TPTP Problem File: SWW472^1.p
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%------------------------------------------------------------------------------
% File : SWW472^1 : TPTP v8.2.0. Released v5.3.0.
% Domain : Software Verification
% Problem : Hoare's Logic with Procedures line 327, 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_l327 [Bla11]
% Status : Theorem
% Rating : 1.00 v5.3.0
% Syntax : Number of formulae : 120 ( 19 unt; 27 typ; 0 def)
% Number of atoms : 406 ( 84 equ; 3 cnn)
% Maximal formula atoms : 9 ( 4 avg)
% Number of connectives : 802 ( 43 ~; 6 |; 18 &; 614 @)
% ( 22 <=>; 99 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 8 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 284 ( 284 >; 0 *; 0 +; 0 <<)
% Number of symbols : 29 ( 24 usr; 7 con; 0-4 aty)
% Number of variables : 273 ( 3 ^; 256 !; 14 ?; 273 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-08-09 19:16:50
%------------------------------------------------------------------------------
%----Should-be-implicit typings (3)
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_Itc__Com__Ostate_J,type,
hoare_1262092251_state: $tType ).
%----Explicit typings (24)
thf(sy_c_Big__Operators_Osemilattice__big_000tc__Hoare____Mirabelle____ghhkfsbqqq__O,type,
big_se1697321605_state: ( hoare_1262092251_state > hoare_1262092251_state > hoare_1262092251_state ) > ( ( hoare_1262092251_state > $o ) > hoare_1262092251_state ) > $o ).
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_1262092251_state > $o ) > $o ).
thf(sy_c_Finite__Set_Ofinite_000tc__Hoare____Mirabelle____ghhkfsbqqq__Otriple_Itc__C,type,
finite1178804552_state: ( hoare_1262092251_state > $o ) > $o ).
thf(sy_c_Finite__Set_Ofold1Set_000tc__Hoare____Mirabelle____ghhkfsbqqq__Otriple_Itc_,type,
finite403475723_state: ( hoare_1262092251_state > hoare_1262092251_state > hoare_1262092251_state ) > ( hoare_1262092251_state > $o ) > hoare_1262092251_state > $o ).
thf(sy_c_Finite__Set_Ofold1_000tc__Hoare____Mirabelle____ghhkfsbqqq__Otriple_Itc__Co,type,
finite1740352635_state: ( hoare_1262092251_state > hoare_1262092251_state > hoare_1262092251_state ) > ( hoare_1262092251_state > $o ) > hoare_1262092251_state ).
thf(sy_c_Finite__Set_Ofold__graph_000tc__Hoare____Mirabelle____ghhkfsbqqq__Otriple_I,type,
finite975744042_state: ( hoare_1262092251_state > hoare_1262092251_state > hoare_1262092251_state ) > hoare_1262092251_state > ( hoare_1262092251_state > $o ) > hoare_1262092251_state > $o ).
thf(sy_c_Finite__Set_Ofolding__one_000tc__Hoare____Mirabelle____ghhkfsbqqq__Otriple_,type,
finite1168661790_state: ( hoare_1262092251_state > hoare_1262092251_state > hoare_1262092251_state ) > ( ( hoare_1262092251_state > $o ) > hoare_1262092251_state ) > $o ).
thf(sy_c_Finite__Set_Ofolding__one__idem_000tc__Hoare____Mirabelle____ghhkfsbqqq__Ot,type,
finite900773345_state: ( hoare_1262092251_state > hoare_1262092251_state > hoare_1262092251_state ) > ( ( hoare_1262092251_state > $o ) > hoare_1262092251_state ) > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_000_062_Itc__Hoare____Mirabelle____ghhkfsbqqq__,type,
minus_2758725tate_o: ( hoare_1262092251_state > $o ) > ( hoare_1262092251_state > $o ) > hoare_1262092251_state > $o ).
thf(sy_c_Hoare__Mirabelle__ghhkfsbqqq_OMGT,type,
hoare_Mirabelle_MGT: com > hoare_1262092251_state ).
thf(sy_c_Hoare__Mirabelle__ghhkfsbqqq_Ohoare__derivs_000tc__Com__Ostate,type,
hoare_930741239_state: ( hoare_1262092251_state > $o ) > ( hoare_1262092251_state > $o ) > $o ).
thf(sy_c_Hoare__Mirabelle__ghhkfsbqqq_Ohoare__valids_000tc__Com__Ostate,type,
hoare_1337152501_state: ( hoare_1262092251_state > $o ) > ( hoare_1262092251_state > $o ) > $o ).
thf(sy_c_Hoare__Mirabelle__ghhkfsbqqq_Otriple_Otriple_000tc__Com__Ostate,type,
hoare_951399329_state: ( state > state > $o ) > com > ( state > state > $o ) > hoare_1262092251_state ).
thf(sy_c_Orderings_Obot__class_Obot_000_062_Itc__Hoare____Mirabelle____ghhkfsbqqq__O,type,
bot_bo113204042tate_o: hoare_1262092251_state > $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_Itc__Com__Ost,type,
collec1121927558_state: ( hoare_1262092251_state > $o ) > hoare_1262092251_state > $o ).
thf(sy_c_Set_Oinsert_000tc__Hoare____Mirabelle____ghhkfsbqqq__Otriple_Itc__Com__Osta,type,
insert81609953_state: hoare_1262092251_state > ( hoare_1262092251_state > $o ) > hoare_1262092251_state > $o ).
thf(sy_c_Set_Othe__elem_000tc__Hoare____Mirabelle____ghhkfsbqqq__Otriple_Itc__Com__O,type,
the_el417915516_state: ( hoare_1262092251_state > $o ) > hoare_1262092251_state ).
thf(sy_c_member_000tc__Hoare____Mirabelle____ghhkfsbqqq__Otriple_Itc__Com__Ostate_J,type,
member5164104_state: hoare_1262092251_state > ( hoare_1262092251_state > $o ) > $o ).
thf(sy_v_P,type,
p: state > state > $o ).
thf(sy_v_Q,type,
q: state > state > $o ).
thf(sy_v_c,type,
c: com ).
%----Relevant facts (90)
thf(fact_0_empty,axiom,
! [G_12: hoare_1262092251_state > $o] : ( hoare_930741239_state @ G_12 @ bot_bo113204042tate_o ) ).
thf(fact_1_triple_Oinject,axiom,
! [Fun1_2: state > state > $o,Com_2: com,Fun2_2: state > state > $o,Fun1_1: state > state > $o,Com_1: com,Fun2_1: state > state > $o] :
( ( ( hoare_951399329_state @ Fun1_2 @ Com_2 @ Fun2_2 )
= ( hoare_951399329_state @ Fun1_1 @ Com_1 @ Fun2_1 ) )
<=> ( ( Fun1_2 = Fun1_1 )
& ( Com_2 = Com_1 )
& ( Fun2_2 = Fun2_1 ) ) ) ).
thf(fact_2_hoare__sound,axiom,
! [G_11: hoare_1262092251_state > $o,Ts_3: hoare_1262092251_state > $o] :
( ( hoare_930741239_state @ G_11 @ Ts_3 )
=> ( hoare_1337152501_state @ G_11 @ Ts_3 ) ) ).
thf(fact_3_cut,axiom,
! [G_10: hoare_1262092251_state > $o,G_9: hoare_1262092251_state > $o,Ts_2: hoare_1262092251_state > $o] :
( ( hoare_930741239_state @ G_9 @ Ts_2 )
=> ( ( hoare_930741239_state @ G_10 @ G_9 )
=> ( hoare_930741239_state @ G_10 @ Ts_2 ) ) ) ).
thf(fact_4_hoare__derivs_Oinsert,axiom,
! [Ts_1: hoare_1262092251_state > $o,G_8: hoare_1262092251_state > $o,T_1: hoare_1262092251_state] :
( ( hoare_930741239_state @ G_8 @ ( insert81609953_state @ T_1 @ bot_bo113204042tate_o ) )
=> ( ( hoare_930741239_state @ G_8 @ Ts_1 )
=> ( hoare_930741239_state @ G_8 @ ( insert81609953_state @ T_1 @ Ts_1 ) ) ) ) ).
thf(fact_5_derivs__insertD,axiom,
! [G_7: hoare_1262092251_state > $o,T: hoare_1262092251_state,Ts: hoare_1262092251_state > $o] :
( ( hoare_930741239_state @ G_7 @ ( insert81609953_state @ T @ Ts ) )
=> ( ( hoare_930741239_state @ G_7 @ ( insert81609953_state @ T @ bot_bo113204042tate_o ) )
& ( hoare_930741239_state @ G_7 @ Ts ) ) ) ).
thf(fact_6_conseq2,axiom,
! [Q_7: state > state > $o,G_6: hoare_1262092251_state > $o,P_14: state > state > $o,C_8: com,Q_6: state > state > $o] :
( ( hoare_930741239_state @ G_6 @ ( insert81609953_state @ ( hoare_951399329_state @ P_14 @ C_8 @ Q_6 ) @ bot_bo113204042tate_o ) )
=> ( ! [Z_5: state,S: state] :
( ( Q_6 @ Z_5 @ S )
=> ( Q_7 @ Z_5 @ S ) )
=> ( hoare_930741239_state @ G_6 @ ( insert81609953_state @ ( hoare_951399329_state @ P_14 @ C_8 @ Q_7 ) @ bot_bo113204042tate_o ) ) ) ) ).
thf(fact_7_conseq1,axiom,
! [P_13: state > state > $o,G_5: hoare_1262092251_state > $o,P_12: state > state > $o,C_7: com,Q_5: state > state > $o] :
( ( hoare_930741239_state @ G_5 @ ( insert81609953_state @ ( hoare_951399329_state @ P_12 @ C_7 @ Q_5 ) @ bot_bo113204042tate_o ) )
=> ( ! [Z_5: state,S: state] :
( ( P_13 @ Z_5 @ S )
=> ( P_12 @ Z_5 @ S ) )
=> ( hoare_930741239_state @ G_5 @ ( insert81609953_state @ ( hoare_951399329_state @ P_13 @ C_7 @ Q_5 ) @ bot_bo113204042tate_o ) ) ) ) ).
thf(fact_8_insertE,axiom,
! [A_71: hoare_1262092251_state,B_20: hoare_1262092251_state,A_70: hoare_1262092251_state > $o] :
( ( member5164104_state @ A_71 @ ( insert81609953_state @ B_20 @ A_70 ) )
=> ( ( A_71 != B_20 )
=> ( member5164104_state @ A_71 @ A_70 ) ) ) ).
thf(fact_9_insertCI,axiom,
! [B_19: hoare_1262092251_state,A_69: hoare_1262092251_state,B_18: hoare_1262092251_state > $o] :
( ( ~ ( member5164104_state @ A_69 @ B_18 )
=> ( A_69 = B_19 ) )
=> ( member5164104_state @ A_69 @ ( insert81609953_state @ B_19 @ B_18 ) ) ) ).
thf(fact_10_conseq12,axiom,
! [Q_4: state > state > $o,P_11: state > state > $o,G_4: hoare_1262092251_state > $o,P_10: state > state > $o,C_6: com,Q_3: state > state > $o] :
( ( hoare_930741239_state @ G_4 @ ( insert81609953_state @ ( hoare_951399329_state @ P_10 @ C_6 @ Q_3 ) @ bot_bo113204042tate_o ) )
=> ( ! [Z_5: state,S: state] :
( ( P_11 @ Z_5 @ S )
=> ! [S_1: state] :
( ! [Z_6: state] :
( ( P_10 @ Z_6 @ S )
=> ( Q_3 @ Z_6 @ S_1 ) )
=> ( Q_4 @ Z_5 @ S_1 ) ) )
=> ( hoare_930741239_state @ G_4 @ ( insert81609953_state @ ( hoare_951399329_state @ P_11 @ C_6 @ Q_4 ) @ bot_bo113204042tate_o ) ) ) ) ).
thf(fact_11_emptyE,axiom,
! [A_68: hoare_1262092251_state] :
~ ( member5164104_state @ A_68 @ bot_bo113204042tate_o ) ).
thf(fact_12_empty__not__insert,axiom,
! [A_67: hoare_1262092251_state,A_66: hoare_1262092251_state > $o] :
( bot_bo113204042tate_o
!= ( insert81609953_state @ A_67 @ A_66 ) ) ).
thf(fact_13_insert__not__empty,axiom,
! [A_65: hoare_1262092251_state,A_64: hoare_1262092251_state > $o] :
( ( insert81609953_state @ A_65 @ A_64 )
!= bot_bo113204042tate_o ) ).
thf(fact_14_singleton__iff,axiom,
! [B_17: hoare_1262092251_state,A_63: hoare_1262092251_state] :
( ( member5164104_state @ B_17 @ ( insert81609953_state @ A_63 @ bot_bo113204042tate_o ) )
<=> ( B_17 = A_63 ) ) ).
thf(fact_15_doubleton__eq__iff,axiom,
! [A_62: hoare_1262092251_state,B_16: hoare_1262092251_state,C_5: hoare_1262092251_state,D_1: hoare_1262092251_state] :
( ( ( insert81609953_state @ A_62 @ ( insert81609953_state @ B_16 @ bot_bo113204042tate_o ) )
= ( insert81609953_state @ C_5 @ ( insert81609953_state @ D_1 @ bot_bo113204042tate_o ) ) )
<=> ( ( ( A_62 = C_5 )
& ( B_16 = D_1 ) )
| ( ( A_62 = D_1 )
& ( B_16 = C_5 ) ) ) ) ).
thf(fact_16_equals0D,axiom,
! [A_61: hoare_1262092251_state,A_60: hoare_1262092251_state > $o] :
( ( A_60 = bot_bo113204042tate_o )
=> ~ ( member5164104_state @ A_61 @ A_60 ) ) ).
thf(fact_17_Collect__empty__eq,axiom,
! [P_9: hoare_1262092251_state > $o] :
( ( ( collec1121927558_state @ P_9 )
= bot_bo113204042tate_o )
<=> ! [X_2: hoare_1262092251_state] :
~ ( P_9 @ X_2 ) ) ).
thf(fact_18_empty__iff,axiom,
! [C_4: hoare_1262092251_state] :
~ ( member5164104_state @ C_4 @ bot_bo113204042tate_o ) ).
thf(fact_19_empty__Collect__eq,axiom,
! [P_8: hoare_1262092251_state > $o] :
( ( bot_bo113204042tate_o
= ( collec1121927558_state @ P_8 ) )
<=> ! [X_2: hoare_1262092251_state] :
~ ( P_8 @ X_2 ) ) ).
thf(fact_20_ex__in__conv,axiom,
! [A_59: hoare_1262092251_state > $o] :
( ? [X_2: hoare_1262092251_state] : ( member5164104_state @ X_2 @ A_59 )
<=> ( A_59 != bot_bo113204042tate_o ) ) ).
thf(fact_21_all__not__in__conv,axiom,
! [A_58: hoare_1262092251_state > $o] :
( ! [X_2: hoare_1262092251_state] :
~ ( member5164104_state @ X_2 @ A_58 )
<=> ( A_58 = bot_bo113204042tate_o ) ) ).
thf(fact_22_empty__def,axiom,
( bot_bo113204042tate_o
= ( collec1121927558_state
@ ^ [X_2: hoare_1262092251_state] : $false ) ) ).
thf(fact_23_insert__absorb,axiom,
! [A_57: hoare_1262092251_state,A_56: hoare_1262092251_state > $o] :
( ( member5164104_state @ A_57 @ A_56 )
=> ( ( insert81609953_state @ A_57 @ A_56 )
= A_56 ) ) ).
thf(fact_24_insertI2,axiom,
! [B_15: hoare_1262092251_state,A_55: hoare_1262092251_state,B_14: hoare_1262092251_state > $o] :
( ( member5164104_state @ A_55 @ B_14 )
=> ( member5164104_state @ A_55 @ ( insert81609953_state @ B_15 @ B_14 ) ) ) ).
thf(fact_25_insert__ident,axiom,
! [B_13: hoare_1262092251_state > $o,X_22: hoare_1262092251_state,A_54: hoare_1262092251_state > $o] :
( ~ ( member5164104_state @ X_22 @ A_54 )
=> ( ~ ( member5164104_state @ X_22 @ B_13 )
=> ( ( ( insert81609953_state @ X_22 @ A_54 )
= ( insert81609953_state @ X_22 @ B_13 ) )
<=> ( A_54 = B_13 ) ) ) ) ).
thf(fact_26_insert__code,axiom,
! [Y_4: hoare_1262092251_state,A_53: hoare_1262092251_state > $o,X_21: hoare_1262092251_state] :
( ( insert81609953_state @ Y_4 @ A_53 @ X_21 )
<=> ( ( Y_4 = X_21 )
| ( A_53 @ X_21 ) ) ) ).
thf(fact_27_insert__iff,axiom,
! [A_52: hoare_1262092251_state,B_12: hoare_1262092251_state,A_51: hoare_1262092251_state > $o] :
( ( member5164104_state @ A_52 @ ( insert81609953_state @ B_12 @ A_51 ) )
<=> ( ( A_52 = B_12 )
| ( member5164104_state @ A_52 @ A_51 ) ) ) ).
thf(fact_28_insert__commute,axiom,
! [X_20: hoare_1262092251_state,Y_3: hoare_1262092251_state,A_50: hoare_1262092251_state > $o] :
( ( insert81609953_state @ X_20 @ ( insert81609953_state @ Y_3 @ A_50 ) )
= ( insert81609953_state @ Y_3 @ ( insert81609953_state @ X_20 @ A_50 ) ) ) ).
thf(fact_29_insert__absorb2,axiom,
! [X_19: hoare_1262092251_state,A_49: hoare_1262092251_state > $o] :
( ( insert81609953_state @ X_19 @ ( insert81609953_state @ X_19 @ A_49 ) )
= ( insert81609953_state @ X_19 @ A_49 ) ) ).
thf(fact_30_insert__Collect,axiom,
! [A_48: hoare_1262092251_state,P_7: hoare_1262092251_state > $o] :
( ( insert81609953_state @ A_48 @ ( collec1121927558_state @ P_7 ) )
= ( collec1121927558_state
@ ^ [U: hoare_1262092251_state] : ( (=>) @ ( (~) @ ( U = A_48 ) ) @ ( P_7 @ U ) ) ) ) ).
thf(fact_31_insert__compr,axiom,
! [A_47: hoare_1262092251_state,B_11: hoare_1262092251_state > $o] :
( ( insert81609953_state @ A_47 @ B_11 )
= ( collec1121927558_state
@ ^ [X_2: hoare_1262092251_state] : ( (|) @ ( X_2 = A_47 ) @ ( member5164104_state @ X_2 @ B_11 ) ) ) ) ).
thf(fact_32_insertI1,axiom,
! [A_46: hoare_1262092251_state,B_10: hoare_1262092251_state > $o] : ( member5164104_state @ A_46 @ ( insert81609953_state @ A_46 @ B_10 ) ) ).
thf(fact_33_singleton__inject,axiom,
! [A_45: hoare_1262092251_state,B_9: hoare_1262092251_state] :
( ( ( insert81609953_state @ A_45 @ bot_bo113204042tate_o )
= ( insert81609953_state @ B_9 @ bot_bo113204042tate_o ) )
=> ( A_45 = B_9 ) ) ).
thf(fact_34_singletonE,axiom,
! [B_8: hoare_1262092251_state,A_44: hoare_1262092251_state] :
( ( member5164104_state @ B_8 @ ( insert81609953_state @ A_44 @ bot_bo113204042tate_o ) )
=> ( B_8 = A_44 ) ) ).
thf(fact_35_the__elem__eq,axiom,
! [X_18: hoare_1262092251_state] :
( ( the_el417915516_state @ ( insert81609953_state @ X_18 @ bot_bo113204042tate_o ) )
= X_18 ) ).
thf(fact_36_bot__apply,axiom,
! [X_17: hoare_1262092251_state] :
( ( bot_bo113204042tate_o @ X_17 )
<=> bot_bot_o ) ).
thf(fact_37_bot__fun__def,axiom,
! [X_2: hoare_1262092251_state] :
( ( bot_bo113204042tate_o @ X_2 )
<=> bot_bot_o ) ).
thf(fact_38_hoare__derivs_OSkip,axiom,
! [G_3: hoare_1262092251_state > $o,P_6: state > state > $o] : ( hoare_930741239_state @ G_3 @ ( insert81609953_state @ ( hoare_951399329_state @ P_6 @ skip @ P_6 ) @ bot_bo113204042tate_o ) ) ).
thf(fact_39_Comp,axiom,
! [D: com,R: state > state > $o,G_2: hoare_1262092251_state > $o,P_5: state > state > $o,C_3: com,Q_2: state > state > $o] :
( ( hoare_930741239_state @ G_2 @ ( insert81609953_state @ ( hoare_951399329_state @ P_5 @ C_3 @ Q_2 ) @ bot_bo113204042tate_o ) )
=> ( ( hoare_930741239_state @ G_2 @ ( insert81609953_state @ ( hoare_951399329_state @ Q_2 @ D @ R ) @ bot_bo113204042tate_o ) )
=> ( hoare_930741239_state @ G_2 @ ( insert81609953_state @ ( hoare_951399329_state @ P_5 @ ( semi @ C_3 @ D ) @ R ) @ bot_bo113204042tate_o ) ) ) ) ).
thf(fact_40_triple_Oexhaust,axiom,
! [Y_2: hoare_1262092251_state] :
~ ! [Fun1: state > state > $o,Com: com,Fun2: state > state > $o] :
( Y_2
!= ( hoare_951399329_state @ Fun1 @ Com @ Fun2 ) ) ).
thf(fact_41_Set_Oset__insert,axiom,
! [X_16: hoare_1262092251_state,A_43: hoare_1262092251_state > $o] :
( ( member5164104_state @ X_16 @ A_43 )
=> ~ ! [B_7: hoare_1262092251_state > $o] :
( ( A_43
= ( insert81609953_state @ X_16 @ B_7 ) )
=> ( member5164104_state @ X_16 @ B_7 ) ) ) ).
thf(fact_42_mk__disjoint__insert,axiom,
! [A_42: hoare_1262092251_state,A_41: hoare_1262092251_state > $o] :
( ( member5164104_state @ A_42 @ A_41 )
=> ? [B_7: hoare_1262092251_state > $o] :
( ( A_41
= ( insert81609953_state @ A_42 @ B_7 ) )
& ~ ( member5164104_state @ A_42 @ B_7 ) ) ) ).
thf(fact_43_com_Osimps_I13_J,axiom,
! [Com1: com,Com2: com] :
( ( semi @ Com1 @ Com2 )
!= skip ) ).
thf(fact_44_com_Osimps_I12_J,axiom,
! [Com1: com,Com2: com] :
( skip
!= ( semi @ Com1 @ Com2 ) ) ).
thf(fact_45_equals0I,axiom,
! [A_40: hoare_1262092251_state > $o] :
( ! [Y: hoare_1262092251_state] :
~ ( member5164104_state @ Y @ A_40 )
=> ( A_40 = bot_bo113204042tate_o ) ) ).
thf(fact_46_conseq,axiom,
! [Q: state > state > $o,G_1: hoare_1262092251_state > $o,C_2: com,P_3: state > state > $o] :
( ! [Z_5: state,S: state] :
( ( P_3 @ Z_5 @ S )
=> ? [P_4: state > state > $o,Q_1: state > state > $o] :
( ( hoare_930741239_state @ G_1 @ ( insert81609953_state @ ( hoare_951399329_state @ P_4 @ C_2 @ Q_1 ) @ bot_bo113204042tate_o ) )
& ! [S_1: state] :
( ! [Z_6: state] :
( ( P_4 @ Z_6 @ S )
=> ( Q_1 @ Z_6 @ S_1 ) )
=> ( Q @ Z_5 @ S_1 ) ) ) )
=> ( hoare_930741239_state @ G_1 @ ( insert81609953_state @ ( hoare_951399329_state @ P_3 @ C_2 @ Q ) @ bot_bo113204042tate_o ) ) ) ).
thf(fact_47_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_48_nonempty__iff,axiom,
! [A_39: hoare_1262092251_state > $o] :
( ( A_39 != bot_bo113204042tate_o )
<=> ? [X_2: hoare_1262092251_state,B_7: hoare_1262092251_state > $o] :
( ( A_39
= ( insert81609953_state @ X_2 @ B_7 ) )
& ~ ( member5164104_state @ X_2 @ B_7 ) ) ) ).
thf(fact_49_bot__empty__eq,axiom,
! [X_2: hoare_1262092251_state] :
( ( bot_bo113204042tate_o @ X_2 )
<=> ( member5164104_state @ X_2 @ bot_bo113204042tate_o ) ) ).
thf(fact_50_fold1Set__sing,axiom,
! [F_34: hoare_1262092251_state > hoare_1262092251_state > hoare_1262092251_state,A_38: hoare_1262092251_state,B_6: hoare_1262092251_state] :
( ( finite403475723_state @ F_34 @ ( insert81609953_state @ A_38 @ bot_bo113204042tate_o ) @ B_6 )
<=> ( A_38 = B_6 ) ) ).
thf(fact_51_folding__one_Osingleton,axiom,
! [X_15: hoare_1262092251_state,F_33: hoare_1262092251_state > hoare_1262092251_state > hoare_1262092251_state,F_32: ( hoare_1262092251_state > $o ) > hoare_1262092251_state] :
( ( finite1168661790_state @ F_33 @ F_32 )
=> ( ( F_32 @ ( insert81609953_state @ X_15 @ bot_bo113204042tate_o ) )
= X_15 ) ) ).
thf(fact_52_fold1__singleton,axiom,
! [F_31: hoare_1262092251_state > hoare_1262092251_state > hoare_1262092251_state,A_37: hoare_1262092251_state] :
( ( finite1740352635_state @ F_31 @ ( insert81609953_state @ A_37 @ bot_bo113204042tate_o ) )
= A_37 ) ).
thf(fact_53_fold1__singleton__def,axiom,
! [A_36: hoare_1262092251_state,G: ( hoare_1262092251_state > $o ) > hoare_1262092251_state,F_30: hoare_1262092251_state > hoare_1262092251_state > hoare_1262092251_state] :
( ( G
= ( finite1740352635_state @ F_30 ) )
=> ( ( G @ ( insert81609953_state @ A_36 @ bot_bo113204042tate_o ) )
= A_36 ) ) ).
thf(fact_54_empty__fold1SetE,axiom,
! [F_29: hoare_1262092251_state > hoare_1262092251_state > hoare_1262092251_state,X_14: hoare_1262092251_state] :
~ ( finite403475723_state @ F_29 @ bot_bo113204042tate_o @ X_14 ) ).
thf(fact_55_fold1Set__nonempty,axiom,
! [F_28: hoare_1262092251_state > hoare_1262092251_state > hoare_1262092251_state,A_35: hoare_1262092251_state > $o,X_13: hoare_1262092251_state] :
( ( finite403475723_state @ F_28 @ A_35 @ X_13 )
=> ( A_35 != bot_bo113204042tate_o ) ) ).
thf(fact_56_fold1Set_Ointros,axiom,
! [F_27: hoare_1262092251_state > hoare_1262092251_state > hoare_1262092251_state,A_34: hoare_1262092251_state,A_33: hoare_1262092251_state > $o,X_12: hoare_1262092251_state] :
( ( finite975744042_state @ F_27 @ A_34 @ A_33 @ X_12 )
=> ( ~ ( member5164104_state @ A_34 @ A_33 )
=> ( finite403475723_state @ F_27 @ ( insert81609953_state @ A_34 @ A_33 ) @ X_12 ) ) ) ).
thf(fact_57_folding__one_Oinsert,axiom,
! [X_11: hoare_1262092251_state,A_32: hoare_1262092251_state > $o,F_26: hoare_1262092251_state > hoare_1262092251_state > hoare_1262092251_state,F_25: ( hoare_1262092251_state > $o ) > hoare_1262092251_state] :
( ( finite1168661790_state @ F_26 @ F_25 )
=> ( ( finite1178804552_state @ A_32 )
=> ( ~ ( member5164104_state @ X_11 @ A_32 )
=> ( ( A_32 != bot_bo113204042tate_o )
=> ( ( F_25 @ ( insert81609953_state @ X_11 @ A_32 ) )
= ( F_26 @ X_11 @ ( F_25 @ A_32 ) ) ) ) ) ) ) ).
thf(fact_58_folding__one_Oeq__fold,axiom,
! [A_31: hoare_1262092251_state > $o,F_24: hoare_1262092251_state > hoare_1262092251_state > hoare_1262092251_state,F_23: ( hoare_1262092251_state > $o ) > hoare_1262092251_state] :
( ( finite1168661790_state @ F_24 @ F_23 )
=> ( ( finite1178804552_state @ A_31 )
=> ( ( F_23 @ A_31 )
= ( finite1740352635_state @ F_24 @ A_31 ) ) ) ) ).
thf(fact_59_finite_OemptyI,axiom,
finite1178804552_state @ bot_bo113204042tate_o ).
thf(fact_60_finite_OinsertI,axiom,
! [A_30: hoare_1262092251_state,A_29: hoare_1262092251_state > $o] :
( ( finite1178804552_state @ A_29 )
=> ( finite1178804552_state @ ( insert81609953_state @ A_30 @ A_29 ) ) ) ).
thf(fact_61_fold__graph_OemptyI,axiom,
! [F_22: hoare_1262092251_state > hoare_1262092251_state > hoare_1262092251_state,Z_4: hoare_1262092251_state] : ( finite975744042_state @ F_22 @ Z_4 @ bot_bo113204042tate_o @ Z_4 ) ).
thf(fact_62_empty__fold__graphE,axiom,
! [F_21: hoare_1262092251_state > hoare_1262092251_state > hoare_1262092251_state,Z_3: hoare_1262092251_state,X_10: hoare_1262092251_state] :
( ( finite975744042_state @ F_21 @ Z_3 @ bot_bo113204042tate_o @ X_10 )
=> ( X_10 = Z_3 ) ) ).
thf(fact_63_fold__graph_OinsertI,axiom,
! [F_20: hoare_1262092251_state > hoare_1262092251_state > hoare_1262092251_state,Z_2: hoare_1262092251_state,Y_1: hoare_1262092251_state,X_9: hoare_1262092251_state,A_28: hoare_1262092251_state > $o] :
( ~ ( member5164104_state @ X_9 @ A_28 )
=> ( ( finite975744042_state @ F_20 @ Z_2 @ A_28 @ Y_1 )
=> ( finite975744042_state @ F_20 @ Z_2 @ ( insert81609953_state @ X_9 @ A_28 ) @ ( F_20 @ X_9 @ Y_1 ) ) ) ) ).
thf(fact_64_finite__insert,axiom,
! [A_27: hoare_1262092251_state,A_26: hoare_1262092251_state > $o] :
( ( finite1178804552_state @ ( insert81609953_state @ A_27 @ A_26 ) )
<=> ( finite1178804552_state @ A_26 ) ) ).
thf(fact_65_folding__one_Oclosed,axiom,
! [A_25: hoare_1262092251_state > $o,F_19: hoare_1262092251_state > hoare_1262092251_state > hoare_1262092251_state,F_18: ( hoare_1262092251_state > $o ) > hoare_1262092251_state] :
( ( finite1168661790_state @ F_19 @ F_18 )
=> ( ( finite1178804552_state @ A_25 )
=> ( ( A_25 != bot_bo113204042tate_o )
=> ( ! [X_2: hoare_1262092251_state,Y: hoare_1262092251_state] : ( member5164104_state @ ( F_19 @ X_2 @ Y ) @ ( insert81609953_state @ X_2 @ ( insert81609953_state @ Y @ bot_bo113204042tate_o ) ) )
=> ( member5164104_state @ ( F_18 @ A_25 ) @ A_25 ) ) ) ) ) ).
thf(fact_66_insert__fold1SetE,axiom,
! [F_17: hoare_1262092251_state > hoare_1262092251_state > hoare_1262092251_state,A_24: hoare_1262092251_state,X_8: hoare_1262092251_state > $o,X_7: hoare_1262092251_state] :
( ( finite403475723_state @ F_17 @ ( insert81609953_state @ A_24 @ X_8 ) @ X_7 )
=> ~ ! [A_19: hoare_1262092251_state,A_18: hoare_1262092251_state > $o] :
( ( ( insert81609953_state @ A_24 @ X_8 )
= ( insert81609953_state @ A_19 @ A_18 ) )
=> ( ( finite975744042_state @ F_17 @ A_19 @ A_18 @ X_7 )
=> ( member5164104_state @ A_19 @ A_18 ) ) ) ) ).
thf(fact_67_finite__nonempty__imp__fold1Set,axiom,
! [F_16: hoare_1262092251_state > hoare_1262092251_state > hoare_1262092251_state,A_23: hoare_1262092251_state > $o] :
( ( finite1178804552_state @ A_23 )
=> ( ( A_23 != bot_bo113204042tate_o )
=> ( ex @ ( finite403475723_state @ F_16 @ A_23 ) ) ) ) ).
thf(fact_68_finite__induct,axiom,
! [P_2: ( hoare_1262092251_state > $o ) > $o,F_15: hoare_1262092251_state > $o] :
( ( finite1178804552_state @ F_15 )
=> ( ( P_2 @ bot_bo113204042tate_o )
=> ( ! [X_2: hoare_1262092251_state,F_5: hoare_1262092251_state > $o] :
( ( finite1178804552_state @ F_5 )
=> ( ~ ( member5164104_state @ X_2 @ F_5 )
=> ( ( P_2 @ F_5 )
=> ( P_2 @ ( insert81609953_state @ X_2 @ F_5 ) ) ) ) )
=> ( P_2 @ F_15 ) ) ) ) ).
thf(fact_69_mem__def,axiom,
! [X_6: hoare_1262092251_state,A_22: hoare_1262092251_state > $o] :
( ( member5164104_state @ X_6 @ A_22 )
<=> ( A_22 @ X_6 ) ) ).
thf(fact_70_Collect__def,axiom,
! [P_1: hoare_1262092251_state > $o] :
( ( collec1121927558_state @ P_1 )
= P_1 ) ).
thf(fact_71_finite_Osimps,axiom,
! [A_21: hoare_1262092251_state > $o] :
( ( finite1178804552_state @ A_21 )
<=> ( ( A_21 = bot_bo113204042tate_o )
| ? [A_18: hoare_1262092251_state > $o,A_19: hoare_1262092251_state] :
( ( A_21
= ( insert81609953_state @ A_19 @ A_18 ) )
& ( finite1178804552_state @ A_18 ) ) ) ) ).
thf(fact_72_finite__imp__fold__graph,axiom,
! [F_14: hoare_1262092251_state > hoare_1262092251_state > hoare_1262092251_state,Z_1: hoare_1262092251_state,A_20: hoare_1262092251_state > $o] :
( ( finite1178804552_state @ A_20 )
=> ( ex @ ( finite975744042_state @ F_14 @ Z_1 @ A_20 ) ) ) ).
thf(fact_73_fold1Set_Osimps,axiom,
! [F_13: hoare_1262092251_state > hoare_1262092251_state > hoare_1262092251_state,A1_1: hoare_1262092251_state > $o,A2_1: hoare_1262092251_state] :
( ( finite403475723_state @ F_13 @ A1_1 @ A2_1 )
<=> ? [A_19: hoare_1262092251_state,A_18: hoare_1262092251_state > $o,X_2: hoare_1262092251_state] :
( ( A1_1
= ( insert81609953_state @ A_19 @ A_18 ) )
& ( A2_1 = X_2 )
& ( finite975744042_state @ F_13 @ A_19 @ A_18 @ X_2 )
& ~ ( member5164104_state @ A_19 @ A_18 ) ) ) ).
thf(fact_74_fold__graph_Osimps,axiom,
! [F_12: hoare_1262092251_state > hoare_1262092251_state > hoare_1262092251_state,Z: hoare_1262092251_state,A1: hoare_1262092251_state > $o,A2: hoare_1262092251_state] :
( ( finite975744042_state @ F_12 @ Z @ A1 @ A2 )
<=> ( ( ( A1 = bot_bo113204042tate_o )
& ( A2 = Z ) )
| ? [X_2: hoare_1262092251_state,A_18: hoare_1262092251_state > $o,Y: hoare_1262092251_state] :
( ( A1
= ( insert81609953_state @ X_2 @ A_18 ) )
& ( A2
= ( F_12 @ X_2 @ Y ) )
& ~ ( member5164104_state @ X_2 @ A_18 )
& ( finite975744042_state @ F_12 @ Z @ A_18 @ Y ) ) ) ) ).
thf(fact_75_folding__one__idem_Oinsert__idem,axiom,
! [X_5: hoare_1262092251_state,A_17: hoare_1262092251_state > $o,F_11: hoare_1262092251_state > hoare_1262092251_state > hoare_1262092251_state,F_10: ( hoare_1262092251_state > $o ) > hoare_1262092251_state] :
( ( finite900773345_state @ F_11 @ F_10 )
=> ( ( finite1178804552_state @ A_17 )
=> ( ( A_17 != bot_bo113204042tate_o )
=> ( ( F_10 @ ( insert81609953_state @ X_5 @ A_17 ) )
= ( F_11 @ X_5 @ ( F_10 @ A_17 ) ) ) ) ) ) ).
thf(fact_76_folding__one__idem_Oidem,axiom,
! [X_4: hoare_1262092251_state,F_9: hoare_1262092251_state > hoare_1262092251_state > hoare_1262092251_state,F_8: ( hoare_1262092251_state > $o ) > hoare_1262092251_state] :
( ( finite900773345_state @ F_9 @ F_8 )
=> ( ( F_9 @ X_4 @ X_4 )
= X_4 ) ) ).
thf(fact_77_folding__one__idem_Oin__idem,axiom,
! [X_3: hoare_1262092251_state,A_16: hoare_1262092251_state > $o,F_7: hoare_1262092251_state > hoare_1262092251_state > hoare_1262092251_state,F_6: ( hoare_1262092251_state > $o ) > hoare_1262092251_state] :
( ( finite900773345_state @ F_7 @ F_6 )
=> ( ( finite1178804552_state @ A_16 )
=> ( ( member5164104_state @ X_3 @ A_16 )
=> ( ( F_7 @ X_3 @ ( F_6 @ A_16 ) )
= ( F_6 @ A_16 ) ) ) ) ) ).
thf(fact_78_finite__ne__induct,axiom,
! [P: ( hoare_1262092251_state > $o ) > $o,F_4: hoare_1262092251_state > $o] :
( ( finite1178804552_state @ F_4 )
=> ( ( F_4 != bot_bo113204042tate_o )
=> ( ! [X_2: hoare_1262092251_state] : ( P @ ( insert81609953_state @ X_2 @ bot_bo113204042tate_o ) )
=> ( ! [X_2: hoare_1262092251_state,F_5: hoare_1262092251_state > $o] :
( ( finite1178804552_state @ F_5 )
=> ( ( F_5 != bot_bo113204042tate_o )
=> ( ~ ( member5164104_state @ X_2 @ F_5 )
=> ( ( P @ F_5 )
=> ( P @ ( insert81609953_state @ X_2 @ F_5 ) ) ) ) ) )
=> ( P @ F_4 ) ) ) ) ) ).
thf(fact_79_semilattice__big_OF__eq,axiom,
! [A_15: hoare_1262092251_state > $o,F_3: hoare_1262092251_state > hoare_1262092251_state > hoare_1262092251_state,F_2: ( hoare_1262092251_state > $o ) > hoare_1262092251_state] :
( ( big_se1697321605_state @ F_3 @ F_2 )
=> ( ( finite1178804552_state @ A_15 )
=> ( ( F_2 @ A_15 )
= ( finite1740352635_state @ F_3 @ A_15 ) ) ) ) ).
thf(fact_80_folding__one_Oremove,axiom,
! [X_1: hoare_1262092251_state,A_14: hoare_1262092251_state > $o,F_1: hoare_1262092251_state > hoare_1262092251_state > hoare_1262092251_state,F: ( hoare_1262092251_state > $o ) > hoare_1262092251_state] :
( ( finite1168661790_state @ F_1 @ F )
=> ( ( finite1178804552_state @ A_14 )
=> ( ( member5164104_state @ X_1 @ A_14 )
=> ( ( ( ( minus_2758725tate_o @ A_14 @ ( insert81609953_state @ X_1 @ bot_bo113204042tate_o ) )
= bot_bo113204042tate_o )
=> ( ( F @ A_14 )
= X_1 ) )
& ( ( ( minus_2758725tate_o @ A_14 @ ( insert81609953_state @ X_1 @ bot_bo113204042tate_o ) )
!= bot_bo113204042tate_o )
=> ( ( F @ A_14 )
= ( F_1 @ X_1 @ ( F @ ( minus_2758725tate_o @ A_14 @ ( insert81609953_state @ X_1 @ bot_bo113204042tate_o ) ) ) ) ) ) ) ) ) ) ).
thf(fact_81_DiffI,axiom,
! [B_5: hoare_1262092251_state > $o,C_1: hoare_1262092251_state,A_13: hoare_1262092251_state > $o] :
( ( member5164104_state @ C_1 @ A_13 )
=> ( ~ ( member5164104_state @ C_1 @ B_5 )
=> ( member5164104_state @ C_1 @ ( minus_2758725tate_o @ A_13 @ B_5 ) ) ) ) ).
thf(fact_82_DiffE,axiom,
! [C: hoare_1262092251_state,A_12: hoare_1262092251_state > $o,B_4: hoare_1262092251_state > $o] :
( ( member5164104_state @ C @ ( minus_2758725tate_o @ A_12 @ B_4 ) )
=> ~ ( ( member5164104_state @ C @ A_12 )
=> ( member5164104_state @ C @ B_4 ) ) ) ).
thf(fact_83_finite__Diff,axiom,
! [B_3: hoare_1262092251_state > $o,A_11: hoare_1262092251_state > $o] :
( ( finite1178804552_state @ A_11 )
=> ( finite1178804552_state @ ( minus_2758725tate_o @ A_11 @ B_3 ) ) ) ).
thf(fact_84_insert__Diff,axiom,
! [A_10: hoare_1262092251_state,A_9: hoare_1262092251_state > $o] :
( ( member5164104_state @ A_10 @ A_9 )
=> ( ( insert81609953_state @ A_10 @ ( minus_2758725tate_o @ A_9 @ ( insert81609953_state @ A_10 @ bot_bo113204042tate_o ) ) )
= A_9 ) ) ).
thf(fact_85_Diff__insert__absorb,axiom,
! [X: hoare_1262092251_state,A_8: hoare_1262092251_state > $o] :
( ~ ( member5164104_state @ X @ A_8 )
=> ( ( minus_2758725tate_o @ ( insert81609953_state @ X @ A_8 ) @ ( insert81609953_state @ X @ bot_bo113204042tate_o ) )
= A_8 ) ) ).
thf(fact_86_insert__Diff__single,axiom,
! [A_7: hoare_1262092251_state,A_6: hoare_1262092251_state > $o] :
( ( insert81609953_state @ A_7 @ ( minus_2758725tate_o @ A_6 @ ( insert81609953_state @ A_7 @ bot_bo113204042tate_o ) ) )
= ( insert81609953_state @ A_7 @ A_6 ) ) ).
thf(fact_87_Diff__insert2,axiom,
! [A_5: hoare_1262092251_state > $o,A_4: hoare_1262092251_state,B_2: hoare_1262092251_state > $o] :
( ( minus_2758725tate_o @ A_5 @ ( insert81609953_state @ A_4 @ B_2 ) )
= ( minus_2758725tate_o @ ( minus_2758725tate_o @ A_5 @ ( insert81609953_state @ A_4 @ bot_bo113204042tate_o ) ) @ B_2 ) ) ).
thf(fact_88_Diff__insert,axiom,
! [A_3: hoare_1262092251_state > $o,A_2: hoare_1262092251_state,B_1: hoare_1262092251_state > $o] :
( ( minus_2758725tate_o @ A_3 @ ( insert81609953_state @ A_2 @ B_1 ) )
= ( minus_2758725tate_o @ ( minus_2758725tate_o @ A_3 @ B_1 ) @ ( insert81609953_state @ A_2 @ bot_bo113204042tate_o ) ) ) ).
thf(fact_89_finite__Diff__insert,axiom,
! [A_1: hoare_1262092251_state > $o,A: hoare_1262092251_state,B: hoare_1262092251_state > $o] :
( ( finite1178804552_state @ ( minus_2758725tate_o @ A_1 @ ( insert81609953_state @ A @ B ) ) )
<=> ( finite1178804552_state @ ( minus_2758725tate_o @ A_1 @ B ) ) ) ).
%----Conjectures (3)
thf(conj_0,hypothesis,
hoare_930741239_state @ bot_bo113204042tate_o @ ( insert81609953_state @ ( hoare_Mirabelle_MGT @ c ) @ bot_bo113204042tate_o ) ).
thf(conj_1,hypothesis,
hoare_1337152501_state @ bot_bo113204042tate_o @ ( insert81609953_state @ ( hoare_951399329_state @ p @ c @ q ) @ bot_bo113204042tate_o ) ).
thf(conj_2,conjecture,
hoare_930741239_state @ bot_bo113204042tate_o @ ( insert81609953_state @ ( hoare_951399329_state @ p @ c @ q ) @ bot_bo113204042tate_o ) ).
%------------------------------------------------------------------------------