TPTP Problem File: SWW524_5.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SWW524_5 : TPTP v8.2.0. Released v6.0.0.
% Domain : Software Verification
% Problem : Hoare's Logic with Procedures line 316
% Version : Especial.
% English :
% Refs : [BN10] Boehme & Nipkow (2010), Sledgehammer: Judgement Day
% : [Bla13] Blanchette (2011), Email to Geoff Sutcliffe
% Source : [Bla13]
% Names : hoare_316 [Bla13]
% Status : Unknown
% Rating : 1.00 v6.4.0
% Syntax : Number of formulae : 169 ( 48 unt; 42 typ; 0 def)
% Number of atoms : 284 ( 105 equ)
% Maximal formula atoms : 10 ( 1 avg)
% Number of connectives : 225 ( 68 ~; 21 |; 22 &)
% ( 18 <=>; 96 =>; 0 <=; 0 <~>)
% Maximal formula depth : 17 ( 6 avg)
% Maximal term depth : 13 ( 2 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 35 ( 21 >; 14 *; 0 +; 0 <<)
% Number of predicates : 6 ( 5 usr; 1 prp; 0-3 aty)
% Number of functors : 34 ( 34 usr; 9 con; 0-5 aty)
% Number of variables : 480 ( 436 !; 11 ?; 480 :)
% ( 33 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_UNK_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:25:23
%------------------------------------------------------------------------------
%----Should-be-implicit typings (5)
tff(ty_tc_Com_Ocom,type,
com: $tType ).
tff(ty_tc_Com_Ostate,type,
state: $tType ).
tff(ty_tc_HOL_Obool,type,
bool: $tType ).
tff(ty_tc_Hoare__Mirabelle__vtrypsmcwp_Otriple,type,
hoare_28830079triple: $tType > $tType ).
tff(ty_tc_fun,type,
fun: ( $tType * $tType ) > $tType ).
%----Explicit typings (37)
tff(sy_cl_Orderings_Obot,type,
bot:
!>[A: $tType] : $o ).
tff(sy_c_COMBB,type,
combb:
!>[B: $tType,C: $tType,A: $tType] : ( fun(B,C) > fun(fun(A,B),fun(A,C)) ) ).
tff(sy_c_COMBC,type,
combc:
!>[A: $tType,B: $tType,C: $tType] : fun(fun(A,fun(B,C)),fun(B,fun(A,C))) ).
tff(sy_c_COMBK,type,
combk:
!>[A: $tType,B: $tType] : ( A > fun(B,A) ) ).
tff(sy_c_COMBS,type,
combs:
!>[A: $tType,B: $tType,C: $tType] : fun(fun(A,fun(B,C)),fun(fun(A,B),fun(A,C))) ).
tff(sy_c_Com_Ocom_OCond,type,
cond: ( fun(state,bool) * com * com ) > com ).
tff(sy_c_Com_Ocom_OSKIP,type,
skip: com ).
tff(sy_c_Com_Ocom_OSemi,type,
semi: ( com * com ) > com ).
tff(sy_c_Com_Ocom_OWhile,type,
while: ( fun(state,bool) * com ) > com ).
tff(sy_c_Fun_Ocomp,type,
comp:
!>[B: $tType,C: $tType,A: $tType] : ( ( fun(B,C) * fun(A,B) ) > fun(A,C) ) ).
tff(sy_c_HOL_OThe,type,
the:
!>[A: $tType] : ( fun(A,bool) > A ) ).
tff(sy_c_Hoare__Mirabelle__vtrypsmcwp_OMGT,type,
hoare_Mirabelle_MGT: com > hoare_28830079triple(state) ).
tff(sy_c_Hoare__Mirabelle__vtrypsmcwp_Ohoare__derivs,type,
hoare_992312373derivs:
!>[A: $tType] : ( ( fun(hoare_28830079triple(A),bool) * fun(hoare_28830079triple(A),bool) ) > $o ) ).
tff(sy_c_Hoare__Mirabelle__vtrypsmcwp_Ohoare__valids,type,
hoare_592965047valids:
!>[A: $tType] : ( ( fun(hoare_28830079triple(A),bool) * fun(hoare_28830079triple(A),bool) ) > $o ) ).
tff(sy_c_Hoare__Mirabelle__vtrypsmcwp_Opeek__and,type,
hoare_1442473487ek_and:
!>[A: $tType] : ( ( fun(A,fun(state,bool)) * fun(state,bool) ) > fun(A,fun(state,bool)) ) ).
tff(sy_c_Hoare__Mirabelle__vtrypsmcwp_Ostate__not__singleton,type,
hoare_531836380gleton: $o ).
tff(sy_c_Hoare__Mirabelle__vtrypsmcwp_Otriple_Otriple,type,
hoare_1841697145triple:
!>[A: $tType] : ( ( fun(A,fun(state,bool)) * com * fun(A,fun(state,bool)) ) > hoare_28830079triple(A) ) ).
tff(sy_c_Hoare__Mirabelle__vtrypsmcwp_Otriple_Otriple__case,type,
hoare_376461865e_case:
!>[A: $tType,T4: $tType] : ( ( fun(fun(A,fun(state,bool)),fun(com,fun(fun(A,fun(state,bool)),T4))) * hoare_28830079triple(A) ) > T4 ) ).
tff(sy_c_Hoare__Mirabelle__vtrypsmcwp_Otriple_Otriple__rec,type,
hoare_678420151le_rec:
!>[A: $tType,T4: $tType] : ( ( fun(fun(A,fun(state,bool)),fun(com,fun(fun(A,fun(state,bool)),T4))) * hoare_28830079triple(A) ) > T4 ) ).
tff(sy_c_Natural_Oevalc,type,
evalc: com > fun(state,fun(state,bool)) ).
tff(sy_c_Orderings_Obot__class_Obot,type,
bot_bot:
!>[A: $tType] : A ).
tff(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( fun(A,bool) > fun(A,bool) ) ).
tff(sy_c_Set_Oinsert,type,
insert:
!>[A: $tType] : fun(A,fun(fun(A,bool),fun(A,bool))) ).
tff(sy_c_Set_Othe__elem,type,
the_elem:
!>[A: $tType] : ( fun(A,bool) > A ) ).
tff(sy_c_aa,type,
aa:
!>[A: $tType,B: $tType] : ( ( fun(A,B) * A ) > B ) ).
tff(sy_c_fAll,type,
fAll:
!>[A: $tType] : fun(fun(A,bool),bool) ).
tff(sy_c_fFalse,type,
fFalse: bool ).
tff(sy_c_fNot,type,
fNot: fun(bool,bool) ).
tff(sy_c_fTrue,type,
fTrue: bool ).
tff(sy_c_fconj,type,
fconj: fun(bool,fun(bool,bool)) ).
tff(sy_c_fdisj,type,
fdisj: fun(bool,fun(bool,bool)) ).
tff(sy_c_fequal,type,
fequal:
!>[A: $tType] : fun(A,fun(A,bool)) ).
tff(sy_c_fimplies,type,
fimplies: fun(bool,fun(bool,bool)) ).
tff(sy_c_member,type,
member:
!>[A: $tType] : fun(A,fun(fun(A,bool),bool)) ).
tff(sy_c_pp,type,
pp: bool > $o ).
tff(sy_v_G,type,
g: fun(hoare_28830079triple(state),bool) ).
tff(sy_v_c,type,
c: com ).
%----Relevant facts (100)
tff(fact_0_triple_Oinject,axiom,
! [A: $tType,Fun22: fun(A,fun(state,bool)),Com: com,Fun13: fun(A,fun(state,bool)),Fun21: fun(A,fun(state,bool)),Com4: com,Fun12: fun(A,fun(state,bool))] :
( ( hoare_1841697145triple(A,Fun12,Com4,Fun21) = hoare_1841697145triple(A,Fun13,Com,Fun22) )
<=> ( ( Fun12 = Fun13 )
& ( Com4 = Com )
& ( Fun21 = Fun22 ) ) ) ).
tff(fact_1_empty,axiom,
! [A: $tType,Ga: fun(hoare_28830079triple(A),bool)] : hoare_992312373derivs(A,Ga,bot_bot(fun(hoare_28830079triple(A),bool))) ).
tff(fact_2_cut,axiom,
! [A: $tType,Ga: fun(hoare_28830079triple(A),bool),Ts: fun(hoare_28830079triple(A),bool),G1: fun(hoare_28830079triple(A),bool)] :
( hoare_992312373derivs(A,G1,Ts)
=> ( hoare_992312373derivs(A,Ga,G1)
=> hoare_992312373derivs(A,Ga,Ts) ) ) ).
tff(fact_3_hoare__derivs_Oinsert,axiom,
! [A: $tType,Ts: fun(hoare_28830079triple(A),bool),T: hoare_28830079triple(A),Ga: fun(hoare_28830079triple(A),bool)] :
( hoare_992312373derivs(A,Ga,aa(fun(hoare_28830079triple(A),bool),fun(hoare_28830079triple(A),bool),aa(hoare_28830079triple(A),fun(fun(hoare_28830079triple(A),bool),fun(hoare_28830079triple(A),bool)),insert(hoare_28830079triple(A)),T),bot_bot(fun(hoare_28830079triple(A),bool))))
=> ( hoare_992312373derivs(A,Ga,Ts)
=> hoare_992312373derivs(A,Ga,aa(fun(hoare_28830079triple(A),bool),fun(hoare_28830079triple(A),bool),aa(hoare_28830079triple(A),fun(fun(hoare_28830079triple(A),bool),fun(hoare_28830079triple(A),bool)),insert(hoare_28830079triple(A)),T),Ts)) ) ) ).
tff(fact_4_derivs__insertD,axiom,
! [A: $tType,Ts: fun(hoare_28830079triple(A),bool),T: hoare_28830079triple(A),Ga: fun(hoare_28830079triple(A),bool)] :
( hoare_992312373derivs(A,Ga,aa(fun(hoare_28830079triple(A),bool),fun(hoare_28830079triple(A),bool),aa(hoare_28830079triple(A),fun(fun(hoare_28830079triple(A),bool),fun(hoare_28830079triple(A),bool)),insert(hoare_28830079triple(A)),T),Ts))
=> ( hoare_992312373derivs(A,Ga,aa(fun(hoare_28830079triple(A),bool),fun(hoare_28830079triple(A),bool),aa(hoare_28830079triple(A),fun(fun(hoare_28830079triple(A),bool),fun(hoare_28830079triple(A),bool)),insert(hoare_28830079triple(A)),T),bot_bot(fun(hoare_28830079triple(A),bool))))
& hoare_992312373derivs(A,Ga,Ts) ) ) ).
tff(fact_5_constant,axiom,
! [A: $tType,Q1: fun(A,fun(state,bool)),Ca: com,P1: fun(A,fun(state,bool)),Ga: fun(hoare_28830079triple(A),bool),C5: bool] :
( ( pp(C5)
=> hoare_992312373derivs(A,Ga,aa(fun(hoare_28830079triple(A),bool),fun(hoare_28830079triple(A),bool),aa(hoare_28830079triple(A),fun(fun(hoare_28830079triple(A),bool),fun(hoare_28830079triple(A),bool)),insert(hoare_28830079triple(A)),hoare_1841697145triple(A,P1,Ca,Q1)),bot_bot(fun(hoare_28830079triple(A),bool)))) )
=> hoare_992312373derivs(A,Ga,aa(fun(hoare_28830079triple(A),bool),fun(hoare_28830079triple(A),bool),aa(hoare_28830079triple(A),fun(fun(hoare_28830079triple(A),bool),fun(hoare_28830079triple(A),bool)),insert(hoare_28830079triple(A)),hoare_1841697145triple(A,aa(bool,fun(A,fun(state,bool)),aa(fun(A,fun(bool,fun(state,bool))),fun(bool,fun(A,fun(state,bool))),combc(A,bool,fun(state,bool)),aa(fun(A,fun(state,fun(bool,bool))),fun(A,fun(bool,fun(state,bool))),combb(fun(state,fun(bool,bool)),fun(bool,fun(state,bool)),A,combc(state,bool,bool)),aa(fun(A,fun(state,bool)),fun(A,fun(state,fun(bool,bool))),combb(fun(state,bool),fun(state,fun(bool,bool)),A,combb(bool,fun(bool,bool),state,fconj)),P1))),C5),Ca,Q1)),bot_bot(fun(hoare_28830079triple(A),bool)))) ) ).
tff(fact_6_MGT__alternI,axiom,
! [Ca: com,Ga: fun(hoare_28830079triple(state),bool)] :
( hoare_992312373derivs(state,Ga,aa(fun(hoare_28830079triple(state),bool),fun(hoare_28830079triple(state),bool),aa(hoare_28830079triple(state),fun(fun(hoare_28830079triple(state),bool),fun(hoare_28830079triple(state),bool)),insert(hoare_28830079triple(state)),hoare_Mirabelle_MGT(Ca)),bot_bot(fun(hoare_28830079triple(state),bool))))
=> hoare_992312373derivs(state,Ga,aa(fun(hoare_28830079triple(state),bool),fun(hoare_28830079triple(state),bool),aa(hoare_28830079triple(state),fun(fun(hoare_28830079triple(state),bool),fun(hoare_28830079triple(state),bool)),insert(hoare_28830079triple(state)),hoare_1841697145triple(state,aa(fun(state,fun(state,fun(state,bool))),fun(state,fun(state,bool)),combb(fun(state,fun(state,bool)),fun(state,bool),state,combb(fun(state,bool),bool,state,fAll(state))),aa(fun(state,fun(state,bool)),fun(state,fun(state,fun(state,bool))),combb(fun(state,bool),fun(state,fun(state,bool)),state,aa(fun(state,fun(fun(state,bool),fun(state,bool))),fun(fun(state,bool),fun(state,fun(state,bool))),combc(state,fun(state,bool),fun(state,bool)),aa(fun(state,fun(state,fun(bool,bool))),fun(state,fun(fun(state,bool),fun(state,bool))),combb(fun(state,fun(bool,bool)),fun(fun(state,bool),fun(state,bool)),state,combs(state,bool,bool)),aa(fun(state,fun(state,bool)),fun(state,fun(state,fun(bool,bool))),combb(fun(state,bool),fun(state,fun(bool,bool)),state,combb(bool,fun(bool,bool),state,fimplies)),evalc(Ca))))),fequal(state))),Ca,fequal(state))),bot_bot(fun(hoare_28830079triple(state),bool)))) ) ).
tff(fact_7_singleton__conv2,axiom,
! [A: $tType,A2: A] : collect(A,aa(A,fun(A,bool),fequal(A),A2)) = aa(fun(A,bool),fun(A,bool),aa(A,fun(fun(A,bool),fun(A,bool)),insert(A),A2),bot_bot(fun(A,bool))) ).
tff(fact_8_singleton__conv,axiom,
! [A: $tType,A2: A] : collect(A,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),combc(A,A,bool),fequal(A)),A2)) = aa(fun(A,bool),fun(A,bool),aa(A,fun(fun(A,bool),fun(A,bool)),insert(A),A2),bot_bot(fun(A,bool))) ).
tff(fact_9_escape,axiom,
! [A: $tType,Q1: fun(A,fun(state,bool)),Ca: com,Ga: fun(hoare_28830079triple(A),bool),P1: fun(A,fun(state,bool))] :
( ! [Z: A,S2: state] :
( pp(aa(state,bool,aa(A,fun(state,bool),P1,Z),S2))
=> hoare_992312373derivs(A,Ga,aa(fun(hoare_28830079triple(A),bool),fun(hoare_28830079triple(A),bool),aa(hoare_28830079triple(A),fun(fun(hoare_28830079triple(A),bool),fun(hoare_28830079triple(A),bool)),insert(hoare_28830079triple(A)),hoare_1841697145triple(A,combk(fun(state,bool),A,aa(state,fun(state,bool),aa(fun(state,fun(state,bool)),fun(state,fun(state,bool)),combc(state,state,bool),fequal(state)),S2)),Ca,combk(fun(state,bool),A,aa(A,fun(state,bool),Q1,Z)))),bot_bot(fun(hoare_28830079triple(A),bool)))) )
=> hoare_992312373derivs(A,Ga,aa(fun(hoare_28830079triple(A),bool),fun(hoare_28830079triple(A),bool),aa(hoare_28830079triple(A),fun(fun(hoare_28830079triple(A),bool),fun(hoare_28830079triple(A),bool)),insert(hoare_28830079triple(A)),hoare_1841697145triple(A,P1,Ca,Q1)),bot_bot(fun(hoare_28830079triple(A),bool)))) ) ).
tff(fact_10_conseq2,axiom,
! [A: $tType,Q1: fun(A,fun(state,bool)),Q3: fun(A,fun(state,bool)),Ca: com,P1: fun(A,fun(state,bool)),Ga: fun(hoare_28830079triple(A),bool)] :
( hoare_992312373derivs(A,Ga,aa(fun(hoare_28830079triple(A),bool),fun(hoare_28830079triple(A),bool),aa(hoare_28830079triple(A),fun(fun(hoare_28830079triple(A),bool),fun(hoare_28830079triple(A),bool)),insert(hoare_28830079triple(A)),hoare_1841697145triple(A,P1,Ca,Q3)),bot_bot(fun(hoare_28830079triple(A),bool))))
=> ( ! [Z: A,S2: state] :
( pp(aa(state,bool,aa(A,fun(state,bool),Q3,Z),S2))
=> pp(aa(state,bool,aa(A,fun(state,bool),Q1,Z),S2)) )
=> hoare_992312373derivs(A,Ga,aa(fun(hoare_28830079triple(A),bool),fun(hoare_28830079triple(A),bool),aa(hoare_28830079triple(A),fun(fun(hoare_28830079triple(A),bool),fun(hoare_28830079triple(A),bool)),insert(hoare_28830079triple(A)),hoare_1841697145triple(A,P1,Ca,Q1)),bot_bot(fun(hoare_28830079triple(A),bool)))) ) ) ).
tff(fact_11_conseq1,axiom,
! [A: $tType,P1: fun(A,fun(state,bool)),Q1: fun(A,fun(state,bool)),Ca: com,P4: fun(A,fun(state,bool)),Ga: fun(hoare_28830079triple(A),bool)] :
( hoare_992312373derivs(A,Ga,aa(fun(hoare_28830079triple(A),bool),fun(hoare_28830079triple(A),bool),aa(hoare_28830079triple(A),fun(fun(hoare_28830079triple(A),bool),fun(hoare_28830079triple(A),bool)),insert(hoare_28830079triple(A)),hoare_1841697145triple(A,P4,Ca,Q1)),bot_bot(fun(hoare_28830079triple(A),bool))))
=> ( ! [Z: A,S2: state] :
( pp(aa(state,bool,aa(A,fun(state,bool),P1,Z),S2))
=> pp(aa(state,bool,aa(A,fun(state,bool),P4,Z),S2)) )
=> hoare_992312373derivs(A,Ga,aa(fun(hoare_28830079triple(A),bool),fun(hoare_28830079triple(A),bool),aa(hoare_28830079triple(A),fun(fun(hoare_28830079triple(A),bool),fun(hoare_28830079triple(A),bool)),insert(hoare_28830079triple(A)),hoare_1841697145triple(A,P1,Ca,Q1)),bot_bot(fun(hoare_28830079triple(A),bool)))) ) ) ).
tff(fact_12_MGT__def,axiom,
! [Ca: com] : hoare_Mirabelle_MGT(Ca) = hoare_1841697145triple(state,fequal(state),Ca,evalc(Ca)) ).
tff(fact_13_state__not__singleton__def,axiom,
( hoare_531836380gleton
<=> ? [S5: state,T3: state] : S5 != T3 ) ).
tff(fact_14_single__stateE,axiom,
( hoare_531836380gleton
=> ! [T2: state] :
~ ! [S2: state] : S2 = T2 ) ).
tff(fact_15_insert__absorb2,axiom,
! [A: $tType,A1: fun(A,bool),X3: A] : aa(fun(A,bool),fun(A,bool),aa(A,fun(fun(A,bool),fun(A,bool)),insert(A),X3),aa(fun(A,bool),fun(A,bool),aa(A,fun(fun(A,bool),fun(A,bool)),insert(A),X3),A1)) = aa(fun(A,bool),fun(A,bool),aa(A,fun(fun(A,bool),fun(A,bool)),insert(A),X3),A1) ).
tff(fact_16_insert__iff,axiom,
! [A: $tType,A1: fun(A,bool),B1: A,A2: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),A2),aa(fun(A,bool),fun(A,bool),aa(A,fun(fun(A,bool),fun(A,bool)),insert(A),B1),A1)))
<=> ( ( A2 = B1 )
| pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),A2),A1)) ) ) ).
tff(fact_17_insertE,axiom,
! [A: $tType,A1: fun(A,bool),B1: A,A2: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),A2),aa(fun(A,bool),fun(A,bool),aa(A,fun(fun(A,bool),fun(A,bool)),insert(A),B1),A1)))
=> ( ( A2 != B1 )
=> pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),A2),A1)) ) ) ).
tff(fact_18_emptyE,axiom,
! [A: $tType,A2: A] : ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),A2),bot_bot(fun(A,bool)))) ).
tff(fact_19_Collect__empty__eq,axiom,
! [A: $tType,P1: fun(A,bool)] :
( ( collect(A,P1) = bot_bot(fun(A,bool)) )
<=> ! [X2: A] : ~ pp(aa(A,bool,P1,X2)) ) ).
tff(fact_20_empty__iff,axiom,
! [A: $tType,Ca: A] : ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Ca),bot_bot(fun(A,bool)))) ).
tff(fact_21_empty__Collect__eq,axiom,
! [A: $tType,P1: fun(A,bool)] :
( ( bot_bot(fun(A,bool)) = collect(A,P1) )
<=> ! [X2: A] : ~ pp(aa(A,bool,P1,X2)) ) ).
tff(fact_22_all__not__in__conv,axiom,
! [A: $tType,A1: fun(A,bool)] :
( ! [X2: A] : ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X2),A1))
<=> ( A1 = bot_bot(fun(A,bool)) ) ) ).
tff(fact_23_insertCI,axiom,
! [A: $tType,B1: A,B4: fun(A,bool),A2: A] :
( ( ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),A2),B4))
=> ( A2 = B1 ) )
=> pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),A2),aa(fun(A,bool),fun(A,bool),aa(A,fun(fun(A,bool),fun(A,bool)),insert(A),B1),B4))) ) ).
tff(fact_24_equals0D,axiom,
! [A: $tType,A2: A,A1: fun(A,bool)] :
( ( A1 = bot_bot(fun(A,bool)) )
=> ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),A2),A1)) ) ).
tff(fact_25_ex__in__conv,axiom,
! [A: $tType,A1: fun(A,bool)] :
( ? [X2: A] : pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X2),A1))
<=> ( A1 != bot_bot(fun(A,bool)) ) ) ).
tff(fact_26_empty__def,axiom,
! [A: $tType] : bot_bot(fun(A,bool)) = collect(A,combk(bool,A,fFalse)) ).
tff(fact_27_insert__absorb,axiom,
! [A: $tType,A1: fun(A,bool),A2: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),A2),A1))
=> ( aa(fun(A,bool),fun(A,bool),aa(A,fun(fun(A,bool),fun(A,bool)),insert(A),A2),A1) = A1 ) ) ).
tff(fact_28_insertI2,axiom,
! [A: $tType,B1: A,B4: fun(A,bool),A2: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),A2),B4))
=> pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),A2),aa(fun(A,bool),fun(A,bool),aa(A,fun(fun(A,bool),fun(A,bool)),insert(A),B1),B4))) ) ).
tff(fact_29_insert__eq__iff,axiom,
! [A: $tType,B4: fun(A,bool),B1: A,A1: fun(A,bool),A2: A] :
( ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),A2),A1))
=> ( ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),B1),B4))
=> ( ( aa(fun(A,bool),fun(A,bool),aa(A,fun(fun(A,bool),fun(A,bool)),insert(A),A2),A1) = aa(fun(A,bool),fun(A,bool),aa(A,fun(fun(A,bool),fun(A,bool)),insert(A),B1),B4) )
<=> ( ( ( A2 = B1 )
=> ( A1 = B4 ) )
& ( ( A2 != B1 )
=> ? [C4: fun(A,bool)] :
( ( A1 = aa(fun(A,bool),fun(A,bool),aa(A,fun(fun(A,bool),fun(A,bool)),insert(A),B1),C4) )
& ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),B1),C4))
& ( B4 = aa(fun(A,bool),fun(A,bool),aa(A,fun(fun(A,bool),fun(A,bool)),insert(A),A2),C4) )
& ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),A2),C4)) ) ) ) ) ) ) ).
tff(fact_30_insert__ident,axiom,
! [A: $tType,B4: fun(A,bool),A1: fun(A,bool),X3: A] :
( ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X3),A1))
=> ( ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X3),B4))
=> ( ( aa(fun(A,bool),fun(A,bool),aa(A,fun(fun(A,bool),fun(A,bool)),insert(A),X3),A1) = aa(fun(A,bool),fun(A,bool),aa(A,fun(fun(A,bool),fun(A,bool)),insert(A),X3),B4) )
<=> ( A1 = B4 ) ) ) ) ).
tff(fact_31_insert__code,axiom,
! [A: $tType,X3: A,A1: fun(A,bool),Y2: A] :
( pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aa(A,fun(fun(A,bool),fun(A,bool)),insert(A),Y2),A1),X3))
<=> ( ( Y2 = X3 )
| pp(aa(A,bool,A1,X3)) ) ) ).
tff(fact_32_insert__commute,axiom,
! [A: $tType,A1: fun(A,bool),Y2: A,X3: A] : aa(fun(A,bool),fun(A,bool),aa(A,fun(fun(A,bool),fun(A,bool)),insert(A),X3),aa(fun(A,bool),fun(A,bool),aa(A,fun(fun(A,bool),fun(A,bool)),insert(A),Y2),A1)) = aa(fun(A,bool),fun(A,bool),aa(A,fun(fun(A,bool),fun(A,bool)),insert(A),Y2),aa(fun(A,bool),fun(A,bool),aa(A,fun(fun(A,bool),fun(A,bool)),insert(A),X3),A1)) ).
tff(fact_33_insert__Collect,axiom,
! [A: $tType,P1: fun(A,bool),A2: A] : aa(fun(A,bool),fun(A,bool),aa(A,fun(fun(A,bool),fun(A,bool)),insert(A),A2),collect(A,P1)) = collect(A,aa(fun(A,bool),fun(A,bool),aa(fun(A,fun(bool,bool)),fun(fun(A,bool),fun(A,bool)),combs(A,bool,bool),aa(fun(A,bool),fun(A,fun(bool,bool)),combb(bool,fun(bool,bool),A,fimplies),aa(fun(A,bool),fun(A,bool),combb(bool,bool,A,fNot),aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),combc(A,A,bool),fequal(A)),A2)))),P1)) ).
tff(fact_34_insert__compr,axiom,
! [A: $tType,B4: fun(A,bool),A2: A] : aa(fun(A,bool),fun(A,bool),aa(A,fun(fun(A,bool),fun(A,bool)),insert(A),A2),B4) = collect(A,aa(fun(A,bool),fun(A,bool),aa(fun(A,fun(bool,bool)),fun(fun(A,bool),fun(A,bool)),combs(A,bool,bool),aa(fun(A,bool),fun(A,fun(bool,bool)),combb(bool,fun(bool,bool),A,fdisj),aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),combc(A,A,bool),fequal(A)),A2))),aa(fun(A,bool),fun(A,bool),aa(fun(A,fun(fun(A,bool),bool)),fun(fun(A,bool),fun(A,bool)),combc(A,fun(A,bool),bool),member(A)),B4))) ).
tff(fact_35_insertI1,axiom,
! [A: $tType,B4: fun(A,bool),A2: A] : pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),A2),aa(fun(A,bool),fun(A,bool),aa(A,fun(fun(A,bool),fun(A,bool)),insert(A),A2),B4))) ).
tff(fact_36_insert__compr__raw,axiom,
! [A: $tType,X1: A,Xa: fun(A,bool)] : aa(fun(A,bool),fun(A,bool),aa(A,fun(fun(A,bool),fun(A,bool)),insert(A),X1),Xa) = collect(A,aa(fun(A,bool),fun(A,bool),aa(fun(A,fun(bool,bool)),fun(fun(A,bool),fun(A,bool)),combs(A,bool,bool),aa(fun(A,bool),fun(A,fun(bool,bool)),combb(bool,fun(bool,bool),A,fdisj),aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),combc(A,A,bool),fequal(A)),X1))),aa(fun(A,bool),fun(A,bool),aa(fun(A,fun(fun(A,bool),bool)),fun(fun(A,bool),fun(A,bool)),combc(A,fun(A,bool),bool),member(A)),Xa))) ).
tff(fact_37_singleton__inject,axiom,
! [A: $tType,B1: A,A2: A] :
( ( aa(fun(A,bool),fun(A,bool),aa(A,fun(fun(A,bool),fun(A,bool)),insert(A),A2),bot_bot(fun(A,bool))) = aa(fun(A,bool),fun(A,bool),aa(A,fun(fun(A,bool),fun(A,bool)),insert(A),B1),bot_bot(fun(A,bool))) )
=> ( A2 = B1 ) ) ).
tff(fact_38_singletonE,axiom,
! [A: $tType,A2: A,B1: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),B1),aa(fun(A,bool),fun(A,bool),aa(A,fun(fun(A,bool),fun(A,bool)),insert(A),A2),bot_bot(fun(A,bool)))))
=> ( B1 = A2 ) ) ).
tff(fact_39_doubleton__eq__iff,axiom,
! [A: $tType,D: A,Ca: A,B1: A,A2: A] :
( ( aa(fun(A,bool),fun(A,bool),aa(A,fun(fun(A,bool),fun(A,bool)),insert(A),A2),aa(fun(A,bool),fun(A,bool),aa(A,fun(fun(A,bool),fun(A,bool)),insert(A),B1),bot_bot(fun(A,bool)))) = aa(fun(A,bool),fun(A,bool),aa(A,fun(fun(A,bool),fun(A,bool)),insert(A),Ca),aa(fun(A,bool),fun(A,bool),aa(A,fun(fun(A,bool),fun(A,bool)),insert(A),D),bot_bot(fun(A,bool)))) )
<=> ( ( ( A2 = Ca )
& ( B1 = D ) )
| ( ( A2 = D )
& ( B1 = Ca ) ) ) ) ).
tff(fact_40_singleton__iff,axiom,
! [A: $tType,A2: A,B1: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),B1),aa(fun(A,bool),fun(A,bool),aa(A,fun(fun(A,bool),fun(A,bool)),insert(A),A2),bot_bot(fun(A,bool)))))
<=> ( B1 = A2 ) ) ).
tff(fact_41_insert__not__empty,axiom,
! [A: $tType,A1: fun(A,bool),A2: A] : aa(fun(A,bool),fun(A,bool),aa(A,fun(fun(A,bool),fun(A,bool)),insert(A),A2),A1) != bot_bot(fun(A,bool)) ).
tff(fact_42_empty__not__insert,axiom,
! [A: $tType,A1: fun(A,bool),A2: A] : bot_bot(fun(A,bool)) != aa(fun(A,bool),fun(A,bool),aa(A,fun(fun(A,bool),fun(A,bool)),insert(A),A2),A1) ).
tff(fact_43_Collect__conv__if,axiom,
! [A: $tType,A2: A,P1: fun(A,bool)] :
( ( pp(aa(A,bool,P1,A2))
=> ( collect(A,aa(fun(A,bool),fun(A,bool),aa(fun(A,fun(bool,bool)),fun(fun(A,bool),fun(A,bool)),combs(A,bool,bool),aa(fun(A,bool),fun(A,fun(bool,bool)),combb(bool,fun(bool,bool),A,fconj),aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),combc(A,A,bool),fequal(A)),A2))),P1)) = aa(fun(A,bool),fun(A,bool),aa(A,fun(fun(A,bool),fun(A,bool)),insert(A),A2),bot_bot(fun(A,bool))) ) )
& ( ~ pp(aa(A,bool,P1,A2))
=> ( collect(A,aa(fun(A,bool),fun(A,bool),aa(fun(A,fun(bool,bool)),fun(fun(A,bool),fun(A,bool)),combs(A,bool,bool),aa(fun(A,bool),fun(A,fun(bool,bool)),combb(bool,fun(bool,bool),A,fconj),aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),combc(A,A,bool),fequal(A)),A2))),P1)) = bot_bot(fun(A,bool)) ) ) ) ).
tff(fact_44_Collect__conv__if2,axiom,
! [A: $tType,A2: A,P1: fun(A,bool)] :
( ( pp(aa(A,bool,P1,A2))
=> ( collect(A,aa(fun(A,bool),fun(A,bool),aa(fun(A,fun(bool,bool)),fun(fun(A,bool),fun(A,bool)),combs(A,bool,bool),aa(fun(A,bool),fun(A,fun(bool,bool)),combb(bool,fun(bool,bool),A,fconj),aa(A,fun(A,bool),fequal(A),A2))),P1)) = aa(fun(A,bool),fun(A,bool),aa(A,fun(fun(A,bool),fun(A,bool)),insert(A),A2),bot_bot(fun(A,bool))) ) )
& ( ~ pp(aa(A,bool,P1,A2))
=> ( collect(A,aa(fun(A,bool),fun(A,bool),aa(fun(A,fun(bool,bool)),fun(fun(A,bool),fun(A,bool)),combs(A,bool,bool),aa(fun(A,bool),fun(A,fun(bool,bool)),combb(bool,fun(bool,bool),A,fconj),aa(A,fun(A,bool),fequal(A),A2))),P1)) = bot_bot(fun(A,bool)) ) ) ) ).
tff(fact_45_conseq12,axiom,
! [A: $tType,Q1: fun(A,fun(state,bool)),P1: fun(A,fun(state,bool)),Q3: fun(A,fun(state,bool)),Ca: com,P4: fun(A,fun(state,bool)),Ga: fun(hoare_28830079triple(A),bool)] :
( hoare_992312373derivs(A,Ga,aa(fun(hoare_28830079triple(A),bool),fun(hoare_28830079triple(A),bool),aa(hoare_28830079triple(A),fun(fun(hoare_28830079triple(A),bool),fun(hoare_28830079triple(A),bool)),insert(hoare_28830079triple(A)),hoare_1841697145triple(A,P4,Ca,Q3)),bot_bot(fun(hoare_28830079triple(A),bool))))
=> ( ! [Z: A,S2: state] :
( pp(aa(state,bool,aa(A,fun(state,bool),P1,Z),S2))
=> ! [S3: state] :
( ! [Z1: A] :
( pp(aa(state,bool,aa(A,fun(state,bool),P4,Z1),S2))
=> pp(aa(state,bool,aa(A,fun(state,bool),Q3,Z1),S3)) )
=> pp(aa(state,bool,aa(A,fun(state,bool),Q1,Z),S3)) ) )
=> hoare_992312373derivs(A,Ga,aa(fun(hoare_28830079triple(A),bool),fun(hoare_28830079triple(A),bool),aa(hoare_28830079triple(A),fun(fun(hoare_28830079triple(A),bool),fun(hoare_28830079triple(A),bool)),insert(hoare_28830079triple(A)),hoare_1841697145triple(A,P1,Ca,Q1)),bot_bot(fun(hoare_28830079triple(A),bool)))) ) ) ).
tff(fact_46_the__elem__eq,axiom,
! [A: $tType,X3: A] : the_elem(A,aa(fun(A,bool),fun(A,bool),aa(A,fun(fun(A,bool),fun(A,bool)),insert(A),X3),bot_bot(fun(A,bool)))) = X3 ).
tff(fact_47_triple_Osimps_I2_J,axiom,
! [A: $tType,B: $tType,Fun21: fun(B,fun(state,bool)),Com4: com,Fun12: fun(B,fun(state,bool)),F1: fun(fun(B,fun(state,bool)),fun(com,fun(fun(B,fun(state,bool)),A)))] : hoare_376461865e_case(B,A,F1,hoare_1841697145triple(B,Fun12,Com4,Fun21)) = aa(fun(B,fun(state,bool)),A,aa(com,fun(fun(B,fun(state,bool)),A),aa(fun(B,fun(state,bool)),fun(com,fun(fun(B,fun(state,bool)),A)),F1,Fun12),Com4),Fun21) ).
tff(fact_48_triple_Orecs,axiom,
! [A: $tType,B: $tType,Fun21: fun(B,fun(state,bool)),Com4: com,Fun12: fun(B,fun(state,bool)),F1: fun(fun(B,fun(state,bool)),fun(com,fun(fun(B,fun(state,bool)),A)))] : hoare_678420151le_rec(B,A,F1,hoare_1841697145triple(B,Fun12,Com4,Fun21)) = aa(fun(B,fun(state,bool)),A,aa(com,fun(fun(B,fun(state,bool)),A),aa(fun(B,fun(state,bool)),fun(com,fun(fun(B,fun(state,bool)),A)),F1,Fun12),Com4),Fun21) ).
tff(fact_49_bot__apply,axiom,
! [B: $tType,A: $tType] :
( bot(A)
=> ! [X3: B] : aa(B,A,bot_bot(fun(B,A)),X3) = bot_bot(A) ) ).
tff(fact_50_bot__fun__def,axiom,
! [A: $tType,B: $tType] :
( bot(B)
=> ! [X1: A] : aa(A,B,bot_bot(fun(A,B)),X1) = bot_bot(B) ) ).
tff(fact_51_hoare__derivs_OSkip,axiom,
! [A: $tType,P1: fun(A,fun(state,bool)),Ga: fun(hoare_28830079triple(A),bool)] : hoare_992312373derivs(A,Ga,aa(fun(hoare_28830079triple(A),bool),fun(hoare_28830079triple(A),bool),aa(hoare_28830079triple(A),fun(fun(hoare_28830079triple(A),bool),fun(hoare_28830079triple(A),bool)),insert(hoare_28830079triple(A)),hoare_1841697145triple(A,P1,skip,P1)),bot_bot(fun(hoare_28830079triple(A),bool)))) ).
tff(fact_52_LoopF,axiom,
! [A: $tType,Ca: com,B1: fun(state,bool),P1: fun(A,fun(state,bool)),Ga: fun(hoare_28830079triple(A),bool)] : hoare_992312373derivs(A,Ga,aa(fun(hoare_28830079triple(A),bool),fun(hoare_28830079triple(A),bool),aa(hoare_28830079triple(A),fun(fun(hoare_28830079triple(A),bool),fun(hoare_28830079triple(A),bool)),insert(hoare_28830079triple(A)),hoare_1841697145triple(A,aa(fun(state,bool),fun(A,fun(state,bool)),aa(fun(A,fun(fun(state,bool),fun(state,bool))),fun(fun(state,bool),fun(A,fun(state,bool))),combc(A,fun(state,bool),fun(state,bool)),aa(fun(A,fun(state,fun(bool,bool))),fun(A,fun(fun(state,bool),fun(state,bool))),combb(fun(state,fun(bool,bool)),fun(fun(state,bool),fun(state,bool)),A,combs(state,bool,bool)),aa(fun(A,fun(state,bool)),fun(A,fun(state,fun(bool,bool))),combb(fun(state,bool),fun(state,fun(bool,bool)),A,combb(bool,fun(bool,bool),state,fconj)),P1))),aa(fun(state,bool),fun(state,bool),combb(bool,bool,state,fNot),B1)),while(B1,Ca),P1)),bot_bot(fun(hoare_28830079triple(A),bool)))) ).
tff(fact_53_evalc_OSkip,axiom,
! [S4: state] : pp(aa(state,bool,aa(state,fun(state,bool),evalc(skip),S4),S4)) ).
tff(fact_54_evalc__elim__cases_I1_J,axiom,
! [T1: state,S4: state] :
( pp(aa(state,bool,aa(state,fun(state,bool),evalc(skip),S4),T1))
=> ( T1 = S4 ) ) ).
tff(fact_55_evalc_OWhileFalse,axiom,
! [Ca: com,S: state,B1: fun(state,bool)] :
( ~ pp(aa(state,bool,B1,S))
=> pp(aa(state,bool,aa(state,fun(state,bool),evalc(while(B1,Ca)),S),S)) ) ).
tff(fact_56_evalc_OWhileTrue,axiom,
! [S22: state,S1: state,Ca: com,S01: state,B1: fun(state,bool)] :
( pp(aa(state,bool,B1,S01))
=> ( pp(aa(state,bool,aa(state,fun(state,bool),evalc(Ca),S01),S1))
=> ( pp(aa(state,bool,aa(state,fun(state,bool),evalc(while(B1,Ca)),S1),S22))
=> pp(aa(state,bool,aa(state,fun(state,bool),evalc(while(B1,Ca)),S01),S22)) ) ) ) ).
tff(fact_57_com__det,axiom,
! [U: state,T1: state,S4: state,C3: com] :
( pp(aa(state,bool,aa(state,fun(state,bool),evalc(C3),S4),T1))
=> ( pp(aa(state,bool,aa(state,fun(state,bool),evalc(C3),S4),U))
=> ( U = T1 ) ) ) ).
tff(fact_58_com_Osimps_I5_J,axiom,
! [Com: com,Fun: fun(state,bool),Com4: com,Fun1: fun(state,bool)] :
( ( while(Fun1,Com4) = while(Fun,Com) )
<=> ( ( Fun1 = Fun )
& ( Com4 = Com ) ) ) ).
tff(fact_59_evalc__WHILE__case,axiom,
! [T: state,S: state,Ca: com,B1: fun(state,bool)] :
( pp(aa(state,bool,aa(state,fun(state,bool),evalc(while(B1,Ca)),S),T))
=> ( ( ( T = S )
=> pp(aa(state,bool,B1,S)) )
=> ~ ( pp(aa(state,bool,B1,S))
=> ! [S11: state] :
( pp(aa(state,bool,aa(state,fun(state,bool),evalc(Ca),S),S11))
=> ~ pp(aa(state,bool,aa(state,fun(state,bool),evalc(while(B1,Ca)),S11),T)) ) ) ) ) ).
tff(fact_60_com_Osimps_I16_J,axiom,
! [Com: com,Fun: fun(state,bool)] : skip != while(Fun,Com) ).
tff(fact_61_com_Osimps_I17_J,axiom,
! [Com: com,Fun: fun(state,bool)] : while(Fun,Com) != skip ).
tff(fact_62_Comp,axiom,
! [A: $tType,R1: fun(A,fun(state,bool)),D: com,Q1: fun(A,fun(state,bool)),Ca: com,P1: fun(A,fun(state,bool)),Ga: fun(hoare_28830079triple(A),bool)] :
( hoare_992312373derivs(A,Ga,aa(fun(hoare_28830079triple(A),bool),fun(hoare_28830079triple(A),bool),aa(hoare_28830079triple(A),fun(fun(hoare_28830079triple(A),bool),fun(hoare_28830079triple(A),bool)),insert(hoare_28830079triple(A)),hoare_1841697145triple(A,P1,Ca,Q1)),bot_bot(fun(hoare_28830079triple(A),bool))))
=> ( hoare_992312373derivs(A,Ga,aa(fun(hoare_28830079triple(A),bool),fun(hoare_28830079triple(A),bool),aa(hoare_28830079triple(A),fun(fun(hoare_28830079triple(A),bool),fun(hoare_28830079triple(A),bool)),insert(hoare_28830079triple(A)),hoare_1841697145triple(A,Q1,D,R1)),bot_bot(fun(hoare_28830079triple(A),bool))))
=> hoare_992312373derivs(A,Ga,aa(fun(hoare_28830079triple(A),bool),fun(hoare_28830079triple(A),bool),aa(hoare_28830079triple(A),fun(fun(hoare_28830079triple(A),bool),fun(hoare_28830079triple(A),bool)),insert(hoare_28830079triple(A)),hoare_1841697145triple(A,P1,semi(Ca,D),R1)),bot_bot(fun(hoare_28830079triple(A),bool)))) ) ) ).
tff(fact_63_the__elem__def,axiom,
! [A: $tType,X5: fun(A,bool)] : the_elem(A,X5) = the(A,aa(fun(A,fun(A,bool)),fun(A,bool),combb(fun(A,bool),bool,A,aa(fun(A,bool),fun(fun(A,bool),bool),fequal(fun(A,bool)),X5)),aa(fun(A,bool),fun(A,fun(A,bool)),aa(fun(A,fun(fun(A,bool),fun(A,bool))),fun(fun(A,bool),fun(A,fun(A,bool))),combc(A,fun(A,bool),fun(A,bool)),insert(A)),bot_bot(fun(A,bool))))) ).
tff(fact_64_triple_Oexhaust,axiom,
! [A: $tType,Y2: hoare_28830079triple(A)] :
~ ! [Fun11: fun(A,fun(state,bool)),Com3: com,Fun2: fun(A,fun(state,bool))] : Y2 != hoare_1841697145triple(A,Fun11,Com3,Fun2) ).
tff(fact_65_com_Osimps_I3_J,axiom,
! [Com21: com,Com11: com,Com2: com,Com1: com] :
( ( semi(Com1,Com2) = semi(Com11,Com21) )
<=> ( ( Com1 = Com11 )
& ( Com2 = Com21 ) ) ) ).
tff(fact_66_evalc_OSemi,axiom,
! [S21: state,C11: com,S12: state,S0: state,C01: com] :
( pp(aa(state,bool,aa(state,fun(state,bool),evalc(C01),S0),S12))
=> ( pp(aa(state,bool,aa(state,fun(state,bool),evalc(C11),S12),S21))
=> pp(aa(state,bool,aa(state,fun(state,bool),evalc(semi(C01,C11)),S0),S21)) ) ) ).
tff(fact_67_com_Osimps_I12_J,axiom,
! [Com22: com,Com12: com] : skip != semi(Com12,Com22) ).
tff(fact_68_com_Osimps_I13_J,axiom,
! [Com22: com,Com12: com] : semi(Com12,Com22) != skip ).
tff(fact_69_com_Osimps_I46_J,axiom,
! [Com: com,Fun: fun(state,bool),Com2: com,Com1: com] : semi(Com1,Com2) != while(Fun,Com) ).
tff(fact_70_com_Osimps_I47_J,axiom,
! [Com2: com,Com1: com,Com: com,Fun: fun(state,bool)] : while(Fun,Com) != semi(Com1,Com2) ).
tff(fact_71_evalc__elim__cases_I4_J,axiom,
! [T1: state,S4: state,C21: com,C11: com] :
( pp(aa(state,bool,aa(state,fun(state,bool),evalc(semi(C11,C21)),S4),T1))
=> ~ ! [S11: state] :
( pp(aa(state,bool,aa(state,fun(state,bool),evalc(C11),S4),S11))
=> ~ pp(aa(state,bool,aa(state,fun(state,bool),evalc(C21),S11),T1)) ) ) ).
tff(fact_72_the__sym__eq__trivial,axiom,
! [A: $tType,X3: A] : the(A,aa(A,fun(A,bool),fequal(A),X3)) = X3 ).
tff(fact_73_the__eq__trivial,axiom,
! [A: $tType,A2: A] : the(A,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),combc(A,A,bool),fequal(A)),A2)) = A2 ).
tff(fact_74_If__def,axiom,
! [A: $tType,Y2: A,X3: A,P1: bool] :
( ( pp(P1)
=> ( X3 = the(A,aa(fun(A,bool),fun(A,bool),aa(fun(A,fun(bool,bool)),fun(fun(A,bool),fun(A,bool)),combs(A,bool,bool),aa(fun(A,bool),fun(A,fun(bool,bool)),combb(bool,fun(bool,bool),A,fconj),aa(fun(A,bool),fun(A,bool),combb(bool,bool,A,aa(bool,fun(bool,bool),fimplies,P1)),aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),combc(A,A,bool),fequal(A)),X3)))),aa(fun(A,bool),fun(A,bool),combb(bool,bool,A,aa(bool,fun(bool,bool),fimplies,aa(bool,bool,fNot,P1))),aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),combc(A,A,bool),fequal(A)),Y2)))) ) )
& ( ~ pp(P1)
=> ( Y2 = the(A,aa(fun(A,bool),fun(A,bool),aa(fun(A,fun(bool,bool)),fun(fun(A,bool),fun(A,bool)),combs(A,bool,bool),aa(fun(A,bool),fun(A,fun(bool,bool)),combb(bool,fun(bool,bool),A,fconj),aa(fun(A,bool),fun(A,bool),combb(bool,bool,A,aa(bool,fun(bool,bool),fimplies,P1)),aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),combc(A,A,bool),fequal(A)),X3)))),aa(fun(A,bool),fun(A,bool),combb(bool,bool,A,aa(bool,fun(bool,bool),fimplies,aa(bool,bool,fNot,P1))),aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),combc(A,A,bool),fequal(A)),Y2)))) ) ) ) ).
tff(fact_75_ext,axiom,
! [B: $tType,A: $tType,G: fun(A,B),F: fun(A,B)] :
( ! [X4: A] : aa(A,B,F,X4) = aa(A,B,G,X4)
=> ( F = G ) ) ).
tff(fact_76_mem__def,axiom,
! [A: $tType,A1: fun(A,bool),X3: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X3),A1))
<=> pp(aa(A,bool,A1,X3)) ) ).
tff(fact_77_Collect__def,axiom,
! [A: $tType,P1: fun(A,bool)] : collect(A,P1) = P1 ).
tff(fact_78_the__equality,axiom,
! [A: $tType,A2: A,P1: fun(A,bool)] :
( pp(aa(A,bool,P1,A2))
=> ( ! [X4: A] :
( pp(aa(A,bool,P1,X4))
=> ( X4 = A2 ) )
=> ( the(A,P1) = A2 ) ) ) ).
tff(fact_79_theI,axiom,
! [A: $tType,A2: A,P1: fun(A,bool)] :
( pp(aa(A,bool,P1,A2))
=> ( ! [X4: A] :
( pp(aa(A,bool,P1,X4))
=> ( X4 = A2 ) )
=> pp(aa(A,bool,P1,the(A,P1))) ) ) ).
tff(fact_80_theI_H,axiom,
! [A: $tType,P1: fun(A,bool)] :
( ? [X1: A] :
( pp(aa(A,bool,P1,X1))
& ! [Y1: A] :
( pp(aa(A,bool,P1,Y1))
=> ( Y1 = X1 ) ) )
=> pp(aa(A,bool,P1,the(A,P1))) ) ).
tff(fact_81_the1__equality,axiom,
! [A: $tType,A2: A,P1: fun(A,bool)] :
( ? [X1: A] :
( pp(aa(A,bool,P1,X1))
& ! [Y1: A] :
( pp(aa(A,bool,P1,Y1))
=> ( Y1 = X1 ) ) )
=> ( pp(aa(A,bool,P1,A2))
=> ( the(A,P1) = A2 ) ) ) ).
tff(fact_82_mk__disjoint__insert,axiom,
! [A: $tType,A1: fun(A,bool),A2: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),A2),A1))
=> ? [B3: fun(A,bool)] :
( ( A1 = aa(fun(A,bool),fun(A,bool),aa(A,fun(fun(A,bool),fun(A,bool)),insert(A),A2),B3) )
& ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),A2),B3)) ) ) ).
tff(fact_83_Set_Oset__insert,axiom,
! [A: $tType,A1: fun(A,bool),X3: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X3),A1))
=> ~ ! [B3: fun(A,bool)] :
( ( A1 = aa(fun(A,bool),fun(A,bool),aa(A,fun(fun(A,bool),fun(A,bool)),insert(A),X3),B3) )
=> pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X3),B3)) ) ) ).
tff(fact_84_equals0I,axiom,
! [A: $tType,A1: fun(A,bool)] :
( ! [Y1: A] : ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Y1),A1))
=> ( A1 = bot_bot(fun(A,bool)) ) ) ).
tff(fact_85_conseq,axiom,
! [A: $tType,Q1: fun(A,fun(state,bool)),Ca: com,Ga: fun(hoare_28830079triple(A),bool),P1: fun(A,fun(state,bool))] :
( ! [Z: A,S2: state] :
( pp(aa(state,bool,aa(A,fun(state,bool),P1,Z),S2))
=> ? [P3: fun(A,fun(state,bool)),Q2: fun(A,fun(state,bool))] :
( hoare_992312373derivs(A,Ga,aa(fun(hoare_28830079triple(A),bool),fun(hoare_28830079triple(A),bool),aa(hoare_28830079triple(A),fun(fun(hoare_28830079triple(A),bool),fun(hoare_28830079triple(A),bool)),insert(hoare_28830079triple(A)),hoare_1841697145triple(A,P3,Ca,Q2)),bot_bot(fun(hoare_28830079triple(A),bool))))
& ! [S3: state] :
( ! [Z1: A] :
( pp(aa(state,bool,aa(A,fun(state,bool),P3,Z1),S2))
=> pp(aa(state,bool,aa(A,fun(state,bool),Q2,Z1),S3)) )
=> pp(aa(state,bool,aa(A,fun(state,bool),Q1,Z),S3)) ) ) )
=> hoare_992312373derivs(A,Ga,aa(fun(hoare_28830079triple(A),bool),fun(hoare_28830079triple(A),bool),aa(hoare_28830079triple(A),fun(fun(hoare_28830079triple(A),bool),fun(hoare_28830079triple(A),bool)),insert(hoare_28830079triple(A)),hoare_1841697145triple(A,P1,Ca,Q1)),bot_bot(fun(hoare_28830079triple(A),bool)))) ) ).
tff(fact_86_Loop,axiom,
! [A: $tType,Ca: com,B1: fun(state,bool),P1: fun(A,fun(state,bool)),Ga: fun(hoare_28830079triple(A),bool)] :
( hoare_992312373derivs(A,Ga,aa(fun(hoare_28830079triple(A),bool),fun(hoare_28830079triple(A),bool),aa(hoare_28830079triple(A),fun(fun(hoare_28830079triple(A),bool),fun(hoare_28830079triple(A),bool)),insert(hoare_28830079triple(A)),hoare_1841697145triple(A,hoare_1442473487ek_and(A,P1,B1),Ca,P1)),bot_bot(fun(hoare_28830079triple(A),bool))))
=> hoare_992312373derivs(A,Ga,aa(fun(hoare_28830079triple(A),bool),fun(hoare_28830079triple(A),bool),aa(hoare_28830079triple(A),fun(fun(hoare_28830079triple(A),bool),fun(hoare_28830079triple(A),bool)),insert(hoare_28830079triple(A)),hoare_1841697145triple(A,P1,while(B1,Ca),hoare_1442473487ek_and(A,P1,comp(bool,bool,state,fNot,B1)))),bot_bot(fun(hoare_28830079triple(A),bool)))) ) ).
tff(fact_87_nonempty__iff,axiom,
! [A: $tType,A1: fun(A,bool)] :
( ( A1 != bot_bot(fun(A,bool)) )
<=> ? [X2: A,B2: fun(A,bool)] :
( ( A1 = aa(fun(A,bool),fun(A,bool),aa(A,fun(fun(A,bool),fun(A,bool)),insert(A),X2),B2) )
& ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X2),B2)) ) ) ).
tff(fact_88_peek__and__def,axiom,
! [A: $tType,P2: fun(state,bool),P1: fun(A,fun(state,bool)),X1: A,Xa: state] :
( pp(aa(state,bool,aa(A,fun(state,bool),hoare_1442473487ek_and(A,P1,P2),X1),Xa))
<=> ( pp(aa(state,bool,aa(A,fun(state,bool),P1,X1),Xa))
& pp(aa(state,bool,P2,Xa)) ) ) ).
tff(fact_89_Loop__sound__lemma,axiom,
! [A: $tType,Ca: com,B1: fun(state,bool),P1: fun(A,fun(state,bool)),Ga: fun(hoare_28830079triple(A),bool)] :
( hoare_592965047valids(A,Ga,aa(fun(hoare_28830079triple(A),bool),fun(hoare_28830079triple(A),bool),aa(hoare_28830079triple(A),fun(fun(hoare_28830079triple(A),bool),fun(hoare_28830079triple(A),bool)),insert(hoare_28830079triple(A)),hoare_1841697145triple(A,hoare_1442473487ek_and(A,P1,B1),Ca,P1)),bot_bot(fun(hoare_28830079triple(A),bool))))
=> hoare_592965047valids(A,Ga,aa(fun(hoare_28830079triple(A),bool),fun(hoare_28830079triple(A),bool),aa(hoare_28830079triple(A),fun(fun(hoare_28830079triple(A),bool),fun(hoare_28830079triple(A),bool)),insert(hoare_28830079triple(A)),hoare_1841697145triple(A,P1,while(B1,Ca),hoare_1442473487ek_and(A,P1,comp(bool,bool,state,fNot,B1)))),bot_bot(fun(hoare_28830079triple(A),bool)))) ) ).
tff(fact_90_hoare__derivs_OIf,axiom,
! [A: $tType,D: com,Q1: fun(A,fun(state,bool)),Ca: com,B1: fun(state,bool),P1: fun(A,fun(state,bool)),Ga: fun(hoare_28830079triple(A),bool)] :
( hoare_992312373derivs(A,Ga,aa(fun(hoare_28830079triple(A),bool),fun(hoare_28830079triple(A),bool),aa(hoare_28830079triple(A),fun(fun(hoare_28830079triple(A),bool),fun(hoare_28830079triple(A),bool)),insert(hoare_28830079triple(A)),hoare_1841697145triple(A,hoare_1442473487ek_and(A,P1,B1),Ca,Q1)),bot_bot(fun(hoare_28830079triple(A),bool))))
=> ( hoare_992312373derivs(A,Ga,aa(fun(hoare_28830079triple(A),bool),fun(hoare_28830079triple(A),bool),aa(hoare_28830079triple(A),fun(fun(hoare_28830079triple(A),bool),fun(hoare_28830079triple(A),bool)),insert(hoare_28830079triple(A)),hoare_1841697145triple(A,hoare_1442473487ek_and(A,P1,comp(bool,bool,state,fNot,B1)),D,Q1)),bot_bot(fun(hoare_28830079triple(A),bool))))
=> hoare_992312373derivs(A,Ga,aa(fun(hoare_28830079triple(A),bool),fun(hoare_28830079triple(A),bool),aa(hoare_28830079triple(A),fun(fun(hoare_28830079triple(A),bool),fun(hoare_28830079triple(A),bool)),insert(hoare_28830079triple(A)),hoare_1841697145triple(A,P1,cond(B1,Ca,D),Q1)),bot_bot(fun(hoare_28830079triple(A),bool)))) ) ) ).
tff(fact_91_com_Osimps_I4_J,axiom,
! [Com21: com,Com11: com,Fun: fun(state,bool),Com2: com,Com1: com,Fun1: fun(state,bool)] :
( ( cond(Fun1,Com1,Com2) = cond(Fun,Com11,Com21) )
<=> ( ( Fun1 = Fun )
& ( Com1 = Com11 )
& ( Com2 = Com21 ) ) ) ).
tff(fact_92_evalc_OIfFalse,axiom,
! [C0: com,S1: state,C1: com,S: state,B1: fun(state,bool)] :
( ~ pp(aa(state,bool,B1,S))
=> ( pp(aa(state,bool,aa(state,fun(state,bool),evalc(C1),S),S1))
=> pp(aa(state,bool,aa(state,fun(state,bool),evalc(cond(B1,C0,C1)),S),S1)) ) ) ).
tff(fact_93_evalc_OIfTrue,axiom,
! [C1: com,S1: state,C0: com,S: state,B1: fun(state,bool)] :
( pp(aa(state,bool,B1,S))
=> ( pp(aa(state,bool,aa(state,fun(state,bool),evalc(C0),S),S1))
=> pp(aa(state,bool,aa(state,fun(state,bool),evalc(cond(B1,C0,C1)),S),S1)) ) ) ).
tff(fact_94_evalc__elim__cases_I5_J,axiom,
! [T: state,S: state,C2: com,C1: com,B1: fun(state,bool)] :
( pp(aa(state,bool,aa(state,fun(state,bool),evalc(cond(B1,C1,C2)),S),T))
=> ( ( pp(aa(state,bool,B1,S))
=> ~ pp(aa(state,bool,aa(state,fun(state,bool),evalc(C1),S),T)) )
=> ~ ( ~ pp(aa(state,bool,B1,S))
=> ~ pp(aa(state,bool,aa(state,fun(state,bool),evalc(C2),S),T)) ) ) ) ).
tff(fact_95_com_Osimps_I45_J,axiom,
! [Com2: com,Com1: com,Com21: com,Com11: com,Fun: fun(state,bool)] : cond(Fun,Com11,Com21) != semi(Com1,Com2) ).
tff(fact_96_com_Osimps_I44_J,axiom,
! [Com21: com,Com11: com,Fun: fun(state,bool),Com2: com,Com1: com] : semi(Com1,Com2) != cond(Fun,Com11,Com21) ).
tff(fact_97_com_Osimps_I52_J,axiom,
! [Com: com,Fun: fun(state,bool),Com2: com,Com1: com,Fun1: fun(state,bool)] : cond(Fun1,Com1,Com2) != while(Fun,Com) ).
tff(fact_98_com_Osimps_I15_J,axiom,
! [Com21: com,Com11: com,Fun: fun(state,bool)] : cond(Fun,Com11,Com21) != skip ).
tff(fact_99_com_Osimps_I53_J,axiom,
! [Com2: com,Com1: com,Fun1: fun(state,bool),Com: com,Fun: fun(state,bool)] : while(Fun,Com) != cond(Fun1,Com1,Com2) ).
%----Arities (2)
tff(arity_fun___Orderings_Obot,axiom,
! [T_1: $tType,T_2: $tType] :
( bot(T_2)
=> bot(fun(T_1,T_2)) ) ).
tff(arity_HOL_Obool___Orderings_Obot,axiom,
bot(bool) ).
%----Helper facts (22)
tff(help_pp_1_1_U,axiom,
~ pp(fFalse) ).
tff(help_pp_2_1_U,axiom,
pp(fTrue) ).
tff(help_fAll_1_1_U,axiom,
! [A: $tType,X: A,P: fun(A,bool)] :
( ~ pp(aa(fun(A,bool),bool,fAll(A),P))
| pp(aa(A,bool,P,X)) ) ).
tff(help_fNot_1_1_U,axiom,
! [P: bool] :
( ~ pp(aa(bool,bool,fNot,P))
| ~ pp(P) ) ).
tff(help_fNot_2_1_U,axiom,
! [P: bool] :
( pp(P)
| pp(aa(bool,bool,fNot,P)) ) ).
tff(help_COMBB_1_1_U,axiom,
! [C: $tType,B: $tType,A: $tType,R: A,Q: fun(A,B),P: fun(B,C)] : aa(A,C,aa(fun(A,B),fun(A,C),combb(B,C,A,P),Q),R) = aa(B,C,P,aa(A,B,Q,R)) ).
tff(help_COMBC_1_1_U,axiom,
! [A: $tType,C: $tType,B: $tType,R: A,Q: B,P: fun(A,fun(B,C))] : aa(A,C,aa(B,fun(A,C),aa(fun(A,fun(B,C)),fun(B,fun(A,C)),combc(A,B,C),P),Q),R) = aa(B,C,aa(A,fun(B,C),P,R),Q) ).
tff(help_COMBK_1_1_U,axiom,
! [B: $tType,A: $tType,Q: B,P: A] : aa(B,A,combk(A,B,P),Q) = P ).
tff(help_COMBS_1_1_U,axiom,
! [C: $tType,B: $tType,A: $tType,R: A,Q: fun(A,B),P: fun(A,fun(B,C))] : aa(A,C,aa(fun(A,B),fun(A,C),aa(fun(A,fun(B,C)),fun(fun(A,B),fun(A,C)),combs(A,B,C),P),Q),R) = aa(B,C,aa(A,fun(B,C),P,R),aa(A,B,Q,R)) ).
tff(help_fconj_1_1_U,axiom,
! [Q: bool,P: bool] :
( ~ pp(P)
| ~ pp(Q)
| pp(aa(bool,bool,aa(bool,fun(bool,bool),fconj,P),Q)) ) ).
tff(help_fconj_2_1_U,axiom,
! [Q: bool,P: bool] :
( ~ pp(aa(bool,bool,aa(bool,fun(bool,bool),fconj,P),Q))
| pp(P) ) ).
tff(help_fconj_3_1_U,axiom,
! [Q: bool,P: bool] :
( ~ pp(aa(bool,bool,aa(bool,fun(bool,bool),fconj,P),Q))
| pp(Q) ) ).
tff(help_fdisj_1_1_U,axiom,
! [Q: bool,P: bool] :
( ~ pp(P)
| pp(aa(bool,bool,aa(bool,fun(bool,bool),fdisj,P),Q)) ) ).
tff(help_fdisj_2_1_U,axiom,
! [P: bool,Q: bool] :
( ~ pp(Q)
| pp(aa(bool,bool,aa(bool,fun(bool,bool),fdisj,P),Q)) ) ).
tff(help_fdisj_3_1_U,axiom,
! [Q: bool,P: bool] :
( ~ pp(aa(bool,bool,aa(bool,fun(bool,bool),fdisj,P),Q))
| pp(P)
| pp(Q) ) ).
tff(help_fFalse_1_1_U,axiom,
~ pp(fFalse) ).
tff(help_fFalse_1_1_T,axiom,
! [P: bool] :
( ( P = fTrue )
| ( P = fFalse ) ) ).
tff(help_fequal_1_1_T,axiom,
! [A: $tType,Y: A,X: A] :
( ~ pp(aa(A,bool,aa(A,fun(A,bool),fequal(A),X),Y))
| ( X = Y ) ) ).
tff(help_fequal_2_1_T,axiom,
! [A: $tType,Y: A,X: A] :
( ( X != Y )
| pp(aa(A,bool,aa(A,fun(A,bool),fequal(A),X),Y)) ) ).
tff(help_fimplies_1_1_U,axiom,
! [Q: bool,P: bool] :
( pp(P)
| pp(aa(bool,bool,aa(bool,fun(bool,bool),fimplies,P),Q)) ) ).
tff(help_fimplies_2_1_U,axiom,
! [P: bool,Q: bool] :
( ~ pp(Q)
| pp(aa(bool,bool,aa(bool,fun(bool,bool),fimplies,P),Q)) ) ).
tff(help_fimplies_3_1_U,axiom,
! [Q: bool,P: bool] :
( ~ pp(aa(bool,bool,aa(bool,fun(bool,bool),fimplies,P),Q))
| ~ pp(P)
| pp(Q) ) ).
%----Conjectures (3)
tff(conj_0,hypothesis,
hoare_531836380gleton ).
tff(conj_1,hypothesis,
hoare_992312373derivs(state,g,aa(fun(hoare_28830079triple(state),bool),fun(hoare_28830079triple(state),bool),aa(hoare_28830079triple(state),fun(fun(hoare_28830079triple(state),bool),fun(hoare_28830079triple(state),bool)),insert(hoare_28830079triple(state)),hoare_1841697145triple(state,aa(fun(state,fun(state,fun(state,bool))),fun(state,fun(state,bool)),combb(fun(state,fun(state,bool)),fun(state,bool),state,combb(fun(state,bool),bool,state,fAll(state))),aa(fun(state,fun(state,bool)),fun(state,fun(state,fun(state,bool))),combb(fun(state,bool),fun(state,fun(state,bool)),state,aa(fun(state,fun(fun(state,bool),fun(state,bool))),fun(fun(state,bool),fun(state,fun(state,bool))),combc(state,fun(state,bool),fun(state,bool)),aa(fun(state,fun(state,fun(bool,bool))),fun(state,fun(fun(state,bool),fun(state,bool))),combb(fun(state,fun(bool,bool)),fun(fun(state,bool),fun(state,bool)),state,combs(state,bool,bool)),aa(fun(state,fun(state,bool)),fun(state,fun(state,fun(bool,bool))),combb(fun(state,bool),fun(state,fun(bool,bool)),state,combb(bool,fun(bool,bool),state,fimplies)),evalc(c))))),fequal(state))),c,fequal(state))),bot_bot(fun(hoare_28830079triple(state),bool)))) ).
tff(conj_2,conjecture,
hoare_992312373derivs(state,g,aa(fun(hoare_28830079triple(state),bool),fun(hoare_28830079triple(state),bool),aa(hoare_28830079triple(state),fun(fun(hoare_28830079triple(state),bool),fun(hoare_28830079triple(state),bool)),insert(hoare_28830079triple(state)),hoare_1841697145triple(state,fequal(state),c,evalc(c))),bot_bot(fun(hoare_28830079triple(state),bool)))) ).
%------------------------------------------------------------------------------