## TPTP Problem File: SWW470+1.p

View Solutions - Solve Problem

```%------------------------------------------------------------------------------
% File     : SWW470+1 : TPTP v8.0.0. Released v5.3.0.
% Domain   : Software Verification
% Problem  : Hoare's Logic with Procedures line 200, 100 axioms selected
% Version  : Especial.
% English  :

% Refs     : [BN10]  Boehme & Nipkow (2010), Sledgehammer: Judgement Day
%          : [Bla11] Blanchette (2011), Email to Geoff Sutcliffe
% Source   : [Bla11]
% Names    : hoare_100_fofmg_l200 [Bla11]

% Status   : Theorem
% Rating   : 0.42 v7.5.0, 0.47 v7.4.0, 0.40 v7.3.0, 0.48 v7.1.0, 0.39 v7.0.0, 0.40 v6.4.0, 0.46 v6.3.0, 0.42 v6.2.0, 0.52 v6.1.0, 0.63 v6.0.0, 0.65 v5.5.0, 0.74 v5.4.0, 0.86 v5.3.0
% Syntax   : Number of formulae    :  144 (  56 unt;   0 def)
%            Number of atoms       :  323 ( 119 equ)
%            Maximal formula atoms :    9 (   2 avg)
%            Number of connectives :  239 (  60   ~;  27   |;  22   &)
%                                         (  22 <=>; 108  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   16 (   5 avg)
%            Maximal term depth    :   11 (   2 avg)
%            Number of predicates  :    3 (   2 usr;   0 prp; 1-2 aty)
%            Number of functors    :   90 (  90 usr;  30 con; 0-3 aty)
%            Number of variables   :  383 ( 365   !;  18   ?)
% SPC      : FOF_THM_RFO_SEQ

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2011-08-09 15:27:58
%          : Encoded with monomorphized guards.
%------------------------------------------------------------------------------
%----Explicit typings (9)
fof(gsy_c_Finite__Set_Ofolding__one_000tc__Hoare____Mirabelle____jfehddehev__Otriple,axiom,
! [B_1_1,B_2] : is_bool(finite520909254iple_a(B_1_1,B_2)) ).

fof(gsy_c_Finite__Set_Ofolding__one__idem_000tc__Hoare____Mirabelle____jfehddehev__O,axiom,
! [B_1_1,B_2] : is_bool(finite1948426435iple_a(B_1_1,B_2)) ).

fof(gsy_c_Orderings_Obot__class_Obot_000tc__HOL__Obool,axiom,
is_bool(bot_bot_bool) ).

fof(gsy_c_fFalse,hypothesis,
is_bool(fFalse) ).

fof(gsy_c_fTrue,axiom,
is_bool(fTrue) ).

fof(gsy_c_hAPP_000tc__Com__Ostate_000tc__HOL__Obool,axiom,
! [B_1_1,B_2] : is_bool(hAPP_state_bool(B_1_1,B_2)) ).

fof(gsy_c_hAPP_000tc__HOL__Obool_000tc__HOL__Obool,axiom,
! [B_1_1,B_2] :
( is_bool(B_2)
=> is_bool(hAPP_bool_bool(B_1_1,B_2)) ) ).

fof(gsy_c_hAPP_000tc__Hoare____Mirabelle____jfehddehev__Otriple_It__a_J_000tc__HOL__,axiom,
! [B_1_1,B_2] : is_bool(hAPP_H1927961489a_bool(B_1_1,B_2)) ).

fof(gsy_c_hAPP_000tc__fun_Itc__Hoare____Mirabelle____jfehddehev__Otriple_It__a_J_Mtc,hypothesis,
! [B_1_1,B_2] : is_bool(hAPP_f1753944735l_bool(B_1_1,B_2)) ).

%----Relevant facts (95)
fof(fact_0_empty,axiom,
! [Ga] : hBOOL(hAPP_f1753944735l_bool(hoare_472868247rivs_a(Ga),bot_bo1687970473a_bool)) ).

fof(fact_1_triple_Oinject,axiom,
! [Fun1_2,Com_2,Fun2_2,Fun1_1,Com_1,Fun2_1] :
( hoare_1050552211iple_a(Fun1_2,Com_2,Fun2_2) = hoare_1050552211iple_a(Fun1_1,Com_1,Fun2_1)
<=> ( Fun1_2 = Fun1_1
& Com_2 = Com_1
& Fun2_2 = Fun2_1 ) ) ).

fof(fact_2_cut,axiom,
! [Ga,G_1,Ts] :
( hBOOL(hAPP_f1753944735l_bool(hoare_472868247rivs_a(G_1),Ts))
=> ( hBOOL(hAPP_f1753944735l_bool(hoare_472868247rivs_a(Ga),G_1))
=> hBOOL(hAPP_f1753944735l_bool(hoare_472868247rivs_a(Ga),Ts)) ) ) ).

fof(fact_3_hoare__derivs_Oinsert,axiom,
! [Ts,Ga,T] :
( hBOOL(hAPP_f1753944735l_bool(hoare_472868247rivs_a(Ga),hAPP_f1400872321a_bool(hAPP_H1816261935a_bool(insert1871499715iple_a,T),bot_bo1687970473a_bool)))
=> ( hBOOL(hAPP_f1753944735l_bool(hoare_472868247rivs_a(Ga),Ts))
=> hBOOL(hAPP_f1753944735l_bool(hoare_472868247rivs_a(Ga),hAPP_f1400872321a_bool(hAPP_H1816261935a_bool(insert1871499715iple_a,T),Ts))) ) ) ).

fof(fact_4_constant,axiom,
! [Ga,Pa,Ca,Q_1,C] :
( ( hBOOL(C)
=> hBOOL(hAPP_f1753944735l_bool(hoare_472868247rivs_a(Ga),hAPP_f1400872321a_bool(hAPP_H1816261935a_bool(insert1871499715iple_a,hoare_1050552211iple_a(Pa,Ca,Q_1)),bot_bo1687970473a_bool))) )
=> hBOOL(hAPP_f1753944735l_bool(hoare_472868247rivs_a(Ga),hAPP_f1400872321a_bool(hAPP_H1816261935a_bool(insert1871499715iple_a,hoare_1050552211iple_a(hAPP_b540892988e_bool(hAPP_f1824947087e_bool(cOMBC_41962815e_bool,hAPP_f340725611e_bool(cOMBB_1348041619bool_a(cOMBC_231445413l_bool),hAPP_f1509969235l_bool(cOMBB_1355796797bool_a(cOMBB_188601460_state(fconj)),Pa))),C),Ca,Q_1)),bot_bo1687970473a_bool))) ) ).

fof(fact_5_escape,axiom,
! [Ga,Ca,Q_1,Pa] :
( ! [Z_1,S] :
( hBOOL(hAPP_state_bool(hAPP_a2036067514e_bool(Pa,Z_1),S))
=> hBOOL(hAPP_f1753944735l_bool(hoare_472868247rivs_a(Ga),hAPP_f1400872321a_bool(hAPP_H1816261935a_bool(insert1871499715iple_a,hoare_1050552211iple_a(cOMBK_1458035955bool_a(hAPP_s1806633685e_bool(hAPP_f817621513e_bool(cOMBC_2027030106e_bool,fequal_state),S)),Ca,cOMBK_1458035955bool_a(hAPP_a2036067514e_bool(Q_1,Z_1)))),bot_bo1687970473a_bool))) )
=> hBOOL(hAPP_f1753944735l_bool(hoare_472868247rivs_a(Ga),hAPP_f1400872321a_bool(hAPP_H1816261935a_bool(insert1871499715iple_a,hoare_1050552211iple_a(Pa,Ca,Q_1)),bot_bo1687970473a_bool))) ) ).

fof(fact_6_conseq2,axiom,
! [Q_1,Ga,Pa,Ca,Q_3] :
( hBOOL(hAPP_f1753944735l_bool(hoare_472868247rivs_a(Ga),hAPP_f1400872321a_bool(hAPP_H1816261935a_bool(insert1871499715iple_a,hoare_1050552211iple_a(Pa,Ca,Q_3)),bot_bo1687970473a_bool)))
=> ( ! [Z_1,S] :
( hBOOL(hAPP_state_bool(hAPP_a2036067514e_bool(Q_3,Z_1),S))
=> hBOOL(hAPP_state_bool(hAPP_a2036067514e_bool(Q_1,Z_1),S)) )
=> hBOOL(hAPP_f1753944735l_bool(hoare_472868247rivs_a(Ga),hAPP_f1400872321a_bool(hAPP_H1816261935a_bool(insert1871499715iple_a,hoare_1050552211iple_a(Pa,Ca,Q_1)),bot_bo1687970473a_bool))) ) ) ).

fof(fact_7_conseq1,axiom,
! [Pa,Ga,P_2,Ca,Q_1] :
( hBOOL(hAPP_f1753944735l_bool(hoare_472868247rivs_a(Ga),hAPP_f1400872321a_bool(hAPP_H1816261935a_bool(insert1871499715iple_a,hoare_1050552211iple_a(P_2,Ca,Q_1)),bot_bo1687970473a_bool)))
=> ( ! [Z_1,S] :
( hBOOL(hAPP_state_bool(hAPP_a2036067514e_bool(Pa,Z_1),S))
=> hBOOL(hAPP_state_bool(hAPP_a2036067514e_bool(P_2,Z_1),S)) )
=> hBOOL(hAPP_f1753944735l_bool(hoare_472868247rivs_a(Ga),hAPP_f1400872321a_bool(hAPP_H1816261935a_bool(insert1871499715iple_a,hoare_1050552211iple_a(Pa,Ca,Q_1)),bot_bo1687970473a_bool))) ) ) ).

fof(fact_8_conseq12,axiom,
! [Q_1,Pa,Ga,P_2,Ca,Q_3] :
( hBOOL(hAPP_f1753944735l_bool(hoare_472868247rivs_a(Ga),hAPP_f1400872321a_bool(hAPP_H1816261935a_bool(insert1871499715iple_a,hoare_1050552211iple_a(P_2,Ca,Q_3)),bot_bo1687970473a_bool)))
=> ( ! [Z_1,S] :
( hBOOL(hAPP_state_bool(hAPP_a2036067514e_bool(Pa,Z_1),S))
=> ! [S_1] :
( ! [Z_2] :
( hBOOL(hAPP_state_bool(hAPP_a2036067514e_bool(P_2,Z_2),S))
=> hBOOL(hAPP_state_bool(hAPP_a2036067514e_bool(Q_3,Z_2),S_1)) )
=> hBOOL(hAPP_state_bool(hAPP_a2036067514e_bool(Q_1,Z_1),S_1)) ) )
=> hBOOL(hAPP_f1753944735l_bool(hoare_472868247rivs_a(Ga),hAPP_f1400872321a_bool(hAPP_H1816261935a_bool(insert1871499715iple_a,hoare_1050552211iple_a(Pa,Ca,Q_1)),bot_bo1687970473a_bool))) ) ) ).

fof(fact_9_insertE,axiom,
! [A_3,Ba,A] :
( hBOOL(hAPP_f1753944735l_bool(hAPP_H1926610125l_bool(member564727580iple_a,A_3),hAPP_f1400872321a_bool(hAPP_H1816261935a_bool(insert1871499715iple_a,Ba),A)))
=> ( A_3 != Ba
=> hBOOL(hAPP_f1753944735l_bool(hAPP_H1926610125l_bool(member564727580iple_a,A_3),A)) ) ) ).

fof(fact_10_insertCI,axiom,
! [Ba,A_3,B_1] :
( ( ~ hBOOL(hAPP_f1753944735l_bool(hAPP_H1926610125l_bool(member564727580iple_a,A_3),B_1))
=> A_3 = Ba )
=> hBOOL(hAPP_f1753944735l_bool(hAPP_H1926610125l_bool(member564727580iple_a,A_3),hAPP_f1400872321a_bool(hAPP_H1816261935a_bool(insert1871499715iple_a,Ba),B_1))) ) ).

fof(fact_11_emptyE,axiom,
! [A_3] : ~ hBOOL(hAPP_f1753944735l_bool(hAPP_H1926610125l_bool(member564727580iple_a,A_3),bot_bo1687970473a_bool)) ).

fof(fact_12_singleton__conv2,axiom,
! [A_3] : collec1266446174iple_a(hAPP_H562195827a_bool(fequal1878252616iple_a,A_3)) = hAPP_f1400872321a_bool(hAPP_H1816261935a_bool(insert1871499715iple_a,A_3),bot_bo1687970473a_bool) ).

fof(fact_13_singleton__conv,axiom,
! [A_3] : collec1266446174iple_a(hAPP_H562195827a_bool(hAPP_f1915402821a_bool(cOMBC_2049287834a_bool,fequal1878252616iple_a),A_3)) = hAPP_f1400872321a_bool(hAPP_H1816261935a_bool(insert1871499715iple_a,A_3),bot_bo1687970473a_bool) ).

fof(fact_14_Collect__conv__if2,axiom,
! [Pa,A_3] :
( ( hBOOL(hAPP_H1927961489a_bool(Pa,A_3))
=> collec1266446174iple_a(hAPP_f1400872321a_bool(hAPP_f1104866853a_bool(cOMBS_213702372l_bool,hAPP_f13210641l_bool(cOMBB_633860163iple_a(fconj),hAPP_H562195827a_bool(fequal1878252616iple_a,A_3))),Pa)) = hAPP_f1400872321a_bool(hAPP_H1816261935a_bool(insert1871499715iple_a,A_3),bot_bo1687970473a_bool) )
& ( ~ hBOOL(hAPP_H1927961489a_bool(Pa,A_3))
=> collec1266446174iple_a(hAPP_f1400872321a_bool(hAPP_f1104866853a_bool(cOMBS_213702372l_bool,hAPP_f13210641l_bool(cOMBB_633860163iple_a(fconj),hAPP_H562195827a_bool(fequal1878252616iple_a,A_3))),Pa)) = bot_bo1687970473a_bool ) ) ).

fof(fact_15_Collect__conv__if,axiom,
! [Pa,A_3] :
( ( hBOOL(hAPP_H1927961489a_bool(Pa,A_3))
=> collec1266446174iple_a(hAPP_f1400872321a_bool(hAPP_f1104866853a_bool(cOMBS_213702372l_bool,hAPP_f13210641l_bool(cOMBB_633860163iple_a(fconj),hAPP_H562195827a_bool(hAPP_f1915402821a_bool(cOMBC_2049287834a_bool,fequal1878252616iple_a),A_3))),Pa)) = hAPP_f1400872321a_bool(hAPP_H1816261935a_bool(insert1871499715iple_a,A_3),bot_bo1687970473a_bool) )
& ( ~ hBOOL(hAPP_H1927961489a_bool(Pa,A_3))
=> collec1266446174iple_a(hAPP_f1400872321a_bool(hAPP_f1104866853a_bool(cOMBS_213702372l_bool,hAPP_f13210641l_bool(cOMBB_633860163iple_a(fconj),hAPP_H562195827a_bool(hAPP_f1915402821a_bool(cOMBC_2049287834a_bool,fequal1878252616iple_a),A_3))),Pa)) = bot_bo1687970473a_bool ) ) ).

fof(fact_16_equals0D,axiom,
! [A_3,A] :
( A = bot_bo1687970473a_bool
=> ~ hBOOL(hAPP_f1753944735l_bool(hAPP_H1926610125l_bool(member564727580iple_a,A_3),A)) ) ).

fof(fact_17_Collect__empty__eq,axiom,
! [Pa] :
( collec1266446174iple_a(Pa) = bot_bo1687970473a_bool
<=> ! [X_2] : ~ hBOOL(hAPP_H1927961489a_bool(Pa,X_2)) ) ).

fof(fact_18_empty__iff,axiom,
! [Ca] : ~ hBOOL(hAPP_f1753944735l_bool(hAPP_H1926610125l_bool(member564727580iple_a,Ca),bot_bo1687970473a_bool)) ).

fof(fact_19_empty__Collect__eq,axiom,
! [Pa] :
( bot_bo1687970473a_bool = collec1266446174iple_a(Pa)
<=> ! [X_2] : ~ hBOOL(hAPP_H1927961489a_bool(Pa,X_2)) ) ).

fof(fact_20_ex__in__conv,axiom,
! [A] :
( ? [X_2] : hBOOL(hAPP_f1753944735l_bool(hAPP_H1926610125l_bool(member564727580iple_a,X_2),A))
<=> A != bot_bo1687970473a_bool ) ).

fof(fact_21_all__not__in__conv,axiom,
! [A] :
( ! [X_2] : ~ hBOOL(hAPP_f1753944735l_bool(hAPP_H1926610125l_bool(member564727580iple_a,X_2),A))
<=> A = bot_bo1687970473a_bool ) ).

fof(fact_22_empty__def,axiom,
bot_bo1687970473a_bool = collec1266446174iple_a(cOMBK_1150238960iple_a(fFalse)) ).

fof(fact_23_insert__absorb,axiom,
! [A_3,A] :
( hBOOL(hAPP_f1753944735l_bool(hAPP_H1926610125l_bool(member564727580iple_a,A_3),A))
=> hAPP_f1400872321a_bool(hAPP_H1816261935a_bool(insert1871499715iple_a,A_3),A) = A ) ).

fof(fact_24_insertI2,axiom,
! [Ba,A_3,B_1] :
( hBOOL(hAPP_f1753944735l_bool(hAPP_H1926610125l_bool(member564727580iple_a,A_3),B_1))
=> hBOOL(hAPP_f1753944735l_bool(hAPP_H1926610125l_bool(member564727580iple_a,A_3),hAPP_f1400872321a_bool(hAPP_H1816261935a_bool(insert1871499715iple_a,Ba),B_1))) ) ).

fof(fact_25_insert__ident,axiom,
! [B_1,X_1,A] :
( ~ hBOOL(hAPP_f1753944735l_bool(hAPP_H1926610125l_bool(member564727580iple_a,X_1),A))
=> ( ~ hBOOL(hAPP_f1753944735l_bool(hAPP_H1926610125l_bool(member564727580iple_a,X_1),B_1))
=> ( hAPP_f1400872321a_bool(hAPP_H1816261935a_bool(insert1871499715iple_a,X_1),A) = hAPP_f1400872321a_bool(hAPP_H1816261935a_bool(insert1871499715iple_a,X_1),B_1)
<=> A = B_1 ) ) ) ).

fof(fact_26_insert__code,axiom,
! [Y_2,A,X_1] :
( hBOOL(hAPP_H1927961489a_bool(hAPP_f1400872321a_bool(hAPP_H1816261935a_bool(insert1871499715iple_a,Y_2),A),X_1))
<=> ( Y_2 = X_1
| hBOOL(hAPP_H1927961489a_bool(A,X_1)) ) ) ).

fof(fact_27_insert__iff,axiom,
! [A_3,Ba,A] :
( hBOOL(hAPP_f1753944735l_bool(hAPP_H1926610125l_bool(member564727580iple_a,A_3),hAPP_f1400872321a_bool(hAPP_H1816261935a_bool(insert1871499715iple_a,Ba),A)))
<=> ( A_3 = Ba
| hBOOL(hAPP_f1753944735l_bool(hAPP_H1926610125l_bool(member564727580iple_a,A_3),A)) ) ) ).

fof(fact_28_insert__commute,axiom,
! [X_1,Y_2,A] : hAPP_f1400872321a_bool(hAPP_H1816261935a_bool(insert1871499715iple_a,X_1),hAPP_f1400872321a_bool(hAPP_H1816261935a_bool(insert1871499715iple_a,Y_2),A)) = hAPP_f1400872321a_bool(hAPP_H1816261935a_bool(insert1871499715iple_a,Y_2),hAPP_f1400872321a_bool(hAPP_H1816261935a_bool(insert1871499715iple_a,X_1),A)) ).

fof(fact_29_insert__absorb2,axiom,
! [X_1,A] : hAPP_f1400872321a_bool(hAPP_H1816261935a_bool(insert1871499715iple_a,X_1),hAPP_f1400872321a_bool(hAPP_H1816261935a_bool(insert1871499715iple_a,X_1),A)) = hAPP_f1400872321a_bool(hAPP_H1816261935a_bool(insert1871499715iple_a,X_1),A) ).

fof(fact_30_insert__Collect,axiom,
! [A_3,Pa] : hAPP_f1400872321a_bool(hAPP_H1816261935a_bool(insert1871499715iple_a,A_3),collec1266446174iple_a(Pa)) = collec1266446174iple_a(hAPP_f1400872321a_bool(hAPP_f1104866853a_bool(cOMBS_213702372l_bool,hAPP_f13210641l_bool(cOMBB_633860163iple_a(fimplies),hAPP_f1400872321a_bool(cOMBB_650444389iple_a(fNot),hAPP_H562195827a_bool(hAPP_f1915402821a_bool(cOMBC_2049287834a_bool,fequal1878252616iple_a),A_3)))),Pa)) ).

fof(fact_31_insert__compr,axiom,
! [A_3,B_1] : hAPP_f1400872321a_bool(hAPP_H1816261935a_bool(insert1871499715iple_a,A_3),B_1) = collec1266446174iple_a(hAPP_f1400872321a_bool(hAPP_f1104866853a_bool(cOMBS_213702372l_bool,hAPP_f13210641l_bool(cOMBB_633860163iple_a(fdisj),hAPP_H562195827a_bool(hAPP_f1915402821a_bool(cOMBC_2049287834a_bool,fequal1878252616iple_a),A_3))),hAPP_f1400872321a_bool(hAPP_f945663555a_bool(cOMBC_2067518550l_bool,member564727580iple_a),B_1))) ).

fof(fact_32_insertI1,axiom,
! [A_3,B_1] : hBOOL(hAPP_f1753944735l_bool(hAPP_H1926610125l_bool(member564727580iple_a,A_3),hAPP_f1400872321a_bool(hAPP_H1816261935a_bool(insert1871499715iple_a,A_3),B_1))) ).

fof(fact_33_insert__compr__raw,axiom,
! [X_2,Xa] : hAPP_f1400872321a_bool(hAPP_H1816261935a_bool(insert1871499715iple_a,X_2),Xa) = collec1266446174iple_a(hAPP_f1400872321a_bool(hAPP_f1104866853a_bool(cOMBS_213702372l_bool,hAPP_f13210641l_bool(cOMBB_633860163iple_a(fdisj),hAPP_H562195827a_bool(hAPP_f1915402821a_bool(cOMBC_2049287834a_bool,fequal1878252616iple_a),X_2))),hAPP_f1400872321a_bool(hAPP_f945663555a_bool(cOMBC_2067518550l_bool,member564727580iple_a),Xa))) ).

fof(fact_34_singleton__inject,axiom,
! [A_3,Ba] :
( hAPP_f1400872321a_bool(hAPP_H1816261935a_bool(insert1871499715iple_a,A_3),bot_bo1687970473a_bool) = hAPP_f1400872321a_bool(hAPP_H1816261935a_bool(insert1871499715iple_a,Ba),bot_bo1687970473a_bool)
=> A_3 = Ba ) ).

fof(fact_35_singletonE,axiom,
! [Ba,A_3] :
( hBOOL(hAPP_f1753944735l_bool(hAPP_H1926610125l_bool(member564727580iple_a,Ba),hAPP_f1400872321a_bool(hAPP_H1816261935a_bool(insert1871499715iple_a,A_3),bot_bo1687970473a_bool)))
=> Ba = A_3 ) ).

fof(fact_36_doubleton__eq__iff,axiom,
! [A_3,Ba,Ca,D] :
( hAPP_f1400872321a_bool(hAPP_H1816261935a_bool(insert1871499715iple_a,A_3),hAPP_f1400872321a_bool(hAPP_H1816261935a_bool(insert1871499715iple_a,Ba),bot_bo1687970473a_bool)) = hAPP_f1400872321a_bool(hAPP_H1816261935a_bool(insert1871499715iple_a,Ca),hAPP_f1400872321a_bool(hAPP_H1816261935a_bool(insert1871499715iple_a,D),bot_bo1687970473a_bool))
<=> ( ( A_3 = Ca
& Ba = D )
| ( A_3 = D
& Ba = Ca ) ) ) ).

fof(fact_37_singleton__iff,axiom,
! [Ba,A_3] :
( hBOOL(hAPP_f1753944735l_bool(hAPP_H1926610125l_bool(member564727580iple_a,Ba),hAPP_f1400872321a_bool(hAPP_H1816261935a_bool(insert1871499715iple_a,A_3),bot_bo1687970473a_bool)))
<=> Ba = A_3 ) ).

fof(fact_38_insert__not__empty,axiom,
! [A_3,A] : hAPP_f1400872321a_bool(hAPP_H1816261935a_bool(insert1871499715iple_a,A_3),A) != bot_bo1687970473a_bool ).

fof(fact_39_empty__not__insert,axiom,
! [A_3,A] : bot_bo1687970473a_bool != hAPP_f1400872321a_bool(hAPP_H1816261935a_bool(insert1871499715iple_a,A_3),A) ).

fof(fact_40_the__elem__eq,axiom,
! [X_1] : hAPP_f1826273671iple_a(the_el287271400iple_a,hAPP_f1400872321a_bool(hAPP_H1816261935a_bool(insert1871499715iple_a,X_1),bot_bo1687970473a_bool)) = X_1 ).

fof(fact_41_bot__apply,axiom,
! [X_1] :
( hBOOL(hAPP_H1927961489a_bool(bot_bo1687970473a_bool,X_1))
<=> hBOOL(bot_bot_bool) ) ).

fof(fact_42_bot__fun__def,axiom,
! [X_2] :
( hBOOL(hAPP_H1927961489a_bool(bot_bo1687970473a_bool,X_2))
<=> hBOOL(bot_bot_bool) ) ).

fof(fact_43_hoare__derivs_OSkip,axiom,
! [Ga,Pa] : hBOOL(hAPP_f1753944735l_bool(hoare_472868247rivs_a(Ga),hAPP_f1400872321a_bool(hAPP_H1816261935a_bool(insert1871499715iple_a,hoare_1050552211iple_a(Pa,skip,Pa)),bot_bo1687970473a_bool))) ).

fof(fact_44_Comp,axiom,
! [D,R_1,Ga,Pa,Ca,Q_1] :
( hBOOL(hAPP_f1753944735l_bool(hoare_472868247rivs_a(Ga),hAPP_f1400872321a_bool(hAPP_H1816261935a_bool(insert1871499715iple_a,hoare_1050552211iple_a(Pa,Ca,Q_1)),bot_bo1687970473a_bool)))
=> ( hBOOL(hAPP_f1753944735l_bool(hoare_472868247rivs_a(Ga),hAPP_f1400872321a_bool(hAPP_H1816261935a_bool(insert1871499715iple_a,hoare_1050552211iple_a(Q_1,D,R_1)),bot_bo1687970473a_bool)))
=> hBOOL(hAPP_f1753944735l_bool(hoare_472868247rivs_a(Ga),hAPP_f1400872321a_bool(hAPP_H1816261935a_bool(insert1871499715iple_a,hoare_1050552211iple_a(Pa,semi(Ca,D),R_1)),bot_bo1687970473a_bool))) ) ) ).

fof(fact_45_triple_Oexhaust,axiom,
! [Y_2] :
~ ! [Fun1,Com,Fun2] : Y_2 != hoare_1050552211iple_a(Fun1,Com,Fun2) ).

fof(fact_46_Set_Oset__insert,axiom,
! [X_1,A] :
( hBOOL(hAPP_f1753944735l_bool(hAPP_H1926610125l_bool(member564727580iple_a,X_1),A))
=> ~ ! [B] :
( A = hAPP_f1400872321a_bool(hAPP_H1816261935a_bool(insert1871499715iple_a,X_1),B)
=> hBOOL(hAPP_f1753944735l_bool(hAPP_H1926610125l_bool(member564727580iple_a,X_1),B)) ) ) ).

fof(fact_47_com_Osimps_I13_J,axiom,
! [Com1_2,Com2_2] : semi(Com1_2,Com2_2) != skip ).

fof(fact_48_com_Osimps_I12_J,axiom,
! [Com1_2,Com2_2] : skip != semi(Com1_2,Com2_2) ).

fof(fact_49_the__elem__def,axiom,
! [X_3] : hAPP_f1826273671iple_a(the_el287271400iple_a,X_3) = hAPP_f1826273671iple_a(the_Ho1745054714iple_a,hAPP_f1447988451a_bool(cOMBB_545742339iple_a(hAPP_f1945881407l_bool(fequal1765155200a_bool,X_3)),hAPP_f1170963427a_bool(hAPP_f1874567875a_bool(cOMBC_1089176504a_bool,insert1871499715iple_a),bot_bo1687970473a_bool))) ).

fof(fact_50_mk__disjoint__insert,axiom,
! [A_3,A] :
( hBOOL(hAPP_f1753944735l_bool(hAPP_H1926610125l_bool(member564727580iple_a,A_3),A))
=> ? [B] :
( A = hAPP_f1400872321a_bool(hAPP_H1816261935a_bool(insert1871499715iple_a,A_3),B)
& ~ hBOOL(hAPP_f1753944735l_bool(hAPP_H1926610125l_bool(member564727580iple_a,A_3),B)) ) ) ).

fof(fact_51_com_Osimps_I3_J,axiom,
! [Com1_1,Com2_1,Com1,Com2] :
( semi(Com1_1,Com2_1) = semi(Com1,Com2)
<=> ( Com1_1 = Com1
& Com2_1 = Com2 ) ) ).

fof(fact_52_the__sym__eq__trivial,axiom,
! [X_1] : hAPP_f1826273671iple_a(the_Ho1745054714iple_a,hAPP_H562195827a_bool(fequal1878252616iple_a,X_1)) = X_1 ).

fof(fact_53_the__eq__trivial,axiom,
! [A_3] : hAPP_f1826273671iple_a(the_Ho1745054714iple_a,hAPP_H562195827a_bool(hAPP_f1915402821a_bool(cOMBC_2049287834a_bool,fequal1878252616iple_a),A_3)) = A_3 ).

fof(fact_54_If__def,axiom,
! [X_1,Y_2,Pa] :
( ( hBOOL(Pa)
=> X_1 = hAPP_f1826273671iple_a(the_Ho1745054714iple_a,hAPP_f1400872321a_bool(hAPP_f1104866853a_bool(cOMBS_213702372l_bool,hAPP_f13210641l_bool(cOMBB_633860163iple_a(fconj),hAPP_f1400872321a_bool(cOMBB_650444389iple_a(hAPP_b589554111l_bool(fimplies,Pa)),hAPP_H562195827a_bool(hAPP_f1915402821a_bool(cOMBC_2049287834a_bool,fequal1878252616iple_a),X_1)))),hAPP_f1400872321a_bool(cOMBB_650444389iple_a(hAPP_b589554111l_bool(fimplies,hAPP_bool_bool(fNot,Pa))),hAPP_H562195827a_bool(hAPP_f1915402821a_bool(cOMBC_2049287834a_bool,fequal1878252616iple_a),Y_2)))) )
& ( ~ hBOOL(Pa)
=> Y_2 = hAPP_f1826273671iple_a(the_Ho1745054714iple_a,hAPP_f1400872321a_bool(hAPP_f1104866853a_bool(cOMBS_213702372l_bool,hAPP_f13210641l_bool(cOMBB_633860163iple_a(fconj),hAPP_f1400872321a_bool(cOMBB_650444389iple_a(hAPP_b589554111l_bool(fimplies,Pa)),hAPP_H562195827a_bool(hAPP_f1915402821a_bool(cOMBC_2049287834a_bool,fequal1878252616iple_a),X_1)))),hAPP_f1400872321a_bool(cOMBB_650444389iple_a(hAPP_b589554111l_bool(fimplies,hAPP_bool_bool(fNot,Pa))),hAPP_H562195827a_bool(hAPP_f1915402821a_bool(cOMBC_2049287834a_bool,fequal1878252616iple_a),Y_2)))) ) ) ).

fof(fact_55_equals0I,axiom,
! [A] :
( ! [Y_1] : ~ hBOOL(hAPP_f1753944735l_bool(hAPP_H1926610125l_bool(member564727580iple_a,Y_1),A))
=> A = bot_bo1687970473a_bool ) ).

fof(fact_56_the__equality,axiom,
! [Pa,A_3] :
( hBOOL(hAPP_H1927961489a_bool(Pa,A_3))
=> ( ! [X_2] :
( hBOOL(hAPP_H1927961489a_bool(Pa,X_2))
=> X_2 = A_3 )
=> hAPP_f1826273671iple_a(the_Ho1745054714iple_a,Pa) = A_3 ) ) ).

fof(fact_57_theI,axiom,
! [Pa,A_3] :
( hBOOL(hAPP_H1927961489a_bool(Pa,A_3))
=> ( ! [X_2] :
( hBOOL(hAPP_H1927961489a_bool(Pa,X_2))
=> X_2 = A_3 )
=> hBOOL(hAPP_H1927961489a_bool(Pa,hAPP_f1826273671iple_a(the_Ho1745054714iple_a,Pa))) ) ) ).

fof(fact_58_the1__equality,axiom,
! [A_3,Pa] :
( ? [X_2] :
( hBOOL(hAPP_H1927961489a_bool(Pa,X_2))
& ! [Y_1] :
( hBOOL(hAPP_H1927961489a_bool(Pa,Y_1))
=> Y_1 = X_2 ) )
=> ( hBOOL(hAPP_H1927961489a_bool(Pa,A_3))
=> hAPP_f1826273671iple_a(the_Ho1745054714iple_a,Pa) = A_3 ) ) ).

fof(fact_59_theI_H,axiom,
! [Pa] :
( ? [X_2] :
( hBOOL(hAPP_H1927961489a_bool(Pa,X_2))
& ! [Y_1] :
( hBOOL(hAPP_H1927961489a_bool(Pa,Y_1))
=> Y_1 = X_2 ) )
=> hBOOL(hAPP_H1927961489a_bool(Pa,hAPP_f1826273671iple_a(the_Ho1745054714iple_a,Pa))) ) ).

fof(fact_60_conseq,axiom,
! [Q_1,Ga,Ca,Pa] :
( ! [Z_1,S] :
( hBOOL(hAPP_state_bool(hAPP_a2036067514e_bool(Pa,Z_1),S))
=> ? [P_1,Q_2] :
( hBOOL(hAPP_f1753944735l_bool(hoare_472868247rivs_a(Ga),hAPP_f1400872321a_bool(hAPP_H1816261935a_bool(insert1871499715iple_a,hoare_1050552211iple_a(P_1,Ca,Q_2)),bot_bo1687970473a_bool)))
& ! [S_1] :
( ! [Z_2] :
( hBOOL(hAPP_state_bool(hAPP_a2036067514e_bool(P_1,Z_2),S))
=> hBOOL(hAPP_state_bool(hAPP_a2036067514e_bool(Q_2,Z_2),S_1)) )
=> hBOOL(hAPP_state_bool(hAPP_a2036067514e_bool(Q_1,Z_1),S_1)) ) ) )
=> hBOOL(hAPP_f1753944735l_bool(hoare_472868247rivs_a(Ga),hAPP_f1400872321a_bool(hAPP_H1816261935a_bool(insert1871499715iple_a,hoare_1050552211iple_a(Pa,Ca,Q_1)),bot_bo1687970473a_bool))) ) ).

fof(fact_61_nonempty__iff,axiom,
! [A] :
( A != bot_bo1687970473a_bool
<=> ? [X_2,B] :
( A = hAPP_f1400872321a_bool(hAPP_H1816261935a_bool(insert1871499715iple_a,X_2),B)
& ~ hBOOL(hAPP_f1753944735l_bool(hAPP_H1926610125l_bool(member564727580iple_a,X_2),B)) ) ) ).

fof(fact_62_fold1Set__sing,axiom,
! [F_1,A_3,Ba] :
( hBOOL(hAPP_H1927961489a_bool(finite388748825iple_a(F_1,hAPP_f1400872321a_bool(hAPP_H1816261935a_bool(insert1871499715iple_a,A_3),bot_bo1687970473a_bool)),Ba))
<=> A_3 = Ba ) ).

fof(fact_63_folding__one_Osingleton,axiom,
! [X_1,F_1,F] :
( hBOOL(finite520909254iple_a(F_1,F))
=> hAPP_f1826273671iple_a(F,hAPP_f1400872321a_bool(hAPP_H1816261935a_bool(insert1871499715iple_a,X_1),bot_bo1687970473a_bool)) = X_1 ) ).

fof(fact_64_bot__empty__eq,axiom,
! [X_2] :
( hBOOL(hAPP_H1927961489a_bool(bot_bo1687970473a_bool,X_2))
<=> hBOOL(hAPP_f1753944735l_bool(hAPP_H1926610125l_bool(member564727580iple_a,X_2),bot_bo1687970473a_bool)) ) ).

fof(fact_65_empty__fold1SetE,axiom,
! [F_1,X_1] : ~ hBOOL(hAPP_H1927961489a_bool(finite388748825iple_a(F_1,bot_bo1687970473a_bool),X_1)) ).

fof(fact_66_fold1Set__nonempty,axiom,
! [F_1,A,X_1] :
( hBOOL(hAPP_H1927961489a_bool(finite388748825iple_a(F_1,A),X_1))
=> A != bot_bo1687970473a_bool ) ).

fof(fact_67_fold1Set_Ointros,axiom,
! [F_1,A_3,A,X_1] :
( hBOOL(hAPP_H1927961489a_bool(finite1734202118iple_a(F_1,A_3,A),X_1))
=> ( ~ hBOOL(hAPP_f1753944735l_bool(hAPP_H1926610125l_bool(member564727580iple_a,A_3),A))
=> hBOOL(hAPP_H1927961489a_bool(finite388748825iple_a(F_1,hAPP_f1400872321a_bool(hAPP_H1816261935a_bool(insert1871499715iple_a,A_3),A)),X_1)) ) ) ).

fof(fact_68_folding__one_Oinsert,axiom,
! [X_1,A,F_1,F] :
( hBOOL(finite520909254iple_a(F_1,F))
=> ( hBOOL(hAPP_f1753944735l_bool(finite506133020iple_a,A))
=> ( ~ hBOOL(hAPP_f1753944735l_bool(hAPP_H1926610125l_bool(member564727580iple_a,X_1),A))
=> ( A != bot_bo1687970473a_bool
=> hAPP_f1826273671iple_a(F,hAPP_f1400872321a_bool(hAPP_H1816261935a_bool(insert1871499715iple_a,X_1),A)) = hAPP_H401672213iple_a(hAPP_H568064713iple_a(F_1,X_1),hAPP_f1826273671iple_a(F,A)) ) ) ) ) ).

fof(fact_69_fold1__def,axiom,
! [F_1,A] : hAPP_f1826273671iple_a(finite233325225iple_a(F_1),A) = hAPP_f1826273671iple_a(the_Ho1745054714iple_a,finite388748825iple_a(F_1,A)) ).

fof(fact_70_finite__Collect__conjI,axiom,
! [Q_1,Pa] :
( ( hBOOL(hAPP_f1753944735l_bool(finite506133020iple_a,collec1266446174iple_a(Pa)))
| hBOOL(hAPP_f1753944735l_bool(finite506133020iple_a,collec1266446174iple_a(Q_1))) )
=> hBOOL(hAPP_f1753944735l_bool(finite506133020iple_a,collec1266446174iple_a(hAPP_f1400872321a_bool(hAPP_f1104866853a_bool(cOMBS_213702372l_bool,hAPP_f13210641l_bool(cOMBB_633860163iple_a(fconj),Pa)),Q_1)))) ) ).

fof(fact_71_finite_OemptyI,axiom,
hBOOL(hAPP_f1753944735l_bool(finite506133020iple_a,bot_bo1687970473a_bool)) ).

fof(fact_72_finite_OinsertI,axiom,
! [A_3,A] :
( hBOOL(hAPP_f1753944735l_bool(finite506133020iple_a,A))
=> hBOOL(hAPP_f1753944735l_bool(finite506133020iple_a,hAPP_f1400872321a_bool(hAPP_H1816261935a_bool(insert1871499715iple_a,A_3),A))) ) ).

fof(fact_73_mem__def,axiom,
! [X_1,A] :
( hBOOL(hAPP_f1753944735l_bool(hAPP_H1926610125l_bool(member564727580iple_a,X_1),A))
<=> hBOOL(hAPP_H1927961489a_bool(A,X_1)) ) ).

fof(fact_74_Collect__def,axiom,
! [Pa] : collec1266446174iple_a(Pa) = Pa ).

fof(fact_75_folding__one_Oeq__fold,axiom,
! [A,F_1,F] :
( hBOOL(finite520909254iple_a(F_1,F))
=> ( hBOOL(hAPP_f1753944735l_bool(finite506133020iple_a,A))
=> hAPP_f1826273671iple_a(F,A) = hAPP_f1826273671iple_a(finite233325225iple_a(F_1),A) ) ) ).

fof(fact_76_fold__graph_OemptyI,axiom,
! [F_1,Z] : hBOOL(hAPP_H1927961489a_bool(finite1734202118iple_a(F_1,Z,bot_bo1687970473a_bool),Z)) ).

fof(fact_77_empty__fold__graphE,axiom,
! [F_1,Z,X_1] :
( hBOOL(hAPP_H1927961489a_bool(finite1734202118iple_a(F_1,Z,bot_bo1687970473a_bool),X_1))
=> X_1 = Z ) ).

fof(fact_78_fold__graph_OinsertI,axiom,
! [F_1,Z,Y_2,X_1,A] :
( ~ hBOOL(hAPP_f1753944735l_bool(hAPP_H1926610125l_bool(member564727580iple_a,X_1),A))
=> ( hBOOL(hAPP_H1927961489a_bool(finite1734202118iple_a(F_1,Z,A),Y_2))
=> hBOOL(hAPP_H1927961489a_bool(finite1734202118iple_a(F_1,Z,hAPP_f1400872321a_bool(hAPP_H1816261935a_bool(insert1871499715iple_a,X_1),A)),hAPP_H401672213iple_a(hAPP_H568064713iple_a(F_1,X_1),Y_2))) ) ) ).

fof(fact_79_finite__Collect__disjI,axiom,
! [Pa,Q_1] :
( hBOOL(hAPP_f1753944735l_bool(finite506133020iple_a,collec1266446174iple_a(hAPP_f1400872321a_bool(hAPP_f1104866853a_bool(cOMBS_213702372l_bool,hAPP_f13210641l_bool(cOMBB_633860163iple_a(fdisj),Pa)),Q_1))))
<=> ( hBOOL(hAPP_f1753944735l_bool(finite506133020iple_a,collec1266446174iple_a(Pa)))
& hBOOL(hAPP_f1753944735l_bool(finite506133020iple_a,collec1266446174iple_a(Q_1))) ) ) ).

fof(fact_80_finite__insert,axiom,
! [A_3,A] :
( hBOOL(hAPP_f1753944735l_bool(finite506133020iple_a,hAPP_f1400872321a_bool(hAPP_H1816261935a_bool(insert1871499715iple_a,A_3),A)))
<=> hBOOL(hAPP_f1753944735l_bool(finite506133020iple_a,A)) ) ).

fof(fact_81_fold1__singleton__def,axiom,
! [A_3,G,F_1] :
( G = finite233325225iple_a(F_1)
=> hAPP_f1826273671iple_a(G,hAPP_f1400872321a_bool(hAPP_H1816261935a_bool(insert1871499715iple_a,A_3),bot_bo1687970473a_bool)) = A_3 ) ).

fof(fact_82_fold1__singleton,axiom,
! [F_1,A_3] : hAPP_f1826273671iple_a(finite233325225iple_a(F_1),hAPP_f1400872321a_bool(hAPP_H1816261935a_bool(insert1871499715iple_a,A_3),bot_bo1687970473a_bool)) = A_3 ).

fof(fact_83_folding__one_Oclosed,axiom,
! [A,F_1,F] :
( hBOOL(finite520909254iple_a(F_1,F))
=> ( hBOOL(hAPP_f1753944735l_bool(finite506133020iple_a,A))
=> ( A != bot_bo1687970473a_bool
=> ( ! [X_2,Y_1] : hBOOL(hAPP_f1753944735l_bool(hAPP_H1926610125l_bool(member564727580iple_a,hAPP_H401672213iple_a(hAPP_H568064713iple_a(F_1,X_2),Y_1)),hAPP_f1400872321a_bool(hAPP_H1816261935a_bool(insert1871499715iple_a,X_2),hAPP_f1400872321a_bool(hAPP_H1816261935a_bool(insert1871499715iple_a,Y_1),bot_bo1687970473a_bool))))
=> hBOOL(hAPP_f1753944735l_bool(hAPP_H1926610125l_bool(member564727580iple_a,hAPP_f1826273671iple_a(F,A)),A)) ) ) ) ) ).

fof(fact_84_insert__fold1SetE,axiom,
! [F_1,A_3,X_3,X_1] :
( hBOOL(hAPP_H1927961489a_bool(finite388748825iple_a(F_1,hAPP_f1400872321a_bool(hAPP_H1816261935a_bool(insert1871499715iple_a,A_3),X_3)),X_1))
=> ~ ! [A_2,A_1] :
( hAPP_f1400872321a_bool(hAPP_H1816261935a_bool(insert1871499715iple_a,A_3),X_3) = hAPP_f1400872321a_bool(hAPP_H1816261935a_bool(insert1871499715iple_a,A_2),A_1)
=> ( hBOOL(hAPP_H1927961489a_bool(finite1734202118iple_a(F_1,A_2,A_1),X_1))
=> hBOOL(hAPP_f1753944735l_bool(hAPP_H1926610125l_bool(member564727580iple_a,A_2),A_1)) ) ) ) ).

fof(fact_85_finite__nonempty__imp__fold1Set,axiom,
! [F_1,A] :
( hBOOL(hAPP_f1753944735l_bool(finite506133020iple_a,A))
=> ( A != bot_bo1687970473a_bool
=> ? [X1] : hBOOL(hAPP_H1927961489a_bool(finite388748825iple_a(F_1,A),X1)) ) ) ).

fof(fact_86_finite__induct,axiom,
! [Pa,F] :
( hBOOL(hAPP_f1753944735l_bool(finite506133020iple_a,F))
=> ( hBOOL(hAPP_f1753944735l_bool(Pa,bot_bo1687970473a_bool))
=> ( ! [X_2,F_2] :
( hBOOL(hAPP_f1753944735l_bool(finite506133020iple_a,F_2))
=> ( ~ hBOOL(hAPP_f1753944735l_bool(hAPP_H1926610125l_bool(member564727580iple_a,X_2),F_2))
=> ( hBOOL(hAPP_f1753944735l_bool(Pa,F_2))
=> hBOOL(hAPP_f1753944735l_bool(Pa,hAPP_f1400872321a_bool(hAPP_H1816261935a_bool(insert1871499715iple_a,X_2),F_2))) ) ) )
=> hBOOL(hAPP_f1753944735l_bool(Pa,F)) ) ) ) ).

fof(fact_87_finite_Osimps,axiom,
! [A_3] :
( hBOOL(hAPP_f1753944735l_bool(finite506133020iple_a,A_3))
<=> ( A_3 = bot_bo1687970473a_bool
| ? [A_1,A_2] :
( A_3 = hAPP_f1400872321a_bool(hAPP_H1816261935a_bool(insert1871499715iple_a,A_2),A_1)
& hBOOL(hAPP_f1753944735l_bool(finite506133020iple_a,A_1)) ) ) ) ).

fof(fact_88_finite__imp__fold__graph,axiom,
! [F_1,Z,A] :
( hBOOL(hAPP_f1753944735l_bool(finite506133020iple_a,A))
=> ? [X1] : hBOOL(hAPP_H1927961489a_bool(finite1734202118iple_a(F_1,Z,A),X1)) ) ).

fof(fact_89_fold1Set_Osimps,axiom,
! [F_1,A1,A2] :
( hBOOL(hAPP_H1927961489a_bool(finite388748825iple_a(F_1,A1),A2))
<=> ? [A_2,A_1,X_2] :
( A1 = hAPP_f1400872321a_bool(hAPP_H1816261935a_bool(insert1871499715iple_a,A_2),A_1)
& A2 = X_2
& hBOOL(hAPP_H1927961489a_bool(finite1734202118iple_a(F_1,A_2,A_1),X_2))
& ~ hBOOL(hAPP_f1753944735l_bool(hAPP_H1926610125l_bool(member564727580iple_a,A_2),A_1)) ) ) ).

fof(fact_90_fold__graph_Osimps,axiom,
! [F_1,Z,A1,A2] :
( hBOOL(hAPP_H1927961489a_bool(finite1734202118iple_a(F_1,Z,A1),A2))
<=> ( ( A1 = bot_bo1687970473a_bool
& A2 = Z )
| ? [X_2,A_1,Y_1] :
( A1 = hAPP_f1400872321a_bool(hAPP_H1816261935a_bool(insert1871499715iple_a,X_2),A_1)
& A2 = hAPP_H401672213iple_a(hAPP_H568064713iple_a(F_1,X_2),Y_1)
& ~ hBOOL(hAPP_f1753944735l_bool(hAPP_H1926610125l_bool(member564727580iple_a,X_2),A_1))
& hBOOL(hAPP_H1927961489a_bool(finite1734202118iple_a(F_1,Z,A_1),Y_1)) ) ) ) ).

fof(fact_91_folding__one__idem_Oinsert__idem,axiom,
! [X_1,A,F_1,F] :
( hBOOL(finite1948426435iple_a(F_1,F))
=> ( hBOOL(hAPP_f1753944735l_bool(finite506133020iple_a,A))
=> ( A != bot_bo1687970473a_bool
=> hAPP_f1826273671iple_a(F,hAPP_f1400872321a_bool(hAPP_H1816261935a_bool(insert1871499715iple_a,X_1),A)) = hAPP_H401672213iple_a(hAPP_H568064713iple_a(F_1,X_1),hAPP_f1826273671iple_a(F,A)) ) ) ) ).

fof(fact_92_finite__ne__induct,axiom,
! [Pa,F] :
( hBOOL(hAPP_f1753944735l_bool(finite506133020iple_a,F))
=> ( F != bot_bo1687970473a_bool
=> ( ! [X_2] : hBOOL(hAPP_f1753944735l_bool(Pa,hAPP_f1400872321a_bool(hAPP_H1816261935a_bool(insert1871499715iple_a,X_2),bot_bo1687970473a_bool)))
=> ( ! [X_2,F_2] :
( hBOOL(hAPP_f1753944735l_bool(finite506133020iple_a,F_2))
=> ( F_2 != bot_bo1687970473a_bool
=> ( ~ hBOOL(hAPP_f1753944735l_bool(hAPP_H1926610125l_bool(member564727580iple_a,X_2),F_2))
=> ( hBOOL(hAPP_f1753944735l_bool(Pa,F_2))
=> hBOOL(hAPP_f1753944735l_bool(Pa,hAPP_f1400872321a_bool(hAPP_H1816261935a_bool(insert1871499715iple_a,X_2),F_2))) ) ) ) )
=> hBOOL(hAPP_f1753944735l_bool(Pa,F)) ) ) ) ) ).

fof(fact_93_folding__one__idem_Oidem,axiom,
! [X_1,F_1,F] :
( hBOOL(finite1948426435iple_a(F_1,F))
=> hAPP_H401672213iple_a(hAPP_H568064713iple_a(F_1,X_1),X_1) = X_1 ) ).

fof(fact_94_folding__one__idem_Oin__idem,axiom,
! [X_1,A,F_1,F] :
( hBOOL(finite1948426435iple_a(F_1,F))
=> ( hBOOL(hAPP_f1753944735l_bool(finite506133020iple_a,A))
=> ( hBOOL(hAPP_f1753944735l_bool(hAPP_H1926610125l_bool(member564727580iple_a,X_1),A))
=> hAPP_H401672213iple_a(hAPP_H568064713iple_a(F_1,X_1),hAPP_f1826273671iple_a(F,A)) = hAPP_f1826273671iple_a(F,A) ) ) ) ).

%----Helper facts (39)
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_fconj_1_1_U,axiom,
! [Q,P] :
( ~ hBOOL(P)
| ~ hBOOL(Q)
| hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fconj,P),Q)) ) ).

fof(help_fconj_2_1_U,axiom,
! [P,Q] :
( ~ hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fconj,P),Q))
| hBOOL(P) ) ).

fof(help_fconj_3_1_U,axiom,
! [P,Q] :
( ~ hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fconj,P),Q))
| hBOOL(Q) ) ).

fof(help_fdisj_1_1_U,axiom,
! [Q,P] :
( ~ hBOOL(P)
| hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fdisj,P),Q)) ) ).

fof(help_fdisj_2_1_U,axiom,
! [P,Q] :
( ~ hBOOL(Q)
| hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fdisj,P),Q)) ) ).

fof(help_fdisj_3_1_U,axiom,
! [P,Q] :
( ~ hBOOL(hAPP_bool_bool(hAPP_b589554111l_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] :
( is_bool(P)
=> ( P = fTrue
| P = fFalse ) ) ).

fof(help_fimplies_1_1_U,axiom,
! [Q,P] :
( hBOOL(P)
| hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fimplies,P),Q)) ) ).

fof(help_fimplies_2_1_U,axiom,
! [P,Q] :
( ~ hBOOL(Q)
| hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fimplies,P),Q)) ) ).

fof(help_fimplies_3_1_U,axiom,
! [P,Q] :
( ~ hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fimplies,P),Q))
| ~ hBOOL(P)
| hBOOL(Q) ) ).

fof(help_fequal_1_1_fequal_000tc__Com__Ostate_T,axiom,
! [X,Y] :
( ~ hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(fequal_state,X),Y))
| X = Y ) ).

fof(help_fequal_2_1_fequal_000tc__Com__Ostate_T,axiom,
! [X,Y] :
( X != Y
| hBOOL(hAPP_state_bool(hAPP_s1806633685e_bool(fequal_state,X),Y)) ) ).

fof(help_COMBK_1_1_COMBK_000tc__HOL__Obool_000tc__Com__Ostate_U,axiom,
! [P,Q] :
( is_bool(P)
=> hAPP_state_bool(cOMBK_bool_state(P),Q) = P ) ).

fof(help_COMBK_1_1_COMBK_000tc__fun_Itc__Com__Ostate_Mtc__HOL__Obool_J_000t__a_U,axiom,
! [P,Q] : hAPP_a2036067514e_bool(cOMBK_1458035955bool_a(P),Q) = P ).

fof(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__HOL__Obool_000tc__Com__Ostate_U,axiom,
! [P,Q,R] : hAPP_state_bool(hAPP_f1759915619e_bool(cOMBB_160679318_state(P),Q),R) = hAPP_bool_bool(P,hAPP_state_bool(Q,R)) ).

fof(help_COMBC_1_1_COMBC_000tc__Com__Ostate_000tc__HOL__Obool_000tc__HOL__Obool_U,axiom,
! [P,Q,R] : hAPP_state_bool(hAPP_b2019457360e_bool(hAPP_f167292325e_bool(cOMBC_231445413l_bool,P),Q),R) = hAPP_bool_bool(hAPP_s58564346l_bool(P,R),Q) ).

fof(help_COMBS_1_1_COMBS_000tc__Com__Ostate_000tc__HOL__Obool_000tc__HOL__Obool_U,axiom,
! [P,Q,R] : hAPP_state_bool(hAPP_f1759915619e_bool(hAPP_f644196280e_bool(cOMBS_1378840469l_bool,P),Q),R) = hAPP_bool_bool(hAPP_s58564346l_bool(P,R),hAPP_state_bool(Q,R)) ).

fof(help_COMBC_1_1_COMBC_000tc__Com__Ostate_000tc__Com__Ostate_000tc__HOL__Obool_U,axiom,
! [P,Q,R] : hAPP_state_bool(hAPP_s1806633685e_bool(hAPP_f817621513e_bool(cOMBC_2027030106e_bool,P),Q),R) = hAPP_state_bool(hAPP_s1806633685e_bool(P,R),Q) ).

fof(help_fequal_1_1_fequal_000tc__Hoare____Mirabelle____jfehddehev__Otriple_It__a_J_,axiom,
! [X,Y] :
( ~ hBOOL(hAPP_H1927961489a_bool(hAPP_H562195827a_bool(fequal1878252616iple_a,X),Y))
| X = Y ) ).

fof(help_fequal_2_1_fequal_000tc__Hoare____Mirabelle____jfehddehev__Otriple_It__a_J_,axiom,
! [X,Y] :
( X != Y
| hBOOL(hAPP_H1927961489a_bool(hAPP_H562195827a_bool(fequal1878252616iple_a,X),Y)) ) ).

fof(help_COMBC_1_1_COMBC_000t__a_000tc__HOL__Obool_000tc__fun_Itc__Com__Ostate_Mtc__,axiom,
! [P,Q,R] : hAPP_a2036067514e_bool(hAPP_b540892988e_bool(hAPP_f1824947087e_bool(cOMBC_41962815e_bool,P),Q),R) = hAPP_b2019457360e_bool(hAPP_a723219176e_bool(P,R),Q) ).

fof(help_COMBK_1_1_COMBK_000tc__HOL__Obool_000tc__Hoare____Mirabelle____jfehddehev__,axiom,
! [P,Q] :
( is_bool(P)
=> hAPP_H1927961489a_bool(cOMBK_1150238960iple_a(P),Q) = P ) ).

fof(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Oboo,axiom,
! [P,Q,R] : hAPP_s58564346l_bool(hAPP_f1259673775l_bool(cOMBB_188601460_state(P),Q),R) = hAPP_b589554111l_bool(P,hAPP_state_bool(Q,R)) ).

fof(help_fequal_1_1_fequal_000tc__fun_Itc__Hoare____Mirabelle____jfehddehev__Otriple,axiom,
! [X,Y] :
( ~ hBOOL(hAPP_f1753944735l_bool(hAPP_f1945881407l_bool(fequal1765155200a_bool,X),Y))
| X = Y ) ).

fof(help_fequal_2_1_fequal_000tc__fun_Itc__Hoare____Mirabelle____jfehddehev__Otriple,axiom,
! [X,Y] :
( X != Y
| hBOOL(hAPP_f1753944735l_bool(hAPP_f1945881407l_bool(fequal1765155200a_bool,X),Y)) ) ).

fof(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__HOL__Obool_000tc__Hoare____Mirabel,axiom,
! [P,Q,R] : hAPP_H1927961489a_bool(hAPP_f1400872321a_bool(cOMBB_650444389iple_a(P),Q),R) = hAPP_bool_bool(P,hAPP_H1927961489a_bool(Q,R)) ).

fof(help_COMBS_1_1_COMBS_000tc__Hoare____Mirabelle____jfehddehev__Otriple_It__a_J_00,axiom,
! [P,Q,R] : hAPP_H1927961489a_bool(hAPP_f1400872321a_bool(hAPP_f1104866853a_bool(cOMBS_213702372l_bool,P),Q),R) = hAPP_bool_bool(hAPP_H1877746411l_bool(P,R),hAPP_H1927961489a_bool(Q,R)) ).

fof(help_COMBC_1_1_COMBC_000t__a_000tc__fun_Itc__Com__Ostate_Mtc__HOL__Obool_J_000tc,axiom,
! [P,Q,R] : hAPP_a2036067514e_bool(hAPP_f762886889e_bool(hAPP_f1261923407e_bool(cOMBC_892787026e_bool,P),Q),R) = hAPP_f1759915619e_bool(hAPP_a1200519163e_bool(P,R),Q) ).

fof(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Oboo_001,axiom,
! [P,Q,R] : hAPP_H1877746411l_bool(hAPP_f13210641l_bool(cOMBB_633860163iple_a(P),Q),R) = hAPP_b589554111l_bool(P,hAPP_H1927961489a_bool(Q,R)) ).

fof(help_COMBB_1_1_COMBB_000tc__fun_Itc__Com__Ostate_Mtc__HOL__Obool_J_000tc__fun_It,axiom,
! [P,Q,R] : hAPP_a849909144l_bool(hAPP_f1509969235l_bool(cOMBB_1355796797bool_a(P),Q),R) = hAPP_f1259673775l_bool(P,hAPP_a2036067514e_bool(Q,R)) ).

fof(help_COMBC_1_1_COMBC_000tc__Hoare____Mirabelle____jfehddehev__Otriple_It__a_J_00,axiom,
! [P,Q,R] : hAPP_H1927961489a_bool(hAPP_H562195827a_bool(hAPP_f1915402821a_bool(cOMBC_2049287834a_bool,P),Q),R) = hAPP_H1927961489a_bool(hAPP_H562195827a_bool(P,R),Q) ).

fof(help_COMBB_1_1_COMBB_000tc__fun_Itc__Com__Ostate_Mtc__fun_Itc__HOL__Obool_Mtc__H,axiom,
! [P,Q,R] : hAPP_a723219176e_bool(hAPP_f340725611e_bool(cOMBB_1348041619bool_a(P),Q),R) = hAPP_f167292325e_bool(P,hAPP_a849909144l_bool(Q,R)) ).

fof(help_COMBB_1_1_COMBB_000tc__fun_Itc__Hoare____Mirabelle____jfehddehev__Otriple_I,axiom,
! [P,Q,R] : hAPP_H1927961489a_bool(hAPP_f1447988451a_bool(cOMBB_545742339iple_a(P),Q),R) = hAPP_f1753944735l_bool(P,hAPP_H562195827a_bool(Q,R)) ).

fof(help_COMBC_1_1_COMBC_000tc__Hoare____Mirabelle____jfehddehev__Otriple_It__a_J_00_002,axiom,
! [P,Q,R] : hAPP_H1927961489a_bool(hAPP_f1400872321a_bool(hAPP_f945663555a_bool(cOMBC_2067518550l_bool,P),Q),R) = hAPP_f1753944735l_bool(hAPP_H1926610125l_bool(P,R),Q) ).

fof(help_COMBB_1_1_COMBB_000tc__fun_Itc__Com__Ostate_Mtc__fun_Itc__HOL__Obool_Mtc__H_003,axiom,
! [P,Q,R] : hAPP_a1200519163e_bool(hAPP_f963367678e_bool(cOMBB_145932198bool_a(P),Q),R) = hAPP_f644196280e_bool(P,hAPP_a849909144l_bool(Q,R)) ).

fof(help_COMBC_1_1_COMBC_000tc__Hoare____Mirabelle____jfehddehev__Otriple_It__a_J_00_004,axiom,
! [P,Q,R] : hAPP_H562195827a_bool(hAPP_f1170963427a_bool(hAPP_f1874567875a_bool(cOMBC_1089176504a_bool,P),Q),R) = hAPP_f1400872321a_bool(hAPP_H1816261935a_bool(P,R),Q) ).

%----Conjectures (1)
fof(conj_0,conjecture,
hBOOL(hAPP_f1753944735l_bool(hoare_472868247rivs_a(g),hAPP_f1400872321a_bool(hAPP_H1816261935a_bool(insert1871499715iple_a,hoare_1050552211iple_a(cOMBK_1458035955bool_a(cOMBK_bool_state(fFalse)),c,hAPP_f762886889e_bool(hAPP_f1261923407e_bool(cOMBC_892787026e_bool,hAPP_f963367678e_bool(cOMBB_145932198bool_a(cOMBS_1378840469l_bool),hAPP_f1509969235l_bool(cOMBB_1355796797bool_a(cOMBB_188601460_state(fconj)),p))),hAPP_f1759915619e_bool(cOMBB_160679318_state(fNot),b)))),bot_bo1687970473a_bool))) ).

%------------------------------------------------------------------------------
```