TPTP Problem File: SWW470^1.p
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%------------------------------------------------------------------------------
% File : SWW470^1 : TPTP v8.2.0. 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_thf_l200 [Bla11]
% Status : Theorem
% Rating : 0.40 v8.2.0, 0.38 v8.1.0, 0.27 v7.5.0, 0.29 v7.4.0, 0.22 v7.2.0, 0.12 v7.1.0, 0.25 v7.0.0, 0.14 v6.4.0, 0.17 v6.3.0, 0.20 v6.2.0, 0.29 v6.0.0, 0.43 v5.5.0, 0.50 v5.4.0, 0.80 v5.3.0
% Syntax : Number of formulae : 125 ( 25 unt; 27 typ; 0 def)
% Number of atoms : 422 ( 108 equ; 21 cnn)
% Maximal formula atoms : 9 ( 4 avg)
% Number of connectives : 854 ( 47 ~; 11 |; 31 &; 634 @)
% ( 22 <=>; 109 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 8 avg)
% Number of types : 5 ( 4 usr)
% Number of type conns : 268 ( 268 >; 0 *; 0 +; 0 <<)
% Number of symbols : 29 ( 23 usr; 8 con; 0-4 aty)
% Number of variables : 307 ( 24 ^; 267 !; 16 ?; 307 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-08-09 19:02:36
%------------------------------------------------------------------------------
%----Should-be-implicit typings (4)
thf(ty_ty_t__a,type,
x_a: $tType ).
thf(ty_ty_tc__Com__Ocom,type,
com: $tType ).
thf(ty_ty_tc__Com__Ostate,type,
state: $tType ).
thf(ty_ty_tc__Hoare____Mirabelle____ghhkfsbqqq__Otriple_It__a_J,type,
hoare_669141180iple_a: $tType ).
%----Explicit typings (23)
thf(sy_c_Com_Ocom_OSKIP,type,
skip: com ).
thf(sy_c_Com_Ocom_OSemi,type,
semi: com > com > com ).
thf(sy_c_Ex,type,
ex: ( hoare_669141180iple_a > $o ) > $o ).
thf(sy_c_Finite__Set_Ofinite_000tc__Hoare____Mirabelle____ghhkfsbqqq__Otriple_It__a_,type,
finite957651855iple_a: ( hoare_669141180iple_a > $o ) > $o ).
thf(sy_c_Finite__Set_Ofold1Set_000tc__Hoare____Mirabelle____ghhkfsbqqq__Otriple_It__,type,
finite840267660iple_a: ( hoare_669141180iple_a > hoare_669141180iple_a > hoare_669141180iple_a ) > ( hoare_669141180iple_a > $o ) > hoare_669141180iple_a > $o ).
thf(sy_c_Finite__Set_Ofold1_000tc__Hoare____Mirabelle____ghhkfsbqqq__Otriple_It__a_J,type,
finite684844060iple_a: ( hoare_669141180iple_a > hoare_669141180iple_a > hoare_669141180iple_a ) > ( hoare_669141180iple_a > $o ) > hoare_669141180iple_a ).
thf(sy_c_Finite__Set_Ofold__graph_000tc__Hoare____Mirabelle____ghhkfsbqqq__Otriple_I,type,
finite590756294iple_a: ( hoare_669141180iple_a > hoare_669141180iple_a > hoare_669141180iple_a ) > hoare_669141180iple_a > ( hoare_669141180iple_a > $o ) > hoare_669141180iple_a > $o ).
thf(sy_c_Finite__Set_Ofolding__one_000tc__Hoare____Mirabelle____ghhkfsbqqq__Otriple_,type,
finite972428089iple_a: ( hoare_669141180iple_a > hoare_669141180iple_a > hoare_669141180iple_a ) > ( ( hoare_669141180iple_a > $o ) > hoare_669141180iple_a ) > $o ).
thf(sy_c_Finite__Set_Ofolding__one__idem_000tc__Hoare____Mirabelle____ghhkfsbqqq__Ot,type,
finite252461622iple_a: ( hoare_669141180iple_a > hoare_669141180iple_a > hoare_669141180iple_a ) > ( ( hoare_669141180iple_a > $o ) > hoare_669141180iple_a ) > $o ).
thf(sy_c_HOL_OThe_000tc__Hoare____Mirabelle____ghhkfsbqqq__Otriple_It__a_J,type,
the_Ho49089901iple_a: ( hoare_669141180iple_a > $o ) > hoare_669141180iple_a ).
thf(sy_c_Hoare__Mirabelle__ghhkfsbqqq_Ohoare__derivs_000t__a,type,
hoare_2128652938rivs_a: ( hoare_669141180iple_a > $o ) > ( hoare_669141180iple_a > $o ) > $o ).
thf(sy_c_Hoare__Mirabelle__ghhkfsbqqq_Otriple_Otriple_000t__a,type,
hoare_1295064928iple_a: ( x_a > state > $o ) > com > ( x_a > state > $o ) > hoare_669141180iple_a ).
thf(sy_c_Orderings_Obot__class_Obot_000_062_Itc__Hoare____Mirabelle____ghhkfsbqqq__O,type,
bot_bo280939947le_a_o: hoare_669141180iple_a > $o ).
thf(sy_c_Orderings_Obot__class_Obot_000_Eo,type,
bot_bot_o: $o ).
thf(sy_c_Set_OCollect_000tc__Hoare____Mirabelle____ghhkfsbqqq__Otriple_It__a_J,type,
collec1717965009iple_a: ( hoare_669141180iple_a > $o ) > hoare_669141180iple_a > $o ).
thf(sy_c_Set_Oinsert_000tc__Hoare____Mirabelle____ghhkfsbqqq__Otriple_It__a_J,type,
insert175534902iple_a: hoare_669141180iple_a > ( hoare_669141180iple_a > $o ) > hoare_669141180iple_a > $o ).
thf(sy_c_Set_Othe__elem_000tc__Hoare____Mirabelle____ghhkfsbqqq__Otriple_It__a_J,type,
the_el738790235iple_a: ( hoare_669141180iple_a > $o ) > hoare_669141180iple_a ).
thf(sy_c_fequal_000tc__Hoare____Mirabelle____ghhkfsbqqq__Otriple_It__a_J,type,
fequal182287803iple_a: hoare_669141180iple_a > hoare_669141180iple_a > $o ).
thf(sy_c_member_000tc__Hoare____Mirabelle____ghhkfsbqqq__Otriple_It__a_J,type,
member1016246415iple_a: hoare_669141180iple_a > ( hoare_669141180iple_a > $o ) > $o ).
thf(sy_v_G,type,
g: hoare_669141180iple_a > $o ).
thf(sy_v_P,type,
p: x_a > state > $o ).
thf(sy_v_b,type,
b: state > $o ).
thf(sy_v_c,type,
c: com ).
%----Relevant facts (95)
thf(fact_0_empty,axiom,
! [G_12: hoare_669141180iple_a > $o] : ( hoare_2128652938rivs_a @ G_12 @ bot_bo280939947le_a_o ) ).
thf(fact_1_triple_Oinject,axiom,
! [Fun1_2: x_a > state > $o,Com_2: com,Fun2_2: x_a > state > $o,Fun1_1: x_a > state > $o,Com_1: com,Fun2_1: x_a > state > $o] :
( ( ( hoare_1295064928iple_a @ Fun1_2 @ Com_2 @ Fun2_2 )
= ( hoare_1295064928iple_a @ Fun1_1 @ Com_1 @ Fun2_1 ) )
<=> ( ( Fun1_2 = Fun1_1 )
& ( Com_2 = Com_1 )
& ( Fun2_2 = Fun2_1 ) ) ) ).
thf(fact_2_cut,axiom,
! [G_11: hoare_669141180iple_a > $o,G_10: hoare_669141180iple_a > $o,Ts_1: hoare_669141180iple_a > $o] :
( ( hoare_2128652938rivs_a @ G_10 @ Ts_1 )
=> ( ( hoare_2128652938rivs_a @ G_11 @ G_10 )
=> ( hoare_2128652938rivs_a @ G_11 @ Ts_1 ) ) ) ).
thf(fact_3_hoare__derivs_Oinsert,axiom,
! [Ts: hoare_669141180iple_a > $o,G_9: hoare_669141180iple_a > $o,T: hoare_669141180iple_a] :
( ( hoare_2128652938rivs_a @ G_9 @ ( insert175534902iple_a @ T @ bot_bo280939947le_a_o ) )
=> ( ( hoare_2128652938rivs_a @ G_9 @ Ts )
=> ( hoare_2128652938rivs_a @ G_9 @ ( insert175534902iple_a @ T @ Ts ) ) ) ) ).
thf(fact_4_constant,axiom,
! [G_8: hoare_669141180iple_a > $o,P_25: x_a > state > $o,C_9: com,Q_11: x_a > state > $o,C_8: $o] :
( ( C_8
=> ( hoare_2128652938rivs_a @ G_8 @ ( insert175534902iple_a @ ( hoare_1295064928iple_a @ P_25 @ C_9 @ Q_11 ) @ bot_bo280939947le_a_o ) ) )
=> ( hoare_2128652938rivs_a @ G_8
@ ( insert175534902iple_a
@ ( hoare_1295064928iple_a
@ ^ [Z: x_a,S: state] : ( (&) @ ( P_25 @ Z @ S ) @ C_8 )
@ C_9
@ Q_11 )
@ bot_bo280939947le_a_o ) ) ) ).
thf(fact_5_escape,axiom,
! [G_7: hoare_669141180iple_a > $o,C_7: com,Q_10: x_a > state > $o,P_24: x_a > state > $o] :
( ! [Z: x_a,S: state] :
( ( P_24 @ Z @ S )
=> ( hoare_2128652938rivs_a @ G_7
@ ( insert175534902iple_a
@ ( hoare_1295064928iple_a
@ ^ [Za: x_a,S_1: state] : S_1 = S
@ C_7
@ ^ [Z_6: x_a] : ( Q_10 @ Z ) )
@ bot_bo280939947le_a_o ) ) )
=> ( hoare_2128652938rivs_a @ G_7 @ ( insert175534902iple_a @ ( hoare_1295064928iple_a @ P_24 @ C_7 @ Q_10 ) @ bot_bo280939947le_a_o ) ) ) ).
thf(fact_6_conseq2,axiom,
! [Q_9: x_a > state > $o,G_6: hoare_669141180iple_a > $o,P_23: x_a > state > $o,C_6: com,Q_8: x_a > state > $o] :
( ( hoare_2128652938rivs_a @ G_6 @ ( insert175534902iple_a @ ( hoare_1295064928iple_a @ P_23 @ C_6 @ Q_8 ) @ bot_bo280939947le_a_o ) )
=> ( ! [Z: x_a,S: state] :
( ( Q_8 @ Z @ S )
=> ( Q_9 @ Z @ S ) )
=> ( hoare_2128652938rivs_a @ G_6 @ ( insert175534902iple_a @ ( hoare_1295064928iple_a @ P_23 @ C_6 @ Q_9 ) @ bot_bo280939947le_a_o ) ) ) ) ).
thf(fact_7_conseq1,axiom,
! [P_22: x_a > state > $o,G_5: hoare_669141180iple_a > $o,P_21: x_a > state > $o,C_5: com,Q_7: x_a > state > $o] :
( ( hoare_2128652938rivs_a @ G_5 @ ( insert175534902iple_a @ ( hoare_1295064928iple_a @ P_21 @ C_5 @ Q_7 ) @ bot_bo280939947le_a_o ) )
=> ( ! [Z: x_a,S: state] :
( ( P_22 @ Z @ S )
=> ( P_21 @ Z @ S ) )
=> ( hoare_2128652938rivs_a @ G_5 @ ( insert175534902iple_a @ ( hoare_1295064928iple_a @ P_22 @ C_5 @ Q_7 ) @ bot_bo280939947le_a_o ) ) ) ) ).
thf(fact_8_conseq12,axiom,
! [Q_6: x_a > state > $o,P_20: x_a > state > $o,G_4: hoare_669141180iple_a > $o,P_19: x_a > state > $o,C_4: com,Q_5: x_a > state > $o] :
( ( hoare_2128652938rivs_a @ G_4 @ ( insert175534902iple_a @ ( hoare_1295064928iple_a @ P_19 @ C_4 @ Q_5 ) @ bot_bo280939947le_a_o ) )
=> ( ! [Z: x_a,S: state] :
( ( P_20 @ Z @ S )
=> ! [S_1: state] :
( ! [Z_6: x_a] :
( ( P_19 @ Z_6 @ S )
=> ( Q_5 @ Z_6 @ S_1 ) )
=> ( Q_6 @ Z @ S_1 ) ) )
=> ( hoare_2128652938rivs_a @ G_4 @ ( insert175534902iple_a @ ( hoare_1295064928iple_a @ P_20 @ C_4 @ Q_6 ) @ bot_bo280939947le_a_o ) ) ) ) ).
thf(fact_9_insertE,axiom,
! [A_64: hoare_669141180iple_a,B_14: hoare_669141180iple_a,A_63: hoare_669141180iple_a > $o] :
( ( member1016246415iple_a @ A_64 @ ( insert175534902iple_a @ B_14 @ A_63 ) )
=> ( ( A_64 != B_14 )
=> ( member1016246415iple_a @ A_64 @ A_63 ) ) ) ).
thf(fact_10_insertCI,axiom,
! [B_13: hoare_669141180iple_a,A_62: hoare_669141180iple_a,B_12: hoare_669141180iple_a > $o] :
( ( ~ ( member1016246415iple_a @ A_62 @ B_12 )
=> ( A_62 = B_13 ) )
=> ( member1016246415iple_a @ A_62 @ ( insert175534902iple_a @ B_13 @ B_12 ) ) ) ).
thf(fact_11_emptyE,axiom,
! [A_61: hoare_669141180iple_a] :
~ ( member1016246415iple_a @ A_61 @ bot_bo280939947le_a_o ) ).
thf(fact_12_singleton__conv2,axiom,
! [A_60: hoare_669141180iple_a] :
( ( collec1717965009iple_a @ ( fequal182287803iple_a @ A_60 ) )
= ( insert175534902iple_a @ A_60 @ bot_bo280939947le_a_o ) ) ).
thf(fact_13_singleton__conv,axiom,
! [A_59: hoare_669141180iple_a] :
( ( collec1717965009iple_a
@ ^ [X_3: hoare_669141180iple_a] : X_3 = A_59 )
= ( insert175534902iple_a @ A_59 @ bot_bo280939947le_a_o ) ) ).
thf(fact_14_Collect__conv__if2,axiom,
! [P_18: hoare_669141180iple_a > $o,A_58: hoare_669141180iple_a] :
( ( ( P_18 @ A_58 )
=> ( ( collec1717965009iple_a
@ ^ [X_3: hoare_669141180iple_a] : ( (&) @ ( A_58 = X_3 ) @ ( P_18 @ X_3 ) ) )
= ( insert175534902iple_a @ A_58 @ bot_bo280939947le_a_o ) ) )
& ( ~ ( P_18 @ A_58 )
=> ( ( collec1717965009iple_a
@ ^ [X_3: hoare_669141180iple_a] : ( (&) @ ( A_58 = X_3 ) @ ( P_18 @ X_3 ) ) )
= bot_bo280939947le_a_o ) ) ) ).
thf(fact_15_Collect__conv__if,axiom,
! [P_17: hoare_669141180iple_a > $o,A_57: hoare_669141180iple_a] :
( ( ( P_17 @ A_57 )
=> ( ( collec1717965009iple_a
@ ^ [X_3: hoare_669141180iple_a] : ( (&) @ ( X_3 = A_57 ) @ ( P_17 @ X_3 ) ) )
= ( insert175534902iple_a @ A_57 @ bot_bo280939947le_a_o ) ) )
& ( ~ ( P_17 @ A_57 )
=> ( ( collec1717965009iple_a
@ ^ [X_3: hoare_669141180iple_a] : ( (&) @ ( X_3 = A_57 ) @ ( P_17 @ X_3 ) ) )
= bot_bo280939947le_a_o ) ) ) ).
thf(fact_16_equals0D,axiom,
! [A_56: hoare_669141180iple_a,A_55: hoare_669141180iple_a > $o] :
( ( A_55 = bot_bo280939947le_a_o )
=> ~ ( member1016246415iple_a @ A_56 @ A_55 ) ) ).
thf(fact_17_Collect__empty__eq,axiom,
! [P_16: hoare_669141180iple_a > $o] :
( ( ( collec1717965009iple_a @ P_16 )
= bot_bo280939947le_a_o )
<=> ! [X_3: hoare_669141180iple_a] :
~ ( P_16 @ X_3 ) ) ).
thf(fact_18_empty__iff,axiom,
! [C_3: hoare_669141180iple_a] :
~ ( member1016246415iple_a @ C_3 @ bot_bo280939947le_a_o ) ).
thf(fact_19_empty__Collect__eq,axiom,
! [P_15: hoare_669141180iple_a > $o] :
( ( bot_bo280939947le_a_o
= ( collec1717965009iple_a @ P_15 ) )
<=> ! [X_3: hoare_669141180iple_a] :
~ ( P_15 @ X_3 ) ) ).
thf(fact_20_ex__in__conv,axiom,
! [A_54: hoare_669141180iple_a > $o] :
( ? [X_3: hoare_669141180iple_a] : ( member1016246415iple_a @ X_3 @ A_54 )
<=> ( A_54 != bot_bo280939947le_a_o ) ) ).
thf(fact_21_all__not__in__conv,axiom,
! [A_53: hoare_669141180iple_a > $o] :
( ! [X_3: hoare_669141180iple_a] :
~ ( member1016246415iple_a @ X_3 @ A_53 )
<=> ( A_53 = bot_bo280939947le_a_o ) ) ).
thf(fact_22_empty__def,axiom,
( bot_bo280939947le_a_o
= ( collec1717965009iple_a
@ ^ [X_3: hoare_669141180iple_a] : $false ) ) ).
thf(fact_23_insert__absorb,axiom,
! [A_52: hoare_669141180iple_a,A_51: hoare_669141180iple_a > $o] :
( ( member1016246415iple_a @ A_52 @ A_51 )
=> ( ( insert175534902iple_a @ A_52 @ A_51 )
= A_51 ) ) ).
thf(fact_24_insertI2,axiom,
! [B_11: hoare_669141180iple_a,A_50: hoare_669141180iple_a,B_10: hoare_669141180iple_a > $o] :
( ( member1016246415iple_a @ A_50 @ B_10 )
=> ( member1016246415iple_a @ A_50 @ ( insert175534902iple_a @ B_11 @ B_10 ) ) ) ).
thf(fact_25_insert__ident,axiom,
! [B_9: hoare_669141180iple_a > $o,X_24: hoare_669141180iple_a,A_49: hoare_669141180iple_a > $o] :
( ~ ( member1016246415iple_a @ X_24 @ A_49 )
=> ( ~ ( member1016246415iple_a @ X_24 @ B_9 )
=> ( ( ( insert175534902iple_a @ X_24 @ A_49 )
= ( insert175534902iple_a @ X_24 @ B_9 ) )
<=> ( A_49 = B_9 ) ) ) ) ).
thf(fact_26_insert__code,axiom,
! [Y_6: hoare_669141180iple_a,A_48: hoare_669141180iple_a > $o,X_23: hoare_669141180iple_a] :
( ( insert175534902iple_a @ Y_6 @ A_48 @ X_23 )
<=> ( ( Y_6 = X_23 )
| ( A_48 @ X_23 ) ) ) ).
thf(fact_27_insert__iff,axiom,
! [A_47: hoare_669141180iple_a,B_8: hoare_669141180iple_a,A_46: hoare_669141180iple_a > $o] :
( ( member1016246415iple_a @ A_47 @ ( insert175534902iple_a @ B_8 @ A_46 ) )
<=> ( ( A_47 = B_8 )
| ( member1016246415iple_a @ A_47 @ A_46 ) ) ) ).
thf(fact_28_insert__commute,axiom,
! [X_22: hoare_669141180iple_a,Y_5: hoare_669141180iple_a,A_45: hoare_669141180iple_a > $o] :
( ( insert175534902iple_a @ X_22 @ ( insert175534902iple_a @ Y_5 @ A_45 ) )
= ( insert175534902iple_a @ Y_5 @ ( insert175534902iple_a @ X_22 @ A_45 ) ) ) ).
thf(fact_29_insert__absorb2,axiom,
! [X_21: hoare_669141180iple_a,A_44: hoare_669141180iple_a > $o] :
( ( insert175534902iple_a @ X_21 @ ( insert175534902iple_a @ X_21 @ A_44 ) )
= ( insert175534902iple_a @ X_21 @ A_44 ) ) ).
thf(fact_30_insert__Collect,axiom,
! [A_43: hoare_669141180iple_a,P_14: hoare_669141180iple_a > $o] :
( ( insert175534902iple_a @ A_43 @ ( collec1717965009iple_a @ P_14 ) )
= ( collec1717965009iple_a
@ ^ [U: hoare_669141180iple_a] : ( (=>) @ ( (~) @ ( U = A_43 ) ) @ ( P_14 @ U ) ) ) ) ).
thf(fact_31_insert__compr,axiom,
! [A_42: hoare_669141180iple_a,B_7: hoare_669141180iple_a > $o] :
( ( insert175534902iple_a @ A_42 @ B_7 )
= ( collec1717965009iple_a
@ ^ [X_3: hoare_669141180iple_a] : ( (|) @ ( X_3 = A_42 ) @ ( member1016246415iple_a @ X_3 @ B_7 ) ) ) ) ).
thf(fact_32_insertI1,axiom,
! [A_41: hoare_669141180iple_a,B_6: hoare_669141180iple_a > $o] : ( member1016246415iple_a @ A_41 @ ( insert175534902iple_a @ A_41 @ B_6 ) ) ).
thf(fact_33_insert__compr__raw,axiom,
! [X_3: hoare_669141180iple_a,Xa: hoare_669141180iple_a > $o] :
( ( insert175534902iple_a @ X_3 @ Xa )
= ( collec1717965009iple_a
@ ^ [Y_1: hoare_669141180iple_a] : ( (|) @ ( Y_1 = X_3 ) @ ( member1016246415iple_a @ Y_1 @ Xa ) ) ) ) ).
thf(fact_34_singleton__inject,axiom,
! [A_40: hoare_669141180iple_a,B_5: hoare_669141180iple_a] :
( ( ( insert175534902iple_a @ A_40 @ bot_bo280939947le_a_o )
= ( insert175534902iple_a @ B_5 @ bot_bo280939947le_a_o ) )
=> ( A_40 = B_5 ) ) ).
thf(fact_35_singletonE,axiom,
! [B_4: hoare_669141180iple_a,A_39: hoare_669141180iple_a] :
( ( member1016246415iple_a @ B_4 @ ( insert175534902iple_a @ A_39 @ bot_bo280939947le_a_o ) )
=> ( B_4 = A_39 ) ) ).
thf(fact_36_doubleton__eq__iff,axiom,
! [A_38: hoare_669141180iple_a,B_3: hoare_669141180iple_a,C_2: hoare_669141180iple_a,D_1: hoare_669141180iple_a] :
( ( ( insert175534902iple_a @ A_38 @ ( insert175534902iple_a @ B_3 @ bot_bo280939947le_a_o ) )
= ( insert175534902iple_a @ C_2 @ ( insert175534902iple_a @ D_1 @ bot_bo280939947le_a_o ) ) )
<=> ( ( ( A_38 = C_2 )
& ( B_3 = D_1 ) )
| ( ( A_38 = D_1 )
& ( B_3 = C_2 ) ) ) ) ).
thf(fact_37_singleton__iff,axiom,
! [B_2: hoare_669141180iple_a,A_37: hoare_669141180iple_a] :
( ( member1016246415iple_a @ B_2 @ ( insert175534902iple_a @ A_37 @ bot_bo280939947le_a_o ) )
<=> ( B_2 = A_37 ) ) ).
thf(fact_38_insert__not__empty,axiom,
! [A_36: hoare_669141180iple_a,A_35: hoare_669141180iple_a > $o] :
( ( insert175534902iple_a @ A_36 @ A_35 )
!= bot_bo280939947le_a_o ) ).
thf(fact_39_empty__not__insert,axiom,
! [A_34: hoare_669141180iple_a,A_33: hoare_669141180iple_a > $o] :
( bot_bo280939947le_a_o
!= ( insert175534902iple_a @ A_34 @ A_33 ) ) ).
thf(fact_40_the__elem__eq,axiom,
! [X_20: hoare_669141180iple_a] :
( ( the_el738790235iple_a @ ( insert175534902iple_a @ X_20 @ bot_bo280939947le_a_o ) )
= X_20 ) ).
thf(fact_41_bot__apply,axiom,
! [X_19: hoare_669141180iple_a] :
( ( bot_bo280939947le_a_o @ X_19 )
<=> bot_bot_o ) ).
thf(fact_42_bot__fun__def,axiom,
! [X_3: hoare_669141180iple_a] :
( ( bot_bo280939947le_a_o @ X_3 )
<=> bot_bot_o ) ).
thf(fact_43_hoare__derivs_OSkip,axiom,
! [G_3: hoare_669141180iple_a > $o,P_13: x_a > state > $o] : ( hoare_2128652938rivs_a @ G_3 @ ( insert175534902iple_a @ ( hoare_1295064928iple_a @ P_13 @ skip @ P_13 ) @ bot_bo280939947le_a_o ) ) ).
thf(fact_44_Comp,axiom,
! [D: com,R: x_a > state > $o,G_2: hoare_669141180iple_a > $o,P_12: x_a > state > $o,C_1: com,Q_4: x_a > state > $o] :
( ( hoare_2128652938rivs_a @ G_2 @ ( insert175534902iple_a @ ( hoare_1295064928iple_a @ P_12 @ C_1 @ Q_4 ) @ bot_bo280939947le_a_o ) )
=> ( ( hoare_2128652938rivs_a @ G_2 @ ( insert175534902iple_a @ ( hoare_1295064928iple_a @ Q_4 @ D @ R ) @ bot_bo280939947le_a_o ) )
=> ( hoare_2128652938rivs_a @ G_2 @ ( insert175534902iple_a @ ( hoare_1295064928iple_a @ P_12 @ ( semi @ C_1 @ D ) @ R ) @ bot_bo280939947le_a_o ) ) ) ) ).
thf(fact_45_triple_Oexhaust,axiom,
! [Y_4: hoare_669141180iple_a] :
~ ! [Fun1: x_a > state > $o,Com: com,Fun2: x_a > state > $o] :
( Y_4
!= ( hoare_1295064928iple_a @ Fun1 @ Com @ Fun2 ) ) ).
thf(fact_46_Set_Oset__insert,axiom,
! [X_18: hoare_669141180iple_a,A_32: hoare_669141180iple_a > $o] :
( ( member1016246415iple_a @ X_18 @ A_32 )
=> ~ ! [B_1: hoare_669141180iple_a > $o] :
( ( A_32
= ( insert175534902iple_a @ X_18 @ B_1 ) )
=> ( member1016246415iple_a @ X_18 @ B_1 ) ) ) ).
thf(fact_47_com_Osimps_I13_J,axiom,
! [Com1: com,Com2: com] :
( ( semi @ Com1 @ Com2 )
!= skip ) ).
thf(fact_48_com_Osimps_I12_J,axiom,
! [Com1: com,Com2: com] :
( skip
!= ( semi @ Com1 @ Com2 ) ) ).
thf(fact_49_the__elem__def,axiom,
! [X_17: hoare_669141180iple_a > $o] :
( ( the_el738790235iple_a @ X_17 )
= ( the_Ho49089901iple_a
@ ^ [X_3: hoare_669141180iple_a] :
( X_17
= ( insert175534902iple_a @ X_3 @ bot_bo280939947le_a_o ) ) ) ) ).
thf(fact_50_mk__disjoint__insert,axiom,
! [A_31: hoare_669141180iple_a,A_30: hoare_669141180iple_a > $o] :
( ( member1016246415iple_a @ A_31 @ A_30 )
=> ? [B_1: hoare_669141180iple_a > $o] :
( ( A_30
= ( insert175534902iple_a @ A_31 @ B_1 ) )
& ~ ( member1016246415iple_a @ A_31 @ B_1 ) ) ) ).
thf(fact_51_com_Osimps_I3_J,axiom,
! [Com1_1: com,Com2_1: com,Com1: com,Com2: com] :
( ( ( semi @ Com1_1 @ Com2_1 )
= ( semi @ Com1 @ Com2 ) )
<=> ( ( Com1_1 = Com1 )
& ( Com2_1 = Com2 ) ) ) ).
thf(fact_52_the__sym__eq__trivial,axiom,
! [X_16: hoare_669141180iple_a] :
( ( the_Ho49089901iple_a @ ( fequal182287803iple_a @ X_16 ) )
= X_16 ) ).
thf(fact_53_the__eq__trivial,axiom,
! [A_29: hoare_669141180iple_a] :
( ( the_Ho49089901iple_a
@ ^ [X_3: hoare_669141180iple_a] : X_3 = A_29 )
= A_29 ) ).
thf(fact_54_If__def,axiom,
! [X_15: hoare_669141180iple_a,Y_3: hoare_669141180iple_a,P_11: $o] :
( ( P_11
=> ( X_15
= ( the_Ho49089901iple_a
@ ^ [Z_7: hoare_669141180iple_a] : ( (&) @ ( (=>) @ P_11 @ ( Z_7 = X_15 ) ) @ ( (=>) @ ( (~) @ P_11 ) @ ( Z_7 = Y_3 ) ) ) ) ) )
& ( ~ P_11
=> ( Y_3
= ( the_Ho49089901iple_a
@ ^ [Z_7: hoare_669141180iple_a] : ( (&) @ ( (=>) @ P_11 @ ( Z_7 = X_15 ) ) @ ( (=>) @ ( (~) @ P_11 ) @ ( Z_7 = Y_3 ) ) ) ) ) ) ) ).
thf(fact_55_equals0I,axiom,
! [A_28: hoare_669141180iple_a > $o] :
( ! [Y_1: hoare_669141180iple_a] :
~ ( member1016246415iple_a @ Y_1 @ A_28 )
=> ( A_28 = bot_bo280939947le_a_o ) ) ).
thf(fact_56_the__equality,axiom,
! [P_10: hoare_669141180iple_a > $o,A_27: hoare_669141180iple_a] :
( ( P_10 @ A_27 )
=> ( ! [X_3: hoare_669141180iple_a] :
( ( P_10 @ X_3 )
=> ( X_3 = A_27 ) )
=> ( ( the_Ho49089901iple_a @ P_10 )
= A_27 ) ) ) ).
thf(fact_57_theI,axiom,
! [P_9: hoare_669141180iple_a > $o,A_26: hoare_669141180iple_a] :
( ( P_9 @ A_26 )
=> ( ! [X_3: hoare_669141180iple_a] :
( ( P_9 @ X_3 )
=> ( X_3 = A_26 ) )
=> ( P_9 @ ( the_Ho49089901iple_a @ P_9 ) ) ) ) ).
thf(fact_58_the1__equality,axiom,
! [A_25: hoare_669141180iple_a,P_8: hoare_669141180iple_a > $o] :
( ? [X_3: hoare_669141180iple_a] :
( ( P_8 @ X_3 )
& ! [Y_1: hoare_669141180iple_a] :
( ( P_8 @ Y_1 )
=> ( Y_1 = X_3 ) ) )
=> ( ( P_8 @ A_25 )
=> ( ( the_Ho49089901iple_a @ P_8 )
= A_25 ) ) ) ).
thf(fact_59_theI_H,axiom,
! [P_7: hoare_669141180iple_a > $o] :
( ? [X_3: hoare_669141180iple_a] :
( ( P_7 @ X_3 )
& ! [Y_1: hoare_669141180iple_a] :
( ( P_7 @ Y_1 )
=> ( Y_1 = X_3 ) ) )
=> ( P_7 @ ( the_Ho49089901iple_a @ P_7 ) ) ) ).
thf(fact_60_conseq,axiom,
! [Q_2: x_a > state > $o,G_1: hoare_669141180iple_a > $o,C: com,P_5: x_a > state > $o] :
( ! [Z: x_a,S: state] :
( ( P_5 @ Z @ S )
=> ? [P_6: x_a > state > $o,Q_3: x_a > state > $o] :
( ( hoare_2128652938rivs_a @ G_1 @ ( insert175534902iple_a @ ( hoare_1295064928iple_a @ P_6 @ C @ Q_3 ) @ bot_bo280939947le_a_o ) )
& ! [S_1: state] :
( ! [Z_6: x_a] :
( ( P_6 @ Z_6 @ S )
=> ( Q_3 @ Z_6 @ S_1 ) )
=> ( Q_2 @ Z @ S_1 ) ) ) )
=> ( hoare_2128652938rivs_a @ G_1 @ ( insert175534902iple_a @ ( hoare_1295064928iple_a @ P_5 @ C @ Q_2 ) @ bot_bo280939947le_a_o ) ) ) ).
thf(fact_61_nonempty__iff,axiom,
! [A_24: hoare_669141180iple_a > $o] :
( ( A_24 != bot_bo280939947le_a_o )
<=> ? [X_3: hoare_669141180iple_a,B_1: hoare_669141180iple_a > $o] :
( ( A_24
= ( insert175534902iple_a @ X_3 @ B_1 ) )
& ~ ( member1016246415iple_a @ X_3 @ B_1 ) ) ) ).
thf(fact_62_fold1Set__sing,axiom,
! [F_31: hoare_669141180iple_a > hoare_669141180iple_a > hoare_669141180iple_a,A_23: hoare_669141180iple_a,B: hoare_669141180iple_a] :
( ( finite840267660iple_a @ F_31 @ ( insert175534902iple_a @ A_23 @ bot_bo280939947le_a_o ) @ B )
<=> ( A_23 = B ) ) ).
thf(fact_63_folding__one_Osingleton,axiom,
! [X_14: hoare_669141180iple_a,F_30: hoare_669141180iple_a > hoare_669141180iple_a > hoare_669141180iple_a,F_29: ( hoare_669141180iple_a > $o ) > hoare_669141180iple_a] :
( ( finite972428089iple_a @ F_30 @ F_29 )
=> ( ( F_29 @ ( insert175534902iple_a @ X_14 @ bot_bo280939947le_a_o ) )
= X_14 ) ) ).
thf(fact_64_bot__empty__eq,axiom,
! [X_3: hoare_669141180iple_a] :
( ( bot_bo280939947le_a_o @ X_3 )
<=> ( member1016246415iple_a @ X_3 @ bot_bo280939947le_a_o ) ) ).
thf(fact_65_empty__fold1SetE,axiom,
! [F_28: hoare_669141180iple_a > hoare_669141180iple_a > hoare_669141180iple_a,X_13: hoare_669141180iple_a] :
~ ( finite840267660iple_a @ F_28 @ bot_bo280939947le_a_o @ X_13 ) ).
thf(fact_66_fold1Set__nonempty,axiom,
! [F_27: hoare_669141180iple_a > hoare_669141180iple_a > hoare_669141180iple_a,A_22: hoare_669141180iple_a > $o,X_12: hoare_669141180iple_a] :
( ( finite840267660iple_a @ F_27 @ A_22 @ X_12 )
=> ( A_22 != bot_bo280939947le_a_o ) ) ).
thf(fact_67_fold1Set_Ointros,axiom,
! [F_26: hoare_669141180iple_a > hoare_669141180iple_a > hoare_669141180iple_a,A_21: hoare_669141180iple_a,A_20: hoare_669141180iple_a > $o,X_11: hoare_669141180iple_a] :
( ( finite590756294iple_a @ F_26 @ A_21 @ A_20 @ X_11 )
=> ( ~ ( member1016246415iple_a @ A_21 @ A_20 )
=> ( finite840267660iple_a @ F_26 @ ( insert175534902iple_a @ A_21 @ A_20 ) @ X_11 ) ) ) ).
thf(fact_68_folding__one_Oinsert,axiom,
! [X_10: hoare_669141180iple_a,A_19: hoare_669141180iple_a > $o,F_25: hoare_669141180iple_a > hoare_669141180iple_a > hoare_669141180iple_a,F_24: ( hoare_669141180iple_a > $o ) > hoare_669141180iple_a] :
( ( finite972428089iple_a @ F_25 @ F_24 )
=> ( ( finite957651855iple_a @ A_19 )
=> ( ~ ( member1016246415iple_a @ X_10 @ A_19 )
=> ( ( A_19 != bot_bo280939947le_a_o )
=> ( ( F_24 @ ( insert175534902iple_a @ X_10 @ A_19 ) )
= ( F_25 @ X_10 @ ( F_24 @ A_19 ) ) ) ) ) ) ) ).
thf(fact_69_fold1__def,axiom,
! [F_23: hoare_669141180iple_a > hoare_669141180iple_a > hoare_669141180iple_a,A_18: hoare_669141180iple_a > $o] :
( ( finite684844060iple_a @ F_23 @ A_18 )
= ( the_Ho49089901iple_a @ ( finite840267660iple_a @ F_23 @ A_18 ) ) ) ).
thf(fact_70_finite__Collect__conjI,axiom,
! [Q_1: hoare_669141180iple_a > $o,P_4: hoare_669141180iple_a > $o] :
( ( ( finite957651855iple_a @ ( collec1717965009iple_a @ P_4 ) )
| ( finite957651855iple_a @ ( collec1717965009iple_a @ Q_1 ) ) )
=> ( finite957651855iple_a
@ ( collec1717965009iple_a
@ ^ [X_3: hoare_669141180iple_a] : ( (&) @ ( P_4 @ X_3 ) @ ( Q_1 @ X_3 ) ) ) ) ) ).
thf(fact_71_finite_OemptyI,axiom,
finite957651855iple_a @ bot_bo280939947le_a_o ).
thf(fact_72_finite_OinsertI,axiom,
! [A_17: hoare_669141180iple_a,A_16: hoare_669141180iple_a > $o] :
( ( finite957651855iple_a @ A_16 )
=> ( finite957651855iple_a @ ( insert175534902iple_a @ A_17 @ A_16 ) ) ) ).
thf(fact_73_mem__def,axiom,
! [X_9: hoare_669141180iple_a,A_15: hoare_669141180iple_a > $o] :
( ( member1016246415iple_a @ X_9 @ A_15 )
<=> ( A_15 @ X_9 ) ) ).
thf(fact_74_Collect__def,axiom,
! [P_3: hoare_669141180iple_a > $o] :
( ( collec1717965009iple_a @ P_3 )
= P_3 ) ).
thf(fact_75_folding__one_Oeq__fold,axiom,
! [A_14: hoare_669141180iple_a > $o,F_22: hoare_669141180iple_a > hoare_669141180iple_a > hoare_669141180iple_a,F_21: ( hoare_669141180iple_a > $o ) > hoare_669141180iple_a] :
( ( finite972428089iple_a @ F_22 @ F_21 )
=> ( ( finite957651855iple_a @ A_14 )
=> ( ( F_21 @ A_14 )
= ( finite684844060iple_a @ F_22 @ A_14 ) ) ) ) ).
thf(fact_76_fold__graph_OemptyI,axiom,
! [F_20: hoare_669141180iple_a > hoare_669141180iple_a > hoare_669141180iple_a,Z_5: hoare_669141180iple_a] : ( finite590756294iple_a @ F_20 @ Z_5 @ bot_bo280939947le_a_o @ Z_5 ) ).
thf(fact_77_empty__fold__graphE,axiom,
! [F_19: hoare_669141180iple_a > hoare_669141180iple_a > hoare_669141180iple_a,Z_4: hoare_669141180iple_a,X_8: hoare_669141180iple_a] :
( ( finite590756294iple_a @ F_19 @ Z_4 @ bot_bo280939947le_a_o @ X_8 )
=> ( X_8 = Z_4 ) ) ).
thf(fact_78_fold__graph_OinsertI,axiom,
! [F_18: hoare_669141180iple_a > hoare_669141180iple_a > hoare_669141180iple_a,Z_3: hoare_669141180iple_a,Y_2: hoare_669141180iple_a,X_7: hoare_669141180iple_a,A_13: hoare_669141180iple_a > $o] :
( ~ ( member1016246415iple_a @ X_7 @ A_13 )
=> ( ( finite590756294iple_a @ F_18 @ Z_3 @ A_13 @ Y_2 )
=> ( finite590756294iple_a @ F_18 @ Z_3 @ ( insert175534902iple_a @ X_7 @ A_13 ) @ ( F_18 @ X_7 @ Y_2 ) ) ) ) ).
thf(fact_79_finite__Collect__disjI,axiom,
! [P_2: hoare_669141180iple_a > $o,Q: hoare_669141180iple_a > $o] :
( ( finite957651855iple_a
@ ( collec1717965009iple_a
@ ^ [X_3: hoare_669141180iple_a] : ( (|) @ ( P_2 @ X_3 ) @ ( Q @ X_3 ) ) ) )
<=> ( ( finite957651855iple_a @ ( collec1717965009iple_a @ P_2 ) )
& ( finite957651855iple_a @ ( collec1717965009iple_a @ Q ) ) ) ) ).
thf(fact_80_finite__insert,axiom,
! [A_12: hoare_669141180iple_a,A_11: hoare_669141180iple_a > $o] :
( ( finite957651855iple_a @ ( insert175534902iple_a @ A_12 @ A_11 ) )
<=> ( finite957651855iple_a @ A_11 ) ) ).
thf(fact_81_fold1__singleton__def,axiom,
! [A_10: hoare_669141180iple_a,G: ( hoare_669141180iple_a > $o ) > hoare_669141180iple_a,F_17: hoare_669141180iple_a > hoare_669141180iple_a > hoare_669141180iple_a] :
( ( G
= ( finite684844060iple_a @ F_17 ) )
=> ( ( G @ ( insert175534902iple_a @ A_10 @ bot_bo280939947le_a_o ) )
= A_10 ) ) ).
thf(fact_82_fold1__singleton,axiom,
! [F_16: hoare_669141180iple_a > hoare_669141180iple_a > hoare_669141180iple_a,A_9: hoare_669141180iple_a] :
( ( finite684844060iple_a @ F_16 @ ( insert175534902iple_a @ A_9 @ bot_bo280939947le_a_o ) )
= A_9 ) ).
thf(fact_83_folding__one_Oclosed,axiom,
! [A_8: hoare_669141180iple_a > $o,F_15: hoare_669141180iple_a > hoare_669141180iple_a > hoare_669141180iple_a,F_14: ( hoare_669141180iple_a > $o ) > hoare_669141180iple_a] :
( ( finite972428089iple_a @ F_15 @ F_14 )
=> ( ( finite957651855iple_a @ A_8 )
=> ( ( A_8 != bot_bo280939947le_a_o )
=> ( ! [X_3: hoare_669141180iple_a,Y_1: hoare_669141180iple_a] : ( member1016246415iple_a @ ( F_15 @ X_3 @ Y_1 ) @ ( insert175534902iple_a @ X_3 @ ( insert175534902iple_a @ Y_1 @ bot_bo280939947le_a_o ) ) )
=> ( member1016246415iple_a @ ( F_14 @ A_8 ) @ A_8 ) ) ) ) ) ).
thf(fact_84_insert__fold1SetE,axiom,
! [F_13: hoare_669141180iple_a > hoare_669141180iple_a > hoare_669141180iple_a,A_7: hoare_669141180iple_a,X_6: hoare_669141180iple_a > $o,X_5: hoare_669141180iple_a] :
( ( finite840267660iple_a @ F_13 @ ( insert175534902iple_a @ A_7 @ X_6 ) @ X_5 )
=> ~ ! [A_3: hoare_669141180iple_a,A_2: hoare_669141180iple_a > $o] :
( ( ( insert175534902iple_a @ A_7 @ X_6 )
= ( insert175534902iple_a @ A_3 @ A_2 ) )
=> ( ( finite590756294iple_a @ F_13 @ A_3 @ A_2 @ X_5 )
=> ( member1016246415iple_a @ A_3 @ A_2 ) ) ) ) ).
thf(fact_85_finite__nonempty__imp__fold1Set,axiom,
! [F_12: hoare_669141180iple_a > hoare_669141180iple_a > hoare_669141180iple_a,A_6: hoare_669141180iple_a > $o] :
( ( finite957651855iple_a @ A_6 )
=> ( ( A_6 != bot_bo280939947le_a_o )
=> ( ex @ ( finite840267660iple_a @ F_12 @ A_6 ) ) ) ) ).
thf(fact_86_finite__induct,axiom,
! [P_1: ( hoare_669141180iple_a > $o ) > $o,F_11: hoare_669141180iple_a > $o] :
( ( finite957651855iple_a @ F_11 )
=> ( ( P_1 @ bot_bo280939947le_a_o )
=> ( ! [X_3: hoare_669141180iple_a,F_5: hoare_669141180iple_a > $o] :
( ( finite957651855iple_a @ F_5 )
=> ( ~ ( member1016246415iple_a @ X_3 @ F_5 )
=> ( ( P_1 @ F_5 )
=> ( P_1 @ ( insert175534902iple_a @ X_3 @ F_5 ) ) ) ) )
=> ( P_1 @ F_11 ) ) ) ) ).
thf(fact_87_finite_Osimps,axiom,
! [A_5: hoare_669141180iple_a > $o] :
( ( finite957651855iple_a @ A_5 )
<=> ( ( A_5 = bot_bo280939947le_a_o )
| ? [A_2: hoare_669141180iple_a > $o,A_3: hoare_669141180iple_a] :
( ( A_5
= ( insert175534902iple_a @ A_3 @ A_2 ) )
& ( finite957651855iple_a @ A_2 ) ) ) ) ).
thf(fact_88_finite__imp__fold__graph,axiom,
! [F_10: hoare_669141180iple_a > hoare_669141180iple_a > hoare_669141180iple_a,Z_2: hoare_669141180iple_a,A_4: hoare_669141180iple_a > $o] :
( ( finite957651855iple_a @ A_4 )
=> ( ex @ ( finite590756294iple_a @ F_10 @ Z_2 @ A_4 ) ) ) ).
thf(fact_89_fold1Set_Osimps,axiom,
! [F_9: hoare_669141180iple_a > hoare_669141180iple_a > hoare_669141180iple_a,A1_1: hoare_669141180iple_a > $o,A2_1: hoare_669141180iple_a] :
( ( finite840267660iple_a @ F_9 @ A1_1 @ A2_1 )
<=> ? [A_3: hoare_669141180iple_a,A_2: hoare_669141180iple_a > $o,X_3: hoare_669141180iple_a] :
( ( A1_1
= ( insert175534902iple_a @ A_3 @ A_2 ) )
& ( A2_1 = X_3 )
& ( finite590756294iple_a @ F_9 @ A_3 @ A_2 @ X_3 )
& ~ ( member1016246415iple_a @ A_3 @ A_2 ) ) ) ).
thf(fact_90_fold__graph_Osimps,axiom,
! [F_8: hoare_669141180iple_a > hoare_669141180iple_a > hoare_669141180iple_a,Z_1: hoare_669141180iple_a,A1: hoare_669141180iple_a > $o,A2: hoare_669141180iple_a] :
( ( finite590756294iple_a @ F_8 @ Z_1 @ A1 @ A2 )
<=> ( ( ( A1 = bot_bo280939947le_a_o )
& ( A2 = Z_1 ) )
| ? [X_3: hoare_669141180iple_a,A_2: hoare_669141180iple_a > $o,Y_1: hoare_669141180iple_a] :
( ( A1
= ( insert175534902iple_a @ X_3 @ A_2 ) )
& ( A2
= ( F_8 @ X_3 @ Y_1 ) )
& ~ ( member1016246415iple_a @ X_3 @ A_2 )
& ( finite590756294iple_a @ F_8 @ Z_1 @ A_2 @ Y_1 ) ) ) ) ).
thf(fact_91_folding__one__idem_Oinsert__idem,axiom,
! [X_4: hoare_669141180iple_a,A_1: hoare_669141180iple_a > $o,F_7: hoare_669141180iple_a > hoare_669141180iple_a > hoare_669141180iple_a,F_6: ( hoare_669141180iple_a > $o ) > hoare_669141180iple_a] :
( ( finite252461622iple_a @ F_7 @ F_6 )
=> ( ( finite957651855iple_a @ A_1 )
=> ( ( A_1 != bot_bo280939947le_a_o )
=> ( ( F_6 @ ( insert175534902iple_a @ X_4 @ A_1 ) )
= ( F_7 @ X_4 @ ( F_6 @ A_1 ) ) ) ) ) ) ).
thf(fact_92_finite__ne__induct,axiom,
! [P: ( hoare_669141180iple_a > $o ) > $o,F_4: hoare_669141180iple_a > $o] :
( ( finite957651855iple_a @ F_4 )
=> ( ( F_4 != bot_bo280939947le_a_o )
=> ( ! [X_3: hoare_669141180iple_a] : ( P @ ( insert175534902iple_a @ X_3 @ bot_bo280939947le_a_o ) )
=> ( ! [X_3: hoare_669141180iple_a,F_5: hoare_669141180iple_a > $o] :
( ( finite957651855iple_a @ F_5 )
=> ( ( F_5 != bot_bo280939947le_a_o )
=> ( ~ ( member1016246415iple_a @ X_3 @ F_5 )
=> ( ( P @ F_5 )
=> ( P @ ( insert175534902iple_a @ X_3 @ F_5 ) ) ) ) ) )
=> ( P @ F_4 ) ) ) ) ) ).
thf(fact_93_folding__one__idem_Oidem,axiom,
! [X_2: hoare_669141180iple_a,F_3: hoare_669141180iple_a > hoare_669141180iple_a > hoare_669141180iple_a,F_2: ( hoare_669141180iple_a > $o ) > hoare_669141180iple_a] :
( ( finite252461622iple_a @ F_3 @ F_2 )
=> ( ( F_3 @ X_2 @ X_2 )
= X_2 ) ) ).
thf(fact_94_folding__one__idem_Oin__idem,axiom,
! [X_1: hoare_669141180iple_a,A: hoare_669141180iple_a > $o,F_1: hoare_669141180iple_a > hoare_669141180iple_a > hoare_669141180iple_a,F: ( hoare_669141180iple_a > $o ) > hoare_669141180iple_a] :
( ( finite252461622iple_a @ F_1 @ F )
=> ( ( finite957651855iple_a @ A )
=> ( ( member1016246415iple_a @ X_1 @ A )
=> ( ( F_1 @ X_1 @ ( F @ A ) )
= ( F @ A ) ) ) ) ) ).
%----Helper facts (2)
thf(help_fequal_1_1_fequal_000tc__Hoare____Mirabelle____ghhkfsbqqq__Otriple_It__a_J_,axiom,
! [X: hoare_669141180iple_a,Y: hoare_669141180iple_a] :
( ~ ( fequal182287803iple_a @ X @ Y )
| ( X = Y ) ) ).
thf(help_fequal_2_1_fequal_000tc__Hoare____Mirabelle____ghhkfsbqqq__Otriple_It__a_J_,axiom,
! [X: hoare_669141180iple_a,Y: hoare_669141180iple_a] :
( ( X != Y )
| ( fequal182287803iple_a @ X @ Y ) ) ).
%----Conjectures (1)
thf(conj_0,conjecture,
( hoare_2128652938rivs_a @ g
@ ( insert175534902iple_a
@ ( hoare_1295064928iple_a
@ ^ [Z: x_a,S: state] : $false
@ c
@ ^ [Z: x_a,S: state] : ( (&) @ ( p @ Z @ S ) @ ( (~) @ ( b @ S ) ) ) )
@ bot_bo280939947le_a_o ) ) ).
%------------------------------------------------------------------------------