TPTP Problem File: SWW512_5.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SWW512_5 : TPTP v8.2.0. Released v6.0.0.
% Domain : Software Verification
% Problem : Hoare's Logic with Procedures line 213
% Version : Especial.
% English :
% Refs : [BN10] Boehme & Nipkow (2010), Sledgehammer: Judgement Day
% : [Bla13] Blanchette (2011), Email to Geoff Sutcliffe
% Source : [Bla13]
% Names : hoare_213 [Bla13]
% Status : Theorem
% Rating : 0.00 v7.5.0, 0.33 v7.4.0, 0.00 v6.4.0
% Syntax : Number of formulae : 160 ( 33 unt; 34 typ; 0 def)
% Number of atoms : 311 ( 63 equ)
% Maximal formula atoms : 7 ( 1 avg)
% Number of connectives : 212 ( 27 ~; 11 |; 13 &)
% ( 25 <=>; 136 =>; 0 <=; 0 <~>)
% Maximal formula depth : 14 ( 6 avg)
% Maximal term depth : 6 ( 2 avg)
% Number of types : 3 ( 2 usr)
% Number of type conns : 26 ( 16 >; 10 *; 0 +; 0 <<)
% Number of predicates : 12 ( 11 usr; 0 prp; 1-6 aty)
% Number of functors : 21 ( 21 usr; 5 con; 0-5 aty)
% Number of variables : 423 ( 387 !; 3 ?; 423 :)
% ( 33 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:16:13
%------------------------------------------------------------------------------
%----Should-be-implicit typings (4)
tff(ty_t_a,type,
a: $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 (30)
tff(sy_cl_Enum_Oenum,type,
enum:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Obot,type,
bot:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Oorder,type,
order:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Opreorder,type,
preorder:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Owellorder,type,
wellorder:
!>[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_Enum_Oenum__class_Oenum__all,type,
enum_enum_all:
!>[A: $tType] : ( fun(A,bool) > $o ) ).
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_Orderings_Obot__class_Obot,type,
bot_bot:
!>[A: $tType] : A ).
tff(sy_c_Orderings_Oord__class_OLeast,type,
ord_Least:
!>[A: $tType] : ( fun(A,bool) > A ) ).
tff(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : fun(A,fun(A,bool)) ).
tff(sy_c_Predicate_OPowp,type,
powp:
!>[A: $tType] : ( fun(A,bool) > fun(fun(A,bool),bool) ) ).
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_OPow,type,
pow:
!>[A: $tType] : ( fun(A,bool) > fun(fun(A,bool),bool) ) ).
tff(sy_c_Set_Oinsert,type,
insert:
!>[A: $tType] : ( ( A * fun(A,bool) ) > fun(A,bool) ) ).
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_fTrue,type,
fTrue: bool ).
tff(sy_c_fconj,type,
fconj: fun(bool,fun(bool,bool)) ).
tff(sy_c_fequal,type,
fequal:
!>[A: $tType] : fun(A,fun(A,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_Ga,type,
ga: fun(hoare_28830079triple(a),bool) ).
%----Relevant facts (100)
tff(fact_0_empty,axiom,
! [B: $tType,G1: fun(hoare_28830079triple(B),bool)] : hoare_992312373derivs(B,G1,bot_bot(fun(hoare_28830079triple(B),bool))) ).
tff(fact_1_asm,axiom,
! [B: $tType,G1: fun(hoare_28830079triple(B),bool),Tsa: fun(hoare_28830079triple(B),bool)] :
( pp(aa(fun(hoare_28830079triple(B),bool),bool,aa(fun(hoare_28830079triple(B),bool),fun(fun(hoare_28830079triple(B),bool),bool),ord_less_eq(fun(hoare_28830079triple(B),bool)),Tsa),G1))
=> hoare_992312373derivs(B,G1,Tsa) ) ).
tff(fact_2_weaken,axiom,
! [B: $tType,Tsa: fun(hoare_28830079triple(B),bool),Ts: fun(hoare_28830079triple(B),bool),G1: fun(hoare_28830079triple(B),bool)] :
( hoare_992312373derivs(B,G1,Ts)
=> ( pp(aa(fun(hoare_28830079triple(B),bool),bool,aa(fun(hoare_28830079triple(B),bool),fun(fun(hoare_28830079triple(B),bool),bool),ord_less_eq(fun(hoare_28830079triple(B),bool)),Tsa),Ts))
=> hoare_992312373derivs(B,G1,Tsa) ) ) ).
tff(fact_3_cut,axiom,
! [B: $tType,G1: fun(hoare_28830079triple(B),bool),Tsa: fun(hoare_28830079triple(B),bool),G_a: fun(hoare_28830079triple(B),bool)] :
( hoare_992312373derivs(B,G_a,Tsa)
=> ( hoare_992312373derivs(B,G1,G_a)
=> hoare_992312373derivs(B,G1,Tsa) ) ) ).
tff(fact_4_empty__subsetI,axiom,
! [B: $tType,A1: fun(B,bool)] : pp(aa(fun(B,bool),bool,aa(fun(B,bool),fun(fun(B,bool),bool),ord_less_eq(fun(B,bool)),bot_bot(fun(B,bool))),A1)) ).
tff(fact_5_subset__empty,axiom,
! [B: $tType,A1: fun(B,bool)] :
( pp(aa(fun(B,bool),bool,aa(fun(B,bool),fun(fun(B,bool),bool),ord_less_eq(fun(B,bool)),A1),bot_bot(fun(B,bool))))
<=> ( A1 = bot_bot(fun(B,bool)) ) ) ).
tff(fact_6_subsetD,axiom,
! [B: $tType,C1: B,B1: fun(B,bool),A1: fun(B,bool)] :
( pp(aa(fun(B,bool),bool,aa(fun(B,bool),fun(fun(B,bool),bool),ord_less_eq(fun(B,bool)),A1),B1))
=> ( pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),C1),A1))
=> pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),C1),B1)) ) ) ).
tff(fact_7_equalityI,axiom,
! [B: $tType,B1: fun(B,bool),A1: fun(B,bool)] :
( pp(aa(fun(B,bool),bool,aa(fun(B,bool),fun(fun(B,bool),bool),ord_less_eq(fun(B,bool)),A1),B1))
=> ( pp(aa(fun(B,bool),bool,aa(fun(B,bool),fun(fun(B,bool),bool),ord_less_eq(fun(B,bool)),B1),A1))
=> ( A1 = B1 ) ) ) ).
tff(fact_8_all__not__in__conv,axiom,
! [B: $tType,A1: fun(B,bool)] :
( ! [X4: B] : ~ pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),X4),A1))
<=> ( A1 = bot_bot(fun(B,bool)) ) ) ).
tff(fact_9_empty__Collect__eq,axiom,
! [B: $tType,P1: fun(B,bool)] :
( ( bot_bot(fun(B,bool)) = collect(B,P1) )
<=> ! [X4: B] : ~ pp(aa(B,bool,P1,X4)) ) ).
tff(fact_10_empty__iff,axiom,
! [B: $tType,C1: B] : ~ pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),C1),bot_bot(fun(B,bool)))) ).
tff(fact_11_Collect__empty__eq,axiom,
! [B: $tType,P1: fun(B,bool)] :
( ( collect(B,P1) = bot_bot(fun(B,bool)) )
<=> ! [X4: B] : ~ pp(aa(B,bool,P1,X4)) ) ).
tff(fact_12_emptyE,axiom,
! [B: $tType,A2: B] : ~ pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),A2),bot_bot(fun(B,bool)))) ).
tff(fact_13_order__refl,axiom,
! [A: $tType] :
( preorder(A)
=> ! [X: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),X)) ) ).
tff(fact_14_bot__fun__def,axiom,
! [B: $tType,C: $tType] :
( bot(C)
=> ! [X3: B] : aa(B,C,bot_bot(fun(B,C)),X3) = bot_bot(C) ) ).
tff(fact_15_bot__apply,axiom,
! [C: $tType,B: $tType] :
( bot(B)
=> ! [X1: C] : aa(C,B,bot_bot(fun(C,B)),X1) = bot_bot(B) ) ).
tff(fact_16_linorder__le__cases,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y: A,X: A] :
( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X)) ) ) ).
tff(fact_17_order__trans,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Z: A,Y: A,X: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),Z))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Z)) ) ) ) ).
tff(fact_18_order__antisym,axiom,
! [A: $tType] :
( order(A)
=> ! [Y: A,X: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X))
=> ( X = Y ) ) ) ) ).
tff(fact_19_ord__le__eq__trans,axiom,
! [A: $tType] :
( ord(A)
=> ! [C3: A,B4: A,A4: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A4),B4))
=> ( ( B4 = C3 )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A4),C3)) ) ) ) ).
tff(fact_20_ord__eq__le__trans,axiom,
! [A: $tType] :
( ord(A)
=> ! [C3: A,B4: A,A4: A] :
( ( A4 = B4 )
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B4),C3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A4),C3)) ) ) ) ).
tff(fact_21_order__antisym__conv,axiom,
! [B: $tType] :
( order(B)
=> ! [X1: B,Y2: B] :
( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),Y2),X1))
=> ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),X1),Y2))
<=> ( X1 = Y2 ) ) ) ) ).
tff(fact_22_order__eq__refl,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Y: A,X: A] :
( ( X = Y )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y)) ) ) ).
tff(fact_23_order__eq__iff,axiom,
! [B: $tType] :
( order(B)
=> ! [Y2: B,X1: B] :
( ( X1 = Y2 )
<=> ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),X1),Y2))
& pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),Y2),X1)) ) ) ) ).
tff(fact_24_linorder__linear,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y: A,X: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
| pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X)) ) ) ).
tff(fact_25_equals0D,axiom,
! [B: $tType,A2: B,A1: fun(B,bool)] :
( ( A1 = bot_bot(fun(B,bool)) )
=> ~ pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),A2),A1)) ) ).
tff(fact_26_ex__in__conv,axiom,
! [B: $tType,A1: fun(B,bool)] :
( ? [X4: B] : pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),X4),A1))
<=> ( A1 != bot_bot(fun(B,bool)) ) ) ).
tff(fact_27_empty__def,axiom,
! [B: $tType] : bot_bot(fun(B,bool)) = collect(B,combk(bool,B,fFalse)) ).
tff(fact_28_equalityE,axiom,
! [B: $tType,B1: fun(B,bool),A1: fun(B,bool)] :
( ( A1 = B1 )
=> ~ ( pp(aa(fun(B,bool),bool,aa(fun(B,bool),fun(fun(B,bool),bool),ord_less_eq(fun(B,bool)),A1),B1))
=> ~ pp(aa(fun(B,bool),bool,aa(fun(B,bool),fun(fun(B,bool),bool),ord_less_eq(fun(B,bool)),B1),A1)) ) ) ).
tff(fact_29_subset__trans,axiom,
! [B: $tType,C2: fun(B,bool),B1: fun(B,bool),A1: fun(B,bool)] :
( pp(aa(fun(B,bool),bool,aa(fun(B,bool),fun(fun(B,bool),bool),ord_less_eq(fun(B,bool)),A1),B1))
=> ( pp(aa(fun(B,bool),bool,aa(fun(B,bool),fun(fun(B,bool),bool),ord_less_eq(fun(B,bool)),B1),C2))
=> pp(aa(fun(B,bool),bool,aa(fun(B,bool),fun(fun(B,bool),bool),ord_less_eq(fun(B,bool)),A1),C2)) ) ) ).
tff(fact_30_set__mp,axiom,
! [B: $tType,X1: B,B1: fun(B,bool),A1: fun(B,bool)] :
( pp(aa(fun(B,bool),bool,aa(fun(B,bool),fun(fun(B,bool),bool),ord_less_eq(fun(B,bool)),A1),B1))
=> ( pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),X1),A1))
=> pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),X1),B1)) ) ) ).
tff(fact_31_set__rev__mp,axiom,
! [B: $tType,B1: fun(B,bool),A1: fun(B,bool),X1: B] :
( pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),X1),A1))
=> ( pp(aa(fun(B,bool),bool,aa(fun(B,bool),fun(fun(B,bool),bool),ord_less_eq(fun(B,bool)),A1),B1))
=> pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),X1),B1)) ) ) ).
tff(fact_32_in__mono,axiom,
! [B: $tType,X1: B,B1: fun(B,bool),A1: fun(B,bool)] :
( pp(aa(fun(B,bool),bool,aa(fun(B,bool),fun(fun(B,bool),bool),ord_less_eq(fun(B,bool)),A1),B1))
=> ( pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),X1),A1))
=> pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),X1),B1)) ) ) ).
tff(fact_33_equalityD2,axiom,
! [B: $tType,B1: fun(B,bool),A1: fun(B,bool)] :
( ( A1 = B1 )
=> pp(aa(fun(B,bool),bool,aa(fun(B,bool),fun(fun(B,bool),bool),ord_less_eq(fun(B,bool)),B1),A1)) ) ).
tff(fact_34_equalityD1,axiom,
! [B: $tType,B1: fun(B,bool),A1: fun(B,bool)] :
( ( A1 = B1 )
=> pp(aa(fun(B,bool),bool,aa(fun(B,bool),fun(fun(B,bool),bool),ord_less_eq(fun(B,bool)),A1),B1)) ) ).
tff(fact_35_set__eq__subset,axiom,
! [B: $tType,B1: fun(B,bool),A1: fun(B,bool)] :
( ( A1 = B1 )
<=> ( pp(aa(fun(B,bool),bool,aa(fun(B,bool),fun(fun(B,bool),bool),ord_less_eq(fun(B,bool)),A1),B1))
& pp(aa(fun(B,bool),bool,aa(fun(B,bool),fun(fun(B,bool),bool),ord_less_eq(fun(B,bool)),B1),A1)) ) ) ).
tff(fact_36_subset__refl,axiom,
! [B: $tType,A1: fun(B,bool)] : pp(aa(fun(B,bool),bool,aa(fun(B,bool),fun(fun(B,bool),bool),ord_less_eq(fun(B,bool)),A1),A1)) ).
tff(fact_37_le__bot,axiom,
! [A: $tType] :
( bot(A)
=> ! [A4: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A4),bot_bot(A)))
=> ( A4 = bot_bot(A) ) ) ) ).
tff(fact_38_bot__unique,axiom,
! [B: $tType] :
( bot(B)
=> ! [A2: B] :
( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),A2),bot_bot(B)))
<=> ( A2 = bot_bot(B) ) ) ) ).
tff(fact_39_bot__least,axiom,
! [A: $tType] :
( bot(A)
=> ! [A4: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),bot_bot(A)),A4)) ) ).
tff(fact_40_le__funE,axiom,
! [C: $tType,B: $tType] :
( ord(C)
=> ! [X1: B,G: fun(B,C),F: fun(B,C)] :
( pp(aa(fun(B,C),bool,aa(fun(B,C),fun(fun(B,C),bool),ord_less_eq(fun(B,C)),F),G))
=> pp(aa(C,bool,aa(C,fun(C,bool),ord_less_eq(C),aa(B,C,F,X1)),aa(B,C,G,X1))) ) ) ).
tff(fact_41_le__funD,axiom,
! [C: $tType,B: $tType] :
( ord(C)
=> ! [X1: B,G: fun(B,C),F: fun(B,C)] :
( pp(aa(fun(B,C),bool,aa(fun(B,C),fun(fun(B,C),bool),ord_less_eq(fun(B,C)),F),G))
=> pp(aa(C,bool,aa(C,fun(C,bool),ord_less_eq(C),aa(B,C,F,X1)),aa(B,C,G,X1))) ) ) ).
tff(fact_42_le__fun__def,axiom,
! [C: $tType,B: $tType] :
( ord(C)
=> ! [G: fun(B,C),F: fun(B,C)] :
( pp(aa(fun(B,C),bool,aa(fun(B,C),fun(fun(B,C),bool),ord_less_eq(fun(B,C)),F),G))
<=> ! [X4: B] : pp(aa(C,bool,aa(C,fun(C,bool),ord_less_eq(C),aa(B,C,F,X4)),aa(B,C,G,X4))) ) ) ).
tff(fact_43_subsetI,axiom,
! [B: $tType,B1: fun(B,bool),A1: fun(B,bool)] :
( ! [X2: B] :
( pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),X2),A1))
=> pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),X2),B1)) )
=> pp(aa(fun(B,bool),bool,aa(fun(B,bool),fun(fun(B,bool),bool),ord_less_eq(fun(B,bool)),A1),B1)) ) ).
tff(fact_44_le__funI,axiom,
! [C: $tType,B: $tType] :
( ord(C)
=> ! [G: fun(B,C),F: fun(B,C)] :
( ! [X2: B] : pp(aa(C,bool,aa(C,fun(C,bool),ord_less_eq(C),aa(B,C,F,X2)),aa(B,C,G,X2)))
=> pp(aa(fun(B,C),bool,aa(fun(B,C),fun(fun(B,C),bool),ord_less_eq(fun(B,C)),F),G)) ) ) ).
tff(fact_45_Collect__mono,axiom,
! [B: $tType,Q1: fun(B,bool),P1: fun(B,bool)] :
( ! [X2: B] :
( pp(aa(B,bool,P1,X2))
=> pp(aa(B,bool,Q1,X2)) )
=> pp(aa(fun(B,bool),bool,aa(fun(B,bool),fun(fun(B,bool),bool),ord_less_eq(fun(B,bool)),collect(B,P1)),collect(B,Q1))) ) ).
tff(fact_46_pred__subset__eq,axiom,
! [B: $tType,S: fun(B,bool),R2: fun(B,bool)] :
( pp(aa(fun(B,bool),bool,aa(fun(B,bool),fun(fun(B,bool),bool),ord_less_eq(fun(B,bool)),combc(B,fun(B,bool),bool,member(B),R2)),combc(B,fun(B,bool),bool,member(B),S)))
<=> pp(aa(fun(B,bool),bool,aa(fun(B,bool),fun(fun(B,bool),bool),ord_less_eq(fun(B,bool)),R2),S)) ) ).
tff(fact_47_equals0I,axiom,
! [B: $tType,A1: fun(B,bool)] :
( ! [Y1: B] : ~ pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),Y1),A1))
=> ( A1 = bot_bot(fun(B,bool)) ) ) ).
tff(fact_48_order__subst1,axiom,
! [B: $tType,C: $tType] :
( ( order(C)
& order(B) )
=> ! [C1: C,B2: C,F: fun(C,B),A2: B] :
( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),A2),aa(C,B,F,B2)))
=> ( pp(aa(C,bool,aa(C,fun(C,bool),ord_less_eq(C),B2),C1))
=> ( ! [X2: C,Y1: C] :
( pp(aa(C,bool,aa(C,fun(C,bool),ord_less_eq(C),X2),Y1))
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(C,B,F,X2)),aa(C,B,F,Y1))) )
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),A2),aa(C,B,F,C1))) ) ) ) ) ).
tff(fact_49_ord__eq__le__subst,axiom,
! [B: $tType,C: $tType] :
( ( ord(C)
& ord(B) )
=> ! [C1: C,B2: C,F: fun(C,B),A2: B] :
( ( A2 = aa(C,B,F,B2) )
=> ( pp(aa(C,bool,aa(C,fun(C,bool),ord_less_eq(C),B2),C1))
=> ( ! [X2: C,Y1: C] :
( pp(aa(C,bool,aa(C,fun(C,bool),ord_less_eq(C),X2),Y1))
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(C,B,F,X2)),aa(C,B,F,Y1))) )
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),A2),aa(C,B,F,C1))) ) ) ) ) ).
tff(fact_50_order__subst2,axiom,
! [B: $tType,C: $tType] :
( ( order(C)
& order(B) )
=> ! [C1: C,F: fun(B,C),B2: B,A2: B] :
( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),A2),B2))
=> ( pp(aa(C,bool,aa(C,fun(C,bool),ord_less_eq(C),aa(B,C,F,B2)),C1))
=> ( ! [X2: B,Y1: B] :
( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),X2),Y1))
=> pp(aa(C,bool,aa(C,fun(C,bool),ord_less_eq(C),aa(B,C,F,X2)),aa(B,C,F,Y1))) )
=> pp(aa(C,bool,aa(C,fun(C,bool),ord_less_eq(C),aa(B,C,F,A2)),C1)) ) ) ) ) ).
tff(fact_51_bot__empty__eq,axiom,
! [B: $tType,X3: B] :
( pp(aa(B,bool,bot_bot(fun(B,bool)),X3))
<=> pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),X3),bot_bot(fun(B,bool)))) ) ).
tff(fact_52_rev__predicate1D,axiom,
! [B: $tType,Q1: fun(B,bool),X1: B,P1: fun(B,bool)] :
( pp(aa(B,bool,P1,X1))
=> ( pp(aa(fun(B,bool),bool,aa(fun(B,bool),fun(fun(B,bool),bool),ord_less_eq(fun(B,bool)),P1),Q1))
=> pp(aa(B,bool,Q1,X1)) ) ) ).
tff(fact_53_predicate1D,axiom,
! [B: $tType,X1: B,Q1: fun(B,bool),P1: fun(B,bool)] :
( pp(aa(fun(B,bool),bool,aa(fun(B,bool),fun(fun(B,bool),bool),ord_less_eq(fun(B,bool)),P1),Q1))
=> ( pp(aa(B,bool,P1,X1))
=> pp(aa(B,bool,Q1,X1)) ) ) ).
tff(fact_54_predicate1I,axiom,
! [B: $tType,Q1: fun(B,bool),P1: fun(B,bool)] :
( ! [X2: B] :
( pp(aa(B,bool,P1,X2))
=> pp(aa(B,bool,Q1,X2)) )
=> pp(aa(fun(B,bool),bool,aa(fun(B,bool),fun(fun(B,bool),bool),ord_less_eq(fun(B,bool)),P1),Q1)) ) ).
tff(fact_55_ord__le__eq__subst,axiom,
! [B: $tType,C: $tType] :
( ( ord(C)
& ord(B) )
=> ! [C1: C,F: fun(B,C),B2: B,A2: B] :
( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),A2),B2))
=> ( ( aa(B,C,F,B2) = C1 )
=> ( ! [X2: B,Y1: B] :
( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),X2),Y1))
=> pp(aa(C,bool,aa(C,fun(C,bool),ord_less_eq(C),aa(B,C,F,X2)),aa(B,C,F,Y1))) )
=> pp(aa(C,bool,aa(C,fun(C,bool),ord_less_eq(C),aa(B,C,F,A2)),C1)) ) ) ) ) ).
tff(fact_56_Powp__mono,axiom,
! [B: $tType,B1: fun(B,bool),A1: fun(B,bool)] :
( pp(aa(fun(B,bool),bool,aa(fun(B,bool),fun(fun(B,bool),bool),ord_less_eq(fun(B,bool)),A1),B1))
=> pp(aa(fun(fun(B,bool),bool),bool,aa(fun(fun(B,bool),bool),fun(fun(fun(B,bool),bool),bool),ord_less_eq(fun(fun(B,bool),bool)),powp(B,A1)),powp(B,B1))) ) ).
tff(fact_57_in__inv__imagep,axiom,
! [B: $tType,C: $tType,Y2: C,X1: C,F: fun(C,B),R1: fun(B,fun(B,bool))] :
( inv_imagep(B,C,R1,F,X1,Y2)
<=> pp(aa(B,bool,aa(B,fun(B,bool),R1,aa(C,B,F,X1)),aa(C,B,F,Y2))) ) ).
tff(fact_58_inv__imagep__def,axiom,
! [C: $tType,B: $tType,F: fun(B,C),R1: fun(C,fun(C,bool)),X3: B,Xa: B] :
( inv_imagep(C,B,R1,F,X3,Xa)
<=> pp(aa(C,bool,aa(C,fun(C,bool),R1,aa(B,C,F,X3)),aa(B,C,F,Xa))) ) ).
tff(fact_59_Pow__mono,axiom,
! [B: $tType,B1: fun(B,bool),A1: fun(B,bool)] :
( pp(aa(fun(B,bool),bool,aa(fun(B,bool),fun(fun(B,bool),bool),ord_less_eq(fun(B,bool)),A1),B1))
=> pp(aa(fun(fun(B,bool),bool),bool,aa(fun(fun(B,bool),bool),fun(fun(fun(B,bool),bool),bool),ord_less_eq(fun(fun(B,bool),bool)),pow(B,A1)),pow(B,B1))) ) ).
tff(fact_60_order__fun_I1_J,axiom,
! [C: $tType,B: $tType] :
( ( enum(B)
& order(C) )
=> ! [G: fun(B,C),F: fun(B,C)] :
( pp(aa(fun(B,C),bool,aa(fun(B,C),fun(fun(B,C),bool),ord_less_eq(fun(B,C)),F),G))
<=> enum_enum_all(B,combs(B,C,bool,combb(C,fun(C,bool),B,ord_less_eq(C),F),G)) ) ) ).
tff(fact_61_Least__le,axiom,
! [B: $tType] :
( wellorder(B)
=> ! [K: B,P1: fun(B,bool)] :
( pp(aa(B,bool,P1,K))
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),ord_Least(B,P1)),K)) ) ) ).
tff(fact_62_Pow__iff,axiom,
! [B: $tType,B1: fun(B,bool),A1: fun(B,bool)] :
( pp(aa(fun(fun(B,bool),bool),bool,aa(fun(B,bool),fun(fun(fun(B,bool),bool),bool),member(fun(B,bool)),A1),pow(B,B1)))
<=> pp(aa(fun(B,bool),bool,aa(fun(B,bool),fun(fun(B,bool),bool),ord_less_eq(fun(B,bool)),A1),B1)) ) ).
tff(fact_63_PowI,axiom,
! [B: $tType,B1: fun(B,bool),A1: fun(B,bool)] :
( pp(aa(fun(B,bool),bool,aa(fun(B,bool),fun(fun(B,bool),bool),ord_less_eq(fun(B,bool)),A1),B1))
=> pp(aa(fun(fun(B,bool),bool),bool,aa(fun(B,bool),fun(fun(fun(B,bool),bool),bool),member(fun(B,bool)),A1),pow(B,B1))) ) ).
tff(fact_64_Pow__def,axiom,
! [B: $tType,A1: fun(B,bool)] : pow(B,A1) = collect(fun(B,bool),combc(fun(B,bool),fun(B,bool),bool,ord_less_eq(fun(B,bool)),A1)) ).
tff(fact_65_PowD,axiom,
! [B: $tType,B1: fun(B,bool),A1: fun(B,bool)] :
( pp(aa(fun(fun(B,bool),bool),bool,aa(fun(B,bool),fun(fun(fun(B,bool),bool),bool),member(fun(B,bool)),A1),pow(B,B1)))
=> pp(aa(fun(B,bool),bool,aa(fun(B,bool),fun(fun(B,bool),bool),ord_less_eq(fun(B,bool)),A1),B1)) ) ).
tff(fact_66_Pow__not__empty,axiom,
! [B: $tType,A1: fun(B,bool)] : pow(B,A1) != bot_bot(fun(fun(B,bool),bool)) ).
tff(fact_67_Pow__top,axiom,
! [B: $tType,A1: fun(B,bool)] : pp(aa(fun(fun(B,bool),bool),bool,aa(fun(B,bool),fun(fun(fun(B,bool),bool),bool),member(fun(B,bool)),A1),pow(B,A1))) ).
tff(fact_68_LeastI__ex,axiom,
! [B: $tType] :
( wellorder(B)
=> ! [P1: fun(B,bool)] :
( ? [X11: B] : pp(aa(B,bool,P1,X11))
=> pp(aa(B,bool,P1,ord_Least(B,P1))) ) ) ).
tff(fact_69_all__code,axiom,
! [B: $tType] :
( enum(B)
=> ! [P1: fun(B,bool)] :
( ! [X12: B] : pp(aa(B,bool,P1,X12))
<=> enum_enum_all(B,P1) ) ) ).
tff(fact_70_enum__all,axiom,
! [B: $tType] :
( enum(B)
=> ! [P1: fun(B,bool)] :
( enum_enum_all(B,P1)
<=> ! [X12: B] : pp(aa(B,bool,P1,X12)) ) ) ).
tff(fact_71_Pow__bottom,axiom,
! [B: $tType,B1: fun(B,bool)] : pp(aa(fun(fun(B,bool),bool),bool,aa(fun(B,bool),fun(fun(fun(B,bool),bool),bool),member(fun(B,bool)),bot_bot(fun(B,bool))),pow(B,B1))) ).
tff(fact_72_LeastI,axiom,
! [B: $tType] :
( wellorder(B)
=> ! [K: B,P1: fun(B,bool)] :
( pp(aa(B,bool,P1,K))
=> pp(aa(B,bool,P1,ord_Least(B,P1))) ) ) ).
tff(fact_73_Powp__Pow__eq,axiom,
! [B: $tType,A1: fun(B,bool),X3: fun(B,bool)] :
( pp(aa(fun(B,bool),bool,powp(B,combc(B,fun(B,bool),bool,member(B),A1)),X3))
<=> pp(aa(fun(fun(B,bool),bool),bool,aa(fun(B,bool),fun(fun(fun(B,bool),bool),bool),member(fun(B,bool)),X3),pow(B,A1))) ) ).
tff(fact_74_Least__equality,axiom,
! [B: $tType] :
( order(B)
=> ! [X1: B,P1: fun(B,bool)] :
( pp(aa(B,bool,P1,X1))
=> ( ! [Y1: B] :
( pp(aa(B,bool,P1,Y1))
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),X1),Y1)) )
=> ( ord_Least(B,P1) = X1 ) ) ) ) ).
tff(fact_75_ext,axiom,
! [C: $tType,B: $tType,G: fun(B,C),F: fun(B,C)] :
( ! [X2: B] : aa(B,C,F,X2) = aa(B,C,G,X2)
=> ( F = G ) ) ).
tff(fact_76_mem__def,axiom,
! [B: $tType,A1: fun(B,bool),X1: B] :
( pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),X1),A1))
<=> pp(aa(B,bool,A1,X1)) ) ).
tff(fact_77_Collect__def,axiom,
! [B: $tType,P1: fun(B,bool)] : collect(B,P1) = P1 ).
tff(fact_78_LeastI2__ex,axiom,
! [B: $tType] :
( wellorder(B)
=> ! [Q1: fun(B,bool),P1: fun(B,bool)] :
( ? [X11: B] : pp(aa(B,bool,P1,X11))
=> ( ! [X2: B] :
( pp(aa(B,bool,P1,X2))
=> pp(aa(B,bool,Q1,X2)) )
=> pp(aa(B,bool,Q1,ord_Least(B,P1))) ) ) ) ).
tff(fact_79_LeastI2,axiom,
! [B: $tType] :
( wellorder(B)
=> ! [Q1: fun(B,bool),A2: B,P1: fun(B,bool)] :
( pp(aa(B,bool,P1,A2))
=> ( ! [X2: B] :
( pp(aa(B,bool,P1,X2))
=> pp(aa(B,bool,Q1,X2)) )
=> pp(aa(B,bool,Q1,ord_Least(B,P1))) ) ) ) ).
tff(fact_80_LeastI2__wellorder,axiom,
! [B: $tType] :
( wellorder(B)
=> ! [Q1: fun(B,bool),A2: B,P1: fun(B,bool)] :
( pp(aa(B,bool,P1,A2))
=> ( ! [A3: B] :
( pp(aa(B,bool,P1,A3))
=> ( ! [B3: B] :
( pp(aa(B,bool,P1,B3))
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),A3),B3)) )
=> pp(aa(B,bool,Q1,A3)) ) )
=> pp(aa(B,bool,Q1,ord_Least(B,P1))) ) ) ) ).
tff(fact_81_Pow__empty,axiom,
! [B: $tType] : pow(B,bot_bot(fun(B,bool))) = insert(fun(B,bool),bot_bot(fun(B,bool)),bot_bot(fun(fun(B,bool),bool))) ).
tff(fact_82_insert__absorb2,axiom,
! [B: $tType,A1: fun(B,bool),X1: B] : insert(B,X1,insert(B,X1,A1)) = insert(B,X1,A1) ).
tff(fact_83_insert__iff,axiom,
! [B: $tType,A1: fun(B,bool),B2: B,A2: B] :
( pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),A2),insert(B,B2,A1)))
<=> ( ( A2 = B2 )
| pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),A2),A1)) ) ) ).
tff(fact_84_insertE,axiom,
! [B: $tType,A1: fun(B,bool),B2: B,A2: B] :
( pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),A2),insert(B,B2,A1)))
=> ( ( A2 != B2 )
=> pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),A2),A1)) ) ) ).
tff(fact_85_insertCI,axiom,
! [B: $tType,B2: B,B1: fun(B,bool),A2: B] :
( ( ~ pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),A2),B1))
=> ( A2 = B2 ) )
=> pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),A2),insert(B,B2,B1))) ) ).
tff(fact_86_insert__subset,axiom,
! [B: $tType,B1: fun(B,bool),A1: fun(B,bool),X1: B] :
( pp(aa(fun(B,bool),bool,aa(fun(B,bool),fun(fun(B,bool),bool),ord_less_eq(fun(B,bool)),insert(B,X1,A1)),B1))
<=> ( pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),X1),B1))
& pp(aa(fun(B,bool),bool,aa(fun(B,bool),fun(fun(B,bool),bool),ord_less_eq(fun(B,bool)),A1),B1)) ) ) ).
tff(fact_87_singleton__conv2,axiom,
! [B: $tType,A2: B] : collect(B,aa(B,fun(B,bool),fequal(B),A2)) = insert(B,A2,bot_bot(fun(B,bool))) ).
tff(fact_88_singleton__conv,axiom,
! [B: $tType,A2: B] : collect(B,combc(B,B,bool,fequal(B),A2)) = insert(B,A2,bot_bot(fun(B,bool))) ).
tff(fact_89_empty__not__insert,axiom,
! [B: $tType,A1: fun(B,bool),A2: B] : bot_bot(fun(B,bool)) != insert(B,A2,A1) ).
tff(fact_90_insert__not__empty,axiom,
! [B: $tType,A1: fun(B,bool),A2: B] : insert(B,A2,A1) != bot_bot(fun(B,bool)) ).
tff(fact_91_singleton__iff,axiom,
! [B: $tType,A2: B,B2: B] :
( pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),B2),insert(B,A2,bot_bot(fun(B,bool)))))
<=> ( B2 = A2 ) ) ).
tff(fact_92_doubleton__eq__iff,axiom,
! [B: $tType,D: B,C1: B,B2: B,A2: B] :
( ( insert(B,A2,insert(B,B2,bot_bot(fun(B,bool)))) = insert(B,C1,insert(B,D,bot_bot(fun(B,bool)))) )
<=> ( ( ( A2 = C1 )
& ( B2 = D ) )
| ( ( A2 = D )
& ( B2 = C1 ) ) ) ) ).
tff(fact_93_singletonE,axiom,
! [B: $tType,A2: B,B2: B] :
( pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),B2),insert(B,A2,bot_bot(fun(B,bool)))))
=> ( B2 = A2 ) ) ).
tff(fact_94_singleton__inject,axiom,
! [B: $tType,B2: B,A2: B] :
( ( insert(B,A2,bot_bot(fun(B,bool))) = insert(B,B2,bot_bot(fun(B,bool))) )
=> ( A2 = B2 ) ) ).
tff(fact_95_Collect__conv__if2,axiom,
! [B: $tType,A2: B,P1: fun(B,bool)] :
( ( pp(aa(B,bool,P1,A2))
=> ( collect(B,combs(B,bool,bool,combb(bool,fun(bool,bool),B,fconj,aa(B,fun(B,bool),fequal(B),A2)),P1)) = insert(B,A2,bot_bot(fun(B,bool))) ) )
& ( ~ pp(aa(B,bool,P1,A2))
=> ( collect(B,combs(B,bool,bool,combb(bool,fun(bool,bool),B,fconj,aa(B,fun(B,bool),fequal(B),A2)),P1)) = bot_bot(fun(B,bool)) ) ) ) ).
tff(fact_96_Collect__conv__if,axiom,
! [B: $tType,A2: B,P1: fun(B,bool)] :
( ( pp(aa(B,bool,P1,A2))
=> ( collect(B,combs(B,bool,bool,combb(bool,fun(bool,bool),B,fconj,combc(B,B,bool,fequal(B),A2)),P1)) = insert(B,A2,bot_bot(fun(B,bool))) ) )
& ( ~ pp(aa(B,bool,P1,A2))
=> ( collect(B,combs(B,bool,bool,combb(bool,fun(bool,bool),B,fconj,combc(B,B,bool,fequal(B),A2)),P1)) = bot_bot(fun(B,bool)) ) ) ) ).
tff(fact_97_subset__singletonD,axiom,
! [B: $tType,X1: B,A1: fun(B,bool)] :
( pp(aa(fun(B,bool),bool,aa(fun(B,bool),fun(fun(B,bool),bool),ord_less_eq(fun(B,bool)),A1),insert(B,X1,bot_bot(fun(B,bool)))))
=> ( ( A1 = bot_bot(fun(B,bool)) )
| ( A1 = insert(B,X1,bot_bot(fun(B,bool))) ) ) ) ).
tff(fact_98_subset__insertI,axiom,
! [B: $tType,A2: B,B1: fun(B,bool)] : pp(aa(fun(B,bool),bool,aa(fun(B,bool),fun(fun(B,bool),bool),ord_less_eq(fun(B,bool)),B1),insert(B,A2,B1))) ).
tff(fact_99_subset__insert,axiom,
! [B: $tType,B1: fun(B,bool),A1: fun(B,bool),X1: B] :
( ~ pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),X1),A1))
=> ( pp(aa(fun(B,bool),bool,aa(fun(B,bool),fun(fun(B,bool),bool),ord_less_eq(fun(B,bool)),A1),insert(B,X1,B1)))
<=> pp(aa(fun(B,bool),bool,aa(fun(B,bool),fun(fun(B,bool),bool),ord_less_eq(fun(B,bool)),A1),B1)) ) ) ).
%----Arities (11)
tff(arity_fun___Orderings_Opreorder,axiom,
! [T_1: $tType,T_2: $tType] :
( preorder(T_2)
=> preorder(fun(T_1,T_2)) ) ).
tff(arity_fun___Orderings_Oorder,axiom,
! [T_1: $tType,T_2: $tType] :
( order(T_2)
=> order(fun(T_1,T_2)) ) ).
tff(arity_fun___Orderings_Oord,axiom,
! [T_1: $tType,T_2: $tType] :
( ord(T_2)
=> ord(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___Enum_Oenum,axiom,
! [T_1: $tType,T_2: $tType] :
( ( enum(T_2)
& enum(T_1) )
=> enum(fun(T_1,T_2)) ) ).
tff(arity_HOL_Obool___Orderings_Opreorder,axiom,
preorder(bool) ).
tff(arity_HOL_Obool___Orderings_Olinorder,axiom,
linorder(bool) ).
tff(arity_HOL_Obool___Orderings_Oorder,axiom,
order(bool) ).
tff(arity_HOL_Obool___Orderings_Oord,axiom,
ord(bool) ).
tff(arity_HOL_Obool___Orderings_Obot,axiom,
bot(bool) ).
tff(arity_HOL_Obool___Enum_Oenum,axiom,
enum(bool) ).
%----Helper facts (13)
tff(help_pp_1_1_U,axiom,
~ pp(fFalse) ).
tff(help_pp_2_1_U,axiom,
pp(fTrue) ).
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,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,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,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_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)) ) ).
%----Conjectures (2)
tff(conj_0,hypothesis,
pp(aa(fun(hoare_28830079triple(a),bool),bool,aa(fun(hoare_28830079triple(a),bool),fun(fun(hoare_28830079triple(a),bool),bool),ord_less_eq(fun(hoare_28830079triple(a),bool)),g),ga)) ).
tff(conj_1,conjecture,
hoare_992312373derivs(a,ga,bot_bot(fun(hoare_28830079triple(a),bool))) ).
%------------------------------------------------------------------------------