TPTP Problem File: SWW528_5.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SWW528_5 : TPTP v8.2.0. Released v6.0.0.
% Domain : Software Verification
% Problem : Huffman's Algorithm line 468
% Version : Especial.
% English :
% Refs : [BN10] Boehme & Nipkow (2010), Sledgehammer: Judgement Day
% : [Bla09] Blanchette (2009), Proof Pearl: Mechanizing the Textbo
% : [Bla13] Blanchette (2011), Email to Geoff Sutcliffe
% Source : [Bla13]
% Names : huff_468 [Bla13]
% Status : Unknown
% Rating : 1.00 v6.4.0
% Syntax : Number of formulae : 172 ( 36 unt; 49 typ; 0 def)
% Number of atoms : 282 ( 45 equ)
% Maximal formula atoms : 7 ( 1 avg)
% Number of connectives : 173 ( 14 ~; 3 |; 9 &)
% ( 18 <=>; 129 =>; 0 <=; 0 <~>)
% Maximal formula depth : 14 ( 5 avg)
% Maximal term depth : 7 ( 1 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 42 ( 29 >; 13 *; 0 +; 0 <<)
% Number of predicates : 18 ( 17 usr; 0 prp; 1-6 aty)
% Number of functors : 29 ( 29 usr; 5 con; 0-4 aty)
% Number of variables : 378 ( 334 !; 1 ?; 378 :)
% ( 43 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_UNK_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:15:30
%------------------------------------------------------------------------------
%----Should-be-implicit typings (7)
tff(ty_t_a,type,
a: $tType ).
tff(ty_tc_HOL_Obool,type,
bool: $tType ).
tff(ty_tc_Huffman__Mirabelle__lalbadcutu_Otree,type,
huffma1450048681e_tree: $tType > $tType ).
tff(ty_tc_List_Olist,type,
list: $tType > $tType ).
tff(ty_tc_Nat_Onat,type,
nat: $tType ).
tff(ty_tc_fun,type,
fun: ( $tType * $tType ) > $tType ).
tff(ty_tc_prod,type,
product_prod: ( $tType * $tType ) > $tType ).
%----Explicit typings (42)
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_Finite__Set_Ofinite,type,
finite_finite:
!>[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_c_COMBK,type,
combk:
!>[A: $tType,B: $tType] : ( A > fun(B,A) ) ).
tff(sy_c_Finite__Set_Ofinite,type,
finite_finite1:
!>[A: $tType] : ( fun(A,bool) > $o ) ).
tff(sy_c_Finite__Set_Ofolding__image__simple__idem,type,
finite908156982e_idem:
!>[A: $tType,B: $tType] : ( ( fun(A,fun(A,A)) * A * fun(B,A) * fun(fun(B,bool),A) ) > $o ) ).
tff(sy_c_Finite__Set_Ofolding__one__idem,type,
finite2073411215e_idem:
!>[A: $tType] : ( ( fun(A,fun(A,A)) * fun(fun(A,bool),A) ) > $o ) ).
tff(sy_c_List_Olenlex,type,
lenlex:
!>[A: $tType] : ( fun(product_prod(A,A),bool) > fun(product_prod(list(A),list(A)),bool) ) ).
tff(sy_c_List_Olex,type,
lex:
!>[A: $tType] : ( fun(product_prod(A,A),bool) > fun(product_prod(list(A),list(A)),bool) ) ).
tff(sy_c_List_Olexn,type,
lexn:
!>[A: $tType] : ( ( fun(product_prod(A,A),bool) * nat ) > fun(product_prod(list(A),list(A)),bool) ) ).
tff(sy_c_List_Olistrel1,type,
listrel1:
!>[A: $tType] : ( fun(product_prod(A,A),bool) > fun(product_prod(list(A),list(A)),bool) ) ).
tff(sy_c_List_Omeasures,type,
measures:
!>[A: $tType] : ( list(fun(A,nat)) > fun(product_prod(A,A),bool) ) ).
tff(sy_c_Nat_Ocompow,type,
compow:
!>[A: $tType,B: $tType] : fun(nat,fun(fun(A,B),fun(A,B))) ).
tff(sy_c_Nat_Ofunpow,type,
funpow:
!>[A: $tType] : fun(nat,fun(fun(A,A),fun(A,A))) ).
tff(sy_c_Nitpick_Ounknown,type,
unknown:
!>[A: $tType] : $o ).
tff(sy_c_Nitpick_Owf_H,type,
wf1:
!>[A: $tType] : ( fun(product_prod(A,A),bool) > $o ) ).
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_Relation_Oconverse,type,
converse:
!>[A: $tType,B: $tType] : ( fun(product_prod(A,B),bool) > fun(product_prod(B,A),bool) ) ).
tff(sy_c_Relation_Oinv__image,type,
inv_image:
!>[B: $tType,A: $tType] : ( fun(product_prod(B,B),bool) > fun(fun(A,B),fun(product_prod(A,A),bool)) ) ).
tff(sy_c_Relation_Ototal__on,type,
total_on:
!>[A: $tType] : ( ( fun(A,bool) * fun(product_prod(A,A),bool) ) > $o ) ).
tff(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( fun(A,bool) > fun(A,bool) ) ).
tff(sy_c_Transitive__Closure_Oacyclic,type,
transitive_acyclic:
!>[A: $tType] : ( fun(product_prod(A,A),bool) > $o ) ).
tff(sy_c_Wellfounded_Oacc,type,
acc:
!>[A: $tType] : ( fun(product_prod(A,A),bool) > fun(A,bool) ) ).
tff(sy_c_Wellfounded_Ofinite__psubset,type,
finite_psubset:
!>[A: $tType] : fun(product_prod(fun(A,bool),fun(A,bool)),bool) ).
tff(sy_c_Wellfounded_Oless__than,type,
less_than: fun(product_prod(nat,nat),bool) ).
tff(sy_c_Wellfounded_Olex__prod,type,
lex_prod:
!>[A: $tType,B: $tType] : ( ( fun(product_prod(A,A),bool) * fun(product_prod(B,B),bool) ) > fun(product_prod(product_prod(A,B),product_prod(A,B)),bool) ) ).
tff(sy_c_Wellfounded_Omax__ext,type,
max_ext:
!>[A: $tType] : ( fun(product_prod(A,A),bool) > fun(product_prod(fun(A,bool),fun(A,bool)),bool) ) ).
tff(sy_c_Wellfounded_Omeasure,type,
measure:
!>[A: $tType] : fun(fun(A,nat),fun(product_prod(A,A),bool)) ).
tff(sy_c_Wellfounded_Omin__ext,type,
min_ext:
!>[A: $tType] : ( fun(product_prod(A,A),bool) > fun(product_prod(fun(A,bool),fun(A,bool)),bool) ) ).
tff(sy_c_Wellfounded_Omlex__prod,type,
mlex_prod:
!>[A: $tType] : ( ( fun(A,nat) * fun(product_prod(A,A),bool) ) > fun(product_prod(A,A),bool) ) ).
tff(sy_c_Wellfounded_Opred__nat,type,
pred_nat: fun(product_prod(nat,nat),bool) ).
tff(sy_c_Wellfounded_Owf,type,
wf:
!>[A: $tType] : ( fun(product_prod(A,A),bool) > $o ) ).
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_ATP_058R__2,type,
aTP_R_2: fun(product_prod(product_prod(huffma1450048681e_tree(a),a),product_prod(huffma1450048681e_tree(a),a)),bool) ).
%----Relevant facts (100)
tff(fact_0_wf__lex__prod,axiom,
! [B: $tType,C: $tType,Rb: fun(product_prod(C,C),bool),Ra: fun(product_prod(B,B),bool)] :
( wf(B,Ra)
=> ( wf(C,Rb)
=> wf(product_prod(B,C),lex_prod(B,C,Ra,Rb)) ) ) ).
tff(fact_1_wf__lenlex,axiom,
! [B: $tType,R: fun(product_prod(B,B),bool)] :
( wf(B,R)
=> wf(list(B),lenlex(B,R)) ) ).
tff(fact_2_wf__measure,axiom,
! [B: $tType,F: fun(B,nat)] : wf(B,aa(fun(B,nat),fun(product_prod(B,B),bool),measure(B),F)) ).
tff(fact_3_min__ext__wf,axiom,
! [B: $tType,R: fun(product_prod(B,B),bool)] :
( wf(B,R)
=> wf(fun(B,bool),min_ext(B,R)) ) ).
tff(fact_4_wf__lex,axiom,
! [B: $tType,R: fun(product_prod(B,B),bool)] :
( wf(B,R)
=> wf(list(B),lex(B,R)) ) ).
tff(fact_5_wf__mlex,axiom,
! [B: $tType,F: fun(B,nat),R1: fun(product_prod(B,B),bool)] :
( wf(B,R1)
=> wf(B,mlex_prod(B,F,R1)) ) ).
tff(fact_6_wf__inv__image,axiom,
! [B: $tType,C: $tType,F: fun(C,B),R: fun(product_prod(B,B),bool)] :
( wf(B,R)
=> wf(C,aa(fun(C,B),fun(product_prod(C,C),bool),inv_image(B,C,R),F)) ) ).
tff(fact_7_wf__lexn,axiom,
! [B: $tType,N: nat,R: fun(product_prod(B,B),bool)] :
( wf(B,R)
=> wf(list(B),lexn(B,R,N)) ) ).
tff(fact_8_wf__measures,axiom,
! [B: $tType,Fs: list(fun(B,nat))] : wf(B,measures(B,Fs)) ).
tff(fact_9_max__ext__wf,axiom,
! [B: $tType,R: fun(product_prod(B,B),bool)] :
( wf(B,R)
=> wf(fun(B,bool),max_ext(B,R)) ) ).
tff(fact_10_wf__acyclic,axiom,
! [B: $tType,R: fun(product_prod(B,B),bool)] :
( wf(B,R)
=> transitive_acyclic(B,R) ) ).
tff(fact_11_wf__listrel1__iff,axiom,
! [B: $tType,R: fun(product_prod(B,B),bool)] :
( wf(list(B),listrel1(B,R))
<=> wf(B,R) ) ).
tff(fact_12_wf__finite__psubset,axiom,
! [B: $tType] : wf(fun(B,bool),finite_psubset(B)) ).
tff(fact_13_measure__def,axiom,
! [B: $tType] : measure(B) = inv_image(nat,B,less_than) ).
tff(fact_14_finite__acyclic__wf,axiom,
! [B: $tType,R: fun(product_prod(B,B),bool)] :
( finite_finite1(product_prod(B,B),R)
=> ( transitive_acyclic(B,R)
=> wf(B,R) ) ) ).
tff(fact_15_wf__iff__acyclic__if__finite,axiom,
! [B: $tType,R: fun(product_prod(B,B),bool)] :
( finite_finite1(product_prod(B,B),R)
=> ( wf(B,R)
<=> transitive_acyclic(B,R) ) ) ).
tff(fact_16_acc__wfD,axiom,
! [B: $tType,X: B,R: fun(product_prod(B,B),bool)] :
( wf(B,R)
=> member(B,X,acc(B,R)) ) ).
tff(fact_17_wf__acc__iff,axiom,
! [B: $tType,R: fun(product_prod(B,B),bool)] :
( wf(B,R)
<=> ! [X2: B] : member(B,X2,acc(B,R)) ) ).
tff(fact_18_wf__empty,axiom,
! [B: $tType] : wf(B,bot_bot(fun(product_prod(B,B),bool))) ).
tff(fact_19_wf__exp,axiom,
! [B: $tType,R1: fun(product_prod(B,B),bool),N: nat] :
( wf(B,aa(fun(product_prod(B,B),bool),fun(product_prod(B,B),bool),aa(nat,fun(fun(product_prod(B,B),bool),fun(product_prod(B,B),bool)),compow(product_prod(B,B),bool),N),R1))
=> wf(B,R1) ) ).
tff(fact_20_wf__subset,axiom,
! [B: $tType,P1: fun(product_prod(B,B),bool),R: fun(product_prod(B,B),bool)] :
( wf(B,R)
=> ( ord_less_eq(fun(product_prod(B,B),bool),P1,R)
=> wf(B,P1) ) ) ).
tff(fact_21_wf__less__than,axiom,
wf(nat,less_than) ).
tff(fact_22_acc__subset,axiom,
! [B: $tType,R2: fun(product_prod(B,B),bool),R11: fun(product_prod(B,B),bool)] :
( ord_less_eq(fun(product_prod(B,B),bool),R11,R2)
=> ord_less_eq(fun(B,bool),acc(B,R2),acc(B,R11)) ) ).
tff(fact_23_listrel1__mono,axiom,
! [B: $tType,S: fun(product_prod(B,B),bool),R: fun(product_prod(B,B),bool)] :
( ord_less_eq(fun(product_prod(B,B),bool),R,S)
=> ord_less_eq(fun(product_prod(list(B),list(B)),bool),listrel1(B,R),listrel1(B,S)) ) ).
tff(fact_24_subset__empty,axiom,
! [B: $tType,A1: fun(B,bool)] :
( ord_less_eq(fun(B,bool),A1,bot_bot(fun(B,bool)))
<=> ( A1 = bot_bot(fun(B,bool)) ) ) ).
tff(fact_25_empty__subsetI,axiom,
! [B: $tType,A1: fun(B,bool)] : ord_less_eq(fun(B,bool),bot_bot(fun(B,bool)),A1) ).
tff(fact_26_finite_OemptyI,axiom,
! [B: $tType] : finite_finite1(B,bot_bot(fun(B,bool))) ).
tff(fact_27_acc__wfI,axiom,
! [B: $tType,R: fun(product_prod(B,B),bool)] :
( ! [X1: B] : member(B,X1,acc(B,R))
=> wf(B,R) ) ).
tff(fact_28_equalityI,axiom,
! [B: $tType,B2: fun(B,bool),A1: fun(B,bool)] :
( ord_less_eq(fun(B,bool),A1,B2)
=> ( ord_less_eq(fun(B,bool),B2,A1)
=> ( A1 = B2 ) ) ) ).
tff(fact_29_subsetD,axiom,
! [B: $tType,C1: B,B2: fun(B,bool),A1: fun(B,bool)] :
( ord_less_eq(fun(B,bool),A1,B2)
=> ( member(B,C1,A1)
=> member(B,C1,B2) ) ) ).
tff(fact_30_finite__code,axiom,
! [B: $tType] :
( finite_finite(B)
=> ! [A1: fun(B,bool)] : finite_finite1(B,A1) ) ).
tff(fact_31_all__not__in__conv,axiom,
! [B: $tType,A1: fun(B,bool)] :
( ! [X2: B] : ~ member(B,X2,A1)
<=> ( A1 = bot_bot(fun(B,bool)) ) ) ).
tff(fact_32_empty__Collect__eq,axiom,
! [B: $tType,Pa: fun(B,bool)] :
( ( bot_bot(fun(B,bool)) = collect(B,Pa) )
<=> ! [X2: B] : ~ pp(aa(B,bool,Pa,X2)) ) ).
tff(fact_33_empty__iff,axiom,
! [B: $tType,C1: B] : ~ member(B,C1,bot_bot(fun(B,bool))) ).
tff(fact_34_Collect__empty__eq,axiom,
! [B: $tType,Pa: fun(B,bool)] :
( ( collect(B,Pa) = bot_bot(fun(B,bool)) )
<=> ! [X2: B] : ~ pp(aa(B,bool,Pa,X2)) ) ).
tff(fact_35_emptyE,axiom,
! [B: $tType,Aa: B] : ~ member(B,Aa,bot_bot(fun(B,bool))) ).
tff(fact_36_empty__def,axiom,
! [B: $tType] : bot_bot(fun(B,bool)) = collect(B,combk(bool,B,fFalse)) ).
tff(fact_37_ex__in__conv,axiom,
! [B: $tType,A1: fun(B,bool)] :
( ? [X2: B] : member(B,X2,A1)
<=> ( A1 != bot_bot(fun(B,bool)) ) ) ).
tff(fact_38_equals0D,axiom,
! [B: $tType,Aa: B,A1: fun(B,bool)] :
( ( A1 = bot_bot(fun(B,bool)) )
=> ~ member(B,Aa,A1) ) ).
tff(fact_39_finite,axiom,
! [B: $tType] :
( finite_finite(B)
=> ! [A1: fun(B,bool)] : finite_finite1(B,A1) ) ).
tff(fact_40_subset__refl,axiom,
! [B: $tType,A1: fun(B,bool)] : ord_less_eq(fun(B,bool),A1,A1) ).
tff(fact_41_set__eq__subset,axiom,
! [B: $tType,B2: fun(B,bool),A1: fun(B,bool)] :
( ( A1 = B2 )
<=> ( ord_less_eq(fun(B,bool),A1,B2)
& ord_less_eq(fun(B,bool),B2,A1) ) ) ).
tff(fact_42_equalityD1,axiom,
! [B: $tType,B2: fun(B,bool),A1: fun(B,bool)] :
( ( A1 = B2 )
=> ord_less_eq(fun(B,bool),A1,B2) ) ).
tff(fact_43_equalityD2,axiom,
! [B: $tType,B2: fun(B,bool),A1: fun(B,bool)] :
( ( A1 = B2 )
=> ord_less_eq(fun(B,bool),B2,A1) ) ).
tff(fact_44_in__mono,axiom,
! [B: $tType,X: B,B2: fun(B,bool),A1: fun(B,bool)] :
( ord_less_eq(fun(B,bool),A1,B2)
=> ( member(B,X,A1)
=> member(B,X,B2) ) ) ).
tff(fact_45_set__rev__mp,axiom,
! [B: $tType,B2: fun(B,bool),A1: fun(B,bool),X: B] :
( member(B,X,A1)
=> ( ord_less_eq(fun(B,bool),A1,B2)
=> member(B,X,B2) ) ) ).
tff(fact_46_set__mp,axiom,
! [B: $tType,X: B,B2: fun(B,bool),A1: fun(B,bool)] :
( ord_less_eq(fun(B,bool),A1,B2)
=> ( member(B,X,A1)
=> member(B,X,B2) ) ) ).
tff(fact_47_subset__trans,axiom,
! [B: $tType,C3: fun(B,bool),B2: fun(B,bool),A1: fun(B,bool)] :
( ord_less_eq(fun(B,bool),A1,B2)
=> ( ord_less_eq(fun(B,bool),B2,C3)
=> ord_less_eq(fun(B,bool),A1,C3) ) ) ).
tff(fact_48_equalityE,axiom,
! [B: $tType,B2: fun(B,bool),A1: fun(B,bool)] :
( ( A1 = B2 )
=> ~ ( ord_less_eq(fun(B,bool),A1,B2)
=> ~ ord_less_eq(fun(B,bool),B2,A1) ) ) ).
tff(fact_49_rev__finite__subset,axiom,
! [B: $tType,A1: fun(B,bool),B2: fun(B,bool)] :
( finite_finite1(B,B2)
=> ( ord_less_eq(fun(B,bool),A1,B2)
=> finite_finite1(B,A1) ) ) ).
tff(fact_50_finite__subset,axiom,
! [B: $tType,B2: fun(B,bool),A1: fun(B,bool)] :
( ord_less_eq(fun(B,bool),A1,B2)
=> ( finite_finite1(B,B2)
=> finite_finite1(B,A1) ) ) ).
tff(fact_51_subsetI,axiom,
! [B: $tType,B2: fun(B,bool),A1: fun(B,bool)] :
( ! [X1: B] :
( member(B,X1,A1)
=> member(B,X1,B2) )
=> ord_less_eq(fun(B,bool),A1,B2) ) ).
tff(fact_52_order__refl,axiom,
! [A: $tType] :
( preorder(A)
=> ! [X3: A] : ord_less_eq(A,X3,X3) ) ).
tff(fact_53_acyclic__subset,axiom,
! [B: $tType,R: fun(product_prod(B,B),bool),S: fun(product_prod(B,B),bool)] :
( transitive_acyclic(B,S)
=> ( ord_less_eq(fun(product_prod(B,B),bool),R,S)
=> transitive_acyclic(B,R) ) ) ).
tff(fact_54_less__by__empty,axiom,
! [B: $tType,B2: fun(product_prod(B,B),bool),A1: fun(product_prod(B,B),bool)] :
( ( A1 = bot_bot(fun(product_prod(B,B),bool)) )
=> ord_less_eq(fun(product_prod(B,B),bool),A1,B2) ) ).
tff(fact_55_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_56_bot__apply,axiom,
! [C: $tType,B: $tType] :
( bot(B)
=> ! [X: C] : aa(C,B,bot_bot(fun(C,B)),X) = bot_bot(B) ) ).
tff(fact_57_linorder__le__cases,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y1: A,X3: A] :
( ~ ord_less_eq(A,X3,Y1)
=> ord_less_eq(A,Y1,X3) ) ) ).
tff(fact_58_order__trans,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Z1: A,Y1: A,X3: A] :
( ord_less_eq(A,X3,Y1)
=> ( ord_less_eq(A,Y1,Z1)
=> ord_less_eq(A,X3,Z1) ) ) ) ).
tff(fact_59_order__antisym,axiom,
! [A: $tType] :
( order(A)
=> ! [Y1: A,X3: A] :
( ord_less_eq(A,X3,Y1)
=> ( ord_less_eq(A,Y1,X3)
=> ( X3 = Y1 ) ) ) ) ).
tff(fact_60_ord__le__eq__trans,axiom,
! [A: $tType] :
( ord(A)
=> ! [C2: A,B3: A,A2: A] :
( ord_less_eq(A,A2,B3)
=> ( ( B3 = C2 )
=> ord_less_eq(A,A2,C2) ) ) ) ).
tff(fact_61_ord__eq__le__trans,axiom,
! [A: $tType] :
( ord(A)
=> ! [C2: A,B3: A,A2: A] :
( ( A2 = B3 )
=> ( ord_less_eq(A,B3,C2)
=> ord_less_eq(A,A2,C2) ) ) ) ).
tff(fact_62_order__antisym__conv,axiom,
! [B: $tType] :
( order(B)
=> ! [X: B,Y2: B] :
( ord_less_eq(B,Y2,X)
=> ( ord_less_eq(B,X,Y2)
<=> ( X = Y2 ) ) ) ) ).
tff(fact_63_order__eq__refl,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Y1: A,X3: A] :
( ( X3 = Y1 )
=> ord_less_eq(A,X3,Y1) ) ) ).
tff(fact_64_order__eq__iff,axiom,
! [B: $tType] :
( order(B)
=> ! [Y2: B,X: B] :
( ( X = Y2 )
<=> ( ord_less_eq(B,X,Y2)
& ord_less_eq(B,Y2,X) ) ) ) ).
tff(fact_65_linorder__linear,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y1: A,X3: A] :
( ord_less_eq(A,X3,Y1)
| ord_less_eq(A,Y1,X3) ) ) ).
tff(fact_66_le__bot,axiom,
! [A: $tType] :
( bot(A)
=> ! [A2: A] :
( ord_less_eq(A,A2,bot_bot(A))
=> ( A2 = bot_bot(A) ) ) ) ).
tff(fact_67_bot__unique,axiom,
! [B: $tType] :
( bot(B)
=> ! [Aa: B] :
( ord_less_eq(B,Aa,bot_bot(B))
<=> ( Aa = bot_bot(B) ) ) ) ).
tff(fact_68_bot__least,axiom,
! [A: $tType] :
( bot(A)
=> ! [A2: A] : ord_less_eq(A,bot_bot(A),A2) ) ).
tff(fact_69_le__funE,axiom,
! [C: $tType,B: $tType] :
( ord(C)
=> ! [X: 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,X),aa(B,C,G,X)) ) ) ).
tff(fact_70_le__funD,axiom,
! [C: $tType,B: $tType] :
( ord(C)
=> ! [X: 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,X),aa(B,C,G,X)) ) ) ).
tff(fact_71_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)
<=> ! [X2: B] : ord_less_eq(C,aa(B,C,F,X2),aa(B,C,G,X2)) ) ) ).
tff(fact_72_le__funI,axiom,
! [C: $tType,B: $tType] :
( ord(C)
=> ! [G: fun(B,C),F: fun(B,C)] :
( ! [X1: B] : ord_less_eq(C,aa(B,C,F,X1),aa(B,C,G,X1))
=> ord_less_eq(fun(B,C),F,G) ) ) ).
tff(fact_73_folding__one__idem_Osubset__idem,axiom,
! [B: $tType,B2: fun(B,bool),A1: fun(B,bool),F1: fun(fun(B,bool),B),F: fun(B,fun(B,B))] :
( finite2073411215e_idem(B,F,F1)
=> ( finite_finite1(B,A1)
=> ( ( B2 != bot_bot(fun(B,bool)) )
=> ( ord_less_eq(fun(B,bool),B2,A1)
=> ( aa(B,B,aa(B,fun(B,B),F,aa(fun(B,bool),B,F1,B2)),aa(fun(B,bool),B,F1,A1)) = aa(fun(B,bool),B,F1,A1) ) ) ) ) ) ).
tff(fact_74_wf__pred__nat,axiom,
wf(nat,pred_nat) ).
tff(fact_75_ext,axiom,
! [C: $tType,B: $tType,G: fun(B,C),F: fun(B,C)] :
( ! [X1: B] : aa(B,C,F,X1) = aa(B,C,G,X1)
=> ( F = G ) ) ).
tff(fact_76_mem__def,axiom,
! [B: $tType,A1: fun(B,bool),X: B] :
( member(B,X,A1)
<=> pp(aa(B,bool,A1,X)) ) ).
tff(fact_77_Collect__def,axiom,
! [B: $tType,Pa: fun(B,bool)] : collect(B,Pa) = Pa ).
tff(fact_78_folding__one__idem_Oidem,axiom,
! [B: $tType,X: B,F1: fun(fun(B,bool),B),F: fun(B,fun(B,B))] :
( finite2073411215e_idem(B,F,F1)
=> ( aa(B,B,aa(B,fun(B,B),F,X),X) = X ) ) ).
tff(fact_79_folding__one__idem_Oin__idem,axiom,
! [B: $tType,X: B,A1: fun(B,bool),F1: fun(fun(B,bool),B),F: fun(B,fun(B,B))] :
( finite2073411215e_idem(B,F,F1)
=> ( finite_finite1(B,A1)
=> ( member(B,X,A1)
=> ( aa(B,B,aa(B,fun(B,B),F,X),aa(fun(B,bool),B,F1,A1)) = aa(fun(B,bool),B,F1,A1) ) ) ) ) ).
tff(fact_80_folding__image__simple__idem_Osubset__idem,axiom,
! [B: $tType,C: $tType,B2: fun(C,bool),A1: fun(C,bool),F1: fun(fun(C,bool),B),G: fun(C,B),Z: B,F: fun(B,fun(B,B))] :
( finite908156982e_idem(B,C,F,Z,G,F1)
=> ( finite_finite1(C,A1)
=> ( ord_less_eq(fun(C,bool),B2,A1)
=> ( aa(B,B,aa(B,fun(B,B),F,aa(fun(C,bool),B,F1,B2)),aa(fun(C,bool),B,F1,A1)) = aa(fun(C,bool),B,F1,A1) ) ) ) ) ).
tff(fact_81_Collect__mono,axiom,
! [B: $tType,Q1: fun(B,bool),Pa: fun(B,bool)] :
( ! [X1: B] :
( pp(aa(B,bool,Pa,X1))
=> pp(aa(B,bool,Q1,X1)) )
=> ord_less_eq(fun(B,bool),collect(B,Pa),collect(B,Q1)) ) ).
tff(fact_82_folding__image__simple__idem_Oidem,axiom,
! [C: $tType,B: $tType,X: B,F1: fun(fun(C,bool),B),G: fun(C,B),Z: B,F: fun(B,fun(B,B))] :
( finite908156982e_idem(B,C,F,Z,G,F1)
=> ( aa(B,B,aa(B,fun(B,B),F,X),X) = X ) ) ).
tff(fact_83_folding__image__simple__idem_Oin__idem,axiom,
! [B: $tType,C: $tType,X: C,A1: fun(C,bool),F1: fun(fun(C,bool),B),G: fun(C,B),Z: B,F: fun(B,fun(B,B))] :
( finite908156982e_idem(B,C,F,Z,G,F1)
=> ( finite_finite1(C,A1)
=> ( member(C,X,A1)
=> ( aa(B,B,aa(B,fun(B,B),F,aa(C,B,G,X)),aa(fun(C,bool),B,F1,A1)) = aa(fun(C,bool),B,F1,A1) ) ) ) ) ).
tff(fact_84_equals0I,axiom,
! [B: $tType,A1: fun(B,bool)] :
( ! [Y: B] : ~ member(B,Y,A1)
=> ( A1 = bot_bot(fun(B,bool)) ) ) ).
tff(fact_85_order__subst1,axiom,
! [B: $tType,C: $tType] :
( ( order(C)
& order(B) )
=> ! [C1: C,B1: C,F: fun(C,B),Aa: B] :
( ord_less_eq(B,Aa,aa(C,B,F,B1))
=> ( ord_less_eq(C,B1,C1)
=> ( ! [X1: C,Y: C] :
( ord_less_eq(C,X1,Y)
=> ord_less_eq(B,aa(C,B,F,X1),aa(C,B,F,Y)) )
=> ord_less_eq(B,Aa,aa(C,B,F,C1)) ) ) ) ) ).
tff(fact_86_ord__le__eq__subst,axiom,
! [B: $tType,C: $tType] :
( ( ord(C)
& ord(B) )
=> ! [C1: C,F: fun(B,C),B1: B,Aa: B] :
( ord_less_eq(B,Aa,B1)
=> ( ( aa(B,C,F,B1) = C1 )
=> ( ! [X1: B,Y: B] :
( ord_less_eq(B,X1,Y)
=> ord_less_eq(C,aa(B,C,F,X1),aa(B,C,F,Y)) )
=> ord_less_eq(C,aa(B,C,F,Aa),C1) ) ) ) ) ).
tff(fact_87_order__subst2,axiom,
! [B: $tType,C: $tType] :
( ( order(C)
& order(B) )
=> ! [C1: C,F: fun(B,C),B1: B,Aa: B] :
( ord_less_eq(B,Aa,B1)
=> ( ord_less_eq(C,aa(B,C,F,B1),C1)
=> ( ! [X1: B,Y: B] :
( ord_less_eq(B,X1,Y)
=> ord_less_eq(C,aa(B,C,F,X1),aa(B,C,F,Y)) )
=> ord_less_eq(C,aa(B,C,F,Aa),C1) ) ) ) ) ).
tff(fact_88_ord__eq__le__subst,axiom,
! [B: $tType,C: $tType] :
( ( ord(C)
& ord(B) )
=> ! [C1: C,B1: C,F: fun(C,B),Aa: B] :
( ( Aa = aa(C,B,F,B1) )
=> ( ord_less_eq(C,B1,C1)
=> ( ! [X1: C,Y: C] :
( ord_less_eq(C,X1,Y)
=> ord_less_eq(B,aa(C,B,F,X1),aa(C,B,F,Y)) )
=> ord_less_eq(B,Aa,aa(C,B,F,C1)) ) ) ) ) ).
tff(fact_89_Nitpick_Owf_H__def,axiom,
! [B: $tType,R: fun(product_prod(B,B),bool)] :
( wf1(B,R)
<=> ( transitive_acyclic(B,R)
& ( finite_finite1(product_prod(B,B),R)
| unknown(bool) ) ) ) ).
tff(fact_90_funpow__swap1,axiom,
! [B: $tType,X: B,N: nat,F: fun(B,B)] : aa(B,B,F,aa(B,B,aa(fun(B,B),fun(B,B),aa(nat,fun(fun(B,B),fun(B,B)),compow(B,B),N),F),X)) = aa(B,B,aa(fun(B,B),fun(B,B),aa(nat,fun(fun(B,B),fun(B,B)),compow(B,B),N),F),aa(B,B,F,X)) ).
tff(fact_91_finite__acyclic__wf__converse,axiom,
! [B: $tType,R: fun(product_prod(B,B),bool)] :
( finite_finite1(product_prod(B,B),R)
=> ( transitive_acyclic(B,R)
=> wf(B,converse(B,B,R)) ) ) ).
tff(fact_92_acyclic__converse,axiom,
! [B: $tType,R: fun(product_prod(B,B),bool)] :
( transitive_acyclic(B,converse(B,B,R))
<=> transitive_acyclic(B,R) ) ).
tff(fact_93_listrel1__converse,axiom,
! [B: $tType,R: fun(product_prod(B,B),bool)] : listrel1(B,converse(B,B,R)) = converse(list(B),list(B),listrel1(B,R)) ).
tff(fact_94_converse__inv__image,axiom,
! [C: $tType,B: $tType,F: fun(B,C),R1: fun(product_prod(C,C),bool)] : converse(B,B,aa(fun(B,C),fun(product_prod(B,B),bool),inv_image(C,B,R1),F)) = aa(fun(B,C),fun(product_prod(B,B),bool),inv_image(C,B,converse(C,C,R1)),F) ).
tff(fact_95_finite__converse,axiom,
! [B: $tType,C: $tType,R: fun(product_prod(C,B),bool)] :
( finite_finite1(product_prod(B,C),converse(C,B,R))
<=> finite_finite1(product_prod(C,B),R) ) ).
tff(fact_96_converse__converse,axiom,
! [C: $tType,B: $tType,R: fun(product_prod(B,C),bool)] : converse(C,B,converse(B,C,R)) = R ).
tff(fact_97_funpow__code__def,axiom,
! [B: $tType] : funpow(B) = compow(B,B) ).
tff(fact_98_total__on__converse,axiom,
! [B: $tType,R: fun(product_prod(B,B),bool),A1: fun(B,bool)] :
( total_on(B,A1,converse(B,B,R))
<=> total_on(B,A1,R) ) ).
tff(fact_99_total__on__empty,axiom,
! [B: $tType,R: fun(product_prod(B,B),bool)] : total_on(B,bot_bot(fun(B,bool)),R) ).
%----Arities (17)
tff(arity_fun___Orderings_Opreorder,axiom,
! [T_1: $tType,T_2: $tType] :
( preorder(T_2)
=> preorder(fun(T_1,T_2)) ) ).
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_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___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___Orderings_Opreorder,axiom,
preorder(bool) ).
tff(arity_HOL_Obool___Orderings_Olinorder,axiom,
linorder(bool) ).
tff(arity_HOL_Obool___Finite__Set_Ofinite,axiom,
finite_finite(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_prod___Finite__Set_Ofinite,axiom,
! [T_1: $tType,T_2: $tType] :
( ( finite_finite(T_2)
& finite_finite(T_1) )
=> finite_finite(product_prod(T_1,T_2)) ) ).
%----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,
wf(product_prod(huffma1450048681e_tree(a),a),aTP_R_2) ).
%------------------------------------------------------------------------------