TPTP Problem File: SWW508_5.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SWW508_5 : TPTP v8.2.0. Released v6.0.0.
% Domain : Software Verification
% Problem : Hoare's Logic with Procedures line 159
% Version : Especial.
% English :
% Refs : [BN10] Boehme & Nipkow (2010), Sledgehammer: Judgement Day
% : [Bla13] Blanchette (2011), Email to Geoff Sutcliffe
% Source : [Bla13]
% Names : hoare_159 [Bla13]
% Status : Theorem
% Rating : 0.00 v7.5.0, 0.33 v7.4.0, 0.50 v7.1.0, 0.67 v6.4.0
% Syntax : Number of formulae : 180 ( 43 unt; 50 typ; 0 def)
% Number of atoms : 302 ( 107 equ)
% Maximal formula atoms : 10 ( 1 avg)
% Number of connectives : 237 ( 65 ~; 24 |; 27 &)
% ( 24 <=>; 97 =>; 0 <=; 0 <~>)
% Maximal formula depth : 17 ( 6 avg)
% Maximal term depth : 8 ( 2 avg)
% Number of types : 9 ( 8 usr)
% Number of type conns : 57 ( 25 >; 32 *; 0 +; 0 <<)
% Number of predicates : 13 ( 12 usr; 0 prp; 1-6 aty)
% Number of functors : 30 ( 30 usr; 11 con; 0-10 aty)
% Number of variables : 501 ( 445 !; 19 ?; 501 :)
% ( 37 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:14:44
%------------------------------------------------------------------------------
%----Should-be-implicit typings (10)
tff(ty_t_a,type,
a: $tType ).
tff(ty_tc_Com_Ocom,type,
com: $tType ).
tff(ty_tc_Com_Oloc,type,
loc: $tType ).
tff(ty_tc_Com_Opname,type,
pname: $tType ).
tff(ty_tc_Com_Ostate,type,
state: $tType ).
tff(ty_tc_Com_Ovname,type,
vname: $tType ).
tff(ty_tc_HOL_Obool,type,
bool: $tType ).
tff(ty_tc_Hoare__Mirabelle__vtrypsmcwp_Otriple,type,
hoare_28830079triple: $tType > $tType ).
tff(ty_tc_Nat_Onat,type,
nat: $tType ).
tff(ty_tc_fun,type,
fun: ( $tType * $tType ) > $tType ).
%----Explicit typings (40)
tff(sy_cl_Groups_Ominus,type,
cl_Groups_Ominus:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Obot,type,
bot:
!>[A: $tType] : $o ).
tff(sy_cl_Finite__Set_Ofinite,type,
finite_finite:
!>[A: $tType] : $o ).
tff(sy_c_COMBB,type,
combb:
!>[B: $tType,C: $tType,A: $tType] : ( ( fun(B,C) * fun(A,B) ) > fun(A,C) ) ).
tff(sy_c_COMBC,type,
combc:
!>[A: $tType,B: $tType,C: $tType] : ( ( fun(A,fun(B,C)) * 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(A,fun(B,C)) * fun(A,B) ) > fun(A,C) ) ).
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_Ocom__case,type,
com_case:
!>[T1: $tType] : ( ( T1 * fun(vname,fun(fun(state,nat),T1)) * fun(loc,fun(fun(state,nat),fun(com,T1))) * fun(com,fun(com,T1)) * fun(fun(state,bool),fun(com,fun(com,T1))) * fun(fun(state,bool),fun(com,T1)) * fun(pname,T1) * fun(vname,fun(pname,fun(fun(state,nat),T1))) * com ) > T1 ) ).
tff(sy_c_Finite__Set_Ofinite,type,
finite_finite1:
!>[A: $tType] : ( fun(A,bool) > $o ) ).
tff(sy_c_Finite__Set_Ofold1Set,type,
finite_fold1Set:
!>[A: $tType] : ( ( fun(A,fun(A,A)) * fun(A,bool) * A ) > $o ) ).
tff(sy_c_Finite__Set_Ofold__graph,type,
finite_fold_graph:
!>[A: $tType,B: $tType] : ( ( fun(A,fun(B,B)) * B * fun(A,bool) * B ) > $o ) ).
tff(sy_c_Finite__Set_Ofolding__one,type,
finite_folding_one:
!>[A: $tType] : ( ( fun(A,fun(A,A)) * fun(fun(A,bool),A) ) > $o ) ).
tff(sy_c_Groups_Ominus__class_Ominus,type,
minus_minus:
!>[A: $tType] : ( ( A * A ) > A ) ).
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_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,T1: $tType] : ( ( fun(fun(A,fun(state,bool)),fun(com,fun(fun(A,fun(state,bool)),T1))) * hoare_28830079triple(A) ) > T1 ) ).
tff(sy_c_Hoare__Mirabelle__vtrypsmcwp_Otriple_Otriple__rec,type,
hoare_678420151le_rec:
!>[A: $tType,T1: $tType] : ( ( fun(fun(A,fun(state,bool)),fun(com,fun(fun(A,fun(state,bool)),T1))) * hoare_28830079triple(A) ) > T1 ) ).
tff(sy_c_Orderings_Obot__class_Obot,type,
bot_bot:
!>[A: $tType] : A ).
tff(sy_c_Predicate_Oinv__imagep,type,
inv_imagep:
!>[B: $tType,A: $tType] : ( ( fun(B,fun(B,bool)) * fun(A,B) * A * A ) > $o ) ).
tff(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( fun(A,bool) > fun(A,bool) ) ).
tff(sy_c_Set_Oinsert,type,
insert:
!>[A: $tType] : ( ( A * 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_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(a),bool) ).
tff(sy_v_P,type,
p: ( a * state ) > $o ).
tff(sy_v_P_H,type,
p1: fun(a,fun(state,bool)) ).
tff(sy_v_Q,type,
q: ( a * state ) > $o ).
tff(sy_v_Q_H,type,
q1: fun(a,fun(state,bool)) ).
tff(sy_v_c,type,
c: com ).
%----Relevant facts (100)
tff(fact_0_triple_Oinject,axiom,
! [B: $tType,Fun22: fun(B,fun(state,bool)),Com4: com,Fun12: fun(B,fun(state,bool)),Fun21: fun(B,fun(state,bool)),Com3: com,Fun11: fun(B,fun(state,bool))] :
( ( hoare_1841697145triple(B,Fun11,Com3,Fun21) = hoare_1841697145triple(B,Fun12,Com4,Fun22) )
<=> ( ( Fun11 = Fun12 )
& ( Com3 = Com4 )
& ( Fun21 = Fun22 ) ) ) ).
tff(fact_1_empty,axiom,
! [B: $tType,Ga: fun(hoare_28830079triple(B),bool)] : hoare_992312373derivs(B,Ga,bot_bot(fun(hoare_28830079triple(B),bool))) ).
tff(fact_2_cut,axiom,
! [B: $tType,Ga: fun(hoare_28830079triple(B),bool),Ts: fun(hoare_28830079triple(B),bool),G1: fun(hoare_28830079triple(B),bool)] :
( hoare_992312373derivs(B,G1,Ts)
=> ( hoare_992312373derivs(B,Ga,G1)
=> hoare_992312373derivs(B,Ga,Ts) ) ) ).
tff(fact_3_hoare__derivs_Oinsert,axiom,
! [B: $tType,Ts: fun(hoare_28830079triple(B),bool),T: hoare_28830079triple(B),Ga: fun(hoare_28830079triple(B),bool)] :
( hoare_992312373derivs(B,Ga,insert(hoare_28830079triple(B),T,bot_bot(fun(hoare_28830079triple(B),bool))))
=> ( hoare_992312373derivs(B,Ga,Ts)
=> hoare_992312373derivs(B,Ga,insert(hoare_28830079triple(B),T,Ts)) ) ) ).
tff(fact_4_insert__absorb2,axiom,
! [B: $tType,A4: fun(B,bool),X2: B] : insert(B,X2,insert(B,X2,A4)) = insert(B,X2,A4) ).
tff(fact_5_insert__iff,axiom,
! [B: $tType,A4: fun(B,bool),B2: B,A1: B] :
( pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),A1),insert(B,B2,A4)))
<=> ( ( A1 = B2 )
| pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),A1),A4)) ) ) ).
tff(fact_6_insertE,axiom,
! [B: $tType,A4: fun(B,bool),B2: B,A1: B] :
( pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),A1),insert(B,B2,A4)))
=> ( ( A1 != B2 )
=> pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),A1),A4)) ) ) ).
tff(fact_7_insertCI,axiom,
! [B: $tType,B2: B,B1: fun(B,bool),A1: B] :
( ( ~ pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),A1),B1))
=> ( A1 = B2 ) )
=> pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),A1),insert(B,B2,B1))) ) ).
tff(fact_8_all__not__in__conv,axiom,
! [B: $tType,A4: fun(B,bool)] :
( ! [X3: B] : ~ pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),X3),A4))
<=> ( A4 = bot_bot(fun(B,bool)) ) ) ).
tff(fact_9_empty__Collect__eq,axiom,
! [B: $tType,Pa: fun(B,bool)] :
( ( bot_bot(fun(B,bool)) = collect(B,Pa) )
<=> ! [X3: B] : ~ pp(aa(B,bool,Pa,X3)) ) ).
tff(fact_10_empty__iff,axiom,
! [B: $tType,Ca: B] : ~ pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),Ca),bot_bot(fun(B,bool)))) ).
tff(fact_11_Collect__empty__eq,axiom,
! [B: $tType,Pa: fun(B,bool)] :
( ( collect(B,Pa) = bot_bot(fun(B,bool)) )
<=> ! [X3: B] : ~ pp(aa(B,bool,Pa,X3)) ) ).
tff(fact_12_emptyE,axiom,
! [B: $tType,A1: B] : ~ pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),A1),bot_bot(fun(B,bool)))) ).
tff(fact_13_triple_Orecs,axiom,
! [B: $tType,C: $tType,Fun21: fun(C,fun(state,bool)),Com3: com,Fun11: fun(C,fun(state,bool)),F11: fun(fun(C,fun(state,bool)),fun(com,fun(fun(C,fun(state,bool)),B)))] : hoare_678420151le_rec(C,B,F11,hoare_1841697145triple(C,Fun11,Com3,Fun21)) = aa(fun(C,fun(state,bool)),B,aa(com,fun(fun(C,fun(state,bool)),B),aa(fun(C,fun(state,bool)),fun(com,fun(fun(C,fun(state,bool)),B)),F11,Fun11),Com3),Fun21) ).
tff(fact_14_triple_Osimps_I2_J,axiom,
! [B: $tType,C: $tType,Fun21: fun(C,fun(state,bool)),Com3: com,Fun11: fun(C,fun(state,bool)),F11: fun(fun(C,fun(state,bool)),fun(com,fun(fun(C,fun(state,bool)),B)))] : hoare_376461865e_case(C,B,F11,hoare_1841697145triple(C,Fun11,Com3,Fun21)) = aa(fun(C,fun(state,bool)),B,aa(com,fun(fun(C,fun(state,bool)),B),aa(fun(C,fun(state,bool)),fun(com,fun(fun(C,fun(state,bool)),B)),F11,Fun11),Com3),Fun21) ).
tff(fact_15_empty__not__insert,axiom,
! [B: $tType,A4: fun(B,bool),A1: B] : bot_bot(fun(B,bool)) != insert(B,A1,A4) ).
tff(fact_16_equals0D,axiom,
! [B: $tType,A1: B,A4: fun(B,bool)] :
( ( A4 = bot_bot(fun(B,bool)) )
=> ~ pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),A1),A4)) ) ).
tff(fact_17_ex__in__conv,axiom,
! [B: $tType,A4: fun(B,bool)] :
( ? [X3: B] : pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),X3),A4))
<=> ( A4 != bot_bot(fun(B,bool)) ) ) ).
tff(fact_18_empty__def,axiom,
! [B: $tType] : bot_bot(fun(B,bool)) = collect(B,combk(bool,B,fFalse)) ).
tff(fact_19_insert__absorb,axiom,
! [B: $tType,A4: fun(B,bool),A1: B] :
( pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),A1),A4))
=> ( insert(B,A1,A4) = A4 ) ) ).
tff(fact_20_insertI2,axiom,
! [B: $tType,B2: B,B1: fun(B,bool),A1: B] :
( pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),A1),B1))
=> pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),A1),insert(B,B2,B1))) ) ).
tff(fact_21_insert__eq__iff,axiom,
! [B: $tType,B1: fun(B,bool),B2: B,A4: fun(B,bool),A1: B] :
( ~ pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),A1),A4))
=> ( ~ pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),B2),B1))
=> ( ( insert(B,A1,A4) = insert(B,B2,B1) )
<=> ( ( ( A1 = B2 )
=> ( A4 = B1 ) )
& ( ( A1 != B2 )
=> ? [C1: fun(B,bool)] :
( ( A4 = insert(B,B2,C1) )
& ~ pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),B2),C1))
& ( B1 = insert(B,A1,C1) )
& ~ pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),A1),C1)) ) ) ) ) ) ) ).
tff(fact_22_insert__ident,axiom,
! [B: $tType,B1: fun(B,bool),A4: fun(B,bool),X2: B] :
( ~ pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),X2),A4))
=> ( ~ pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),X2),B1))
=> ( ( insert(B,X2,A4) = insert(B,X2,B1) )
<=> ( A4 = B1 ) ) ) ) ).
tff(fact_23_insert__code,axiom,
! [B: $tType,X2: B,A4: fun(B,bool),Y3: B] :
( pp(aa(B,bool,insert(B,Y3,A4),X2))
<=> ( ( Y3 = X2 )
| pp(aa(B,bool,A4,X2)) ) ) ).
tff(fact_24_insert__commute,axiom,
! [B: $tType,A4: fun(B,bool),Y3: B,X2: B] : insert(B,X2,insert(B,Y3,A4)) = insert(B,Y3,insert(B,X2,A4)) ).
tff(fact_25_insert__Collect,axiom,
! [B: $tType,Pa: fun(B,bool),A1: B] : insert(B,A1,collect(B,Pa)) = collect(B,combs(B,bool,bool,combb(bool,fun(bool,bool),B,fimplies,combb(bool,bool,B,fNot,combc(B,B,bool,fequal(B),A1))),Pa)) ).
tff(fact_26_insert__compr,axiom,
! [B: $tType,B1: fun(B,bool),A1: B] : insert(B,A1,B1) = collect(B,combs(B,bool,bool,combb(bool,fun(bool,bool),B,fdisj,combc(B,B,bool,fequal(B),A1)),combc(B,fun(B,bool),bool,member(B),B1))) ).
tff(fact_27_insertI1,axiom,
! [B: $tType,B1: fun(B,bool),A1: B] : pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),A1),insert(B,A1,B1))) ).
tff(fact_28_singleton__inject,axiom,
! [B: $tType,B2: B,A1: B] :
( ( insert(B,A1,bot_bot(fun(B,bool))) = insert(B,B2,bot_bot(fun(B,bool))) )
=> ( A1 = B2 ) ) ).
tff(fact_29_singletonE,axiom,
! [B: $tType,A1: B,B2: B] :
( pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),B2),insert(B,A1,bot_bot(fun(B,bool)))))
=> ( B2 = A1 ) ) ).
tff(fact_30_doubleton__eq__iff,axiom,
! [B: $tType,D: B,Ca: B,B2: B,A1: B] :
( ( insert(B,A1,insert(B,B2,bot_bot(fun(B,bool)))) = insert(B,Ca,insert(B,D,bot_bot(fun(B,bool)))) )
<=> ( ( ( A1 = Ca )
& ( B2 = D ) )
| ( ( A1 = D )
& ( B2 = Ca ) ) ) ) ).
tff(fact_31_singleton__iff,axiom,
! [B: $tType,A1: B,B2: B] :
( pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),B2),insert(B,A1,bot_bot(fun(B,bool)))))
<=> ( B2 = A1 ) ) ).
tff(fact_32_insert__not__empty,axiom,
! [B: $tType,A4: fun(B,bool),A1: B] : insert(B,A1,A4) != bot_bot(fun(B,bool)) ).
tff(fact_33_the__elem__eq,axiom,
! [B: $tType,X2: B] : the_elem(B,insert(B,X2,bot_bot(fun(B,bool)))) = X2 ).
tff(fact_34_bot__apply,axiom,
! [C: $tType,B: $tType] :
( bot(B)
=> ! [X2: C] : aa(C,B,bot_bot(fun(C,B)),X2) = bot_bot(B) ) ).
tff(fact_35_bot__fun__def,axiom,
! [B: $tType,C: $tType] :
( bot(C)
=> ! [X4: B] : aa(B,C,bot_bot(fun(B,C)),X4) = bot_bot(C) ) ).
tff(fact_36_hoare__derivs_OSkip,axiom,
! [B: $tType,Pa: fun(B,fun(state,bool)),Ga: fun(hoare_28830079triple(B),bool)] : hoare_992312373derivs(B,Ga,insert(hoare_28830079triple(B),hoare_1841697145triple(B,Pa,skip,Pa),bot_bot(fun(hoare_28830079triple(B),bool)))) ).
tff(fact_37_Comp,axiom,
! [B: $tType,R2: fun(B,fun(state,bool)),D: com,Qa: fun(B,fun(state,bool)),Ca: com,Pa: fun(B,fun(state,bool)),Ga: fun(hoare_28830079triple(B),bool)] :
( hoare_992312373derivs(B,Ga,insert(hoare_28830079triple(B),hoare_1841697145triple(B,Pa,Ca,Qa),bot_bot(fun(hoare_28830079triple(B),bool))))
=> ( hoare_992312373derivs(B,Ga,insert(hoare_28830079triple(B),hoare_1841697145triple(B,Qa,D,R2),bot_bot(fun(hoare_28830079triple(B),bool))))
=> hoare_992312373derivs(B,Ga,insert(hoare_28830079triple(B),hoare_1841697145triple(B,Pa,semi(Ca,D),R2),bot_bot(fun(hoare_28830079triple(B),bool)))) ) ) ).
tff(fact_38_triple_Oexhaust,axiom,
! [B: $tType,Y3: hoare_28830079triple(B)] :
~ ! [Fun1: fun(B,fun(state,bool)),Com: com,Fun2: fun(B,fun(state,bool))] : Y3 != hoare_1841697145triple(B,Fun1,Com,Fun2) ).
tff(fact_39_Set_Oset__insert,axiom,
! [B: $tType,A4: fun(B,bool),X2: B] :
( pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),X2),A4))
=> ~ ! [B4: fun(B,bool)] :
( ( A4 = insert(B,X2,B4) )
=> pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),X2),B4)) ) ) ).
tff(fact_40_mk__disjoint__insert,axiom,
! [B: $tType,A4: fun(B,bool),A1: B] :
( pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),A1),A4))
=> ? [B4: fun(B,bool)] :
( ( A4 = insert(B,A1,B4) )
& ~ pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),A1),B4)) ) ) ).
tff(fact_41_com_Osimps_I3_J,axiom,
! [Com22: com,Com12: com,Com2: com,Com1: com] :
( ( semi(Com1,Com2) = semi(Com12,Com22) )
<=> ( ( Com1 = Com12 )
& ( Com2 = Com22 ) ) ) ).
tff(fact_42_com_Osimps_I13_J,axiom,
! [Com21: com,Com11: com] : semi(Com11,Com21) != skip ).
tff(fact_43_com_Osimps_I12_J,axiom,
! [Com21: com,Com11: com] : skip != semi(Com11,Com21) ).
tff(fact_44_equals0I,axiom,
! [B: $tType,A4: fun(B,bool)] :
( ! [Y1: B] : ~ pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),Y1),A4))
=> ( A4 = bot_bot(fun(B,bool)) ) ) ).
tff(fact_45_conseq,axiom,
! [B: $tType,Qa: fun(B,fun(state,bool)),Ca: com,Ga: fun(hoare_28830079triple(B),bool),Pa: fun(B,fun(state,bool))] :
( ! [Z: B,S: state] :
( pp(aa(state,bool,aa(B,fun(state,bool),Pa,Z),S))
=> ? [P: fun(B,fun(state,bool)),Q: fun(B,fun(state,bool))] :
( hoare_992312373derivs(B,Ga,insert(hoare_28830079triple(B),hoare_1841697145triple(B,P,Ca,Q),bot_bot(fun(hoare_28830079triple(B),bool))))
& ! [S1: state] :
( ! [Z1: B] :
( pp(aa(state,bool,aa(B,fun(state,bool),P,Z1),S))
=> pp(aa(state,bool,aa(B,fun(state,bool),Q,Z1),S1)) )
=> pp(aa(state,bool,aa(B,fun(state,bool),Qa,Z),S1)) ) ) )
=> hoare_992312373derivs(B,Ga,insert(hoare_28830079triple(B),hoare_1841697145triple(B,Pa,Ca,Qa),bot_bot(fun(hoare_28830079triple(B),bool)))) ) ).
tff(fact_46_nonempty__iff,axiom,
! [B: $tType,A4: fun(B,bool)] :
( ( A4 != bot_bot(fun(B,bool)) )
<=> ? [X3: B,B3: fun(B,bool)] :
( ( A4 = insert(B,X3,B3) )
& ~ pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),X3),B3)) ) ) ).
tff(fact_47_bot__empty__eq,axiom,
! [B: $tType,X4: B] :
( pp(aa(B,bool,bot_bot(fun(B,bool)),X4))
<=> pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),X4),bot_bot(fun(B,bool)))) ) ).
tff(fact_48_fold1Set__sing,axiom,
! [B: $tType,B2: B,A1: B,F2: fun(B,fun(B,B))] :
( finite_fold1Set(B,F2,insert(B,A1,bot_bot(fun(B,bool))),B2)
<=> ( A1 = B2 ) ) ).
tff(fact_49_com_Osimps_I64_J,axiom,
! [B: $tType,F8: fun(vname,fun(pname,fun(fun(state,nat),B))),F7: fun(pname,B),F6: fun(fun(state,bool),fun(com,B)),F5: fun(fun(state,bool),fun(com,fun(com,B))),F4: fun(com,fun(com,B)),F3: fun(loc,fun(fun(state,nat),fun(com,B))),F21: fun(vname,fun(fun(state,nat),B)),F11: B] : com_case(B,F11,F21,F3,F4,F5,F6,F7,F8,skip) = F11 ).
tff(fact_50_fold1Set__nonempty,axiom,
! [B: $tType,X2: B,A4: fun(B,bool),F2: fun(B,fun(B,B))] :
( finite_fold1Set(B,F2,A4,X2)
=> ( A4 != bot_bot(fun(B,bool)) ) ) ).
tff(fact_51_empty__fold1SetE,axiom,
! [B: $tType,X2: B,F2: fun(B,fun(B,B))] : ~ finite_fold1Set(B,F2,bot_bot(fun(B,bool)),X2) ).
tff(fact_52_com_Osimps_I67_J,axiom,
! [B: $tType,Com2: com,Com1: com,F8: fun(vname,fun(pname,fun(fun(state,nat),B))),F7: fun(pname,B),F6: fun(fun(state,bool),fun(com,B)),F5: fun(fun(state,bool),fun(com,fun(com,B))),F4: fun(com,fun(com,B)),F3: fun(loc,fun(fun(state,nat),fun(com,B))),F21: fun(vname,fun(fun(state,nat),B)),F11: B] : com_case(B,F11,F21,F3,F4,F5,F6,F7,F8,semi(Com1,Com2)) = aa(com,B,aa(com,fun(com,B),F4,Com1),Com2) ).
tff(fact_53_fold1Set_Ointros,axiom,
! [B: $tType,X2: B,A4: fun(B,bool),A1: B,F2: fun(B,fun(B,B))] :
( finite_fold_graph(B,B,F2,A1,A4,X2)
=> ( ~ pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),A1),A4))
=> finite_fold1Set(B,F2,insert(B,A1,A4),X2) ) ) ).
tff(fact_54_folding__one_Osingleton,axiom,
! [B: $tType,X2: B,F: fun(fun(B,bool),B),F2: fun(B,fun(B,B))] :
( finite_folding_one(B,F2,F)
=> ( aa(fun(B,bool),B,F,insert(B,X2,bot_bot(fun(B,bool)))) = X2 ) ) ).
tff(fact_55_in__inv__imagep,axiom,
! [B: $tType,C: $tType,Y3: C,X2: C,F2: fun(C,B),R1: fun(B,fun(B,bool))] :
( inv_imagep(B,C,R1,F2,X2,Y3)
<=> pp(aa(B,bool,aa(B,fun(B,bool),R1,aa(C,B,F2,X2)),aa(C,B,F2,Y3))) ) ).
tff(fact_56_insert__Diff__single,axiom,
! [B: $tType,A4: fun(B,bool),A1: B] : insert(B,A1,minus_minus(fun(B,bool),A4,insert(B,A1,bot_bot(fun(B,bool))))) = insert(B,A1,A4) ).
tff(fact_57_DiffE,axiom,
! [B: $tType,B1: fun(B,bool),A4: fun(B,bool),Ca: B] :
( pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),Ca),minus_minus(fun(B,bool),A4,B1)))
=> ~ ( pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),Ca),A4))
=> pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),Ca),B1)) ) ) ).
tff(fact_58_DiffI,axiom,
! [B: $tType,B1: fun(B,bool),A4: fun(B,bool),Ca: B] :
( pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),Ca),A4))
=> ( ~ pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),Ca),B1))
=> pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),Ca),minus_minus(fun(B,bool),A4,B1))) ) ) ).
tff(fact_59_Diff__idemp,axiom,
! [B: $tType,B1: fun(B,bool),A4: fun(B,bool)] : minus_minus(fun(B,bool),minus_minus(fun(B,bool),A4,B1),B1) = minus_minus(fun(B,bool),A4,B1) ).
tff(fact_60_Diff__iff,axiom,
! [B: $tType,B1: fun(B,bool),A4: fun(B,bool),Ca: B] :
( pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),Ca),minus_minus(fun(B,bool),A4,B1)))
<=> ( pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),Ca),A4))
& ~ pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),Ca),B1)) ) ) ).
tff(fact_61_empty__Diff,axiom,
! [B: $tType,A4: fun(B,bool)] : minus_minus(fun(B,bool),bot_bot(fun(B,bool)),A4) = bot_bot(fun(B,bool)) ).
tff(fact_62_Diff__cancel,axiom,
! [B: $tType,A4: fun(B,bool)] : minus_minus(fun(B,bool),A4,A4) = bot_bot(fun(B,bool)) ).
tff(fact_63_Diff__empty,axiom,
! [B: $tType,A4: fun(B,bool)] : minus_minus(fun(B,bool),A4,bot_bot(fun(B,bool))) = A4 ).
tff(fact_64_insert__Diff__if,axiom,
! [B: $tType,A4: fun(B,bool),B1: fun(B,bool),X2: B] :
( ( pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),X2),B1))
=> ( minus_minus(fun(B,bool),insert(B,X2,A4),B1) = minus_minus(fun(B,bool),A4,B1) ) )
& ( ~ pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),X2),B1))
=> ( minus_minus(fun(B,bool),insert(B,X2,A4),B1) = insert(B,X2,minus_minus(fun(B,bool),A4,B1)) ) ) ) ).
tff(fact_65_insert__Diff1,axiom,
! [B: $tType,A4: fun(B,bool),B1: fun(B,bool),X2: B] :
( pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),X2),B1))
=> ( minus_minus(fun(B,bool),insert(B,X2,A4),B1) = minus_minus(fun(B,bool),A4,B1) ) ) ).
tff(fact_66_DiffD2,axiom,
! [B: $tType,B1: fun(B,bool),A4: fun(B,bool),Ca: B] :
( pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),Ca),minus_minus(fun(B,bool),A4,B1)))
=> ~ pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),Ca),B1)) ) ).
tff(fact_67_DiffD1,axiom,
! [B: $tType,B1: fun(B,bool),A4: fun(B,bool),Ca: B] :
( pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),Ca),minus_minus(fun(B,bool),A4,B1)))
=> pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),Ca),A4)) ) ).
tff(fact_68_set__diff__eq,axiom,
! [B: $tType,B1: fun(B,bool),A4: fun(B,bool)] : minus_minus(fun(B,bool),A4,B1) = collect(B,combs(B,bool,bool,combb(bool,fun(bool,bool),B,fconj,combc(B,fun(B,bool),bool,member(B),A4)),combb(bool,bool,B,fNot,combc(B,fun(B,bool),bool,member(B),B1)))) ).
tff(fact_69_fold__graph_OemptyI,axiom,
! [B: $tType,C: $tType,Z4: C,F2: fun(B,fun(C,C))] : finite_fold_graph(B,C,F2,Z4,bot_bot(fun(B,bool)),Z4) ).
tff(fact_70_empty__fold__graphE,axiom,
! [B: $tType,C: $tType,X2: C,Z4: C,F2: fun(B,fun(C,C))] :
( finite_fold_graph(B,C,F2,Z4,bot_bot(fun(B,bool)),X2)
=> ( X2 = Z4 ) ) ).
tff(fact_71_fold__graph_OinsertI,axiom,
! [B: $tType,C: $tType,Y3: C,Z4: C,F2: fun(B,fun(C,C)),A4: fun(B,bool),X2: B] :
( ~ pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),X2),A4))
=> ( finite_fold_graph(B,C,F2,Z4,A4,Y3)
=> finite_fold_graph(B,C,F2,Z4,insert(B,X2,A4),aa(C,C,aa(B,fun(C,C),F2,X2),Y3)) ) ) ).
tff(fact_72_insert__Diff,axiom,
! [B: $tType,A4: fun(B,bool),A1: B] :
( pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),A1),A4))
=> ( insert(B,A1,minus_minus(fun(B,bool),A4,insert(B,A1,bot_bot(fun(B,bool))))) = A4 ) ) ).
tff(fact_73_Diff__insert__absorb,axiom,
! [B: $tType,A4: fun(B,bool),X2: B] :
( ~ pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),X2),A4))
=> ( minus_minus(fun(B,bool),insert(B,X2,A4),insert(B,X2,bot_bot(fun(B,bool)))) = A4 ) ) ).
tff(fact_74_Diff__insert2,axiom,
! [B: $tType,B1: fun(B,bool),A1: B,A4: fun(B,bool)] : minus_minus(fun(B,bool),A4,insert(B,A1,B1)) = minus_minus(fun(B,bool),minus_minus(fun(B,bool),A4,insert(B,A1,bot_bot(fun(B,bool)))),B1) ).
tff(fact_75_ext,axiom,
! [C: $tType,B: $tType,G: fun(B,C),F2: fun(B,C)] :
( ! [X1: B] : aa(B,C,F2,X1) = aa(B,C,G,X1)
=> ( F2 = G ) ) ).
tff(fact_76_mem__def,axiom,
! [B: $tType,A4: fun(B,bool),X2: B] :
( pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),X2),A4))
<=> pp(aa(B,bool,A4,X2)) ) ).
tff(fact_77_Collect__def,axiom,
! [B: $tType,Pa: fun(B,bool)] : collect(B,Pa) = Pa ).
tff(fact_78_Diff__insert,axiom,
! [B: $tType,B1: fun(B,bool),A1: B,A4: fun(B,bool)] : minus_minus(fun(B,bool),A4,insert(B,A1,B1)) = minus_minus(fun(B,bool),minus_minus(fun(B,bool),A4,B1),insert(B,A1,bot_bot(fun(B,bool)))) ).
tff(fact_79_insert__fold1SetE,axiom,
! [B: $tType,X2: B,X5: fun(B,bool),A1: B,F2: fun(B,fun(B,B))] :
( finite_fold1Set(B,F2,insert(B,A1,X5),X2)
=> ~ ! [A6: B,A5: fun(B,bool)] :
( ( insert(B,A1,X5) = insert(B,A6,A5) )
=> ( finite_fold_graph(B,B,F2,A6,A5,X2)
=> pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),A6),A5)) ) ) ) ).
tff(fact_80_fold1Set_Osimps,axiom,
! [B: $tType,A21: B,A11: fun(B,bool),F2: fun(B,fun(B,B))] :
( finite_fold1Set(B,F2,A11,A21)
<=> ? [A2: B,A3: fun(B,bool),X3: B] :
( ( A11 = insert(B,A2,A3) )
& ( A21 = X3 )
& finite_fold_graph(B,B,F2,A2,A3,X3)
& ~ pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),A2),A3)) ) ) ).
tff(fact_81_fun__diff__def,axiom,
! [C: $tType,B: $tType] :
( cl_Groups_Ominus(C)
=> ! [B1: fun(B,C),A4: fun(B,C),X4: B] : aa(B,C,minus_minus(fun(B,C),A4,B1),X4) = minus_minus(C,aa(B,C,A4,X4),aa(B,C,B1,X4)) ) ).
tff(fact_82_minus__apply,axiom,
! [B: $tType,C: $tType] :
( cl_Groups_Ominus(B)
=> ! [X2: C,B1: fun(C,B),A4: fun(C,B)] : aa(C,B,minus_minus(fun(C,B),A4,B1),X2) = minus_minus(B,aa(C,B,A4,X2),aa(C,B,B1,X2)) ) ).
tff(fact_83_fold__graph_Osimps,axiom,
! [B: $tType,C: $tType,A21: C,A11: fun(B,bool),Z4: C,F2: fun(B,fun(C,C))] :
( finite_fold_graph(B,C,F2,Z4,A11,A21)
<=> ( ( ( A11 = bot_bot(fun(B,bool)) )
& ( A21 = Z4 ) )
| ? [X3: B,A3: fun(B,bool),Y2: C] :
( ( A11 = insert(B,X3,A3) )
& ( A21 = aa(C,C,aa(B,fun(C,C),F2,X3),Y2) )
& ~ pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),X3),A3))
& finite_fold_graph(B,C,F2,Z4,A3,Y2) ) ) ) ).
tff(fact_84_folding__one_Oremove,axiom,
! [B: $tType,X2: B,A4: fun(B,bool),F: fun(fun(B,bool),B),F2: fun(B,fun(B,B))] :
( finite_folding_one(B,F2,F)
=> ( finite_finite1(B,A4)
=> ( pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),X2),A4))
=> ( ( ( minus_minus(fun(B,bool),A4,insert(B,X2,bot_bot(fun(B,bool)))) = bot_bot(fun(B,bool)) )
=> ( aa(fun(B,bool),B,F,A4) = X2 ) )
& ( ( minus_minus(fun(B,bool),A4,insert(B,X2,bot_bot(fun(B,bool)))) != bot_bot(fun(B,bool)) )
=> ( aa(fun(B,bool),B,F,A4) = aa(B,B,aa(B,fun(B,B),F2,X2),aa(fun(B,bool),B,F,minus_minus(fun(B,bool),A4,insert(B,X2,bot_bot(fun(B,bool)))))) ) ) ) ) ) ) ).
tff(fact_85_finite__code,axiom,
! [B: $tType] :
( finite_finite(B)
=> ! [A4: fun(B,bool)] : finite_finite1(B,A4) ) ).
tff(fact_86_finite_OemptyI,axiom,
! [B: $tType] : finite_finite1(B,bot_bot(fun(B,bool))) ).
tff(fact_87_finite_OinsertI,axiom,
! [B: $tType,A1: B,A4: fun(B,bool)] :
( finite_finite1(B,A4)
=> finite_finite1(B,insert(B,A1,A4)) ) ).
tff(fact_88_finite__insert,axiom,
! [B: $tType,A4: fun(B,bool),A1: B] :
( finite_finite1(B,insert(B,A1,A4))
<=> finite_finite1(B,A4) ) ).
tff(fact_89_finite__Diff,axiom,
! [B: $tType,B1: fun(B,bool),A4: fun(B,bool)] :
( finite_finite1(B,A4)
=> finite_finite1(B,minus_minus(fun(B,bool),A4,B1)) ) ).
tff(fact_90_finite__Diff__insert,axiom,
! [B: $tType,B1: fun(B,bool),A1: B,A4: fun(B,bool)] :
( finite_finite1(B,minus_minus(fun(B,bool),A4,insert(B,A1,B1)))
<=> finite_finite1(B,minus_minus(fun(B,bool),A4,B1)) ) ).
tff(fact_91_finite__Diff2,axiom,
! [B: $tType,A4: fun(B,bool),B1: fun(B,bool)] :
( finite_finite1(B,B1)
=> ( finite_finite1(B,minus_minus(fun(B,bool),A4,B1))
<=> finite_finite1(B,A4) ) ) ).
tff(fact_92_finite,axiom,
! [B: $tType] :
( finite_finite(B)
=> ! [A4: fun(B,bool)] : finite_finite1(B,A4) ) ).
tff(fact_93_folding__one_Oinsert,axiom,
! [B: $tType,X2: B,A4: fun(B,bool),F: fun(fun(B,bool),B),F2: fun(B,fun(B,B))] :
( finite_folding_one(B,F2,F)
=> ( finite_finite1(B,A4)
=> ( ~ pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),X2),A4))
=> ( ( A4 != bot_bot(fun(B,bool)) )
=> ( aa(fun(B,bool),B,F,insert(B,X2,A4)) = aa(B,B,aa(B,fun(B,B),F2,X2),aa(fun(B,bool),B,F,A4)) ) ) ) ) ) ).
tff(fact_94_folding__one_Oinsert__remove,axiom,
! [B: $tType,X2: B,A4: fun(B,bool),F: fun(fun(B,bool),B),F2: fun(B,fun(B,B))] :
( finite_folding_one(B,F2,F)
=> ( finite_finite1(B,A4)
=> ( ( ( minus_minus(fun(B,bool),A4,insert(B,X2,bot_bot(fun(B,bool)))) = bot_bot(fun(B,bool)) )
=> ( aa(fun(B,bool),B,F,insert(B,X2,A4)) = X2 ) )
& ( ( minus_minus(fun(B,bool),A4,insert(B,X2,bot_bot(fun(B,bool)))) != bot_bot(fun(B,bool)) )
=> ( aa(fun(B,bool),B,F,insert(B,X2,A4)) = aa(B,B,aa(B,fun(B,B),F2,X2),aa(fun(B,bool),B,F,minus_minus(fun(B,bool),A4,insert(B,X2,bot_bot(fun(B,bool)))))) ) ) ) ) ) ).
tff(fact_95_folding__one_Oclosed,axiom,
! [B: $tType,A4: fun(B,bool),F: fun(fun(B,bool),B),F2: fun(B,fun(B,B))] :
( finite_folding_one(B,F2,F)
=> ( finite_finite1(B,A4)
=> ( ( A4 != bot_bot(fun(B,bool)) )
=> ( ! [X1: B,Y1: B] : pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),aa(B,B,aa(B,fun(B,B),F2,X1),Y1)),insert(B,X1,insert(B,Y1,bot_bot(fun(B,bool))))))
=> pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),aa(fun(B,bool),B,F,A4)),A4)) ) ) ) ) ).
tff(fact_96_finite__empty__induct,axiom,
! [B: $tType,Pa: fun(fun(B,bool),bool),A4: fun(B,bool)] :
( finite_finite1(B,A4)
=> ( pp(aa(fun(B,bool),bool,Pa,A4))
=> ( ! [A6: B,A5: fun(B,bool)] :
( finite_finite1(B,A5)
=> ( pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),A6),A5))
=> ( pp(aa(fun(B,bool),bool,Pa,A5))
=> pp(aa(fun(B,bool),bool,Pa,minus_minus(fun(B,bool),A5,insert(B,A6,bot_bot(fun(B,bool)))))) ) ) )
=> pp(aa(fun(B,bool),bool,Pa,bot_bot(fun(B,bool)))) ) ) ) ).
tff(fact_97_finite__nonempty__imp__fold1Set,axiom,
! [B: $tType,F2: fun(B,fun(B,B)),A4: fun(B,bool)] :
( finite_finite1(B,A4)
=> ( ( A4 != bot_bot(fun(B,bool)) )
=> ? [X11: B] : finite_fold1Set(B,F2,A4,X11) ) ) ).
tff(fact_98_finite_Osimps,axiom,
! [B: $tType,A1: fun(B,bool)] :
( finite_finite1(B,A1)
<=> ( ( A1 = bot_bot(fun(B,bool)) )
| ? [A3: fun(B,bool),A2: B] :
( ( A1 = insert(B,A2,A3) )
& finite_finite1(B,A3) ) ) ) ).
tff(fact_99_finite__induct,axiom,
! [B: $tType,Pa: fun(fun(B,bool),bool),F: fun(B,bool)] :
( finite_finite1(B,F)
=> ( pp(aa(fun(B,bool),bool,Pa,bot_bot(fun(B,bool))))
=> ( ! [X1: B,F1: fun(B,bool)] :
( finite_finite1(B,F1)
=> ( ~ pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),X1),F1))
=> ( pp(aa(fun(B,bool),bool,Pa,F1))
=> pp(aa(fun(B,bool),bool,Pa,insert(B,X1,F1))) ) ) )
=> pp(aa(fun(B,bool),bool,Pa,F)) ) ) ) ).
%----Arities (6)
tff(arity_fun___Finite__Set_Ofinite,axiom,
! [T_1: $tType,T_2: $tType] :
( ( finite_finite(T_2)
& finite_finite(T_1) )
=> finite_finite(fun(T_1,T_2)) ) ).
tff(arity_fun___Orderings_Obot,axiom,
! [T_1: $tType,T_2: $tType] :
( bot(T_2)
=> bot(fun(T_1,T_2)) ) ).
tff(arity_fun___Groups_Ominus,axiom,
! [T_1: $tType,T_2: $tType] :
( cl_Groups_Ominus(T_2)
=> cl_Groups_Ominus(fun(T_1,T_2)) ) ).
tff(arity_HOL_Obool___Finite__Set_Ofinite,axiom,
finite_finite(bool) ).
tff(arity_HOL_Obool___Orderings_Obot,axiom,
bot(bool) ).
tff(arity_HOL_Obool___Groups_Ominus,axiom,
cl_Groups_Ominus(bool) ).
%----Helper facts (21)
tff(help_pp_1_1_U,axiom,
~ pp(fFalse) ).
tff(help_pp_2_1_U,axiom,
pp(fTrue) ).
tff(help_fNot_1_1_U,axiom,
! [P1: bool] :
( ~ pp(aa(bool,bool,fNot,P1))
| ~ pp(P1) ) ).
tff(help_fNot_2_1_U,axiom,
! [P1: bool] :
( pp(P1)
| pp(aa(bool,bool,fNot,P1)) ) ).
tff(help_COMBB_1_1_U,axiom,
! [C: $tType,B: $tType,A: $tType,R: A,Q1: fun(A,B),P1: fun(B,C)] : aa(A,C,combb(B,C,A,P1,Q1),R) = aa(B,C,P1,aa(A,B,Q1,R)) ).
tff(help_COMBC_1_1_U,axiom,
! [A: $tType,C: $tType,B: $tType,R: A,Q1: B,P1: fun(A,fun(B,C))] : aa(A,C,combc(A,B,C,P1,Q1),R) = aa(B,C,aa(A,fun(B,C),P1,R),Q1) ).
tff(help_COMBK_1_1_U,axiom,
! [B: $tType,A: $tType,Q1: B,P1: A] : aa(B,A,combk(A,B,P1),Q1) = P1 ).
tff(help_COMBS_1_1_U,axiom,
! [C: $tType,B: $tType,A: $tType,R: A,Q1: fun(A,B),P1: fun(A,fun(B,C))] : aa(A,C,combs(A,B,C,P1,Q1),R) = aa(B,C,aa(A,fun(B,C),P1,R),aa(A,B,Q1,R)) ).
tff(help_fconj_1_1_U,axiom,
! [Q1: bool,P1: bool] :
( ~ pp(P1)
| ~ pp(Q1)
| pp(aa(bool,bool,aa(bool,fun(bool,bool),fconj,P1),Q1)) ) ).
tff(help_fconj_2_1_U,axiom,
! [Q1: bool,P1: bool] :
( ~ pp(aa(bool,bool,aa(bool,fun(bool,bool),fconj,P1),Q1))
| pp(P1) ) ).
tff(help_fconj_3_1_U,axiom,
! [Q1: bool,P1: bool] :
( ~ pp(aa(bool,bool,aa(bool,fun(bool,bool),fconj,P1),Q1))
| pp(Q1) ) ).
tff(help_fdisj_1_1_U,axiom,
! [Q1: bool,P1: bool] :
( ~ pp(P1)
| pp(aa(bool,bool,aa(bool,fun(bool,bool),fdisj,P1),Q1)) ) ).
tff(help_fdisj_2_1_U,axiom,
! [P1: bool,Q1: bool] :
( ~ pp(Q1)
| pp(aa(bool,bool,aa(bool,fun(bool,bool),fdisj,P1),Q1)) ) ).
tff(help_fdisj_3_1_U,axiom,
! [Q1: bool,P1: bool] :
( ~ pp(aa(bool,bool,aa(bool,fun(bool,bool),fdisj,P1),Q1))
| pp(P1)
| pp(Q1) ) ).
tff(help_fFalse_1_1_U,axiom,
~ pp(fFalse) ).
tff(help_fFalse_1_1_T,axiom,
! [P1: bool] :
( ( P1 = fTrue )
| ( P1 = 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,
! [Q1: bool,P1: bool] :
( pp(P1)
| pp(aa(bool,bool,aa(bool,fun(bool,bool),fimplies,P1),Q1)) ) ).
tff(help_fimplies_2_1_U,axiom,
! [P1: bool,Q1: bool] :
( ~ pp(Q1)
| pp(aa(bool,bool,aa(bool,fun(bool,bool),fimplies,P1),Q1)) ) ).
tff(help_fimplies_3_1_U,axiom,
! [Q1: bool,P1: bool] :
( ~ pp(aa(bool,bool,aa(bool,fun(bool,bool),fimplies,P1),Q1))
| ~ pp(P1)
| pp(Q1) ) ).
%----Conjectures (3)
tff(conj_0,hypothesis,
hoare_992312373derivs(a,g,insert(hoare_28830079triple(a),hoare_1841697145triple(a,p1,c,q1),bot_bot(fun(hoare_28830079triple(a),bool)))) ).
tff(conj_1,hypothesis,
! [Z2: a,S2: state] :
( p(Z2,S2)
=> ! [S3: state] :
( ! [Z3: a] :
( pp(aa(state,bool,aa(a,fun(state,bool),p1,Z3),S2))
=> pp(aa(state,bool,aa(a,fun(state,bool),q1,Z3),S3)) )
=> q(Z2,S3) ) ) ).
tff(conj_2,conjecture,
! [Z: a,S: state] :
( ~ p(Z,S)
| ? [P: fun(a,fun(state,bool)),Q: fun(a,fun(state,bool))] :
( hoare_992312373derivs(a,g,insert(hoare_28830079triple(a),hoare_1841697145triple(a,P,c,Q),bot_bot(fun(hoare_28830079triple(a),bool))))
& ! [S1: state] :
( ? [Z1: a] :
( pp(aa(state,bool,aa(a,fun(state,bool),P,Z1),S))
& ~ pp(aa(state,bool,aa(a,fun(state,bool),Q,Z1),S1)) )
| q(Z,S1) ) ) ) ).
%------------------------------------------------------------------------------