TPTP Problem File: SWW471^1.p
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%------------------------------------------------------------------------------
% File : SWW471^1 : TPTP v8.2.0. Released v5.3.0.
% Domain : Software Verification
% Problem : Hoare's Logic with Procedures line 269, 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_l269 [Bla11]
% Status : Theorem
% Rating : 1.00 v5.3.0
% Syntax : Number of formulae : 235 ( 88 unt; 39 typ; 0 def)
% Number of atoms : 791 ( 174 equ; 20 cnn)
% Maximal formula atoms : 14 ( 4 avg)
% Number of connectives : 1500 ( 52 ~; 22 |; 31 &;1230 @)
% ( 54 <=>; 111 =>; 0 <=; 0 <~>)
% Maximal formula depth : 17 ( 6 avg)
% Number of types : 8 ( 7 usr)
% Number of type conns : 387 ( 387 >; 0 *; 0 +; 0 <<)
% Number of symbols : 38 ( 32 usr; 7 con; 0-3 aty)
% Number of variables : 549 ( 44 ^; 495 !; 10 ?; 549 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-08-09 19:09:52
%------------------------------------------------------------------------------
%----Should-be-implicit typings (7)
thf(ty_ty_t__a,type,
x_a: $tType ).
thf(ty_ty_tc__Com__Ocom,type,
com: $tType ).
thf(ty_ty_tc__Com__Opname,type,
pname: $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 ).
thf(ty_ty_tc__Nat__Onat,type,
nat: $tType ).
thf(ty_ty_tc__Option__Ooption_Itc__Com__Ocom_J,type,
option_com: $tType ).
%----Explicit typings (32)
thf(sy_c_Com_Obody,type,
body_1: pname > option_com ).
thf(sy_c_Com_Ocom_OBODY,type,
body: pname > com ).
thf(sy_c_Groups_Ozero__class_Ozero_000tc__Nat__Onat,type,
zero_zero_nat: nat ).
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_Ohoare__valids_000t__a,type,
hoare_319002636lids_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_Hoare__Mirabelle__ghhkfsbqqq_Otriple__valid_000t__a,type,
hoare_2082685510alid_a: nat > hoare_669141180iple_a > $o ).
thf(sy_c_Lattices_Osemilattice__sup__class_Osup_000_062_Itc__Com__Opname_M_Eo_J,type,
semila1780557381name_o: ( pname > $o ) > ( pname > $o ) > pname > $o ).
thf(sy_c_Lattices_Osemilattice__sup__class_Osup_000_062_Itc__Hoare____Mirabelle____g,type,
semila1689936973le_a_o: ( hoare_669141180iple_a > $o ) > ( hoare_669141180iple_a > $o ) > hoare_669141180iple_a > $o ).
thf(sy_c_Lattices_Osemilattice__sup__class_Osup_000_Eo,type,
semila10642723_sup_o: $o > $o > $o ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Natural_Oevalc,type,
evalc: com > state > state > $o ).
thf(sy_c_Option_Othe_000tc__Com__Ocom,type,
the_com: option_com > com ).
thf(sy_c_Orderings_Obot__class_Obot_000_062_Itc__Com__Opname_M_Eo_J,type,
bot_bot_pname_o: pname > $o ).
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_Set_OCollect_000tc__Com__Opname,type,
collect_pname: ( pname > $o ) > pname > $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_Oimage_000tc__Com__Opname_000tc__Com__Opname,type,
image_pname_pname: ( pname > pname ) > ( pname > $o ) > pname > $o ).
thf(sy_c_Set_Oimage_000tc__Com__Opname_000tc__Hoare____Mirabelle____ghhkfsbqqq__Otri,type,
image_957198589iple_a: ( pname > hoare_669141180iple_a ) > ( pname > $o ) > hoare_669141180iple_a > $o ).
thf(sy_c_Set_Oimage_000tc__Hoare____Mirabelle____ghhkfsbqqq__Otriple_It__a_J_000tc__,type,
image_225123213_pname: ( hoare_669141180iple_a > pname ) > ( hoare_669141180iple_a > $o ) > pname > $o ).
thf(sy_c_Set_Oimage_000tc__Hoare____Mirabelle____ghhkfsbqqq__Otriple_It__a_J_000tc___001,type,
image_1033305477iple_a: ( hoare_669141180iple_a > hoare_669141180iple_a ) > ( hoare_669141180iple_a > $o ) > hoare_669141180iple_a > $o ).
thf(sy_c_Set_Oinsert_000tc__Com__Opname,type,
insert_pname: pname > ( pname > $o ) > pname > $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_fequal_000tc__Com__Opname,type,
fequal_pname: pname > pname > $o ).
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__Com__Opname,type,
member_pname: pname > ( pname > $o ) > $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: pname > x_a > state > $o ).
thf(sy_v_Procs,type,
procs: pname > $o ).
thf(sy_v_Q,type,
q: pname > x_a > state > $o ).
thf(sy_v_n,type,
n: nat ).
%----Relevant facts (190)
thf(fact_0_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_1_hoare__valids__def,axiom,
! [G_10: hoare_669141180iple_a > $o,Ts_3: hoare_669141180iple_a > $o] :
( ( hoare_319002636lids_a @ G_10 @ Ts_3 )
<=> ! [N: nat] :
( ! [X: hoare_669141180iple_a] :
( ( member1016246415iple_a @ X @ G_10 )
=> ( hoare_2082685510alid_a @ N @ X ) )
=> ! [X: hoare_669141180iple_a] :
( ( member1016246415iple_a @ X @ Ts_3 )
=> ( hoare_2082685510alid_a @ N @ X ) ) ) ) ).
thf(fact_2_hoare__derivs_OBody,axiom,
! [G_9: hoare_669141180iple_a > $o,P_14: pname > x_a > state > $o,Q_4: pname > x_a > state > $o,Procs_1: pname > $o] :
( ( hoare_2128652938rivs_a
@ ( semila1689936973le_a_o @ G_9
@ ( image_957198589iple_a
@ ^ [P_9: pname] : ( hoare_1295064928iple_a @ ( P_14 @ P_9 ) @ ( body @ P_9 ) @ ( Q_4 @ P_9 ) )
@ Procs_1 ) )
@ ( image_957198589iple_a
@ ^ [P_9: pname] : ( hoare_1295064928iple_a @ ( P_14 @ P_9 ) @ ( the_com @ ( body_1 @ P_9 ) ) @ ( Q_4 @ P_9 ) )
@ Procs_1 ) )
=> ( hoare_2128652938rivs_a @ G_9
@ ( image_957198589iple_a
@ ^ [P_9: pname] : ( hoare_1295064928iple_a @ ( P_14 @ P_9 ) @ ( body @ P_9 ) @ ( Q_4 @ P_9 ) )
@ Procs_1 ) ) ) ).
thf(fact_3_UnE,axiom,
! [C_13: hoare_669141180iple_a,A_65: hoare_669141180iple_a > $o,B_36: hoare_669141180iple_a > $o] :
( ( member1016246415iple_a @ C_13 @ ( semila1689936973le_a_o @ A_65 @ B_36 ) )
=> ( ~ ( member1016246415iple_a @ C_13 @ A_65 )
=> ( member1016246415iple_a @ C_13 @ B_36 ) ) ) ).
thf(fact_4_UnE,axiom,
! [C_13: pname,A_65: pname > $o,B_36: pname > $o] :
( ( member_pname @ C_13 @ ( semila1780557381name_o @ A_65 @ B_36 ) )
=> ( ~ ( member_pname @ C_13 @ A_65 )
=> ( member_pname @ C_13 @ B_36 ) ) ) ).
thf(fact_5_sup1E,axiom,
! [A_64: hoare_669141180iple_a > $o,B_35: hoare_669141180iple_a > $o,X_24: hoare_669141180iple_a] :
( ( semila1689936973le_a_o @ A_64 @ B_35 @ X_24 )
=> ( ~ ( A_64 @ X_24 )
=> ( B_35 @ X_24 ) ) ) ).
thf(fact_6_sup1E,axiom,
! [A_64: pname > $o,B_35: pname > $o,X_24: pname] :
( ( semila1780557381name_o @ A_64 @ B_35 @ X_24 )
=> ( ~ ( A_64 @ X_24 )
=> ( B_35 @ X_24 ) ) ) ).
thf(fact_7_sup1CI,axiom,
! [A_63: hoare_669141180iple_a > $o,B_34: hoare_669141180iple_a > $o,X_23: hoare_669141180iple_a] :
( ( ~ ( B_34 @ X_23 )
=> ( A_63 @ X_23 ) )
=> ( semila1689936973le_a_o @ A_63 @ B_34 @ X_23 ) ) ).
thf(fact_8_sup1CI,axiom,
! [A_63: pname > $o,B_34: pname > $o,X_23: pname] :
( ( ~ ( B_34 @ X_23 )
=> ( A_63 @ X_23 ) )
=> ( semila1780557381name_o @ A_63 @ B_34 @ X_23 ) ) ).
thf(fact_9_UnCI,axiom,
! [A_62: hoare_669141180iple_a > $o,C_12: hoare_669141180iple_a,B_33: hoare_669141180iple_a > $o] :
( ( ~ ( member1016246415iple_a @ C_12 @ B_33 )
=> ( member1016246415iple_a @ C_12 @ A_62 ) )
=> ( member1016246415iple_a @ C_12 @ ( semila1689936973le_a_o @ A_62 @ B_33 ) ) ) ).
thf(fact_10_UnCI,axiom,
! [A_62: pname > $o,C_12: pname,B_33: pname > $o] :
( ( ~ ( member_pname @ C_12 @ B_33 )
=> ( member_pname @ C_12 @ A_62 ) )
=> ( member_pname @ C_12 @ ( semila1780557381name_o @ A_62 @ B_33 ) ) ) ).
thf(fact_11_image__eqI,axiom,
! [A_61: pname > $o,B_32: hoare_669141180iple_a,F_9: pname > hoare_669141180iple_a,X_22: pname] :
( ( B_32
= ( F_9 @ X_22 ) )
=> ( ( member_pname @ X_22 @ A_61 )
=> ( member1016246415iple_a @ B_32 @ ( image_957198589iple_a @ F_9 @ A_61 ) ) ) ) ).
thf(fact_12_image__eqI,axiom,
! [A_61: hoare_669141180iple_a > $o,B_32: pname,F_9: hoare_669141180iple_a > pname,X_22: hoare_669141180iple_a] :
( ( B_32
= ( F_9 @ X_22 ) )
=> ( ( member1016246415iple_a @ X_22 @ A_61 )
=> ( member_pname @ B_32 @ ( image_225123213_pname @ F_9 @ A_61 ) ) ) ) ).
thf(fact_13_image__Un,axiom,
! [F_8: pname > hoare_669141180iple_a,A_60: pname > $o,B_31: pname > $o] :
( ( image_957198589iple_a @ F_8 @ ( semila1780557381name_o @ A_60 @ B_31 ) )
= ( semila1689936973le_a_o @ ( image_957198589iple_a @ F_8 @ A_60 ) @ ( image_957198589iple_a @ F_8 @ B_31 ) ) ) ).
thf(fact_14_image__Un,axiom,
! [F_8: hoare_669141180iple_a > pname,A_60: hoare_669141180iple_a > $o,B_31: hoare_669141180iple_a > $o] :
( ( image_225123213_pname @ F_8 @ ( semila1689936973le_a_o @ A_60 @ B_31 ) )
= ( semila1780557381name_o @ ( image_225123213_pname @ F_8 @ A_60 ) @ ( image_225123213_pname @ F_8 @ B_31 ) ) ) ).
thf(fact_15_sup__fun__def,axiom,
! [F_7: hoare_669141180iple_a > $o,G_8: hoare_669141180iple_a > $o,X: hoare_669141180iple_a] :
( ( semila1689936973le_a_o @ F_7 @ G_8 @ X )
<=> ( semila10642723_sup_o @ ( F_7 @ X ) @ ( G_8 @ X ) ) ) ).
thf(fact_16_sup__fun__def,axiom,
! [F_7: pname > $o,G_8: pname > $o,X: pname] :
( ( semila1780557381name_o @ F_7 @ G_8 @ X )
<=> ( semila10642723_sup_o @ ( F_7 @ X ) @ ( G_8 @ X ) ) ) ).
thf(fact_17_sup__apply,axiom,
! [F_6: hoare_669141180iple_a > $o,G_7: hoare_669141180iple_a > $o,X_21: hoare_669141180iple_a] :
( ( semila1689936973le_a_o @ F_6 @ G_7 @ X_21 )
<=> ( semila10642723_sup_o @ ( F_6 @ X_21 ) @ ( G_7 @ X_21 ) ) ) ).
thf(fact_18_sup__apply,axiom,
! [F_6: pname > $o,G_7: pname > $o,X_21: pname] :
( ( semila1780557381name_o @ F_6 @ G_7 @ X_21 )
<=> ( semila10642723_sup_o @ ( F_6 @ X_21 ) @ ( G_7 @ X_21 ) ) ) ).
thf(fact_19_cut,axiom,
! [G_6: hoare_669141180iple_a > $o,G_5: hoare_669141180iple_a > $o,Ts_2: hoare_669141180iple_a > $o] :
( ( hoare_2128652938rivs_a @ G_5 @ Ts_2 )
=> ( ( hoare_2128652938rivs_a @ G_6 @ G_5 )
=> ( hoare_2128652938rivs_a @ G_6 @ Ts_2 ) ) ) ).
thf(fact_20_sup__assoc,axiom,
! [X_20: hoare_669141180iple_a > $o,Y_12: hoare_669141180iple_a > $o,Z_4: hoare_669141180iple_a > $o] :
( ( semila1689936973le_a_o @ ( semila1689936973le_a_o @ X_20 @ Y_12 ) @ Z_4 )
= ( semila1689936973le_a_o @ X_20 @ ( semila1689936973le_a_o @ Y_12 @ Z_4 ) ) ) ).
thf(fact_21_sup__assoc,axiom,
! [X_20: pname > $o,Y_12: pname > $o,Z_4: pname > $o] :
( ( semila1780557381name_o @ ( semila1780557381name_o @ X_20 @ Y_12 ) @ Z_4 )
= ( semila1780557381name_o @ X_20 @ ( semila1780557381name_o @ Y_12 @ Z_4 ) ) ) ).
thf(fact_22_sup__assoc,axiom,
! [X_20: $o,Y_12: $o,Z_4: $o] :
( ( semila10642723_sup_o @ ( semila10642723_sup_o @ X_20 @ Y_12 ) @ Z_4 )
<=> ( semila10642723_sup_o @ X_20 @ ( semila10642723_sup_o @ Y_12 @ Z_4 ) ) ) ).
thf(fact_23_inf__sup__aci_I6_J,axiom,
! [X_19: hoare_669141180iple_a > $o,Y_11: hoare_669141180iple_a > $o,Z_3: hoare_669141180iple_a > $o] :
( ( semila1689936973le_a_o @ ( semila1689936973le_a_o @ X_19 @ Y_11 ) @ Z_3 )
= ( semila1689936973le_a_o @ X_19 @ ( semila1689936973le_a_o @ Y_11 @ Z_3 ) ) ) ).
thf(fact_24_inf__sup__aci_I6_J,axiom,
! [X_19: pname > $o,Y_11: pname > $o,Z_3: pname > $o] :
( ( semila1780557381name_o @ ( semila1780557381name_o @ X_19 @ Y_11 ) @ Z_3 )
= ( semila1780557381name_o @ X_19 @ ( semila1780557381name_o @ Y_11 @ Z_3 ) ) ) ).
thf(fact_25_inf__sup__aci_I6_J,axiom,
! [X_19: $o,Y_11: $o,Z_3: $o] :
( ( semila10642723_sup_o @ ( semila10642723_sup_o @ X_19 @ Y_11 ) @ Z_3 )
<=> ( semila10642723_sup_o @ X_19 @ ( semila10642723_sup_o @ Y_11 @ Z_3 ) ) ) ).
thf(fact_26_sup_Oassoc,axiom,
! [A_59: hoare_669141180iple_a > $o,B_30: hoare_669141180iple_a > $o,C_11: hoare_669141180iple_a > $o] :
( ( semila1689936973le_a_o @ ( semila1689936973le_a_o @ A_59 @ B_30 ) @ C_11 )
= ( semila1689936973le_a_o @ A_59 @ ( semila1689936973le_a_o @ B_30 @ C_11 ) ) ) ).
thf(fact_27_sup_Oassoc,axiom,
! [A_59: pname > $o,B_30: pname > $o,C_11: pname > $o] :
( ( semila1780557381name_o @ ( semila1780557381name_o @ A_59 @ B_30 ) @ C_11 )
= ( semila1780557381name_o @ A_59 @ ( semila1780557381name_o @ B_30 @ C_11 ) ) ) ).
thf(fact_28_sup_Oassoc,axiom,
! [A_59: $o,B_30: $o,C_11: $o] :
( ( semila10642723_sup_o @ ( semila10642723_sup_o @ A_59 @ B_30 ) @ C_11 )
<=> ( semila10642723_sup_o @ A_59 @ ( semila10642723_sup_o @ B_30 @ C_11 ) ) ) ).
thf(fact_29_sup__left__commute,axiom,
! [X_18: hoare_669141180iple_a > $o,Y_10: hoare_669141180iple_a > $o,Z_2: hoare_669141180iple_a > $o] :
( ( semila1689936973le_a_o @ X_18 @ ( semila1689936973le_a_o @ Y_10 @ Z_2 ) )
= ( semila1689936973le_a_o @ Y_10 @ ( semila1689936973le_a_o @ X_18 @ Z_2 ) ) ) ).
thf(fact_30_sup__left__commute,axiom,
! [X_18: pname > $o,Y_10: pname > $o,Z_2: pname > $o] :
( ( semila1780557381name_o @ X_18 @ ( semila1780557381name_o @ Y_10 @ Z_2 ) )
= ( semila1780557381name_o @ Y_10 @ ( semila1780557381name_o @ X_18 @ Z_2 ) ) ) ).
thf(fact_31_sup__left__commute,axiom,
! [X_18: $o,Y_10: $o,Z_2: $o] :
( ( semila10642723_sup_o @ X_18 @ ( semila10642723_sup_o @ Y_10 @ Z_2 ) )
<=> ( semila10642723_sup_o @ Y_10 @ ( semila10642723_sup_o @ X_18 @ Z_2 ) ) ) ).
thf(fact_32_inf__sup__aci_I7_J,axiom,
! [X_17: hoare_669141180iple_a > $o,Y_9: hoare_669141180iple_a > $o,Z_1: hoare_669141180iple_a > $o] :
( ( semila1689936973le_a_o @ X_17 @ ( semila1689936973le_a_o @ Y_9 @ Z_1 ) )
= ( semila1689936973le_a_o @ Y_9 @ ( semila1689936973le_a_o @ X_17 @ Z_1 ) ) ) ).
thf(fact_33_inf__sup__aci_I7_J,axiom,
! [X_17: pname > $o,Y_9: pname > $o,Z_1: pname > $o] :
( ( semila1780557381name_o @ X_17 @ ( semila1780557381name_o @ Y_9 @ Z_1 ) )
= ( semila1780557381name_o @ Y_9 @ ( semila1780557381name_o @ X_17 @ Z_1 ) ) ) ).
thf(fact_34_inf__sup__aci_I7_J,axiom,
! [X_17: $o,Y_9: $o,Z_1: $o] :
( ( semila10642723_sup_o @ X_17 @ ( semila10642723_sup_o @ Y_9 @ Z_1 ) )
<=> ( semila10642723_sup_o @ Y_9 @ ( semila10642723_sup_o @ X_17 @ Z_1 ) ) ) ).
thf(fact_35_sup_Oleft__commute,axiom,
! [B_29: hoare_669141180iple_a > $o,A_58: hoare_669141180iple_a > $o,C_10: hoare_669141180iple_a > $o] :
( ( semila1689936973le_a_o @ B_29 @ ( semila1689936973le_a_o @ A_58 @ C_10 ) )
= ( semila1689936973le_a_o @ A_58 @ ( semila1689936973le_a_o @ B_29 @ C_10 ) ) ) ).
thf(fact_36_sup_Oleft__commute,axiom,
! [B_29: pname > $o,A_58: pname > $o,C_10: pname > $o] :
( ( semila1780557381name_o @ B_29 @ ( semila1780557381name_o @ A_58 @ C_10 ) )
= ( semila1780557381name_o @ A_58 @ ( semila1780557381name_o @ B_29 @ C_10 ) ) ) ).
thf(fact_37_sup_Oleft__commute,axiom,
! [B_29: $o,A_58: $o,C_10: $o] :
( ( semila10642723_sup_o @ B_29 @ ( semila10642723_sup_o @ A_58 @ C_10 ) )
<=> ( semila10642723_sup_o @ A_58 @ ( semila10642723_sup_o @ B_29 @ C_10 ) ) ) ).
thf(fact_38_sup__left__idem,axiom,
! [X_16: hoare_669141180iple_a > $o,Y_8: hoare_669141180iple_a > $o] :
( ( semila1689936973le_a_o @ X_16 @ ( semila1689936973le_a_o @ X_16 @ Y_8 ) )
= ( semila1689936973le_a_o @ X_16 @ Y_8 ) ) ).
thf(fact_39_sup__left__idem,axiom,
! [X_16: pname > $o,Y_8: pname > $o] :
( ( semila1780557381name_o @ X_16 @ ( semila1780557381name_o @ X_16 @ Y_8 ) )
= ( semila1780557381name_o @ X_16 @ Y_8 ) ) ).
thf(fact_40_sup__left__idem,axiom,
! [X_16: $o,Y_8: $o] :
( ( semila10642723_sup_o @ X_16 @ ( semila10642723_sup_o @ X_16 @ Y_8 ) )
<=> ( semila10642723_sup_o @ X_16 @ Y_8 ) ) ).
thf(fact_41_inf__sup__aci_I8_J,axiom,
! [X_15: hoare_669141180iple_a > $o,Y_7: hoare_669141180iple_a > $o] :
( ( semila1689936973le_a_o @ X_15 @ ( semila1689936973le_a_o @ X_15 @ Y_7 ) )
= ( semila1689936973le_a_o @ X_15 @ Y_7 ) ) ).
thf(fact_42_inf__sup__aci_I8_J,axiom,
! [X_15: pname > $o,Y_7: pname > $o] :
( ( semila1780557381name_o @ X_15 @ ( semila1780557381name_o @ X_15 @ Y_7 ) )
= ( semila1780557381name_o @ X_15 @ Y_7 ) ) ).
thf(fact_43_inf__sup__aci_I8_J,axiom,
! [X_15: $o,Y_7: $o] :
( ( semila10642723_sup_o @ X_15 @ ( semila10642723_sup_o @ X_15 @ Y_7 ) )
<=> ( semila10642723_sup_o @ X_15 @ Y_7 ) ) ).
thf(fact_44_sup_Oleft__idem,axiom,
! [A_57: hoare_669141180iple_a > $o,B_28: hoare_669141180iple_a > $o] :
( ( semila1689936973le_a_o @ A_57 @ ( semila1689936973le_a_o @ A_57 @ B_28 ) )
= ( semila1689936973le_a_o @ A_57 @ B_28 ) ) ).
thf(fact_45_sup_Oleft__idem,axiom,
! [A_57: pname > $o,B_28: pname > $o] :
( ( semila1780557381name_o @ A_57 @ ( semila1780557381name_o @ A_57 @ B_28 ) )
= ( semila1780557381name_o @ A_57 @ B_28 ) ) ).
thf(fact_46_sup_Oleft__idem,axiom,
! [A_57: $o,B_28: $o] :
( ( semila10642723_sup_o @ A_57 @ ( semila10642723_sup_o @ A_57 @ B_28 ) )
<=> ( semila10642723_sup_o @ A_57 @ B_28 ) ) ).
thf(fact_47_sup__commute,axiom,
! [X_14: hoare_669141180iple_a > $o,Y_6: hoare_669141180iple_a > $o] :
( ( semila1689936973le_a_o @ X_14 @ Y_6 )
= ( semila1689936973le_a_o @ Y_6 @ X_14 ) ) ).
thf(fact_48_sup__commute,axiom,
! [X_14: pname > $o,Y_6: pname > $o] :
( ( semila1780557381name_o @ X_14 @ Y_6 )
= ( semila1780557381name_o @ Y_6 @ X_14 ) ) ).
thf(fact_49_sup__commute,axiom,
! [X_14: $o,Y_6: $o] :
( ( semila10642723_sup_o @ X_14 @ Y_6 )
<=> ( semila10642723_sup_o @ Y_6 @ X_14 ) ) ).
thf(fact_50_inf__sup__aci_I5_J,axiom,
! [X_13: hoare_669141180iple_a > $o,Y_5: hoare_669141180iple_a > $o] :
( ( semila1689936973le_a_o @ X_13 @ Y_5 )
= ( semila1689936973le_a_o @ Y_5 @ X_13 ) ) ).
thf(fact_51_inf__sup__aci_I5_J,axiom,
! [X_13: pname > $o,Y_5: pname > $o] :
( ( semila1780557381name_o @ X_13 @ Y_5 )
= ( semila1780557381name_o @ Y_5 @ X_13 ) ) ).
thf(fact_52_inf__sup__aci_I5_J,axiom,
! [X_13: $o,Y_5: $o] :
( ( semila10642723_sup_o @ X_13 @ Y_5 )
<=> ( semila10642723_sup_o @ Y_5 @ X_13 ) ) ).
thf(fact_53_sup_Ocommute,axiom,
! [A_56: hoare_669141180iple_a > $o,B_27: hoare_669141180iple_a > $o] :
( ( semila1689936973le_a_o @ A_56 @ B_27 )
= ( semila1689936973le_a_o @ B_27 @ A_56 ) ) ).
thf(fact_54_sup_Ocommute,axiom,
! [A_56: pname > $o,B_27: pname > $o] :
( ( semila1780557381name_o @ A_56 @ B_27 )
= ( semila1780557381name_o @ B_27 @ A_56 ) ) ).
thf(fact_55_sup_Ocommute,axiom,
! [A_56: $o,B_27: $o] :
( ( semila10642723_sup_o @ A_56 @ B_27 )
<=> ( semila10642723_sup_o @ B_27 @ A_56 ) ) ).
thf(fact_56_sup__idem,axiom,
! [X_12: hoare_669141180iple_a > $o] :
( ( semila1689936973le_a_o @ X_12 @ X_12 )
= X_12 ) ).
thf(fact_57_sup__idem,axiom,
! [X_12: pname > $o] :
( ( semila1780557381name_o @ X_12 @ X_12 )
= X_12 ) ).
thf(fact_58_sup__idem,axiom,
! [X_12: $o] :
( ( semila10642723_sup_o @ X_12 @ X_12 )
<=> X_12 ) ).
thf(fact_59_sup_Oidem,axiom,
! [A_55: hoare_669141180iple_a > $o] :
( ( semila1689936973le_a_o @ A_55 @ A_55 )
= A_55 ) ).
thf(fact_60_sup_Oidem,axiom,
! [A_55: pname > $o] :
( ( semila1780557381name_o @ A_55 @ A_55 )
= A_55 ) ).
thf(fact_61_sup_Oidem,axiom,
! [A_55: $o] :
( ( semila10642723_sup_o @ A_55 @ A_55 )
<=> A_55 ) ).
thf(fact_62_rev__image__eqI,axiom,
! [B_26: pname,F_5: hoare_669141180iple_a > pname,X_11: hoare_669141180iple_a,A_54: hoare_669141180iple_a > $o] :
( ( member1016246415iple_a @ X_11 @ A_54 )
=> ( ( B_26
= ( F_5 @ X_11 ) )
=> ( member_pname @ B_26 @ ( image_225123213_pname @ F_5 @ A_54 ) ) ) ) ).
thf(fact_63_rev__image__eqI,axiom,
! [B_26: hoare_669141180iple_a,F_5: pname > hoare_669141180iple_a,X_11: pname,A_54: pname > $o] :
( ( member_pname @ X_11 @ A_54 )
=> ( ( B_26
= ( F_5 @ X_11 ) )
=> ( member1016246415iple_a @ B_26 @ ( image_957198589iple_a @ F_5 @ A_54 ) ) ) ) ).
thf(fact_64_imageI,axiom,
! [F_4: hoare_669141180iple_a > pname,X_10: hoare_669141180iple_a,A_53: hoare_669141180iple_a > $o] :
( ( member1016246415iple_a @ X_10 @ A_53 )
=> ( member_pname @ ( F_4 @ X_10 ) @ ( image_225123213_pname @ F_4 @ A_53 ) ) ) ).
thf(fact_65_imageI,axiom,
! [F_4: pname > hoare_669141180iple_a,X_10: pname,A_53: pname > $o] :
( ( member_pname @ X_10 @ A_53 )
=> ( member1016246415iple_a @ ( F_4 @ X_10 ) @ ( image_957198589iple_a @ F_4 @ A_53 ) ) ) ).
thf(fact_66_image__iff,axiom,
! [Z: pname,F_3: hoare_669141180iple_a > pname,A_52: hoare_669141180iple_a > $o] :
( ( member_pname @ Z @ ( image_225123213_pname @ F_3 @ A_52 ) )
<=> ? [X: hoare_669141180iple_a] :
( ( member1016246415iple_a @ X @ A_52 )
& ( Z
= ( F_3 @ X ) ) ) ) ).
thf(fact_67_image__iff,axiom,
! [Z: hoare_669141180iple_a,F_3: pname > hoare_669141180iple_a,A_52: pname > $o] :
( ( member1016246415iple_a @ Z @ ( image_957198589iple_a @ F_3 @ A_52 ) )
<=> ? [X: pname] :
( ( member_pname @ X @ A_52 )
& ( Z
= ( F_3 @ X ) ) ) ) ).
thf(fact_68_UnI2,axiom,
! [A_51: hoare_669141180iple_a > $o,C_9: hoare_669141180iple_a,B_25: hoare_669141180iple_a > $o] :
( ( member1016246415iple_a @ C_9 @ B_25 )
=> ( member1016246415iple_a @ C_9 @ ( semila1689936973le_a_o @ A_51 @ B_25 ) ) ) ).
thf(fact_69_UnI2,axiom,
! [A_51: pname > $o,C_9: pname,B_25: pname > $o] :
( ( member_pname @ C_9 @ B_25 )
=> ( member_pname @ C_9 @ ( semila1780557381name_o @ A_51 @ B_25 ) ) ) ).
thf(fact_70_UnI1,axiom,
! [B_24: hoare_669141180iple_a > $o,C_8: hoare_669141180iple_a,A_50: hoare_669141180iple_a > $o] :
( ( member1016246415iple_a @ C_8 @ A_50 )
=> ( member1016246415iple_a @ C_8 @ ( semila1689936973le_a_o @ A_50 @ B_24 ) ) ) ).
thf(fact_71_UnI1,axiom,
! [B_24: pname > $o,C_8: pname,A_50: pname > $o] :
( ( member_pname @ C_8 @ A_50 )
=> ( member_pname @ C_8 @ ( semila1780557381name_o @ A_50 @ B_24 ) ) ) ).
thf(fact_72_sup1I2,axiom,
! [A_49: hoare_669141180iple_a > $o,B_23: hoare_669141180iple_a > $o,X_9: hoare_669141180iple_a] :
( ( B_23 @ X_9 )
=> ( semila1689936973le_a_o @ A_49 @ B_23 @ X_9 ) ) ).
thf(fact_73_sup1I2,axiom,
! [A_49: pname > $o,B_23: pname > $o,X_9: pname] :
( ( B_23 @ X_9 )
=> ( semila1780557381name_o @ A_49 @ B_23 @ X_9 ) ) ).
thf(fact_74_sup1I1,axiom,
! [B_22: hoare_669141180iple_a > $o,A_48: hoare_669141180iple_a > $o,X_8: hoare_669141180iple_a] :
( ( A_48 @ X_8 )
=> ( semila1689936973le_a_o @ A_48 @ B_22 @ X_8 ) ) ).
thf(fact_75_sup1I1,axiom,
! [B_22: pname > $o,A_48: pname > $o,X_8: pname] :
( ( A_48 @ X_8 )
=> ( semila1780557381name_o @ A_48 @ B_22 @ X_8 ) ) ).
thf(fact_76_ball__Un,axiom,
! [P_13: hoare_669141180iple_a > $o,A_47: hoare_669141180iple_a > $o,B_21: hoare_669141180iple_a > $o] :
( ! [X: hoare_669141180iple_a] :
( ( member1016246415iple_a @ X @ ( semila1689936973le_a_o @ A_47 @ B_21 ) )
=> ( P_13 @ X ) )
<=> ( ! [X: hoare_669141180iple_a] :
( ( member1016246415iple_a @ X @ A_47 )
=> ( P_13 @ X ) )
& ! [X: hoare_669141180iple_a] :
( ( member1016246415iple_a @ X @ B_21 )
=> ( P_13 @ X ) ) ) ) ).
thf(fact_77_ball__Un,axiom,
! [P_13: pname > $o,A_47: pname > $o,B_21: pname > $o] :
( ! [X: pname] :
( ( member_pname @ X @ ( semila1780557381name_o @ A_47 @ B_21 ) )
=> ( P_13 @ X ) )
<=> ( ! [X: pname] :
( ( member_pname @ X @ A_47 )
=> ( P_13 @ X ) )
& ! [X: pname] :
( ( member_pname @ X @ B_21 )
=> ( P_13 @ X ) ) ) ) ).
thf(fact_78_bex__Un,axiom,
! [P_12: hoare_669141180iple_a > $o,A_46: hoare_669141180iple_a > $o,B_20: hoare_669141180iple_a > $o] :
( ? [X: hoare_669141180iple_a] :
( ( member1016246415iple_a @ X @ ( semila1689936973le_a_o @ A_46 @ B_20 ) )
& ( P_12 @ X ) )
<=> ( ? [X: hoare_669141180iple_a] :
( ( member1016246415iple_a @ X @ A_46 )
& ( P_12 @ X ) )
| ? [X: hoare_669141180iple_a] :
( ( member1016246415iple_a @ X @ B_20 )
& ( P_12 @ X ) ) ) ) ).
thf(fact_79_bex__Un,axiom,
! [P_12: pname > $o,A_46: pname > $o,B_20: pname > $o] :
( ? [X: pname] :
( ( member_pname @ X @ ( semila1780557381name_o @ A_46 @ B_20 ) )
& ( P_12 @ X ) )
<=> ( ? [X: pname] :
( ( member_pname @ X @ A_46 )
& ( P_12 @ X ) )
| ? [X: pname] :
( ( member_pname @ X @ B_20 )
& ( P_12 @ X ) ) ) ) ).
thf(fact_80_Un__assoc,axiom,
! [A_45: hoare_669141180iple_a > $o,B_19: hoare_669141180iple_a > $o,C_7: hoare_669141180iple_a > $o] :
( ( semila1689936973le_a_o @ ( semila1689936973le_a_o @ A_45 @ B_19 ) @ C_7 )
= ( semila1689936973le_a_o @ A_45 @ ( semila1689936973le_a_o @ B_19 @ C_7 ) ) ) ).
thf(fact_81_Un__assoc,axiom,
! [A_45: pname > $o,B_19: pname > $o,C_7: pname > $o] :
( ( semila1780557381name_o @ ( semila1780557381name_o @ A_45 @ B_19 ) @ C_7 )
= ( semila1780557381name_o @ A_45 @ ( semila1780557381name_o @ B_19 @ C_7 ) ) ) ).
thf(fact_82_Un__iff,axiom,
! [C_6: hoare_669141180iple_a,A_44: hoare_669141180iple_a > $o,B_18: hoare_669141180iple_a > $o] :
( ( member1016246415iple_a @ C_6 @ ( semila1689936973le_a_o @ A_44 @ B_18 ) )
<=> ( ( member1016246415iple_a @ C_6 @ A_44 )
| ( member1016246415iple_a @ C_6 @ B_18 ) ) ) ).
thf(fact_83_Un__iff,axiom,
! [C_6: pname,A_44: pname > $o,B_18: pname > $o] :
( ( member_pname @ C_6 @ ( semila1780557381name_o @ A_44 @ B_18 ) )
<=> ( ( member_pname @ C_6 @ A_44 )
| ( member_pname @ C_6 @ B_18 ) ) ) ).
thf(fact_84_Un__left__commute,axiom,
! [A_43: hoare_669141180iple_a > $o,B_17: hoare_669141180iple_a > $o,C_5: hoare_669141180iple_a > $o] :
( ( semila1689936973le_a_o @ A_43 @ ( semila1689936973le_a_o @ B_17 @ C_5 ) )
= ( semila1689936973le_a_o @ B_17 @ ( semila1689936973le_a_o @ A_43 @ C_5 ) ) ) ).
thf(fact_85_Un__left__commute,axiom,
! [A_43: pname > $o,B_17: pname > $o,C_5: pname > $o] :
( ( semila1780557381name_o @ A_43 @ ( semila1780557381name_o @ B_17 @ C_5 ) )
= ( semila1780557381name_o @ B_17 @ ( semila1780557381name_o @ A_43 @ C_5 ) ) ) ).
thf(fact_86_Un__left__absorb,axiom,
! [A_42: hoare_669141180iple_a > $o,B_16: hoare_669141180iple_a > $o] :
( ( semila1689936973le_a_o @ A_42 @ ( semila1689936973le_a_o @ A_42 @ B_16 ) )
= ( semila1689936973le_a_o @ A_42 @ B_16 ) ) ).
thf(fact_87_Un__left__absorb,axiom,
! [A_42: pname > $o,B_16: pname > $o] :
( ( semila1780557381name_o @ A_42 @ ( semila1780557381name_o @ A_42 @ B_16 ) )
= ( semila1780557381name_o @ A_42 @ B_16 ) ) ).
thf(fact_88_Un__commute,axiom,
! [A_41: hoare_669141180iple_a > $o,B_15: hoare_669141180iple_a > $o] :
( ( semila1689936973le_a_o @ A_41 @ B_15 )
= ( semila1689936973le_a_o @ B_15 @ A_41 ) ) ).
thf(fact_89_Un__commute,axiom,
! [A_41: pname > $o,B_15: pname > $o] :
( ( semila1780557381name_o @ A_41 @ B_15 )
= ( semila1780557381name_o @ B_15 @ A_41 ) ) ).
thf(fact_90_Un__def,axiom,
! [A_40: hoare_669141180iple_a > $o,B_14: hoare_669141180iple_a > $o] :
( ( semila1689936973le_a_o @ A_40 @ B_14 )
= ( collec1717965009iple_a
@ ^ [X: hoare_669141180iple_a] : ( (|) @ ( member1016246415iple_a @ X @ A_40 ) @ ( member1016246415iple_a @ X @ B_14 ) ) ) ) ).
thf(fact_91_Un__def,axiom,
! [A_40: pname > $o,B_14: pname > $o] :
( ( semila1780557381name_o @ A_40 @ B_14 )
= ( collect_pname
@ ^ [X: pname] : ( (|) @ ( member_pname @ X @ A_40 ) @ ( member_pname @ X @ B_14 ) ) ) ) ).
thf(fact_92_Un__absorb,axiom,
! [A_39: hoare_669141180iple_a > $o] :
( ( semila1689936973le_a_o @ A_39 @ A_39 )
= A_39 ) ).
thf(fact_93_Un__absorb,axiom,
! [A_39: pname > $o] :
( ( semila1780557381name_o @ A_39 @ A_39 )
= A_39 ) ).
thf(fact_94_image__image,axiom,
! [F_2: pname > hoare_669141180iple_a,G_4: hoare_669141180iple_a > pname,A_38: hoare_669141180iple_a > $o] :
( ( image_957198589iple_a @ F_2 @ ( image_225123213_pname @ G_4 @ A_38 ) )
= ( image_1033305477iple_a
@ ^ [X: hoare_669141180iple_a] : ( F_2 @ ( G_4 @ X ) )
@ A_38 ) ) ).
thf(fact_95_image__image,axiom,
! [F_2: hoare_669141180iple_a > pname,G_4: pname > hoare_669141180iple_a,A_38: pname > $o] :
( ( image_225123213_pname @ F_2 @ ( image_957198589iple_a @ G_4 @ A_38 ) )
= ( image_pname_pname
@ ^ [X: pname] : ( F_2 @ ( G_4 @ X ) )
@ A_38 ) ) ).
thf(fact_96_sup__Un__eq,axiom,
! [R: hoare_669141180iple_a > $o,S_1: hoare_669141180iple_a > $o,X: hoare_669141180iple_a] :
( ( semila1689936973le_a_o
@ ^ [Y_1: hoare_669141180iple_a] : ( member1016246415iple_a @ Y_1 @ R )
@ ^ [Y_1: hoare_669141180iple_a] : ( member1016246415iple_a @ Y_1 @ S_1 )
@ X )
<=> ( member1016246415iple_a @ X @ ( semila1689936973le_a_o @ R @ S_1 ) ) ) ).
thf(fact_97_sup__Un__eq,axiom,
! [R: pname > $o,S_1: pname > $o,X: pname] :
( ( semila1780557381name_o
@ ^ [Y_1: pname] : ( member_pname @ Y_1 @ R )
@ ^ [Y_1: pname] : ( member_pname @ Y_1 @ S_1 )
@ X )
<=> ( member_pname @ X @ ( semila1780557381name_o @ R @ S_1 ) ) ) ).
thf(fact_98_Collect__disj__eq,axiom,
! [P_11: hoare_669141180iple_a > $o,Q_3: hoare_669141180iple_a > $o] :
( ( collec1717965009iple_a
@ ^ [X: hoare_669141180iple_a] : ( (|) @ ( P_11 @ X ) @ ( Q_3 @ X ) ) )
= ( semila1689936973le_a_o @ ( collec1717965009iple_a @ P_11 ) @ ( collec1717965009iple_a @ Q_3 ) ) ) ).
thf(fact_99_Collect__disj__eq,axiom,
! [P_11: pname > $o,Q_3: pname > $o] :
( ( collect_pname
@ ^ [X: pname] : ( (|) @ ( P_11 @ X ) @ ( Q_3 @ X ) ) )
= ( semila1780557381name_o @ ( collect_pname @ P_11 ) @ ( collect_pname @ Q_3 ) ) ) ).
thf(fact_100_imageE,axiom,
! [B_13: pname,F_1: hoare_669141180iple_a > pname,A_37: hoare_669141180iple_a > $o] :
( ( member_pname @ B_13 @ ( image_225123213_pname @ F_1 @ A_37 ) )
=> ~ ! [X: hoare_669141180iple_a] :
( ( B_13
= ( F_1 @ X ) )
=> ~ ( member1016246415iple_a @ X @ A_37 ) ) ) ).
thf(fact_101_imageE,axiom,
! [B_13: hoare_669141180iple_a,F_1: pname > hoare_669141180iple_a,A_37: pname > $o] :
( ( member1016246415iple_a @ B_13 @ ( image_957198589iple_a @ F_1 @ A_37 ) )
=> ~ ! [X: pname] :
( ( B_13
= ( F_1 @ X ) )
=> ~ ( member_pname @ X @ A_37 ) ) ) ).
thf(fact_102_Body__triple__valid__Suc,axiom,
! [N_2: nat,P_10: x_a > state > $o,Pn_4: pname,Q_2: x_a > state > $o] :
( ( hoare_2082685510alid_a @ N_2 @ ( hoare_1295064928iple_a @ P_10 @ ( the_com @ ( body_1 @ Pn_4 ) ) @ Q_2 ) )
<=> ( hoare_2082685510alid_a @ ( suc @ N_2 ) @ ( hoare_1295064928iple_a @ P_10 @ ( body @ Pn_4 ) @ Q_2 ) ) ) ).
thf(fact_103_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_104_Body1,axiom,
! [Pn_3: pname,G_3: hoare_669141180iple_a > $o,P_8: pname > x_a > state > $o,Q_1: pname > x_a > state > $o,Procs: pname > $o] :
( ( hoare_2128652938rivs_a
@ ( semila1689936973le_a_o @ G_3
@ ( image_957198589iple_a
@ ^ [P_9: pname] : ( hoare_1295064928iple_a @ ( P_8 @ P_9 ) @ ( body @ P_9 ) @ ( Q_1 @ P_9 ) )
@ Procs ) )
@ ( image_957198589iple_a
@ ^ [P_9: pname] : ( hoare_1295064928iple_a @ ( P_8 @ P_9 ) @ ( the_com @ ( body_1 @ P_9 ) ) @ ( Q_1 @ P_9 ) )
@ Procs ) )
=> ( ( member_pname @ Pn_3 @ Procs )
=> ( hoare_2128652938rivs_a @ G_3 @ ( insert175534902iple_a @ ( hoare_1295064928iple_a @ ( P_8 @ Pn_3 ) @ ( body @ Pn_3 ) @ ( Q_1 @ Pn_3 ) ) @ bot_bo280939947le_a_o ) ) ) ) ).
thf(fact_105_image__cong,axiom,
! [F: hoare_669141180iple_a > pname,G_2: hoare_669141180iple_a > pname,M: hoare_669141180iple_a > $o,N_1: hoare_669141180iple_a > $o] :
( ( M = N_1 )
=> ( ! [X: hoare_669141180iple_a] :
( ( member1016246415iple_a @ X @ N_1 )
=> ( ( F @ X )
= ( G_2 @ X ) ) )
=> ( ( image_225123213_pname @ F @ M )
= ( image_225123213_pname @ G_2 @ N_1 ) ) ) ) ).
thf(fact_106_image__cong,axiom,
! [F: pname > hoare_669141180iple_a,G_2: pname > hoare_669141180iple_a,M: pname > $o,N_1: pname > $o] :
( ( M = N_1 )
=> ( ! [X: pname] :
( ( member_pname @ X @ N_1 )
=> ( ( F @ X )
= ( G_2 @ X ) ) )
=> ( ( image_957198589iple_a @ F @ M )
= ( image_957198589iple_a @ G_2 @ N_1 ) ) ) ) ).
thf(fact_107_Body__triple__valid__0,axiom,
! [P_7: x_a > state > $o,Pn_2: pname,Q: x_a > state > $o] : ( hoare_2082685510alid_a @ zero_zero_nat @ ( hoare_1295064928iple_a @ P_7 @ ( body @ Pn_2 ) @ Q ) ) ).
thf(fact_108_com_Osimps_I6_J,axiom,
! [Pname_1: pname,Pname: pname] :
( ( ( body @ Pname_1 )
= ( body @ Pname ) )
<=> ( Pname_1 = Pname ) ) ).
thf(fact_109_evalc_OBody,axiom,
! [Pn_1: pname,S0: state,S1: state] :
( ( evalc @ ( the_com @ ( body_1 @ Pn_1 ) ) @ S0 @ S1 )
=> ( evalc @ ( body @ Pn_1 ) @ S0 @ S1 ) ) ).
thf(fact_110_emptyE,axiom,
! [A_36: hoare_669141180iple_a] :
~ ( member1016246415iple_a @ A_36 @ bot_bo280939947le_a_o ) ).
thf(fact_111_emptyE,axiom,
! [A_36: pname] :
~ ( member_pname @ A_36 @ bot_bot_pname_o ) ).
thf(fact_112_insertE,axiom,
! [A_35: hoare_669141180iple_a,B_12: hoare_669141180iple_a,A_34: hoare_669141180iple_a > $o] :
( ( member1016246415iple_a @ A_35 @ ( insert175534902iple_a @ B_12 @ A_34 ) )
=> ( ( A_35 != B_12 )
=> ( member1016246415iple_a @ A_35 @ A_34 ) ) ) ).
thf(fact_113_insertE,axiom,
! [A_35: pname,B_12: pname,A_34: pname > $o] :
( ( member_pname @ A_35 @ ( insert_pname @ B_12 @ A_34 ) )
=> ( ( A_35 != B_12 )
=> ( member_pname @ A_35 @ A_34 ) ) ) ).
thf(fact_114_insertCI,axiom,
! [B_11: hoare_669141180iple_a,A_33: hoare_669141180iple_a,B_10: hoare_669141180iple_a > $o] :
( ( ~ ( member1016246415iple_a @ A_33 @ B_10 )
=> ( A_33 = B_11 ) )
=> ( member1016246415iple_a @ A_33 @ ( insert175534902iple_a @ B_11 @ B_10 ) ) ) ).
thf(fact_115_insertCI,axiom,
! [B_11: pname,A_33: pname,B_10: pname > $o] :
( ( ~ ( member_pname @ A_33 @ B_10 )
=> ( A_33 = B_11 ) )
=> ( member_pname @ A_33 @ ( insert_pname @ B_11 @ B_10 ) ) ) ).
thf(fact_116_evalc__elim__cases_I6_J,axiom,
! [P_6: pname,S: state,S1: state] :
( ( evalc @ ( body @ P_6 ) @ S @ S1 )
=> ( evalc @ ( the_com @ ( body_1 @ P_6 ) ) @ S @ S1 ) ) ).
thf(fact_117_empty__not__insert,axiom,
! [A_32: hoare_669141180iple_a,A_31: hoare_669141180iple_a > $o] :
( bot_bo280939947le_a_o
!= ( insert175534902iple_a @ A_32 @ A_31 ) ) ).
thf(fact_118_empty__not__insert,axiom,
! [A_32: pname,A_31: pname > $o] :
( bot_bot_pname_o
!= ( insert_pname @ A_32 @ A_31 ) ) ).
thf(fact_119_insert__not__empty,axiom,
! [A_30: hoare_669141180iple_a,A_29: hoare_669141180iple_a > $o] :
( ( insert175534902iple_a @ A_30 @ A_29 )
!= bot_bo280939947le_a_o ) ).
thf(fact_120_insert__not__empty,axiom,
! [A_30: pname,A_29: pname > $o] :
( ( insert_pname @ A_30 @ A_29 )
!= bot_bot_pname_o ) ).
thf(fact_121_bot__empty__eq,axiom,
! [X: hoare_669141180iple_a] :
( ( bot_bo280939947le_a_o @ X )
<=> ( member1016246415iple_a @ X @ bot_bo280939947le_a_o ) ) ).
thf(fact_122_bot__empty__eq,axiom,
! [X: pname] :
( ( bot_bot_pname_o @ X )
<=> ( member_pname @ X @ bot_bot_pname_o ) ) ).
thf(fact_123_empty__def,axiom,
( bot_bot_pname_o
= ( collect_pname
@ ^ [X: pname] : $false ) ) ).
thf(fact_124_empty__def,axiom,
( bot_bo280939947le_a_o
= ( collec1717965009iple_a
@ ^ [X: hoare_669141180iple_a] : $false ) ) ).
thf(fact_125_insertI1,axiom,
! [A_28: hoare_669141180iple_a,B_9: hoare_669141180iple_a > $o] : ( member1016246415iple_a @ A_28 @ ( insert175534902iple_a @ A_28 @ B_9 ) ) ).
thf(fact_126_insertI1,axiom,
! [A_28: pname,B_9: pname > $o] : ( member_pname @ A_28 @ ( insert_pname @ A_28 @ B_9 ) ) ).
thf(fact_127_all__not__in__conv,axiom,
! [A_27: hoare_669141180iple_a > $o] :
( ! [X: hoare_669141180iple_a] :
~ ( member1016246415iple_a @ X @ A_27 )
<=> ( A_27 = bot_bo280939947le_a_o ) ) ).
thf(fact_128_all__not__in__conv,axiom,
! [A_27: pname > $o] :
( ! [X: pname] :
~ ( member_pname @ X @ A_27 )
<=> ( A_27 = bot_bot_pname_o ) ) ).
thf(fact_129_singleton__conv2,axiom,
! [A_26: hoare_669141180iple_a] :
( ( collec1717965009iple_a @ ( fequal182287803iple_a @ A_26 ) )
= ( insert175534902iple_a @ A_26 @ bot_bo280939947le_a_o ) ) ).
thf(fact_130_singleton__conv2,axiom,
! [A_26: pname] :
( ( collect_pname @ ( fequal_pname @ A_26 ) )
= ( insert_pname @ A_26 @ bot_bot_pname_o ) ) ).
thf(fact_131_ex__in__conv,axiom,
! [A_25: hoare_669141180iple_a > $o] :
( ? [X: hoare_669141180iple_a] : ( member1016246415iple_a @ X @ A_25 )
<=> ( A_25 != bot_bo280939947le_a_o ) ) ).
thf(fact_132_ex__in__conv,axiom,
! [A_25: pname > $o] :
( ? [X: pname] : ( member_pname @ X @ A_25 )
<=> ( A_25 != bot_bot_pname_o ) ) ).
thf(fact_133_singleton__conv,axiom,
! [A_24: hoare_669141180iple_a] :
( ( collec1717965009iple_a
@ ^ [X: hoare_669141180iple_a] : X = A_24 )
= ( insert175534902iple_a @ A_24 @ bot_bo280939947le_a_o ) ) ).
thf(fact_134_singleton__conv,axiom,
! [A_24: pname] :
( ( collect_pname
@ ^ [X: pname] : X = A_24 )
= ( insert_pname @ A_24 @ bot_bot_pname_o ) ) ).
thf(fact_135_Collect__conv__if2,axiom,
! [P_5: hoare_669141180iple_a > $o,A_23: hoare_669141180iple_a] :
( ( ( P_5 @ A_23 )
=> ( ( collec1717965009iple_a
@ ^ [X: hoare_669141180iple_a] : ( (&) @ ( A_23 = X ) @ ( P_5 @ X ) ) )
= ( insert175534902iple_a @ A_23 @ bot_bo280939947le_a_o ) ) )
& ( ~ ( P_5 @ A_23 )
=> ( ( collec1717965009iple_a
@ ^ [X: hoare_669141180iple_a] : ( (&) @ ( A_23 = X ) @ ( P_5 @ X ) ) )
= bot_bo280939947le_a_o ) ) ) ).
thf(fact_136_Collect__conv__if2,axiom,
! [P_5: pname > $o,A_23: pname] :
( ( ( P_5 @ A_23 )
=> ( ( collect_pname
@ ^ [X: pname] : ( (&) @ ( A_23 = X ) @ ( P_5 @ X ) ) )
= ( insert_pname @ A_23 @ bot_bot_pname_o ) ) )
& ( ~ ( P_5 @ A_23 )
=> ( ( collect_pname
@ ^ [X: pname] : ( (&) @ ( A_23 = X ) @ ( P_5 @ X ) ) )
= bot_bot_pname_o ) ) ) ).
thf(fact_137_Collect__conv__if,axiom,
! [P_4: hoare_669141180iple_a > $o,A_22: hoare_669141180iple_a] :
( ( ( P_4 @ A_22 )
=> ( ( collec1717965009iple_a
@ ^ [X: hoare_669141180iple_a] : ( (&) @ ( X = A_22 ) @ ( P_4 @ X ) ) )
= ( insert175534902iple_a @ A_22 @ bot_bo280939947le_a_o ) ) )
& ( ~ ( P_4 @ A_22 )
=> ( ( collec1717965009iple_a
@ ^ [X: hoare_669141180iple_a] : ( (&) @ ( X = A_22 ) @ ( P_4 @ X ) ) )
= bot_bo280939947le_a_o ) ) ) ).
thf(fact_138_Collect__conv__if,axiom,
! [P_4: pname > $o,A_22: pname] :
( ( ( P_4 @ A_22 )
=> ( ( collect_pname
@ ^ [X: pname] : ( (&) @ ( X = A_22 ) @ ( P_4 @ X ) ) )
= ( insert_pname @ A_22 @ bot_bot_pname_o ) ) )
& ( ~ ( P_4 @ A_22 )
=> ( ( collect_pname
@ ^ [X: pname] : ( (&) @ ( X = A_22 ) @ ( P_4 @ X ) ) )
= bot_bot_pname_o ) ) ) ).
thf(fact_139_empty__Collect__eq,axiom,
! [P_3: pname > $o] :
( ( bot_bot_pname_o
= ( collect_pname @ P_3 ) )
<=> ! [X: pname] :
~ ( P_3 @ X ) ) ).
thf(fact_140_empty__Collect__eq,axiom,
! [P_3: hoare_669141180iple_a > $o] :
( ( bot_bo280939947le_a_o
= ( collec1717965009iple_a @ P_3 ) )
<=> ! [X: hoare_669141180iple_a] :
~ ( P_3 @ X ) ) ).
thf(fact_141_empty__iff,axiom,
! [C_4: hoare_669141180iple_a] :
~ ( member1016246415iple_a @ C_4 @ bot_bo280939947le_a_o ) ).
thf(fact_142_empty__iff,axiom,
! [C_4: pname] :
~ ( member_pname @ C_4 @ bot_bot_pname_o ) ).
thf(fact_143_mem__def,axiom,
! [X_7: hoare_669141180iple_a,A_21: hoare_669141180iple_a > $o] :
( ( member1016246415iple_a @ X_7 @ A_21 )
<=> ( A_21 @ X_7 ) ) ).
thf(fact_144_mem__def,axiom,
! [X_7: pname,A_21: pname > $o] :
( ( member_pname @ X_7 @ A_21 )
<=> ( A_21 @ X_7 ) ) ).
thf(fact_145_Collect__def,axiom,
! [P_2: pname > $o] :
( ( collect_pname @ P_2 )
= P_2 ) ).
thf(fact_146_Collect__def,axiom,
! [P_2: hoare_669141180iple_a > $o] :
( ( collec1717965009iple_a @ P_2 )
= P_2 ) ).
thf(fact_147_insert__compr,axiom,
! [A_20: hoare_669141180iple_a,B_8: hoare_669141180iple_a > $o] :
( ( insert175534902iple_a @ A_20 @ B_8 )
= ( collec1717965009iple_a
@ ^ [X: hoare_669141180iple_a] : ( (|) @ ( X = A_20 ) @ ( member1016246415iple_a @ X @ B_8 ) ) ) ) ).
thf(fact_148_insert__compr,axiom,
! [A_20: pname,B_8: pname > $o] :
( ( insert_pname @ A_20 @ B_8 )
= ( collect_pname
@ ^ [X: pname] : ( (|) @ ( X = A_20 ) @ ( member_pname @ X @ B_8 ) ) ) ) ).
thf(fact_149_insert__is__Un,axiom,
! [A_19: hoare_669141180iple_a,A_18: hoare_669141180iple_a > $o] :
( ( insert175534902iple_a @ A_19 @ A_18 )
= ( semila1689936973le_a_o @ ( insert175534902iple_a @ A_19 @ bot_bo280939947le_a_o ) @ A_18 ) ) ).
thf(fact_150_insert__is__Un,axiom,
! [A_19: pname,A_18: pname > $o] :
( ( insert_pname @ A_19 @ A_18 )
= ( semila1780557381name_o @ ( insert_pname @ A_19 @ bot_bot_pname_o ) @ A_18 ) ) ).
thf(fact_151_insert__Collect,axiom,
! [A_17: hoare_669141180iple_a,P_1: hoare_669141180iple_a > $o] :
( ( insert175534902iple_a @ A_17 @ ( collec1717965009iple_a @ P_1 ) )
= ( collec1717965009iple_a
@ ^ [U_1: hoare_669141180iple_a] : ( (=>) @ ( (~) @ ( U_1 = A_17 ) ) @ ( P_1 @ U_1 ) ) ) ) ).
thf(fact_152_insert__Collect,axiom,
! [A_17: pname,P_1: pname > $o] :
( ( insert_pname @ A_17 @ ( collect_pname @ P_1 ) )
= ( collect_pname
@ ^ [U_1: pname] : ( (=>) @ ( (~) @ ( U_1 = A_17 ) ) @ ( P_1 @ U_1 ) ) ) ) ).
thf(fact_153_singleton__iff,axiom,
! [B_7: hoare_669141180iple_a,A_16: hoare_669141180iple_a] :
( ( member1016246415iple_a @ B_7 @ ( insert175534902iple_a @ A_16 @ bot_bo280939947le_a_o ) )
<=> ( B_7 = A_16 ) ) ).
thf(fact_154_singleton__iff,axiom,
! [B_7: pname,A_16: pname] :
( ( member_pname @ B_7 @ ( insert_pname @ A_16 @ bot_bot_pname_o ) )
<=> ( B_7 = A_16 ) ) ).
thf(fact_155_insert__absorb2,axiom,
! [X_6: hoare_669141180iple_a,A_15: hoare_669141180iple_a > $o] :
( ( insert175534902iple_a @ X_6 @ ( insert175534902iple_a @ X_6 @ A_15 ) )
= ( insert175534902iple_a @ X_6 @ A_15 ) ) ).
thf(fact_156_insert__absorb2,axiom,
! [X_6: pname,A_15: pname > $o] :
( ( insert_pname @ X_6 @ ( insert_pname @ X_6 @ A_15 ) )
= ( insert_pname @ X_6 @ A_15 ) ) ).
thf(fact_157_insert__commute,axiom,
! [X_5: hoare_669141180iple_a,Y_3: hoare_669141180iple_a,A_14: hoare_669141180iple_a > $o] :
( ( insert175534902iple_a @ X_5 @ ( insert175534902iple_a @ Y_3 @ A_14 ) )
= ( insert175534902iple_a @ Y_3 @ ( insert175534902iple_a @ X_5 @ A_14 ) ) ) ).
thf(fact_158_insert__commute,axiom,
! [X_5: pname,Y_3: pname,A_14: pname > $o] :
( ( insert_pname @ X_5 @ ( insert_pname @ Y_3 @ A_14 ) )
= ( insert_pname @ Y_3 @ ( insert_pname @ X_5 @ A_14 ) ) ) ).
thf(fact_159_insert__iff,axiom,
! [A_13: hoare_669141180iple_a,B_6: hoare_669141180iple_a,A_12: hoare_669141180iple_a > $o] :
( ( member1016246415iple_a @ A_13 @ ( insert175534902iple_a @ B_6 @ A_12 ) )
<=> ( ( A_13 = B_6 )
| ( member1016246415iple_a @ A_13 @ A_12 ) ) ) ).
thf(fact_160_insert__iff,axiom,
! [A_13: pname,B_6: pname,A_12: pname > $o] :
( ( member_pname @ A_13 @ ( insert_pname @ B_6 @ A_12 ) )
<=> ( ( A_13 = B_6 )
| ( member_pname @ A_13 @ A_12 ) ) ) ).
thf(fact_161_Collect__empty__eq,axiom,
! [P: pname > $o] :
( ( ( collect_pname @ P )
= bot_bot_pname_o )
<=> ! [X: pname] :
~ ( P @ X ) ) ).
thf(fact_162_Collect__empty__eq,axiom,
! [P: hoare_669141180iple_a > $o] :
( ( ( collec1717965009iple_a @ P )
= bot_bo280939947le_a_o )
<=> ! [X: hoare_669141180iple_a] :
~ ( P @ X ) ) ).
thf(fact_163_doubleton__eq__iff,axiom,
! [A_11: hoare_669141180iple_a,B_5: hoare_669141180iple_a,C_3: hoare_669141180iple_a,D: hoare_669141180iple_a] :
( ( ( insert175534902iple_a @ A_11 @ ( insert175534902iple_a @ B_5 @ bot_bo280939947le_a_o ) )
= ( insert175534902iple_a @ C_3 @ ( insert175534902iple_a @ D @ bot_bo280939947le_a_o ) ) )
<=> ( ( ( A_11 = C_3 )
& ( B_5 = D ) )
| ( ( A_11 = D )
& ( B_5 = C_3 ) ) ) ) ).
thf(fact_164_doubleton__eq__iff,axiom,
! [A_11: pname,B_5: pname,C_3: pname,D: pname] :
( ( ( insert_pname @ A_11 @ ( insert_pname @ B_5 @ bot_bot_pname_o ) )
= ( insert_pname @ C_3 @ ( insert_pname @ D @ bot_bot_pname_o ) ) )
<=> ( ( ( A_11 = C_3 )
& ( B_5 = D ) )
| ( ( A_11 = D )
& ( B_5 = C_3 ) ) ) ) ).
thf(fact_165_insert__code,axiom,
! [Y_2: hoare_669141180iple_a,A_10: hoare_669141180iple_a > $o,X_4: hoare_669141180iple_a] :
( ( insert175534902iple_a @ Y_2 @ A_10 @ X_4 )
<=> ( ( Y_2 = X_4 )
| ( A_10 @ X_4 ) ) ) ).
thf(fact_166_insert__code,axiom,
! [Y_2: pname,A_10: pname > $o,X_4: pname] :
( ( insert_pname @ Y_2 @ A_10 @ X_4 )
<=> ( ( Y_2 = X_4 )
| ( A_10 @ X_4 ) ) ) ).
thf(fact_167_insert__compr__raw,axiom,
! [X: hoare_669141180iple_a,Xa: hoare_669141180iple_a > $o] :
( ( insert175534902iple_a @ X @ Xa )
= ( collec1717965009iple_a
@ ^ [Y_1: hoare_669141180iple_a] : ( (|) @ ( Y_1 = X ) @ ( member1016246415iple_a @ Y_1 @ Xa ) ) ) ) ).
thf(fact_168_insert__compr__raw,axiom,
! [X: pname,Xa: pname > $o] :
( ( insert_pname @ X @ Xa )
= ( collect_pname
@ ^ [Y_1: pname] : ( (|) @ ( Y_1 = X ) @ ( member_pname @ Y_1 @ Xa ) ) ) ) ).
thf(fact_169_insert__ident,axiom,
! [B_4: hoare_669141180iple_a > $o,X_3: hoare_669141180iple_a,A_9: hoare_669141180iple_a > $o] :
( ~ ( member1016246415iple_a @ X_3 @ A_9 )
=> ( ~ ( member1016246415iple_a @ X_3 @ B_4 )
=> ( ( ( insert175534902iple_a @ X_3 @ A_9 )
= ( insert175534902iple_a @ X_3 @ B_4 ) )
<=> ( A_9 = B_4 ) ) ) ) ).
thf(fact_170_insert__ident,axiom,
! [B_4: pname > $o,X_3: pname,A_9: pname > $o] :
( ~ ( member_pname @ X_3 @ A_9 )
=> ( ~ ( member_pname @ X_3 @ B_4 )
=> ( ( ( insert_pname @ X_3 @ A_9 )
= ( insert_pname @ X_3 @ B_4 ) )
<=> ( A_9 = B_4 ) ) ) ) ).
thf(fact_171_equals0D,axiom,
! [A_8: hoare_669141180iple_a,A_7: hoare_669141180iple_a > $o] :
( ( A_7 = bot_bo280939947le_a_o )
=> ~ ( member1016246415iple_a @ A_8 @ A_7 ) ) ).
thf(fact_172_equals0D,axiom,
! [A_8: pname,A_7: pname > $o] :
( ( A_7 = bot_bot_pname_o )
=> ~ ( member_pname @ A_8 @ A_7 ) ) ).
thf(fact_173_insertI2,axiom,
! [B_3: hoare_669141180iple_a,A_6: hoare_669141180iple_a,B_2: hoare_669141180iple_a > $o] :
( ( member1016246415iple_a @ A_6 @ B_2 )
=> ( member1016246415iple_a @ A_6 @ ( insert175534902iple_a @ B_3 @ B_2 ) ) ) ).
thf(fact_174_insertI2,axiom,
! [B_3: pname,A_6: pname,B_2: pname > $o] :
( ( member_pname @ A_6 @ B_2 )
=> ( member_pname @ A_6 @ ( insert_pname @ B_3 @ B_2 ) ) ) ).
thf(fact_175_insert__absorb,axiom,
! [A_5: hoare_669141180iple_a,A_4: hoare_669141180iple_a > $o] :
( ( member1016246415iple_a @ A_5 @ A_4 )
=> ( ( insert175534902iple_a @ A_5 @ A_4 )
= A_4 ) ) ).
thf(fact_176_insert__absorb,axiom,
! [A_5: pname,A_4: pname > $o] :
( ( member_pname @ A_5 @ A_4 )
=> ( ( insert_pname @ A_5 @ A_4 )
= A_4 ) ) ).
thf(fact_177_hoare__derivs_Oinsert,axiom,
! [Ts_1: hoare_669141180iple_a > $o,G_1: hoare_669141180iple_a > $o,T_2: hoare_669141180iple_a] :
( ( hoare_2128652938rivs_a @ G_1 @ ( insert175534902iple_a @ T_2 @ bot_bo280939947le_a_o ) )
=> ( ( hoare_2128652938rivs_a @ G_1 @ Ts_1 )
=> ( hoare_2128652938rivs_a @ G_1 @ ( insert175534902iple_a @ T_2 @ Ts_1 ) ) ) ) ).
thf(fact_178_singletonE,axiom,
! [B_1: hoare_669141180iple_a,A_3: hoare_669141180iple_a] :
( ( member1016246415iple_a @ B_1 @ ( insert175534902iple_a @ A_3 @ bot_bo280939947le_a_o ) )
=> ( B_1 = A_3 ) ) ).
thf(fact_179_singletonE,axiom,
! [B_1: pname,A_3: pname] :
( ( member_pname @ B_1 @ ( insert_pname @ A_3 @ bot_bot_pname_o ) )
=> ( B_1 = A_3 ) ) ).
thf(fact_180_derivs__insertD,axiom,
! [G: hoare_669141180iple_a > $o,T_1: hoare_669141180iple_a,Ts: hoare_669141180iple_a > $o] :
( ( hoare_2128652938rivs_a @ G @ ( insert175534902iple_a @ T_1 @ Ts ) )
=> ( ( hoare_2128652938rivs_a @ G @ ( insert175534902iple_a @ T_1 @ bot_bo280939947le_a_o ) )
& ( hoare_2128652938rivs_a @ G @ Ts ) ) ) ).
thf(fact_181_singleton__inject,axiom,
! [A_2: hoare_669141180iple_a,B: hoare_669141180iple_a] :
( ( ( insert175534902iple_a @ A_2 @ bot_bo280939947le_a_o )
= ( insert175534902iple_a @ B @ bot_bo280939947le_a_o ) )
=> ( A_2 = B ) ) ).
thf(fact_182_singleton__inject,axiom,
! [A_2: pname,B: pname] :
( ( ( insert_pname @ A_2 @ bot_bot_pname_o )
= ( insert_pname @ B @ bot_bot_pname_o ) )
=> ( A_2 = B ) ) ).
thf(fact_183_com__det,axiom,
! [U: state,C_2: com,S: state,T: state] :
( ( evalc @ C_2 @ S @ T )
=> ( ( evalc @ C_2 @ S @ U )
=> ( U = T ) ) ) ).
thf(fact_184_image__constant__conv,axiom,
! [C_1: pname,A_1: hoare_669141180iple_a > $o] :
( ( ( A_1 = bot_bo280939947le_a_o )
=> ( ( image_225123213_pname
@ ^ [X: hoare_669141180iple_a] : C_1
@ A_1 )
= bot_bot_pname_o ) )
& ( ( A_1 != bot_bo280939947le_a_o )
=> ( ( image_225123213_pname
@ ^ [X: hoare_669141180iple_a] : C_1
@ A_1 )
= ( insert_pname @ C_1 @ bot_bot_pname_o ) ) ) ) ).
thf(fact_185_image__constant__conv,axiom,
! [C_1: hoare_669141180iple_a,A_1: pname > $o] :
( ( ( A_1 = bot_bot_pname_o )
=> ( ( image_957198589iple_a
@ ^ [X: pname] : C_1
@ A_1 )
= bot_bo280939947le_a_o ) )
& ( ( A_1 != bot_bot_pname_o )
=> ( ( image_957198589iple_a
@ ^ [X: pname] : C_1
@ A_1 )
= ( insert175534902iple_a @ C_1 @ bot_bo280939947le_a_o ) ) ) ) ).
thf(fact_186_image__constant,axiom,
! [C: pname,X_2: hoare_669141180iple_a,A: hoare_669141180iple_a > $o] :
( ( member1016246415iple_a @ X_2 @ A )
=> ( ( image_225123213_pname
@ ^ [X: hoare_669141180iple_a] : C
@ A )
= ( insert_pname @ C @ bot_bot_pname_o ) ) ) ).
thf(fact_187_image__constant,axiom,
! [C: hoare_669141180iple_a,X_2: hoare_669141180iple_a,A: hoare_669141180iple_a > $o] :
( ( member1016246415iple_a @ X_2 @ A )
=> ( ( image_1033305477iple_a
@ ^ [X: hoare_669141180iple_a] : C
@ A )
= ( insert175534902iple_a @ C @ bot_bo280939947le_a_o ) ) ) ).
thf(fact_188_image__constant,axiom,
! [C: pname,X_2: pname,A: pname > $o] :
( ( member_pname @ X_2 @ A )
=> ( ( image_pname_pname
@ ^ [X: pname] : C
@ A )
= ( insert_pname @ C @ bot_bot_pname_o ) ) ) ).
thf(fact_189_image__constant,axiom,
! [C: hoare_669141180iple_a,X_2: pname,A: pname > $o] :
( ( member_pname @ X_2 @ A )
=> ( ( image_957198589iple_a
@ ^ [X: pname] : C
@ A )
= ( insert175534902iple_a @ C @ bot_bo280939947le_a_o ) ) ) ).
%----Helper facts (4)
thf(help_fequal_1_1_fequal_000tc__Com__Opname_T,axiom,
! [X_1: pname,Y: pname] :
( ~ ( fequal_pname @ X_1 @ Y )
| ( X_1 = Y ) ) ).
thf(help_fequal_2_1_fequal_000tc__Com__Opname_T,axiom,
! [X_1: pname,Y: pname] :
( ( X_1 != Y )
| ( fequal_pname @ X_1 @ Y ) ) ).
thf(help_fequal_1_1_fequal_000tc__Hoare____Mirabelle____ghhkfsbqqq__Otriple_It__a_J_,axiom,
! [X_1: hoare_669141180iple_a,Y: hoare_669141180iple_a] :
( ~ ( fequal182287803iple_a @ X_1 @ Y )
| ( X_1 = Y ) ) ).
thf(help_fequal_2_1_fequal_000tc__Hoare____Mirabelle____ghhkfsbqqq__Otriple_It__a_J_,axiom,
! [X_1: hoare_669141180iple_a,Y: hoare_669141180iple_a] :
( ( X_1 != Y )
| ( fequal182287803iple_a @ X_1 @ Y ) ) ).
%----Conjectures (2)
thf(conj_0,hypothesis,
! [N: nat] :
( ! [X: hoare_669141180iple_a] :
( ( member1016246415iple_a @ X
@ ( semila1689936973le_a_o @ g
@ ( image_957198589iple_a
@ ^ [Pn: pname] : ( hoare_1295064928iple_a @ ( p @ Pn ) @ ( body @ Pn ) @ ( q @ Pn ) )
@ procs ) ) )
=> ( hoare_2082685510alid_a @ N @ X ) )
=> ! [X: hoare_669141180iple_a] :
( ( member1016246415iple_a @ X
@ ( image_957198589iple_a
@ ^ [Pn: pname] : ( hoare_1295064928iple_a @ ( p @ Pn ) @ ( the_com @ ( body_1 @ Pn ) ) @ ( q @ Pn ) )
@ procs ) )
=> ( hoare_2082685510alid_a @ N @ X ) ) ) ).
thf(conj_1,conjecture,
( ! [X: hoare_669141180iple_a] :
( ( member1016246415iple_a @ X @ g )
=> ( hoare_2082685510alid_a @ n @ X ) )
=> ! [X: hoare_669141180iple_a] :
( ( member1016246415iple_a @ X
@ ( image_957198589iple_a
@ ^ [Pn: pname] : ( hoare_1295064928iple_a @ ( p @ Pn ) @ ( body @ Pn ) @ ( q @ Pn ) )
@ procs ) )
=> ( hoare_2082685510alid_a @ n @ X ) ) ) ).
%------------------------------------------------------------------------------