_card(X_b),A_3))) ) ) ).
fof(fact_823_less__zeroE,axiom,
! [N_2] : ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N_2),zero_zero(nat))) ).
fof(fact_824_lessI,axiom,
! [N_2] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N_2),hAPP(nat,nat,suc,N_2))) ).
fof(fact_825_Suc__mono,axiom,
! [M,N_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N_2))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,suc,M)),hAPP(nat,nat,suc,N_2))) ) ).
fof(fact_826_finite__Collect__less__nat,axiom,
! [K_1] : hBOOL(hAPP(fun(nat,bool),bool,finite_finite_1(nat),hAPP(fun(nat,bool),fun(nat,bool),collect(nat),hAPP(nat,fun(nat,bool),hAPP(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool)),combc(nat,nat,bool),ord_less(nat)),K_1)))) ).
fof(fact_827_zero__less__Suc,axiom,
! [N_2] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(nat,nat,suc,N_2))) ).
fof(fact_828_finite__M__bounded__by__nat,axiom,
! [Pa,I_1] : hBOOL(hAPP(fun(nat,bool),bool,finite_finite_1(nat),hAPP(fun(nat,bool),fun(nat,bool),collect(nat),hAPP(fun(nat,bool),fun(nat,bool),hAPP(fun(nat,fun(bool,bool)),fun(fun(nat,bool),fun(nat,bool)),combs(nat,bool,bool),hAPP(fun(nat,bool),fun(nat,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(nat,bool),fun(nat,fun(bool,bool))),combb(bool,fun(bool,bool),nat),fconj),Pa)),hAPP(nat,fun(nat,bool),hAPP(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool)),combc(nat,nat,bool),ord_less(nat)),I_1))))) ).
fof(fact_829_finite__nat__set__iff__bounded,axiom,
! [N_3] :
( hBOOL(hAPP(fun(nat,bool),bool,finite_finite_1(nat),N_3))
<=> ? [M_2] :
! [X] :
( hBOOL(hAPP(fun(nat,bool),bool,hAPP(nat,fun(fun(nat,bool),bool),member(nat),X),N_3))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),X),M_2)) ) ) ).
fof(fact_830_xt1_I8_J,axiom,
! [X_a] :
( order(X_a)
=> ! [Z,Y,X_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),X_1))
=> ( 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_1)) ) ) ) ).
fof(fact_831_order__le__less__trans,axiom,
! [X_a] :
( preorder(X_a)
=> ! [Z,X_1,Y] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_1),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_1),Z)) ) ) ) ).
fof(fact_832_xt1_I7_J,axiom,
! [X_a] :
( order(X_a)
=> ! [Z,Y,X_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y),X_1))
=> ( 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_1)) ) ) ) ).
fof(fact_833_order__less__le__trans,axiom,
! [X_a] :
( preorder(X_a)
=> ! [Z,X_1,Y] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_1),Y))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),Z))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_1),Z)) ) ) ) ).
fof(fact_834_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_835_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_836_order__le__imp__less__or__eq,axiom,
! [X_a] :
( order(X_a)
=> ! [X_1,Y] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_1),Y))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_1),Y))
| ti(X_a,X_1) = ti(X_a,Y) ) ) ) ).
fof(fact_837_linorder__antisym__conv2,axiom,
! [X_b] :
( linorder(X_b)
=> ! [X_2,Y_1] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),X_2),Y_1))
=> ( ~ hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),X_2),Y_1))
<=> ti(X_b,X_2) = ti(X_b,Y_1) ) ) ) ).
fof(fact_838_order__less__imp__le,axiom,
! [X_a] :
( preorder(X_a)
=> ! [X_1,Y] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_1),Y))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_1),Y)) ) ) ).
fof(fact_839_leD,axiom,
! [X_a] :
( linorder(X_a)
=> ! [Y,X_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),X_1))
=> ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_1),Y)) ) ) ).
fof(fact_840_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_841_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_842_linorder__antisym__conv1,axiom,
! [X_b] :
( linorder(X_b)
=> ! [X_2,Y_1] :
( ~ hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),X_2),Y_1))
=> ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),X_2),Y_1))
<=> ti(X_b,X_2) = ti(X_b,Y_1) ) ) ) ).
fof(fact_843_not__leE,axiom,
! [X_a] :
( linorder(X_a)
=> ! [Y,X_1] :
( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),X_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_1),Y)) ) ) ).
fof(fact_844_leI,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X_1,Y] :
( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_1),Y))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),X_1)) ) ) ).
fof(fact_845_order__le__less,axiom,
! [X_b] :
( order(X_b)
=> ! [X_2,Y_1] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),X_2),Y_1))
<=> ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),X_2),Y_1))
| ti(X_b,X_2) = ti(X_b,Y_1) ) ) ) ).
fof(fact_846_less__le__not__le,axiom,
! [X_b] :
( preorder(X_b)
=> ! [X_2,Y_1] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),X_2),Y_1))
<=> ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),X_2),Y_1))
& ~ hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),Y_1),X_2)) ) ) ) ).
fof(fact_847_order__less__le,axiom,
! [X_b] :
( order(X_b)
=> ! [X_2,Y_1] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),X_2),Y_1))
<=> ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),X_2),Y_1))
& ti(X_b,X_2) != ti(X_b,Y_1) ) ) ) ).
fof(fact_848_linorder__le__less__linear,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X_1,Y] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_1),Y))
| hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y),X_1)) ) ) ).
fof(fact_849_linorder__not__le,axiom,
! [X_b] :
( linorder(X_b)
=> ! [X_2,Y_1] :
( ~ hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),X_2),Y_1))
<=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),Y_1),X_2)) ) ) ).
fof(fact_850_linorder__not__less,axiom,
! [X_b] :
( linorder(X_b)
=> ! [X_2,Y_1] :
( ~ hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),X_2),Y_1))
<=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),Y_1),X_2)) ) ) ).
fof(fact_851_less__or__eq__imp__le,axiom,
! [M,N_2] :
( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N_2))
| M = N_2 )
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N_2)) ) ).
fof(fact_852_le__neq__implies__less,axiom,
! [M,N_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N_2))
=> ( M != N_2
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N_2)) ) ) ).
fof(fact_853_less__imp__le__nat,axiom,
! [M,N_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N_2))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N_2)) ) ).
fof(fact_854_le__eq__less__or__eq,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),N_1))
| M_1 = N_1 ) ) ).
fof(fact_855_nat__less__le,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))
& M_1 != N_1 ) ) ).
fof(fact_856_nat__seg__image__imp__finite,axiom,
! [X_b,A_3,F,N_1] :
( ti(fun(X_b,bool),A_3) = hAPP(fun(nat,bool),fun(X_b,bool),hAPP(fun(nat,X_b),fun(fun(nat,bool),fun(X_b,bool)),image(nat,X_b),F),hAPP(fun(nat,bool),fun(nat,bool),collect(nat),hAPP(nat,fun(nat,bool),hAPP(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool)),combc(nat,nat,bool),ord_less(nat)),N_1)))
=> hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3)) ) ).
fof(fact_857_add__less__le__mono,axiom,
! [X_a] :
( ordere223160158up_add(X_a)
=> ! [C,D_1,A_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C),D_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),C)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),B),D_1))) ) ) ) ).
fof(fact_858_add__le__less__mono,axiom,
! [X_a] :
( ordere223160158up_add(X_a)
=> ! [C,D_1,A_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C),D_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),C)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),B),D_1))) ) ) ) ).
fof(fact_859_Suc__le__lessD,axiom,
! [M,N_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,suc,M)),N_2))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N_2)) ) ).
fof(fact_860_le__less__Suc__eq,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),N_1),hAPP(nat,nat,suc,M_1)))
<=> N_1 = M_1 ) ) ).
fof(fact_861_Suc__leI,axiom,
! [M,N_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N_2))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,suc,M)),N_2)) ) ).
fof(fact_862_le__imp__less__Suc,axiom,
! [M,N_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N_2))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),hAPP(nat,nat,suc,N_2))) ) ).
fof(fact_863_Suc__le__eq,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_864_less__Suc__eq__le,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_eq(nat),M_1),N_1)) ) ).
fof(fact_865_less__eq__Suc__le,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_eq(nat),hAPP(nat,nat,suc,N_1)),M_1)) ) ).
fof(fact_866_less__diff__iff,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))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M_1),K_1)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N_1),K_1)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_1),N_1)) ) ) ) ).
fof(fact_867_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_868_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_869_bot__less,axiom,
! [X_b] :
( bot(X_b)
=> ! [A_2] :
( ti(X_b,A_2) != bot_bot(X_b)
<=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),bot_bot(X_b)),A_2)) ) ) ).
fof(fact_870_not__add__less1,axiom,
! [I,J_1] : ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),I),J_1)),I)) ).
fof(fact_871_not__add__less2,axiom,
! [J_1,I] : ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),J_1),I)),I)) ).
fof(fact_872_nat__add__left__cancel__less,axiom,
! [K_1,M_1,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),K_1),M_1)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),K_1),N_1)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_1),N_1)) ) ).
fof(fact_873_trans__less__add1,axiom,
! [M,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),I),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),J_1),M))) ) ).
fof(fact_874_trans__less__add2,axiom,
! [M,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),I),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),J_1))) ) ).
fof(fact_875_add__less__mono1,axiom,
! [K_2,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,hAPP(nat,fun(nat,nat),plus_plus(nat),I),K_2)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),J_1),K_2))) ) ).
fof(fact_876_add__less__mono,axiom,
! [K_2,L,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),K_2),L))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),I),K_2)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),J_1),L))) ) ) ).
fof(fact_877_less__add__eq__less,axiom,
! [M,N_2,K_2,L] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),K_2),L))
=> ( hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),L) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),K_2),N_2)
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N_2)) ) ) ).
fof(fact_878_add__lessD1,axiom,
! [I,J_1,K_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),I),J_1)),K_2))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I),K_2)) ) ).
fof(fact_879_add__less__cancel__right,axiom,
! [X_b] :
( ordere236663937imp_le(X_b)
=> ! [A_2,C_1,B_1] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),A_2),C_1)),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),B_1),C_1)))
<=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),A_2),B_1)) ) ) ).
fof(fact_880_add__less__cancel__left,axiom,
! [X_b] :
( ordere236663937imp_le(X_b)
=> ! [C_1,A_2,B_1] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),C_1),A_2)),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),C_1),B_1)))
<=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),A_2),B_1)) ) ) ).
fof(fact_881_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_882_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_883_add__strict__mono,axiom,
! [X_a] :
( ordere223160158up_add(X_a)
=> ! [C,D_1,A_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C),D_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),C)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),B),D_1))) ) ) ) ).
fof(fact_884_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_885_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_886_diff__less__mono2,axiom,
! [L,M,N_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N_2))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),L))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),L),N_2)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),L),M))) ) ) ).
fof(fact_887_less__imp__diff__less,axiom,
! [N_2,J_1,K_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),J_1),K_2))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),J_1),N_2)),K_2)) ) ).
fof(fact_888_diff__eq__diff__less,axiom,
! [X_b] :
( ordered_ab_group_add(X_b)
=> ! [A_2,B_1,C_1,D] :
( hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),minus_minus(X_b),A_2),B_1) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),minus_minus(X_b),C_1),D)
=> ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),A_2),B_1))
<=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),C_1),D)) ) ) ) ).
fof(fact_889_less__supI2,axiom,
! [X_a] :
( semilattice_sup(X_a)
=> ! [A_1,X_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_1),B))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),A_1),B))) ) ) ).
fof(fact_890_less__supI1,axiom,
! [X_a] :
( semilattice_sup(X_a)
=> ! [B,X_1,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_1),A_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),semilattice_sup_sup(X_a),A_1),B))) ) ) ).
fof(fact_891_less__infI2,axiom,
! [X_a] :
( semilattice_inf(X_a)
=> ! [A_1,B,X_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),X_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),semilattice_inf_inf(X_a),A_1),B)),X_1)) ) ) ).
fof(fact_892_less__infI1,axiom,
! [X_a] :
( semilattice_inf(X_a)
=> ! [B,A_1,X_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),X_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),semilattice_inf_inf(X_a),A_1),B)),X_1)) ) ) ).
fof(fact_893_not__less0,axiom,
! [N_2] : ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N_2),zero_zero(nat))) ).
fof(fact_894_neq0__conv,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_895_less__nat__zero__code,axiom,
! [N_2] : ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N_2),zero_zero(nat))) ).
fof(fact_896_gr__implies__not0,axiom,
! [M,N_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N_2))
=> N_2 != zero_zero(nat) ) ).
fof(fact_897_gr0I,axiom,
! [N_2] :
( N_2 != zero_zero(nat)
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N_2)) ) ).
fof(fact_898_not__less__eq,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),N_1),hAPP(nat,nat,suc,M_1))) ) ).
fof(fact_899_less__Suc__eq,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_900_Suc__less__eq,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_901_not__less__less__Suc__eq,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)))
<=> N_1 = M_1 ) ) ).
fof(fact_902_less__antisym,axiom,
! [N_2,M] :
( ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N_2),M))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N_2),hAPP(nat,nat,suc,M)))
=> M = N_2 ) ) ).
fof(fact_903_less__SucI,axiom,
! [M,N_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N_2))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),hAPP(nat,nat,suc,N_2))) ) ).
fof(fact_904_Suc__lessI,axiom,
! [M,N_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N_2))
=> ( hAPP(nat,nat,suc,M) != N_2
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,suc,M)),N_2)) ) ) ).
fof(fact_905_less__trans__Suc,axiom,
! [K_2,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),J_1),K_2))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,suc,I)),K_2)) ) ) ).
fof(fact_906_less__SucE,axiom,
! [M,N_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),hAPP(nat,nat,suc,N_2)))
=> ( ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N_2))
=> M = N_2 ) ) ).
fof(fact_907_Suc__lessD,axiom,
! [M,N_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,suc,M)),N_2))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N_2)) ) ).
fof(fact_908_Suc__less__SucD,axiom,
! [M,N_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,suc,M)),hAPP(nat,nat,suc,N_2)))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N_2)) ) ).
fof(fact_909_less__not__refl,axiom,
! [N_2] : ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N_2),N_2)) ).
fof(fact_910_nat__neq__iff,axiom,
! [M_1,N_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),N_1),M_1)) ) ) ).
fof(fact_911_linorder__neqE__nat,axiom,
! [X_1,Y] :
( X_1 != Y
=> ( ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),X_1),Y))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Y),X_1)) ) ) ).
fof(fact_912_less__irrefl__nat,axiom,
! [N_2] : ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N_2),N_2)) ).
fof(fact_913_less__not__refl2,axiom,
! [N_2,M] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N_2),M))
=> M != N_2 ) ).
fof(fact_914_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_915_nat__less__cases,axiom,
! [Pa,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),Pa,N_1),M_1)) )
=> ( ( M_1 = N_1
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),Pa,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),Pa,N_1),M_1)) )
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),Pa,N_1),M_1)) ) ) ) ).
fof(fact_916_order__less__irrefl,axiom,
! [X_a] :
( preorder(X_a)
=> ! [X_1] : ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_1),X_1)) ) ).
fof(fact_917_linorder__neq__iff,axiom,
! [X_b] :
( linorder(X_b)
=> ! [X_2,Y_1] :
( ti(X_b,X_2) != ti(X_b,Y_1)
<=> ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),X_2),Y_1))
| hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),Y_1),X_2)) ) ) ) ).
fof(fact_918_not__less__iff__gr__or__eq,axiom,
! [X_b] :
( linorder(X_b)
=> ! [X_2,Y_1] :
( ~ hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),X_2),Y_1))
<=> ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),Y_1),X_2))
| ti(X_b,X_2) = ti(X_b,Y_1) ) ) ) ).
fof(fact_919_linorder__less__linear,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X_1,Y] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_1),Y))
| ti(X_a,X_1) = ti(X_a,Y)
| hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y),X_1)) ) ) ).
fof(fact_920_linorder__antisym__conv3,axiom,
! [X_b] :
( linorder(X_b)
=> ! [Y_1,X_2] :
( ~ hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),Y_1),X_2))
=> ( ~ hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),X_2),Y_1))
<=> ti(X_b,X_2) = ti(X_b,Y_1) ) ) ) ).
fof(fact_921_linorder__neqE,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X_1,Y] :
( ti(X_a,X_1) != ti(X_a,Y)
=> ( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_1),Y))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y),X_1)) ) ) ) ).
fof(fact_922_less__imp__neq,axiom,
! [X_a] :
( order(X_a)
=> ! [X_1,Y] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_1),Y))
=> ti(X_a,X_1) != ti(X_a,Y) ) ) ).
fof(fact_923_order__less__not__sym,axiom,
! [X_a] :
( preorder(X_a)
=> ! [X_1,Y] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_1),Y))
=> ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y),X_1)) ) ) ).
fof(fact_924_order__less__imp__not__less,axiom,
! [X_a] :
( preorder(X_a)
=> ! [X_1,Y] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_1),Y))
=> ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y),X_1)) ) ) ).
fof(fact_925_order__less__imp__not__eq,axiom,
! [X_a] :
( order(X_a)
=> ! [X_1,Y] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_1),Y))
=> ti(X_a,X_1) != ti(X_a,Y) ) ) ).
fof(fact_926_order__less__imp__not__eq2,axiom,
! [X_a] :
( order(X_a)
=> ! [X_1,Y] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_1),Y))
=> ti(X_a,Y) != ti(X_a,X_1) ) ) ).
fof(fact_927_order__less__imp__triv,axiom,
! [X_b] :
( preorder(X_b)
=> ! [Pa,X_2,Y_1] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),X_2),Y_1))
=> ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),Y_1),X_2))
=> hBOOL(Pa) ) ) ) ).
fof(fact_928_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_929_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_930_ord__eq__less__trans,axiom,
! [X_a] :
( ord(X_a)
=> ! [C,A_1,B] :
( A_1 = 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_931_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_932_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))
=> ( B = C
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),C)) ) ) ) ).
fof(fact_933_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_934_order__less__trans,axiom,
! [X_a] :
( preorder(X_a)
=> ! [Z,X_1,Y] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_1),Y))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y),Z))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_1),Z)) ) ) ) ).
fof(fact_935_xt1_I10_J,axiom,
! [X_a] :
( order(X_a)
=> ! [Z,Y,X_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y),X_1))
=> ( 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_1)) ) ) ) ).
fof(fact_936_order__less__asym,axiom,
! [X_a] :
( preorder(X_a)
=> ! [X_1,Y] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_1),Y))
=> ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y),X_1)) ) ) ).
fof(fact_937_linorder__cases,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X_1,Y] :
( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_1),Y))
=> ( ti(X_a,X_1) != ti(X_a,Y)
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y),X_1)) ) ) ) ).
fof(fact_938_linorder__neqE__linordered__idom,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [X_1,Y] :
( ti(X_a,X_1) != ti(X_a,Y)
=> ( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_1),Y))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y),X_1)) ) ) ) ).
fof(fact_939_less__diff__conv,axiom,
! [I_1,J_2,K_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),J_2),K_1)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),I_1),K_1)),J_2)) ) ).
fof(fact_940_add__diff__inverse,axiom,
! [M,N_2] :
( ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N_2))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N_2),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),N_2)) = M ) ).
fof(fact_941_Suc__mult__less__cancel1,axiom,
! [K_1,M_1,N_1] :
( 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_1)),M_1)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,suc,K_1)),N_1)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_1),N_1)) ) ).
fof(fact_942_diff__less__Suc,axiom,
! [M,N_2] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),N_2)),hAPP(nat,nat,suc,M))) ).
fof(fact_943_nat__0__less__mult__iff,axiom,
! [M_1,N_1] :
( 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_1),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),zero_zero(nat)),N_1)) ) ) ).
fof(fact_944_mult__less__cancel1,axiom,
! [K_1,M_1,N_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),M_1)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K_1),N_1)))
<=> ( 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),M_1),N_1)) ) ) ).
fof(fact_945_mult__less__cancel2,axiom,
! [M_1,K_1,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(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)))
<=> ( 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),M_1),N_1)) ) ) ).
fof(fact_946_mult__less__mono1,axiom,
! [K_2,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),zero_zero(nat)),K_2))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),I),K_2)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),J_1),K_2))) ) ) ).
fof(fact_947_mult__less__mono2,axiom,
! [K_2,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),zero_zero(nat)),K_2))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K_2),I)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K_2),J_1))) ) ) ).
fof(fact_948_nat__mult__eq__cancel1,axiom,
! [M_1,N_1,K_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K_1))
=> ( 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)
<=> M_1 = N_1 ) ) ).
fof(fact_949_nat__mult__less__cancel1,axiom,
! [M_1,N_1,K_1] :
( 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),M_1)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K_1),N_1)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_1),N_1)) ) ) ).
fof(fact_950_diff__less,axiom,
! [M,N_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N_2))
=> ( 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),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),N_2)),M)) ) ) ).
fof(fact_951_zero__less__diff,axiom,
! [N_1,M_1] :
( 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_1),M_1)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_1),N_1)) ) ).
fof(fact_952_less__add__Suc1,axiom,
! [I,M] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I),hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),I),M)))) ).
fof(fact_953_less__add__Suc2,axiom,
! [I,M] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I),hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),I)))) ).
fof(fact_954_less__iff__Suc__add,axiom,
! [M_1,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_1),N_1))
<=> ? [K] : N_1 = hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M_1),K)) ) ).
fof(fact_955_add__gr__0,axiom,
! [M_1,N_1] :
( 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_1),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),zero_zero(nat)),N_1)) ) ) ).
fof(fact_956_gr0__conv__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_957_less__Suc0,axiom,
! [N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N_1),hAPP(nat,nat,suc,zero_zero(nat))))
<=> N_1 = zero_zero(nat) ) ).
fof(fact_958_less__Suc__eq__0__disj,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)))
<=> ( M_1 = zero_zero(nat)
| ? [J] :
( M_1 = hAPP(nat,nat,suc,J)
& hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),J),N_1)) ) ) ) ).
fof(fact_959_less__1__mult,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [N_2,M] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),M))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),N_2))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),M),N_2))) ) ) ) ).
fof(fact_960_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_961_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_962_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_963_less__iff__diff__less__0,axiom,
! [X_b] :
( ordered_ab_group_add(X_b)
=> ! [A_2,B_1] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),A_2),B_1))
<=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),minus_minus(X_b),A_2),B_1)),zero_zero(X_b))) ) ) ).
fof(fact_964_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_965_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_966_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_967_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_968_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_969_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_970_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_971_mult__less__cancel__left__neg,axiom,
! [X_b] :
( linord581940658strict(X_b)
=> ! [A_2,B_1,C_1] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),C_1),zero_zero(X_b)))
=> ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),C_1),A_2)),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),C_1),B_1)))
<=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),B_1),A_2)) ) ) ) ).
fof(fact_972_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_973_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_974_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_975_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_976_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_977_mult__less__cancel__left__pos,axiom,
! [X_b] :
( linord581940658strict(X_b)
=> ! [A_2,B_1,C_1] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),zero_zero(X_b)),C_1))
=> ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),C_1),A_2)),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),C_1),B_1)))
<=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),A_2),B_1)) ) ) ) ).
fof(fact_978_mult__less__cancel__left__disj,axiom,
! [X_b] :
( linord581940658strict(X_b)
=> ! [C_1,A_2,B_1] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),C_1),A_2)),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),C_1),B_1)))
<=> ( ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),zero_zero(X_b)),C_1))
& hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),A_2),B_1)) )
| ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),C_1),zero_zero(X_b)))
& hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),B_1),A_2)) ) ) ) ) ).
fof(fact_979_mult__less__cancel__right__disj,axiom,
! [X_b] :
( linord581940658strict(X_b)
=> ! [A_2,C_1,B_1] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),A_2),C_1)),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),B_1),C_1)))
<=> ( ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),zero_zero(X_b)),C_1))
& hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),A_2),B_1)) )
| ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),C_1),zero_zero(X_b)))
& hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),B_1),A_2)) ) ) ) ) ).
fof(fact_980_not__square__less__zero,axiom,
! [X_a] :
( linordered_ring(X_a)
=> ! [A_1] : ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),A_1)),zero_zero(X_a))) ) ).
fof(fact_981_even__less__0__iff,axiom,
! [X_b] :
( linordered_idom(X_b)
=> ! [A_2] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),A_2),A_2)),zero_zero(X_b)))
<=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),A_2),zero_zero(X_b))) ) ) ).
fof(fact_982_zero__less__double__add__iff__zero__less__single__add,axiom,
! [X_b] :
( linord219039673up_add(X_b)
=> ! [A_2] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),zero_zero(X_b)),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),A_2),A_2)))
<=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),zero_zero(X_b)),A_2)) ) ) ).
fof(fact_983_double__add__less__zero__iff__single__add__less__zero,axiom,
! [X_b] :
( linord219039673up_add(X_b)
=> ! [A_2] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),A_2),A_2)),zero_zero(X_b)))
<=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),A_2),zero_zero(X_b))) ) ) ).
fof(fact_984_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_985_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_986_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_987_card__Collect__less__nat,axiom,
! [N_1] : hAPP(fun(nat,bool),nat,finite_card(nat),hAPP(fun(nat,bool),fun(nat,bool),collect(nat),hAPP(nat,fun(nat,bool),hAPP(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool)),combc(nat,nat,bool),ord_less(nat)),N_1))) = N_1 ).
fof(fact_988_less__eq__Suc__le__raw,axiom,
! [X] : hAPP(nat,fun(nat,bool),ord_less(nat),X) = hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,suc,X)) ).
fof(fact_989_termination__basic__simps_I1_J,axiom,
! [Z,X_1,Y] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),X_1),Y))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),X_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),Y),Z))) ) ).
fof(fact_990_termination__basic__simps_I2_J,axiom,
! [Y,X_1,Z] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),X_1),Z))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),X_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),Y),Z))) ) ).
fof(fact_991_termination__basic__simps_I5_J,axiom,
! [X_1,Y] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),X_1),Y))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),X_1),Y)) ) ).
fof(fact_992_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_993_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_994_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_995_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))) ) ) ) ).
%----Arities (62)
fof(arity_HOL_Obool___Lattices_Obounded__lattice,axiom,
bounded_lattice(bool) ).
fof(arity_fun___Lattices_Obounded__lattice,axiom,
! [T_2,T_1] :
( bounded_lattice(T_1)
=> bounded_lattice(fun(T_2,T_1)) ) ).
fof(arity_fun___Lattices_Obounded__lattice__bot,axiom,
! [T_2,T_1] :
( bounded_lattice(T_1)
=> bounded_lattice_bot(fun(T_2,T_1)) ) ).
fof(arity_fun___Lattices_Osemilattice__sup,axiom,
! [T_2,T_1] :
( lattice(T_1)
=> semilattice_sup(fun(T_2,T_1)) ) ).
fof(arity_fun___Lattices_Osemilattice__inf,axiom,
! [T_2,T_1] :
( lattice(T_1)
=> semilattice_inf(fun(T_2,T_1)) ) ).
fof(arity_fun___Lattices_Odistrib__lattice,axiom,
! [T_2,T_1] :
( distrib_lattice(T_1)
=> distrib_lattice(fun(T_2,T_1)) ) ).
fof(arity_fun___Orderings_Opreorder,axiom,
! [T_2,T_1] :
( preorder(T_1)
=> preorder(fun(T_2,T_1)) ) ).
fof(arity_fun___Finite__Set_Ofinite,axiom,
! [T_2,T_1] :
( ( finite_finite(T_1)
& finite_finite(T_2) )
=> finite_finite(fun(T_2,T_1)) ) ).
fof(arity_fun___Lattices_Olattice,axiom,
! [T_2,T_1] :
( lattice(T_1)
=> lattice(fun(T_2,T_1)) ) ).
fof(arity_fun___Orderings_Oorder,axiom,
! [T_2,T_1] :
( order(T_1)
=> order(fun(T_2,T_1)) ) ).
fof(arity_fun___Orderings_Oord,axiom,
! [T_2,T_1] :
( ord(T_1)
=> ord(fun(T_2,T_1)) ) ).
fof(arity_fun___Orderings_Obot,axiom,
! [T_2,T_1] :
( bot(T_1)
=> bot(fun(T_2,T_1)) ) ).
fof(arity_fun___Groups_Ominus,axiom,
! [T_2,T_1] :
( minus(T_1)
=> minus(fun(T_2,T_1)) ) ).
fof(arity_Nat_Onat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduc,axiom,
semiri456707255roduct(nat) ).
fof(arity_Nat_Onat___Groups_Oordered__cancel__ab__semigroup__add,axiom,
ordere223160158up_add(nat) ).
fof(arity_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
ordere236663937imp_le(nat) ).
fof(arity_Nat_Onat___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__semidom,axiom,
linordered_semidom(nat) ).
fof(arity_Nat_Onat___Lattices_Osemilattice__sup,axiom,
semilattice_sup(nat) ).
fof(arity_Nat_Onat___Lattices_Osemilattice__inf,axiom,
semilattice_inf(nat) ).
fof(arity_Nat_Onat___Lattices_Odistrib__lattice,axiom,
distrib_lattice(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___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___Lattices_Olattice,axiom,
lattice(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___Groups_Oone,axiom,
one(nat) ).
fof(arity_HOL_Obool___Lattices_Obounded__lattice__bot,axiom,
bounded_lattice_bot(bool) ).
fof(arity_HOL_Obool___Lattices_Osemilattice__sup,axiom,
semilattice_sup(bool) ).
fof(arity_HOL_Obool___Lattices_Osemilattice__inf,axiom,
semilattice_inf(bool) ).
fof(arity_HOL_Obool___Lattices_Odistrib__lattice,axiom,
distrib_lattice(bool) ).
fof(arity_HOL_Obool___Orderings_Opreorder,axiom,
preorder(bool) ).
fof(arity_HOL_Obool___Finite__Set_Ofinite,axiom,
finite_finite(bool) ).
fof(arity_HOL_Obool___Lattices_Olattice,axiom,
lattice(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 (24)
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_1,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_1),Y) = ti(X_a,X_1) ).
fof(help_If_2_1_T,axiom,
! [X_a,X_1,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_1),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_COMBI_1_1_U,axiom,
! [X_a,P] : hAPP(X_a,X_a,combi(X_a),P) = ti(X_a,P) ).
fof(help_COMBK_1_1_U,axiom,
! [X_b,X_a,P,Q] : hAPP(X_b,X_a,hAPP(X_a,fun(X_b,X_a),combk(X_a,X_b),P),Q) = ti(X_a,P) ).
fof(help_COMBS_1_1_U,axiom,
! [X_a,X_c,X_b,P,Q,R] : hAPP(X_a,X_c,hAPP(fun(X_a,X_b),fun(X_a,X_c),hAPP(fun(X_a,fun(X_b,X_c)),fun(fun(X_a,X_b),fun(X_a,X_c)),combs(X_a,X_b,X_c),P),Q),R) = hAPP(X_b,X_c,hAPP(X_a,fun(X_b,X_c),P,R),hAPP(X_a,X_b,Q,R)) ).
fof(help_fconj_1_1_U,axiom,
! [Q,P] :
( ~ hBOOL(P)
| ~ hBOOL(Q)
| hBOOL(hAPP(bool,bool,hAPP(bool,fun(bool,bool),fconj,P),Q)) ) ).
fof(help_fconj_2_1_U,axiom,
! [P,Q] :
( ~ hBOOL(hAPP(bool,bool,hAPP(bool,fun(bool,bool),fconj,P),Q))
| hBOOL(P) ) ).
fof(help_fconj_3_1_U,axiom,
! [P,Q] :
( ~ hBOOL(hAPP(bool,bool,hAPP(bool,fun(bool,bool),fconj,P),Q))
| hBOOL(Q) ) ).
fof(help_fdisj_1_1_U,axiom,
! [Q,P] :
( ~ hBOOL(P)
| hBOOL(hAPP(bool,bool,hAPP(bool,fun(bool,bool),fdisj,P),Q)) ) ).
fof(help_fdisj_2_1_U,axiom,
! [P,Q] :
( ~ hBOOL(Q)
| hBOOL(hAPP(bool,bool,hAPP(bool,fun(bool,bool),fdisj,P),Q)) ) ).
fof(help_fdisj_3_1_U,axiom,
! [P,Q] :
( ~ hBOOL(hAPP(bool,bool,hAPP(bool,fun(bool,bool),fdisj,P),Q))
| hBOOL(P)
| hBOOL(Q) ) ).
fof(help_fFalse_1_1_U,axiom,
~ hBOOL(fFalse) ).
fof(help_fFalse_1_1_T,axiom,
! [P] :
( ti(bool,P) = fTrue
| ti(bool,P) = fFalse ) ).
fof(help_fequal_1_1_T,axiom,
! [X_a,X_1,Y] :
( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),fequal(X_a),X_1),Y))
| ti(X_a,X_1) = ti(X_a,Y) ) ).
fof(help_fequal_2_1_T,axiom,
! [X_a,X_1,Y] :
( ti(X_a,X_1) != ti(X_a,Y)
| hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),fequal(X_a),X_1),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 (2)
fof(conj_0,hypothesis,
! [N] :
( ! [X] :
( hBOOL(hAPP(fun(hoare_2118899576triple(x_a),bool),bool,hAPP(hoare_2118899576triple(x_a),fun(fun(hoare_2118899576triple(x_a),bool),bool),member(hoare_2118899576triple(x_a)),X),hAPP(fun(hoare_2118899576triple(x_a),bool),fun(hoare_2118899576triple(x_a),bool),hAPP(fun(hoare_2118899576triple(x_a),bool),fun(fun(hoare_2118899576triple(x_a),bool),fun(hoare_2118899576triple(x_a),bool)),semilattice_sup_sup(fun(hoare_2118899576triple(x_a),bool)),g),hAPP(fun(pname,bool),fun(hoare_2118899576triple(x_a),bool),hAPP(fun(pname,hoare_2118899576triple(x_a)),fun(fun(pname,bool),fun(hoare_2118899576triple(x_a),bool)),image(pname,hoare_2118899576triple(x_a)),hAPP(fun(pname,fun(x_a,fun(state,bool))),fun(pname,hoare_2118899576triple(x_a)),hAPP(fun(pname,fun(fun(x_a,fun(state,bool)),hoare_2118899576triple(x_a))),fun(fun(pname,fun(x_a,fun(state,bool))),fun(pname,hoare_2118899576triple(x_a))),combs(pname,fun(x_a,fun(state,bool)),hoare_2118899576triple(x_a)),hAPP(fun(pname,com),fun(pname,fun(fun(x_a,fun(state,bool)),hoare_2118899576triple(x_a))),hAPP(fun(pname,fun(com,fun(fun(x_a,fun(state,bool)),hoare_2118899576triple(x_a)))),fun(fun(pname,com),fun(pname,fun(fun(x_a,fun(state,bool)),hoare_2118899576triple(x_a)))),combs(pname,com,fun(fun(x_a,fun(state,bool)),hoare_2118899576triple(x_a))),hAPP(fun(pname,fun(x_a,fun(state,bool))),fun(pname,fun(com,fun(fun(x_a,fun(state,bool)),hoare_2118899576triple(x_a)))),hAPP(fun(fun(x_a,fun(state,bool)),fun(com,fun(fun(x_a,fun(state,bool)),hoare_2118899576triple(x_a)))),fun(fun(pname,fun(x_a,fun(state,bool))),fun(pname,fun(com,fun(fun(x_a,fun(state,bool)),hoare_2118899576triple(x_a))))),combb(fun(x_a,fun(state,bool)),fun(com,fun(fun(x_a,fun(state,bool)),hoare_2118899576triple(x_a))),pname),hoare_759811442triple(x_a)),p)),body)),q)),procs))))
=> hBOOL(hAPP(hoare_2118899576triple(x_a),bool,hAPP(nat,fun(hoare_2118899576triple(x_a),bool),hoare_1942962616_valid(x_a),N),X)) )
=> ! [X] :
( hBOOL(hAPP(fun(hoare_2118899576triple(x_a),bool),bool,hAPP(hoare_2118899576triple(x_a),fun(fun(hoare_2118899576triple(x_a),bool),bool),member(hoare_2118899576triple(x_a)),X),hAPP(fun(pname,bool),fun(hoare_2118899576triple(x_a),bool),hAPP(fun(pname,hoare_2118899576triple(x_a)),fun(fun(pname,bool),fun(hoare_2118899576triple(x_a),bool)),image(pname,hoare_2118899576triple(x_a)),hAPP(fun(pname,fun(x_a,fun(state,bool))),fun(pname,hoare_2118899576triple(x_a)),hAPP(fun(pname,fun(fun(x_a,fun(state,bool)),hoare_2118899576triple(x_a))),fun(fun(pname,fun(x_a,fun(state,bool))),fun(pname,hoare_2118899576triple(x_a))),combs(pname,fun(x_a,fun(state,bool)),hoare_2118899576triple(x_a)),hAPP(fun(pname,com),fun(pname,fun(fun(x_a,fun(state,bool)),hoare_2118899576triple(x_a))),hAPP(fun(pname,fun(com,fun(fun(x_a,fun(state,bool)),hoare_2118899576triple(x_a)))),fun(fun(pname,com),fun(pname,fun(fun(x_a,fun(state,bool)),hoare_2118899576triple(x_a)))),combs(pname,com,fun(fun(x_a,fun(state,bool)),hoare_2118899576triple(x_a))),hAPP(fun(pname,fun(x_a,fun(state,bool))),fun(pname,fun(com,fun(fun(x_a,fun(state,bool)),hoare_2118899576triple(x_a)))),hAPP(fun(fun(x_a,fun(state,bool)),fun(com,fun(fun(x_a,fun(state,bool)),hoare_2118899576triple(x_a)))),fun(fun(pname,fun(x_a,fun(state,bool))),fun(pname,fun(com,fun(fun(x_a,fun(state,bool)),hoare_2118899576triple(x_a))))),combb(fun(x_a,fun(state,bool)),fun(com,fun(fun(x_a,fun(state,bool)),hoare_2118899576triple(x_a))),pname),hoare_759811442triple(x_a)),p)),hAPP(fun(pname,option(com)),fun(pname,com),hAPP(fun(option(com),com),fun(fun(pname,option(com)),fun(pname,com)),combb(option(com),com,pname),the(com)),body_1))),q)),procs)))
=> hBOOL(hAPP(hoare_2118899576triple(x_a),bool,hAPP(nat,fun(hoare_2118899576triple(x_a),bool),hoare_1942962616_valid(x_a),N),X)) ) ) ).
fof(conj_1,conjecture,
( ! [X] :
( hBOOL(hAPP(fun(hoare_2118899576triple(x_a),bool),bool,hAPP(hoare_2118899576triple(x_a),fun(fun(hoare_2118899576triple(x_a),bool),bool),member(hoare_2118899576triple(x_a)),X),g))
=> hBOOL(hAPP(hoare_2118899576triple(x_a),bool,hAPP(nat,fun(hoare_2118899576triple(x_a),bool),hoare_1942962616_valid(x_a),n),X)) )
=> ! [X] :
( hBOOL(hAPP(fun(hoare_2118899576triple(x_a),bool),bool,hAPP(hoare_2118899576triple(x_a),fun(fun(hoare_2118899576triple(x_a),bool),bool),member(hoare_2118899576triple(x_a)),X),hAPP(fun(pname,bool),fun(hoare_2118899576triple(x_a),bool),hAPP(fun(pname,hoare_2118899576triple(x_a)),fun(fun(pname,bool),fun(hoare_2118899576triple(x_a),bool)),image(pname,hoare_2118899576triple(x_a)),hAPP(fun(pname,fun(x_a,fun(state,bool))),fun(pname,hoare_2118899576triple(x_a)),hAPP(fun(pname,fun(fun(x_a,fun(state,bool)),hoare_2118899576triple(x_a))),fun(fun(pname,fun(x_a,fun(state,bool))),fun(pname,hoare_2118899576triple(x_a))),combs(pname,fun(x_a,fun(state,bool)),hoare_2118899576triple(x_a)),hAPP(fun(pname,com),fun(pname,fun(fun(x_a,fun(state,bool)),hoare_2118899576triple(x_a))),hAPP(fun(pname,fun(com,fun(fun(x_a,fun(state,bool)),hoare_2118899576triple(x_a)))),fun(fun(pname,com),fun(pname,fun(fun(x_a,fun(state,bool)),hoare_2118899576triple(x_a)))),combs(pname,com,fun(fun(x_a,fun(state,bool)),hoare_2118899576triple(x_a))),hAPP(fun(pname,fun(x_a,fun(state,bool))),fun(pname,fun(com,fun(fun(x_a,fun(state,bool)),hoare_2118899576triple(x_a)))),hAPP(fun(fun(x_a,fun(state,bool)),fun(com,fun(fun(x_a,fun(state,bool)),hoare_2118899576triple(x_a)))),fun(fun(pname,fun(x_a,fun(state,bool))),fun(pname,fun(com,fun(fun(x_a,fun(state,bool)),hoare_2118899576triple(x_a))))),combb(fun(x_a,fun(state,bool)),fun(com,fun(fun(x_a,fun(state,bool)),hoare_2118899576triple(x_a))),pname),hoare_759811442triple(x_a)),p)),body)),q)),procs)))
=> hBOOL(hAPP(hoare_2118899576triple(x_a),bool,hAPP(nat,fun(hoare_2118899576triple(x_a),bool),hoare_1942962616_valid(x_a),n),X)) ) ) ).
%------------------------------------------------------------------------------