TPTP Problem File: SWW520_5.p
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%------------------------------------------------------------------------------
% File : SWW520_5 : TPTP v8.2.0. Released v6.0.0.
% Domain : Software Verification
% Problem : Hoare's Logic with Procedures line 283
% Version : Especial.
% English :
% Refs : [BN10] Boehme & Nipkow (2010), Sledgehammer: Judgement Day
% : [Bla13] Blanchette (2011), Email to Geoff Sutcliffe
% Source : [Bla13]
% Names : hoare_283 [Bla13]
% Status : Theorem
% Rating : 0.00 v6.4.0
% Syntax : Number of formulae : 159 ( 36 unt; 33 typ; 0 def)
% Number of atoms : 302 ( 52 equ)
% Maximal formula atoms : 7 ( 1 avg)
% Number of connectives : 200 ( 24 ~; 5 |; 7 &)
% ( 20 <=>; 144 =>; 0 <=; 0 <~>)
% Maximal formula depth : 14 ( 5 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 32 ( 19 >; 13 *; 0 +; 0 <<)
% Number of predicates : 14 ( 13 usr; 0 prp; 1-6 aty)
% Number of functors : 17 ( 17 usr; 3 con; 0-4 aty)
% Number of variables : 373 ( 343 !; 3 ?; 373 :)
% ( 27 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:23:44
%------------------------------------------------------------------------------
%----Should-be-implicit typings (6)
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_Nat_Onat,type,
nat: $tType ).
tff(ty_tc_Predicate_Opred,type,
pred: $tType > $tType ).
tff(ty_tc_fun,type,
fun: ( $tType * $tType ) > $tType ).
%----Explicit typings (27)
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_Rings_Osemiring__1,type,
semiring_1:
!>[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_Complete__Lattices_Ocomplete__lattice,type,
comple187826305attice:
!>[A: $tType] : $o ).
tff(sy_c_COMBK,type,
combk:
!>[A: $tType,B: $tType] : ( A > fun(B,A) ) ).
tff(sy_c_Complete__Lattices_OSup__class_OSup,type,
complete_Sup_Sup:
!>[A: $tType] : ( fun(A,bool) > A ) ).
tff(sy_c_Hoare__Mirabelle__vtrypsmcwp_Ohoare__valids,type,
hoare_592965047valids:
!>[A: $tType] : ( ( fun(hoare_28830079triple(A),bool) * fun(hoare_28830079triple(A),bool) ) > $o ) ).
tff(sy_c_Hoare__Mirabelle__vtrypsmcwp_Otriple__valid,type,
hoare_1633586161_valid:
!>[A: $tType] : ( ( nat * hoare_28830079triple(A) ) > $o ) ).
tff(sy_c_Nat_OSuc,type,
suc: nat > nat ).
tff(sy_c_Nat_Osemiring__1__class_Oof__nat__aux,type,
semiri532925092at_aux:
!>[A: $tType] : ( ( fun(A,A) * nat * A ) > A ) ).
tff(sy_c_Orderings_Obot__class_Obot,type,
bot_bot:
!>[A: $tType] : A ).
tff(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( ( A * A ) > $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_Predicate_Opred_OPred,type,
pred1:
!>[A: $tType] : ( fun(A,bool) > pred(A) ) ).
tff(sy_c_Predicate_Opred_Opred__case,type,
pred_case:
!>[A: $tType,T1: $tType] : ( ( fun(fun(A,bool),T1) * pred(A) ) > T1 ) ).
tff(sy_c_Predicate_Opred_Opred__rec,type,
pred_rec:
!>[A: $tType,T1: $tType] : ( ( fun(fun(A,bool),T1) * pred(A) ) > T1 ) ).
tff(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( 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_member,type,
member:
!>[A: $tType] : ( ( A * fun(A,bool) ) > $o ) ).
tff(sy_c_pp,type,
pp: bool > $o ).
tff(sy_v_Ga,type,
ga: fun(hoare_28830079triple(a),bool) ).
%----Relevant facts (98)
tff(fact_0_ball__empty,axiom,
! [B: $tType,P1: fun(B,bool),X1: B] :
( member(B,X1,bot_bot(fun(B,bool)))
=> pp(aa(B,bool,P1,X1)) ) ).
tff(fact_1_all__not__in__conv,axiom,
! [B: $tType,A2: fun(B,bool)] :
( ! [X6: B] : ~ member(B,X6,A2)
<=> ( A2 = bot_bot(fun(B,bool)) ) ) ).
tff(fact_2_empty__Collect__eq,axiom,
! [B: $tType,P1: fun(B,bool)] :
( ( bot_bot(fun(B,bool)) = collect(B,P1) )
<=> ! [X6: B] : ~ pp(aa(B,bool,P1,X6)) ) ).
tff(fact_3_empty__iff,axiom,
! [B: $tType,C1: B] : ~ member(B,C1,bot_bot(fun(B,bool))) ).
tff(fact_4_Collect__empty__eq,axiom,
! [B: $tType,P1: fun(B,bool)] :
( ( collect(B,P1) = bot_bot(fun(B,bool)) )
<=> ! [X6: B] : ~ pp(aa(B,bool,P1,X6)) ) ).
tff(fact_5_emptyE,axiom,
! [B: $tType,A1: B] : ~ member(B,A1,bot_bot(fun(B,bool))) ).
tff(fact_6_hoare__valids__def,axiom,
! [B: $tType,Tsa: fun(hoare_28830079triple(B),bool),Ga: fun(hoare_28830079triple(B),bool)] :
( hoare_592965047valids(B,Ga,Tsa)
<=> ! [N4: nat] :
( ! [X6: hoare_28830079triple(B)] :
( member(hoare_28830079triple(B),X6,Ga)
=> hoare_1633586161_valid(B,N4,X6) )
=> ! [X6: hoare_28830079triple(B)] :
( member(hoare_28830079triple(B),X6,Tsa)
=> hoare_1633586161_valid(B,N4,X6) ) ) ) ).
tff(fact_7_bot__fun__def,axiom,
! [B: $tType,C: $tType] :
( bot(C)
=> ! [X1: B] : aa(B,C,bot_bot(fun(B,C)),X1) = bot_bot(C) ) ).
tff(fact_8_bot__apply,axiom,
! [C: $tType,B: $tType] :
( bot(B)
=> ! [X4: C] : aa(C,B,bot_bot(fun(C,B)),X4) = bot_bot(B) ) ).
tff(fact_9_bot__empty__eq,axiom,
! [B: $tType,X1: B] :
( pp(aa(B,bool,bot_bot(fun(B,bool)),X1))
<=> member(B,X1,bot_bot(fun(B,bool))) ) ).
tff(fact_10_empty__def,axiom,
! [B: $tType] : bot_bot(fun(B,bool)) = collect(B,combk(bool,B,fFalse)) ).
tff(fact_11_equals0D,axiom,
! [B: $tType,A1: B,A2: fun(B,bool)] :
( ( A2 = bot_bot(fun(B,bool)) )
=> ~ member(B,A1,A2) ) ).
tff(fact_12_ex__in__conv,axiom,
! [B: $tType,A2: fun(B,bool)] :
( ? [X6: B] : member(B,X6,A2)
<=> ( A2 != bot_bot(fun(B,bool)) ) ) ).
tff(fact_13_equals0I,axiom,
! [B: $tType,A2: fun(B,bool)] :
( ! [Y: B] : ~ member(B,Y,A2)
=> ( A2 = bot_bot(fun(B,bool)) ) ) ).
tff(fact_14_in__inv__imagep,axiom,
! [B: $tType,C: $tType,Y1: C,X4: C,F: fun(C,B),R1: fun(B,fun(B,bool))] :
( inv_imagep(B,C,R1,F,X4,Y1)
<=> pp(aa(B,bool,aa(B,fun(B,bool),R1,aa(C,B,F,X4)),aa(C,B,F,Y1))) ) ).
tff(fact_15_triples__valid__Suc,axiom,
! [B: $tType,N1: nat,Tsa: fun(hoare_28830079triple(B),bool)] :
( ! [X: hoare_28830079triple(B)] :
( member(hoare_28830079triple(B),X,Tsa)
=> hoare_1633586161_valid(B,suc(N1),X) )
=> ! [X1: hoare_28830079triple(B)] :
( member(hoare_28830079triple(B),X1,Tsa)
=> hoare_1633586161_valid(B,N1,X1) ) ) ).
tff(fact_16_Sup__empty,axiom,
! [B: $tType] :
( comple187826305attice(B)
=> ( complete_Sup_Sup(B,bot_bot(fun(B,bool))) = bot_bot(B) ) ) ).
tff(fact_17_bot__pred__def,axiom,
! [B: $tType] : bot_bot(pred(B)) = pred1(B,bot_bot(fun(B,bool))) ).
tff(fact_18_ball__reg,axiom,
! [B: $tType,Q1: fun(B,bool),P1: fun(B,bool),R: fun(B,bool)] :
( ! [X: B] :
( member(B,X,R)
=> ( pp(aa(B,bool,P1,X))
=> pp(aa(B,bool,Q1,X)) ) )
=> ( ! [X: B] :
( member(B,X,R)
=> pp(aa(B,bool,P1,X)) )
=> ! [X1: B] :
( member(B,X1,R)
=> pp(aa(B,bool,Q1,X1)) ) ) ) ).
tff(fact_19_triple__valid__Suc,axiom,
! [A: $tType,T: hoare_28830079triple(A),N2: nat] :
( hoare_1633586161_valid(A,suc(N2),T)
=> hoare_1633586161_valid(A,N2,T) ) ).
tff(fact_20_subset__empty,axiom,
! [B: $tType,A2: fun(B,bool)] :
( ord_less_eq(fun(B,bool),A2,bot_bot(fun(B,bool)))
<=> ( A2 = bot_bot(fun(B,bool)) ) ) ).
tff(fact_21_empty__subsetI,axiom,
! [B: $tType,A2: fun(B,bool)] : ord_less_eq(fun(B,bool),bot_bot(fun(B,bool)),A2) ).
tff(fact_22_order__refl,axiom,
! [A: $tType] :
( preorder(A)
=> ! [X3: A] : ord_less_eq(A,X3,X3) ) ).
tff(fact_23_subsetD,axiom,
! [B: $tType,C1: B,B2: fun(B,bool),A2: fun(B,bool)] :
( ord_less_eq(fun(B,bool),A2,B2)
=> ( member(B,C1,A2)
=> member(B,C1,B2) ) ) ).
tff(fact_24_equalityI,axiom,
! [B: $tType,B2: fun(B,bool),A2: fun(B,bool)] :
( ord_less_eq(fun(B,bool),A2,B2)
=> ( ord_less_eq(fun(B,bool),B2,A2)
=> ( A2 = B2 ) ) ) ).
tff(fact_25_pred_Oinject,axiom,
! [B: $tType,Fun2: fun(B,bool),Fun1: fun(B,bool)] :
( ( pred1(B,Fun1) = pred1(B,Fun2) )
<=> ( Fun1 = Fun2 ) ) ).
tff(fact_26_subset__refl,axiom,
! [B: $tType,A2: fun(B,bool)] : ord_less_eq(fun(B,bool),A2,A2) ).
tff(fact_27_le__fun__def,axiom,
! [C: $tType,B: $tType] :
( ord(C)
=> ! [G: fun(B,C),F: fun(B,C)] :
( ord_less_eq(fun(B,C),F,G)
<=> ! [X6: B] : ord_less_eq(C,aa(B,C,F,X6),aa(B,C,G,X6)) ) ) ).
tff(fact_28_linorder__linear,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y2: A,X3: A] :
( ord_less_eq(A,X3,Y2)
| ord_less_eq(A,Y2,X3) ) ) ).
tff(fact_29_set__eq__subset,axiom,
! [B: $tType,B2: fun(B,bool),A2: fun(B,bool)] :
( ( A2 = B2 )
<=> ( ord_less_eq(fun(B,bool),A2,B2)
& ord_less_eq(fun(B,bool),B2,A2) ) ) ).
tff(fact_30_order__eq__iff,axiom,
! [B: $tType] :
( order(B)
=> ! [Y1: B,X4: B] :
( ( X4 = Y1 )
<=> ( ord_less_eq(B,X4,Y1)
& ord_less_eq(B,Y1,X4) ) ) ) ).
tff(fact_31_rev__predicate1D,axiom,
! [B: $tType,Q1: fun(B,bool),X4: B,P1: fun(B,bool)] :
( pp(aa(B,bool,P1,X4))
=> ( ord_less_eq(fun(B,bool),P1,Q1)
=> pp(aa(B,bool,Q1,X4)) ) ) ).
tff(fact_32_equalityD1,axiom,
! [B: $tType,B2: fun(B,bool),A2: fun(B,bool)] :
( ( A2 = B2 )
=> ord_less_eq(fun(B,bool),A2,B2) ) ).
tff(fact_33_equalityD2,axiom,
! [B: $tType,B2: fun(B,bool),A2: fun(B,bool)] :
( ( A2 = B2 )
=> ord_less_eq(fun(B,bool),B2,A2) ) ).
tff(fact_34_order__eq__refl,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Y2: A,X3: A] :
( ( X3 = Y2 )
=> ord_less_eq(A,X3,Y2) ) ) ).
tff(fact_35_le__funD,axiom,
! [C: $tType,B: $tType] :
( ord(C)
=> ! [X4: B,G: fun(B,C),F: fun(B,C)] :
( ord_less_eq(fun(B,C),F,G)
=> ord_less_eq(C,aa(B,C,F,X4),aa(B,C,G,X4)) ) ) ).
tff(fact_36_in__mono,axiom,
! [B: $tType,X4: B,B2: fun(B,bool),A2: fun(B,bool)] :
( ord_less_eq(fun(B,bool),A2,B2)
=> ( member(B,X4,A2)
=> member(B,X4,B2) ) ) ).
tff(fact_37_order__antisym__conv,axiom,
! [B: $tType] :
( order(B)
=> ! [X4: B,Y1: B] :
( ord_less_eq(B,Y1,X4)
=> ( ord_less_eq(B,X4,Y1)
<=> ( X4 = Y1 ) ) ) ) ).
tff(fact_38_predicate1D,axiom,
! [B: $tType,X4: B,Q1: fun(B,bool),P1: fun(B,bool)] :
( ord_less_eq(fun(B,bool),P1,Q1)
=> ( pp(aa(B,bool,P1,X4))
=> pp(aa(B,bool,Q1,X4)) ) ) ).
tff(fact_39_ord__eq__le__trans,axiom,
! [A: $tType] :
( ord(A)
=> ! [C3: A,B3: A,A3: A] :
( ( A3 = B3 )
=> ( ord_less_eq(A,B3,C3)
=> ord_less_eq(A,A3,C3) ) ) ) ).
tff(fact_40_set__rev__mp,axiom,
! [B: $tType,B2: fun(B,bool),A2: fun(B,bool),X4: B] :
( member(B,X4,A2)
=> ( ord_less_eq(fun(B,bool),A2,B2)
=> member(B,X4,B2) ) ) ).
tff(fact_41_set__mp,axiom,
! [B: $tType,X4: B,B2: fun(B,bool),A2: fun(B,bool)] :
( ord_less_eq(fun(B,bool),A2,B2)
=> ( member(B,X4,A2)
=> member(B,X4,B2) ) ) ).
tff(fact_42_subset__trans,axiom,
! [B: $tType,C2: fun(B,bool),B2: fun(B,bool),A2: fun(B,bool)] :
( ord_less_eq(fun(B,bool),A2,B2)
=> ( ord_less_eq(fun(B,bool),B2,C2)
=> ord_less_eq(fun(B,bool),A2,C2) ) ) ).
tff(fact_43_ord__le__eq__trans,axiom,
! [A: $tType] :
( ord(A)
=> ! [C3: A,B3: A,A3: A] :
( ord_less_eq(A,A3,B3)
=> ( ( B3 = C3 )
=> ord_less_eq(A,A3,C3) ) ) ) ).
tff(fact_44_order__antisym,axiom,
! [A: $tType] :
( order(A)
=> ! [Y2: A,X3: A] :
( ord_less_eq(A,X3,Y2)
=> ( ord_less_eq(A,Y2,X3)
=> ( X3 = Y2 ) ) ) ) ).
tff(fact_45_order__trans,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Z1: A,Y2: A,X3: A] :
( ord_less_eq(A,X3,Y2)
=> ( ord_less_eq(A,Y2,Z1)
=> ord_less_eq(A,X3,Z1) ) ) ) ).
tff(fact_46_equalityE,axiom,
! [B: $tType,B2: fun(B,bool),A2: fun(B,bool)] :
( ( A2 = B2 )
=> ~ ( ord_less_eq(fun(B,bool),A2,B2)
=> ~ ord_less_eq(fun(B,bool),B2,A2) ) ) ).
tff(fact_47_le__funE,axiom,
! [C: $tType,B: $tType] :
( ord(C)
=> ! [X4: B,G: fun(B,C),F: fun(B,C)] :
( ord_less_eq(fun(B,C),F,G)
=> ord_less_eq(C,aa(B,C,F,X4),aa(B,C,G,X4)) ) ) ).
tff(fact_48_linorder__le__cases,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y2: A,X3: A] :
( ~ ord_less_eq(A,X3,Y2)
=> ord_less_eq(A,Y2,X3) ) ) ).
tff(fact_49_Union__empty,axiom,
! [B: $tType] : complete_Sup_Sup(fun(B,bool),bot_bot(fun(fun(B,bool),bool))) = bot_bot(fun(B,bool)) ).
tff(fact_50_Sup__upper2,axiom,
! [B: $tType] :
( comple187826305attice(B)
=> ! [V1: B,A2: fun(B,bool),U: B] :
( member(B,U,A2)
=> ( ord_less_eq(B,V1,U)
=> ord_less_eq(B,V1,complete_Sup_Sup(B,A2)) ) ) ) ).
tff(fact_51_Union__mono,axiom,
! [B: $tType,B2: fun(fun(B,bool),bool),A2: fun(fun(B,bool),bool)] :
( ord_less_eq(fun(fun(B,bool),bool),A2,B2)
=> ord_less_eq(fun(B,bool),complete_Sup_Sup(fun(B,bool),A2),complete_Sup_Sup(fun(B,bool),B2)) ) ).
tff(fact_52_Sup__subset__mono,axiom,
! [B: $tType] :
( comple187826305attice(B)
=> ! [B2: fun(B,bool),A2: fun(B,bool)] :
( ord_less_eq(fun(B,bool),A2,B2)
=> ord_less_eq(B,complete_Sup_Sup(B,A2),complete_Sup_Sup(B,B2)) ) ) ).
tff(fact_53_Sup__upper,axiom,
! [B: $tType] :
( comple187826305attice(B)
=> ! [A2: fun(B,bool),X4: B] :
( member(B,X4,A2)
=> ord_less_eq(B,X4,complete_Sup_Sup(B,A2)) ) ) ).
tff(fact_54_Union__upper,axiom,
! [B: $tType,A2: fun(fun(B,bool),bool),B2: fun(B,bool)] :
( member(fun(B,bool),B2,A2)
=> ord_less_eq(fun(B,bool),B2,complete_Sup_Sup(fun(B,bool),A2)) ) ).
tff(fact_55_Sup__le__iff,axiom,
! [B: $tType] :
( comple187826305attice(B)
=> ! [B1: B,A2: fun(B,bool)] :
( ord_less_eq(B,complete_Sup_Sup(B,A2),B1)
<=> ! [X6: B] :
( member(B,X6,A2)
=> ord_less_eq(B,X6,B1) ) ) ) ).
tff(fact_56_bot__least,axiom,
! [A: $tType] :
( bot(A)
=> ! [A3: A] : ord_less_eq(A,bot_bot(A),A3) ) ).
tff(fact_57_bot__unique,axiom,
! [B: $tType] :
( bot(B)
=> ! [A1: B] :
( ord_less_eq(B,A1,bot_bot(B))
<=> ( A1 = bot_bot(B) ) ) ) ).
tff(fact_58_le__bot,axiom,
! [A: $tType] :
( bot(A)
=> ! [A3: A] :
( ord_less_eq(A,A3,bot_bot(A))
=> ( A3 = bot_bot(A) ) ) ) ).
tff(fact_59_subsetI,axiom,
! [B: $tType,B2: fun(B,bool),A2: fun(B,bool)] :
( ! [X: B] :
( member(B,X,A2)
=> member(B,X,B2) )
=> ord_less_eq(fun(B,bool),A2,B2) ) ).
tff(fact_60_less__eq__Sup,axiom,
! [B: $tType] :
( comple187826305attice(B)
=> ! [U: B,A2: fun(B,bool)] :
( ! [V: B] :
( member(B,V,A2)
=> ord_less_eq(B,U,V) )
=> ( ( A2 != bot_bot(fun(B,bool)) )
=> ord_less_eq(B,U,complete_Sup_Sup(B,A2)) ) ) ) ).
tff(fact_61_pred_Orecs,axiom,
! [B: $tType,C: $tType,Fun1: fun(C,bool),F1: fun(fun(C,bool),B)] : pred_rec(C,B,F1,pred1(C,Fun1)) = aa(fun(C,bool),B,F1,Fun1) ).
tff(fact_62_pred_Osimps_I2_J,axiom,
! [B: $tType,C: $tType,Fun1: fun(C,bool),F1: fun(fun(C,bool),B)] : pred_case(C,B,F1,pred1(C,Fun1)) = aa(fun(C,bool),B,F1,Fun1) ).
tff(fact_63_nat_Oinject,axiom,
! [Nat1: nat,Nat: nat] :
( ( suc(Nat) = suc(Nat1) )
<=> ( Nat = Nat1 ) ) ).
tff(fact_64_Sup__least,axiom,
! [B: $tType] :
( comple187826305attice(B)
=> ! [Z: B,A2: fun(B,bool)] :
( ! [X: B] :
( member(B,X,A2)
=> ord_less_eq(B,X,Z) )
=> ord_less_eq(B,complete_Sup_Sup(B,A2),Z) ) ) ).
tff(fact_65_Suc__le__mono,axiom,
! [M3: nat,N1: nat] :
( ord_less_eq(nat,suc(N1),suc(M3))
<=> ord_less_eq(nat,N1,M3) ) ).
tff(fact_66_UnionI,axiom,
! [B: $tType,A2: B,C2: fun(fun(B,bool),bool),X5: fun(B,bool)] :
( member(fun(B,bool),X5,C2)
=> ( member(B,A2,X5)
=> member(B,A2,complete_Sup_Sup(fun(B,bool),C2)) ) ) ).
tff(fact_67_Suc__leD,axiom,
! [N2: nat,M2: nat] :
( ord_less_eq(nat,suc(M2),N2)
=> ord_less_eq(nat,M2,N2) ) ).
tff(fact_68_le__SucE,axiom,
! [N2: nat,M2: nat] :
( ord_less_eq(nat,M2,suc(N2))
=> ( ~ ord_less_eq(nat,M2,N2)
=> ( M2 = suc(N2) ) ) ) ).
tff(fact_69_le__SucI,axiom,
! [N2: nat,M2: nat] :
( ord_less_eq(nat,M2,N2)
=> ord_less_eq(nat,M2,suc(N2)) ) ).
tff(fact_70_le__Suc__eq,axiom,
! [N1: nat,M3: nat] :
( ord_less_eq(nat,M3,suc(N1))
<=> ( ord_less_eq(nat,M3,N1)
| ( M3 = suc(N1) ) ) ) ).
tff(fact_71_not__less__eq__eq,axiom,
! [N1: nat,M3: nat] :
( ~ ord_less_eq(nat,M3,N1)
<=> ord_less_eq(nat,suc(N1),M3) ) ).
tff(fact_72_Suc__n__not__le__n,axiom,
! [N2: nat] : ~ ord_less_eq(nat,suc(N2),N2) ).
tff(fact_73_ext,axiom,
! [C: $tType,B: $tType,G: fun(B,C),F: fun(B,C)] :
( ! [X: B] : aa(B,C,F,X) = aa(B,C,G,X)
=> ( F = G ) ) ).
tff(fact_74_mem__def,axiom,
! [B: $tType,A2: fun(B,bool),X4: B] :
( member(B,X4,A2)
<=> pp(aa(B,bool,A2,X4)) ) ).
tff(fact_75_Collect__def,axiom,
! [B: $tType,P1: fun(B,bool)] : collect(B,P1) = P1 ).
tff(fact_76_n__not__Suc__n,axiom,
! [N2: nat] : N2 != suc(N2) ).
tff(fact_77_Suc__n__not__n,axiom,
! [N2: nat] : suc(N2) != N2 ).
tff(fact_78_Suc__inject,axiom,
! [Y2: nat,X3: nat] :
( ( suc(X3) = suc(Y2) )
=> ( X3 = Y2 ) ) ).
tff(fact_79_UnionE,axiom,
! [B: $tType,C2: fun(fun(B,bool),bool),A2: B] :
( member(B,A2,complete_Sup_Sup(fun(B,bool),C2))
=> ~ ! [X2: fun(B,bool)] :
( member(B,A2,X2)
=> ~ member(fun(B,bool),X2,C2) ) ) ).
tff(fact_80_Union__least,axiom,
! [B: $tType,C2: fun(B,bool),A2: fun(fun(B,bool),bool)] :
( ! [X2: fun(B,bool)] :
( member(fun(B,bool),X2,A2)
=> ord_less_eq(fun(B,bool),X2,C2) )
=> ord_less_eq(fun(B,bool),complete_Sup_Sup(fun(B,bool),A2),C2) ) ).
tff(fact_81_le__funI,axiom,
! [C: $tType,B: $tType] :
( ord(C)
=> ! [G: fun(B,C),F: fun(B,C)] :
( ! [X: B] : ord_less_eq(C,aa(B,C,F,X),aa(B,C,G,X))
=> ord_less_eq(fun(B,C),F,G) ) ) ).
tff(fact_82_le__antisym,axiom,
! [N2: nat,M2: nat] :
( ord_less_eq(nat,M2,N2)
=> ( ord_less_eq(nat,N2,M2)
=> ( M2 = N2 ) ) ) ).
tff(fact_83_le__trans,axiom,
! [K: nat,J: nat,I1: nat] :
( ord_less_eq(nat,I1,J)
=> ( ord_less_eq(nat,J,K)
=> ord_less_eq(nat,I1,K) ) ) ).
tff(fact_84_eq__imp__le,axiom,
! [N2: nat,M2: nat] :
( ( M2 = N2 )
=> ord_less_eq(nat,M2,N2) ) ).
tff(fact_85_nat__le__linear,axiom,
! [N2: nat,M2: nat] :
( ord_less_eq(nat,M2,N2)
| ord_less_eq(nat,N2,M2) ) ).
tff(fact_86_le__refl,axiom,
! [N2: nat] : ord_less_eq(nat,N2,N2) ).
tff(fact_87_lift__Suc__mono__le,axiom,
! [B: $tType] :
( order(B)
=> ! [N3: nat,N1: nat,F: fun(nat,B)] :
( ! [N: nat] : ord_less_eq(B,aa(nat,B,F,N),aa(nat,B,F,suc(N)))
=> ( ord_less_eq(nat,N1,N3)
=> ord_less_eq(B,aa(nat,B,F,N1),aa(nat,B,F,N3)) ) ) ) ).
tff(fact_88_Suc__le__D,axiom,
! [M: nat,N2: nat] :
( ord_less_eq(nat,suc(N2),M)
=> ? [M1: nat] : M = suc(M1) ) ).
tff(fact_89_Powp__mono,axiom,
! [B: $tType,B2: fun(B,bool),A2: fun(B,bool)] :
( ord_less_eq(fun(B,bool),A2,B2)
=> ord_less_eq(fun(fun(B,bool),bool),powp(B,A2),powp(B,B2)) ) ).
tff(fact_90_pred_Oexhaust,axiom,
! [B: $tType,Y1: pred(B)] :
~ ! [Fun: fun(B,bool)] : Y1 != pred1(B,Fun) ).
tff(fact_91_of__nat__aux_Osimps_I2_J,axiom,
! [B: $tType] :
( semiring_1(B)
=> ! [I: B,N1: nat,Inc: fun(B,B)] : semiri532925092at_aux(B,Inc,suc(N1),I) = semiri532925092at_aux(B,Inc,N1,aa(B,B,Inc,I)) ) ).
tff(fact_92_Collect__mono,axiom,
! [B: $tType,Q1: fun(B,bool),P1: fun(B,bool)] :
( ! [X: B] :
( pp(aa(B,bool,P1,X))
=> pp(aa(B,bool,Q1,X)) )
=> ord_less_eq(fun(B,bool),collect(B,P1),collect(B,Q1)) ) ).
tff(fact_93_predicate1I,axiom,
! [B: $tType,Q1: fun(B,bool),P1: fun(B,bool)] :
( ! [X: B] :
( pp(aa(B,bool,P1,X))
=> pp(aa(B,bool,Q1,X)) )
=> ord_less_eq(fun(B,bool),P1,Q1) ) ).
tff(fact_94_order__subst1,axiom,
! [B: $tType,C: $tType] :
( ( order(C)
& order(B) )
=> ! [C1: C,B1: C,F: fun(C,B),A1: B] :
( ord_less_eq(B,A1,aa(C,B,F,B1))
=> ( ord_less_eq(C,B1,C1)
=> ( ! [X: C,Y: C] :
( ord_less_eq(C,X,Y)
=> ord_less_eq(B,aa(C,B,F,X),aa(C,B,F,Y)) )
=> ord_less_eq(B,A1,aa(C,B,F,C1)) ) ) ) ) ).
tff(fact_95_ord__le__eq__subst,axiom,
! [B: $tType,C: $tType] :
( ( ord(C)
& ord(B) )
=> ! [C1: C,F: fun(B,C),B1: B,A1: B] :
( ord_less_eq(B,A1,B1)
=> ( ( aa(B,C,F,B1) = C1 )
=> ( ! [X: B,Y: B] :
( ord_less_eq(B,X,Y)
=> ord_less_eq(C,aa(B,C,F,X),aa(B,C,F,Y)) )
=> ord_less_eq(C,aa(B,C,F,A1),C1) ) ) ) ) ).
tff(fact_96_order__subst2,axiom,
! [B: $tType,C: $tType] :
( ( order(C)
& order(B) )
=> ! [C1: C,F: fun(B,C),B1: B,A1: B] :
( ord_less_eq(B,A1,B1)
=> ( ord_less_eq(C,aa(B,C,F,B1),C1)
=> ( ! [X: B,Y: B] :
( ord_less_eq(B,X,Y)
=> ord_less_eq(C,aa(B,C,F,X),aa(B,C,F,Y)) )
=> ord_less_eq(C,aa(B,C,F,A1),C1) ) ) ) ) ).
tff(fact_97_ord__eq__le__subst,axiom,
! [B: $tType,C: $tType] :
( ( ord(C)
& ord(B) )
=> ! [C1: C,B1: C,F: fun(C,B),A1: B] :
( ( A1 = aa(C,B,F,B1) )
=> ( ord_less_eq(C,B1,C1)
=> ( ! [X: C,Y: C] :
( ord_less_eq(C,X,Y)
=> ord_less_eq(B,aa(C,B,F,X),aa(C,B,F,Y)) )
=> ord_less_eq(B,A1,aa(C,B,F,C1)) ) ) ) ) ).
%----Arities (22)
tff(arity_fun___Complete__Lattices_Ocomplete__lattice,axiom,
! [T_1: $tType,T_2: $tType] :
( comple187826305attice(T_2)
=> comple187826305attice(fun(T_1,T_2)) ) ).
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_Nat_Onat___Orderings_Opreorder,axiom,
preorder(nat) ).
tff(arity_Nat_Onat___Orderings_Olinorder,axiom,
linorder(nat) ).
tff(arity_Nat_Onat___Rings_Osemiring__1,axiom,
semiring_1(nat) ).
tff(arity_Nat_Onat___Orderings_Oorder,axiom,
order(nat) ).
tff(arity_Nat_Onat___Orderings_Oord,axiom,
ord(nat) ).
tff(arity_Nat_Onat___Orderings_Obot,axiom,
bot(nat) ).
tff(arity_HOL_Obool___Complete__Lattices_Ocomplete__lattice,axiom,
comple187826305attice(bool) ).
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_Predicate_Opred___Complete__Lattices_Ocomplete__lattice,axiom,
! [T_1: $tType] : comple187826305attice(pred(T_1)) ).
tff(arity_Predicate_Opred___Orderings_Opreorder,axiom,
! [T_1: $tType] : preorder(pred(T_1)) ).
tff(arity_Predicate_Opred___Orderings_Oorder,axiom,
! [T_1: $tType] : order(pred(T_1)) ).
tff(arity_Predicate_Opred___Orderings_Oord,axiom,
! [T_1: $tType] : ord(pred(T_1)) ).
tff(arity_Predicate_Opred___Orderings_Obot,axiom,
! [T_1: $tType] : bot(pred(T_1)) ).
%----Helper facts (5)
tff(help_pp_1_1_U,axiom,
~ pp(fFalse) ).
tff(help_pp_2_1_U,axiom,
pp(fTrue) ).
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_fFalse_1_1_U,axiom,
~ pp(fFalse) ).
tff(help_fFalse_1_1_T,axiom,
! [P: bool] :
( ( P = fTrue )
| ( P = fFalse ) ) ).
%----Conjectures (1)
tff(conj_0,conjecture,
! [N: nat] :
( ? [X1: hoare_28830079triple(a)] :
( member(hoare_28830079triple(a),X1,ga)
& ~ hoare_1633586161_valid(a,N,X1) )
| ! [X: hoare_28830079triple(a)] :
( member(hoare_28830079triple(a),X,bot_bot(fun(hoare_28830079triple(a),bool)))
=> hoare_1633586161_valid(a,N,X) ) ) ).
%------------------------------------------------------------------------------