TPTP Problem File: SWW513_5.p
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%------------------------------------------------------------------------------
% File : SWW513_5 : TPTP v8.2.0. Released v6.0.0.
% Domain : Software Verification
% Problem : Hoare's Logic with Procedures line 217
% Version : Especial.
% English :
% Refs : [BN10] Boehme & Nipkow (2010), Sledgehammer: Judgement Day
% : [Bla13] Blanchette (2011), Email to Geoff Sutcliffe
% Source : [Bla13]
% Names : hoare_217 [Bla13]
% Status : Theorem
% Rating : 0.00 v7.5.0, 0.33 v7.4.0, 0.00 v7.1.0, 0.33 v6.4.0
% Syntax : Number of formulae : 163 ( 22 unt; 42 typ; 0 def)
% Number of atoms : 335 ( 49 equ)
% Maximal formula atoms : 9 ( 2 avg)
% Number of connectives : 222 ( 8 ~; 3 |; 17 &)
% ( 30 <=>; 164 =>; 0 <=; 0 <~>)
% Maximal formula depth : 14 ( 6 avg)
% Maximal term depth : 6 ( 2 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 34 ( 22 >; 12 *; 0 +; 0 <<)
% Number of predicates : 15 ( 14 usr; 0 prp; 1-6 aty)
% Number of functors : 25 ( 25 usr; 7 con; 0-5 aty)
% Number of variables : 441 ( 397 !; 4 ?; 441 :)
% ( 40 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:16:41
%------------------------------------------------------------------------------
%----Should-be-implicit typings (5)
tff(ty_t_a,type,
a: $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_Enum_Oenum,type,
enum:
!>[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_cl_Groups_Oordered__ab__group__add,type,
ordered_ab_group_add:
!>[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_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_Groups_Ouminus__class_Ouminus,type,
uminus_uminus:
!>[A: $tType] : fun(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_Opeek__and,type,
hoare_1442473487ek_and:
!>[A: $tType] : ( ( fun(A,fun(state,bool)) * fun(state,bool) * A * state ) > $o ) ).
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_Orderings_Oorder__class_Omono,type,
order_mono:
!>[A: $tType,B: $tType] : ( fun(A,B) > $o ) ).
tff(sy_c_Orderings_Oorder__class_Ostrict__mono,type,
order_strict_mono:
!>[A: $tType,B: $tType] : ( fun(A,B) > $o ) ).
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_SetInterval_Oord__class_OatLeast,type,
ord_atLeast:
!>[A: $tType] : ( A > fun(A,bool) ) ).
tff(sy_c_SetInterval_Oord__class_OatMost,type,
ord_atMost:
!>[A: $tType] : ( A > fun(A,bool) ) ).
tff(sy_c_SetInterval_Oord__class_OgreaterThan,type,
ord_greaterThan:
!>[A: $tType] : ( A > fun(A,bool) ) ).
tff(sy_c_SetInterval_Oord__class_OlessThan,type,
ord_lessThan:
!>[A: $tType] : ( A > fun(A,bool) ) ).
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_Oimage,type,
image:
!>[A: $tType,B: $tType] : ( fun(A,B) > fun(fun(A,bool),fun(B,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_fNot,type,
fNot: fun(bool,bool) ).
tff(sy_c_fTrue,type,
fTrue: 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) ).
tff(sy_v_ts_H,type,
ts: fun(hoare_28830079triple(a),bool) ).
tff(sy_v_tsa,type,
tsa: fun(hoare_28830079triple(a),bool) ).
%----Relevant facts (100)
tff(fact_0_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_1_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_2_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_3_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_4_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_5_order__refl,axiom,
! [A: $tType] :
( preorder(A)
=> ! [X3: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),X3)) ) ).
tff(fact_6_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_7_le__funD,axiom,
! [C: $tType,B: $tType] :
( ord(C)
=> ! [X: 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,X)),aa(B,C,G,X))) ) ) ).
tff(fact_8_le__funE,axiom,
! [C: $tType,B: $tType] :
( ord(C)
=> ! [X: 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,X)),aa(B,C,G,X))) ) ) ).
tff(fact_9_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_10_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_11_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_12_rev__predicate1D,axiom,
! [B: $tType,Q1: fun(B,bool),X: B,P1: fun(B,bool)] :
( pp(aa(B,bool,P1,X))
=> ( 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,X)) ) ) ).
tff(fact_13_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_14_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_15_linorder__le__cases,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y3: A,X3: A] :
( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Y3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y3),X3)) ) ) ).
tff(fact_16_order__trans,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Z: A,Y3: A,X3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Y3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y3),Z))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Z)) ) ) ) ).
tff(fact_17_order__antisym,axiom,
! [A: $tType] :
( order(A)
=> ! [Y3: A,X3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Y3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y3),X3))
=> ( X3 = Y3 ) ) ) ) ).
tff(fact_18_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_19_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_20_order__antisym__conv,axiom,
! [B: $tType] :
( order(B)
=> ! [X: B,Y: B] :
( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),Y),X))
=> ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),X),Y))
<=> ( X = Y ) ) ) ) ).
tff(fact_21_order__eq__refl,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Y3: A,X3: A] :
( ( X3 = Y3 )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Y3)) ) ) ).
tff(fact_22_order__eq__iff,axiom,
! [B: $tType] :
( order(B)
=> ! [Y: B,X: B] :
( ( X = Y )
<=> ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),X),Y))
& pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),Y),X)) ) ) ) ).
tff(fact_23_linorder__linear,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y3: A,X3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Y3))
| pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y3),X3)) ) ) ).
tff(fact_24_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_25_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_26_set__mp,axiom,
! [B: $tType,X: 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),X),A1))
=> pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),X),B1)) ) ) ).
tff(fact_27_set__rev__mp,axiom,
! [B: $tType,B1: fun(B,bool),A1: fun(B,bool),X: B] :
( pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),X),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),X),B1)) ) ) ).
tff(fact_28_predicate1D,axiom,
! [B: $tType,X: 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,X))
=> pp(aa(B,bool,Q1,X)) ) ) ).
tff(fact_29_in__mono,axiom,
! [B: $tType,X: 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),X),A1))
=> pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),X),B1)) ) ) ).
tff(fact_30_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_31_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_32_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_33_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_34_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_35_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_36_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_37_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_38_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_39_in__inv__imagep,axiom,
! [B: $tType,C: $tType,Y: C,X: C,F: fun(C,B),R1: fun(B,fun(B,bool))] :
( inv_imagep(B,C,R1,F,X,Y)
<=> pp(aa(B,bool,aa(B,fun(B,bool),R1,aa(C,B,F,X)),aa(C,B,F,Y))) ) ).
tff(fact_40_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_41_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_42_inv__imagep__def,axiom,
! [C: $tType,B: $tType,F: fun(B,C),R1: fun(C,fun(C,bool)),X1: B,Xa1: B] :
( inv_imagep(C,B,R1,F,X1,Xa1)
<=> pp(aa(C,bool,aa(C,fun(C,bool),R1,aa(B,C,F,X1)),aa(B,C,F,Xa1))) ) ).
tff(fact_43_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_44_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_45_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_46_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_47_Least__equality,axiom,
! [B: $tType] :
( order(B)
=> ! [X: B,P1: fun(B,bool)] :
( pp(aa(B,bool,P1,X))
=> ( ! [Y1: B] :
( pp(aa(B,bool,P1,Y1))
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),X),Y1)) )
=> ( ord_Least(B,P1) = X ) ) ) ) ).
tff(fact_48_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_49_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_50_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_51_LeastI2__order,axiom,
! [B: $tType] :
( order(B)
=> ! [Q1: fun(B,bool),X: B,P1: fun(B,bool)] :
( pp(aa(B,bool,P1,X))
=> ( ! [Y1: B] :
( pp(aa(B,bool,P1,Y1))
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),X),Y1)) )
=> ( ! [X2: B] :
( pp(aa(B,bool,P1,X2))
=> ( ! [Y2: B] :
( pp(aa(B,bool,P1,Y2))
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),X2),Y2)) )
=> pp(aa(B,bool,Q1,X2)) ) )
=> pp(aa(B,bool,Q1,ord_Least(B,P1))) ) ) ) ) ).
tff(fact_52_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_53_Powp__Pow__eq,axiom,
! [B: $tType,A1: fun(B,bool),X1: fun(B,bool)] :
( pp(aa(fun(B,bool),bool,powp(B,combc(B,fun(B,bool),bool,member(B),A1)),X1))
<=> pp(aa(fun(fun(B,bool),bool),bool,aa(fun(B,bool),fun(fun(fun(B,bool),bool),bool),member(fun(B,bool)),X1),pow(B,A1))) ) ).
tff(fact_54_strict__mono__less__eq,axiom,
! [C: $tType,B: $tType] :
( ( linorder(B)
& order(C) )
=> ! [Y: B,X: B,F: fun(B,C)] :
( order_strict_mono(B,C,F)
=> ( pp(aa(C,bool,aa(C,fun(C,bool),ord_less_eq(C),aa(B,C,F,X)),aa(B,C,F,Y)))
<=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),X),Y)) ) ) ) ).
tff(fact_55_peek__and__def,axiom,
! [B: $tType,P2: fun(state,bool),P1: fun(B,fun(state,bool)),X1: B,Xa1: state] :
( hoare_1442473487ek_and(B,P1,P2,X1,Xa1)
<=> ( pp(aa(state,bool,aa(B,fun(state,bool),P1,X1),Xa1))
& pp(aa(state,bool,P2,Xa1)) ) ) ).
tff(fact_56_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_57_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_58_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_59_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_60_strict__mono__eq,axiom,
! [C: $tType,B: $tType] :
( ( linorder(B)
& order(C) )
=> ! [Y: B,X: B,F: fun(B,C)] :
( order_strict_mono(B,C,F)
=> ( ( aa(B,C,F,X) = aa(B,C,F,Y) )
<=> ( X = Y ) ) ) ) ).
tff(fact_61_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_62_greaterThan__subset__iff,axiom,
! [B: $tType] :
( linorder(B)
=> ! [Y: B,X: B] :
( pp(aa(fun(B,bool),bool,aa(fun(B,bool),fun(fun(B,bool),bool),ord_less_eq(fun(B,bool)),ord_greaterThan(B,X)),ord_greaterThan(B,Y)))
<=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),Y),X)) ) ) ).
tff(fact_63_strict__mono__mono,axiom,
! [C: $tType,B: $tType] :
( ( order(B)
& order(C) )
=> ! [F: fun(B,C)] :
( order_strict_mono(B,C,F)
=> order_mono(B,C,F) ) ) ).
tff(fact_64_lessThan__subset__iff,axiom,
! [B: $tType] :
( linorder(B)
=> ! [Y: B,X: B] :
( pp(aa(fun(B,bool),bool,aa(fun(B,bool),fun(fun(B,bool),bool),ord_less_eq(fun(B,bool)),ord_lessThan(B,X)),ord_lessThan(B,Y)))
<=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),X),Y)) ) ) ).
tff(fact_65_lessThan__eq__iff,axiom,
! [B: $tType] :
( linorder(B)
=> ! [Y: B,X: B] :
( ( ord_lessThan(B,X) = ord_lessThan(B,Y) )
<=> ( X = Y ) ) ) ).
tff(fact_66_greaterThan__eq__iff,axiom,
! [B: $tType] :
( linorder(B)
=> ! [Y: B,X: B] :
( ( ord_greaterThan(B,X) = ord_greaterThan(B,Y) )
<=> ( X = Y ) ) ) ).
tff(fact_67_monoD,axiom,
! [C: $tType,B: $tType] :
( ( order(B)
& order(C) )
=> ! [Y: B,X: B,F: fun(B,C)] :
( order_mono(B,C,F)
=> ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),X),Y))
=> pp(aa(C,bool,aa(C,fun(C,bool),ord_less_eq(C),aa(B,C,F,X)),aa(B,C,F,Y))) ) ) ) ).
tff(fact_68_monoI,axiom,
! [C: $tType,B: $tType] :
( ( order(B)
& order(C) )
=> ! [F: fun(B,C)] :
( ! [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))) )
=> order_mono(B,C,F) ) ) ).
tff(fact_69_atMost__subset__iff,axiom,
! [B: $tType] :
( order(B)
=> ! [Y: B,X: B] :
( pp(aa(fun(B,bool),bool,aa(fun(B,bool),fun(fun(B,bool),bool),ord_less_eq(fun(B,bool)),ord_atMost(B,X)),ord_atMost(B,Y)))
<=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),X),Y)) ) ) ).
tff(fact_70_atLeast__subset__iff,axiom,
! [B: $tType] :
( order(B)
=> ! [Y: B,X: B] :
( pp(aa(fun(B,bool),bool,aa(fun(B,bool),fun(fun(B,bool),bool),ord_less_eq(fun(B,bool)),ord_atLeast(B,X)),ord_atLeast(B,Y)))
<=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),Y),X)) ) ) ).
tff(fact_71_atMost__eq__iff,axiom,
! [B: $tType] :
( linorder(B)
=> ! [Y: B,X: B] :
( ( ord_atMost(B,X) = ord_atMost(B,Y) )
<=> ( X = Y ) ) ) ).
tff(fact_72_atLeast__eq__iff,axiom,
! [B: $tType] :
( linorder(B)
=> ! [Y: B,X: B] :
( ( ord_atLeast(B,X) = ord_atLeast(B,Y) )
<=> ( X = Y ) ) ) ).
tff(fact_73_atMost__iff,axiom,
! [B: $tType] :
( ord(B)
=> ! [K: B,I: B] :
( pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),I),ord_atMost(B,K)))
<=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),I),K)) ) ) ).
tff(fact_74_atLeast__iff,axiom,
! [B: $tType] :
( ord(B)
=> ! [K: B,I: B] :
( pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),I),ord_atLeast(B,K)))
<=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),K),I)) ) ) ).
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),X: B] :
( pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),X),A1))
<=> pp(aa(B,bool,A1,X)) ) ).
tff(fact_77_Collect__def,axiom,
! [B: $tType,P1: fun(B,bool)] : collect(B,P1) = P1 ).
tff(fact_78_Least__mono,axiom,
! [C: $tType,B: $tType] :
( ( order(B)
& order(C) )
=> ! [S: fun(B,bool),F: fun(B,C)] :
( order_mono(B,C,F)
=> ( ? [X1: B] :
( pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),X1),S))
& ! [Xa: B] :
( pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),Xa),S))
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),X1),Xa)) ) )
=> ( ord_Least(C,combc(C,fun(C,bool),bool,member(C),aa(fun(B,bool),fun(C,bool),image(B,C,F),S))) = aa(B,C,F,ord_Least(B,combc(B,fun(B,bool),bool,member(B),S))) ) ) ) ) ).
tff(fact_79_Compl__greaterThan,axiom,
! [B: $tType] :
( linorder(B)
=> ! [K: B] : aa(fun(B,bool),fun(B,bool),uminus_uminus(fun(B,bool)),ord_greaterThan(B,K)) = ord_atMost(B,K) ) ).
tff(fact_80_Compl__atMost,axiom,
! [B: $tType] :
( linorder(B)
=> ! [K: B] : aa(fun(B,bool),fun(B,bool),uminus_uminus(fun(B,bool)),ord_atMost(B,K)) = ord_greaterThan(B,K) ) ).
tff(fact_81_image__eqI,axiom,
! [B: $tType,C: $tType,A1: fun(C,bool),X: C,F: fun(C,B),B2: B] :
( ( B2 = aa(C,B,F,X) )
=> ( pp(aa(fun(C,bool),bool,aa(C,fun(fun(C,bool),bool),member(C),X),A1))
=> pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),B2),aa(fun(C,bool),fun(B,bool),image(C,B,F),A1))) ) ) ).
tff(fact_82_Compl__iff,axiom,
! [B: $tType,A1: fun(B,bool),C1: B] :
( pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),C1),aa(fun(B,bool),fun(B,bool),uminus_uminus(fun(B,bool)),A1)))
<=> ~ pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),C1),A1)) ) ).
tff(fact_83_Compl__eq__Compl__iff,axiom,
! [B: $tType,B1: fun(B,bool),A1: fun(B,bool)] :
( ( aa(fun(B,bool),fun(B,bool),uminus_uminus(fun(B,bool)),A1) = aa(fun(B,bool),fun(B,bool),uminus_uminus(fun(B,bool)),B1) )
<=> ( A1 = B1 ) ) ).
tff(fact_84_ComplI,axiom,
! [B: $tType,A1: fun(B,bool),C1: B] :
( ~ 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),aa(fun(B,bool),fun(B,bool),uminus_uminus(fun(B,bool)),A1))) ) ).
tff(fact_85_Compl__subset__Compl__iff,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)),aa(fun(B,bool),fun(B,bool),uminus_uminus(fun(B,bool)),A1)),aa(fun(B,bool),fun(B,bool),uminus_uminus(fun(B,bool)),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_86_Compl__atLeast,axiom,
! [B: $tType] :
( linorder(B)
=> ! [K: B] : aa(fun(B,bool),fun(B,bool),uminus_uminus(fun(B,bool)),ord_atLeast(B,K)) = ord_lessThan(B,K) ) ).
tff(fact_87_Compl__lessThan,axiom,
! [B: $tType] :
( linorder(B)
=> ! [K: B] : aa(fun(B,bool),fun(B,bool),uminus_uminus(fun(B,bool)),ord_lessThan(B,K)) = ord_atLeast(B,K) ) ).
tff(fact_88_image__uminus__atLeast,axiom,
! [B: $tType] :
( ordered_ab_group_add(B)
=> ! [X: B] : aa(fun(B,bool),fun(B,bool),image(B,B,uminus_uminus(B)),ord_atLeast(B,X)) = ord_atMost(B,aa(B,B,uminus_uminus(B),X)) ) ).
tff(fact_89_image__uminus__atMost,axiom,
! [B: $tType] :
( ordered_ab_group_add(B)
=> ! [X: B] : aa(fun(B,bool),fun(B,bool),image(B,B,uminus_uminus(B)),ord_atMost(B,X)) = ord_atLeast(B,aa(B,B,uminus_uminus(B),X)) ) ).
tff(fact_90_image__uminus__lessThan,axiom,
! [B: $tType] :
( ordered_ab_group_add(B)
=> ! [X: B] : aa(fun(B,bool),fun(B,bool),image(B,B,uminus_uminus(B)),ord_lessThan(B,X)) = ord_greaterThan(B,aa(B,B,uminus_uminus(B),X)) ) ).
tff(fact_91_image__uminus__greaterThan,axiom,
! [B: $tType] :
( ordered_ab_group_add(B)
=> ! [X: B] : aa(fun(B,bool),fun(B,bool),image(B,B,uminus_uminus(B)),ord_greaterThan(B,X)) = ord_lessThan(B,aa(B,B,uminus_uminus(B),X)) ) ).
tff(fact_92_image__Pow__surj,axiom,
! [C: $tType,B: $tType,B1: fun(B,bool),A1: fun(C,bool),F: fun(C,B)] :
( ( aa(fun(C,bool),fun(B,bool),image(C,B,F),A1) = B1 )
=> ( aa(fun(fun(C,bool),bool),fun(fun(B,bool),bool),image(fun(C,bool),fun(B,bool),image(C,B,F)),pow(C,A1)) = pow(B,B1) ) ) ).
tff(fact_93_image__Pow__mono,axiom,
! [C: $tType,B: $tType,B1: fun(B,bool),A1: fun(C,bool),F: fun(C,B)] :
( pp(aa(fun(B,bool),bool,aa(fun(B,bool),fun(fun(B,bool),bool),ord_less_eq(fun(B,bool)),aa(fun(C,bool),fun(B,bool),image(C,B,F),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)),aa(fun(fun(C,bool),bool),fun(fun(B,bool),bool),image(fun(C,bool),fun(B,bool),image(C,B,F)),pow(C,A1))),pow(B,B1))) ) ).
tff(fact_94_subset__image__iff,axiom,
! [B: $tType,C: $tType,A1: fun(C,bool),F: fun(C,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),aa(fun(C,bool),fun(B,bool),image(C,B,F),A1)))
<=> ? [AA: fun(C,bool)] :
( pp(aa(fun(C,bool),bool,aa(fun(C,bool),fun(fun(C,bool),bool),ord_less_eq(fun(C,bool)),AA),A1))
& ( B1 = aa(fun(C,bool),fun(B,bool),image(C,B,F),AA) ) ) ) ).
tff(fact_95_image__mono,axiom,
! [C: $tType,B: $tType,F: fun(B,C),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(C,bool),bool,aa(fun(C,bool),fun(fun(C,bool),bool),ord_less_eq(fun(C,bool)),aa(fun(B,bool),fun(C,bool),image(B,C,F),A1)),aa(fun(B,bool),fun(C,bool),image(B,C,F),B1))) ) ).
tff(fact_96_Compl__anti__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(B,bool),bool,aa(fun(B,bool),fun(fun(B,bool),bool),ord_less_eq(fun(B,bool)),aa(fun(B,bool),fun(B,bool),uminus_uminus(fun(B,bool)),B1)),aa(fun(B,bool),fun(B,bool),uminus_uminus(fun(B,bool)),A1))) ) ).
tff(fact_97_Compl__eq,axiom,
! [B: $tType,A1: fun(B,bool)] : aa(fun(B,bool),fun(B,bool),uminus_uminus(fun(B,bool)),A1) = collect(B,combb(bool,bool,B,fNot,combc(B,fun(B,bool),bool,member(B),A1))) ).
tff(fact_98_Collect__neg__eq,axiom,
! [B: $tType,P1: fun(B,bool)] : collect(B,combb(bool,bool,B,fNot,P1)) = aa(fun(B,bool),fun(B,bool),uminus_uminus(fun(B,bool)),collect(B,P1)) ).
tff(fact_99_double__complement,axiom,
! [B: $tType,A1: fun(B,bool)] : aa(fun(B,bool),fun(B,bool),uminus_uminus(fun(B,bool)),aa(fun(B,bool),fun(B,bool),uminus_uminus(fun(B,bool)),A1)) = A1 ).
%----Arities (9)
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___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___Enum_Oenum,axiom,
enum(bool) ).
%----Helper facts (7)
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,
! [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,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_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)) ).
%----Conjectures (5)
tff(conj_0,hypothesis,
hoare_992312373derivs(a,g,ts) ).
tff(conj_1,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)),tsa),ts)) ).
tff(conj_2,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_3,hypothesis,
hoare_992312373derivs(a,ga,ts) ).
tff(conj_4,conjecture,
hoare_992312373derivs(a,ga,tsa) ).
%------------------------------------------------------------------------------