TPTP Problem File: SWW471+6.p
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%------------------------------------------------------------------------------
% File : SWW471+6 : TPTP v8.2.0. Released v5.3.0.
% Domain : Software Verification
% Problem : Hoare's Logic with Procedures line 269, 500 axioms selected
% Version : Especial.
% English :
% Refs : [BN10] Boehme & Nipkow (2010), Sledgehammer: Judgement Day
% : [Bla11] Blanchette (2011), Email to Geoff Sutcliffe
% Source : [Bla11]
% Names : hoare_500_fofpt_l269 [Bla11]
% Status : Theorem
% Rating : 1.00 v5.3.0
% Syntax : Number of formulae : 633 ( 243 unt; 0 def)
% Number of atoms : 1408 ( 628 equ)
% Maximal formula atoms : 9 ( 2 avg)
% Number of connectives : 947 ( 172 ~; 30 |; 64 &)
% ( 64 <=>; 617 =>; 0 <=; 0 <~>)
% Maximal formula depth : 17 ( 5 avg)
% Maximal term depth : 18 ( 2 avg)
% Number of predicates : 26 ( 25 usr; 0 prp; 1-2 aty)
% Number of functors : 79 ( 79 usr; 28 con; 0-4 aty)
% Number of variables : 1949 (1926 !; 23 ?)
% SPC : FOF_THM_RFO_SEQ
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-08-09 14:11:14
% : Encoded with polymorphic tags.
%------------------------------------------------------------------------------
%----Explicit typings (74)
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_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_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_b] :
( minus(X_b)
=> ti(fun(X_b,fun(X_b,X_b)),minus_minus(X_b)) = minus_minus(X_b) ) ).
fof(tsy_c_Groups_Oone__class_Oone_res,axiom,
! [X_b] :
( one(X_b)
=> ti(X_b,one_one(X_b)) = one_one(X_b) ) ).
fof(tsy_c_Groups_Oplus__class_Oplus_0_res,axiom,
! [X_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_1_res,axiom,
! [X_a] :
( ab_semigroup_add(X_a)
=> ti(fun(X_a,fun(X_a,X_a)),plus_plus(X_a)) = plus_plus(X_a) ) ).
fof(tsy_c_Groups_Oplus__class_Oplus_2_res,axiom,
! [X_a] :
( 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_Otimes__class_Otimes_res,axiom,
! [X_c] :
( ab_semigroup_mult(X_c)
=> ti(fun(X_c,fun(X_c,X_c)),times_times(X_c)) = times_times(X_c) ) ).
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__nrugjuseim_OMGT_res,axiom,
ti(fun(com,hoare_1656922687triple(state)),hoare_Mirabelle_MGT) = hoare_Mirabelle_MGT ).
fof(tsy_c_Hoare__Mirabelle__nrugjuseim_Ohoare__derivs_res,axiom,
! [X_b] : ti(fun(fun(hoare_1656922687triple(X_b),bool),fun(fun(hoare_1656922687triple(X_b),bool),bool)),hoare_279057269derivs(X_b)) = hoare_279057269derivs(X_b) ).
fof(tsy_c_Hoare__Mirabelle__nrugjuseim_Ohoare__valids_res,axiom,
! [X_b] : ti(fun(fun(hoare_1656922687triple(X_b),bool),fun(fun(hoare_1656922687triple(X_b),bool),bool)),hoare_2027193591valids(X_b)) = hoare_2027193591valids(X_b) ).
fof(tsy_c_Hoare__Mirabelle__nrugjuseim_Otriple_Otriple_res,axiom,
! [X_b] : ti(fun(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b)))),hoare_246368825triple(X_b)) = hoare_246368825triple(X_b) ).
fof(tsy_c_Hoare__Mirabelle__nrugjuseim_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_1656922687triple(X_c),X_b)),hoare_1312322281e_case(X_c,X_b)) = hoare_1312322281e_case(X_c,X_b) ).
fof(tsy_c_Hoare__Mirabelle__nrugjuseim_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_1656922687triple(X_c),X_b)),hoare_1632998903le_rec(X_c,X_b)) = hoare_1632998903le_rec(X_c,X_b) ).
fof(tsy_c_Hoare__Mirabelle__nrugjuseim_Otriple_Otriple__size_res,axiom,
! [X_b] : ti(fun(fun(X_b,nat),fun(hoare_1656922687triple(X_b),nat)),hoare_983366810e_size(X_b)) = hoare_983366810e_size(X_b) ).
fof(tsy_c_Hoare__Mirabelle__nrugjuseim_Otriple__valid_res,axiom,
! [X_b] : ti(fun(nat,fun(hoare_1656922687triple(X_b),bool)),hoare_920331057_valid(X_b)) = hoare_920331057_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_1656922687triple(X_b),nat),size_size(hoare_1656922687triple(X_b))) = size_size(hoare_1656922687triple(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_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_1656922687triple(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 (496)
fof(fact_0_triple_Oinject,axiom,
! [X_b,Fun1,Com,Fun2,Fun1_2,Com_1,Fun2_2] :
( hAPP(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b),hAPP(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b)),hAPP(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b))),hoare_246368825triple(X_b),Fun1),Com),Fun2) = hAPP(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b),hAPP(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b)),hAPP(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b))),hoare_246368825triple(X_b),Fun1_2),Com_1),Fun2_2)
<=> ( Fun1 = Fun1_2
& Com = Com_1
& Fun2 = Fun2_2 ) ) ).
fof(fact_1_hoare__valids__def,axiom,
! [X_b,Ga,Ts] :
( hBOOL(hAPP(fun(hoare_1656922687triple(X_b),bool),bool,hAPP(fun(hoare_1656922687triple(X_b),bool),fun(fun(hoare_1656922687triple(X_b),bool),bool),hoare_2027193591valids(X_b),Ga),Ts))
<=> ! [N] :
( ! [X] :
( hBOOL(hAPP(fun(hoare_1656922687triple(X_b),bool),bool,hAPP(hoare_1656922687triple(X_b),fun(fun(hoare_1656922687triple(X_b),bool),bool),member(hoare_1656922687triple(X_b)),X),Ga))
=> hBOOL(hAPP(hoare_1656922687triple(X_b),bool,hAPP(nat,fun(hoare_1656922687triple(X_b),bool),hoare_920331057_valid(X_b),N),X)) )
=> ! [X] :
( hBOOL(hAPP(fun(hoare_1656922687triple(X_b),bool),bool,hAPP(hoare_1656922687triple(X_b),fun(fun(hoare_1656922687triple(X_b),bool),bool),member(hoare_1656922687triple(X_b)),X),Ts))
=> hBOOL(hAPP(hoare_1656922687triple(X_b),bool,hAPP(nat,fun(hoare_1656922687triple(X_b),bool),hoare_920331057_valid(X_b),N),X)) ) ) ) ).
fof(fact_2_hoare__derivs_OBody,axiom,
! [X_b,Ga,Pa,Qa,Procsa] :
( hBOOL(hAPP(fun(hoare_1656922687triple(X_b),bool),bool,hAPP(fun(hoare_1656922687triple(X_b),bool),fun(fun(hoare_1656922687triple(X_b),bool),bool),hoare_279057269derivs(X_b),hAPP(fun(hoare_1656922687triple(X_b),bool),fun(hoare_1656922687triple(X_b),bool),hAPP(fun(hoare_1656922687triple(X_b),bool),fun(fun(hoare_1656922687triple(X_b),bool),fun(hoare_1656922687triple(X_b),bool)),semilattice_sup_sup(fun(hoare_1656922687triple(X_b),bool)),Ga),hAPP(fun(pname,bool),fun(hoare_1656922687triple(X_b),bool),hAPP(fun(pname,hoare_1656922687triple(X_b)),fun(fun(pname,bool),fun(hoare_1656922687triple(X_b),bool)),image(pname,hoare_1656922687triple(X_b)),hAPP(fun(pname,fun(X_b,fun(state,bool))),fun(pname,hoare_1656922687triple(X_b)),hAPP(fun(pname,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b))),fun(fun(pname,fun(X_b,fun(state,bool))),fun(pname,hoare_1656922687triple(X_b))),combs(pname,fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b)),hAPP(fun(pname,com),fun(pname,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b))),hAPP(fun(pname,fun(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b)))),fun(fun(pname,com),fun(pname,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b)))),combs(pname,com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b))),hAPP(fun(pname,fun(X_b,fun(state,bool))),fun(pname,fun(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b)))),hAPP(fun(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b)))),fun(fun(pname,fun(X_b,fun(state,bool))),fun(pname,fun(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b))))),combb(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b))),pname),hoare_246368825triple(X_b)),Pa)),body)),Qa)),Procsa))),hAPP(fun(pname,bool),fun(hoare_1656922687triple(X_b),bool),hAPP(fun(pname,hoare_1656922687triple(X_b)),fun(fun(pname,bool),fun(hoare_1656922687triple(X_b),bool)),image(pname,hoare_1656922687triple(X_b)),hAPP(fun(pname,fun(X_b,fun(state,bool))),fun(pname,hoare_1656922687triple(X_b)),hAPP(fun(pname,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b))),fun(fun(pname,fun(X_b,fun(state,bool))),fun(pname,hoare_1656922687triple(X_b))),combs(pname,fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b)),hAPP(fun(pname,com),fun(pname,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b))),hAPP(fun(pname,fun(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b)))),fun(fun(pname,com),fun(pname,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b)))),combs(pname,com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b))),hAPP(fun(pname,fun(X_b,fun(state,bool))),fun(pname,fun(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b)))),hAPP(fun(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b)))),fun(fun(pname,fun(X_b,fun(state,bool))),fun(pname,fun(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b))))),combb(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b))),pname),hoare_246368825triple(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_1656922687triple(X_b),bool),bool,hAPP(fun(hoare_1656922687triple(X_b),bool),fun(fun(hoare_1656922687triple(X_b),bool),bool),hoare_279057269derivs(X_b),Ga),hAPP(fun(pname,bool),fun(hoare_1656922687triple(X_b),bool),hAPP(fun(pname,hoare_1656922687triple(X_b)),fun(fun(pname,bool),fun(hoare_1656922687triple(X_b),bool)),image(pname,hoare_1656922687triple(X_b)),hAPP(fun(pname,fun(X_b,fun(state,bool))),fun(pname,hoare_1656922687triple(X_b)),hAPP(fun(pname,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b))),fun(fun(pname,fun(X_b,fun(state,bool))),fun(pname,hoare_1656922687triple(X_b))),combs(pname,fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b)),hAPP(fun(pname,com),fun(pname,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b))),hAPP(fun(pname,fun(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b)))),fun(fun(pname,com),fun(pname,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b)))),combs(pname,com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b))),hAPP(fun(pname,fun(X_b,fun(state,bool))),fun(pname,fun(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b)))),hAPP(fun(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b)))),fun(fun(pname,fun(X_b,fun(state,bool))),fun(pname,fun(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b))))),combb(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b))),pname),hoare_246368825triple(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,Fun2] : hAPP(hoare_1656922687triple(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_1656922687triple(X_c),X_b),hoare_1632998903le_rec(X_c,X_b),F1),hAPP(fun(X_c,fun(state,bool)),hoare_1656922687triple(X_c),hAPP(com,fun(fun(X_c,fun(state,bool)),hoare_1656922687triple(X_c)),hAPP(fun(X_c,fun(state,bool)),fun(com,fun(fun(X_c,fun(state,bool)),hoare_1656922687triple(X_c))),hoare_246368825triple(X_c),Fun1),Com),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),Fun2) ).
fof(fact_9_triple_Osimps_I2_J,axiom,
! [X_c,X_b,F1,Fun1,Com,Fun2] : hAPP(hoare_1656922687triple(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_1656922687triple(X_c),X_b),hoare_1312322281e_case(X_c,X_b),F1),hAPP(fun(X_c,fun(state,bool)),hoare_1656922687triple(X_c),hAPP(com,fun(fun(X_c,fun(state,bool)),hoare_1656922687triple(X_c)),hAPP(fun(X_c,fun(state,bool)),fun(com,fun(fun(X_c,fun(state,bool)),hoare_1656922687triple(X_c))),hoare_246368825triple(X_c),Fun1),Com),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),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_1656922687triple(X_b),bool),bool,hAPP(fun(hoare_1656922687triple(X_b),bool),fun(fun(hoare_1656922687triple(X_b),bool),bool),hoare_279057269derivs(X_b),G_1),Ts))
=> ( hBOOL(hAPP(fun(hoare_1656922687triple(X_b),bool),bool,hAPP(fun(hoare_1656922687triple(X_b),bool),fun(fun(hoare_1656922687triple(X_b),bool),bool),hoare_279057269derivs(X_b),Ga),G_1))
=> hBOOL(hAPP(fun(hoare_1656922687triple(X_b),bool),bool,hAPP(fun(hoare_1656922687triple(X_b),bool),fun(fun(hoare_1656922687triple(X_b),bool),bool),hoare_279057269derivs(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_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)),semilattice_sup_sup(fun(X_b,bool)),A_3),B_2)),C_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)),B_2),C_2)) ).
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_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)),B_2),C_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),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_2)) ).
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_2,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_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)),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_2),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_1656922687triple(X_b),bool,hAPP(nat,fun(hoare_1656922687triple(X_b),bool),hoare_920331057_valid(X_b),N_1),hAPP(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b),hAPP(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b)),hAPP(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b))),hoare_246368825triple(X_b),Pa),hAPP(option(com),com,the(com),hAPP(pname,option(com),body_1,Pn_1))),Qa)))
<=> hBOOL(hAPP(hoare_1656922687triple(X_b),bool,hAPP(nat,fun(hoare_1656922687triple(X_b),bool),hoare_920331057_valid(X_b),hAPP(nat,nat,suc,N_1)),hAPP(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b),hAPP(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b)),hAPP(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b))),hoare_246368825triple(X_b),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_1656922687triple(X_b),hAPP(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b)),hAPP(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b))),hoare_246368825triple(X_b),Fun1_1),Com_2),Fun2_1) ).
fof(fact_51_Body1,axiom,
! [X_b,Pn_1,Ga,Pa,Qa,Procsa] :
( hBOOL(hAPP(fun(hoare_1656922687triple(X_b),bool),bool,hAPP(fun(hoare_1656922687triple(X_b),bool),fun(fun(hoare_1656922687triple(X_b),bool),bool),hoare_279057269derivs(X_b),hAPP(fun(hoare_1656922687triple(X_b),bool),fun(hoare_1656922687triple(X_b),bool),hAPP(fun(hoare_1656922687triple(X_b),bool),fun(fun(hoare_1656922687triple(X_b),bool),fun(hoare_1656922687triple(X_b),bool)),semilattice_sup_sup(fun(hoare_1656922687triple(X_b),bool)),Ga),hAPP(fun(pname,bool),fun(hoare_1656922687triple(X_b),bool),hAPP(fun(pname,hoare_1656922687triple(X_b)),fun(fun(pname,bool),fun(hoare_1656922687triple(X_b),bool)),image(pname,hoare_1656922687triple(X_b)),hAPP(fun(pname,fun(X_b,fun(state,bool))),fun(pname,hoare_1656922687triple(X_b)),hAPP(fun(pname,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b))),fun(fun(pname,fun(X_b,fun(state,bool))),fun(pname,hoare_1656922687triple(X_b))),combs(pname,fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b)),hAPP(fun(pname,com),fun(pname,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b))),hAPP(fun(pname,fun(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b)))),fun(fun(pname,com),fun(pname,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b)))),combs(pname,com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b))),hAPP(fun(pname,fun(X_b,fun(state,bool))),fun(pname,fun(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b)))),hAPP(fun(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b)))),fun(fun(pname,fun(X_b,fun(state,bool))),fun(pname,fun(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b))))),combb(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b))),pname),hoare_246368825triple(X_b)),Pa)),body)),Qa)),Procsa))),hAPP(fun(pname,bool),fun(hoare_1656922687triple(X_b),bool),hAPP(fun(pname,hoare_1656922687triple(X_b)),fun(fun(pname,bool),fun(hoare_1656922687triple(X_b),bool)),image(pname,hoare_1656922687triple(X_b)),hAPP(fun(pname,fun(X_b,fun(state,bool))),fun(pname,hoare_1656922687triple(X_b)),hAPP(fun(pname,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b))),fun(fun(pname,fun(X_b,fun(state,bool))),fun(pname,hoare_1656922687triple(X_b))),combs(pname,fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b)),hAPP(fun(pname,com),fun(pname,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b))),hAPP(fun(pname,fun(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b)))),fun(fun(pname,com),fun(pname,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b)))),combs(pname,com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b))),hAPP(fun(pname,fun(X_b,fun(state,bool))),fun(pname,fun(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b)))),hAPP(fun(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b)))),fun(fun(pname,fun(X_b,fun(state,bool))),fun(pname,fun(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b))))),combb(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b))),pname),hoare_246368825triple(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_1656922687triple(X_b),bool),bool,hAPP(fun(hoare_1656922687triple(X_b),bool),fun(fun(hoare_1656922687triple(X_b),bool),bool),hoare_279057269derivs(X_b),Ga),hAPP(fun(hoare_1656922687triple(X_b),bool),fun(hoare_1656922687triple(X_b),bool),hAPP(hoare_1656922687triple(X_b),fun(fun(hoare_1656922687triple(X_b),bool),fun(hoare_1656922687triple(X_b),bool)),insert(hoare_1656922687triple(X_b)),hAPP(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b),hAPP(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b)),hAPP(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b))),hoare_246368825triple(X_b),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_1656922687triple(X_b),bool))))) ) ) ).
fof(fact_52_image__cong,axiom,
! [X_c,X_b,F,G,M_3,N_3] :
( ti(fun(X_b,bool),M_3) = ti(fun(X_b,bool),N_3)
=> ( ! [X] :
( 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_3) = hAPP(fun(X_b,bool),fun(X_c,bool),hAPP(fun(X_b,X_c),fun(fun(X_b,bool),fun(X_c,bool)),image(X_b,X_c),G),N_3) ) ) ).
fof(fact_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_1656922687triple(X_b),bool,hAPP(nat,fun(hoare_1656922687triple(X_b),bool),hoare_920331057_valid(X_b),zero_zero(nat)),hAPP(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b),hAPP(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b)),hAPP(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b))),hoare_246368825triple(X_b),Pa),hAPP(pname,com,body,Pn_1)),Qa))) ).
fof(fact_55_com_Osimps_I6_J,axiom,
! [Pname_3,Pname_2] :
( hAPP(pname,com,body,Pname_3) = hAPP(pname,com,body,Pname_2)
<=> ti(pname,Pname_3) = ti(pname,Pname_2) ) ).
fof(fact_56_evalc_OBody,axiom,
! [Pn,S0,S1_1] :
( 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_1))
=> hBOOL(hAPP(state,bool,hAPP(state,fun(state,bool),hAPP(com,fun(state,fun(state,bool)),evalc,hAPP(pname,com,body,Pn)),S0),S1_1)) ) ).
fof(fact_57_evalc__elim__cases_I6_J,axiom,
! [P,S_2,S1_1] :
( hBOOL(hAPP(state,bool,hAPP(state,fun(state,bool),hAPP(com,fun(state,fun(state,bool)),evalc,hAPP(pname,com,body,P)),S_2),S1_1))
=> 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_2),S1_1)) ) ).
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_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_60_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_61_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_62_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_63_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_64_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_65_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_66_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_67_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_68_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_69_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_70_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_71_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_72_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_73_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_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__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_78_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_79_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_80_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_81_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_82_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_83_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_84_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_85_doubleton__eq__iff,axiom,
! [X_b,A_2,B_1,C_1,D_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),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_1),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_1) )
| ( ti(X_b,A_2) = ti(X_b,D_1)
& ti(X_b,B_1) = ti(X_b,C_1) ) ) ) ).
fof(fact_86_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_87_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_88_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_89_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_90_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_91_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_92_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_93_com__det,axiom,
! [U_1,C,S_2,T_4] :
( hBOOL(hAPP(state,bool,hAPP(state,fun(state,bool),hAPP(com,fun(state,fun(state,bool)),evalc,C),S_2),T_4))
=> ( hBOOL(hAPP(state,bool,hAPP(state,fun(state,bool),hAPP(com,fun(state,fun(state,bool)),evalc,C),S_2),U_1))
=> U_1 = T_4 ) ) ).
fof(fact_94_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_95_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_96_derivs__insertD,axiom,
! [X_b,Ga,T_3,Ts] :
( hBOOL(hAPP(fun(hoare_1656922687triple(X_b),bool),bool,hAPP(fun(hoare_1656922687triple(X_b),bool),fun(fun(hoare_1656922687triple(X_b),bool),bool),hoare_279057269derivs(X_b),Ga),hAPP(fun(hoare_1656922687triple(X_b),bool),fun(hoare_1656922687triple(X_b),bool),hAPP(hoare_1656922687triple(X_b),fun(fun(hoare_1656922687triple(X_b),bool),fun(hoare_1656922687triple(X_b),bool)),insert(hoare_1656922687triple(X_b)),T_3),Ts)))
=> ( hBOOL(hAPP(fun(hoare_1656922687triple(X_b),bool),bool,hAPP(fun(hoare_1656922687triple(X_b),bool),fun(fun(hoare_1656922687triple(X_b),bool),bool),hoare_279057269derivs(X_b),Ga),hAPP(fun(hoare_1656922687triple(X_b),bool),fun(hoare_1656922687triple(X_b),bool),hAPP(hoare_1656922687triple(X_b),fun(fun(hoare_1656922687triple(X_b),bool),fun(hoare_1656922687triple(X_b),bool)),insert(hoare_1656922687triple(X_b)),T_3),bot_bot(fun(hoare_1656922687triple(X_b),bool)))))
& hBOOL(hAPP(fun(hoare_1656922687triple(X_b),bool),bool,hAPP(fun(hoare_1656922687triple(X_b),bool),fun(fun(hoare_1656922687triple(X_b),bool),bool),hoare_279057269derivs(X_b),Ga),Ts)) ) ) ).
fof(fact_97_hoare__derivs_Oinsert,axiom,
! [X_b,Ts,Ga,T_3] :
( hBOOL(hAPP(fun(hoare_1656922687triple(X_b),bool),bool,hAPP(fun(hoare_1656922687triple(X_b),bool),fun(fun(hoare_1656922687triple(X_b),bool),bool),hoare_279057269derivs(X_b),Ga),hAPP(fun(hoare_1656922687triple(X_b),bool),fun(hoare_1656922687triple(X_b),bool),hAPP(hoare_1656922687triple(X_b),fun(fun(hoare_1656922687triple(X_b),bool),fun(hoare_1656922687triple(X_b),bool)),insert(hoare_1656922687triple(X_b)),T_3),bot_bot(fun(hoare_1656922687triple(X_b),bool)))))
=> ( hBOOL(hAPP(fun(hoare_1656922687triple(X_b),bool),bool,hAPP(fun(hoare_1656922687triple(X_b),bool),fun(fun(hoare_1656922687triple(X_b),bool),bool),hoare_279057269derivs(X_b),Ga),Ts))
=> hBOOL(hAPP(fun(hoare_1656922687triple(X_b),bool),bool,hAPP(fun(hoare_1656922687triple(X_b),bool),fun(fun(hoare_1656922687triple(X_b),bool),bool),hoare_279057269derivs(X_b),Ga),hAPP(fun(hoare_1656922687triple(X_b),bool),fun(hoare_1656922687triple(X_b),bool),hAPP(hoare_1656922687triple(X_b),fun(fun(hoare_1656922687triple(X_b),bool),fun(hoare_1656922687triple(X_b),bool)),insert(hoare_1656922687triple(X_b)),T_3),Ts))) ) ) ).
fof(fact_98_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_99_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_100_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_101_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_102_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_103_Un__insert__left,axiom,
! [X_b,A_2,B_2,C_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(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_2),B_2)),C_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)),B_2),C_2)) ).
fof(fact_104_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_105_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_106_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_107_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_108_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_109_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_110_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_111_constant,axiom,
! [X_b,Ga,Pa,C_1,Qa,C_2] :
( ( hBOOL(C_2)
=> hBOOL(hAPP(fun(hoare_1656922687triple(X_b),bool),bool,hAPP(fun(hoare_1656922687triple(X_b),bool),fun(fun(hoare_1656922687triple(X_b),bool),bool),hoare_279057269derivs(X_b),Ga),hAPP(fun(hoare_1656922687triple(X_b),bool),fun(hoare_1656922687triple(X_b),bool),hAPP(hoare_1656922687triple(X_b),fun(fun(hoare_1656922687triple(X_b),bool),fun(hoare_1656922687triple(X_b),bool)),insert(hoare_1656922687triple(X_b)),hAPP(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b),hAPP(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b)),hAPP(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b))),hoare_246368825triple(X_b),Pa),C_1),Qa)),bot_bot(fun(hoare_1656922687triple(X_b),bool))))) )
=> hBOOL(hAPP(fun(hoare_1656922687triple(X_b),bool),bool,hAPP(fun(hoare_1656922687triple(X_b),bool),fun(fun(hoare_1656922687triple(X_b),bool),bool),hoare_279057269derivs(X_b),Ga),hAPP(fun(hoare_1656922687triple(X_b),bool),fun(hoare_1656922687triple(X_b),bool),hAPP(hoare_1656922687triple(X_b),fun(fun(hoare_1656922687triple(X_b),bool),fun(hoare_1656922687triple(X_b),bool)),insert(hoare_1656922687triple(X_b)),hAPP(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b),hAPP(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b)),hAPP(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b))),hoare_246368825triple(X_b),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_2)),C_1),Qa)),bot_bot(fun(hoare_1656922687triple(X_b),bool))))) ) ).
fof(fact_112_empty,axiom,
! [X_b,Ga] : hBOOL(hAPP(fun(hoare_1656922687triple(X_b),bool),bool,hAPP(fun(hoare_1656922687triple(X_b),bool),fun(fun(hoare_1656922687triple(X_b),bool),bool),hoare_279057269derivs(X_b),Ga),bot_bot(fun(hoare_1656922687triple(X_b),bool)))) ).
fof(fact_113_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_114_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_115_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_116_triple__valid__Suc,axiom,
! [X_a,N_2,T_4] :
( hBOOL(hAPP(hoare_1656922687triple(X_a),bool,hAPP(nat,fun(hoare_1656922687triple(X_a),bool),hoare_920331057_valid(X_a),hAPP(nat,nat,suc,N_2)),T_4))
=> hBOOL(hAPP(hoare_1656922687triple(X_a),bool,hAPP(nat,fun(hoare_1656922687triple(X_a),bool),hoare_920331057_valid(X_a),N_2),T_4)) ) ).
fof(fact_117_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_118_weak__Body,axiom,
! [X_b,Ga,Pa,Pn_1,Qa] :
( hBOOL(hAPP(fun(hoare_1656922687triple(X_b),bool),bool,hAPP(fun(hoare_1656922687triple(X_b),bool),fun(fun(hoare_1656922687triple(X_b),bool),bool),hoare_279057269derivs(X_b),Ga),hAPP(fun(hoare_1656922687triple(X_b),bool),fun(hoare_1656922687triple(X_b),bool),hAPP(hoare_1656922687triple(X_b),fun(fun(hoare_1656922687triple(X_b),bool),fun(hoare_1656922687triple(X_b),bool)),insert(hoare_1656922687triple(X_b)),hAPP(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b),hAPP(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b)),hAPP(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b))),hoare_246368825triple(X_b),Pa),hAPP(option(com),com,the(com),hAPP(pname,option(com),body_1,Pn_1))),Qa)),bot_bot(fun(hoare_1656922687triple(X_b),bool)))))
=> hBOOL(hAPP(fun(hoare_1656922687triple(X_b),bool),bool,hAPP(fun(hoare_1656922687triple(X_b),bool),fun(fun(hoare_1656922687triple(X_b),bool),bool),hoare_279057269derivs(X_b),Ga),hAPP(fun(hoare_1656922687triple(X_b),bool),fun(hoare_1656922687triple(X_b),bool),hAPP(hoare_1656922687triple(X_b),fun(fun(hoare_1656922687triple(X_b),bool),fun(hoare_1656922687triple(X_b),bool)),insert(hoare_1656922687triple(X_b)),hAPP(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b),hAPP(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b)),hAPP(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b))),hoare_246368825triple(X_b),Pa),hAPP(pname,com,body,Pn_1)),Qa)),bot_bot(fun(hoare_1656922687triple(X_b),bool))))) ) ).
fof(fact_119_BodyN,axiom,
! [X_b,Pa,Pn_1,Qa,Ga] :
( hBOOL(hAPP(fun(hoare_1656922687triple(X_b),bool),bool,hAPP(fun(hoare_1656922687triple(X_b),bool),fun(fun(hoare_1656922687triple(X_b),bool),bool),hoare_279057269derivs(X_b),hAPP(fun(hoare_1656922687triple(X_b),bool),fun(hoare_1656922687triple(X_b),bool),hAPP(hoare_1656922687triple(X_b),fun(fun(hoare_1656922687triple(X_b),bool),fun(hoare_1656922687triple(X_b),bool)),insert(hoare_1656922687triple(X_b)),hAPP(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b),hAPP(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b)),hAPP(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b))),hoare_246368825triple(X_b),Pa),hAPP(pname,com,body,Pn_1)),Qa)),Ga)),hAPP(fun(hoare_1656922687triple(X_b),bool),fun(hoare_1656922687triple(X_b),bool),hAPP(hoare_1656922687triple(X_b),fun(fun(hoare_1656922687triple(X_b),bool),fun(hoare_1656922687triple(X_b),bool)),insert(hoare_1656922687triple(X_b)),hAPP(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b),hAPP(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b)),hAPP(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b))),hoare_246368825triple(X_b),Pa),hAPP(option(com),com,the(com),hAPP(pname,option(com),body_1,Pn_1))),Qa)),bot_bot(fun(hoare_1656922687triple(X_b),bool)))))
=> hBOOL(hAPP(fun(hoare_1656922687triple(X_b),bool),bool,hAPP(fun(hoare_1656922687triple(X_b),bool),fun(fun(hoare_1656922687triple(X_b),bool),bool),hoare_279057269derivs(X_b),Ga),hAPP(fun(hoare_1656922687triple(X_b),bool),fun(hoare_1656922687triple(X_b),bool),hAPP(hoare_1656922687triple(X_b),fun(fun(hoare_1656922687triple(X_b),bool),fun(hoare_1656922687triple(X_b),bool)),insert(hoare_1656922687triple(X_b)),hAPP(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b),hAPP(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b)),hAPP(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b))),hoare_246368825triple(X_b),Pa),hAPP(pname,com,body,Pn_1)),Qa)),bot_bot(fun(hoare_1656922687triple(X_b),bool))))) ) ).
fof(fact_120_triples__valid__Suc,axiom,
! [X_b,N_1,Ts] :
( ! [X] :
( hBOOL(hAPP(fun(hoare_1656922687triple(X_b),bool),bool,hAPP(hoare_1656922687triple(X_b),fun(fun(hoare_1656922687triple(X_b),bool),bool),member(hoare_1656922687triple(X_b)),X),Ts))
=> hBOOL(hAPP(hoare_1656922687triple(X_b),bool,hAPP(nat,fun(hoare_1656922687triple(X_b),bool),hoare_920331057_valid(X_b),hAPP(nat,nat,suc,N_1)),X)) )
=> ! [X] :
( hBOOL(hAPP(fun(hoare_1656922687triple(X_b),bool),bool,hAPP(hoare_1656922687triple(X_b),fun(fun(hoare_1656922687triple(X_b),bool),bool),member(hoare_1656922687triple(X_b)),X),Ts))
=> hBOOL(hAPP(hoare_1656922687triple(X_b),bool,hAPP(nat,fun(hoare_1656922687triple(X_b),bool),hoare_920331057_valid(X_b),N_1),X)) ) ) ).
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_1656922687triple(X_b),bool),bool,hAPP(fun(hoare_1656922687triple(X_b),bool),fun(fun(hoare_1656922687triple(X_b),bool),bool),hoare_279057269derivs(X_b),Ga),hAPP(fun(hoare_1656922687triple(X_b),bool),fun(hoare_1656922687triple(X_b),bool),hAPP(hoare_1656922687triple(X_b),fun(fun(hoare_1656922687triple(X_b),bool),fun(hoare_1656922687triple(X_b),bool)),insert(hoare_1656922687triple(X_b)),hAPP(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b),hAPP(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b)),hAPP(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b))),hoare_246368825triple(X_b),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_1656922687triple(X_b),bool))))) )
=> hBOOL(hAPP(fun(hoare_1656922687triple(X_b),bool),bool,hAPP(fun(hoare_1656922687triple(X_b),bool),fun(fun(hoare_1656922687triple(X_b),bool),bool),hoare_279057269derivs(X_b),Ga),hAPP(fun(hoare_1656922687triple(X_b),bool),fun(hoare_1656922687triple(X_b),bool),hAPP(hoare_1656922687triple(X_b),fun(fun(hoare_1656922687triple(X_b),bool),fun(hoare_1656922687triple(X_b),bool)),insert(hoare_1656922687triple(X_b)),hAPP(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b),hAPP(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b)),hAPP(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b))),hoare_246368825triple(X_b),Pa),C_1),Qa)),bot_bot(fun(hoare_1656922687triple(X_b),bool))))) ) ).
fof(fact_122_conseq1,axiom,
! [X_b,Pa,Ga,P_1,C_1,Qa] :
( hBOOL(hAPP(fun(hoare_1656922687triple(X_b),bool),bool,hAPP(fun(hoare_1656922687triple(X_b),bool),fun(fun(hoare_1656922687triple(X_b),bool),bool),hoare_279057269derivs(X_b),Ga),hAPP(fun(hoare_1656922687triple(X_b),bool),fun(hoare_1656922687triple(X_b),bool),hAPP(hoare_1656922687triple(X_b),fun(fun(hoare_1656922687triple(X_b),bool),fun(hoare_1656922687triple(X_b),bool)),insert(hoare_1656922687triple(X_b)),hAPP(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b),hAPP(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b)),hAPP(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b))),hoare_246368825triple(X_b),P_1),C_1),Qa)),bot_bot(fun(hoare_1656922687triple(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_1656922687triple(X_b),bool),bool,hAPP(fun(hoare_1656922687triple(X_b),bool),fun(fun(hoare_1656922687triple(X_b),bool),bool),hoare_279057269derivs(X_b),Ga),hAPP(fun(hoare_1656922687triple(X_b),bool),fun(hoare_1656922687triple(X_b),bool),hAPP(hoare_1656922687triple(X_b),fun(fun(hoare_1656922687triple(X_b),bool),fun(hoare_1656922687triple(X_b),bool)),insert(hoare_1656922687triple(X_b)),hAPP(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b),hAPP(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b)),hAPP(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b))),hoare_246368825triple(X_b),Pa),C_1),Qa)),bot_bot(fun(hoare_1656922687triple(X_b),bool))))) ) ) ).
fof(fact_123_conseq2,axiom,
! [X_b,Qa,Ga,Pa,C_1,Q_1] :
( hBOOL(hAPP(fun(hoare_1656922687triple(X_b),bool),bool,hAPP(fun(hoare_1656922687triple(X_b),bool),fun(fun(hoare_1656922687triple(X_b),bool),bool),hoare_279057269derivs(X_b),Ga),hAPP(fun(hoare_1656922687triple(X_b),bool),fun(hoare_1656922687triple(X_b),bool),hAPP(hoare_1656922687triple(X_b),fun(fun(hoare_1656922687triple(X_b),bool),fun(hoare_1656922687triple(X_b),bool)),insert(hoare_1656922687triple(X_b)),hAPP(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b),hAPP(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b)),hAPP(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b))),hoare_246368825triple(X_b),Pa),C_1),Q_1)),bot_bot(fun(hoare_1656922687triple(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_1656922687triple(X_b),bool),bool,hAPP(fun(hoare_1656922687triple(X_b),bool),fun(fun(hoare_1656922687triple(X_b),bool),bool),hoare_279057269derivs(X_b),Ga),hAPP(fun(hoare_1656922687triple(X_b),bool),fun(hoare_1656922687triple(X_b),bool),hAPP(hoare_1656922687triple(X_b),fun(fun(hoare_1656922687triple(X_b),bool),fun(hoare_1656922687triple(X_b),bool)),insert(hoare_1656922687triple(X_b)),hAPP(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b),hAPP(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b)),hAPP(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b))),hoare_246368825triple(X_b),Pa),C_1),Qa)),bot_bot(fun(hoare_1656922687triple(X_b),bool))))) ) ) ).
fof(fact_124_triple_Osize_I1_J,axiom,
! [X_b,Fa,Fun1,Com,Fun2] : hAPP(hoare_1656922687triple(X_b),nat,hAPP(fun(X_b,nat),fun(hoare_1656922687triple(X_b),nat),hoare_983366810e_size(X_b),Fa),hAPP(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b),hAPP(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b)),hAPP(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b))),hoare_246368825triple(X_b),Fun1),Com),Fun2)) = zero_zero(nat) ).
fof(fact_125_MGT__def,axiom,
! [C_1] : hAPP(com,hoare_1656922687triple(state),hoare_Mirabelle_MGT,C_1) = hAPP(fun(state,fun(state,bool)),hoare_1656922687triple(state),hAPP(com,fun(fun(state,fun(state,bool)),hoare_1656922687triple(state)),hAPP(fun(state,fun(state,bool)),fun(com,fun(fun(state,fun(state,bool)),hoare_1656922687triple(state))),hoare_246368825triple(state),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,Fun2] : hAPP(hoare_1656922687triple(X_b),nat,size_size(hoare_1656922687triple(X_b)),hAPP(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b),hAPP(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b)),hAPP(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b))),hoare_246368825triple(X_b),Fun1),Com),Fun2)) = zero_zero(nat) ).
fof(fact_127_conseq12,axiom,
! [X_b,Qa,Pa,Ga,P_1,C_1,Q_1] :
( hBOOL(hAPP(fun(hoare_1656922687triple(X_b),bool),bool,hAPP(fun(hoare_1656922687triple(X_b),bool),fun(fun(hoare_1656922687triple(X_b),bool),bool),hoare_279057269derivs(X_b),Ga),hAPP(fun(hoare_1656922687triple(X_b),bool),fun(hoare_1656922687triple(X_b),bool),hAPP(hoare_1656922687triple(X_b),fun(fun(hoare_1656922687triple(X_b),bool),fun(hoare_1656922687triple(X_b),bool)),insert(hoare_1656922687triple(X_b)),hAPP(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b),hAPP(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b)),hAPP(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b))),hoare_246368825triple(X_b),P_1),C_1),Q_1)),bot_bot(fun(hoare_1656922687triple(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_1656922687triple(X_b),bool),bool,hAPP(fun(hoare_1656922687triple(X_b),bool),fun(fun(hoare_1656922687triple(X_b),bool),bool),hoare_279057269derivs(X_b),Ga),hAPP(fun(hoare_1656922687triple(X_b),bool),fun(hoare_1656922687triple(X_b),bool),hAPP(hoare_1656922687triple(X_b),fun(fun(hoare_1656922687triple(X_b),bool),fun(hoare_1656922687triple(X_b),bool)),insert(hoare_1656922687triple(X_b)),hAPP(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b),hAPP(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b)),hAPP(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b))),hoare_246368825triple(X_b),Pa),C_1),Qa)),bot_bot(fun(hoare_1656922687triple(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_Zero__not__Suc,axiom,
! [M] : zero_zero(nat) != hAPP(nat,nat,suc,M) ).
fof(fact_130_nat_Osimps_I2_J,axiom,
! [Nat_4] : zero_zero(nat) != hAPP(nat,nat,suc,Nat_4) ).
fof(fact_131_Suc__not__Zero,axiom,
! [M] : hAPP(nat,nat,suc,M) != zero_zero(nat) ).
fof(fact_132_nat_Osimps_I3_J,axiom,
! [Nat_3] : hAPP(nat,nat,suc,Nat_3) != zero_zero(nat) ).
fof(fact_133_Zero__neq__Suc,axiom,
! [M] : zero_zero(nat) != hAPP(nat,nat,suc,M) ).
fof(fact_134_Suc__neq__Zero,axiom,
! [M] : hAPP(nat,nat,suc,M) != zero_zero(nat) ).
fof(fact_135_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_136_bot__nat__def,axiom,
bot_bot(nat) = zero_zero(nat) ).
fof(fact_137_Suc__inject,axiom,
! [X_1,Y] :
( hAPP(nat,nat,suc,X_1) = hAPP(nat,nat,suc,Y)
=> X_1 = Y ) ).
fof(fact_138_nat_Oinject,axiom,
! [Nat,Nat_2] :
( hAPP(nat,nat,suc,Nat) = hAPP(nat,nat,suc,Nat_2)
<=> Nat = Nat_2 ) ).
fof(fact_139_Suc__n__not__n,axiom,
! [N_2] : hAPP(nat,nat,suc,N_2) != N_2 ).
fof(fact_140_n__not__Suc__n,axiom,
! [N_2] : N_2 != hAPP(nat,nat,suc,N_2) ).
fof(fact_141_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_142_nat_Oexhaust,axiom,
! [Y] :
( Y != zero_zero(nat)
=> ~ ! [Nat_1] : Y != hAPP(nat,nat,suc,Nat_1) ) ).
fof(fact_143_zero__induct,axiom,
! [Pa,K] :
( hBOOL(hAPP(nat,bool,Pa,K))
=> ( ! [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_144_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_145_not0__implies__Suc,axiom,
! [N_2] :
( N_2 != zero_zero(nat)
=> ? [M_2] : N_2 = hAPP(nat,nat,suc,M_2) ) ).
fof(fact_146_evaln_OBody,axiom,
! [Pn,S0,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,hAPP(option(com),com,the(com),hAPP(pname,option(com),body_1,Pn))),S0),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,hAPP(pname,com,body,Pn)),S0),hAPP(nat,nat,suc,N_2)),S1_1)) ) ).
fof(fact_147_hoare__derivs_OSkip,axiom,
! [X_b,Ga,Pa] : hBOOL(hAPP(fun(hoare_1656922687triple(X_b),bool),bool,hAPP(fun(hoare_1656922687triple(X_b),bool),fun(fun(hoare_1656922687triple(X_b),bool),bool),hoare_279057269derivs(X_b),Ga),hAPP(fun(hoare_1656922687triple(X_b),bool),fun(hoare_1656922687triple(X_b),bool),hAPP(hoare_1656922687triple(X_b),fun(fun(hoare_1656922687triple(X_b),bool),fun(hoare_1656922687triple(X_b),bool)),insert(hoare_1656922687triple(X_b)),hAPP(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b),hAPP(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b)),hAPP(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b))),hoare_246368825triple(X_b),Pa),skip),Pa)),bot_bot(fun(hoare_1656922687triple(X_b),bool))))) ).
fof(fact_148_evaln__elim__cases_I1_J,axiom,
! [S_2,N_2,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,skip),S_2),N_2),T_4))
=> T_4 = S_2 ) ).
fof(fact_149_evaln_OSkip,axiom,
! [S_2,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_2),N_2),S_2)) ).
fof(fact_150_evalc_OSkip,axiom,
! [S_2] : hBOOL(hAPP(state,bool,hAPP(state,fun(state,bool),hAPP(com,fun(state,fun(state,bool)),evalc,skip),S_2),S_2)) ).
fof(fact_151_evalc__elim__cases_I1_J,axiom,
! [S_2,T_4] :
( hBOOL(hAPP(state,bool,hAPP(state,fun(state,bool),hAPP(com,fun(state,fun(state,bool)),evalc,skip),S_2),T_4))
=> T_4 = S_2 ) ).
fof(fact_152_evaln__Suc,axiom,
! [C,S_2,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_2),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_2),hAPP(nat,nat,suc,N_2)),S_5)) ) ).
fof(fact_153_eval__eq,axiom,
! [C_1,S,T_3] :
( hBOOL(hAPP(state,bool,hAPP(state,fun(state,bool),hAPP(com,fun(state,fun(state,bool)),evalc,C_1),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_1),S),N),T_3)) ) ).
fof(fact_154_evaln__evalc,axiom,
! [C,S_2,N_2,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,C),S_2),N_2),T_4))
=> hBOOL(hAPP(state,bool,hAPP(state,fun(state,bool),hAPP(com,fun(state,fun(state,bool)),evalc,C),S_2),T_4)) ) ).
fof(fact_155_com_Osimps_I19_J,axiom,
! [Pname_1] : hAPP(pname,com,body,Pname_1) != skip ).
fof(fact_156_com_Osimps_I18_J,axiom,
! [Pname_1] : skip != hAPP(pname,com,body,Pname_1) ).
fof(fact_157_triple__valid__def2,axiom,
! [X_b,N_1,Pa,C_1,Qa] :
( hBOOL(hAPP(hoare_1656922687triple(X_b),bool,hAPP(nat,fun(hoare_1656922687triple(X_b),bool),hoare_920331057_valid(X_b),N_1),hAPP(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b),hAPP(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b)),hAPP(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b))),hoare_246368825triple(X_b),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_158_evaln__elim__cases_I6_J,axiom,
! [P,S_2,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,hAPP(pname,com,body,P)),S_2),N_2),S1_1))
=> ~ ! [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_2),N),S1_1)) ) ) ).
fof(fact_159_evalc__evaln,axiom,
! [C,S_2,T_4] :
( hBOOL(hAPP(state,bool,hAPP(state,fun(state,bool),hAPP(com,fun(state,fun(state,bool)),evalc,C),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),S_2),N),T_4)) ) ).
fof(fact_160_LoopF,axiom,
! [X_b,Ga,Pa,B_1,C_1] : hBOOL(hAPP(fun(hoare_1656922687triple(X_b),bool),bool,hAPP(fun(hoare_1656922687triple(X_b),bool),fun(fun(hoare_1656922687triple(X_b),bool),bool),hoare_279057269derivs(X_b),Ga),hAPP(fun(hoare_1656922687triple(X_b),bool),fun(hoare_1656922687triple(X_b),bool),hAPP(hoare_1656922687triple(X_b),fun(fun(hoare_1656922687triple(X_b),bool),fun(hoare_1656922687triple(X_b),bool)),insert(hoare_1656922687triple(X_b)),hAPP(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b),hAPP(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b)),hAPP(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b))),hoare_246368825triple(X_b),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_1656922687triple(X_b),bool))))) ).
fof(fact_161_Comp,axiom,
! [X_b,D_1,R_2,Ga,Pa,C_1,Qa] :
( hBOOL(hAPP(fun(hoare_1656922687triple(X_b),bool),bool,hAPP(fun(hoare_1656922687triple(X_b),bool),fun(fun(hoare_1656922687triple(X_b),bool),bool),hoare_279057269derivs(X_b),Ga),hAPP(fun(hoare_1656922687triple(X_b),bool),fun(hoare_1656922687triple(X_b),bool),hAPP(hoare_1656922687triple(X_b),fun(fun(hoare_1656922687triple(X_b),bool),fun(hoare_1656922687triple(X_b),bool)),insert(hoare_1656922687triple(X_b)),hAPP(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b),hAPP(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b)),hAPP(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b))),hoare_246368825triple(X_b),Pa),C_1),Qa)),bot_bot(fun(hoare_1656922687triple(X_b),bool)))))
=> ( hBOOL(hAPP(fun(hoare_1656922687triple(X_b),bool),bool,hAPP(fun(hoare_1656922687triple(X_b),bool),fun(fun(hoare_1656922687triple(X_b),bool),bool),hoare_279057269derivs(X_b),Ga),hAPP(fun(hoare_1656922687triple(X_b),bool),fun(hoare_1656922687triple(X_b),bool),hAPP(hoare_1656922687triple(X_b),fun(fun(hoare_1656922687triple(X_b),bool),fun(hoare_1656922687triple(X_b),bool)),insert(hoare_1656922687triple(X_b)),hAPP(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b),hAPP(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b)),hAPP(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b))),hoare_246368825triple(X_b),Qa),D_1),R_2)),bot_bot(fun(hoare_1656922687triple(X_b),bool)))))
=> hBOOL(hAPP(fun(hoare_1656922687triple(X_b),bool),bool,hAPP(fun(hoare_1656922687triple(X_b),bool),fun(fun(hoare_1656922687triple(X_b),bool),bool),hoare_279057269derivs(X_b),Ga),hAPP(fun(hoare_1656922687triple(X_b),bool),fun(hoare_1656922687triple(X_b),bool),hAPP(hoare_1656922687triple(X_b),fun(fun(hoare_1656922687triple(X_b),bool),fun(hoare_1656922687triple(X_b),bool)),insert(hoare_1656922687triple(X_b)),hAPP(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b),hAPP(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b)),hAPP(fun(X_b,fun(state,bool)),fun(com,fun(fun(X_b,fun(state,bool)),hoare_1656922687triple(X_b))),hoare_246368825triple(X_b),Pa),hAPP(com,com,hAPP(com,fun(com,com),semi,C_1),D_1)),R_2)),bot_bot(fun(hoare_1656922687triple(X_b),bool))))) ) ) ).
fof(fact_162_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_163_finite__pointwise,axiom,
! [X_c,X_b,Pa,Qa,Ga,P_1,C0,Q_1,U] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),U))
=> ( ! [P_2] :
( hBOOL(hAPP(fun(hoare_1656922687triple(X_c),bool),bool,hAPP(fun(hoare_1656922687triple(X_c),bool),fun(fun(hoare_1656922687triple(X_c),bool),bool),hoare_279057269derivs(X_c),Ga),hAPP(fun(hoare_1656922687triple(X_c),bool),fun(hoare_1656922687triple(X_c),bool),hAPP(hoare_1656922687triple(X_c),fun(fun(hoare_1656922687triple(X_c),bool),fun(hoare_1656922687triple(X_c),bool)),insert(hoare_1656922687triple(X_c)),hAPP(fun(X_c,fun(state,bool)),hoare_1656922687triple(X_c),hAPP(com,fun(fun(X_c,fun(state,bool)),hoare_1656922687triple(X_c)),hAPP(fun(X_c,fun(state,bool)),fun(com,fun(fun(X_c,fun(state,bool)),hoare_1656922687triple(X_c))),hoare_246368825triple(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_1656922687triple(X_c),bool)))))
=> hBOOL(hAPP(fun(hoare_1656922687triple(X_c),bool),bool,hAPP(fun(hoare_1656922687triple(X_c),bool),fun(fun(hoare_1656922687triple(X_c),bool),bool),hoare_279057269derivs(X_c),Ga),hAPP(fun(hoare_1656922687triple(X_c),bool),fun(hoare_1656922687triple(X_c),bool),hAPP(hoare_1656922687triple(X_c),fun(fun(hoare_1656922687triple(X_c),bool),fun(hoare_1656922687triple(X_c),bool)),insert(hoare_1656922687triple(X_c)),hAPP(fun(X_c,fun(state,bool)),hoare_1656922687triple(X_c),hAPP(com,fun(fun(X_c,fun(state,bool)),hoare_1656922687triple(X_c)),hAPP(fun(X_c,fun(state,bool)),fun(com,fun(fun(X_c,fun(state,bool)),hoare_1656922687triple(X_c))),hoare_246368825triple(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_1656922687triple(X_c),bool))))) )
=> ( hBOOL(hAPP(fun(hoare_1656922687triple(X_c),bool),bool,hAPP(fun(hoare_1656922687triple(X_c),bool),fun(fun(hoare_1656922687triple(X_c),bool),bool),hoare_279057269derivs(X_c),Ga),hAPP(fun(X_b,bool),fun(hoare_1656922687triple(X_c),bool),hAPP(fun(X_b,hoare_1656922687triple(X_c)),fun(fun(X_b,bool),fun(hoare_1656922687triple(X_c),bool)),image(X_b,hoare_1656922687triple(X_c)),hAPP(fun(X_b,fun(X_c,fun(state,bool))),fun(X_b,hoare_1656922687triple(X_c)),hAPP(fun(X_b,fun(fun(X_c,fun(state,bool)),hoare_1656922687triple(X_c))),fun(fun(X_b,fun(X_c,fun(state,bool))),fun(X_b,hoare_1656922687triple(X_c))),combs(X_b,fun(X_c,fun(state,bool)),hoare_1656922687triple(X_c)),hAPP(fun(X_b,com),fun(X_b,fun(fun(X_c,fun(state,bool)),hoare_1656922687triple(X_c))),hAPP(fun(X_b,fun(com,fun(fun(X_c,fun(state,bool)),hoare_1656922687triple(X_c)))),fun(fun(X_b,com),fun(X_b,fun(fun(X_c,fun(state,bool)),hoare_1656922687triple(X_c)))),combs(X_b,com,fun(fun(X_c,fun(state,bool)),hoare_1656922687triple(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_1656922687triple(X_c)))),hAPP(fun(fun(X_c,fun(state,bool)),fun(com,fun(fun(X_c,fun(state,bool)),hoare_1656922687triple(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_1656922687triple(X_c))))),combb(fun(X_c,fun(state,bool)),fun(com,fun(fun(X_c,fun(state,bool)),hoare_1656922687triple(X_c))),X_b),hoare_246368825triple(X_c)),P_1)),C0)),Q_1)),U)))
=> hBOOL(hAPP(fun(hoare_1656922687triple(X_c),bool),bool,hAPP(fun(hoare_1656922687triple(X_c),bool),fun(fun(hoare_1656922687triple(X_c),bool),bool),hoare_279057269derivs(X_c),Ga),hAPP(fun(X_b,bool),fun(hoare_1656922687triple(X_c),bool),hAPP(fun(X_b,hoare_1656922687triple(X_c)),fun(fun(X_b,bool),fun(hoare_1656922687triple(X_c),bool)),image(X_b,hoare_1656922687triple(X_c)),hAPP(fun(X_b,fun(X_c,fun(state,bool))),fun(X_b,hoare_1656922687triple(X_c)),hAPP(fun(X_b,fun(fun(X_c,fun(state,bool)),hoare_1656922687triple(X_c))),fun(fun(X_b,fun(X_c,fun(state,bool))),fun(X_b,hoare_1656922687triple(X_c))),combs(X_b,fun(X_c,fun(state,bool)),hoare_1656922687triple(X_c)),hAPP(fun(X_b,com),fun(X_b,fun(fun(X_c,fun(state,bool)),hoare_1656922687triple(X_c))),hAPP(fun(X_b,fun(com,fun(fun(X_c,fun(state,bool)),hoare_1656922687triple(X_c)))),fun(fun(X_b,com),fun(X_b,fun(fun(X_c,fun(state,bool)),hoare_1656922687triple(X_c)))),combs(X_b,com,fun(fun(X_c,fun(state,bool)),hoare_1656922687triple(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_1656922687triple(X_c)))),hAPP(fun(fun(X_c,fun(state,bool)),fun(com,fun(fun(X_c,fun(state,bool)),hoare_1656922687triple(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_1656922687triple(X_c))))),combb(fun(X_c,fun(state,bool)),fun(com,fun(fun(X_c,fun(state,bool)),hoare_1656922687triple(X_c))),X_b),hoare_246368825triple(X_c)),Pa)),C0)),Qa)),U))) ) ) ) ).
fof(fact_164_evaln_OWhileFalse,axiom,
! [C_1,N_1,B_1,S] :
( ~ hBOOL(hAPP(state,bool,B_1,S))
=> 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),N_1),S)) ) ).
fof(fact_165_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_166_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_167_evalc_OWhileFalse,axiom,
! [C_1,B_1,S] :
( ~ hBOOL(hAPP(state,bool,B_1,S))
=> 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),S)) ) ).
fof(fact_168_evaln_OSemi,axiom,
! [C1_1,S2,C0_1,S0,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,C0_1),S0),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,C1_1),S1_1),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_169_evalc_OSemi,axiom,
! [C1_1,S2,C0_1,S0,S1_1] :
( hBOOL(hAPP(state,bool,hAPP(state,fun(state,bool),hAPP(com,fun(state,fun(state,bool)),evalc,C0_1),S0),S1_1))
=> ( hBOOL(hAPP(state,bool,hAPP(state,fun(state,bool),hAPP(com,fun(state,fun(state,bool)),evalc,C1_1),S1_1),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_170_com_Osimps_I46_J,axiom,
! [Com1,Com2,Fun_1,Com_1] : hAPP(com,com,hAPP(com,fun(com,com),semi,Com1),Com2) != hAPP(com,com,hAPP(fun(state,bool),fun(com,com),while,Fun_1),Com_1) ).
fof(fact_171_com_Osimps_I47_J,axiom,
! [Fun_1,Com_1,Com1,Com2] : hAPP(com,com,hAPP(fun(state,bool),fun(com,com),while,Fun_1),Com_1) != hAPP(com,com,hAPP(com,fun(com,com),semi,Com1),Com2) ).
fof(fact_172_com_Osimps_I3_J,axiom,
! [Com1,Com2,Com1_3,Com2_3] :
( hAPP(com,com,hAPP(com,fun(com,com),semi,Com1),Com2) = hAPP(com,com,hAPP(com,fun(com,com),semi,Com1_3),Com2_3)
<=> ( Com1 = Com1_3
& Com2 = Com2_3 ) ) ).
fof(fact_173_com_Osimps_I5_J,axiom,
! [Fun,Com,Fun_1,Com_1] :
( hAPP(com,com,hAPP(fun(state,bool),fun(com,com),while,Fun),Com) = hAPP(com,com,hAPP(fun(state,bool),fun(com,com),while,Fun_1),Com_1)
<=> ( Fun = Fun_1
& Com = Com_1 ) ) ).
fof(fact_174_com_Osimps_I59_J,axiom,
! [Pname_2,Fun,Com] : hAPP(pname,com,body,Pname_2) != hAPP(com,com,hAPP(fun(state,bool),fun(com,com),while,Fun),Com) ).
fof(fact_175_com_Osimps_I58_J,axiom,
! [Fun,Com,Pname_2] : hAPP(com,com,hAPP(fun(state,bool),fun(com,com),while,Fun),Com) != hAPP(pname,com,body,Pname_2) ).
fof(fact_176_com_Osimps_I16_J,axiom,
! [Fun_1,Com_1] : skip != hAPP(com,com,hAPP(fun(state,bool),fun(com,com),while,Fun_1),Com_1) ).
fof(fact_177_com_Osimps_I17_J,axiom,
! [Fun_1,Com_1] : hAPP(com,com,hAPP(fun(state,bool),fun(com,com),while,Fun_1),Com_1) != skip ).
fof(fact_178_com_Osimps_I49_J,axiom,
! [Pname_1,Com1_1,Com2_1] : hAPP(pname,com,body,Pname_1) != hAPP(com,com,hAPP(com,fun(com,com),semi,Com1_1),Com2_1) ).
fof(fact_179_com_Osimps_I48_J,axiom,
! [Com1_1,Com2_1,Pname_1] : hAPP(com,com,hAPP(com,fun(com,com),semi,Com1_1),Com2_1) != hAPP(pname,com,body,Pname_1) ).
fof(fact_180_com_Osimps_I12_J,axiom,
! [Com1_2,Com2_2] : skip != hAPP(com,com,hAPP(com,fun(com,com),semi,Com1_2),Com2_2) ).
fof(fact_181_com_Osimps_I13_J,axiom,
! [Com1_2,Com2_2] : hAPP(com,com,hAPP(com,fun(com,com),semi,Com1_2),Com2_2) != skip ).
fof(fact_182_evalc__elim__cases_I4_J,axiom,
! [C1_1,C2_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(com,fun(com,com),semi,C1_1),C2_1)),S_2),T_4))
=> ~ ! [S1_2] :
( hBOOL(hAPP(state,bool,hAPP(state,fun(state,bool),hAPP(com,fun(state,fun(state,bool)),evalc,C1_1),S_2),S1_2))
=> ~ hBOOL(hAPP(state,bool,hAPP(state,fun(state,bool),hAPP(com,fun(state,fun(state,bool)),evalc,C2_1),S1_2),T_4)) ) ) ).
fof(fact_183_evaln__elim__cases_I4_J,axiom,
! [C1_1,C2_1,S_2,N_2,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),semi,C1_1),C2_1)),S_2),N_2),T_4))
=> ~ ! [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),S_2),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,C2_1),S1_2),N_2),T_4)) ) ) ).
fof(fact_184_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_185_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_186_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_187_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_188_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_189_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_190_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_191_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_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_Osimps,axiom,
! [X_b,A_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_2))
<=> ( ti(fun(X_b,bool),A_2) = bot_bot(fun(X_b,bool))
| ? [A_4,A_5] :
( ti(fun(X_b,bool),A_2) = hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_5),A_4)
& hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_4)) ) ) ) ).
fof(fact_195_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_196_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_197_evalc__WHILE__case,axiom,
! [B_1,C_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(fun(state,bool),fun(com,com),while,B_1),C_1)),S),T_3))
=> ( ( T_3 = S
=> hBOOL(hAPP(state,bool,B_1,S)) )
=> ~ ( hBOOL(hAPP(state,bool,B_1,S))
=> ! [S1_2] :
( hBOOL(hAPP(state,bool,hAPP(state,fun(state,bool),hAPP(com,fun(state,fun(state,bool)),evalc,C_1),S),S1_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)),S1_2),T_3)) ) ) ) ) ).
fof(fact_198_evaln__WHILE__case,axiom,
! [B_1,C_1,S,N_1,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(fun(state,bool),fun(com,com),while,B_1),C_1)),S),N_1),T_3))
=> ( ( T_3 = S
=> hBOOL(hAPP(state,bool,B_1,S)) )
=> ~ ( hBOOL(hAPP(state,bool,B_1,S))
=> ! [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,C_1),S),N_1),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(com,com,hAPP(fun(state,bool),fun(com,com),while,B_1),C_1)),S1_2),N_1),T_3)) ) ) ) ) ).
fof(fact_199_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_200_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_201_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_202_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_203_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_204_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_205_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_206_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_207_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_208_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_209_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_210_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_211_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_212_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_213_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_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_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_216_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_217_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_218_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_219_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_220_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_221_evaln__max2,axiom,
! [C2_1,S2,N2,T2,C1_1,S1_1,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_1),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_1),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_222_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_223_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_224_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_225_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_226_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_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_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_231_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_232_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_233_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_234_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_235_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_236_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_237_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_238_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_239_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_240_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_241_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_242_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_243_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_244_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_245_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_246_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_247_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_248_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_249_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_250_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_251_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_252_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_253_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_254_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_255_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_256_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_257_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_258_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_259_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_260_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_261_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_262_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_263_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_264_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_265_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_266_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_267_Int__left__commute,axiom,
! [X_b,A_3,B_2,C_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)),B_2),C_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),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_2)) ).
fof(fact_268_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_269_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_270_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_271_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_272_Int__assoc,axiom,
! [X_b,A_3,B_2,C_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(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_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)),B_2),C_2)) ).
fof(fact_273_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_274_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_275_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_276_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_277_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_278_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_279_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_280_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_281_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_282_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_283_inf__Int__eq,axiom,
! [X_b,R_2,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_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)),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_2),S_1))) ) ).
fof(fact_284_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_285_Un__Int__crazy,axiom,
! [X_b,A_3,B_2,C_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)),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_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_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)),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_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_2),A_3)) ).
fof(fact_286_Un__Int__distrib2,axiom,
! [X_b,B_2,C_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)),semilattice_inf_inf(fun(X_b,bool)),B_2),C_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)),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_2),A_3)) ).
fof(fact_287_Int__Un__distrib2,axiom,
! [X_b,B_2,C_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)),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_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)),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_2),A_3)) ).
fof(fact_288_Un__Int__distrib,axiom,
! [X_b,A_3,B_2,C_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_inf_inf(fun(X_b,bool)),B_2),C_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(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_2)) ).
fof(fact_289_Int__Un__distrib,axiom,
! [X_b,A_3,B_2,C_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_sup_sup(fun(X_b,bool)),B_2),C_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)),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_2)) ).
fof(fact_290_Int__insert__left__if1,axiom,
! [X_b,B_2,A_2,C_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),A_2),C_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),B_2)),C_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)),B_2),C_2)) ) ).
fof(fact_291_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_292_Int__insert__left__if0,axiom,
! [X_b,B_2,A_2,C_2] :
( ~ hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),A_2),C_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),B_2)),C_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_2) ) ).
fof(fact_293_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_294_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_295_Int__insert__left,axiom,
! [X_b,B_2,A_2,C_2] :
( ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),A_2),C_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),B_2)),C_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)),B_2),C_2)) )
& ( ~ hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),A_2),C_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),B_2)),C_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_2) ) ) ).
fof(fact_296_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_297_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_298_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_299_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_300_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_301_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_302_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_303_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_304_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_305_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_306_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_307_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_308_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_309_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_310_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_311_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_312_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_313_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_314_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_315_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_316_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_317_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_318_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_319_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_320_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_321_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_322_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_323_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_324_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_325_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_326_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_327_Un__Diff,axiom,
! [X_b,A_3,B_2,C_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_sup_sup(fun(X_b,bool)),A_3),B_2)),C_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),C_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)),B_2),C_2)) ).
fof(fact_328_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_329_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_330_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_331_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_332_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_333_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_334_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_335_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_336_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_337_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_338_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_339_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_340_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_341_Diff__Int2,axiom,
! [X_b,A_3,C_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)),semilattice_inf_inf(fun(X_b,bool)),A_3),C_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_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_2)),B_2) ).
fof(fact_342_Diff__Int__distrib2,axiom,
! [X_b,A_3,B_2,C_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(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_3),B_2)),C_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_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_2)) ).
fof(fact_343_Int__Diff,axiom,
! [X_b,A_3,B_2,C_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),B_2)),C_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),C_2)) ).
fof(fact_344_Diff__Int__distrib,axiom,
! [X_b,C_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)),C_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),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_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_2),B_2)) ).
fof(fact_345_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_346_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_347_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_348_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_349_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_350_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_351_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_352_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_353_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_354_Diff__Un,axiom,
! [X_b,A_3,B_2,C_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(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_sup_sup(fun(X_b,bool)),B_2),C_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(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_2)) ).
fof(fact_355_Diff__Int,axiom,
! [X_b,A_3,B_2,C_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(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),semilattice_inf_inf(fun(X_b,bool)),B_2),C_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)),minus_minus(fun(X_b,bool)),A_3),C_2)) ).
fof(fact_356_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_357_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_358_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_359_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_360_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_361_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_362_finite__empty__induct,axiom,
! [X_b,Pa,A_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_3))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,Pa,A_3))
=> ( ! [A_5,A_4] :
( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_4))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),A_5),A_4))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,Pa,A_4))
=> hBOOL(hAPP(fun(X_b,bool),bool,Pa,hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_4),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(X_b,fun(fun(X_b,bool),fun(X_b,bool)),insert(X_b),A_5),bot_bot(fun(X_b,bool)))))) ) ) )
=> hBOOL(hAPP(fun(X_b,bool),bool,Pa,bot_bot(fun(X_b,bool)))) ) ) ) ).
fof(fact_363_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_364_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_365_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_366_Suc__diff__diff,axiom,
! [M,N_2,K_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,suc,M)),N_2)),hAPP(nat,nat,suc,K_1)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),N_2)),K_1) ).
fof(fact_367_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_368_minus__nat_Odiff__0,axiom,
! [M] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),zero_zero(nat)) = M ).
fof(fact_369_diff__self__eq__0,axiom,
! [M] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),M) = zero_zero(nat) ).
fof(fact_370_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_371_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_372_diff__eq__diff__eq,axiom,
! [X_b] :
( ab_group_add(X_b)
=> ! [A_2,B_1,C_1,D_1] :
( 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_1)
=> ( ti(X_b,A_2) = ti(X_b,B_1)
<=> ti(X_b,C_1) = ti(X_b,D_1) ) ) ) ).
fof(fact_373_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_374_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_375_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_376_zero__induct__lemma,axiom,
! [I_1,Pa,K] :
( hBOOL(hAPP(nat,bool,Pa,K))
=> ( ! [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),I_1))) ) ) ).
fof(fact_377_diff__commute,axiom,
! [I,J,K_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),I),J)),K_1) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),I),K_1)),J) ).
fof(fact_378_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_379_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_380_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_381_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_382_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_383_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_384_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_385_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_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__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_390_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_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] :
( hAPP(fun(X_b,bool),nat,finite_card(X_b),A_3) = hAPP(nat,nat,suc,K)
<=> ? [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
& ( K = 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] :
( hAPP(fun(X_b,bool),nat,finite_card(X_b),A_3) = hAPP(nat,nat,suc,K)
=> ? [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
& ( K = zero_zero(nat)
=> ti(fun(X_b,bool),B_3) = bot_bot(fun(X_b,bool)) ) ) ) ).
fof(fact_395_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_396_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_397_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_398_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_399_One__nat__def,axiom,
one_one(nat) = hAPP(nat,nat,suc,zero_zero(nat)) ).
fof(fact_400_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_401_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_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_zero__neq__one,axiom,
! [X_a] :
( zero_neq_one(X_a)
=> zero_zero(X_a) != one_one(X_a) ) ).
fof(fact_404_one__neq__zero,axiom,
! [X_a] :
( zero_neq_one(X_a)
=> one_one(X_a) != zero_zero(X_a) ) ).
fof(fact_405_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_406_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_407_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_408_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_409_diff__cancel2,axiom,
! [M,K_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),M),K_1)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N_2),K_1)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),N_2) ).
fof(fact_410_diff__cancel,axiom,
! [K_1,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_1),M)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),K_1),N_2)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),N_2) ).
fof(fact_411_diff__diff__left,axiom,
! [I,J,K_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),I),J)),K_1) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),I),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),J),K_1)) ).
fof(fact_412_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_413_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_414_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_415_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_416_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_417_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_418_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_419_nat__add__assoc,axiom,
! [M,N_2,K_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),N_2)),K_1) = 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_1)) ).
fof(fact_420_nat__add__left__cancel,axiom,
! [K,M_1,N_1] :
( hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),K),M_1) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),K),N_1)
<=> M_1 = N_1 ) ).
fof(fact_421_nat__add__right__cancel,axiom,
! [M_1,K,N_1] :
( hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M_1),K) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N_1),K)
<=> M_1 = N_1 ) ).
fof(fact_422_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_423_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_424_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_425_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_426_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_427_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_428_Nat_Oadd__0__right,axiom,
! [M] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),zero_zero(nat)) = M ).
fof(fact_429_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_430_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_431_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_432_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_433_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_434_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_435_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_436_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_437_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_438_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_439_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_440_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_441_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_442_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_443_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_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__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_447_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_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_1,Com2_1] : hAPP(com,nat,com_size,hAPP(com,com,hAPP(com,fun(com,com),semi,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_451_com_Osize_I7_J,axiom,
! [Pname] : hAPP(com,nat,com_size,hAPP(pname,com,body,Pname)) = zero_zero(nat) ).
fof(fact_452_com_Osize_I1_J,axiom,
hAPP(com,nat,com_size,skip) = zero_zero(nat) ).
fof(fact_453_com_Osize_I6_J,axiom,
! [Fun,Com] : hAPP(com,nat,com_size,hAPP(com,com,hAPP(fun(state,bool),fun(com,com),while,Fun),Com)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(com,nat,com_size,Com)),hAPP(nat,nat,suc,zero_zero(nat))) ).
fof(fact_454_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_455_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_456_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_1,B,C,D] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),C),D)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),C)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),B),D)) ) ).
fof(fact_457_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_458_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_459_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_1,C,D] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),C),D)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),C)),D) ) ).
fof(fact_460_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_1,C,D] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),C),D)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),C),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),D)) ) ).
fof(fact_461_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_462_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_463_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_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_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_466_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_467_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_468_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_469_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_470_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_471_com_Osize_I12_J,axiom,
! [Com1_1,Com2_1] : hAPP(com,nat,size_size(com),hAPP(com,com,hAPP(com,fun(com,com),semi,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_472_com_Osize_I15_J,axiom,
! [Pname] : hAPP(com,nat,size_size(com),hAPP(pname,com,body,Pname)) = zero_zero(nat) ).
fof(fact_473_com_Osize_I9_J,axiom,
hAPP(com,nat,size_size(com),skip) = zero_zero(nat) ).
fof(fact_474_com_Osize_I14_J,axiom,
! [Fun,Com] : hAPP(com,nat,size_size(com),hAPP(com,com,hAPP(fun(state,bool),fun(com,com),while,Fun),Com)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(com,nat,size_size(com),Com)),hAPP(nat,nat,suc,zero_zero(nat))) ).
fof(fact_475_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_476_com_Osize_I13_J,axiom,
! [Fun,Com1,Com2] : hAPP(com,nat,size_size(com),hAPP(com,com,hAPP(com,fun(com,com),hAPP(fun(state,bool),fun(com,fun(com,com)),cond,Fun),Com1),Com2)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(com,nat,size_size(com),Com1)),hAPP(com,nat,size_size(com),Com2))),hAPP(nat,nat,suc,zero_zero(nat))) ).
fof(fact_477_evaln_OIfFalse,axiom,
! [C0,C1,N_1,S1,B_1,S] :
( ~ hBOOL(hAPP(state,bool,B_1,S))
=> ( 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),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),N_1),S1)) ) ) ).
fof(fact_478_evaln_OIfTrue,axiom,
! [C1,C0,N_1,S1,B_1,S] :
( hBOOL(hAPP(state,bool,B_1,S))
=> ( 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),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),N_1),S1)) ) ) ).
fof(fact_479_evaln__elim__cases_I5_J,axiom,
! [B_1,C1,C2,S,N_1,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),hAPP(fun(state,bool),fun(com,fun(com,com)),cond,B_1),C1),C2)),S),N_1),T_3))
=> ( ( hBOOL(hAPP(state,bool,B_1,S))
=> ~ 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),N_1),T_3)) )
=> ~ ( ~ hBOOL(hAPP(state,bool,B_1,S))
=> ~ 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),N_1),T_3)) ) ) ) ).
fof(fact_480_evalc__elim__cases_I5_J,axiom,
! [B_1,C1,C2,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),hAPP(fun(state,bool),fun(com,fun(com,com)),cond,B_1),C1),C2)),S),T_3))
=> ( ( hBOOL(hAPP(state,bool,B_1,S))
=> ~ hBOOL(hAPP(state,bool,hAPP(state,fun(state,bool),hAPP(com,fun(state,fun(state,bool)),evalc,C1),S),T_3)) )
=> ~ ( ~ hBOOL(hAPP(state,bool,B_1,S))
=> ~ hBOOL(hAPP(state,bool,hAPP(state,fun(state,bool),hAPP(com,fun(state,fun(state,bool)),evalc,C2),S),T_3)) ) ) ) ).
fof(fact_481_evalc_OIfTrue,axiom,
! [C1,C0,S1,B_1,S] :
( hBOOL(hAPP(state,bool,B_1,S))
=> ( hBOOL(hAPP(state,bool,hAPP(state,fun(state,bool),hAPP(com,fun(state,fun(state,bool)),evalc,C0),S),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),S1)) ) ) ).
fof(fact_482_evalc_OIfFalse,axiom,
! [C0,C1,S1,B_1,S] :
( ~ hBOOL(hAPP(state,bool,B_1,S))
=> ( hBOOL(hAPP(state,bool,hAPP(state,fun(state,bool),hAPP(com,fun(state,fun(state,bool)),evalc,C1),S),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),S1)) ) ) ).
fof(fact_483_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_484_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_485_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_486_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_487_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_488_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_489_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_490_crossproduct__noteq,axiom,
! [X_b] :
( semiri456707255roduct(X_b)
=> ! [C_1,D_1,A_2,B_1] :
( ( ti(X_b,A_2) != ti(X_b,B_1)
& ti(X_b,C_1) != ti(X_b,D_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),A_2),C_1)),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),B_1),D_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),A_2),D_1)),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),B_1),C_1)) ) ) ).
fof(fact_491_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_492_add__scale__eq__noteq,axiom,
! [X_a] :
( semiri456707255roduct(X_a)
=> ! [C,D,A_1,B,R_1] :
( ti(X_a,R_1) != zero_zero(X_a)
=> ( ( ti(X_a,A_1) = ti(X_a,B)
& ti(X_a,C) != ti(X_a,D) )
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),R_1),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_1),D)) ) ) ) ).
fof(fact_493_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_494_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_495_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)) ) ).
%----Arities (37)
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___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_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_Ocancel__ab__semigroup__add,axiom,
cancel146912293up_add(nat) ).
fof(arity_Nat_Onat___Groups_Ocancel__semigroup__add,axiom,
cancel_semigroup_add(nat) ).
fof(arity_Nat_Onat___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___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_Ozero__neq__one,axiom,
zero_neq_one(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___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___Finite__Set_Ofinite,axiom,
finite_finite(bool) ).
fof(arity_HOL_Obool___Lattices_Olattice,axiom,
lattice(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_1656922687triple(x_a),bool),bool,hAPP(hoare_1656922687triple(x_a),fun(fun(hoare_1656922687triple(x_a),bool),bool),member(hoare_1656922687triple(x_a)),X),hAPP(fun(hoare_1656922687triple(x_a),bool),fun(hoare_1656922687triple(x_a),bool),hAPP(fun(hoare_1656922687triple(x_a),bool),fun(fun(hoare_1656922687triple(x_a),bool),fun(hoare_1656922687triple(x_a),bool)),semilattice_sup_sup(fun(hoare_1656922687triple(x_a),bool)),g),hAPP(fun(pname,bool),fun(hoare_1656922687triple(x_a),bool),hAPP(fun(pname,hoare_1656922687triple(x_a)),fun(fun(pname,bool),fun(hoare_1656922687triple(x_a),bool)),image(pname,hoare_1656922687triple(x_a)),hAPP(fun(pname,fun(x_a,fun(state,bool))),fun(pname,hoare_1656922687triple(x_a)),hAPP(fun(pname,fun(fun(x_a,fun(state,bool)),hoare_1656922687triple(x_a))),fun(fun(pname,fun(x_a,fun(state,bool))),fun(pname,hoare_1656922687triple(x_a))),combs(pname,fun(x_a,fun(state,bool)),hoare_1656922687triple(x_a)),hAPP(fun(pname,com),fun(pname,fun(fun(x_a,fun(state,bool)),hoare_1656922687triple(x_a))),hAPP(fun(pname,fun(com,fun(fun(x_a,fun(state,bool)),hoare_1656922687triple(x_a)))),fun(fun(pname,com),fun(pname,fun(fun(x_a,fun(state,bool)),hoare_1656922687triple(x_a)))),combs(pname,com,fun(fun(x_a,fun(state,bool)),hoare_1656922687triple(x_a))),hAPP(fun(pname,fun(x_a,fun(state,bool))),fun(pname,fun(com,fun(fun(x_a,fun(state,bool)),hoare_1656922687triple(x_a)))),hAPP(fun(fun(x_a,fun(state,bool)),fun(com,fun(fun(x_a,fun(state,bool)),hoare_1656922687triple(x_a)))),fun(fun(pname,fun(x_a,fun(state,bool))),fun(pname,fun(com,fun(fun(x_a,fun(state,bool)),hoare_1656922687triple(x_a))))),combb(fun(x_a,fun(state,bool)),fun(com,fun(fun(x_a,fun(state,bool)),hoare_1656922687triple(x_a))),pname),hoare_246368825triple(x_a)),p)),body)),q)),procs))))
=> hBOOL(hAPP(hoare_1656922687triple(x_a),bool,hAPP(nat,fun(hoare_1656922687triple(x_a),bool),hoare_920331057_valid(x_a),N),X)) )
=> ! [X] :
( hBOOL(hAPP(fun(hoare_1656922687triple(x_a),bool),bool,hAPP(hoare_1656922687triple(x_a),fun(fun(hoare_1656922687triple(x_a),bool),bool),member(hoare_1656922687triple(x_a)),X),hAPP(fun(pname,bool),fun(hoare_1656922687triple(x_a),bool),hAPP(fun(pname,hoare_1656922687triple(x_a)),fun(fun(pname,bool),fun(hoare_1656922687triple(x_a),bool)),image(pname,hoare_1656922687triple(x_a)),hAPP(fun(pname,fun(x_a,fun(state,bool))),fun(pname,hoare_1656922687triple(x_a)),hAPP(fun(pname,fun(fun(x_a,fun(state,bool)),hoare_1656922687triple(x_a))),fun(fun(pname,fun(x_a,fun(state,bool))),fun(pname,hoare_1656922687triple(x_a))),combs(pname,fun(x_a,fun(state,bool)),hoare_1656922687triple(x_a)),hAPP(fun(pname,com),fun(pname,fun(fun(x_a,fun(state,bool)),hoare_1656922687triple(x_a))),hAPP(fun(pname,fun(com,fun(fun(x_a,fun(state,bool)),hoare_1656922687triple(x_a)))),fun(fun(pname,com),fun(pname,fun(fun(x_a,fun(state,bool)),hoare_1656922687triple(x_a)))),combs(pname,com,fun(fun(x_a,fun(state,bool)),hoare_1656922687triple(x_a))),hAPP(fun(pname,fun(x_a,fun(state,bool))),fun(pname,fun(com,fun(fun(x_a,fun(state,bool)),hoare_1656922687triple(x_a)))),hAPP(fun(fun(x_a,fun(state,bool)),fun(com,fun(fun(x_a,fun(state,bool)),hoare_1656922687triple(x_a)))),fun(fun(pname,fun(x_a,fun(state,bool))),fun(pname,fun(com,fun(fun(x_a,fun(state,bool)),hoare_1656922687triple(x_a))))),combb(fun(x_a,fun(state,bool)),fun(com,fun(fun(x_a,fun(state,bool)),hoare_1656922687triple(x_a))),pname),hoare_246368825triple(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_1656922687triple(x_a),bool,hAPP(nat,fun(hoare_1656922687triple(x_a),bool),hoare_920331057_valid(x_a),N),X)) ) ) ).
fof(conj_1,conjecture,
( ! [X] :
( hBOOL(hAPP(fun(hoare_1656922687triple(x_a),bool),bool,hAPP(hoare_1656922687triple(x_a),fun(fun(hoare_1656922687triple(x_a),bool),bool),member(hoare_1656922687triple(x_a)),X),g))
=> hBOOL(hAPP(hoare_1656922687triple(x_a),bool,hAPP(nat,fun(hoare_1656922687triple(x_a),bool),hoare_920331057_valid(x_a),n),X)) )
=> ! [X] :
( hBOOL(hAPP(fun(hoare_1656922687triple(x_a),bool),bool,hAPP(hoare_1656922687triple(x_a),fun(fun(hoare_1656922687triple(x_a),bool),bool),member(hoare_1656922687triple(x_a)),X),hAPP(fun(pname,bool),fun(hoare_1656922687triple(x_a),bool),hAPP(fun(pname,hoare_1656922687triple(x_a)),fun(fun(pname,bool),fun(hoare_1656922687triple(x_a),bool)),image(pname,hoare_1656922687triple(x_a)),hAPP(fun(pname,fun(x_a,fun(state,bool))),fun(pname,hoare_1656922687triple(x_a)),hAPP(fun(pname,fun(fun(x_a,fun(state,bool)),hoare_1656922687triple(x_a))),fun(fun(pname,fun(x_a,fun(state,bool))),fun(pname,hoare_1656922687triple(x_a))),combs(pname,fun(x_a,fun(state,bool)),hoare_1656922687triple(x_a)),hAPP(fun(pname,com),fun(pname,fun(fun(x_a,fun(state,bool)),hoare_1656922687triple(x_a))),hAPP(fun(pname,fun(com,fun(fun(x_a,fun(state,bool)),hoare_1656922687triple(x_a)))),fun(fun(pname,com),fun(pname,fun(fun(x_a,fun(state,bool)),hoare_1656922687triple(x_a)))),combs(pname,com,fun(fun(x_a,fun(state,bool)),hoare_1656922687triple(x_a))),hAPP(fun(pname,fun(x_a,fun(state,bool))),fun(pname,fun(com,fun(fun(x_a,fun(state,bool)),hoare_1656922687triple(x_a)))),hAPP(fun(fun(x_a,fun(state,bool)),fun(com,fun(fun(x_a,fun(state,bool)),hoare_1656922687triple(x_a)))),fun(fun(pname,fun(x_a,fun(state,bool))),fun(pname,fun(com,fun(fun(x_a,fun(state,bool)),hoare_1656922687triple(x_a))))),combb(fun(x_a,fun(state,bool)),fun(com,fun(fun(x_a,fun(state,bool)),hoare_1656922687triple(x_a))),pname),hoare_246368825triple(x_a)),p)),body)),q)),procs)))
=> hBOOL(hAPP(hoare_1656922687triple(x_a),bool,hAPP(nat,fun(hoare_1656922687triple(x_a),bool),hoare_920331057_valid(x_a),n),X)) ) ) ).
%------------------------------------------------------------------------------