TPTP Problem File: ITP084^2.p
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%------------------------------------------------------------------------------
% File : ITP084^2 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer KnowledgeKeysSecrets problem prob_119__3292052_1
% Version : Especial.
% English :
% Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% Source : [Des21]
% Names : KnowledgeKeysSecrets/prob_119__3292052_1 [Des21]
% Status : Theorem
% Rating : 0.33 v8.1.0, 0.25 v7.5.0
% Syntax : Number of formulae : 344 ( 139 unt; 78 typ; 0 def)
% Number of atoms : 744 ( 433 equ; 0 cnn)
% Maximal formula atoms : 13 ( 2 avg)
% Number of connectives : 4042 ( 141 ~; 45 |; 86 &;3444 @)
% ( 0 <=>; 326 =>; 0 <=; 0 <~>)
% Maximal formula depth : 23 ( 7 avg)
% Number of types : 6 ( 5 usr)
% Number of type conns : 284 ( 284 >; 0 *; 0 +; 0 <<)
% Number of symbols : 76 ( 73 usr; 4 con; 0-5 aty)
% Number of variables : 1123 ( 30 ^; 965 !; 72 ?;1123 :)
% ( 56 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 16:18:54.283
%------------------------------------------------------------------------------
% Could-be-implicit typings (7)
thf(ty_t_Secrecy__types_OExpression,type,
secrecy_Expression: $tType ).
thf(ty_t_Secrecy__types_OSecrets,type,
secrecy_Secrets: $tType ).
thf(ty_t_Secrecy__types_OspecID,type,
secrecy_specID: $tType ).
thf(ty_t_Secrecy__types_OKeys,type,
secrecy_Keys: $tType ).
thf(ty_t_Secrecy__types_OKS,type,
secrecy_KS: $tType ).
thf(ty_t_List_Olist,type,
list: $tType > $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
% Explicit typings (71)
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder,type,
order:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : $o ).
thf(sy_c_BNF__Greatest__Fixpoint_OShift,type,
bNF_Greatest_Shift:
!>[A: $tType] : ( ( set @ ( list @ A ) ) > A > ( set @ ( list @ A ) ) ) ).
thf(sy_c_BNF__Greatest__Fixpoint_OSucc,type,
bNF_Greatest_Succ:
!>[A: $tType] : ( ( set @ ( list @ A ) ) > ( list @ A ) > ( set @ A ) ) ).
thf(sy_c_BNF__Greatest__Fixpoint_Oshift,type,
bNF_Greatest_shift:
!>[A: $tType,B: $tType] : ( ( ( list @ A ) > B ) > A > ( list @ A ) > B ) ).
thf(sy_c_CompLocalSecrets_OLocalSecrets,type,
localSecrets: secrecy_specID > ( set @ secrecy_KS ) ).
thf(sy_c_Groups_Omonoid,type,
monoid:
!>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
thf(sy_c_If,type,
if:
!>[A: $tType] : ( $o > A > A > A ) ).
thf(sy_c_KnowledgeKeysSecrets__Mirabelle__khazhniute_OeoutKnowCorrect,type,
knowle1667810483orrect: secrecy_specID > secrecy_KS > $o ).
thf(sy_c_KnowledgeKeysSecrets__Mirabelle__khazhniute_OeoutKnowsECorrect,type,
knowle1893229345orrect: secrecy_specID > secrecy_Expression > $o ).
thf(sy_c_KnowledgeKeysSecrets__Mirabelle__khazhniute_Oknow,type,
knowle1437066798e_know: secrecy_specID > secrecy_KS > $o ).
thf(sy_c_KnowledgeKeysSecrets__Mirabelle__khazhniute_Oknows,type,
knowle2055185861_knows: secrecy_specID > ( list @ secrecy_Expression ) > $o ).
thf(sy_c_ListExtras_Odisjoint,type,
disjoint:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > $o ) ).
thf(sy_c_ListExtras_Omem,type,
mem:
!>[A: $tType] : ( A > ( list @ A ) > $o ) ).
thf(sy_c_ListExtras_OmemS,type,
memS:
!>[A: $tType] : ( A > ( list @ A ) > $o ) ).
thf(sy_c_List_Oappend,type,
append:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Oarg__min__list,type,
arg_min_list:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( list @ A ) > A ) ).
thf(sy_c_List_Obind,type,
bind:
!>[A: $tType,B: $tType] : ( ( list @ A ) > ( A > ( list @ B ) ) > ( list @ B ) ) ).
thf(sy_c_List_Obutlast,type,
butlast:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Oconcat,type,
concat:
!>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ A ) ) ).
thf(sy_c_List_Odistinct__adj,type,
distinct_adj:
!>[A: $tType] : ( ( list @ A ) > $o ) ).
thf(sy_c_List_Ofoldr,type,
foldr:
!>[A: $tType,B: $tType] : ( ( A > B > B ) > ( list @ A ) > B > B ) ).
thf(sy_c_List_Oinsert,type,
insert:
!>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Olast,type,
last:
!>[A: $tType] : ( ( list @ A ) > A ) ).
thf(sy_c_List_Olist_OCons,type,
cons:
!>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Olist_ONil,type,
nil:
!>[A: $tType] : ( list @ A ) ).
thf(sy_c_List_Olist_Ocase__list,type,
case_list:
!>[B: $tType,A: $tType] : ( B > ( A > ( list @ A ) > B ) > ( list @ A ) > B ) ).
thf(sy_c_List_Olist_Ohd,type,
hd:
!>[A: $tType] : ( ( list @ A ) > A ) ).
thf(sy_c_List_Olist_Otl,type,
tl:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Olist__ex1,type,
list_ex1:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Olistrelp,type,
listrelp:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( list @ A ) > ( list @ B ) > $o ) ).
thf(sy_c_List_Omap__tailrec,type,
map_tailrec:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( list @ A ) > ( list @ B ) ) ).
thf(sy_c_List_Omap__tailrec__rev,type,
map_tailrec_rev:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( list @ A ) > ( list @ B ) > ( list @ B ) ) ).
thf(sy_c_List_Omaps,type,
maps:
!>[A: $tType,B: $tType] : ( ( A > ( list @ B ) ) > ( list @ A ) > ( list @ B ) ) ).
thf(sy_c_List_Omember,type,
member:
!>[A: $tType] : ( ( list @ A ) > A > $o ) ).
thf(sy_c_List_Oord_Olexordp__eq,type,
lexordp_eq:
!>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Oord__class_Olexordp,type,
ord_lexordp:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Oord__class_Olexordp__eq,type,
ord_lexordp_eq:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Oproduct__lists,type,
product_lists:
!>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ ( list @ A ) ) ) ).
thf(sy_c_List_Orev,type,
rev:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Orotate1,type,
rotate1:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Oshuffles,type,
shuffles:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( set @ ( list @ A ) ) ) ).
thf(sy_c_List_Osplice,type,
splice:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Osubseqs,type,
subseqs:
!>[A: $tType] : ( ( list @ A ) > ( list @ ( list @ A ) ) ) ).
thf(sy_c_List_Osuccessively,type,
successively:
!>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > $o ) ).
thf(sy_c_Orderings_Oord__class_Oless,type,
ord_less:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Secrecy_ODecr,type,
decr: secrecy_Keys > ( list @ secrecy_Expression ) > ( list @ secrecy_Expression ) ).
thf(sy_c_Secrecy_OEnc,type,
enc: secrecy_Keys > ( list @ secrecy_Expression ) > ( list @ secrecy_Expression ) ).
thf(sy_c_Secrecy_OEncrDecrKeys,type,
encrDecrKeys: secrecy_Keys > secrecy_Keys > $o ).
thf(sy_c_Secrecy_OExt,type,
ext: secrecy_Keys > ( list @ secrecy_Expression ) > ( list @ secrecy_Expression ) ).
thf(sy_c_Secrecy_OSign,type,
sign: secrecy_Keys > ( list @ secrecy_Expression ) > ( list @ secrecy_Expression ) ).
thf(sy_c_Secrecy_Oeout,type,
eout: secrecy_specID > secrecy_Expression > $o ).
thf(sy_c_Secrecy_Oine,type,
ine: secrecy_specID > secrecy_Expression > $o ).
thf(sy_c_Secrecy_OnotSpecKeysSecretsExpr,type,
notSpe1586871396tsExpr: secrecy_specID > ( list @ secrecy_Expression ) > $o ).
thf(sy_c_Secrecy_OspecKeys,type,
specKeys: secrecy_specID > ( set @ secrecy_Keys ) ).
thf(sy_c_Secrecy_OspecKeysSecrets,type,
specKeysSecrets: secrecy_specID > ( set @ secrecy_KS ) ).
thf(sy_c_Secrecy_OspecSecrets,type,
specSecrets: secrecy_specID > ( set @ secrecy_Secrets ) ).
thf(sy_c_Secrecy__types_OExpression2KSL,type,
secrec1371625429on2KSL: ( list @ secrecy_Expression ) > ( list @ secrecy_KS ) ).
thf(sy_c_Secrecy__types_OExpression_OkE,type,
secrecy_kE: secrecy_Keys > secrecy_Expression ).
thf(sy_c_Secrecy__types_OExpression_OsE,type,
secrecy_sE: secrecy_Secrets > secrecy_Expression ).
thf(sy_c_Secrecy__types_OKS2Expression,type,
secrec1016742679ession: secrecy_KS > secrecy_Expression ).
thf(sy_c_Secrecy__types_OKS_Ocase__KS,type,
secrecy_case_KS:
!>[A: $tType] : ( ( secrecy_Keys > A ) > ( secrecy_Secrets > A ) > secrecy_KS > A ) ).
thf(sy_c_Secrecy__types_OKS_OkKS,type,
secrecy_kKS: secrecy_Keys > secrecy_KS ).
thf(sy_c_Secrecy__types_OKS_Orec__KS,type,
secrecy_rec_KS:
!>[A: $tType] : ( ( secrecy_Keys > A ) > ( secrecy_Secrets > A ) > secrecy_KS > A ) ).
thf(sy_c_Secrecy__types_OKS_OsKS,type,
secrecy_sKS: secrecy_Secrets > secrecy_KS ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_member,type,
member2:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_A,type,
a: secrecy_specID ).
thf(sy_v_m,type,
m: secrecy_Keys ).
% Relevant facts (255)
thf(fact_0_assms,axiom,
knowle2055185861_knows @ a @ ( cons @ secrecy_Expression @ ( secrecy_kE @ m ) @ ( nil @ secrecy_Expression ) ) ).
% assms
thf(fact_1_KS_Oinject_I1_J,axiom,
! [X1: secrecy_Keys,Y1: secrecy_Keys] :
( ( ( secrecy_kKS @ X1 )
= ( secrecy_kKS @ Y1 ) )
= ( X1 = Y1 ) ) ).
% KS.inject(1)
thf(fact_2_knows4,axiom,
! [IncrDecrKeys: secrecy_Keys > secrecy_Keys > $o,K1: secrecy_Keys,K2: secrecy_Keys,A2: secrecy_specID,E: list @ secrecy_Expression] :
( ( ( IncrDecrKeys @ K1 @ K2 )
& ( knowle1437066798e_know @ A2 @ ( secrecy_kKS @ K2 ) )
& ( knowle2055185861_knows @ A2 @ ( enc @ K1 @ E ) ) )
=> ( knowle2055185861_knows @ A2 @ E ) ) ).
% knows4
thf(fact_3_knows5,axiom,
! [IncrDecrKeys: secrecy_Keys > secrecy_Keys > $o,K1: secrecy_Keys,K2: secrecy_Keys,A2: secrecy_specID,E: list @ secrecy_Expression] :
( ( ( IncrDecrKeys @ K1 @ K2 )
& ( knowle1437066798e_know @ A2 @ ( secrecy_kKS @ K1 ) )
& ( knowle2055185861_knows @ A2 @ ( sign @ K2 @ E ) ) )
=> ( knowle2055185861_knows @ A2 @ E ) ) ).
% knows5
thf(fact_4_knows6,axiom,
! [A2: secrecy_specID,K: secrecy_Keys,E1: list @ secrecy_Expression] :
( ( ( knowle1437066798e_know @ A2 @ ( secrecy_kKS @ K ) )
& ( knowle2055185861_knows @ A2 @ E1 ) )
=> ( knowle2055185861_knows @ A2 @ ( enc @ K @ E1 ) ) ) ).
% knows6
thf(fact_5_knows7,axiom,
! [A2: secrecy_specID,K: secrecy_Keys,E1: list @ secrecy_Expression] :
( ( ( knowle1437066798e_know @ A2 @ ( secrecy_kKS @ K ) )
& ( knowle2055185861_knows @ A2 @ E1 ) )
=> ( knowle2055185861_knows @ A2 @ ( sign @ K @ E1 ) ) ) ).
% knows7
thf(fact_6_know2knows__k,axiom,
! [A2: secrecy_specID,M: secrecy_Keys] :
( ( knowle1437066798e_know @ A2 @ ( secrecy_kKS @ M ) )
=> ( knowle2055185861_knows @ A2 @ ( cons @ secrecy_Expression @ ( secrecy_kE @ M ) @ ( nil @ secrecy_Expression ) ) ) ) ).
% know2knows_k
thf(fact_7_know1k,axiom,
! [C: secrecy_specID,M1: secrecy_Keys] :
( ( knowle2055185861_knows @ C @ ( cons @ secrecy_Expression @ ( secrec1016742679ession @ ( secrecy_kKS @ M1 ) ) @ ( nil @ secrecy_Expression ) ) )
= ( knowle1437066798e_know @ C @ ( secrecy_kKS @ M1 ) ) ) ).
% know1k
thf(fact_8_KS_Osimps_I7_J,axiom,
! [A: $tType,F1: secrecy_Keys > A,F2: secrecy_Secrets > A,X1: secrecy_Keys] :
( ( secrecy_rec_KS @ A @ F1 @ F2 @ ( secrecy_kKS @ X1 ) )
= ( F1 @ X1 ) ) ).
% KS.simps(7)
thf(fact_9_KS_Osimps_I5_J,axiom,
! [A: $tType,F1: secrecy_Keys > A,F2: secrecy_Secrets > A,X1: secrecy_Keys] :
( ( secrecy_case_KS @ A @ F1 @ F2 @ ( secrecy_kKS @ X1 ) )
= ( F1 @ X1 ) ) ).
% KS.simps(5)
thf(fact_10_eoutKnowCorrect__L1k,axiom,
! [C: secrecy_specID,M: secrecy_Keys] :
( ( knowle1667810483orrect @ C @ ( secrecy_kKS @ M ) )
=> ( ( eout @ C @ ( secrecy_kE @ M ) )
=> ( ( member2 @ secrecy_Keys @ M @ ( specKeys @ C ) )
| ( knowle1437066798e_know @ C @ ( secrecy_kKS @ M ) ) ) ) ) ).
% eoutKnowCorrect_L1k
thf(fact_11_eout__know__k,axiom,
! [C: secrecy_specID,M: secrecy_Keys] :
( ( knowle1667810483orrect @ C @ ( secrecy_kKS @ M ) )
= ( ( eout @ C @ ( secrecy_kE @ M ) )
= ( ( member2 @ secrecy_Keys @ M @ ( specKeys @ C ) )
| ( knowle1437066798e_know @ C @ ( secrecy_kKS @ M ) ) ) ) ) ).
% eout_know_k
thf(fact_12_know_Osimps_I1_J,axiom,
! [A2: secrecy_specID,M: secrecy_Keys] :
( ( knowle1437066798e_know @ A2 @ ( secrecy_kKS @ M ) )
= ( ( ine @ A2 @ ( secrecy_kE @ M ) )
| ( member2 @ secrecy_KS @ ( secrecy_kKS @ M ) @ ( localSecrets @ A2 ) ) ) ) ).
% know.simps(1)
thf(fact_13_Expression_Oinject_I1_J,axiom,
! [X1: secrecy_Keys,Y1: secrecy_Keys] :
( ( ( secrecy_kE @ X1 )
= ( secrecy_kE @ Y1 ) )
= ( X1 = Y1 ) ) ).
% Expression.inject(1)
thf(fact_14_knows__emptyexpression,axiom,
! [C: secrecy_specID] : ( knowle2055185861_knows @ C @ ( nil @ secrecy_Expression ) ) ).
% knows_emptyexpression
thf(fact_15_KS2Expression_Osimps_I1_J,axiom,
! [M: secrecy_Keys] :
( ( secrec1016742679ession @ ( secrecy_kKS @ M ) )
= ( secrecy_kE @ M ) ) ).
% KS2Expression.simps(1)
thf(fact_16_list_Oinject,axiom,
! [A: $tType,X21: A,X22: list @ A,Y21: A,Y22: list @ A] :
( ( ( cons @ A @ X21 @ X22 )
= ( cons @ A @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_17_know1s,axiom,
! [C: secrecy_specID,M2: secrecy_Secrets] :
( ( knowle2055185861_knows @ C @ ( cons @ secrecy_Expression @ ( secrec1016742679ession @ ( secrecy_sKS @ M2 ) ) @ ( nil @ secrecy_Expression ) ) )
= ( knowle1437066798e_know @ C @ ( secrecy_sKS @ M2 ) ) ) ).
% know1s
thf(fact_18_list_Odistinct_I1_J,axiom,
! [A: $tType,X21: A,X22: list @ A] :
( ( nil @ A )
!= ( cons @ A @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_19_list_OdiscI,axiom,
! [A: $tType,List: list @ A,X21: A,X22: list @ A] :
( ( List
= ( cons @ A @ X21 @ X22 ) )
=> ( List
!= ( nil @ A ) ) ) ).
% list.discI
thf(fact_20_list_Oexhaust,axiom,
! [A: $tType,Y: list @ A] :
( ( Y
!= ( nil @ A ) )
=> ~ ! [X212: A,X222: list @ A] :
( Y
!= ( cons @ A @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_21_list_Oinducts,axiom,
! [A: $tType,P: ( list @ A ) > $o,List: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [X12: A,X2: list @ A] :
( ( P @ X2 )
=> ( P @ ( cons @ A @ X12 @ X2 ) ) )
=> ( P @ List ) ) ) ).
% list.inducts
thf(fact_22_neq__Nil__conv,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
= ( ? [Y2: A,Ys: list @ A] :
( Xs
= ( cons @ A @ Y2 @ Ys ) ) ) ) ).
% neq_Nil_conv
thf(fact_23_list__induct2_H,axiom,
! [A: $tType,B: $tType,P: ( list @ A ) > ( list @ B ) > $o,Xs: list @ A,Ys2: list @ B] :
( ( P @ ( nil @ A ) @ ( nil @ B ) )
=> ( ! [X: A,Xs2: list @ A] : ( P @ ( cons @ A @ X @ Xs2 ) @ ( nil @ B ) )
=> ( ! [Y3: B,Ys3: list @ B] : ( P @ ( nil @ A ) @ ( cons @ B @ Y3 @ Ys3 ) )
=> ( ! [X: A,Xs2: list @ A,Y3: B,Ys3: list @ B] :
( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons @ A @ X @ Xs2 ) @ ( cons @ B @ Y3 @ Ys3 ) ) )
=> ( P @ Xs @ Ys2 ) ) ) ) ) ).
% list_induct2'
thf(fact_24_splice_Oinduct,axiom,
! [A: $tType,P: ( list @ A ) > ( list @ A ) > $o,A0: list @ A,A1: list @ A] :
( ! [X_1: list @ A] : ( P @ ( nil @ A ) @ X_1 )
=> ( ! [X: A,Xs2: list @ A,Ys3: list @ A] :
( ( P @ Ys3 @ Xs2 )
=> ( P @ ( cons @ A @ X @ Xs2 ) @ Ys3 ) )
=> ( P @ A0 @ A1 ) ) ) ).
% splice.induct
thf(fact_25_induct__list012,axiom,
! [A: $tType,P: ( list @ A ) > $o,Xs: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [X: A] : ( P @ ( cons @ A @ X @ ( nil @ A ) ) )
=> ( ! [X: A,Y3: A,Zs: list @ A] :
( ( P @ Zs )
=> ( ( P @ ( cons @ A @ Y3 @ Zs ) )
=> ( P @ ( cons @ A @ X @ ( cons @ A @ Y3 @ Zs ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% induct_list012
thf(fact_26_min__list_Ocases,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X3: list @ A] :
( ! [X: A,Xs2: list @ A] :
( X3
!= ( cons @ A @ X @ Xs2 ) )
=> ( X3
= ( nil @ A ) ) ) ) ).
% min_list.cases
thf(fact_27_KS_Oinject_I2_J,axiom,
! [X23: secrecy_Secrets,Y23: secrecy_Secrets] :
( ( ( secrecy_sKS @ X23 )
= ( secrecy_sKS @ Y23 ) )
= ( X23 = Y23 ) ) ).
% KS.inject(2)
thf(fact_28_KS_Odistinct_I1_J,axiom,
! [X1: secrecy_Keys,X23: secrecy_Secrets] :
( ( secrecy_kKS @ X1 )
!= ( secrecy_sKS @ X23 ) ) ).
% KS.distinct(1)
thf(fact_29_KS_Oinduct,axiom,
! [P: secrecy_KS > $o,KS: secrecy_KS] :
( ! [X: secrecy_Keys] : ( P @ ( secrecy_kKS @ X ) )
=> ( ! [X: secrecy_Secrets] : ( P @ ( secrecy_sKS @ X ) )
=> ( P @ KS ) ) ) ).
% KS.induct
thf(fact_30_KS_Oexhaust,axiom,
! [Y: secrecy_KS] :
( ! [X12: secrecy_Keys] :
( Y
!= ( secrecy_kKS @ X12 ) )
=> ~ ! [X2: secrecy_Secrets] :
( Y
!= ( secrecy_sKS @ X2 ) ) ) ).
% KS.exhaust
thf(fact_31_KS_Osimps_I6_J,axiom,
! [A: $tType,F1: secrecy_Keys > A,F2: secrecy_Secrets > A,X23: secrecy_Secrets] :
( ( secrecy_case_KS @ A @ F1 @ F2 @ ( secrecy_sKS @ X23 ) )
= ( F2 @ X23 ) ) ).
% KS.simps(6)
thf(fact_32_KS_Osimps_I8_J,axiom,
! [A: $tType,F1: secrecy_Keys > A,F2: secrecy_Secrets > A,X23: secrecy_Secrets] :
( ( secrecy_rec_KS @ A @ F1 @ F2 @ ( secrecy_sKS @ X23 ) )
= ( F2 @ X23 ) ) ).
% KS.simps(8)
thf(fact_33_not__Cons__self2,axiom,
! [A: $tType,X3: A,Xs: list @ A] :
( ( cons @ A @ X3 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_34_strict__sorted_Oinduct,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: ( list @ A ) > $o,A0: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [X: A,Ys3: list @ A] :
( ( P @ Ys3 )
=> ( P @ ( cons @ A @ X @ Ys3 ) ) )
=> ( P @ A0 ) ) ) ) ).
% strict_sorted.induct
thf(fact_35_strict__sorted_Ocases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: list @ A] :
( ( X3
!= ( nil @ A ) )
=> ~ ! [X: A,Ys3: list @ A] :
( X3
!= ( cons @ A @ X @ Ys3 ) ) ) ) ).
% strict_sorted.cases
thf(fact_36_map__tailrec__rev_Oinduct,axiom,
! [A: $tType,B: $tType,P: ( A > B ) > ( list @ A ) > ( list @ B ) > $o,A0: A > B,A1: list @ A,A22: list @ B] :
( ! [F: A > B,X_1: list @ B] : ( P @ F @ ( nil @ A ) @ X_1 )
=> ( ! [F: A > B,A3: A,As: list @ A,Bs: list @ B] :
( ( P @ F @ As @ ( cons @ B @ ( F @ A3 ) @ Bs ) )
=> ( P @ F @ ( cons @ A @ A3 @ As ) @ Bs ) )
=> ( P @ A0 @ A1 @ A22 ) ) ) ).
% map_tailrec_rev.induct
thf(fact_37_list__nonempty__induct,axiom,
! [A: $tType,Xs: list @ A,P: ( list @ A ) > $o] :
( ( Xs
!= ( nil @ A ) )
=> ( ! [X: A] : ( P @ ( cons @ A @ X @ ( nil @ A ) ) )
=> ( ! [X: A,Xs2: list @ A] :
( ( Xs2
!= ( nil @ A ) )
=> ( ( P @ Xs2 )
=> ( P @ ( cons @ A @ X @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_38_successively_Oinduct,axiom,
! [A: $tType,P: ( A > A > $o ) > ( list @ A ) > $o,A0: A > A > $o,A1: list @ A] :
( ! [P2: A > A > $o] : ( P @ P2 @ ( nil @ A ) )
=> ( ! [P2: A > A > $o,X: A] : ( P @ P2 @ ( cons @ A @ X @ ( nil @ A ) ) )
=> ( ! [P2: A > A > $o,X: A,Y3: A,Xs2: list @ A] :
( ( P @ P2 @ ( cons @ A @ Y3 @ Xs2 ) )
=> ( P @ P2 @ ( cons @ A @ X @ ( cons @ A @ Y3 @ Xs2 ) ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% successively.induct
thf(fact_39_arg__min__list_Oinduct,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [P: ( A > B ) > ( list @ A ) > $o,A0: A > B,A1: list @ A] :
( ! [F: A > B,X: A] : ( P @ F @ ( cons @ A @ X @ ( nil @ A ) ) )
=> ( ! [F: A > B,X: A,Y3: A,Zs: list @ A] :
( ( P @ F @ ( cons @ A @ Y3 @ Zs ) )
=> ( P @ F @ ( cons @ A @ X @ ( cons @ A @ Y3 @ Zs ) ) ) )
=> ( ! [A3: A > B] : ( P @ A3 @ ( nil @ A ) )
=> ( P @ A0 @ A1 ) ) ) ) ) ).
% arg_min_list.induct
thf(fact_40_remdups__adj_Oinduct,axiom,
! [A: $tType,P: ( list @ A ) > $o,A0: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [X: A] : ( P @ ( cons @ A @ X @ ( nil @ A ) ) )
=> ( ! [X: A,Y3: A,Xs2: list @ A] :
( ( ( X = Y3 )
=> ( P @ ( cons @ A @ X @ Xs2 ) ) )
=> ( ( ( X != Y3 )
=> ( P @ ( cons @ A @ Y3 @ Xs2 ) ) )
=> ( P @ ( cons @ A @ X @ ( cons @ A @ Y3 @ Xs2 ) ) ) ) )
=> ( P @ A0 ) ) ) ) ).
% remdups_adj.induct
thf(fact_41_sorted__wrt_Oinduct,axiom,
! [A: $tType,P: ( A > A > $o ) > ( list @ A ) > $o,A0: A > A > $o,A1: list @ A] :
( ! [P2: A > A > $o] : ( P @ P2 @ ( nil @ A ) )
=> ( ! [P2: A > A > $o,X: A,Ys3: list @ A] :
( ( P @ P2 @ Ys3 )
=> ( P @ P2 @ ( cons @ A @ X @ Ys3 ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% sorted_wrt.induct
thf(fact_42_remdups__adj_Ocases,axiom,
! [A: $tType,X3: list @ A] :
( ( X3
!= ( nil @ A ) )
=> ( ! [X: A] :
( X3
!= ( cons @ A @ X @ ( nil @ A ) ) )
=> ~ ! [X: A,Y3: A,Xs2: list @ A] :
( X3
!= ( cons @ A @ X @ ( cons @ A @ Y3 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_43_transpose_Ocases,axiom,
! [A: $tType,X3: list @ ( list @ A )] :
( ( X3
!= ( nil @ ( list @ A ) ) )
=> ( ! [Xss: list @ ( list @ A )] :
( X3
!= ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss ) )
=> ~ ! [X: A,Xs2: list @ A,Xss: list @ ( list @ A )] :
( X3
!= ( cons @ ( list @ A ) @ ( cons @ A @ X @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_44_mem__Collect__eq,axiom,
! [A: $tType,A4: A,P: A > $o] :
( ( member2 @ A @ A4 @ ( collect @ A @ P ) )
= ( P @ A4 ) ) ).
% mem_Collect_eq
thf(fact_45_Collect__mem__eq,axiom,
! [A: $tType,A2: set @ A] :
( ( collect @ A
@ ^ [X4: A] : ( member2 @ A @ X4 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_46_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X: A] :
( ( P @ X )
= ( Q @ X ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_47_ext,axiom,
! [B: $tType,A: $tType,F3: A > B,G: A > B] :
( ! [X: A] :
( ( F3 @ X )
= ( G @ X ) )
=> ( F3 = G ) ) ).
% ext
thf(fact_48_shuffles_Oinduct,axiom,
! [A: $tType,P: ( list @ A ) > ( list @ A ) > $o,A0: list @ A,A1: list @ A] :
( ! [X_1: list @ A] : ( P @ ( nil @ A ) @ X_1 )
=> ( ! [Xs2: list @ A] : ( P @ Xs2 @ ( nil @ A ) )
=> ( ! [X: A,Xs2: list @ A,Y3: A,Ys3: list @ A] :
( ( P @ Xs2 @ ( cons @ A @ Y3 @ Ys3 ) )
=> ( ( P @ ( cons @ A @ X @ Xs2 ) @ Ys3 )
=> ( P @ ( cons @ A @ X @ Xs2 ) @ ( cons @ A @ Y3 @ Ys3 ) ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% shuffles.induct
thf(fact_49_min__list_Oinduct,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [P: ( list @ A ) > $o,A0: list @ A] :
( ! [X: A,Xs2: list @ A] :
( ! [X213: A,X223: list @ A] :
( ( Xs2
= ( cons @ A @ X213 @ X223 ) )
=> ( P @ Xs2 ) )
=> ( P @ ( cons @ A @ X @ Xs2 ) ) )
=> ( ( P @ ( nil @ A ) )
=> ( P @ A0 ) ) ) ) ).
% min_list.induct
thf(fact_50_know_Osimps_I2_J,axiom,
! [A2: secrecy_specID,M: secrecy_Secrets] :
( ( knowle1437066798e_know @ A2 @ ( secrecy_sKS @ M ) )
= ( ( ine @ A2 @ ( secrecy_sE @ M ) )
| ( member2 @ secrecy_KS @ ( secrecy_sKS @ M ) @ ( localSecrets @ A2 ) ) ) ) ).
% know.simps(2)
thf(fact_51_Expression2KSL_Osimps_I1_J,axiom,
( ( secrec1371625429on2KSL @ ( nil @ secrecy_Expression ) )
= ( nil @ secrecy_KS ) ) ).
% Expression2KSL.simps(1)
thf(fact_52_insert__Nil,axiom,
! [A: $tType,X3: A] :
( ( insert @ A @ X3 @ ( nil @ A ) )
= ( cons @ A @ X3 @ ( nil @ A ) ) ) ).
% insert_Nil
thf(fact_53_list__ex1__simps_I1_J,axiom,
! [A: $tType,P: A > $o] :
~ ( list_ex1 @ A @ P @ ( nil @ A ) ) ).
% list_ex1_simps(1)
thf(fact_54_eoutKnowsECorrect__def,axiom,
( knowle1893229345orrect
= ( ^ [C2: secrecy_specID,E2: secrecy_Expression] :
( ( eout @ C2 @ E2 )
= ( ? [K3: secrecy_Keys] :
( ( E2
= ( secrecy_kE @ K3 ) )
& ( member2 @ secrecy_Keys @ K3 @ ( specKeys @ C2 ) ) )
| ? [S: secrecy_Secrets] :
( ( E2
= ( secrecy_sE @ S ) )
& ( member2 @ secrecy_Secrets @ S @ ( specSecrets @ C2 ) ) )
| ( knowle2055185861_knows @ C2 @ ( cons @ secrecy_Expression @ E2 @ ( nil @ secrecy_Expression ) ) ) ) ) ) ) ).
% eoutKnowsECorrect_def
thf(fact_55_eoutKnowsECorrect__L1,axiom,
! [C: secrecy_specID,E: secrecy_Expression] :
( ( knowle1893229345orrect @ C @ E )
=> ( ( eout @ C @ E )
=> ( ? [K4: secrecy_Keys] :
( ( E
= ( secrecy_kE @ K4 ) )
& ( member2 @ secrecy_Keys @ K4 @ ( specKeys @ C ) ) )
| ? [S2: secrecy_Secrets] :
( ( E
= ( secrecy_sE @ S2 ) )
& ( member2 @ secrecy_Secrets @ S2 @ ( specSecrets @ C ) ) )
| ( knowle2055185861_knows @ C @ ( cons @ secrecy_Expression @ E @ ( nil @ secrecy_Expression ) ) ) ) ) ) ).
% eoutKnowsECorrect_L1
thf(fact_56_map__tailrec__rev_Oelims,axiom,
! [A: $tType,B: $tType,X3: A > B,Xa: list @ A,Xb: list @ B,Y: list @ B] :
( ( ( map_tailrec_rev @ A @ B @ X3 @ Xa @ Xb )
= Y )
=> ( ( ( Xa
= ( nil @ A ) )
=> ( Y != Xb ) )
=> ~ ! [A3: A,As: list @ A] :
( ( Xa
= ( cons @ A @ A3 @ As ) )
=> ( Y
!= ( map_tailrec_rev @ A @ B @ X3 @ As @ ( cons @ B @ ( X3 @ A3 ) @ Xb ) ) ) ) ) ) ).
% map_tailrec_rev.elims
thf(fact_57_arg__min__list_Osimps_I1_J,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [F3: A > B,X3: A] :
( ( arg_min_list @ A @ B @ F3 @ ( cons @ A @ X3 @ ( nil @ A ) ) )
= X3 ) ) ).
% arg_min_list.simps(1)
thf(fact_58_notSpecKeysSecretsExpr__L1,axiom,
! [P: secrecy_specID,A4: secrecy_Expression,L: list @ secrecy_Expression] :
( ( notSpe1586871396tsExpr @ P @ ( cons @ secrecy_Expression @ A4 @ L ) )
=> ( notSpe1586871396tsExpr @ P @ ( cons @ secrecy_Expression @ A4 @ ( nil @ secrecy_Expression ) ) ) ) ).
% notSpecKeysSecretsExpr_L1
thf(fact_59_Expression_Oinject_I2_J,axiom,
! [X23: secrecy_Secrets,Y23: secrecy_Secrets] :
( ( ( secrecy_sE @ X23 )
= ( secrecy_sE @ Y23 ) )
= ( X23 = Y23 ) ) ).
% Expression.inject(2)
thf(fact_60_Expression_Odistinct_I1_J,axiom,
! [X1: secrecy_Keys,X23: secrecy_Secrets] :
( ( secrecy_kE @ X1 )
!= ( secrecy_sE @ X23 ) ) ).
% Expression.distinct(1)
thf(fact_61_notSpecKeysSecretsExpr__L2,axiom,
! [P: secrecy_specID,A4: secrecy_Expression,L: list @ secrecy_Expression] :
( ( notSpe1586871396tsExpr @ P @ ( cons @ secrecy_Expression @ A4 @ L ) )
=> ( notSpe1586871396tsExpr @ P @ L ) ) ).
% notSpecKeysSecretsExpr_L2
thf(fact_62_map__tailrec__rev_Osimps_I2_J,axiom,
! [A: $tType,B: $tType,F3: A > B,A4: A,As2: list @ A,Bs2: list @ B] :
( ( map_tailrec_rev @ A @ B @ F3 @ ( cons @ A @ A4 @ As2 ) @ Bs2 )
= ( map_tailrec_rev @ A @ B @ F3 @ As2 @ ( cons @ B @ ( F3 @ A4 ) @ Bs2 ) ) ) ).
% map_tailrec_rev.simps(2)
thf(fact_63_map__tailrec__rev_Osimps_I1_J,axiom,
! [A: $tType,B: $tType,F3: A > B,Bs2: list @ B] :
( ( map_tailrec_rev @ A @ B @ F3 @ ( nil @ A ) @ Bs2 )
= Bs2 ) ).
% map_tailrec_rev.simps(1)
thf(fact_64_KS2Expression_Osimps_I2_J,axiom,
! [M: secrecy_Secrets] :
( ( secrec1016742679ession @ ( secrecy_sKS @ M ) )
= ( secrecy_sE @ M ) ) ).
% KS2Expression.simps(2)
thf(fact_65_eoutKnowCorrect__L1s,axiom,
! [C: secrecy_specID,M: secrecy_Secrets] :
( ( knowle1667810483orrect @ C @ ( secrecy_sKS @ M ) )
=> ( ( eout @ C @ ( secrecy_sE @ M ) )
=> ( ( member2 @ secrecy_Secrets @ M @ ( specSecrets @ C ) )
| ( knowle1437066798e_know @ C @ ( secrecy_sKS @ M ) ) ) ) ) ).
% eoutKnowCorrect_L1s
thf(fact_66_eout__know__s,axiom,
! [C: secrecy_specID,M: secrecy_Secrets] :
( ( knowle1667810483orrect @ C @ ( secrecy_sKS @ M ) )
= ( ( eout @ C @ ( secrecy_sE @ M ) )
= ( ( member2 @ secrecy_Secrets @ M @ ( specSecrets @ C ) )
| ( knowle1437066798e_know @ C @ ( secrecy_sKS @ M ) ) ) ) ) ).
% eout_know_s
thf(fact_67_product__lists_Osimps_I1_J,axiom,
! [A: $tType] :
( ( product_lists @ A @ ( nil @ ( list @ A ) ) )
= ( cons @ ( list @ A ) @ ( nil @ A ) @ ( nil @ ( list @ A ) ) ) ) ).
% product_lists.simps(1)
thf(fact_68_subseqs_Osimps_I1_J,axiom,
! [A: $tType] :
( ( subseqs @ A @ ( nil @ A ) )
= ( cons @ ( list @ A ) @ ( nil @ A ) @ ( nil @ ( list @ A ) ) ) ) ).
% subseqs.simps(1)
thf(fact_69_DecrEnc,axiom,
! [K12: secrecy_Keys,K22: secrecy_Keys,E3: list @ secrecy_Expression] :
( ( encrDecrKeys @ K12 @ K22 )
=> ( ( decr @ K22 @ ( enc @ K12 @ E3 ) )
= E3 ) ) ).
% DecrEnc
thf(fact_70_ExtSign,axiom,
! [K12: secrecy_Keys,K22: secrecy_Keys,E3: list @ secrecy_Expression] :
( ( encrDecrKeys @ K12 @ K22 )
=> ( ( ext @ K12 @ ( sign @ K22 @ E3 ) )
= E3 ) ) ).
% ExtSign
thf(fact_71_notSpecKeysSecretsExpr__def,axiom,
( notSpe1586871396tsExpr
= ( ^ [P3: secrecy_specID,E2: list @ secrecy_Expression] :
( ! [X4: secrecy_Keys] :
( ( mem @ secrecy_Expression @ ( secrecy_kE @ X4 ) @ E2 )
=> ~ ( member2 @ secrecy_KS @ ( secrecy_kKS @ X4 ) @ ( specKeysSecrets @ P3 ) ) )
& ! [Y2: secrecy_Secrets] :
( ( mem @ secrecy_Expression @ ( secrecy_sE @ Y2 ) @ E2 )
=> ~ ( member2 @ secrecy_KS @ ( secrecy_sKS @ Y2 ) @ ( specKeysSecrets @ P3 ) ) ) ) ) ) ).
% notSpecKeysSecretsExpr_def
thf(fact_72_ord_Olexordp__eq__simps_I3_J,axiom,
! [A: $tType,Less: A > A > $o,X3: A,Xs: list @ A] :
~ ( lexordp_eq @ A @ Less @ ( cons @ A @ X3 @ Xs ) @ ( nil @ A ) ) ).
% ord.lexordp_eq_simps(3)
thf(fact_73_bind__simps_I1_J,axiom,
! [B: $tType,A: $tType,F3: B > ( list @ A )] :
( ( bind @ B @ A @ ( nil @ B ) @ F3 )
= ( nil @ A ) ) ).
% bind_simps(1)
thf(fact_74_ord_Olexordp__eq__simps_I4_J,axiom,
! [A: $tType,Less: A > A > $o,X3: A,Xs: list @ A,Y: A,Ys2: list @ A] :
( ( lexordp_eq @ A @ Less @ ( cons @ A @ X3 @ Xs ) @ ( cons @ A @ Y @ Ys2 ) )
= ( ( Less @ X3 @ Y )
| ( ~ ( Less @ Y @ X3 )
& ( lexordp_eq @ A @ Less @ Xs @ Ys2 ) ) ) ) ).
% ord.lexordp_eq_simps(4)
thf(fact_75_ord_Olexordp__eq__simps_I1_J,axiom,
! [A: $tType,Less: A > A > $o,Ys2: list @ A] : ( lexordp_eq @ A @ Less @ ( nil @ A ) @ Ys2 ) ).
% ord.lexordp_eq_simps(1)
thf(fact_76_ord_Olexordp__eq__simps_I2_J,axiom,
! [A: $tType,Less: A > A > $o,Xs: list @ A] :
( ( lexordp_eq @ A @ Less @ Xs @ ( nil @ A ) )
= ( Xs
= ( nil @ A ) ) ) ).
% ord.lexordp_eq_simps(2)
thf(fact_77_ord_Olexordp__eq_Ocong,axiom,
! [A: $tType] :
( ( lexordp_eq @ A )
= ( lexordp_eq @ A ) ) ).
% ord.lexordp_eq.cong
thf(fact_78_ord_Olexordp__eq__refl,axiom,
! [A: $tType,Less: A > A > $o,Xs: list @ A] : ( lexordp_eq @ A @ Less @ Xs @ Xs ) ).
% ord.lexordp_eq_refl
thf(fact_79_ord_Olexordp__eq_OCons__eq,axiom,
! [A: $tType,Less: A > A > $o,X3: A,Y: A,Xs: list @ A,Ys2: list @ A] :
( ~ ( Less @ X3 @ Y )
=> ( ~ ( Less @ Y @ X3 )
=> ( ( lexordp_eq @ A @ Less @ Xs @ Ys2 )
=> ( lexordp_eq @ A @ Less @ ( cons @ A @ X3 @ Xs ) @ ( cons @ A @ Y @ Ys2 ) ) ) ) ) ).
% ord.lexordp_eq.Cons_eq
thf(fact_80_ord_Olexordp__eq_OCons,axiom,
! [A: $tType,Less: A > A > $o,X3: A,Y: A,Xs: list @ A,Ys2: list @ A] :
( ( Less @ X3 @ Y )
=> ( lexordp_eq @ A @ Less @ ( cons @ A @ X3 @ Xs ) @ ( cons @ A @ Y @ Ys2 ) ) ) ).
% ord.lexordp_eq.Cons
thf(fact_81_ord_Olexordp__eq_ONil,axiom,
! [A: $tType,Less: A > A > $o,Ys2: list @ A] : ( lexordp_eq @ A @ Less @ ( nil @ A ) @ Ys2 ) ).
% ord.lexordp_eq.Nil
thf(fact_82_ord_Olexordp__eq_Oinducts,axiom,
! [A: $tType,Less: A > A > $o,X1: list @ A,X23: list @ A,P: ( list @ A ) > ( list @ A ) > $o] :
( ( lexordp_eq @ A @ Less @ X1 @ X23 )
=> ( ! [X_1: list @ A] : ( P @ ( nil @ A ) @ X_1 )
=> ( ! [X: A,Y3: A,Xs2: list @ A,Ys3: list @ A] :
( ( Less @ X @ Y3 )
=> ( P @ ( cons @ A @ X @ Xs2 ) @ ( cons @ A @ Y3 @ Ys3 ) ) )
=> ( ! [X: A,Y3: A,Xs2: list @ A,Ys3: list @ A] :
( ~ ( Less @ X @ Y3 )
=> ( ~ ( Less @ Y3 @ X )
=> ( ( lexordp_eq @ A @ Less @ Xs2 @ Ys3 )
=> ( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons @ A @ X @ Xs2 ) @ ( cons @ A @ Y3 @ Ys3 ) ) ) ) ) )
=> ( P @ X1 @ X23 ) ) ) ) ) ).
% ord.lexordp_eq.inducts
thf(fact_83_ord_Olexordp__eq_Osimps,axiom,
! [A: $tType] :
( ( lexordp_eq @ A )
= ( ^ [Less2: A > A > $o,A12: list @ A,A23: list @ A] :
( ? [Ys: list @ A] :
( ( A12
= ( nil @ A ) )
& ( A23 = Ys ) )
| ? [X4: A,Y2: A,Xs3: list @ A,Ys: list @ A] :
( ( A12
= ( cons @ A @ X4 @ Xs3 ) )
& ( A23
= ( cons @ A @ Y2 @ Ys ) )
& ( Less2 @ X4 @ Y2 ) )
| ? [X4: A,Y2: A,Xs3: list @ A,Ys: list @ A] :
( ( A12
= ( cons @ A @ X4 @ Xs3 ) )
& ( A23
= ( cons @ A @ Y2 @ Ys ) )
& ~ ( Less2 @ X4 @ Y2 )
& ~ ( Less2 @ Y2 @ X4 )
& ( lexordp_eq @ A @ Less2 @ Xs3 @ Ys ) ) ) ) ) ).
% ord.lexordp_eq.simps
thf(fact_84_ord_Olexordp__eq_Ocases,axiom,
! [A: $tType,Less: A > A > $o,A1: list @ A,A22: list @ A] :
( ( lexordp_eq @ A @ Less @ A1 @ A22 )
=> ( ( A1
!= ( nil @ A ) )
=> ( ! [X: A] :
( ? [Xs2: list @ A] :
( A1
= ( cons @ A @ X @ Xs2 ) )
=> ! [Y3: A] :
( ? [Ys3: list @ A] :
( A22
= ( cons @ A @ Y3 @ Ys3 ) )
=> ~ ( Less @ X @ Y3 ) ) )
=> ~ ! [X: A,Y3: A,Xs2: list @ A] :
( ( A1
= ( cons @ A @ X @ Xs2 ) )
=> ! [Ys3: list @ A] :
( ( A22
= ( cons @ A @ Y3 @ Ys3 ) )
=> ( ~ ( Less @ X @ Y3 )
=> ( ~ ( Less @ Y3 @ X )
=> ~ ( lexordp_eq @ A @ Less @ Xs2 @ Ys3 ) ) ) ) ) ) ) ) ).
% ord.lexordp_eq.cases
thf(fact_85_mem_Osimps_I1_J,axiom,
! [A: $tType,X3: A] :
~ ( mem @ A @ X3 @ ( nil @ A ) ) ).
% mem.simps(1)
thf(fact_86_mem_Osimps_I2_J,axiom,
! [A: $tType,X3: A,Y: A,L: list @ A] :
( ( mem @ A @ X3 @ ( cons @ A @ Y @ L ) )
= ( ( X3 = Y )
| ( mem @ A @ X3 @ L ) ) ) ).
% mem.simps(2)
thf(fact_87_listrelp_Oinducts,axiom,
! [A: $tType,B: $tType,R: A > B > $o,X1: list @ A,X23: list @ B,P: ( list @ A ) > ( list @ B ) > $o] :
( ( listrelp @ A @ B @ R @ X1 @ X23 )
=> ( ( P @ ( nil @ A ) @ ( nil @ B ) )
=> ( ! [X: A,Y3: B,Xs2: list @ A,Ys3: list @ B] :
( ( R @ X @ Y3 )
=> ( ( listrelp @ A @ B @ R @ Xs2 @ Ys3 )
=> ( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons @ A @ X @ Xs2 ) @ ( cons @ B @ Y3 @ Ys3 ) ) ) ) )
=> ( P @ X1 @ X23 ) ) ) ) ).
% listrelp.inducts
thf(fact_88_listrelp_Osimps,axiom,
! [B: $tType,A: $tType] :
( ( listrelp @ A @ B )
= ( ^ [R2: A > B > $o,A12: list @ A,A23: list @ B] :
( ( ( A12
= ( nil @ A ) )
& ( A23
= ( nil @ B ) ) )
| ? [X4: A,Y2: B,Xs3: list @ A,Ys: list @ B] :
( ( A12
= ( cons @ A @ X4 @ Xs3 ) )
& ( A23
= ( cons @ B @ Y2 @ Ys ) )
& ( R2 @ X4 @ Y2 )
& ( listrelp @ A @ B @ R2 @ Xs3 @ Ys ) ) ) ) ) ).
% listrelp.simps
thf(fact_89_listrelp_Ocases,axiom,
! [A: $tType,B: $tType,R: A > B > $o,A1: list @ A,A22: list @ B] :
( ( listrelp @ A @ B @ R @ A1 @ A22 )
=> ( ( ( A1
= ( nil @ A ) )
=> ( A22
!= ( nil @ B ) ) )
=> ~ ! [X: A,Y3: B,Xs2: list @ A] :
( ( A1
= ( cons @ A @ X @ Xs2 ) )
=> ! [Ys3: list @ B] :
( ( A22
= ( cons @ B @ Y3 @ Ys3 ) )
=> ( ( R @ X @ Y3 )
=> ~ ( listrelp @ A @ B @ R @ Xs2 @ Ys3 ) ) ) ) ) ) ).
% listrelp.cases
thf(fact_90_lexordp__eq__simps_I3_J,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X3: A,Xs: list @ A] :
~ ( ord_lexordp_eq @ A @ ( cons @ A @ X3 @ Xs ) @ ( nil @ A ) ) ) ).
% lexordp_eq_simps(3)
thf(fact_91_lexordp__eq__simps_I2_J,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [Xs: list @ A] :
( ( ord_lexordp_eq @ A @ Xs @ ( nil @ A ) )
= ( Xs
= ( nil @ A ) ) ) ) ).
% lexordp_eq_simps(2)
thf(fact_92_lexordp__eq__simps_I1_J,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [Ys2: list @ A] : ( ord_lexordp_eq @ A @ ( nil @ A ) @ Ys2 ) ) ).
% lexordp_eq_simps(1)
thf(fact_93_lexordp__eq__antisym,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,Ys2: list @ A] :
( ( ord_lexordp_eq @ A @ Xs @ Ys2 )
=> ( ( ord_lexordp_eq @ A @ Ys2 @ Xs )
=> ( Xs = Ys2 ) ) ) ) ).
% lexordp_eq_antisym
thf(fact_94_lexordp__eq__linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,Ys2: list @ A] :
( ( ord_lexordp_eq @ A @ Xs @ Ys2 )
| ( ord_lexordp_eq @ A @ Ys2 @ Xs ) ) ) ).
% lexordp_eq_linear
thf(fact_95_lexordp__eq__trans,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,Ys2: list @ A,Zs2: list @ A] :
( ( ord_lexordp_eq @ A @ Xs @ Ys2 )
=> ( ( ord_lexordp_eq @ A @ Ys2 @ Zs2 )
=> ( ord_lexordp_eq @ A @ Xs @ Zs2 ) ) ) ) ).
% lexordp_eq_trans
thf(fact_96_lexordp__eq__refl,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [Xs: list @ A] : ( ord_lexordp_eq @ A @ Xs @ Xs ) ) ).
% lexordp_eq_refl
thf(fact_97_lexordp__eq_ONil,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [Ys2: list @ A] : ( ord_lexordp_eq @ A @ ( nil @ A ) @ Ys2 ) ) ).
% lexordp_eq.Nil
thf(fact_98_listrelp_OCons,axiom,
! [A: $tType,B: $tType,R: A > B > $o,X3: A,Y: B,Xs: list @ A,Ys2: list @ B] :
( ( R @ X3 @ Y )
=> ( ( listrelp @ A @ B @ R @ Xs @ Ys2 )
=> ( listrelp @ A @ B @ R @ ( cons @ A @ X3 @ Xs ) @ ( cons @ B @ Y @ Ys2 ) ) ) ) ).
% listrelp.Cons
thf(fact_99_listrelp_ONil,axiom,
! [A: $tType,B: $tType,R: A > B > $o] : ( listrelp @ A @ B @ R @ ( nil @ A ) @ ( nil @ B ) ) ).
% listrelp.Nil
thf(fact_100_mem__memS__eq,axiom,
! [A: $tType] :
( ( mem @ A )
= ( memS @ A ) ) ).
% mem_memS_eq
thf(fact_101_mem__notdisjoint,axiom,
! [A: $tType,X3: A,L1: list @ A,L2: list @ A] :
( ( mem @ A @ X3 @ L1 )
=> ( ( mem @ A @ X3 @ L2 )
=> ~ ( disjoint @ A @ L1 @ L2 ) ) ) ).
% mem_notdisjoint
thf(fact_102_mem__notdisjoint2,axiom,
! [A: $tType,B: $tType,Schedule: B > ( list @ A ),A2: B,B2: B,X3: A] :
( ( disjoint @ A @ ( Schedule @ A2 ) @ ( Schedule @ B2 ) )
=> ( ( mem @ A @ X3 @ ( Schedule @ A2 ) )
=> ~ ( mem @ A @ X3 @ ( Schedule @ B2 ) ) ) ) ).
% mem_notdisjoint2
thf(fact_103_lexordp__eq_Ocases,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A1: list @ A,A22: list @ A] :
( ( ord_lexordp_eq @ A @ A1 @ A22 )
=> ( ( A1
!= ( nil @ A ) )
=> ( ! [X: A] :
( ? [Xs2: list @ A] :
( A1
= ( cons @ A @ X @ Xs2 ) )
=> ! [Y3: A] :
( ? [Ys3: list @ A] :
( A22
= ( cons @ A @ Y3 @ Ys3 ) )
=> ~ ( ord_less @ A @ X @ Y3 ) ) )
=> ~ ! [X: A,Y3: A,Xs2: list @ A] :
( ( A1
= ( cons @ A @ X @ Xs2 ) )
=> ! [Ys3: list @ A] :
( ( A22
= ( cons @ A @ Y3 @ Ys3 ) )
=> ( ~ ( ord_less @ A @ X @ Y3 )
=> ( ~ ( ord_less @ A @ Y3 @ X )
=> ~ ( ord_lexordp_eq @ A @ Xs2 @ Ys3 ) ) ) ) ) ) ) ) ) ).
% lexordp_eq.cases
thf(fact_104_lexordp__eq_Osimps,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_lexordp_eq @ A )
= ( ^ [A12: list @ A,A23: list @ A] :
( ? [Ys: list @ A] :
( ( A12
= ( nil @ A ) )
& ( A23 = Ys ) )
| ? [X4: A,Y2: A,Xs3: list @ A,Ys: list @ A] :
( ( A12
= ( cons @ A @ X4 @ Xs3 ) )
& ( A23
= ( cons @ A @ Y2 @ Ys ) )
& ( ord_less @ A @ X4 @ Y2 ) )
| ? [X4: A,Y2: A,Xs3: list @ A,Ys: list @ A] :
( ( A12
= ( cons @ A @ X4 @ Xs3 ) )
& ( A23
= ( cons @ A @ Y2 @ Ys ) )
& ~ ( ord_less @ A @ X4 @ Y2 )
& ~ ( ord_less @ A @ Y2 @ X4 )
& ( ord_lexordp_eq @ A @ Xs3 @ Ys ) ) ) ) ) ) ).
% lexordp_eq.simps
thf(fact_105_lexordp__eq__simps_I4_J,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X3: A,Xs: list @ A,Y: A,Ys2: list @ A] :
( ( ord_lexordp_eq @ A @ ( cons @ A @ X3 @ Xs ) @ ( cons @ A @ Y @ Ys2 ) )
= ( ( ord_less @ A @ X3 @ Y )
| ( ~ ( ord_less @ A @ Y @ X3 )
& ( ord_lexordp_eq @ A @ Xs @ Ys2 ) ) ) ) ) ).
% lexordp_eq_simps(4)
thf(fact_106_lexordp__eq_OCons__eq,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X3: A,Y: A,Xs: list @ A,Ys2: list @ A] :
( ~ ( ord_less @ A @ X3 @ Y )
=> ( ~ ( ord_less @ A @ Y @ X3 )
=> ( ( ord_lexordp_eq @ A @ Xs @ Ys2 )
=> ( ord_lexordp_eq @ A @ ( cons @ A @ X3 @ Xs ) @ ( cons @ A @ Y @ Ys2 ) ) ) ) ) ) ).
% lexordp_eq.Cons_eq
thf(fact_107_lexordp__eq_OCons,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X3: A,Y: A,Xs: list @ A,Ys2: list @ A] :
( ( ord_less @ A @ X3 @ Y )
=> ( ord_lexordp_eq @ A @ ( cons @ A @ X3 @ Xs ) @ ( cons @ A @ Y @ Ys2 ) ) ) ) ).
% lexordp_eq.Cons
thf(fact_108_lexordp__eq_Oinducts,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X1: list @ A,X23: list @ A,P: ( list @ A ) > ( list @ A ) > $o] :
( ( ord_lexordp_eq @ A @ X1 @ X23 )
=> ( ! [X_1: list @ A] : ( P @ ( nil @ A ) @ X_1 )
=> ( ! [X: A,Y3: A,Xs2: list @ A,Ys3: list @ A] :
( ( ord_less @ A @ X @ Y3 )
=> ( P @ ( cons @ A @ X @ Xs2 ) @ ( cons @ A @ Y3 @ Ys3 ) ) )
=> ( ! [X: A,Y3: A,Xs2: list @ A,Ys3: list @ A] :
( ~ ( ord_less @ A @ X @ Y3 )
=> ( ~ ( ord_less @ A @ Y3 @ X )
=> ( ( ord_lexordp_eq @ A @ Xs2 @ Ys3 )
=> ( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons @ A @ X @ Xs2 ) @ ( cons @ A @ Y3 @ Ys3 ) ) ) ) ) )
=> ( P @ X1 @ X23 ) ) ) ) ) ) ).
% lexordp_eq.inducts
thf(fact_109_lexordp_Ocases,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A1: list @ A,A22: list @ A] :
( ( ord_lexordp @ A @ A1 @ A22 )
=> ( ( ( A1
= ( nil @ A ) )
=> ! [Y3: A,Ys3: list @ A] :
( A22
!= ( cons @ A @ Y3 @ Ys3 ) ) )
=> ( ! [X: A] :
( ? [Xs2: list @ A] :
( A1
= ( cons @ A @ X @ Xs2 ) )
=> ! [Y3: A] :
( ? [Ys3: list @ A] :
( A22
= ( cons @ A @ Y3 @ Ys3 ) )
=> ~ ( ord_less @ A @ X @ Y3 ) ) )
=> ~ ! [X: A,Y3: A,Xs2: list @ A] :
( ( A1
= ( cons @ A @ X @ Xs2 ) )
=> ! [Ys3: list @ A] :
( ( A22
= ( cons @ A @ Y3 @ Ys3 ) )
=> ( ~ ( ord_less @ A @ X @ Y3 )
=> ( ~ ( ord_less @ A @ Y3 @ X )
=> ~ ( ord_lexordp @ A @ Xs2 @ Ys3 ) ) ) ) ) ) ) ) ) ).
% lexordp.cases
thf(fact_110_lexordp_Osimps,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_lexordp @ A )
= ( ^ [A12: list @ A,A23: list @ A] :
( ? [Y2: A,Ys: list @ A] :
( ( A12
= ( nil @ A ) )
& ( A23
= ( cons @ A @ Y2 @ Ys ) ) )
| ? [X4: A,Y2: A,Xs3: list @ A,Ys: list @ A] :
( ( A12
= ( cons @ A @ X4 @ Xs3 ) )
& ( A23
= ( cons @ A @ Y2 @ Ys ) )
& ( ord_less @ A @ X4 @ Y2 ) )
| ? [X4: A,Y2: A,Xs3: list @ A,Ys: list @ A] :
( ( A12
= ( cons @ A @ X4 @ Xs3 ) )
& ( A23
= ( cons @ A @ Y2 @ Ys ) )
& ~ ( ord_less @ A @ X4 @ Y2 )
& ~ ( ord_less @ A @ Y2 @ X4 )
& ( ord_lexordp @ A @ Xs3 @ Ys ) ) ) ) ) ) ).
% lexordp.simps
thf(fact_111_lexordp_Oinducts,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X1: list @ A,X23: list @ A,P: ( list @ A ) > ( list @ A ) > $o] :
( ( ord_lexordp @ A @ X1 @ X23 )
=> ( ! [Y3: A,Ys3: list @ A] : ( P @ ( nil @ A ) @ ( cons @ A @ Y3 @ Ys3 ) )
=> ( ! [X: A,Y3: A,Xs2: list @ A,Ys3: list @ A] :
( ( ord_less @ A @ X @ Y3 )
=> ( P @ ( cons @ A @ X @ Xs2 ) @ ( cons @ A @ Y3 @ Ys3 ) ) )
=> ( ! [X: A,Y3: A,Xs2: list @ A,Ys3: list @ A] :
( ~ ( ord_less @ A @ X @ Y3 )
=> ( ~ ( ord_less @ A @ Y3 @ X )
=> ( ( ord_lexordp @ A @ Xs2 @ Ys3 )
=> ( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons @ A @ X @ Xs2 ) @ ( cons @ A @ Y3 @ Ys3 ) ) ) ) ) )
=> ( P @ X1 @ X23 ) ) ) ) ) ) ).
% lexordp.inducts
thf(fact_112_lexordp__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,Ys2: list @ A] :
( ( ord_lexordp @ A @ Xs @ Ys2 )
=> ( ( ( Xs
= ( nil @ A ) )
=> ! [Y3: A,Ys4: list @ A] :
( Ys2
!= ( cons @ A @ Y3 @ Ys4 ) ) )
=> ( ! [X: A] :
( ? [Xs4: list @ A] :
( Xs
= ( cons @ A @ X @ Xs4 ) )
=> ! [Y3: A] :
( ? [Ys4: list @ A] :
( Ys2
= ( cons @ A @ Y3 @ Ys4 ) )
=> ~ ( ord_less @ A @ X @ Y3 ) ) )
=> ~ ! [X: A,Xs4: list @ A] :
( ( Xs
= ( cons @ A @ X @ Xs4 ) )
=> ! [Ys4: list @ A] :
( ( Ys2
= ( cons @ A @ X @ Ys4 ) )
=> ~ ( ord_lexordp @ A @ Xs4 @ Ys4 ) ) ) ) ) ) ) ).
% lexordp_cases
thf(fact_113_lexordp__induct,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,Ys2: list @ A,P: ( list @ A ) > ( list @ A ) > $o] :
( ( ord_lexordp @ A @ Xs @ Ys2 )
=> ( ! [Y3: A,Ys3: list @ A] : ( P @ ( nil @ A ) @ ( cons @ A @ Y3 @ Ys3 ) )
=> ( ! [X: A,Xs2: list @ A,Y3: A,Ys3: list @ A] :
( ( ord_less @ A @ X @ Y3 )
=> ( P @ ( cons @ A @ X @ Xs2 ) @ ( cons @ A @ Y3 @ Ys3 ) ) )
=> ( ! [X: A,Xs2: list @ A,Ys3: list @ A] :
( ( ord_lexordp @ A @ Xs2 @ Ys3 )
=> ( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons @ A @ X @ Xs2 ) @ ( cons @ A @ X @ Ys3 ) ) ) )
=> ( P @ Xs @ Ys2 ) ) ) ) ) ) ).
% lexordp_induct
thf(fact_114_lexordp__simps_I2_J,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [Xs: list @ A] :
~ ( ord_lexordp @ A @ Xs @ ( nil @ A ) ) ) ).
% lexordp_simps(2)
thf(fact_115_lexordp__simps_I1_J,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [Ys2: list @ A] :
( ( ord_lexordp @ A @ ( nil @ A ) @ Ys2 )
= ( Ys2
!= ( nil @ A ) ) ) ) ).
% lexordp_simps(1)
thf(fact_116_lexordp__simps_I3_J,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X3: A,Xs: list @ A,Y: A,Ys2: list @ A] :
( ( ord_lexordp @ A @ ( cons @ A @ X3 @ Xs ) @ ( cons @ A @ Y @ Ys2 ) )
= ( ( ord_less @ A @ X3 @ Y )
| ( ~ ( ord_less @ A @ Y @ X3 )
& ( ord_lexordp @ A @ Xs @ Ys2 ) ) ) ) ) ).
% lexordp_simps(3)
thf(fact_117_lexordp__irreflexive_H,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Xs: list @ A] :
~ ( ord_lexordp @ A @ Xs @ Xs ) ) ).
% lexordp_irreflexive'
thf(fact_118_lexordp__linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,Ys2: list @ A] :
( ( ord_lexordp @ A @ Xs @ Ys2 )
| ( Xs = Ys2 )
| ( ord_lexordp @ A @ Ys2 @ Xs ) ) ) ).
% lexordp_linear
thf(fact_119_lexordp__trans,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,Ys2: list @ A,Zs2: list @ A] :
( ( ord_lexordp @ A @ Xs @ Ys2 )
=> ( ( ord_lexordp @ A @ Ys2 @ Zs2 )
=> ( ord_lexordp @ A @ Xs @ Zs2 ) ) ) ) ).
% lexordp_trans
thf(fact_120_lexordp__antisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Xs: list @ A,Ys2: list @ A] :
( ( ord_lexordp @ A @ Xs @ Ys2 )
=> ~ ( ord_lexordp @ A @ Ys2 @ Xs ) ) ) ).
% lexordp_antisym
thf(fact_121_lexordp__irreflexive,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [Xs: list @ A] :
( ! [X: A] :
~ ( ord_less @ A @ X @ X )
=> ~ ( ord_lexordp @ A @ Xs @ Xs ) ) ) ).
% lexordp_irreflexive
thf(fact_122_lexordp__conv__lexordp__eq,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_lexordp @ A )
= ( ^ [Xs3: list @ A,Ys: list @ A] :
( ( ord_lexordp_eq @ A @ Xs3 @ Ys )
& ~ ( ord_lexordp_eq @ A @ Ys @ Xs3 ) ) ) ) ) ).
% lexordp_conv_lexordp_eq
thf(fact_123_lexordp__eq__conv__lexord,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_lexordp_eq @ A )
= ( ^ [Xs3: list @ A,Ys: list @ A] :
( ( Xs3 = Ys )
| ( ord_lexordp @ A @ Xs3 @ Ys ) ) ) ) ) ).
% lexordp_eq_conv_lexord
thf(fact_124_lexordp__into__lexordp__eq,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [Xs: list @ A,Ys2: list @ A] :
( ( ord_lexordp @ A @ Xs @ Ys2 )
=> ( ord_lexordp_eq @ A @ Xs @ Ys2 ) ) ) ).
% lexordp_into_lexordp_eq
thf(fact_125_lexordp_OCons__eq,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X3: A,Y: A,Xs: list @ A,Ys2: list @ A] :
( ~ ( ord_less @ A @ X3 @ Y )
=> ( ~ ( ord_less @ A @ Y @ X3 )
=> ( ( ord_lexordp @ A @ Xs @ Ys2 )
=> ( ord_lexordp @ A @ ( cons @ A @ X3 @ Xs ) @ ( cons @ A @ Y @ Ys2 ) ) ) ) ) ) ).
% lexordp.Cons_eq
thf(fact_126_lexordp_OCons,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X3: A,Y: A,Xs: list @ A,Ys2: list @ A] :
( ( ord_less @ A @ X3 @ Y )
=> ( ord_lexordp @ A @ ( cons @ A @ X3 @ Xs ) @ ( cons @ A @ Y @ Ys2 ) ) ) ) ).
% lexordp.Cons
thf(fact_127_lexordp_ONil,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [Y: A,Ys2: list @ A] : ( ord_lexordp @ A @ ( nil @ A ) @ ( cons @ A @ Y @ Ys2 ) ) ) ).
% lexordp.Nil
thf(fact_128_member__rec_I2_J,axiom,
! [A: $tType,Y: A] :
~ ( member @ A @ ( nil @ A ) @ Y ) ).
% member_rec(2)
thf(fact_129_member__rec_I1_J,axiom,
! [A: $tType,X3: A,Xs: list @ A,Y: A] :
( ( member @ A @ ( cons @ A @ X3 @ Xs ) @ Y )
= ( ( X3 = Y )
| ( member @ A @ Xs @ Y ) ) ) ).
% member_rec(1)
thf(fact_130_splice_Oelims,axiom,
! [A: $tType,X3: list @ A,Xa: list @ A,Y: list @ A] :
( ( ( splice @ A @ X3 @ Xa )
= Y )
=> ( ( ( X3
= ( nil @ A ) )
=> ( Y != Xa ) )
=> ~ ! [X: A,Xs2: list @ A] :
( ( X3
= ( cons @ A @ X @ Xs2 ) )
=> ( Y
!= ( cons @ A @ X @ ( splice @ A @ Xa @ Xs2 ) ) ) ) ) ) ).
% splice.elims
thf(fact_131_lexordp__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_lexordp @ A )
= ( ^ [Xs3: list @ A,Ys: list @ A] :
( ? [X4: A,Vs: list @ A] :
( Ys
= ( append @ A @ Xs3 @ ( cons @ A @ X4 @ Vs ) ) )
| ? [Us: list @ A,A5: A,B3: A,Vs: list @ A,Ws: list @ A] :
( ( ord_less @ A @ A5 @ B3 )
& ( Xs3
= ( append @ A @ Us @ ( cons @ A @ A5 @ Vs ) ) )
& ( Ys
= ( append @ A @ Us @ ( cons @ A @ B3 @ Ws ) ) ) ) ) ) ) ) ).
% lexordp_iff
thf(fact_132_append_Oassoc,axiom,
! [A: $tType,A4: list @ A,B4: list @ A,C3: list @ A] :
( ( append @ A @ ( append @ A @ A4 @ B4 ) @ C3 )
= ( append @ A @ A4 @ ( append @ A @ B4 @ C3 ) ) ) ).
% append.assoc
thf(fact_133_append__assoc,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A,Zs2: list @ A] :
( ( append @ A @ ( append @ A @ Xs @ Ys2 ) @ Zs2 )
= ( append @ A @ Xs @ ( append @ A @ Ys2 @ Zs2 ) ) ) ).
% append_assoc
thf(fact_134_append__same__eq,axiom,
! [A: $tType,Ys2: list @ A,Xs: list @ A,Zs2: list @ A] :
( ( ( append @ A @ Ys2 @ Xs )
= ( append @ A @ Zs2 @ Xs ) )
= ( Ys2 = Zs2 ) ) ).
% append_same_eq
thf(fact_135_same__append__eq,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A,Zs2: list @ A] :
( ( ( append @ A @ Xs @ Ys2 )
= ( append @ A @ Xs @ Zs2 ) )
= ( Ys2 = Zs2 ) ) ).
% same_append_eq
thf(fact_136_append_Oright__neutral,axiom,
! [A: $tType,A4: list @ A] :
( ( append @ A @ A4 @ ( nil @ A ) )
= A4 ) ).
% append.right_neutral
thf(fact_137_append__is__Nil__conv,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A] :
( ( ( append @ A @ Xs @ Ys2 )
= ( nil @ A ) )
= ( ( Xs
= ( nil @ A ) )
& ( Ys2
= ( nil @ A ) ) ) ) ).
% append_is_Nil_conv
thf(fact_138_Nil__is__append__conv,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A] :
( ( ( nil @ A )
= ( append @ A @ Xs @ Ys2 ) )
= ( ( Xs
= ( nil @ A ) )
& ( Ys2
= ( nil @ A ) ) ) ) ).
% Nil_is_append_conv
thf(fact_139_self__append__conv2,axiom,
! [A: $tType,Ys2: list @ A,Xs: list @ A] :
( ( Ys2
= ( append @ A @ Xs @ Ys2 ) )
= ( Xs
= ( nil @ A ) ) ) ).
% self_append_conv2
thf(fact_140_append__self__conv2,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A] :
( ( ( append @ A @ Xs @ Ys2 )
= Ys2 )
= ( Xs
= ( nil @ A ) ) ) ).
% append_self_conv2
thf(fact_141_self__append__conv,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A] :
( ( Xs
= ( append @ A @ Xs @ Ys2 ) )
= ( Ys2
= ( nil @ A ) ) ) ).
% self_append_conv
thf(fact_142_append__self__conv,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A] :
( ( ( append @ A @ Xs @ Ys2 )
= Xs )
= ( Ys2
= ( nil @ A ) ) ) ).
% append_self_conv
thf(fact_143_append__Nil2,axiom,
! [A: $tType,Xs: list @ A] :
( ( append @ A @ Xs @ ( nil @ A ) )
= Xs ) ).
% append_Nil2
thf(fact_144_split__Nil__iff,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A] :
( ( ( splice @ A @ Xs @ Ys2 )
= ( nil @ A ) )
= ( ( Xs
= ( nil @ A ) )
& ( Ys2
= ( nil @ A ) ) ) ) ).
% split_Nil_iff
thf(fact_145_splice__Nil2,axiom,
! [A: $tType,Xs: list @ A] :
( ( splice @ A @ Xs @ ( nil @ A ) )
= Xs ) ).
% splice_Nil2
thf(fact_146_append1__eq__conv,axiom,
! [A: $tType,Xs: list @ A,X3: A,Ys2: list @ A,Y: A] :
( ( ( append @ A @ Xs @ ( cons @ A @ X3 @ ( nil @ A ) ) )
= ( append @ A @ Ys2 @ ( cons @ A @ Y @ ( nil @ A ) ) ) )
= ( ( Xs = Ys2 )
& ( X3 = Y ) ) ) ).
% append1_eq_conv
thf(fact_147_bind__simps_I2_J,axiom,
! [A: $tType,B: $tType,X3: B,Xs: list @ B,F3: B > ( list @ A )] :
( ( bind @ B @ A @ ( cons @ B @ X3 @ Xs ) @ F3 )
= ( append @ A @ ( F3 @ X3 ) @ ( bind @ B @ A @ Xs @ F3 ) ) ) ).
% bind_simps(2)
thf(fact_148_lexordp__append__leftI,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [Us2: list @ A,Vs2: list @ A,Xs: list @ A] :
( ( ord_lexordp @ A @ Us2 @ Vs2 )
=> ( ord_lexordp @ A @ ( append @ A @ Xs @ Us2 ) @ ( append @ A @ Xs @ Vs2 ) ) ) ) ).
% lexordp_append_leftI
thf(fact_149_append__eq__appendI,axiom,
! [A: $tType,Xs: list @ A,Xs1: list @ A,Zs2: list @ A,Ys2: list @ A,Us2: list @ A] :
( ( ( append @ A @ Xs @ Xs1 )
= Zs2 )
=> ( ( Ys2
= ( append @ A @ Xs1 @ Us2 ) )
=> ( ( append @ A @ Xs @ Ys2 )
= ( append @ A @ Zs2 @ Us2 ) ) ) ) ).
% append_eq_appendI
thf(fact_150_append__eq__append__conv2,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A,Zs2: list @ A,Ts: list @ A] :
( ( ( append @ A @ Xs @ Ys2 )
= ( append @ A @ Zs2 @ Ts ) )
= ( ? [Us: list @ A] :
( ( ( Xs
= ( append @ A @ Zs2 @ Us ) )
& ( ( append @ A @ Us @ Ys2 )
= Ts ) )
| ( ( ( append @ A @ Xs @ Us )
= Zs2 )
& ( Ys2
= ( append @ A @ Us @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_151_eq__Nil__appendI,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A] :
( ( Xs = Ys2 )
=> ( Xs
= ( append @ A @ ( nil @ A ) @ Ys2 ) ) ) ).
% eq_Nil_appendI
thf(fact_152_append__Nil,axiom,
! [A: $tType,Ys2: list @ A] :
( ( append @ A @ ( nil @ A ) @ Ys2 )
= Ys2 ) ).
% append_Nil
thf(fact_153_append_Oleft__neutral,axiom,
! [A: $tType,A4: list @ A] :
( ( append @ A @ ( nil @ A ) @ A4 )
= A4 ) ).
% append.left_neutral
thf(fact_154_Cons__eq__appendI,axiom,
! [A: $tType,X3: A,Xs1: list @ A,Ys2: list @ A,Xs: list @ A,Zs2: list @ A] :
( ( ( cons @ A @ X3 @ Xs1 )
= Ys2 )
=> ( ( Xs
= ( append @ A @ Xs1 @ Zs2 ) )
=> ( ( cons @ A @ X3 @ Xs )
= ( append @ A @ Ys2 @ Zs2 ) ) ) ) ).
% Cons_eq_appendI
thf(fact_155_append__Cons,axiom,
! [A: $tType,X3: A,Xs: list @ A,Ys2: list @ A] :
( ( append @ A @ ( cons @ A @ X3 @ Xs ) @ Ys2 )
= ( cons @ A @ X3 @ ( append @ A @ Xs @ Ys2 ) ) ) ).
% append_Cons
thf(fact_156_lexordp__eq__pref,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [U: list @ A,V: list @ A] : ( ord_lexordp_eq @ A @ U @ ( append @ A @ U @ V ) ) ) ).
% lexordp_eq_pref
thf(fact_157_ord_Olexordp__eq__pref,axiom,
! [A: $tType,Less: A > A > $o,U: list @ A,V: list @ A] : ( lexordp_eq @ A @ Less @ U @ ( append @ A @ U @ V ) ) ).
% ord.lexordp_eq_pref
thf(fact_158_knows2b,axiom,
! [A2: secrecy_specID,E: list @ secrecy_Expression,E1: list @ secrecy_Expression] :
( ( knowle2055185861_knows @ A2 @ ( append @ secrecy_Expression @ E @ E1 ) )
=> ( knowle2055185861_knows @ A2 @ E ) ) ).
% knows2b
thf(fact_159_knows2a,axiom,
! [A2: secrecy_specID,E1: list @ secrecy_Expression,E: list @ secrecy_Expression] :
( ( knowle2055185861_knows @ A2 @ ( append @ secrecy_Expression @ E1 @ E ) )
=> ( knowle2055185861_knows @ A2 @ E ) ) ).
% knows2a
thf(fact_160_knows3,axiom,
! [A2: secrecy_specID,E1: list @ secrecy_Expression,E22: list @ secrecy_Expression] :
( ( ( knowle2055185861_knows @ A2 @ E1 )
& ( knowle2055185861_knows @ A2 @ E22 ) )
=> ( knowle2055185861_knows @ A2 @ ( append @ secrecy_Expression @ E1 @ E22 ) ) ) ).
% knows3
thf(fact_161_splice_Osimps_I2_J,axiom,
! [A: $tType,X3: A,Xs: list @ A,Ys2: list @ A] :
( ( splice @ A @ ( cons @ A @ X3 @ Xs ) @ Ys2 )
= ( cons @ A @ X3 @ ( splice @ A @ Ys2 @ Xs ) ) ) ).
% splice.simps(2)
thf(fact_162_splice_Osimps_I1_J,axiom,
! [A: $tType,Ys2: list @ A] :
( ( splice @ A @ ( nil @ A ) @ Ys2 )
= Ys2 ) ).
% splice.simps(1)
thf(fact_163_rev__nonempty__induct,axiom,
! [A: $tType,Xs: list @ A,P: ( list @ A ) > $o] :
( ( Xs
!= ( nil @ A ) )
=> ( ! [X: A] : ( P @ ( cons @ A @ X @ ( nil @ A ) ) )
=> ( ! [X: A,Xs2: list @ A] :
( ( Xs2
!= ( nil @ A ) )
=> ( ( P @ Xs2 )
=> ( P @ ( append @ A @ Xs2 @ ( cons @ A @ X @ ( nil @ A ) ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_164_append__eq__Cons__conv,axiom,
! [A: $tType,Ys2: list @ A,Zs2: list @ A,X3: A,Xs: list @ A] :
( ( ( append @ A @ Ys2 @ Zs2 )
= ( cons @ A @ X3 @ Xs ) )
= ( ( ( Ys2
= ( nil @ A ) )
& ( Zs2
= ( cons @ A @ X3 @ Xs ) ) )
| ? [Ys5: list @ A] :
( ( Ys2
= ( cons @ A @ X3 @ Ys5 ) )
& ( ( append @ A @ Ys5 @ Zs2 )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_165_Cons__eq__append__conv,axiom,
! [A: $tType,X3: A,Xs: list @ A,Ys2: list @ A,Zs2: list @ A] :
( ( ( cons @ A @ X3 @ Xs )
= ( append @ A @ Ys2 @ Zs2 ) )
= ( ( ( Ys2
= ( nil @ A ) )
& ( ( cons @ A @ X3 @ Xs )
= Zs2 ) )
| ? [Ys5: list @ A] :
( ( ( cons @ A @ X3 @ Ys5 )
= Ys2 )
& ( Xs
= ( append @ A @ Ys5 @ Zs2 ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_166_rev__exhaust,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ~ ! [Ys3: list @ A,Y3: A] :
( Xs
!= ( append @ A @ Ys3 @ ( cons @ A @ Y3 @ ( nil @ A ) ) ) ) ) ).
% rev_exhaust
thf(fact_167_rev__induct,axiom,
! [A: $tType,P: ( list @ A ) > $o,Xs: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [X: A,Xs2: list @ A] :
( ( P @ Xs2 )
=> ( P @ ( append @ A @ Xs2 @ ( cons @ A @ X @ ( nil @ A ) ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_168_lexordp__append__leftD,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [Xs: list @ A,Us2: list @ A,Vs2: list @ A] :
( ( ord_lexordp @ A @ ( append @ A @ Xs @ Us2 ) @ ( append @ A @ Xs @ Vs2 ) )
=> ( ! [A3: A] :
~ ( ord_less @ A @ A3 @ A3 )
=> ( ord_lexordp @ A @ Us2 @ Vs2 ) ) ) ) ).
% lexordp_append_leftD
thf(fact_169_lexordp__append__rightI,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [Ys2: list @ A,Xs: list @ A] :
( ( Ys2
!= ( nil @ A ) )
=> ( ord_lexordp @ A @ Xs @ ( append @ A @ Xs @ Ys2 ) ) ) ) ).
% lexordp_append_rightI
thf(fact_170_lexordp__append__left__rightI,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X3: A,Y: A,Us2: list @ A,Xs: list @ A,Ys2: list @ A] :
( ( ord_less @ A @ X3 @ Y )
=> ( ord_lexordp @ A @ ( append @ A @ Us2 @ ( cons @ A @ X3 @ Xs ) ) @ ( append @ A @ Us2 @ ( cons @ A @ Y @ Ys2 ) ) ) ) ) ).
% lexordp_append_left_rightI
thf(fact_171_maps__simps_I1_J,axiom,
! [A: $tType,B: $tType,F3: B > ( list @ A ),X3: B,Xs: list @ B] :
( ( maps @ B @ A @ F3 @ ( cons @ B @ X3 @ Xs ) )
= ( append @ A @ ( F3 @ X3 ) @ ( maps @ B @ A @ F3 @ Xs ) ) ) ).
% maps_simps(1)
thf(fact_172_concat__eq__append__conv,axiom,
! [A: $tType,Xss2: list @ ( list @ A ),Ys2: list @ A,Zs2: list @ A] :
( ( ( concat @ A @ Xss2 )
= ( append @ A @ Ys2 @ Zs2 ) )
= ( ( ( Xss2
= ( nil @ ( list @ A ) ) )
=> ( ( Ys2
= ( nil @ A ) )
& ( Zs2
= ( nil @ A ) ) ) )
& ( ( Xss2
!= ( nil @ ( list @ A ) ) )
=> ? [Xss1: list @ ( list @ A ),Xs3: list @ A,Xs5: list @ A,Xss22: list @ ( list @ A )] :
( ( Xss2
= ( append @ ( list @ A ) @ Xss1 @ ( cons @ ( list @ A ) @ ( append @ A @ Xs3 @ Xs5 ) @ Xss22 ) ) )
& ( Ys2
= ( append @ A @ ( concat @ A @ Xss1 ) @ Xs3 ) )
& ( Zs2
= ( append @ A @ Xs5 @ ( concat @ A @ Xss22 ) ) ) ) ) ) ) ).
% concat_eq_append_conv
thf(fact_173_rotate1_Osimps_I2_J,axiom,
! [A: $tType,X3: A,Xs: list @ A] :
( ( rotate1 @ A @ ( cons @ A @ X3 @ Xs ) )
= ( append @ A @ Xs @ ( cons @ A @ X3 @ ( nil @ A ) ) ) ) ).
% rotate1.simps(2)
thf(fact_174_butlast__snoc,axiom,
! [A: $tType,Xs: list @ A,X3: A] :
( ( butlast @ A @ ( append @ A @ Xs @ ( cons @ A @ X3 @ ( nil @ A ) ) ) )
= Xs ) ).
% butlast_snoc
thf(fact_175_rotate1__is__Nil__conv,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( rotate1 @ A @ Xs )
= ( nil @ A ) )
= ( Xs
= ( nil @ A ) ) ) ).
% rotate1_is_Nil_conv
thf(fact_176_concat__append,axiom,
! [A: $tType,Xs: list @ ( list @ A ),Ys2: list @ ( list @ A )] :
( ( concat @ A @ ( append @ ( list @ A ) @ Xs @ Ys2 ) )
= ( append @ A @ ( concat @ A @ Xs ) @ ( concat @ A @ Ys2 ) ) ) ).
% concat_append
thf(fact_177_butlast_Osimps_I1_J,axiom,
! [A: $tType] :
( ( butlast @ A @ ( nil @ A ) )
= ( nil @ A ) ) ).
% butlast.simps(1)
thf(fact_178_rotate1_Osimps_I1_J,axiom,
! [A: $tType] :
( ( rotate1 @ A @ ( nil @ A ) )
= ( nil @ A ) ) ).
% rotate1.simps(1)
thf(fact_179_butlast_Osimps_I2_J,axiom,
! [A: $tType,Xs: list @ A,X3: A] :
( ( ( Xs
= ( nil @ A ) )
=> ( ( butlast @ A @ ( cons @ A @ X3 @ Xs ) )
= ( nil @ A ) ) )
& ( ( Xs
!= ( nil @ A ) )
=> ( ( butlast @ A @ ( cons @ A @ X3 @ Xs ) )
= ( cons @ A @ X3 @ ( butlast @ A @ Xs ) ) ) ) ) ).
% butlast.simps(2)
thf(fact_180_concat_Osimps_I1_J,axiom,
! [A: $tType] :
( ( concat @ A @ ( nil @ ( list @ A ) ) )
= ( nil @ A ) ) ).
% concat.simps(1)
thf(fact_181_butlast__append,axiom,
! [A: $tType,Ys2: list @ A,Xs: list @ A] :
( ( ( Ys2
= ( nil @ A ) )
=> ( ( butlast @ A @ ( append @ A @ Xs @ Ys2 ) )
= ( butlast @ A @ Xs ) ) )
& ( ( Ys2
!= ( nil @ A ) )
=> ( ( butlast @ A @ ( append @ A @ Xs @ Ys2 ) )
= ( append @ A @ Xs @ ( butlast @ A @ Ys2 ) ) ) ) ) ).
% butlast_append
thf(fact_182_concat_Osimps_I2_J,axiom,
! [A: $tType,X3: list @ A,Xs: list @ ( list @ A )] :
( ( concat @ A @ ( cons @ ( list @ A ) @ X3 @ Xs ) )
= ( append @ A @ X3 @ ( concat @ A @ Xs ) ) ) ).
% concat.simps(2)
thf(fact_183_maps__simps_I2_J,axiom,
! [B: $tType,A: $tType,F3: B > ( list @ A )] :
( ( maps @ B @ A @ F3 @ ( nil @ B ) )
= ( nil @ A ) ) ).
% maps_simps(2)
thf(fact_184_concat__eq__appendD,axiom,
! [A: $tType,Xss2: list @ ( list @ A ),Ys2: list @ A,Zs2: list @ A] :
( ( ( concat @ A @ Xss2 )
= ( append @ A @ Ys2 @ Zs2 ) )
=> ( ( Xss2
!= ( nil @ ( list @ A ) ) )
=> ? [Xss12: list @ ( list @ A ),Xs2: list @ A,Xs4: list @ A,Xss23: list @ ( list @ A )] :
( ( Xss2
= ( append @ ( list @ A ) @ Xss12 @ ( cons @ ( list @ A ) @ ( append @ A @ Xs2 @ Xs4 ) @ Xss23 ) ) )
& ( Ys2
= ( append @ A @ ( concat @ A @ Xss12 ) @ Xs2 ) )
& ( Zs2
= ( append @ A @ Xs4 @ ( concat @ A @ Xss23 ) ) ) ) ) ) ).
% concat_eq_appendD
thf(fact_185_append__butlast__last__id,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( append @ A @ ( butlast @ A @ Xs ) @ ( cons @ A @ ( last @ A @ Xs ) @ ( nil @ A ) ) )
= Xs ) ) ).
% append_butlast_last_id
thf(fact_186_snoc__eq__iff__butlast,axiom,
! [A: $tType,Xs: list @ A,X3: A,Ys2: list @ A] :
( ( ( append @ A @ Xs @ ( cons @ A @ X3 @ ( nil @ A ) ) )
= Ys2 )
= ( ( Ys2
!= ( nil @ A ) )
& ( ( butlast @ A @ Ys2 )
= Xs )
& ( ( last @ A @ Ys2 )
= X3 ) ) ) ).
% snoc_eq_iff_butlast
thf(fact_187_last__snoc,axiom,
! [A: $tType,Xs: list @ A,X3: A] :
( ( last @ A @ ( append @ A @ Xs @ ( cons @ A @ X3 @ ( nil @ A ) ) ) )
= X3 ) ).
% last_snoc
thf(fact_188_last__appendR,axiom,
! [A: $tType,Ys2: list @ A,Xs: list @ A] :
( ( Ys2
!= ( nil @ A ) )
=> ( ( last @ A @ ( append @ A @ Xs @ Ys2 ) )
= ( last @ A @ Ys2 ) ) ) ).
% last_appendR
thf(fact_189_last__appendL,axiom,
! [A: $tType,Ys2: list @ A,Xs: list @ A] :
( ( Ys2
= ( nil @ A ) )
=> ( ( last @ A @ ( append @ A @ Xs @ Ys2 ) )
= ( last @ A @ Xs ) ) ) ).
% last_appendL
thf(fact_190_last__ConsR,axiom,
! [A: $tType,Xs: list @ A,X3: A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( last @ A @ ( cons @ A @ X3 @ Xs ) )
= ( last @ A @ Xs ) ) ) ).
% last_ConsR
thf(fact_191_last__ConsL,axiom,
! [A: $tType,Xs: list @ A,X3: A] :
( ( Xs
= ( nil @ A ) )
=> ( ( last @ A @ ( cons @ A @ X3 @ Xs ) )
= X3 ) ) ).
% last_ConsL
thf(fact_192_last_Osimps,axiom,
! [A: $tType,Xs: list @ A,X3: A] :
( ( ( Xs
= ( nil @ A ) )
=> ( ( last @ A @ ( cons @ A @ X3 @ Xs ) )
= X3 ) )
& ( ( Xs
!= ( nil @ A ) )
=> ( ( last @ A @ ( cons @ A @ X3 @ Xs ) )
= ( last @ A @ Xs ) ) ) ) ).
% last.simps
thf(fact_193_last__append,axiom,
! [A: $tType,Ys2: list @ A,Xs: list @ A] :
( ( ( Ys2
= ( nil @ A ) )
=> ( ( last @ A @ ( append @ A @ Xs @ Ys2 ) )
= ( last @ A @ Xs ) ) )
& ( ( Ys2
!= ( nil @ A ) )
=> ( ( last @ A @ ( append @ A @ Xs @ Ys2 ) )
= ( last @ A @ Ys2 ) ) ) ) ).
% last_append
thf(fact_194_longest__common__suffix,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A] :
? [Ss: list @ A,Xs4: list @ A,Ys4: list @ A] :
( ( Xs
= ( append @ A @ Xs4 @ Ss ) )
& ( Ys2
= ( append @ A @ Ys4 @ Ss ) )
& ( ( Xs4
= ( nil @ A ) )
| ( Ys4
= ( nil @ A ) )
| ( ( last @ A @ Xs4 )
!= ( last @ A @ Ys4 ) ) ) ) ).
% longest_common_suffix
thf(fact_195_concat__conv__foldr,axiom,
! [A: $tType] :
( ( concat @ A )
= ( ^ [Xss3: list @ ( list @ A )] : ( foldr @ ( list @ A ) @ ( list @ A ) @ ( append @ A ) @ Xss3 @ ( nil @ A ) ) ) ) ).
% concat_conv_foldr
thf(fact_196_SuccD,axiom,
! [A: $tType,K: A,Kl: set @ ( list @ A ),Kl2: list @ A] :
( ( member2 @ A @ K @ ( bNF_Greatest_Succ @ A @ Kl @ Kl2 ) )
=> ( member2 @ ( list @ A ) @ ( append @ A @ Kl2 @ ( cons @ A @ K @ ( nil @ A ) ) ) @ Kl ) ) ).
% SuccD
thf(fact_197_foldr__append,axiom,
! [B: $tType,A: $tType,F3: B > A > A,Xs: list @ B,Ys2: list @ B,A4: A] :
( ( foldr @ B @ A @ F3 @ ( append @ B @ Xs @ Ys2 ) @ A4 )
= ( foldr @ B @ A @ F3 @ Xs @ ( foldr @ B @ A @ F3 @ Ys2 @ A4 ) ) ) ).
% foldr_append
thf(fact_198_SuccI,axiom,
! [A: $tType,Kl2: list @ A,K: A,Kl: set @ ( list @ A )] :
( ( member2 @ ( list @ A ) @ ( append @ A @ Kl2 @ ( cons @ A @ K @ ( nil @ A ) ) ) @ Kl )
=> ( member2 @ A @ K @ ( bNF_Greatest_Succ @ A @ Kl @ Kl2 ) ) ) ).
% SuccI
thf(fact_199_shift__def,axiom,
! [B: $tType,A: $tType] :
( ( bNF_Greatest_shift @ A @ B )
= ( ^ [Lab: ( list @ A ) > B,K3: A,Kl3: list @ A] : ( Lab @ ( cons @ A @ K3 @ Kl3 ) ) ) ) ).
% shift_def
thf(fact_200_empty__Shift,axiom,
! [A: $tType,Kl: set @ ( list @ A ),K: A] :
( ( member2 @ ( list @ A ) @ ( nil @ A ) @ Kl )
=> ( ( member2 @ A @ K @ ( bNF_Greatest_Succ @ A @ Kl @ ( nil @ A ) ) )
=> ( member2 @ ( list @ A ) @ ( nil @ A ) @ ( bNF_Greatest_Shift @ A @ Kl @ K ) ) ) ) ).
% empty_Shift
thf(fact_201_ShiftD,axiom,
! [A: $tType,Kl2: list @ A,Kl: set @ ( list @ A ),K: A] :
( ( member2 @ ( list @ A ) @ Kl2 @ ( bNF_Greatest_Shift @ A @ Kl @ K ) )
=> ( member2 @ ( list @ A ) @ ( cons @ A @ K @ Kl2 ) @ Kl ) ) ).
% ShiftD
thf(fact_202_Succ__Shift,axiom,
! [A: $tType,Kl: set @ ( list @ A ),K: A,Kl2: list @ A] :
( ( bNF_Greatest_Succ @ A @ ( bNF_Greatest_Shift @ A @ Kl @ K ) @ Kl2 )
= ( bNF_Greatest_Succ @ A @ Kl @ ( cons @ A @ K @ Kl2 ) ) ) ).
% Succ_Shift
thf(fact_203_rev__eq__Cons__iff,axiom,
! [A: $tType,Xs: list @ A,Y: A,Ys2: list @ A] :
( ( ( rev @ A @ Xs )
= ( cons @ A @ Y @ Ys2 ) )
= ( Xs
= ( append @ A @ ( rev @ A @ Ys2 ) @ ( cons @ A @ Y @ ( nil @ A ) ) ) ) ) ).
% rev_eq_Cons_iff
thf(fact_204_append_Omonoid__axioms,axiom,
! [A: $tType] : ( monoid @ ( list @ A ) @ ( append @ A ) @ ( nil @ A ) ) ).
% append.monoid_axioms
thf(fact_205_rev__is__rev__conv,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A] :
( ( ( rev @ A @ Xs )
= ( rev @ A @ Ys2 ) )
= ( Xs = Ys2 ) ) ).
% rev_is_rev_conv
thf(fact_206_rev__rev__ident,axiom,
! [A: $tType,Xs: list @ A] :
( ( rev @ A @ ( rev @ A @ Xs ) )
= Xs ) ).
% rev_rev_ident
thf(fact_207_rev__is__Nil__conv,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( rev @ A @ Xs )
= ( nil @ A ) )
= ( Xs
= ( nil @ A ) ) ) ).
% rev_is_Nil_conv
thf(fact_208_Nil__is__rev__conv,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( nil @ A )
= ( rev @ A @ Xs ) )
= ( Xs
= ( nil @ A ) ) ) ).
% Nil_is_rev_conv
thf(fact_209_rev__append,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A] :
( ( rev @ A @ ( append @ A @ Xs @ Ys2 ) )
= ( append @ A @ ( rev @ A @ Ys2 ) @ ( rev @ A @ Xs ) ) ) ).
% rev_append
thf(fact_210_singleton__rev__conv,axiom,
! [A: $tType,X3: A,Xs: list @ A] :
( ( ( cons @ A @ X3 @ ( nil @ A ) )
= ( rev @ A @ Xs ) )
= ( Xs
= ( cons @ A @ X3 @ ( nil @ A ) ) ) ) ).
% singleton_rev_conv
thf(fact_211_rev__singleton__conv,axiom,
! [A: $tType,Xs: list @ A,X3: A] :
( ( ( rev @ A @ Xs )
= ( cons @ A @ X3 @ ( nil @ A ) ) )
= ( Xs
= ( cons @ A @ X3 @ ( nil @ A ) ) ) ) ).
% rev_singleton_conv
thf(fact_212_rev_Osimps_I1_J,axiom,
! [A: $tType] :
( ( rev @ A @ ( nil @ A ) )
= ( nil @ A ) ) ).
% rev.simps(1)
thf(fact_213_rev__swap,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A] :
( ( ( rev @ A @ Xs )
= Ys2 )
= ( Xs
= ( rev @ A @ Ys2 ) ) ) ).
% rev_swap
thf(fact_214_rev_Osimps_I2_J,axiom,
! [A: $tType,X3: A,Xs: list @ A] :
( ( rev @ A @ ( cons @ A @ X3 @ Xs ) )
= ( append @ A @ ( rev @ A @ Xs ) @ ( cons @ A @ X3 @ ( nil @ A ) ) ) ) ).
% rev.simps(2)
thf(fact_215_map__tailrec__def,axiom,
! [B: $tType,A: $tType] :
( ( map_tailrec @ A @ B )
= ( ^ [F4: A > B,As3: list @ A] : ( rev @ B @ ( map_tailrec_rev @ A @ B @ F4 @ As3 @ ( nil @ B ) ) ) ) ) ).
% map_tailrec_def
thf(fact_216_hd__rev,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( hd @ A @ ( rev @ A @ Xs ) )
= ( last @ A @ Xs ) ) ) ).
% hd_rev
thf(fact_217_hd__append2,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( hd @ A @ ( append @ A @ Xs @ Ys2 ) )
= ( hd @ A @ Xs ) ) ) ).
% hd_append2
thf(fact_218_hd__concat,axiom,
! [A: $tType,Xs: list @ ( list @ A )] :
( ( Xs
!= ( nil @ ( list @ A ) ) )
=> ( ( ( hd @ ( list @ A ) @ Xs )
!= ( nil @ A ) )
=> ( ( hd @ A @ ( concat @ A @ Xs ) )
= ( hd @ A @ ( hd @ ( list @ A ) @ Xs ) ) ) ) ) ).
% hd_concat
thf(fact_219_list_Osel_I1_J,axiom,
! [A: $tType,X21: A,X22: list @ A] :
( ( hd @ A @ ( cons @ A @ X21 @ X22 ) )
= X21 ) ).
% list.sel(1)
thf(fact_220_longest__common__prefix,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A] :
? [Ps: list @ A,Xs4: list @ A,Ys4: list @ A] :
( ( Xs
= ( append @ A @ Ps @ Xs4 ) )
& ( Ys2
= ( append @ A @ Ps @ Ys4 ) )
& ( ( Xs4
= ( nil @ A ) )
| ( Ys4
= ( nil @ A ) )
| ( ( hd @ A @ Xs4 )
!= ( hd @ A @ Ys4 ) ) ) ) ).
% longest_common_prefix
thf(fact_221_hd__append,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A] :
( ( ( Xs
= ( nil @ A ) )
=> ( ( hd @ A @ ( append @ A @ Xs @ Ys2 ) )
= ( hd @ A @ Ys2 ) ) )
& ( ( Xs
!= ( nil @ A ) )
=> ( ( hd @ A @ ( append @ A @ Xs @ Ys2 ) )
= ( hd @ A @ Xs ) ) ) ) ).
% hd_append
thf(fact_222_last__rev,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( last @ A @ ( rev @ A @ Xs ) )
= ( hd @ A @ Xs ) ) ) ).
% last_rev
thf(fact_223_rotate1__hd__tl,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( rotate1 @ A @ Xs )
= ( append @ A @ ( tl @ A @ Xs ) @ ( cons @ A @ ( hd @ A @ Xs ) @ ( nil @ A ) ) ) ) ) ).
% rotate1_hd_tl
thf(fact_224_distinct__adj__append__iff,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A] :
( ( distinct_adj @ A @ ( append @ A @ Xs @ Ys2 ) )
= ( ( distinct_adj @ A @ Xs )
& ( distinct_adj @ A @ Ys2 )
& ( ( Xs
= ( nil @ A ) )
| ( Ys2
= ( nil @ A ) )
| ( ( last @ A @ Xs )
!= ( hd @ A @ Ys2 ) ) ) ) ) ).
% distinct_adj_append_iff
thf(fact_225_distinct__adj__Cons__Cons,axiom,
! [B: $tType,X3: B,Y: B,Xs: list @ B] :
( ( distinct_adj @ B @ ( cons @ B @ X3 @ ( cons @ B @ Y @ Xs ) ) )
= ( ( X3 != Y )
& ( distinct_adj @ B @ ( cons @ B @ Y @ Xs ) ) ) ) ).
% distinct_adj_Cons_Cons
thf(fact_226_distinct__adj__rev,axiom,
! [A: $tType,Xs: list @ A] :
( ( distinct_adj @ A @ ( rev @ A @ Xs ) )
= ( distinct_adj @ A @ Xs ) ) ).
% distinct_adj_rev
thf(fact_227_tl__append2,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( tl @ A @ ( append @ A @ Xs @ Ys2 ) )
= ( append @ A @ ( tl @ A @ Xs ) @ Ys2 ) ) ) ).
% tl_append2
thf(fact_228_butlast__rev,axiom,
! [A: $tType,Xs: list @ A] :
( ( butlast @ A @ ( rev @ A @ Xs ) )
= ( rev @ A @ ( tl @ A @ Xs ) ) ) ).
% butlast_rev
thf(fact_229_list_Ocollapse,axiom,
! [A: $tType,List: list @ A] :
( ( List
!= ( nil @ A ) )
=> ( ( cons @ A @ ( hd @ A @ List ) @ ( tl @ A @ List ) )
= List ) ) ).
% list.collapse
thf(fact_230_hd__Cons__tl,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( cons @ A @ ( hd @ A @ Xs ) @ ( tl @ A @ Xs ) )
= Xs ) ) ).
% hd_Cons_tl
thf(fact_231_list_Oexpand,axiom,
! [A: $tType,List: list @ A,List2: list @ A] :
( ( ( List
= ( nil @ A ) )
= ( List2
= ( nil @ A ) ) )
=> ( ( ( List
!= ( nil @ A ) )
=> ( ( List2
!= ( nil @ A ) )
=> ( ( ( hd @ A @ List )
= ( hd @ A @ List2 ) )
& ( ( tl @ A @ List )
= ( tl @ A @ List2 ) ) ) ) )
=> ( List = List2 ) ) ) ).
% list.expand
thf(fact_232_distinct__adj__singleton,axiom,
! [B: $tType,X3: B] : ( distinct_adj @ B @ ( cons @ B @ X3 @ ( nil @ B ) ) ) ).
% distinct_adj_singleton
thf(fact_233_tl__Nil,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( tl @ A @ Xs )
= ( nil @ A ) )
= ( ( Xs
= ( nil @ A ) )
| ? [X4: A] :
( Xs
= ( cons @ A @ X4 @ ( nil @ A ) ) ) ) ) ).
% tl_Nil
thf(fact_234_Nil__tl,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( nil @ A )
= ( tl @ A @ Xs ) )
= ( ( Xs
= ( nil @ A ) )
| ? [X4: A] :
( Xs
= ( cons @ A @ X4 @ ( nil @ A ) ) ) ) ) ).
% Nil_tl
thf(fact_235_distinct__adj__ConsD,axiom,
! [A: $tType,X3: A,Xs: list @ A] :
( ( distinct_adj @ A @ ( cons @ A @ X3 @ Xs ) )
=> ( distinct_adj @ A @ Xs ) ) ).
% distinct_adj_ConsD
thf(fact_236_distinct__adj__Nil,axiom,
! [A: $tType] : ( distinct_adj @ A @ ( nil @ A ) ) ).
% distinct_adj_Nil
thf(fact_237_list_Osel_I3_J,axiom,
! [A: $tType,X21: A,X22: list @ A] :
( ( tl @ A @ ( cons @ A @ X21 @ X22 ) )
= X22 ) ).
% list.sel(3)
thf(fact_238_list_Osel_I2_J,axiom,
! [A: $tType] :
( ( tl @ A @ ( nil @ A ) )
= ( nil @ A ) ) ).
% list.sel(2)
thf(fact_239_distinct__adj__appendD1,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A] :
( ( distinct_adj @ A @ ( append @ A @ Xs @ Ys2 ) )
=> ( distinct_adj @ A @ Xs ) ) ).
% distinct_adj_appendD1
thf(fact_240_distinct__adj__appendD2,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A] :
( ( distinct_adj @ A @ ( append @ A @ Xs @ Ys2 ) )
=> ( distinct_adj @ A @ Ys2 ) ) ).
% distinct_adj_appendD2
thf(fact_241_butlast__tl,axiom,
! [A: $tType,Xs: list @ A] :
( ( butlast @ A @ ( tl @ A @ Xs ) )
= ( tl @ A @ ( butlast @ A @ Xs ) ) ) ).
% butlast_tl
thf(fact_242_last__tl,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( Xs
= ( nil @ A ) )
| ( ( tl @ A @ Xs )
!= ( nil @ A ) ) )
=> ( ( last @ A @ ( tl @ A @ Xs ) )
= ( last @ A @ Xs ) ) ) ).
% last_tl
thf(fact_243_list_Oexhaust__sel,axiom,
! [A: $tType,List: list @ A] :
( ( List
!= ( nil @ A ) )
=> ( List
= ( cons @ A @ ( hd @ A @ List ) @ ( tl @ A @ List ) ) ) ) ).
% list.exhaust_sel
thf(fact_244_distinct__adj__Cons,axiom,
! [A: $tType,X3: A,Xs: list @ A] :
( ( distinct_adj @ A @ ( cons @ A @ X3 @ Xs ) )
= ( ( Xs
= ( nil @ A ) )
| ( ( X3
!= ( hd @ A @ Xs ) )
& ( distinct_adj @ A @ Xs ) ) ) ) ).
% distinct_adj_Cons
thf(fact_245_list_Osplit__sel,axiom,
! [B: $tType,A: $tType,P: B > $o,F1: B,F2: A > ( list @ A ) > B,List: list @ A] :
( ( P @ ( case_list @ B @ A @ F1 @ F2 @ List ) )
= ( ( ( List
= ( nil @ A ) )
=> ( P @ F1 ) )
& ( ( List
= ( cons @ A @ ( hd @ A @ List ) @ ( tl @ A @ List ) ) )
=> ( P @ ( F2 @ ( hd @ A @ List ) @ ( tl @ A @ List ) ) ) ) ) ) ).
% list.split_sel
thf(fact_246_list_Osplit__sel__asm,axiom,
! [B: $tType,A: $tType,P: B > $o,F1: B,F2: A > ( list @ A ) > B,List: list @ A] :
( ( P @ ( case_list @ B @ A @ F1 @ F2 @ List ) )
= ( ~ ( ( ( List
= ( nil @ A ) )
& ~ ( P @ F1 ) )
| ( ( List
= ( cons @ A @ ( hd @ A @ List ) @ ( tl @ A @ List ) ) )
& ~ ( P @ ( F2 @ ( hd @ A @ List ) @ ( tl @ A @ List ) ) ) ) ) ) ) ).
% list.split_sel_asm
thf(fact_247_list_Osimps_I5_J,axiom,
! [B: $tType,A: $tType,F1: B,F2: A > ( list @ A ) > B,X21: A,X22: list @ A] :
( ( case_list @ B @ A @ F1 @ F2 @ ( cons @ A @ X21 @ X22 ) )
= ( F2 @ X21 @ X22 ) ) ).
% list.simps(5)
thf(fact_248_list_Osimps_I4_J,axiom,
! [A: $tType,B: $tType,F1: B,F2: A > ( list @ A ) > B] :
( ( case_list @ B @ A @ F1 @ F2 @ ( nil @ A ) )
= F1 ) ).
% list.simps(4)
thf(fact_249_list_Ocase__eq__if,axiom,
! [A: $tType,B: $tType] :
( ( case_list @ B @ A )
= ( ^ [F12: B,F22: A > ( list @ A ) > B,List3: list @ A] :
( if @ B
@ ( List3
= ( nil @ A ) )
@ F12
@ ( F22 @ ( hd @ A @ List3 ) @ ( tl @ A @ List3 ) ) ) ) ) ).
% list.case_eq_if
thf(fact_250_successively__append__iff,axiom,
! [A: $tType,P: A > A > $o,Xs: list @ A,Ys2: list @ A] :
( ( successively @ A @ P @ ( append @ A @ Xs @ Ys2 ) )
= ( ( successively @ A @ P @ Xs )
& ( successively @ A @ P @ Ys2 )
& ( ( Xs
= ( nil @ A ) )
| ( Ys2
= ( nil @ A ) )
| ( P @ ( last @ A @ Xs ) @ ( hd @ A @ Ys2 ) ) ) ) ) ).
% successively_append_iff
thf(fact_251_Cons__in__shuffles__iff,axiom,
! [A: $tType,Z: A,Zs2: list @ A,Xs: list @ A,Ys2: list @ A] :
( ( member2 @ ( list @ A ) @ ( cons @ A @ Z @ Zs2 ) @ ( shuffles @ A @ Xs @ Ys2 ) )
= ( ( ( Xs
!= ( nil @ A ) )
& ( ( hd @ A @ Xs )
= Z )
& ( member2 @ ( list @ A ) @ Zs2 @ ( shuffles @ A @ ( tl @ A @ Xs ) @ Ys2 ) ) )
| ( ( Ys2
!= ( nil @ A ) )
& ( ( hd @ A @ Ys2 )
= Z )
& ( member2 @ ( list @ A ) @ Zs2 @ ( shuffles @ A @ Xs @ ( tl @ A @ Ys2 ) ) ) ) ) ) ).
% Cons_in_shuffles_iff
thf(fact_252_Nil__in__shuffles,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A] :
( ( member2 @ ( list @ A ) @ ( nil @ A ) @ ( shuffles @ A @ Xs @ Ys2 ) )
= ( ( Xs
= ( nil @ A ) )
& ( Ys2
= ( nil @ A ) ) ) ) ).
% Nil_in_shuffles
thf(fact_253_splice__in__shuffles,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A] : ( member2 @ ( list @ A ) @ ( splice @ A @ Xs @ Ys2 ) @ ( shuffles @ A @ Xs @ Ys2 ) ) ).
% splice_in_shuffles
thf(fact_254_shufflesE,axiom,
! [A: $tType,Zs2: list @ A,Xs: list @ A,Ys2: list @ A] :
( ( member2 @ ( list @ A ) @ Zs2 @ ( shuffles @ A @ Xs @ Ys2 ) )
=> ( ( ( Zs2 = Xs )
=> ( Ys2
!= ( nil @ A ) ) )
=> ( ( ( Zs2 = Ys2 )
=> ( Xs
!= ( nil @ A ) ) )
=> ( ! [X: A,Xs4: list @ A] :
( ( Xs
= ( cons @ A @ X @ Xs4 ) )
=> ! [Z2: A,Zs3: list @ A] :
( ( Zs2
= ( cons @ A @ Z2 @ Zs3 ) )
=> ( ( X = Z2 )
=> ~ ( member2 @ ( list @ A ) @ Zs3 @ ( shuffles @ A @ Xs4 @ Ys2 ) ) ) ) )
=> ~ ! [Y3: A,Ys4: list @ A] :
( ( Ys2
= ( cons @ A @ Y3 @ Ys4 ) )
=> ! [Z2: A,Zs3: list @ A] :
( ( Zs2
= ( cons @ A @ Z2 @ Zs3 ) )
=> ( ( Y3 = Z2 )
=> ~ ( member2 @ ( list @ A ) @ Zs3 @ ( shuffles @ A @ Xs @ Ys4 ) ) ) ) ) ) ) ) ) ).
% shufflesE
% Type constructors (7)
thf(tcon_fun___Orderings_Oorder,axiom,
! [A6: $tType,A7: $tType] :
( ( order @ A7 )
=> ( order @ ( A6 > A7 ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A6: $tType,A7: $tType] :
( ( ord @ A7 )
=> ( ord @ ( A6 > A7 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_1,axiom,
! [A6: $tType] : ( order @ ( set @ A6 ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_2,axiom,
! [A6: $tType] : ( ord @ ( set @ A6 ) ) ).
thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
linorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder_3,axiom,
order @ $o ).
thf(tcon_HOL_Obool___Orderings_Oord_4,axiom,
ord @ $o ).
% Helper facts (3)
thf(help_If_3_1_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_T,axiom,
! [A: $tType,X3: A,Y: A] :
( ( if @ A @ $false @ X3 @ Y )
= Y ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X3: A,Y: A] :
( ( if @ A @ $true @ X3 @ Y )
= X3 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
knowle1437066798e_know @ a @ ( secrecy_kKS @ m ) ).
%------------------------------------------------------------------------------