TPTP Problem File: ITP092^2.p
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%------------------------------------------------------------------------------
% File : ITP092^2 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Kuratowski problem prob_158__5524402_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 : Kuratowski/prob_158__5524402_1 [Des21]
% Status : Theorem
% Rating : 0.67 v8.1.0, 0.75 v7.5.0
% Syntax : Number of formulae : 322 ( 119 unt; 55 typ; 0 def)
% Number of atoms : 749 ( 473 equ; 0 cnn)
% Maximal formula atoms : 26 ( 2 avg)
% Number of connectives : 8153 ( 211 ~; 22 |; 91 &;7441 @)
% ( 0 <=>; 388 =>; 0 <=; 0 <~>)
% Maximal formula depth : 34 ( 10 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 234 ( 234 >; 0 *; 0 +; 0 <<)
% Number of symbols : 55 ( 52 usr; 4 con; 0-5 aty)
% Number of variables : 1498 ( 7 ^;1363 !; 72 ?;1498 :)
% ( 56 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 16:29:25.897
%------------------------------------------------------------------------------
% Could-be-implicit typings (7)
thf(ty_t_Pair__Digraph_Opair__pre__digraph_Opair__pre__digraph__ext,type,
pair_p1731315293ph_ext: $tType > $tType > $tType ).
thf(ty_t_Product__Type_Ounit,type,
product_unit: $tType ).
thf(ty_t_Product__Type_Oprod,type,
product_prod: $tType > $tType > $tType ).
thf(ty_t_List_Olist,type,
list: $tType > $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_t_Nat_Onat,type,
nat: $tType ).
thf(ty_tf_a,type,
a: $tType ).
% Explicit typings (48)
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : $o ).
thf(sy_c_Groups_Ozero__class_Ozero,type,
zero_zero:
!>[A: $tType] : A ).
thf(sy_c_If,type,
if:
!>[A: $tType] : ( $o > A > A > A ) ).
thf(sy_c_Kuratowski__Mirabelle__hkvioyikiw_Oprogressing,type,
kurato990191779essing:
!>[A: $tType] : ( ( list @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_List_Oappend,type,
append:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Obind,type,
bind:
!>[A: $tType,B: $tType] : ( ( list @ A ) > ( A > ( list @ B ) ) > ( list @ B ) ) ).
thf(sy_c_List_Oconcat,type,
concat:
!>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ A ) ) ).
thf(sy_c_List_Oinsert,type,
insert:
!>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Olenlex,type,
lenlex:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ).
thf(sy_c_List_Olex,type,
lex:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ).
thf(sy_c_List_Olexord,type,
lexord:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ 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_Oset,type,
set2:
!>[A: $tType] : ( ( list @ A ) > ( set @ A ) ) ).
thf(sy_c_List_Olistrel,type,
listrel:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) ) ).
thf(sy_c_List_Olistrel1,type,
listrel1:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ).
thf(sy_c_List_Omaps,type,
maps:
!>[A: $tType,B: $tType] : ( ( A > ( list @ B ) ) > ( list @ A ) > ( list @ B ) ) ).
thf(sy_c_List_Omeasures,type,
measures:
!>[A: $tType] : ( ( list @ ( A > nat ) ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_List_Omember,type,
member:
!>[A: $tType] : ( ( list @ A ) > A > $o ) ).
thf(sy_c_List_On__lists,type,
n_lists:
!>[A: $tType] : ( nat > ( list @ A ) > ( list @ ( list @ A ) ) ) ).
thf(sy_c_List_Oproduct__lists,type,
product_lists:
!>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ ( list @ A ) ) ) ).
thf(sy_c_List_Osubseqs,type,
subseqs:
!>[A: $tType] : ( ( list @ A ) > ( list @ ( list @ A ) ) ) ).
thf(sy_c_List_Ozip,type,
zip:
!>[A: $tType,B: $tType] : ( ( list @ A ) > ( list @ B ) > ( list @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_Multiset_Olinorder__class_Opart,type,
linorder_part:
!>[B: $tType,A: $tType] : ( ( B > A ) > A > ( list @ B ) > ( product_prod @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) ) ) ) ).
thf(sy_c_Nat_Osize__class_Osize,type,
size_size:
!>[A: $tType] : ( A > nat ) ).
thf(sy_c_Orderings_Oord__class_Oless,type,
ord_less:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Pair__Digraph_Oco__path,type,
pair_co_path:
!>[A: $tType] : ( ( product_prod @ A @ A ) > A > ( list @ ( product_prod @ A @ A ) ) > ( list @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_Pair__Digraph_Oco__path__rel,type,
pair_co_path_rel:
!>[A: $tType] : ( ( product_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ ( list @ ( product_prod @ A @ A ) ) ) ) > ( product_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ ( list @ ( product_prod @ A @ A ) ) ) ) > $o ) ).
thf(sy_c_Pair__Digraph_Opawalk__verts,type,
pair_pawalk_verts:
!>[A: $tType] : ( A > ( list @ ( product_prod @ A @ A ) ) > ( list @ A ) ) ).
thf(sy_c_Pair__Digraph_Osd__path,type,
pair_sd_path:
!>[A: $tType] : ( ( product_prod @ A @ A ) > A > ( list @ ( product_prod @ A @ A ) ) > ( list @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_Pair__Digraph_Osd__path__rel,type,
pair_sd_path_rel:
!>[A: $tType] : ( ( product_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ ( list @ ( product_prod @ A @ A ) ) ) ) > ( product_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ ( list @ ( product_prod @ A @ A ) ) ) ) > $o ) ).
thf(sy_c_Permutations_Oinverse__permutation__of__list,type,
invers965609412f_list:
!>[A: $tType] : ( ( list @ ( product_prod @ A @ A ) ) > A > A ) ).
thf(sy_c_Permutations_Oinverse__permutation__of__list__rel,type,
invers1507583541st_rel:
!>[A: $tType] : ( ( product_prod @ ( list @ ( product_prod @ A @ A ) ) @ A ) > ( product_prod @ ( list @ ( product_prod @ A @ A ) ) @ A ) > $o ) ).
thf(sy_c_Permutations_Olist__permutes,type,
list_permutes:
!>[A: $tType] : ( ( list @ ( product_prod @ A @ A ) ) > ( set @ A ) > $o ) ).
thf(sy_c_Permutations_Opermutation__of__list,type,
permutation_of_list:
!>[A: $tType] : ( ( list @ ( product_prod @ A @ A ) ) > A > A ) ).
thf(sy_c_Product__Type_OPair,type,
product_Pair:
!>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
thf(sy_c_Product__Type_Ointernal__case__prod,type,
produc2004651681e_prod:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
product_rec_prod:
!>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
thf(sy_c_Relation_Oasym,type,
asym:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Relation_Oirrefl,type,
irrefl:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_Set_Othe__elem,type,
the_elem:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Wellfounded_Oaccp,type,
accp:
!>[A: $tType] : ( ( A > A > $o ) > A > $o ) ).
thf(sy_c_member,type,
member2:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_e,type,
e: product_prod @ a @ a ).
thf(sy_v_es,type,
es: list @ ( product_prod @ a @ a ) ).
% Relevant facts (255)
thf(fact_0_assms,axiom,
kurato990191779essing @ a @ ( cons @ ( product_prod @ a @ a ) @ e @ es ) ).
% assms
thf(fact_1_progressing__def,axiom,
! [A: $tType] :
( ( kurato990191779essing @ A )
= ( ^ [P: list @ ( product_prod @ A @ A )] :
! [Xs: list @ ( product_prod @ A @ A ),X: A,Y: A,Ys: list @ ( product_prod @ A @ A )] :
( P
!= ( append @ ( product_prod @ A @ A ) @ Xs @ ( cons @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( cons @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X ) @ Ys ) ) ) ) ) ) ).
% progressing_def
thf(fact_2_append_Oassoc,axiom,
! [A: $tType,A2: list @ A,B2: list @ A,C2: list @ A] :
( ( append @ A @ ( append @ A @ A2 @ B2 ) @ C2 )
= ( append @ A @ A2 @ ( append @ A @ B2 @ C2 ) ) ) ).
% append.assoc
thf(fact_3_append__assoc,axiom,
! [A: $tType,Xs2: list @ A,Ys2: list @ A,Zs: list @ A] :
( ( append @ A @ ( append @ A @ Xs2 @ Ys2 ) @ Zs )
= ( append @ A @ Xs2 @ ( append @ A @ Ys2 @ Zs ) ) ) ).
% append_assoc
thf(fact_4_append__same__eq,axiom,
! [A: $tType,Ys2: list @ A,Xs2: list @ A,Zs: list @ A] :
( ( ( append @ A @ Ys2 @ Xs2 )
= ( append @ A @ Zs @ Xs2 ) )
= ( Ys2 = Zs ) ) ).
% append_same_eq
thf(fact_5_same__append__eq,axiom,
! [A: $tType,Xs2: list @ A,Ys2: list @ A,Zs: list @ A] :
( ( ( append @ A @ Xs2 @ Ys2 )
= ( append @ A @ Xs2 @ Zs ) )
= ( Ys2 = Zs ) ) ).
% same_append_eq
thf(fact_6_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_7_prod_Oinject,axiom,
! [A: $tType,B: $tType,X1: A,X2: B,Y1: A,Y2: B] :
( ( ( product_Pair @ A @ B @ X1 @ X2 )
= ( product_Pair @ A @ B @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_8_old_Oprod_Oinject,axiom,
! [A: $tType,B: $tType,A2: A,B2: B,A3: A,B3: B] :
( ( ( product_Pair @ A @ B @ A2 @ B2 )
= ( product_Pair @ A @ B @ A3 @ B3 ) )
= ( ( A2 = A3 )
& ( B2 = B3 ) ) ) ).
% old.prod.inject
thf(fact_9_Cons__eq__appendI,axiom,
! [A: $tType,X3: A,Xs1: list @ A,Ys2: list @ A,Xs2: list @ A,Zs: list @ A] :
( ( ( cons @ A @ X3 @ Xs1 )
= Ys2 )
=> ( ( Xs2
= ( append @ A @ Xs1 @ Zs ) )
=> ( ( cons @ A @ X3 @ Xs2 )
= ( append @ A @ Ys2 @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_10_append__Cons,axiom,
! [A: $tType,X3: A,Xs2: list @ A,Ys2: list @ A] :
( ( append @ A @ ( cons @ A @ X3 @ Xs2 ) @ Ys2 )
= ( cons @ A @ X3 @ ( append @ A @ Xs2 @ Ys2 ) ) ) ).
% append_Cons
thf(fact_11_append__eq__appendI,axiom,
! [A: $tType,Xs2: list @ A,Xs1: list @ A,Zs: list @ A,Ys2: list @ A,Us: list @ A] :
( ( ( append @ A @ Xs2 @ Xs1 )
= Zs )
=> ( ( Ys2
= ( append @ A @ Xs1 @ Us ) )
=> ( ( append @ A @ Xs2 @ Ys2 )
= ( append @ A @ Zs @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_12_append__eq__append__conv2,axiom,
! [A: $tType,Xs2: list @ A,Ys2: list @ A,Zs: list @ A,Ts: list @ A] :
( ( ( append @ A @ Xs2 @ Ys2 )
= ( append @ A @ Zs @ Ts ) )
= ( ? [Us2: list @ A] :
( ( ( Xs2
= ( append @ A @ Zs @ Us2 ) )
& ( ( append @ A @ Us2 @ Ys2 )
= Ts ) )
| ( ( ( append @ A @ Xs2 @ Us2 )
= Zs )
& ( Ys2
= ( append @ A @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_13_old_Oprod_Oinducts,axiom,
! [B: $tType,A: $tType,P2: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
( ! [A4: A,B4: B] : ( P2 @ ( product_Pair @ A @ B @ A4 @ B4 ) )
=> ( P2 @ Prod ) ) ).
% old.prod.inducts
thf(fact_14_old_Oprod_Oexhaust,axiom,
! [A: $tType,B: $tType,Y3: product_prod @ A @ B] :
~ ! [A4: A,B4: B] :
( Y3
!= ( product_Pair @ A @ B @ A4 @ B4 ) ) ).
% old.prod.exhaust
thf(fact_15_prod__induct7,axiom,
! [G: $tType,F: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) )] :
( ! [A4: A,B4: B,C3: C,D2: D,E2: E,F2: F,G2: G] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F @ G ) @ E2 @ ( product_Pair @ F @ G @ F2 @ G2 ) ) ) ) ) ) )
=> ( P2 @ X3 ) ) ).
% prod_induct7
thf(fact_16_prod__induct6,axiom,
! [F: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) )] :
( ! [A4: A,B4: B,C3: C,D2: D,E2: E,F2: F] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ F ) @ D2 @ ( product_Pair @ E @ F @ E2 @ F2 ) ) ) ) ) )
=> ( P2 @ X3 ) ) ).
% prod_induct6
thf(fact_17_prod__induct5,axiom,
! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
( ! [A4: A,B4: B,C3: C,D2: D,E2: E] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C3 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) )
=> ( P2 @ X3 ) ) ).
% prod_induct5
thf(fact_18_prod__induct4,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
( ! [A4: A,B4: B,C3: C,D2: D] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B4 @ ( product_Pair @ C @ D @ C3 @ D2 ) ) ) )
=> ( P2 @ X3 ) ) ).
% prod_induct4
thf(fact_19_prod__induct3,axiom,
! [C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ C )] :
( ! [A4: A,B4: B,C3: C] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A4 @ ( product_Pair @ B @ C @ B4 @ C3 ) ) )
=> ( P2 @ X3 ) ) ).
% prod_induct3
thf(fact_20_prod__cases7,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F: $tType,G: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) )] :
~ ! [A4: A,B4: B,C3: C,D2: D,E2: E,F2: F,G2: G] :
( Y3
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F @ G ) @ E2 @ ( product_Pair @ F @ G @ F2 @ G2 ) ) ) ) ) ) ) ).
% prod_cases7
thf(fact_21_prod__cases6,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) )] :
~ ! [A4: A,B4: B,C3: C,D2: D,E2: E,F2: F] :
( Y3
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ F ) @ D2 @ ( product_Pair @ E @ F @ E2 @ F2 ) ) ) ) ) ) ).
% prod_cases6
thf(fact_22_prod__cases5,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
~ ! [A4: A,B4: B,C3: C,D2: D,E2: E] :
( Y3
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C3 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) ) ).
% prod_cases5
thf(fact_23_prod__cases4,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
~ ! [A4: A,B4: B,C3: C,D2: D] :
( Y3
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B4 @ ( product_Pair @ C @ D @ C3 @ D2 ) ) ) ) ).
% prod_cases4
thf(fact_24_prod__cases3,axiom,
! [A: $tType,B: $tType,C: $tType,Y3: product_prod @ A @ ( product_prod @ B @ C )] :
~ ! [A4: A,B4: B,C3: C] :
( Y3
!= ( product_Pair @ A @ ( product_prod @ B @ C ) @ A4 @ ( product_Pair @ B @ C @ B4 @ C3 ) ) ) ).
% prod_cases3
thf(fact_25_Pair__inject,axiom,
! [A: $tType,B: $tType,A2: A,B2: B,A3: A,B3: B] :
( ( ( product_Pair @ A @ B @ A2 @ B2 )
= ( product_Pair @ A @ B @ A3 @ B3 ) )
=> ~ ( ( A2 = A3 )
=> ( B2 != B3 ) ) ) ).
% Pair_inject
thf(fact_26_prod__cases,axiom,
! [B: $tType,A: $tType,P2: ( product_prod @ A @ B ) > $o,P3: product_prod @ A @ B] :
( ! [A4: A,B4: B] : ( P2 @ ( product_Pair @ A @ B @ A4 @ B4 ) )
=> ( P2 @ P3 ) ) ).
% prod_cases
thf(fact_27_surj__pair,axiom,
! [A: $tType,B: $tType,P3: product_prod @ A @ B] :
? [X4: A,Y4: B] :
( P3
= ( product_Pair @ A @ B @ X4 @ Y4 ) ) ).
% surj_pair
thf(fact_28_not__Cons__self2,axiom,
! [A: $tType,X3: A,Xs2: list @ A] :
( ( cons @ A @ X3 @ Xs2 )
!= Xs2 ) ).
% not_Cons_self2
thf(fact_29_old_Oprod_Orec,axiom,
! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A2: A,B2: B] :
( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A2 @ B2 ) )
= ( F1 @ A2 @ B2 ) ) ).
% old.prod.rec
thf(fact_30_bind__simps_I2_J,axiom,
! [A: $tType,B: $tType,X3: B,Xs2: list @ B,F3: B > ( list @ A )] :
( ( bind @ B @ A @ ( cons @ B @ X3 @ Xs2 ) @ F3 )
= ( append @ A @ ( F3 @ X3 ) @ ( bind @ B @ A @ Xs2 @ F3 ) ) ) ).
% bind_simps(2)
thf(fact_31_maps__simps_I1_J,axiom,
! [A: $tType,B: $tType,F3: B > ( list @ A ),X3: B,Xs2: list @ B] :
( ( maps @ B @ A @ F3 @ ( cons @ B @ X3 @ Xs2 ) )
= ( append @ A @ ( F3 @ X3 ) @ ( maps @ B @ A @ F3 @ Xs2 ) ) ) ).
% maps_simps(1)
thf(fact_32_internal__case__prod__conv,axiom,
! [B: $tType,A: $tType,C: $tType,C2: B > C > A,A2: B,B2: C] :
( ( produc2004651681e_prod @ B @ C @ A @ C2 @ ( product_Pair @ B @ C @ A2 @ B2 ) )
= ( C2 @ A2 @ B2 ) ) ).
% internal_case_prod_conv
thf(fact_33_lexord__append__left__rightI,axiom,
! [A: $tType,A2: A,B2: A,R: set @ ( product_prod @ A @ A ),U: list @ A,X3: list @ A,Y3: list @ A] :
( ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ R )
=> ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ U @ ( cons @ A @ A2 @ X3 ) ) @ ( append @ A @ U @ ( cons @ A @ B2 @ Y3 ) ) ) @ ( lexord @ A @ R ) ) ) ).
% lexord_append_left_rightI
thf(fact_34_zip__Cons__Cons,axiom,
! [A: $tType,B: $tType,X3: A,Xs2: list @ A,Y3: B,Ys2: list @ B] :
( ( zip @ A @ B @ ( cons @ A @ X3 @ Xs2 ) @ ( cons @ B @ Y3 @ Ys2 ) )
= ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ ( zip @ A @ B @ Xs2 @ Ys2 ) ) ) ).
% zip_Cons_Cons
thf(fact_35_listrel1I,axiom,
! [A: $tType,X3: A,Y3: A,R: set @ ( product_prod @ A @ A ),Xs2: list @ A,Us: list @ A,Vs: list @ A,Ys2: list @ A] :
( ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R )
=> ( ( Xs2
= ( append @ A @ Us @ ( cons @ A @ X3 @ Vs ) ) )
=> ( ( Ys2
= ( append @ A @ Us @ ( cons @ A @ Y3 @ Vs ) ) )
=> ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys2 ) @ ( listrel1 @ A @ R ) ) ) ) ) ).
% listrel1I
thf(fact_36_listrel1E,axiom,
! [A: $tType,Xs2: list @ A,Ys2: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys2 ) @ ( listrel1 @ A @ R ) )
=> ~ ! [X4: A,Y4: A] :
( ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ R )
=> ! [Us3: list @ A,Vs2: list @ A] :
( ( Xs2
= ( append @ A @ Us3 @ ( cons @ A @ X4 @ Vs2 ) ) )
=> ( Ys2
!= ( append @ A @ Us3 @ ( cons @ A @ Y4 @ Vs2 ) ) ) ) ) ) ).
% listrel1E
thf(fact_37_member__rec_I1_J,axiom,
! [A: $tType,X3: A,Xs2: list @ A,Y3: A] :
( ( member @ A @ ( cons @ A @ X3 @ Xs2 ) @ Y3 )
= ( ( X3 = Y3 )
| ( member @ A @ Xs2 @ Y3 ) ) ) ).
% member_rec(1)
thf(fact_38_ssubst__Pair__rhs,axiom,
! [B: $tType,A: $tType,R: A,S: B,R2: set @ ( product_prod @ A @ B ),S2: B] :
( ( member2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R @ S ) @ R2 )
=> ( ( S2 = S )
=> ( member2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R @ S2 ) @ R2 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_39_subdivide_Oinduct,axiom,
! [A: $tType,P2: ( pair_p1731315293ph_ext @ A @ product_unit ) > ( product_prod @ A @ A ) > A > $o,A0: pair_p1731315293ph_ext @ A @ product_unit,A1: product_prod @ A @ A,A22: A] :
( ! [G3: pair_p1731315293ph_ext @ A @ product_unit,U2: A,V: A,X_1: A] : ( P2 @ G3 @ ( product_Pair @ A @ A @ U2 @ V ) @ X_1 )
=> ( P2 @ A0 @ A1 @ A22 ) ) ).
% subdivide.induct
thf(fact_40_Cons__listrel1__Cons,axiom,
! [A: $tType,X3: A,Xs2: list @ A,Y3: A,Ys2: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 @ Xs2 ) @ ( cons @ A @ Y3 @ Ys2 ) ) @ ( listrel1 @ A @ R ) )
= ( ( ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R )
& ( Xs2 = Ys2 ) )
| ( ( X3 = Y3 )
& ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys2 ) @ ( listrel1 @ A @ R ) ) ) ) ) ).
% Cons_listrel1_Cons
thf(fact_41_lexord__cons__cons,axiom,
! [A: $tType,A2: A,X3: list @ A,B2: A,Y3: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ A2 @ X3 ) @ ( cons @ A @ B2 @ Y3 ) ) @ ( lexord @ A @ R ) )
= ( ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ R )
| ( ( A2 = B2 )
& ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X3 @ Y3 ) @ ( lexord @ A @ R ) ) ) ) ) ).
% lexord_cons_cons
thf(fact_42_listrel1I2,axiom,
! [A: $tType,Xs2: list @ A,Ys2: list @ A,R: set @ ( product_prod @ A @ A ),X3: A] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys2 ) @ ( listrel1 @ A @ R ) )
=> ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 @ Xs2 ) @ ( cons @ A @ X3 @ Ys2 ) ) @ ( listrel1 @ A @ R ) ) ) ).
% listrel1I2
thf(fact_43_append__listrel1I,axiom,
! [A: $tType,Xs2: list @ A,Ys2: list @ A,R: set @ ( product_prod @ A @ A ),Us: list @ A,Vs: list @ A] :
( ( ( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys2 ) @ ( listrel1 @ A @ R ) )
& ( Us = Vs ) )
| ( ( Xs2 = Ys2 )
& ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Us @ Vs ) @ ( listrel1 @ A @ R ) ) ) )
=> ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs2 @ Us ) @ ( append @ A @ Ys2 @ Vs ) ) @ ( listrel1 @ A @ R ) ) ) ).
% append_listrel1I
thf(fact_44_lexord__linear,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),X3: list @ A,Y3: list @ A] :
( ! [A4: A,B4: A] :
( ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B4 ) @ R )
| ( A4 = B4 )
| ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B4 @ A4 ) @ R ) )
=> ( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X3 @ Y3 ) @ ( lexord @ A @ R ) )
| ( X3 = Y3 )
| ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Y3 @ X3 ) @ ( lexord @ A @ R ) ) ) ) ).
% lexord_linear
thf(fact_45_mem__Collect__eq,axiom,
! [A: $tType,A2: A,P2: A > $o] :
( ( member2 @ A @ A2 @ ( collect @ A @ P2 ) )
= ( P2 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: $tType,A5: set @ A] :
( ( collect @ A
@ ^ [X: A] : ( member2 @ A @ X @ A5 ) )
= A5 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [A: $tType,P2: A > $o,Q: A > $o] :
( ! [X4: A] :
( ( P2 @ X4 )
= ( Q @ X4 ) )
=> ( ( collect @ A @ P2 )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_48_lexord__irreflexive,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),Xs2: list @ A] :
( ! [X4: A] :
~ ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ X4 ) @ R )
=> ~ ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Xs2 ) @ ( lexord @ A @ R ) ) ) ).
% lexord_irreflexive
thf(fact_49_lexord__append__leftI,axiom,
! [A: $tType,U: list @ A,V2: list @ A,R: set @ ( product_prod @ A @ A ),X3: list @ A] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ U @ V2 ) @ ( lexord @ A @ R ) )
=> ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ X3 @ U ) @ ( append @ A @ X3 @ V2 ) ) @ ( lexord @ A @ R ) ) ) ).
% lexord_append_leftI
thf(fact_50_listrel1I1,axiom,
! [A: $tType,X3: A,Y3: A,R: set @ ( product_prod @ A @ A ),Xs2: list @ A] :
( ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R )
=> ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 @ Xs2 ) @ ( cons @ A @ Y3 @ Xs2 ) ) @ ( listrel1 @ A @ R ) ) ) ).
% listrel1I1
thf(fact_51_Cons__listrel1E1,axiom,
! [A: $tType,X3: A,Xs2: list @ A,Ys2: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 @ Xs2 ) @ Ys2 ) @ ( listrel1 @ A @ R ) )
=> ( ! [Y4: A] :
( ( Ys2
= ( cons @ A @ Y4 @ Xs2 ) )
=> ~ ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y4 ) @ R ) )
=> ~ ! [Zs2: list @ A] :
( ( Ys2
= ( cons @ A @ X3 @ Zs2 ) )
=> ~ ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Zs2 ) @ ( listrel1 @ A @ R ) ) ) ) ) ).
% Cons_listrel1E1
thf(fact_52_Cons__listrel1E2,axiom,
! [A: $tType,Xs2: list @ A,Y3: A,Ys2: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ ( cons @ A @ Y3 @ Ys2 ) ) @ ( listrel1 @ A @ R ) )
=> ( ! [X4: A] :
( ( Xs2
= ( cons @ A @ X4 @ Ys2 ) )
=> ~ ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y3 ) @ R ) )
=> ~ ! [Zs2: list @ A] :
( ( Xs2
= ( cons @ A @ Y3 @ Zs2 ) )
=> ~ ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Zs2 @ Ys2 ) @ ( listrel1 @ A @ R ) ) ) ) ) ).
% Cons_listrel1E2
thf(fact_53_lexord__append__leftD,axiom,
! [A: $tType,X3: list @ A,U: list @ A,V2: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ X3 @ U ) @ ( append @ A @ X3 @ V2 ) ) @ ( lexord @ A @ R ) )
=> ( ! [A4: A] :
~ ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ A4 ) @ R )
=> ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ U @ V2 ) @ ( lexord @ A @ R ) ) ) ) ).
% lexord_append_leftD
thf(fact_54_lexord__append__rightI,axiom,
! [A: $tType,Y3: list @ A,X3: list @ A,R: set @ ( product_prod @ A @ A )] :
( ? [B5: A,Z: list @ A] :
( Y3
= ( cons @ A @ B5 @ Z ) )
=> ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X3 @ ( append @ A @ X3 @ Y3 ) ) @ ( lexord @ A @ R ) ) ) ).
% lexord_append_rightI
thf(fact_55_zip__eq__ConsE,axiom,
! [A: $tType,B: $tType,Xs2: list @ A,Ys2: list @ B,Xy: product_prod @ A @ B,Xys: list @ ( product_prod @ A @ B )] :
( ( ( zip @ A @ B @ Xs2 @ Ys2 )
= ( cons @ ( product_prod @ A @ B ) @ Xy @ Xys ) )
=> ~ ! [X4: A,Xs3: list @ A] :
( ( Xs2
= ( cons @ A @ X4 @ Xs3 ) )
=> ! [Y4: B,Ys3: list @ B] :
( ( Ys2
= ( cons @ B @ Y4 @ Ys3 ) )
=> ( ( Xy
= ( product_Pair @ A @ B @ X4 @ Y4 ) )
=> ( Xys
!= ( zip @ A @ B @ Xs3 @ Ys3 ) ) ) ) ) ) ).
% zip_eq_ConsE
thf(fact_56_subdivide_Ocases,axiom,
! [A: $tType,X3: product_prod @ ( pair_p1731315293ph_ext @ A @ product_unit ) @ ( product_prod @ ( product_prod @ A @ A ) @ A )] :
~ ! [G3: pair_p1731315293ph_ext @ A @ product_unit,U2: A,V: A,W: A] :
( X3
!= ( product_Pair @ ( pair_p1731315293ph_ext @ A @ product_unit ) @ ( product_prod @ ( product_prod @ A @ A ) @ A ) @ G3 @ ( product_Pair @ ( product_prod @ A @ A ) @ A @ ( product_Pair @ A @ A @ U2 @ V ) @ W ) ) ) ).
% subdivide.cases
thf(fact_57_lexord__same__pref__if__irrefl,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),Xs2: list @ A,Ys2: list @ A,Zs: list @ A] :
( ( irrefl @ A @ R )
=> ( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs2 @ Ys2 ) @ ( append @ A @ Xs2 @ Zs ) ) @ ( lexord @ A @ R ) )
= ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys2 @ Zs ) @ ( lexord @ A @ R ) ) ) ) ).
% lexord_same_pref_if_irrefl
thf(fact_58_snoc__listrel1__snoc__iff,axiom,
! [A: $tType,Xs2: list @ A,X3: A,Ys2: list @ A,Y3: A,R: set @ ( product_prod @ A @ A )] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs2 @ ( cons @ A @ X3 @ ( nil @ A ) ) ) @ ( append @ A @ Ys2 @ ( cons @ A @ Y3 @ ( nil @ A ) ) ) ) @ ( listrel1 @ A @ R ) )
= ( ( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys2 ) @ ( listrel1 @ A @ R ) )
& ( X3 = Y3 ) )
| ( ( Xs2 = Ys2 )
& ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R ) ) ) ) ).
% snoc_listrel1_snoc_iff
thf(fact_59_lexord__Nil__left,axiom,
! [A: $tType,Y3: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Y3 ) @ ( lexord @ A @ R ) )
= ( ? [A6: A,X: list @ A] :
( Y3
= ( cons @ A @ A6 @ X ) ) ) ) ).
% lexord_Nil_left
thf(fact_60_lexord__same__pref__iff,axiom,
! [A: $tType,Xs2: list @ A,Ys2: list @ A,Zs: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs2 @ Ys2 ) @ ( append @ A @ Xs2 @ Zs ) ) @ ( lexord @ A @ R ) )
= ( ? [X: A] :
( ( member2 @ A @ X @ ( set2 @ A @ Xs2 ) )
& ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ X ) @ R ) )
| ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys2 @ Zs ) @ ( lexord @ A @ R ) ) ) ) ).
% lexord_same_pref_iff
thf(fact_61_zip__append,axiom,
! [A: $tType,B: $tType,Xs2: list @ A,Us: list @ B,Ys2: list @ A,Vs: list @ B] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ B ) @ Us ) )
=> ( ( zip @ A @ B @ ( append @ A @ Xs2 @ Ys2 ) @ ( append @ B @ Us @ Vs ) )
= ( append @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs2 @ Us ) @ ( zip @ A @ B @ Ys2 @ Vs ) ) ) ) ).
% zip_append
thf(fact_62_lex__append__leftD,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),Xs2: list @ A,Ys2: list @ A,Zs: list @ A] :
( ! [X4: A] :
~ ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ X4 ) @ R )
=> ( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs2 @ Ys2 ) @ ( append @ A @ Xs2 @ Zs ) ) @ ( lex @ A @ R ) )
=> ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys2 @ Zs ) @ ( lex @ A @ R ) ) ) ) ).
% lex_append_leftD
thf(fact_63_lex__append__left__iff,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),Xs2: list @ A,Ys2: list @ A,Zs: list @ A] :
( ! [X4: A] :
~ ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ X4 ) @ R )
=> ( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs2 @ Ys2 ) @ ( append @ A @ Xs2 @ Zs ) ) @ ( lex @ A @ R ) )
= ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys2 @ Zs ) @ ( lex @ A @ R ) ) ) ) ).
% lex_append_left_iff
thf(fact_64_lexord__asymmetric,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A2: list @ A,B2: list @ A] :
( ( asym @ A @ R2 )
=> ( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ A2 @ B2 ) @ ( lexord @ A @ R2 ) )
=> ~ ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ B2 @ A2 ) @ ( lexord @ A @ R2 ) ) ) ) ).
% lexord_asymmetric
thf(fact_65_append_Oright__neutral,axiom,
! [A: $tType,A2: list @ A] :
( ( append @ A @ A2 @ ( nil @ A ) )
= A2 ) ).
% append.right_neutral
thf(fact_66_append__is__Nil__conv,axiom,
! [A: $tType,Xs2: list @ A,Ys2: list @ A] :
( ( ( append @ A @ Xs2 @ Ys2 )
= ( nil @ A ) )
= ( ( Xs2
= ( nil @ A ) )
& ( Ys2
= ( nil @ A ) ) ) ) ).
% append_is_Nil_conv
thf(fact_67_Nil__is__append__conv,axiom,
! [A: $tType,Xs2: list @ A,Ys2: list @ A] :
( ( ( nil @ A )
= ( append @ A @ Xs2 @ Ys2 ) )
= ( ( Xs2
= ( nil @ A ) )
& ( Ys2
= ( nil @ A ) ) ) ) ).
% Nil_is_append_conv
thf(fact_68_self__append__conv2,axiom,
! [A: $tType,Ys2: list @ A,Xs2: list @ A] :
( ( Ys2
= ( append @ A @ Xs2 @ Ys2 ) )
= ( Xs2
= ( nil @ A ) ) ) ).
% self_append_conv2
thf(fact_69_append__self__conv2,axiom,
! [A: $tType,Xs2: list @ A,Ys2: list @ A] :
( ( ( append @ A @ Xs2 @ Ys2 )
= Ys2 )
= ( Xs2
= ( nil @ A ) ) ) ).
% append_self_conv2
thf(fact_70_self__append__conv,axiom,
! [A: $tType,Xs2: list @ A,Ys2: list @ A] :
( ( Xs2
= ( append @ A @ Xs2 @ Ys2 ) )
= ( Ys2
= ( nil @ A ) ) ) ).
% self_append_conv
thf(fact_71_append__self__conv,axiom,
! [A: $tType,Xs2: list @ A,Ys2: list @ A] :
( ( ( append @ A @ Xs2 @ Ys2 )
= Xs2 )
= ( Ys2
= ( nil @ A ) ) ) ).
% append_self_conv
thf(fact_72_append__Nil2,axiom,
! [A: $tType,Xs2: list @ A] :
( ( append @ A @ Xs2 @ ( nil @ A ) )
= Xs2 ) ).
% append_Nil2
thf(fact_73_append__eq__append__conv,axiom,
! [A: $tType,Xs2: list @ A,Ys2: list @ A,Us: list @ A,Vs: list @ A] :
( ( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ A ) @ Ys2 ) )
| ( ( size_size @ ( list @ A ) @ Us )
= ( size_size @ ( list @ A ) @ Vs ) ) )
=> ( ( ( append @ A @ Xs2 @ Us )
= ( append @ A @ Ys2 @ Vs ) )
= ( ( Xs2 = Ys2 )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_74_zip__Nil,axiom,
! [B: $tType,A: $tType,Ys2: list @ B] :
( ( zip @ A @ B @ ( nil @ A ) @ Ys2 )
= ( nil @ ( product_prod @ A @ B ) ) ) ).
% zip_Nil
thf(fact_75_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_76_append1__eq__conv,axiom,
! [A: $tType,Xs2: list @ A,X3: A,Ys2: list @ A,Y3: A] :
( ( ( append @ A @ Xs2 @ ( cons @ A @ X3 @ ( nil @ A ) ) )
= ( append @ A @ Ys2 @ ( cons @ A @ Y3 @ ( nil @ A ) ) ) )
= ( ( Xs2 = Ys2 )
& ( X3 = Y3 ) ) ) ).
% append1_eq_conv
thf(fact_77_Cons__in__lex,axiom,
! [A: $tType,X3: A,Xs2: list @ A,Y3: A,Ys2: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 @ Xs2 ) @ ( cons @ A @ Y3 @ Ys2 ) ) @ ( lex @ A @ R ) )
= ( ( ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R )
& ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ A ) @ Ys2 ) ) )
| ( ( X3 = Y3 )
& ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys2 ) @ ( lex @ A @ R ) ) ) ) ) ).
% Cons_in_lex
thf(fact_78_zip_Osimps_I1_J,axiom,
! [B: $tType,A: $tType,Xs2: list @ A] :
( ( zip @ A @ B @ Xs2 @ ( nil @ B ) )
= ( nil @ ( product_prod @ A @ B ) ) ) ).
% zip.simps(1)
thf(fact_79_Nil__notin__lex,axiom,
! [A: $tType,Ys2: list @ A,R: set @ ( product_prod @ A @ A )] :
~ ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys2 ) @ ( lex @ A @ R ) ) ).
% Nil_notin_lex
thf(fact_80_Nil2__notin__lex,axiom,
! [A: $tType,Xs2: list @ A,R: set @ ( product_prod @ A @ A )] :
~ ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ ( nil @ A ) ) @ ( lex @ A @ R ) ) ).
% Nil2_notin_lex
thf(fact_81_zip__eq__Nil__iff,axiom,
! [A: $tType,B: $tType,Xs2: list @ A,Ys2: list @ B] :
( ( ( zip @ A @ B @ Xs2 @ Ys2 )
= ( nil @ ( product_prod @ A @ B ) ) )
= ( ( Xs2
= ( nil @ A ) )
| ( Ys2
= ( nil @ B ) ) ) ) ).
% zip_eq_Nil_iff
thf(fact_82_in__set__impl__in__set__zip1,axiom,
! [A: $tType,B: $tType,Xs2: list @ A,Ys2: list @ B,X3: A] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ B ) @ Ys2 ) )
=> ( ( member2 @ A @ X3 @ ( set2 @ A @ Xs2 ) )
=> ~ ! [Y4: B] :
~ ( member2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y4 ) @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs2 @ Ys2 ) ) ) ) ) ).
% in_set_impl_in_set_zip1
thf(fact_83_in__set__impl__in__set__zip2,axiom,
! [A: $tType,B: $tType,Xs2: list @ A,Ys2: list @ B,Y3: B] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ B ) @ Ys2 ) )
=> ( ( member2 @ B @ Y3 @ ( set2 @ B @ Ys2 ) )
=> ~ ! [X4: A] :
~ ( member2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y3 ) @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs2 @ Ys2 ) ) ) ) ) ).
% in_set_impl_in_set_zip2
thf(fact_84_list__induct4,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,Xs2: list @ A,Ys2: list @ B,Zs: list @ C,Ws: list @ D,P2: ( list @ A ) > ( list @ B ) > ( list @ C ) > ( list @ D ) > $o] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ B ) @ Ys2 ) )
=> ( ( ( size_size @ ( list @ B ) @ Ys2 )
= ( size_size @ ( list @ C ) @ Zs ) )
=> ( ( ( size_size @ ( list @ C ) @ Zs )
= ( size_size @ ( list @ D ) @ Ws ) )
=> ( ( P2 @ ( nil @ A ) @ ( nil @ B ) @ ( nil @ C ) @ ( nil @ D ) )
=> ( ! [X4: A,Xs4: list @ A,Y4: B,Ys4: list @ B,Z2: C,Zs2: list @ C,W: D,Ws2: list @ D] :
( ( ( size_size @ ( list @ A ) @ Xs4 )
= ( size_size @ ( list @ B ) @ Ys4 ) )
=> ( ( ( size_size @ ( list @ B ) @ Ys4 )
= ( size_size @ ( list @ C ) @ Zs2 ) )
=> ( ( ( size_size @ ( list @ C ) @ Zs2 )
= ( size_size @ ( list @ D ) @ Ws2 ) )
=> ( ( P2 @ Xs4 @ Ys4 @ Zs2 @ Ws2 )
=> ( P2 @ ( cons @ A @ X4 @ Xs4 ) @ ( cons @ B @ Y4 @ Ys4 ) @ ( cons @ C @ Z2 @ Zs2 ) @ ( cons @ D @ W @ Ws2 ) ) ) ) ) )
=> ( P2 @ Xs2 @ Ys2 @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_85_list__induct3,axiom,
! [B: $tType,A: $tType,C: $tType,Xs2: list @ A,Ys2: list @ B,Zs: list @ C,P2: ( list @ A ) > ( list @ B ) > ( list @ C ) > $o] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ B ) @ Ys2 ) )
=> ( ( ( size_size @ ( list @ B ) @ Ys2 )
= ( size_size @ ( list @ C ) @ Zs ) )
=> ( ( P2 @ ( nil @ A ) @ ( nil @ B ) @ ( nil @ C ) )
=> ( ! [X4: A,Xs4: list @ A,Y4: B,Ys4: list @ B,Z2: C,Zs2: list @ C] :
( ( ( size_size @ ( list @ A ) @ Xs4 )
= ( size_size @ ( list @ B ) @ Ys4 ) )
=> ( ( ( size_size @ ( list @ B ) @ Ys4 )
= ( size_size @ ( list @ C ) @ Zs2 ) )
=> ( ( P2 @ Xs4 @ Ys4 @ Zs2 )
=> ( P2 @ ( cons @ A @ X4 @ Xs4 ) @ ( cons @ B @ Y4 @ Ys4 ) @ ( cons @ C @ Z2 @ Zs2 ) ) ) ) )
=> ( P2 @ Xs2 @ Ys2 @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_86_list__induct2,axiom,
! [A: $tType,B: $tType,Xs2: list @ A,Ys2: list @ B,P2: ( list @ A ) > ( list @ B ) > $o] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ B ) @ Ys2 ) )
=> ( ( P2 @ ( nil @ A ) @ ( nil @ B ) )
=> ( ! [X4: A,Xs4: list @ A,Y4: B,Ys4: list @ B] :
( ( ( size_size @ ( list @ A ) @ Xs4 )
= ( size_size @ ( list @ B ) @ Ys4 ) )
=> ( ( P2 @ Xs4 @ Ys4 )
=> ( P2 @ ( cons @ A @ X4 @ Xs4 ) @ ( cons @ B @ Y4 @ Ys4 ) ) ) )
=> ( P2 @ Xs2 @ Ys2 ) ) ) ) ).
% list_induct2
thf(fact_87_transpose_Ocases,axiom,
! [A: $tType,X3: list @ ( list @ A )] :
( ( X3
!= ( nil @ ( list @ A ) ) )
=> ( ! [Xss: list @ ( list @ A )] :
( X3
!= ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss ) )
=> ~ ! [X4: A,Xs4: list @ A,Xss: list @ ( list @ A )] :
( X3
!= ( cons @ ( list @ A ) @ ( cons @ A @ X4 @ Xs4 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_88_neq__if__length__neq,axiom,
! [A: $tType,Xs2: list @ A,Ys2: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
!= ( size_size @ ( list @ A ) @ Ys2 ) )
=> ( Xs2 != Ys2 ) ) ).
% neq_if_length_neq
thf(fact_89_Ex__list__of__length,axiom,
! [A: $tType,N: nat] :
? [Xs4: list @ A] :
( ( size_size @ ( list @ A ) @ Xs4 )
= N ) ).
% Ex_list_of_length
thf(fact_90_irrefl__lex,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( irrefl @ A @ R )
=> ( irrefl @ ( list @ A ) @ ( lex @ A @ R ) ) ) ).
% irrefl_lex
thf(fact_91_asym__lex,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( asym @ A @ R2 )
=> ( asym @ ( list @ A ) @ ( lex @ A @ R2 ) ) ) ).
% asym_lex
thf(fact_92_lexl__not__refl,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),X3: list @ A] :
( ( irrefl @ A @ R )
=> ~ ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X3 @ X3 ) @ ( lex @ A @ R ) ) ) ).
% lexl_not_refl
thf(fact_93_lexord__irrefl,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( irrefl @ A @ R2 )
=> ( irrefl @ ( list @ A ) @ ( lexord @ A @ R2 ) ) ) ).
% lexord_irrefl
thf(fact_94_lexord__asym,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( asym @ A @ R2 )
=> ( asym @ ( list @ A ) @ ( lexord @ A @ R2 ) ) ) ).
% lexord_asym
thf(fact_95_set__zip__rightD,axiom,
! [A: $tType,B: $tType,X3: A,Y3: B,Xs2: list @ A,Ys2: list @ B] :
( ( member2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs2 @ Ys2 ) ) )
=> ( member2 @ B @ Y3 @ ( set2 @ B @ Ys2 ) ) ) ).
% set_zip_rightD
thf(fact_96_set__zip__leftD,axiom,
! [B: $tType,A: $tType,X3: A,Y3: B,Xs2: list @ A,Ys2: list @ B] :
( ( member2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs2 @ Ys2 ) ) )
=> ( member2 @ A @ X3 @ ( set2 @ A @ Xs2 ) ) ) ).
% set_zip_leftD
thf(fact_97_in__set__zipE,axiom,
! [A: $tType,B: $tType,X3: A,Y3: B,Xs2: list @ A,Ys2: list @ B] :
( ( member2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs2 @ Ys2 ) ) )
=> ~ ( ( member2 @ A @ X3 @ ( set2 @ A @ Xs2 ) )
=> ~ ( member2 @ B @ Y3 @ ( set2 @ B @ Ys2 ) ) ) ) ).
% in_set_zipE
thf(fact_98_zip__same,axiom,
! [A: $tType,A2: A,B2: A,Xs2: list @ A] :
( ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ ( set2 @ ( product_prod @ A @ A ) @ ( zip @ A @ A @ Xs2 @ Xs2 ) ) )
= ( ( member2 @ A @ A2 @ ( set2 @ A @ Xs2 ) )
& ( A2 = B2 ) ) ) ).
% zip_same
thf(fact_99_same__length__different,axiom,
! [A: $tType,Xs2: list @ A,Ys2: list @ A] :
( ( Xs2 != Ys2 )
=> ( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ A ) @ Ys2 ) )
=> ? [Pre: list @ A,X4: A,Xs3: list @ A,Y4: A,Ys3: list @ A] :
( ( X4 != Y4 )
& ( Xs2
= ( append @ A @ Pre @ ( append @ A @ ( cons @ A @ X4 @ ( nil @ A ) ) @ Xs3 ) ) )
& ( Ys2
= ( append @ A @ Pre @ ( append @ A @ ( cons @ A @ Y4 @ ( nil @ A ) ) @ Ys3 ) ) ) ) ) ) ).
% same_length_different
thf(fact_100_list_Oset__cases,axiom,
! [A: $tType,E3: A,A2: list @ A] :
( ( member2 @ A @ E3 @ ( set2 @ A @ A2 ) )
=> ( ! [Z22: list @ A] :
( A2
!= ( cons @ A @ E3 @ Z22 ) )
=> ~ ! [Z1: A,Z22: list @ A] :
( ( A2
= ( cons @ A @ Z1 @ Z22 ) )
=> ~ ( member2 @ A @ E3 @ ( set2 @ A @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_101_set__ConsD,axiom,
! [A: $tType,Y3: A,X3: A,Xs2: list @ A] :
( ( member2 @ A @ Y3 @ ( set2 @ A @ ( cons @ A @ X3 @ Xs2 ) ) )
=> ( ( Y3 = X3 )
| ( member2 @ A @ Y3 @ ( set2 @ A @ Xs2 ) ) ) ) ).
% set_ConsD
thf(fact_102_list_Oset__intros_I1_J,axiom,
! [A: $tType,X21: A,X22: list @ A] : ( member2 @ A @ X21 @ ( set2 @ A @ ( cons @ A @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_103_list_Oset__intros_I2_J,axiom,
! [A: $tType,Y3: A,X22: list @ A,X21: A] :
( ( member2 @ A @ Y3 @ ( set2 @ A @ X22 ) )
=> ( member2 @ A @ Y3 @ ( set2 @ A @ ( cons @ A @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_104_lex__append__rightI,axiom,
! [A: $tType,Xs2: list @ A,Ys2: list @ A,R: set @ ( product_prod @ A @ A ),Vs: list @ A,Us: list @ A] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys2 ) @ ( lex @ A @ R ) )
=> ( ( ( size_size @ ( list @ A ) @ Vs )
= ( size_size @ ( list @ A ) @ Us ) )
=> ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs2 @ Us ) @ ( append @ A @ Ys2 @ Vs ) ) @ ( lex @ A @ R ) ) ) ) ).
% lex_append_rightI
thf(fact_105_sorted__wrt_Ocases,axiom,
! [A: $tType,X3: product_prod @ ( A > A > $o ) @ ( list @ A )] :
( ! [P4: A > A > $o] :
( X3
!= ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P4 @ ( nil @ A ) ) )
=> ~ ! [P4: A > A > $o,X4: A,Ys4: list @ A] :
( X3
!= ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P4 @ ( cons @ A @ X4 @ Ys4 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_106_arg__min__list_Ocases,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [X3: product_prod @ ( A > B ) @ ( list @ A )] :
( ! [F2: A > B,X4: A] :
( X3
!= ( product_Pair @ ( A > B ) @ ( list @ A ) @ F2 @ ( cons @ A @ X4 @ ( nil @ A ) ) ) )
=> ( ! [F2: A > B,X4: A,Y4: A,Zs2: list @ A] :
( X3
!= ( product_Pair @ ( A > B ) @ ( list @ A ) @ F2 @ ( cons @ A @ X4 @ ( cons @ A @ Y4 @ Zs2 ) ) ) )
=> ~ ! [A4: A > B] :
( X3
!= ( product_Pair @ ( A > B ) @ ( list @ A ) @ A4 @ ( nil @ A ) ) ) ) ) ) ).
% arg_min_list.cases
thf(fact_107_successively_Ocases,axiom,
! [A: $tType,X3: product_prod @ ( A > A > $o ) @ ( list @ A )] :
( ! [P4: A > A > $o] :
( X3
!= ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P4 @ ( nil @ A ) ) )
=> ( ! [P4: A > A > $o,X4: A] :
( X3
!= ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P4 @ ( cons @ A @ X4 @ ( nil @ A ) ) ) )
=> ~ ! [P4: A > A > $o,X4: A,Y4: A,Xs4: list @ A] :
( X3
!= ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P4 @ ( cons @ A @ X4 @ ( cons @ A @ Y4 @ Xs4 ) ) ) ) ) ) ).
% successively.cases
thf(fact_108_strict__sorted_Oinduct,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P2: ( list @ A ) > $o,A0: list @ A] :
( ( P2 @ ( nil @ A ) )
=> ( ! [X4: A,Ys4: list @ A] :
( ( P2 @ Ys4 )
=> ( P2 @ ( cons @ A @ X4 @ Ys4 ) ) )
=> ( P2 @ A0 ) ) ) ) ).
% strict_sorted.induct
thf(fact_109_strict__sorted_Ocases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: list @ A] :
( ( X3
!= ( nil @ A ) )
=> ~ ! [X4: A,Ys4: list @ A] :
( X3
!= ( cons @ A @ X4 @ Ys4 ) ) ) ) ).
% strict_sorted.cases
thf(fact_110_map__tailrec__rev_Oinduct,axiom,
! [A: $tType,B: $tType,P2: ( A > B ) > ( list @ A ) > ( list @ B ) > $o,A0: A > B,A1: list @ A,A22: list @ B] :
( ! [F2: A > B,X_1: list @ B] : ( P2 @ F2 @ ( nil @ A ) @ X_1 )
=> ( ! [F2: A > B,A4: A,As: list @ A,Bs: list @ B] :
( ( P2 @ F2 @ As @ ( cons @ B @ ( F2 @ A4 ) @ Bs ) )
=> ( P2 @ F2 @ ( cons @ A @ A4 @ As ) @ Bs ) )
=> ( P2 @ A0 @ A1 @ A22 ) ) ) ).
% map_tailrec_rev.induct
thf(fact_111_list__nonempty__induct,axiom,
! [A: $tType,Xs2: list @ A,P2: ( list @ A ) > $o] :
( ( Xs2
!= ( nil @ A ) )
=> ( ! [X4: A] : ( P2 @ ( cons @ A @ X4 @ ( nil @ A ) ) )
=> ( ! [X4: A,Xs4: list @ A] :
( ( Xs4
!= ( nil @ A ) )
=> ( ( P2 @ Xs4 )
=> ( P2 @ ( cons @ A @ X4 @ Xs4 ) ) ) )
=> ( P2 @ Xs2 ) ) ) ) ).
% list_nonempty_induct
thf(fact_112_successively_Oinduct,axiom,
! [A: $tType,P2: ( A > A > $o ) > ( list @ A ) > $o,A0: A > A > $o,A1: list @ A] :
( ! [P4: A > A > $o] : ( P2 @ P4 @ ( nil @ A ) )
=> ( ! [P4: A > A > $o,X4: A] : ( P2 @ P4 @ ( cons @ A @ X4 @ ( nil @ A ) ) )
=> ( ! [P4: A > A > $o,X4: A,Y4: A,Xs4: list @ A] :
( ( P2 @ P4 @ ( cons @ A @ Y4 @ Xs4 ) )
=> ( P2 @ P4 @ ( cons @ A @ X4 @ ( cons @ A @ Y4 @ Xs4 ) ) ) )
=> ( P2 @ A0 @ A1 ) ) ) ) ).
% successively.induct
thf(fact_113_arg__min__list_Oinduct,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [P2: ( A > B ) > ( list @ A ) > $o,A0: A > B,A1: list @ A] :
( ! [F2: A > B,X4: A] : ( P2 @ F2 @ ( cons @ A @ X4 @ ( nil @ A ) ) )
=> ( ! [F2: A > B,X4: A,Y4: A,Zs2: list @ A] :
( ( P2 @ F2 @ ( cons @ A @ Y4 @ Zs2 ) )
=> ( P2 @ F2 @ ( cons @ A @ X4 @ ( cons @ A @ Y4 @ Zs2 ) ) ) )
=> ( ! [A4: A > B] : ( P2 @ A4 @ ( nil @ A ) )
=> ( P2 @ A0 @ A1 ) ) ) ) ) ).
% arg_min_list.induct
thf(fact_114_remdups__adj_Oinduct,axiom,
! [A: $tType,P2: ( list @ A ) > $o,A0: list @ A] :
( ( P2 @ ( nil @ A ) )
=> ( ! [X4: A] : ( P2 @ ( cons @ A @ X4 @ ( nil @ A ) ) )
=> ( ! [X4: A,Y4: A,Xs4: list @ A] :
( ( ( X4 = Y4 )
=> ( P2 @ ( cons @ A @ X4 @ Xs4 ) ) )
=> ( ( ( X4 != Y4 )
=> ( P2 @ ( cons @ A @ Y4 @ Xs4 ) ) )
=> ( P2 @ ( cons @ A @ X4 @ ( cons @ A @ Y4 @ Xs4 ) ) ) ) )
=> ( P2 @ A0 ) ) ) ) ).
% remdups_adj.induct
thf(fact_115_sorted__wrt_Oinduct,axiom,
! [A: $tType,P2: ( A > A > $o ) > ( list @ A ) > $o,A0: A > A > $o,A1: list @ A] :
( ! [P4: A > A > $o] : ( P2 @ P4 @ ( nil @ A ) )
=> ( ! [P4: A > A > $o,X4: A,Ys4: list @ A] :
( ( P2 @ P4 @ Ys4 )
=> ( P2 @ P4 @ ( cons @ A @ X4 @ Ys4 ) ) )
=> ( P2 @ A0 @ A1 ) ) ) ).
% sorted_wrt.induct
thf(fact_116_remdups__adj_Ocases,axiom,
! [A: $tType,X3: list @ A] :
( ( X3
!= ( nil @ A ) )
=> ( ! [X4: A] :
( X3
!= ( cons @ A @ X4 @ ( nil @ A ) ) )
=> ~ ! [X4: A,Y4: A,Xs4: list @ A] :
( X3
!= ( cons @ A @ X4 @ ( cons @ A @ Y4 @ Xs4 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_117_shuffles_Oinduct,axiom,
! [A: $tType,P2: ( list @ A ) > ( list @ A ) > $o,A0: list @ A,A1: list @ A] :
( ! [X_1: list @ A] : ( P2 @ ( nil @ A ) @ X_1 )
=> ( ! [Xs4: list @ A] : ( P2 @ Xs4 @ ( nil @ A ) )
=> ( ! [X4: A,Xs4: list @ A,Y4: A,Ys4: list @ A] :
( ( P2 @ Xs4 @ ( cons @ A @ Y4 @ Ys4 ) )
=> ( ( P2 @ ( cons @ A @ X4 @ Xs4 ) @ Ys4 )
=> ( P2 @ ( cons @ A @ X4 @ Xs4 ) @ ( cons @ A @ Y4 @ Ys4 ) ) ) )
=> ( P2 @ A0 @ A1 ) ) ) ) ).
% shuffles.induct
thf(fact_118_min__list_Oinduct,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [P2: ( list @ A ) > $o,A0: list @ A] :
( ! [X4: A,Xs4: list @ A] :
( ! [X212: A,X222: list @ A] :
( ( Xs4
= ( cons @ A @ X212 @ X222 ) )
=> ( P2 @ Xs4 ) )
=> ( P2 @ ( cons @ A @ X4 @ Xs4 ) ) )
=> ( ( P2 @ ( nil @ A ) )
=> ( P2 @ A0 ) ) ) ) ).
% min_list.induct
thf(fact_119_min__list_Ocases,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X3: list @ A] :
( ! [X4: A,Xs4: list @ A] :
( X3
!= ( cons @ A @ X4 @ Xs4 ) )
=> ( X3
= ( nil @ A ) ) ) ) ).
% min_list.cases
thf(fact_120_induct__list012,axiom,
! [A: $tType,P2: ( list @ A ) > $o,Xs2: list @ A] :
( ( P2 @ ( nil @ A ) )
=> ( ! [X4: A] : ( P2 @ ( cons @ A @ X4 @ ( nil @ A ) ) )
=> ( ! [X4: A,Y4: A,Zs2: list @ A] :
( ( P2 @ Zs2 )
=> ( ( P2 @ ( cons @ A @ Y4 @ Zs2 ) )
=> ( P2 @ ( cons @ A @ X4 @ ( cons @ A @ Y4 @ Zs2 ) ) ) ) )
=> ( P2 @ Xs2 ) ) ) ) ).
% induct_list012
thf(fact_121_splice_Oinduct,axiom,
! [A: $tType,P2: ( list @ A ) > ( list @ A ) > $o,A0: list @ A,A1: list @ A] :
( ! [X_1: list @ A] : ( P2 @ ( nil @ A ) @ X_1 )
=> ( ! [X4: A,Xs4: list @ A,Ys4: list @ A] :
( ( P2 @ Ys4 @ Xs4 )
=> ( P2 @ ( cons @ A @ X4 @ Xs4 ) @ Ys4 ) )
=> ( P2 @ A0 @ A1 ) ) ) ).
% splice.induct
thf(fact_122_list__induct2_H,axiom,
! [A: $tType,B: $tType,P2: ( list @ A ) > ( list @ B ) > $o,Xs2: list @ A,Ys2: list @ B] :
( ( P2 @ ( nil @ A ) @ ( nil @ B ) )
=> ( ! [X4: A,Xs4: list @ A] : ( P2 @ ( cons @ A @ X4 @ Xs4 ) @ ( nil @ B ) )
=> ( ! [Y4: B,Ys4: list @ B] : ( P2 @ ( nil @ A ) @ ( cons @ B @ Y4 @ Ys4 ) )
=> ( ! [X4: A,Xs4: list @ A,Y4: B,Ys4: list @ B] :
( ( P2 @ Xs4 @ Ys4 )
=> ( P2 @ ( cons @ A @ X4 @ Xs4 ) @ ( cons @ B @ Y4 @ Ys4 ) ) )
=> ( P2 @ Xs2 @ Ys2 ) ) ) ) ) ).
% list_induct2'
thf(fact_123_neq__Nil__conv,axiom,
! [A: $tType,Xs2: list @ A] :
( ( Xs2
!= ( nil @ A ) )
= ( ? [Y: A,Ys: list @ A] :
( Xs2
= ( cons @ A @ Y @ Ys ) ) ) ) ).
% neq_Nil_conv
thf(fact_124_list_Oinducts,axiom,
! [A: $tType,P2: ( list @ A ) > $o,List: list @ A] :
( ( P2 @ ( nil @ A ) )
=> ( ! [X12: A,X23: list @ A] :
( ( P2 @ X23 )
=> ( P2 @ ( cons @ A @ X12 @ X23 ) ) )
=> ( P2 @ List ) ) ) ).
% list.inducts
thf(fact_125_list_Oexhaust,axiom,
! [A: $tType,Y3: list @ A] :
( ( Y3
!= ( nil @ A ) )
=> ~ ! [X213: A,X223: list @ A] :
( Y3
!= ( cons @ A @ X213 @ X223 ) ) ) ).
% list.exhaust
thf(fact_126_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_127_list_Odistinct_I1_J,axiom,
! [A: $tType,X21: A,X22: list @ A] :
( ( nil @ A )
!= ( cons @ A @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_128_append_Oleft__neutral,axiom,
! [A: $tType,A2: list @ A] :
( ( append @ A @ ( nil @ A ) @ A2 )
= A2 ) ).
% append.left_neutral
thf(fact_129_append__Nil,axiom,
! [A: $tType,Ys2: list @ A] :
( ( append @ A @ ( nil @ A ) @ Ys2 )
= Ys2 ) ).
% append_Nil
thf(fact_130_eq__Nil__appendI,axiom,
! [A: $tType,Xs2: list @ A,Ys2: list @ A] :
( ( Xs2 = Ys2 )
=> ( Xs2
= ( append @ A @ ( nil @ A ) @ Ys2 ) ) ) ).
% eq_Nil_appendI
thf(fact_131_lexord__lex,axiom,
! [A: $tType,X3: list @ A,Y3: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X3 @ Y3 ) @ ( lex @ A @ R ) )
= ( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X3 @ Y3 ) @ ( lexord @ A @ R ) )
& ( ( size_size @ ( list @ A ) @ X3 )
= ( size_size @ ( list @ A ) @ Y3 ) ) ) ) ).
% lexord_lex
thf(fact_132_pcas_Ocases,axiom,
! [A: $tType,X3: product_prod @ A @ ( product_prod @ ( list @ ( product_prod @ A @ A ) ) @ A )] :
( ! [U2: A,V: A] :
( X3
!= ( product_Pair @ A @ ( product_prod @ ( list @ ( product_prod @ A @ A ) ) @ A ) @ U2 @ ( product_Pair @ ( list @ ( product_prod @ A @ A ) ) @ A @ ( nil @ ( product_prod @ A @ A ) ) @ V ) ) )
=> ~ ! [U2: A,E2: product_prod @ A @ A,Es: list @ ( product_prod @ A @ A ),V: A] :
( X3
!= ( product_Pair @ A @ ( product_prod @ ( list @ ( product_prod @ A @ A ) ) @ A ) @ U2 @ ( product_Pair @ ( list @ ( product_prod @ A @ A ) ) @ A @ ( cons @ ( product_prod @ A @ A ) @ E2 @ Es ) @ V ) ) ) ) ).
% pcas.cases
thf(fact_133_list__bind__cong,axiom,
! [B: $tType,A: $tType,Xs2: list @ A,Ys2: list @ A,F3: A > ( list @ B ),G4: A > ( list @ B )] :
( ( Xs2 = Ys2 )
=> ( ! [X4: A] :
( ( member2 @ A @ X4 @ ( set2 @ A @ Xs2 ) )
=> ( ( F3 @ X4 )
= ( G4 @ X4 ) ) )
=> ( ( bind @ A @ B @ Xs2 @ F3 )
= ( bind @ A @ B @ Ys2 @ G4 ) ) ) ) ).
% list_bind_cong
thf(fact_134_in__set__member,axiom,
! [A: $tType,X3: A,Xs2: list @ A] :
( ( member2 @ A @ X3 @ ( set2 @ A @ Xs2 ) )
= ( member @ A @ Xs2 @ X3 ) ) ).
% in_set_member
thf(fact_135_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_136_member__rec_I2_J,axiom,
! [A: $tType,Y3: A] :
~ ( member @ A @ ( nil @ A ) @ Y3 ) ).
% member_rec(2)
thf(fact_137_progressing__Nil,axiom,
! [A: $tType] : ( kurato990191779essing @ A @ ( nil @ ( product_prod @ A @ A ) ) ) ).
% progressing_Nil
thf(fact_138_split__list__first__prop__iff,axiom,
! [A: $tType,Xs2: list @ A,P2: A > $o] :
( ( ? [X: A] :
( ( member2 @ A @ X @ ( set2 @ A @ Xs2 ) )
& ( P2 @ X ) ) )
= ( ? [Ys: list @ A,X: A] :
( ? [Zs3: list @ A] :
( Xs2
= ( append @ A @ Ys @ ( cons @ A @ X @ Zs3 ) ) )
& ( P2 @ X )
& ! [Y: A] :
( ( member2 @ A @ Y @ ( set2 @ A @ Ys ) )
=> ~ ( P2 @ Y ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_139_split__list__last__prop__iff,axiom,
! [A: $tType,Xs2: list @ A,P2: A > $o] :
( ( ? [X: A] :
( ( member2 @ A @ X @ ( set2 @ A @ Xs2 ) )
& ( P2 @ X ) ) )
= ( ? [Ys: list @ A,X: A,Zs3: list @ A] :
( ( Xs2
= ( append @ A @ Ys @ ( cons @ A @ X @ Zs3 ) ) )
& ( P2 @ X )
& ! [Y: A] :
( ( member2 @ A @ Y @ ( set2 @ A @ Zs3 ) )
=> ~ ( P2 @ Y ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_140_in__set__conv__decomp__first,axiom,
! [A: $tType,X3: A,Xs2: list @ A] :
( ( member2 @ A @ X3 @ ( set2 @ A @ Xs2 ) )
= ( ? [Ys: list @ A,Zs3: list @ A] :
( ( Xs2
= ( append @ A @ Ys @ ( cons @ A @ X3 @ Zs3 ) ) )
& ~ ( member2 @ A @ X3 @ ( set2 @ A @ Ys ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_141_in__set__conv__decomp__last,axiom,
! [A: $tType,X3: A,Xs2: list @ A] :
( ( member2 @ A @ X3 @ ( set2 @ A @ Xs2 ) )
= ( ? [Ys: list @ A,Zs3: list @ A] :
( ( Xs2
= ( append @ A @ Ys @ ( cons @ A @ X3 @ Zs3 ) ) )
& ~ ( member2 @ A @ X3 @ ( set2 @ A @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_142_split__list__first__propE,axiom,
! [A: $tType,Xs2: list @ A,P2: A > $o] :
( ? [X5: A] :
( ( member2 @ A @ X5 @ ( set2 @ A @ Xs2 ) )
& ( P2 @ X5 ) )
=> ~ ! [Ys4: list @ A,X4: A] :
( ? [Zs2: list @ A] :
( Xs2
= ( append @ A @ Ys4 @ ( cons @ A @ X4 @ Zs2 ) ) )
=> ( ( P2 @ X4 )
=> ~ ! [Xa: A] :
( ( member2 @ A @ Xa @ ( set2 @ A @ Ys4 ) )
=> ~ ( P2 @ Xa ) ) ) ) ) ).
% split_list_first_propE
thf(fact_143_split__list__last__propE,axiom,
! [A: $tType,Xs2: list @ A,P2: A > $o] :
( ? [X5: A] :
( ( member2 @ A @ X5 @ ( set2 @ A @ Xs2 ) )
& ( P2 @ X5 ) )
=> ~ ! [Ys4: list @ A,X4: A,Zs2: list @ A] :
( ( Xs2
= ( append @ A @ Ys4 @ ( cons @ A @ X4 @ Zs2 ) ) )
=> ( ( P2 @ X4 )
=> ~ ! [Xa: A] :
( ( member2 @ A @ Xa @ ( set2 @ A @ Zs2 ) )
=> ~ ( P2 @ Xa ) ) ) ) ) ).
% split_list_last_propE
thf(fact_144_split__list__first__prop,axiom,
! [A: $tType,Xs2: list @ A,P2: A > $o] :
( ? [X5: A] :
( ( member2 @ A @ X5 @ ( set2 @ A @ Xs2 ) )
& ( P2 @ X5 ) )
=> ? [Ys4: list @ A,X4: A] :
( ? [Zs2: list @ A] :
( Xs2
= ( append @ A @ Ys4 @ ( cons @ A @ X4 @ Zs2 ) ) )
& ( P2 @ X4 )
& ! [Xa: A] :
( ( member2 @ A @ Xa @ ( set2 @ A @ Ys4 ) )
=> ~ ( P2 @ Xa ) ) ) ) ).
% split_list_first_prop
thf(fact_145_split__list__last__prop,axiom,
! [A: $tType,Xs2: list @ A,P2: A > $o] :
( ? [X5: A] :
( ( member2 @ A @ X5 @ ( set2 @ A @ Xs2 ) )
& ( P2 @ X5 ) )
=> ? [Ys4: list @ A,X4: A,Zs2: list @ A] :
( ( Xs2
= ( append @ A @ Ys4 @ ( cons @ A @ X4 @ Zs2 ) ) )
& ( P2 @ X4 )
& ! [Xa: A] :
( ( member2 @ A @ Xa @ ( set2 @ A @ Zs2 ) )
=> ~ ( P2 @ Xa ) ) ) ) ).
% split_list_last_prop
thf(fact_146_in__set__conv__decomp,axiom,
! [A: $tType,X3: A,Xs2: list @ A] :
( ( member2 @ A @ X3 @ ( set2 @ A @ Xs2 ) )
= ( ? [Ys: list @ A,Zs3: list @ A] :
( Xs2
= ( append @ A @ Ys @ ( cons @ A @ X3 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_147_append__Cons__eq__iff,axiom,
! [A: $tType,X3: A,Xs2: list @ A,Ys2: list @ A,Xs5: list @ A,Ys5: list @ A] :
( ~ ( member2 @ A @ X3 @ ( set2 @ A @ Xs2 ) )
=> ( ~ ( member2 @ A @ X3 @ ( set2 @ A @ Ys2 ) )
=> ( ( ( append @ A @ Xs2 @ ( cons @ A @ X3 @ Ys2 ) )
= ( append @ A @ Xs5 @ ( cons @ A @ X3 @ Ys5 ) ) )
= ( ( Xs2 = Xs5 )
& ( Ys2 = Ys5 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_148_split__list__propE,axiom,
! [A: $tType,Xs2: list @ A,P2: A > $o] :
( ? [X5: A] :
( ( member2 @ A @ X5 @ ( set2 @ A @ Xs2 ) )
& ( P2 @ X5 ) )
=> ~ ! [Ys4: list @ A,X4: A] :
( ? [Zs2: list @ A] :
( Xs2
= ( append @ A @ Ys4 @ ( cons @ A @ X4 @ Zs2 ) ) )
=> ~ ( P2 @ X4 ) ) ) ).
% split_list_propE
thf(fact_149_split__list__first,axiom,
! [A: $tType,X3: A,Xs2: list @ A] :
( ( member2 @ A @ X3 @ ( set2 @ A @ Xs2 ) )
=> ? [Ys4: list @ A,Zs2: list @ A] :
( ( Xs2
= ( append @ A @ Ys4 @ ( cons @ A @ X3 @ Zs2 ) ) )
& ~ ( member2 @ A @ X3 @ ( set2 @ A @ Ys4 ) ) ) ) ).
% split_list_first
thf(fact_150_split__list__prop,axiom,
! [A: $tType,Xs2: list @ A,P2: A > $o] :
( ? [X5: A] :
( ( member2 @ A @ X5 @ ( set2 @ A @ Xs2 ) )
& ( P2 @ X5 ) )
=> ? [Ys4: list @ A,X4: A] :
( ? [Zs2: list @ A] :
( Xs2
= ( append @ A @ Ys4 @ ( cons @ A @ X4 @ Zs2 ) ) )
& ( P2 @ X4 ) ) ) ).
% split_list_prop
thf(fact_151_split__list__last,axiom,
! [A: $tType,X3: A,Xs2: list @ A] :
( ( member2 @ A @ X3 @ ( set2 @ A @ Xs2 ) )
=> ? [Ys4: list @ A,Zs2: list @ A] :
( ( Xs2
= ( append @ A @ Ys4 @ ( cons @ A @ X3 @ Zs2 ) ) )
& ~ ( member2 @ A @ X3 @ ( set2 @ A @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_152_split__list,axiom,
! [A: $tType,X3: A,Xs2: list @ A] :
( ( member2 @ A @ X3 @ ( set2 @ A @ Xs2 ) )
=> ? [Ys4: list @ A,Zs2: list @ A] :
( Xs2
= ( append @ A @ Ys4 @ ( cons @ A @ X3 @ Zs2 ) ) ) ) ).
% split_list
thf(fact_153_rev__nonempty__induct,axiom,
! [A: $tType,Xs2: list @ A,P2: ( list @ A ) > $o] :
( ( Xs2
!= ( nil @ A ) )
=> ( ! [X4: A] : ( P2 @ ( cons @ A @ X4 @ ( nil @ A ) ) )
=> ( ! [X4: A,Xs4: list @ A] :
( ( Xs4
!= ( nil @ A ) )
=> ( ( P2 @ Xs4 )
=> ( P2 @ ( append @ A @ Xs4 @ ( cons @ A @ X4 @ ( nil @ A ) ) ) ) ) )
=> ( P2 @ Xs2 ) ) ) ) ).
% rev_nonempty_induct
thf(fact_154_append__eq__Cons__conv,axiom,
! [A: $tType,Ys2: list @ A,Zs: list @ A,X3: A,Xs2: list @ A] :
( ( ( append @ A @ Ys2 @ Zs )
= ( cons @ A @ X3 @ Xs2 ) )
= ( ( ( Ys2
= ( nil @ A ) )
& ( Zs
= ( cons @ A @ X3 @ Xs2 ) ) )
| ? [Ys6: list @ A] :
( ( Ys2
= ( cons @ A @ X3 @ Ys6 ) )
& ( ( append @ A @ Ys6 @ Zs )
= Xs2 ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_155_Cons__eq__append__conv,axiom,
! [A: $tType,X3: A,Xs2: list @ A,Ys2: list @ A,Zs: list @ A] :
( ( ( cons @ A @ X3 @ Xs2 )
= ( append @ A @ Ys2 @ Zs ) )
= ( ( ( Ys2
= ( nil @ A ) )
& ( ( cons @ A @ X3 @ Xs2 )
= Zs ) )
| ? [Ys6: list @ A] :
( ( ( cons @ A @ X3 @ Ys6 )
= Ys2 )
& ( Xs2
= ( append @ A @ Ys6 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_156_rev__exhaust,axiom,
! [A: $tType,Xs2: list @ A] :
( ( Xs2
!= ( nil @ A ) )
=> ~ ! [Ys4: list @ A,Y4: A] :
( Xs2
!= ( append @ A @ Ys4 @ ( cons @ A @ Y4 @ ( nil @ A ) ) ) ) ) ).
% rev_exhaust
thf(fact_157_rev__induct,axiom,
! [A: $tType,P2: ( list @ A ) > $o,Xs2: list @ A] :
( ( P2 @ ( nil @ A ) )
=> ( ! [X4: A,Xs4: list @ A] :
( ( P2 @ Xs4 )
=> ( P2 @ ( append @ A @ Xs4 @ ( cons @ A @ X4 @ ( nil @ A ) ) ) ) )
=> ( P2 @ Xs2 ) ) ) ).
% rev_induct
thf(fact_158_shuffles_Ocases,axiom,
! [A: $tType,X3: product_prod @ ( list @ A ) @ ( list @ A )] :
( ! [Ys4: list @ A] :
( X3
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys4 ) )
=> ( ! [Xs4: list @ A] :
( X3
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs4 @ ( nil @ A ) ) )
=> ~ ! [X4: A,Xs4: list @ A,Y4: A,Ys4: list @ A] :
( X3
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X4 @ Xs4 ) @ ( cons @ A @ Y4 @ Ys4 ) ) ) ) ) ).
% shuffles.cases
thf(fact_159_map__tailrec__rev_Ocases,axiom,
! [A: $tType,B: $tType,X3: product_prod @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) )] :
( ! [F2: A > B,Bs: list @ B] :
( X3
!= ( product_Pair @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ F2 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ Bs ) ) )
=> ~ ! [F2: A > B,A4: A,As: list @ A,Bs: list @ B] :
( X3
!= ( product_Pair @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ F2 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ A4 @ As ) @ Bs ) ) ) ) ).
% map_tailrec_rev.cases
thf(fact_160_sd__path_Oinduct,axiom,
! [A: $tType,P2: ( product_prod @ A @ A ) > A > ( list @ ( product_prod @ A @ A ) ) > $o,A0: product_prod @ A @ A,A1: A,A22: list @ ( product_prod @ A @ A )] :
( ! [Uu: product_prod @ A @ A,Uv: A] : ( P2 @ Uu @ Uv @ ( nil @ ( product_prod @ A @ A ) ) )
=> ( ! [U2: A,V: A,W: A,E2: product_prod @ A @ A,Es: list @ ( product_prod @ A @ A )] :
( ( P2 @ ( product_Pair @ A @ A @ U2 @ V ) @ W @ Es )
=> ( P2 @ ( product_Pair @ A @ A @ U2 @ V ) @ W @ ( cons @ ( product_prod @ A @ A ) @ E2 @ Es ) ) )
=> ( P2 @ A0 @ A1 @ A22 ) ) ) ).
% sd_path.induct
thf(fact_161_co__path_Oinduct,axiom,
! [A: $tType,P2: ( product_prod @ A @ A ) > A > ( list @ ( product_prod @ A @ A ) ) > $o,A0: product_prod @ A @ A,A1: A,A22: list @ ( product_prod @ A @ A )] :
( ! [Uu: product_prod @ A @ A,Uv: A] : ( P2 @ Uu @ Uv @ ( nil @ ( product_prod @ A @ A ) ) )
=> ( ! [Uw: product_prod @ A @ A,Ux: A,E2: product_prod @ A @ A] : ( P2 @ Uw @ Ux @ ( cons @ ( product_prod @ A @ A ) @ E2 @ ( nil @ ( product_prod @ A @ A ) ) ) )
=> ( ! [U2: A,V: A,W: A,E1: product_prod @ A @ A,E22: product_prod @ A @ A,Es: list @ ( product_prod @ A @ A )] :
( ( ( ( E1
= ( product_Pair @ A @ A @ U2 @ W ) )
& ( E22
= ( product_Pair @ A @ A @ W @ V ) ) )
=> ( P2 @ ( product_Pair @ A @ A @ U2 @ V ) @ W @ Es ) )
=> ( ( ~ ( ( E1
= ( product_Pair @ A @ A @ U2 @ W ) )
& ( E22
= ( product_Pair @ A @ A @ W @ V ) ) )
=> ( ( ( E1
= ( product_Pair @ A @ A @ V @ W ) )
& ( E22
= ( product_Pair @ A @ A @ W @ U2 ) ) )
=> ( P2 @ ( product_Pair @ A @ A @ U2 @ V ) @ W @ Es ) ) )
=> ( ( ~ ( ( E1
= ( product_Pair @ A @ A @ U2 @ W ) )
& ( E22
= ( product_Pair @ A @ A @ W @ V ) ) )
=> ( ~ ( ( E1
= ( product_Pair @ A @ A @ V @ W ) )
& ( E22
= ( product_Pair @ A @ A @ W @ U2 ) ) )
=> ( P2 @ ( product_Pair @ A @ A @ U2 @ V ) @ W @ ( cons @ ( product_prod @ A @ A ) @ E22 @ Es ) ) ) )
=> ( P2 @ ( product_Pair @ A @ A @ U2 @ V ) @ W @ ( cons @ ( product_prod @ A @ A ) @ E1 @ ( cons @ ( product_prod @ A @ A ) @ E22 @ Es ) ) ) ) ) )
=> ( P2 @ A0 @ A1 @ A22 ) ) ) ) ).
% co_path.induct
thf(fact_162_sd__path_Ocases,axiom,
! [A: $tType,X3: product_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ ( list @ ( product_prod @ A @ A ) ) )] :
( ! [Uu: product_prod @ A @ A,Uv: A] :
( X3
!= ( product_Pair @ ( product_prod @ A @ A ) @ ( product_prod @ A @ ( list @ ( product_prod @ A @ A ) ) ) @ Uu @ ( product_Pair @ A @ ( list @ ( product_prod @ A @ A ) ) @ Uv @ ( nil @ ( product_prod @ A @ A ) ) ) ) )
=> ~ ! [U2: A,V: A,W: A,E2: product_prod @ A @ A,Es: list @ ( product_prod @ A @ A )] :
( X3
!= ( product_Pair @ ( product_prod @ A @ A ) @ ( product_prod @ A @ ( list @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ A @ A @ U2 @ V ) @ ( product_Pair @ A @ ( list @ ( product_prod @ A @ A ) ) @ W @ ( cons @ ( product_prod @ A @ A ) @ E2 @ Es ) ) ) ) ) ).
% sd_path.cases
thf(fact_163_co__path_Ocases,axiom,
! [A: $tType,X3: product_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ ( list @ ( product_prod @ A @ A ) ) )] :
( ! [Uu: product_prod @ A @ A,Uv: A] :
( X3
!= ( product_Pair @ ( product_prod @ A @ A ) @ ( product_prod @ A @ ( list @ ( product_prod @ A @ A ) ) ) @ Uu @ ( product_Pair @ A @ ( list @ ( product_prod @ A @ A ) ) @ Uv @ ( nil @ ( product_prod @ A @ A ) ) ) ) )
=> ( ! [Uw: product_prod @ A @ A,Ux: A,E2: product_prod @ A @ A] :
( X3
!= ( product_Pair @ ( product_prod @ A @ A ) @ ( product_prod @ A @ ( list @ ( product_prod @ A @ A ) ) ) @ Uw @ ( product_Pair @ A @ ( list @ ( product_prod @ A @ A ) ) @ Ux @ ( cons @ ( product_prod @ A @ A ) @ E2 @ ( nil @ ( product_prod @ A @ A ) ) ) ) ) )
=> ~ ! [U2: A,V: A,W: A,E1: product_prod @ A @ A,E22: product_prod @ A @ A,Es: list @ ( product_prod @ A @ A )] :
( X3
!= ( product_Pair @ ( product_prod @ A @ A ) @ ( product_prod @ A @ ( list @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ A @ A @ U2 @ V ) @ ( product_Pair @ A @ ( list @ ( product_prod @ A @ A ) ) @ W @ ( cons @ ( product_prod @ A @ A ) @ E1 @ ( cons @ ( product_prod @ A @ A ) @ E22 @ Es ) ) ) ) ) ) ) ).
% co_path.cases
thf(fact_164_listrel1__eq__len,axiom,
! [A: $tType,Xs2: list @ A,Ys2: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys2 ) @ ( listrel1 @ A @ R ) )
=> ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ A ) @ Ys2 ) ) ) ).
% listrel1_eq_len
thf(fact_165_not__listrel1__Nil,axiom,
! [A: $tType,Xs2: list @ A,R: set @ ( product_prod @ A @ A )] :
~ ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ ( nil @ A ) ) @ ( listrel1 @ A @ R ) ) ).
% not_listrel1_Nil
thf(fact_166_not__Nil__listrel1,axiom,
! [A: $tType,Xs2: list @ A,R: set @ ( product_prod @ A @ A )] :
~ ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Xs2 ) @ ( listrel1 @ A @ R ) ) ).
% not_Nil_listrel1
thf(fact_167_lexord__Nil__right,axiom,
! [A: $tType,X3: list @ A,R: set @ ( product_prod @ A @ A )] :
~ ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X3 @ ( nil @ A ) ) @ ( lexord @ A @ R ) ) ).
% lexord_Nil_right
thf(fact_168_lex__append__leftI,axiom,
! [A: $tType,Ys2: list @ A,Zs: list @ A,R: set @ ( product_prod @ A @ A ),Xs2: list @ A] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys2 @ Zs ) @ ( lex @ A @ R ) )
=> ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs2 @ Ys2 ) @ ( append @ A @ Xs2 @ Zs ) ) @ ( lex @ A @ R ) ) ) ).
% lex_append_leftI
thf(fact_169_progressing__single,axiom,
! [A: $tType,E3: product_prod @ A @ A] : ( kurato990191779essing @ A @ ( cons @ ( product_prod @ A @ A ) @ E3 @ ( nil @ ( product_prod @ A @ A ) ) ) ) ).
% progressing_single
thf(fact_170_lexord__partial__trans,axiom,
! [A: $tType,Xs2: list @ A,R: set @ ( product_prod @ A @ A ),Ys2: list @ A,Zs: list @ A] :
( ! [X4: A,Y4: A,Z2: A] :
( ( member2 @ A @ X4 @ ( set2 @ A @ Xs2 ) )
=> ( ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ R )
=> ( ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z2 ) @ R )
=> ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Z2 ) @ R ) ) ) )
=> ( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys2 ) @ ( lexord @ A @ R ) )
=> ( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys2 @ Zs ) @ ( lexord @ A @ R ) )
=> ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Zs ) @ ( lexord @ A @ R ) ) ) ) ) ).
% lexord_partial_trans
thf(fact_171_lexord__sufE,axiom,
! [A: $tType,Xs2: list @ A,Zs: list @ A,Ys2: list @ A,Qs: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs2 @ Zs ) @ ( append @ A @ Ys2 @ Qs ) ) @ ( lexord @ A @ R ) )
=> ( ( Xs2 != Ys2 )
=> ( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ A ) @ Ys2 ) )
=> ( ( ( size_size @ ( list @ A ) @ Zs )
= ( size_size @ ( list @ A ) @ Qs ) )
=> ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys2 ) @ ( lexord @ A @ R ) ) ) ) ) ) ).
% lexord_sufE
thf(fact_172_subset__eq__mset__impl_Ocases,axiom,
! [A: $tType,X3: product_prod @ ( list @ A ) @ ( list @ A )] :
( ! [Ys4: list @ A] :
( X3
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys4 ) )
=> ~ ! [X4: A,Xs4: list @ A,Ys4: list @ A] :
( X3
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X4 @ Xs4 ) @ Ys4 ) ) ) ).
% subset_eq_mset_impl.cases
thf(fact_173_pre__digraph_Ocas_Ocases,axiom,
! [B: $tType,A: $tType,X3: product_prod @ A @ ( product_prod @ ( list @ B ) @ A )] :
( ! [U2: A,V: A] :
( X3
!= ( product_Pair @ A @ ( product_prod @ ( list @ B ) @ A ) @ U2 @ ( product_Pair @ ( list @ B ) @ A @ ( nil @ B ) @ V ) ) )
=> ~ ! [U2: A,E2: B,Es: list @ B,V: A] :
( X3
!= ( product_Pair @ A @ ( product_prod @ ( list @ B ) @ A ) @ U2 @ ( product_Pair @ ( list @ B ) @ A @ ( cons @ B @ E2 @ Es ) @ V ) ) ) ) ).
% pre_digraph.cas.cases
thf(fact_174_asym_Oinducts,axiom,
! [A: $tType,X3: set @ ( product_prod @ A @ A ),P2: ( set @ ( product_prod @ A @ A ) ) > $o] :
( ( asym @ A @ X3 )
=> ( ! [R3: set @ ( product_prod @ A @ A )] :
( ! [A7: A,B5: A] :
( ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A7 @ B5 ) @ R3 )
=> ~ ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B5 @ A7 ) @ R3 ) )
=> ( P2 @ R3 ) )
=> ( P2 @ X3 ) ) ) ).
% asym.inducts
thf(fact_175_asym_Ointros,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ! [A4: A,B4: A] :
( ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B4 ) @ R2 )
=> ~ ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B4 @ A4 ) @ R2 ) )
=> ( asym @ A @ R2 ) ) ).
% asym.intros
thf(fact_176_asym_Osimps,axiom,
! [A: $tType] :
( ( asym @ A )
= ( ^ [A6: set @ ( product_prod @ A @ A )] :
? [R4: set @ ( product_prod @ A @ A )] :
( ( A6 = R4 )
& ! [X: A,Y: A] :
( ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R4 )
=> ~ ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X ) @ R4 ) ) ) ) ) ).
% asym.simps
thf(fact_177_asym_Ocases,axiom,
! [A: $tType,A2: set @ ( product_prod @ A @ A )] :
( ( asym @ A @ A2 )
=> ! [A7: A,B5: A] :
( ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A7 @ B5 ) @ A2 )
=> ~ ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B5 @ A7 ) @ A2 ) ) ) ).
% asym.cases
thf(fact_178_asym__iff,axiom,
! [A: $tType] :
( ( asym @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A )] :
! [X: A,Y: A] :
( ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R4 )
=> ~ ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X ) @ R4 ) ) ) ) ).
% asym_iff
thf(fact_179_irreflI,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ! [A4: A] :
~ ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ A4 ) @ R2 )
=> ( irrefl @ A @ R2 ) ) ).
% irreflI
thf(fact_180_irrefl__def,axiom,
! [A: $tType] :
( ( irrefl @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
! [A6: A] :
~ ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A6 @ A6 ) @ R5 ) ) ) ).
% irrefl_def
thf(fact_181_asymD,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),X3: A,Y3: A] :
( ( asym @ A @ R2 )
=> ( ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R2 )
=> ~ ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X3 ) @ R2 ) ) ) ).
% asymD
thf(fact_182_inverse__permutation__of__list_Ocases,axiom,
! [A: $tType,X3: product_prod @ ( list @ ( product_prod @ A @ A ) ) @ A] :
( ! [X4: A] :
( X3
!= ( product_Pair @ ( list @ ( product_prod @ A @ A ) ) @ A @ ( nil @ ( product_prod @ A @ A ) ) @ X4 ) )
=> ~ ! [Y4: A,X6: A,Xs4: list @ ( product_prod @ A @ A ),X4: A] :
( X3
!= ( product_Pair @ ( list @ ( product_prod @ A @ A ) ) @ A @ ( cons @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ X6 ) @ Xs4 ) @ X4 ) ) ) ).
% inverse_permutation_of_list.cases
thf(fact_183_inverse__permutation__of__list_Oinduct,axiom,
! [A: $tType,P2: ( list @ ( product_prod @ A @ A ) ) > A > $o,A0: list @ ( product_prod @ A @ A ),A1: A] :
( ! [X_1: A] : ( P2 @ ( nil @ ( product_prod @ A @ A ) ) @ X_1 )
=> ( ! [Y4: A,X6: A,Xs4: list @ ( product_prod @ A @ A ),X4: A] :
( ( ( X4 != X6 )
=> ( P2 @ Xs4 @ X4 ) )
=> ( P2 @ ( cons @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ X6 ) @ Xs4 ) @ X4 ) )
=> ( P2 @ A0 @ A1 ) ) ) ).
% inverse_permutation_of_list.induct
thf(fact_184_sd__path_Osimps_I2_J,axiom,
! [A: $tType,U: A,V2: A,W2: A,E3: product_prod @ A @ A,Es2: list @ ( product_prod @ A @ A )] :
( ( pair_sd_path @ A @ ( product_Pair @ A @ A @ U @ V2 ) @ W2 @ ( cons @ ( product_prod @ A @ A ) @ E3 @ Es2 ) )
= ( append @ ( product_prod @ A @ A )
@ ( if @ ( list @ ( product_prod @ A @ A ) )
@ ( E3
= ( product_Pair @ A @ A @ U @ V2 ) )
@ ( cons @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ W2 ) @ ( cons @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ W2 @ V2 ) @ ( nil @ ( product_prod @ A @ A ) ) ) )
@ ( if @ ( list @ ( product_prod @ A @ A ) )
@ ( E3
= ( product_Pair @ A @ A @ V2 @ U ) )
@ ( cons @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V2 @ W2 ) @ ( cons @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ W2 @ U ) @ ( nil @ ( product_prod @ A @ A ) ) ) )
@ ( cons @ ( product_prod @ A @ A ) @ E3 @ ( nil @ ( product_prod @ A @ A ) ) ) ) )
@ ( pair_sd_path @ A @ ( product_Pair @ A @ A @ U @ V2 ) @ W2 @ Es2 ) ) ) ).
% sd_path.simps(2)
thf(fact_185_sd__path_Osimps_I1_J,axiom,
! [A: $tType,Uu2: product_prod @ A @ A,Uv2: A] :
( ( pair_sd_path @ A @ Uu2 @ Uv2 @ ( nil @ ( product_prod @ A @ A ) ) )
= ( nil @ ( product_prod @ A @ A ) ) ) ).
% sd_path.simps(1)
thf(fact_186_sd__path__id,axiom,
! [A: $tType,X3: A,Y3: A,P3: list @ ( product_prod @ A @ A ),W2: A] :
( ~ ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ ( set2 @ ( product_prod @ A @ A ) @ P3 ) )
=> ( ~ ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X3 ) @ ( set2 @ ( product_prod @ A @ A ) @ P3 ) )
=> ( ( pair_sd_path @ A @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ W2 @ P3 )
= P3 ) ) ) ).
% sd_path_id
thf(fact_187_sd__path_Oelims,axiom,
! [A: $tType,X3: product_prod @ A @ A,Xa2: A,Xb: list @ ( product_prod @ A @ A ),Y3: list @ ( product_prod @ A @ A )] :
( ( ( pair_sd_path @ A @ X3 @ Xa2 @ Xb )
= Y3 )
=> ( ( ( Xb
= ( nil @ ( product_prod @ A @ A ) ) )
=> ( Y3
!= ( nil @ ( product_prod @ A @ A ) ) ) )
=> ~ ! [U2: A,V: A] :
( ( X3
= ( product_Pair @ A @ A @ U2 @ V ) )
=> ! [E2: product_prod @ A @ A,Es: list @ ( product_prod @ A @ A )] :
( ( Xb
= ( cons @ ( product_prod @ A @ A ) @ E2 @ Es ) )
=> ( Y3
!= ( append @ ( product_prod @ A @ A )
@ ( if @ ( list @ ( product_prod @ A @ A ) )
@ ( E2
= ( product_Pair @ A @ A @ U2 @ V ) )
@ ( cons @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U2 @ Xa2 ) @ ( cons @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Xa2 @ V ) @ ( nil @ ( product_prod @ A @ A ) ) ) )
@ ( if @ ( list @ ( product_prod @ A @ A ) )
@ ( E2
= ( product_Pair @ A @ A @ V @ U2 ) )
@ ( cons @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V @ Xa2 ) @ ( cons @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Xa2 @ U2 ) @ ( nil @ ( product_prod @ A @ A ) ) ) )
@ ( cons @ ( product_prod @ A @ A ) @ E2 @ ( nil @ ( product_prod @ A @ A ) ) ) ) )
@ ( pair_sd_path @ A @ ( product_Pair @ A @ A @ U2 @ V ) @ Xa2 @ Es ) ) ) ) ) ) ) ).
% sd_path.elims
thf(fact_188_the__elem__set,axiom,
! [A: $tType,X3: A] :
( ( the_elem @ A @ ( set2 @ A @ ( cons @ A @ X3 @ ( nil @ A ) ) ) )
= X3 ) ).
% the_elem_set
thf(fact_189_sd__path_Opelims,axiom,
! [A: $tType,X3: product_prod @ A @ A,Xa2: A,Xb: list @ ( product_prod @ A @ A ),Y3: list @ ( product_prod @ A @ A )] :
( ( ( pair_sd_path @ A @ X3 @ Xa2 @ Xb )
= Y3 )
=> ( ( accp @ ( product_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ ( list @ ( product_prod @ A @ A ) ) ) ) @ ( pair_sd_path_rel @ A ) @ ( product_Pair @ ( product_prod @ A @ A ) @ ( product_prod @ A @ ( list @ ( product_prod @ A @ A ) ) ) @ X3 @ ( product_Pair @ A @ ( list @ ( product_prod @ A @ A ) ) @ Xa2 @ Xb ) ) )
=> ( ( ( Xb
= ( nil @ ( product_prod @ A @ A ) ) )
=> ( ( Y3
= ( nil @ ( product_prod @ A @ A ) ) )
=> ~ ( accp @ ( product_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ ( list @ ( product_prod @ A @ A ) ) ) ) @ ( pair_sd_path_rel @ A ) @ ( product_Pair @ ( product_prod @ A @ A ) @ ( product_prod @ A @ ( list @ ( product_prod @ A @ A ) ) ) @ X3 @ ( product_Pair @ A @ ( list @ ( product_prod @ A @ A ) ) @ Xa2 @ ( nil @ ( product_prod @ A @ A ) ) ) ) ) ) )
=> ~ ! [U2: A,V: A] :
( ( X3
= ( product_Pair @ A @ A @ U2 @ V ) )
=> ! [E2: product_prod @ A @ A,Es: list @ ( product_prod @ A @ A )] :
( ( Xb
= ( cons @ ( product_prod @ A @ A ) @ E2 @ Es ) )
=> ( ( Y3
= ( append @ ( product_prod @ A @ A )
@ ( if @ ( list @ ( product_prod @ A @ A ) )
@ ( E2
= ( product_Pair @ A @ A @ U2 @ V ) )
@ ( cons @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U2 @ Xa2 ) @ ( cons @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Xa2 @ V ) @ ( nil @ ( product_prod @ A @ A ) ) ) )
@ ( if @ ( list @ ( product_prod @ A @ A ) )
@ ( E2
= ( product_Pair @ A @ A @ V @ U2 ) )
@ ( cons @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V @ Xa2 ) @ ( cons @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Xa2 @ U2 ) @ ( nil @ ( product_prod @ A @ A ) ) ) )
@ ( cons @ ( product_prod @ A @ A ) @ E2 @ ( nil @ ( product_prod @ A @ A ) ) ) ) )
@ ( pair_sd_path @ A @ ( product_Pair @ A @ A @ U2 @ V ) @ Xa2 @ Es ) ) )
=> ~ ( accp @ ( product_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ ( list @ ( product_prod @ A @ A ) ) ) ) @ ( pair_sd_path_rel @ A ) @ ( product_Pair @ ( product_prod @ A @ A ) @ ( product_prod @ A @ ( list @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ A @ A @ U2 @ V ) @ ( product_Pair @ A @ ( list @ ( product_prod @ A @ A ) ) @ Xa2 @ ( cons @ ( product_prod @ A @ A ) @ E2 @ Es ) ) ) ) ) ) ) ) ) ) ).
% sd_path.pelims
thf(fact_190_list__exhaust2,axiom,
! [A: $tType,B: $tType,Y3: list @ A,Ya: list @ B] :
( ( ( Y3
= ( nil @ A ) )
=> ( Ya
!= ( nil @ B ) ) )
=> ( ( ( Y3
= ( nil @ A ) )
=> ! [X213: B,X223: list @ B] :
( Ya
!= ( cons @ B @ X213 @ X223 ) ) )
=> ( ( ? [X213: A,X223: list @ A] :
( Y3
= ( cons @ A @ X213 @ X223 ) )
=> ( Ya
!= ( nil @ B ) ) )
=> ~ ( ? [X213: A,X223: list @ A] :
( Y3
= ( cons @ A @ X213 @ X223 ) )
=> ! [X21a: B,X22a: list @ B] :
( Ya
!= ( cons @ B @ X21a @ X22a ) ) ) ) ) ) ).
% list_exhaust2
thf(fact_191_list__exhaust__NSC,axiom,
! [A: $tType,Xs2: list @ A] :
( ( Xs2
!= ( nil @ A ) )
=> ( ! [X4: A] :
( Xs2
!= ( cons @ A @ X4 @ ( nil @ A ) ) )
=> ~ ! [X4: A,Y4: A,Ys4: list @ A] :
( Xs2
!= ( cons @ A @ X4 @ ( cons @ A @ Y4 @ Ys4 ) ) ) ) ) ).
% list_exhaust_NSC
thf(fact_192_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_193_lenlex__append2,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),Us: list @ A,Xs2: list @ A,Ys2: list @ A] :
( ( irrefl @ A @ R2 )
=> ( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Us @ Xs2 ) @ ( append @ A @ Us @ Ys2 ) ) @ ( lenlex @ A @ R2 ) )
= ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys2 ) @ ( lenlex @ A @ R2 ) ) ) ) ).
% lenlex_append2
thf(fact_194_Nil__lenlex__iff1,axiom,
! [A: $tType,Ns: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ns ) @ ( lenlex @ A @ R ) )
= ( Ns
!= ( nil @ A ) ) ) ).
% Nil_lenlex_iff1
thf(fact_195_in__set__product__lists__length,axiom,
! [A: $tType,Xs2: list @ A,Xss2: list @ ( list @ A )] :
( ( member2 @ ( list @ A ) @ Xs2 @ ( set2 @ ( list @ A ) @ ( product_lists @ A @ Xss2 ) ) )
=> ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ ( list @ A ) ) @ Xss2 ) ) ) ).
% in_set_product_lists_length
thf(fact_196_lenlex__irreflexive,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),Xs2: list @ A] :
( ! [X4: A] :
~ ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ X4 ) @ R )
=> ~ ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Xs2 ) @ ( lenlex @ A @ R ) ) ) ).
% lenlex_irreflexive
thf(fact_197_Nil__lenlex__iff2,axiom,
! [A: $tType,Ns: list @ A,R: set @ ( product_prod @ A @ A )] :
~ ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ns @ ( nil @ A ) ) @ ( lenlex @ A @ R ) ) ).
% Nil_lenlex_iff2
thf(fact_198_lenlex__append1,axiom,
! [A: $tType,Us: list @ A,Xs2: list @ A,R2: set @ ( product_prod @ A @ A ),Vs: list @ A,Ys2: list @ A] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Us @ Xs2 ) @ ( lenlex @ A @ R2 ) )
=> ( ( ( size_size @ ( list @ A ) @ Vs )
= ( size_size @ ( list @ A ) @ Ys2 ) )
=> ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Us @ Vs ) @ ( append @ A @ Xs2 @ Ys2 ) ) @ ( lenlex @ A @ R2 ) ) ) ) ).
% lenlex_append1
thf(fact_199_asym__lenlex,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( asym @ A @ R2 )
=> ( asym @ ( list @ A ) @ ( lenlex @ A @ R2 ) ) ) ).
% asym_lenlex
thf(fact_200_co__path_Opelims,axiom,
! [A: $tType,X3: product_prod @ A @ A,Xa2: A,Xb: list @ ( product_prod @ A @ A ),Y3: list @ ( product_prod @ A @ A )] :
( ( ( pair_co_path @ A @ X3 @ Xa2 @ Xb )
= Y3 )
=> ( ( accp @ ( product_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ ( list @ ( product_prod @ A @ A ) ) ) ) @ ( pair_co_path_rel @ A ) @ ( product_Pair @ ( product_prod @ A @ A ) @ ( product_prod @ A @ ( list @ ( product_prod @ A @ A ) ) ) @ X3 @ ( product_Pair @ A @ ( list @ ( product_prod @ A @ A ) ) @ Xa2 @ Xb ) ) )
=> ( ( ( Xb
= ( nil @ ( product_prod @ A @ A ) ) )
=> ( ( Y3
= ( nil @ ( product_prod @ A @ A ) ) )
=> ~ ( accp @ ( product_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ ( list @ ( product_prod @ A @ A ) ) ) ) @ ( pair_co_path_rel @ A ) @ ( product_Pair @ ( product_prod @ A @ A ) @ ( product_prod @ A @ ( list @ ( product_prod @ A @ A ) ) ) @ X3 @ ( product_Pair @ A @ ( list @ ( product_prod @ A @ A ) ) @ Xa2 @ ( nil @ ( product_prod @ A @ A ) ) ) ) ) ) )
=> ( ! [E2: product_prod @ A @ A] :
( ( Xb
= ( cons @ ( product_prod @ A @ A ) @ E2 @ ( nil @ ( product_prod @ A @ A ) ) ) )
=> ( ( Y3
= ( cons @ ( product_prod @ A @ A ) @ E2 @ ( nil @ ( product_prod @ A @ A ) ) ) )
=> ~ ( accp @ ( product_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ ( list @ ( product_prod @ A @ A ) ) ) ) @ ( pair_co_path_rel @ A ) @ ( product_Pair @ ( product_prod @ A @ A ) @ ( product_prod @ A @ ( list @ ( product_prod @ A @ A ) ) ) @ X3 @ ( product_Pair @ A @ ( list @ ( product_prod @ A @ A ) ) @ Xa2 @ ( cons @ ( product_prod @ A @ A ) @ E2 @ ( nil @ ( product_prod @ A @ A ) ) ) ) ) ) ) )
=> ~ ! [U2: A,V: A] :
( ( X3
= ( product_Pair @ A @ A @ U2 @ V ) )
=> ! [E1: product_prod @ A @ A,E22: product_prod @ A @ A,Es: list @ ( product_prod @ A @ A )] :
( ( Xb
= ( cons @ ( product_prod @ A @ A ) @ E1 @ ( cons @ ( product_prod @ A @ A ) @ E22 @ Es ) ) )
=> ( ( ( ( ( E1
= ( product_Pair @ A @ A @ U2 @ Xa2 ) )
& ( E22
= ( product_Pair @ A @ A @ Xa2 @ V ) ) )
=> ( Y3
= ( cons @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U2 @ V ) @ ( pair_co_path @ A @ ( product_Pair @ A @ A @ U2 @ V ) @ Xa2 @ Es ) ) ) )
& ( ~ ( ( E1
= ( product_Pair @ A @ A @ U2 @ Xa2 ) )
& ( E22
= ( product_Pair @ A @ A @ Xa2 @ V ) ) )
=> ( ( ( ( E1
= ( product_Pair @ A @ A @ V @ Xa2 ) )
& ( E22
= ( product_Pair @ A @ A @ Xa2 @ U2 ) ) )
=> ( Y3
= ( cons @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V @ U2 ) @ ( pair_co_path @ A @ ( product_Pair @ A @ A @ U2 @ V ) @ Xa2 @ Es ) ) ) )
& ( ~ ( ( E1
= ( product_Pair @ A @ A @ V @ Xa2 ) )
& ( E22
= ( product_Pair @ A @ A @ Xa2 @ U2 ) ) )
=> ( Y3
= ( cons @ ( product_prod @ A @ A ) @ E1 @ ( pair_co_path @ A @ ( product_Pair @ A @ A @ U2 @ V ) @ Xa2 @ ( cons @ ( product_prod @ A @ A ) @ E22 @ Es ) ) ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ ( list @ ( product_prod @ A @ A ) ) ) ) @ ( pair_co_path_rel @ A ) @ ( product_Pair @ ( product_prod @ A @ A ) @ ( product_prod @ A @ ( list @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ A @ A @ U2 @ V ) @ ( product_Pair @ A @ ( list @ ( product_prod @ A @ A ) ) @ Xa2 @ ( cons @ ( product_prod @ A @ A ) @ E1 @ ( cons @ ( product_prod @ A @ A ) @ E22 @ Es ) ) ) ) ) ) ) ) ) ) ) ) ).
% co_path.pelims
thf(fact_201_pawalk__verts_Osimps_I1_J,axiom,
! [A: $tType,U: A] :
( ( pair_pawalk_verts @ A @ U @ ( nil @ ( product_prod @ A @ A ) ) )
= ( cons @ A @ U @ ( nil @ A ) ) ) ).
% pawalk_verts.simps(1)
thf(fact_202_co__path__nonempty,axiom,
! [A: $tType,E3: product_prod @ A @ A,W2: A,P3: list @ ( product_prod @ A @ A )] :
( ( ( pair_co_path @ A @ E3 @ W2 @ P3 )
= ( nil @ ( product_prod @ A @ A ) ) )
= ( P3
= ( nil @ ( product_prod @ A @ A ) ) ) ) ).
% co_path_nonempty
thf(fact_203_co__path_Osimps_I1_J,axiom,
! [A: $tType,Uu2: product_prod @ A @ A,Uv2: A] :
( ( pair_co_path @ A @ Uu2 @ Uv2 @ ( nil @ ( product_prod @ A @ A ) ) )
= ( nil @ ( product_prod @ A @ A ) ) ) ).
% co_path.simps(1)
thf(fact_204_co__path_Osimps_I3_J,axiom,
! [A: $tType,E12: product_prod @ A @ A,U: A,W2: A,E23: product_prod @ A @ A,V2: A,Es2: list @ ( product_prod @ A @ A )] :
( ( ( ( E12
= ( product_Pair @ A @ A @ U @ W2 ) )
& ( E23
= ( product_Pair @ A @ A @ W2 @ V2 ) ) )
=> ( ( pair_co_path @ A @ ( product_Pair @ A @ A @ U @ V2 ) @ W2 @ ( cons @ ( product_prod @ A @ A ) @ E12 @ ( cons @ ( product_prod @ A @ A ) @ E23 @ Es2 ) ) )
= ( cons @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ V2 ) @ ( pair_co_path @ A @ ( product_Pair @ A @ A @ U @ V2 ) @ W2 @ Es2 ) ) ) )
& ( ~ ( ( E12
= ( product_Pair @ A @ A @ U @ W2 ) )
& ( E23
= ( product_Pair @ A @ A @ W2 @ V2 ) ) )
=> ( ( ( ( E12
= ( product_Pair @ A @ A @ V2 @ W2 ) )
& ( E23
= ( product_Pair @ A @ A @ W2 @ U ) ) )
=> ( ( pair_co_path @ A @ ( product_Pair @ A @ A @ U @ V2 ) @ W2 @ ( cons @ ( product_prod @ A @ A ) @ E12 @ ( cons @ ( product_prod @ A @ A ) @ E23 @ Es2 ) ) )
= ( cons @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V2 @ U ) @ ( pair_co_path @ A @ ( product_Pair @ A @ A @ U @ V2 ) @ W2 @ Es2 ) ) ) )
& ( ~ ( ( E12
= ( product_Pair @ A @ A @ V2 @ W2 ) )
& ( E23
= ( product_Pair @ A @ A @ W2 @ U ) ) )
=> ( ( pair_co_path @ A @ ( product_Pair @ A @ A @ U @ V2 ) @ W2 @ ( cons @ ( product_prod @ A @ A ) @ E12 @ ( cons @ ( product_prod @ A @ A ) @ E23 @ Es2 ) ) )
= ( cons @ ( product_prod @ A @ A ) @ E12 @ ( pair_co_path @ A @ ( product_Pair @ A @ A @ U @ V2 ) @ W2 @ ( cons @ ( product_prod @ A @ A ) @ E23 @ Es2 ) ) ) ) ) ) ) ) ).
% co_path.simps(3)
thf(fact_205_co__path_Osimps_I2_J,axiom,
! [A: $tType,Uw2: product_prod @ A @ A,Ux2: A,E3: product_prod @ A @ A] :
( ( pair_co_path @ A @ Uw2 @ Ux2 @ ( cons @ ( product_prod @ A @ A ) @ E3 @ ( nil @ ( product_prod @ A @ A ) ) ) )
= ( cons @ ( product_prod @ A @ A ) @ E3 @ ( nil @ ( product_prod @ A @ A ) ) ) ) ).
% co_path.simps(2)
thf(fact_206_co__path_Oelims,axiom,
! [A: $tType,X3: product_prod @ A @ A,Xa2: A,Xb: list @ ( product_prod @ A @ A ),Y3: list @ ( product_prod @ A @ A )] :
( ( ( pair_co_path @ A @ X3 @ Xa2 @ Xb )
= Y3 )
=> ( ( ( Xb
= ( nil @ ( product_prod @ A @ A ) ) )
=> ( Y3
!= ( nil @ ( product_prod @ A @ A ) ) ) )
=> ( ! [E2: product_prod @ A @ A] :
( ( Xb
= ( cons @ ( product_prod @ A @ A ) @ E2 @ ( nil @ ( product_prod @ A @ A ) ) ) )
=> ( Y3
!= ( cons @ ( product_prod @ A @ A ) @ E2 @ ( nil @ ( product_prod @ A @ A ) ) ) ) )
=> ~ ! [U2: A,V: A] :
( ( X3
= ( product_Pair @ A @ A @ U2 @ V ) )
=> ! [E1: product_prod @ A @ A,E22: product_prod @ A @ A,Es: list @ ( product_prod @ A @ A )] :
( ( Xb
= ( cons @ ( product_prod @ A @ A ) @ E1 @ ( cons @ ( product_prod @ A @ A ) @ E22 @ Es ) ) )
=> ~ ( ( ( ( E1
= ( product_Pair @ A @ A @ U2 @ Xa2 ) )
& ( E22
= ( product_Pair @ A @ A @ Xa2 @ V ) ) )
=> ( Y3
= ( cons @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U2 @ V ) @ ( pair_co_path @ A @ ( product_Pair @ A @ A @ U2 @ V ) @ Xa2 @ Es ) ) ) )
& ( ~ ( ( E1
= ( product_Pair @ A @ A @ U2 @ Xa2 ) )
& ( E22
= ( product_Pair @ A @ A @ Xa2 @ V ) ) )
=> ( ( ( ( E1
= ( product_Pair @ A @ A @ V @ Xa2 ) )
& ( E22
= ( product_Pair @ A @ A @ Xa2 @ U2 ) ) )
=> ( Y3
= ( cons @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V @ U2 ) @ ( pair_co_path @ A @ ( product_Pair @ A @ A @ U2 @ V ) @ Xa2 @ Es ) ) ) )
& ( ~ ( ( E1
= ( product_Pair @ A @ A @ V @ Xa2 ) )
& ( E22
= ( product_Pair @ A @ A @ Xa2 @ U2 ) ) )
=> ( Y3
= ( cons @ ( product_prod @ A @ A ) @ E1 @ ( pair_co_path @ A @ ( product_Pair @ A @ A @ U2 @ V ) @ Xa2 @ ( cons @ ( product_prod @ A @ A ) @ E22 @ Es ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% co_path.elims
thf(fact_207_co__sd__id,axiom,
! [A: $tType,U: A,W2: A,P3: list @ ( product_prod @ A @ A ),V2: A] :
( ~ ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ W2 ) @ ( set2 @ ( product_prod @ A @ A ) @ P3 ) )
=> ( ~ ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V2 @ W2 ) @ ( set2 @ ( product_prod @ A @ A ) @ P3 ) )
=> ( ( pair_co_path @ A @ ( product_Pair @ A @ A @ U @ V2 ) @ W2 @ ( pair_sd_path @ A @ ( product_Pair @ A @ A @ U @ V2 ) @ W2 @ P3 ) )
= P3 ) ) ) ).
% co_sd_id
thf(fact_208_subseqs_Osimps_I1_J,axiom,
! [A: $tType] :
( ( subseqs @ A @ ( nil @ A ) )
= ( cons @ ( list @ A ) @ ( nil @ A ) @ ( nil @ ( list @ A ) ) ) ) ).
% subseqs.simps(1)
thf(fact_209_Cons__lenlex__iff,axiom,
! [A: $tType,M: A,Ms: list @ A,N: A,Ns: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ M @ Ms ) @ ( cons @ A @ N @ Ns ) ) @ ( lenlex @ A @ R ) )
= ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Ms ) @ ( size_size @ ( list @ A ) @ Ns ) )
| ( ( ( size_size @ ( list @ A ) @ Ms )
= ( size_size @ ( list @ A ) @ Ns ) )
& ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ M @ N ) @ R ) )
| ( ( M = N )
& ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ms @ Ns ) @ ( lenlex @ A @ R ) ) ) ) ) ).
% Cons_lenlex_iff
thf(fact_210_length__induct,axiom,
! [A: $tType,P2: ( list @ A ) > $o,Xs2: list @ A] :
( ! [Xs4: list @ A] :
( ! [Ys7: list @ A] :
( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Ys7 ) @ ( size_size @ ( list @ A ) @ Xs4 ) )
=> ( P2 @ Ys7 ) )
=> ( P2 @ Xs4 ) )
=> ( P2 @ Xs2 ) ) ).
% length_induct
thf(fact_211_subseqs__refl,axiom,
! [A: $tType,Xs2: list @ A] : ( member2 @ ( list @ A ) @ Xs2 @ ( set2 @ ( list @ A ) @ ( subseqs @ A @ Xs2 ) ) ) ).
% subseqs_refl
thf(fact_212_Cons__in__subseqsD,axiom,
! [A: $tType,Y3: A,Ys2: list @ A,Xs2: list @ A] :
( ( member2 @ ( list @ A ) @ ( cons @ A @ Y3 @ Ys2 ) @ ( set2 @ ( list @ A ) @ ( subseqs @ A @ Xs2 ) ) )
=> ( member2 @ ( list @ A ) @ Ys2 @ ( set2 @ ( list @ A ) @ ( subseqs @ A @ Xs2 ) ) ) ) ).
% Cons_in_subseqsD
thf(fact_213_in__measures_I2_J,axiom,
! [A: $tType,X3: A,Y3: A,F3: A > nat,Fs: list @ ( A > nat )] :
( ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ ( measures @ A @ ( cons @ ( A > nat ) @ F3 @ Fs ) ) )
= ( ( ord_less @ nat @ ( F3 @ X3 ) @ ( F3 @ Y3 ) )
| ( ( ( F3 @ X3 )
= ( F3 @ Y3 ) )
& ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ ( measures @ A @ Fs ) ) ) ) ) ).
% in_measures(2)
thf(fact_214_inverse__permutation__of__list_Oelims,axiom,
! [A: $tType,X3: list @ ( product_prod @ A @ A ),Xa2: A,Y3: A] :
( ( ( invers965609412f_list @ A @ X3 @ Xa2 )
= Y3 )
=> ( ( ( X3
= ( nil @ ( product_prod @ A @ A ) ) )
=> ( Y3 != Xa2 ) )
=> ~ ! [Y4: A,X6: A,Xs4: list @ ( product_prod @ A @ A )] :
( ( X3
= ( cons @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ X6 ) @ Xs4 ) )
=> ~ ( ( ( Xa2 = X6 )
=> ( Y3 = Y4 ) )
& ( ( Xa2 != X6 )
=> ( Y3
= ( invers965609412f_list @ A @ Xs4 @ Xa2 ) ) ) ) ) ) ) ).
% inverse_permutation_of_list.elims
thf(fact_215_in__measures_I1_J,axiom,
! [A: $tType,X3: A,Y3: A] :
~ ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ ( measures @ A @ ( nil @ ( A > nat ) ) ) ) ).
% in_measures(1)
thf(fact_216_eval__inverse__permutation__of__list_I3_J,axiom,
! [A: $tType,X3: A,X7: A,Y5: A,Xs2: list @ ( product_prod @ A @ A )] :
( ( X3 != X7 )
=> ( ( invers965609412f_list @ A @ ( cons @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ X7 ) @ Xs2 ) @ X3 )
= ( invers965609412f_list @ A @ Xs2 @ X3 ) ) ) ).
% eval_inverse_permutation_of_list(3)
thf(fact_217_eval__inverse__permutation__of__list_I2_J,axiom,
! [A: $tType,X3: A,X7: A,Y3: A,Xs2: list @ ( product_prod @ A @ A )] :
( ( X3 = X7 )
=> ( ( invers965609412f_list @ A @ ( cons @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X7 ) @ Xs2 ) @ X3 )
= Y3 ) ) ).
% eval_inverse_permutation_of_list(2)
thf(fact_218_inverse__permutation__of__list_Osimps_I1_J,axiom,
! [A: $tType,X3: A] :
( ( invers965609412f_list @ A @ ( nil @ ( product_prod @ A @ A ) ) @ X3 )
= X3 ) ).
% inverse_permutation_of_list.simps(1)
thf(fact_219_inverse__permutation__of__list_Osimps_I2_J,axiom,
! [A: $tType,X3: A,X7: A,Y3: A,Xs2: list @ ( product_prod @ A @ A )] :
( ( ( X3 = X7 )
=> ( ( invers965609412f_list @ A @ ( cons @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X7 ) @ Xs2 ) @ X3 )
= Y3 ) )
& ( ( X3 != X7 )
=> ( ( invers965609412f_list @ A @ ( cons @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X7 ) @ Xs2 ) @ X3 )
= ( invers965609412f_list @ A @ Xs2 @ X3 ) ) ) ) ).
% inverse_permutation_of_list.simps(2)
thf(fact_220_measures__less,axiom,
! [A: $tType,F3: A > nat,X3: A,Y3: A,Fs: list @ ( A > nat )] :
( ( ord_less @ nat @ ( F3 @ X3 ) @ ( F3 @ Y3 ) )
=> ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ ( measures @ A @ ( cons @ ( A > nat ) @ F3 @ Fs ) ) ) ) ).
% measures_less
thf(fact_221_inverse__permutation__of__list__unique,axiom,
! [A: $tType,Xs2: list @ ( product_prod @ A @ A ),A5: set @ A,X3: A,Y3: A] :
( ( list_permutes @ A @ Xs2 @ A5 )
=> ( ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ ( set2 @ ( product_prod @ A @ A ) @ Xs2 ) )
=> ( ( invers965609412f_list @ A @ Xs2 @ Y3 )
= X3 ) ) ) ).
% inverse_permutation_of_list_unique
thf(fact_222_eval__permutation__of__list_I3_J,axiom,
! [A: $tType,X3: A,X7: A,Y5: A,Xs2: list @ ( product_prod @ A @ A )] :
( ( X3 != X7 )
=> ( ( permutation_of_list @ A @ ( cons @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X7 @ Y5 ) @ Xs2 ) @ X3 )
= ( permutation_of_list @ A @ Xs2 @ X3 ) ) ) ).
% eval_permutation_of_list(3)
thf(fact_223_eval__permutation__of__list_I1_J,axiom,
! [A: $tType,X3: A] :
( ( permutation_of_list @ A @ ( nil @ ( product_prod @ A @ A ) ) @ X3 )
= X3 ) ).
% eval_permutation_of_list(1)
thf(fact_224_eval__permutation__of__list_I2_J,axiom,
! [A: $tType,X3: A,X7: A,Y3: A,Xs2: list @ ( product_prod @ A @ A )] :
( ( X3 = X7 )
=> ( ( permutation_of_list @ A @ ( cons @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X7 @ Y3 ) @ Xs2 ) @ X3 )
= Y3 ) ) ).
% eval_permutation_of_list(2)
thf(fact_225_permutation__of__list__unique,axiom,
! [A: $tType,Xs2: list @ ( product_prod @ A @ A ),A5: set @ A,X3: A,Y3: A] :
( ( list_permutes @ A @ Xs2 @ A5 )
=> ( ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ ( set2 @ ( product_prod @ A @ A ) @ Xs2 ) )
=> ( ( permutation_of_list @ A @ Xs2 @ X3 )
= Y3 ) ) ) ).
% permutation_of_list_unique
thf(fact_226_permutation__of__list__Cons,axiom,
! [A: $tType,X3: A,X7: A,Y3: A,Xs2: list @ ( product_prod @ A @ A )] :
( ( ( X3 = X7 )
=> ( ( permutation_of_list @ A @ ( cons @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ Xs2 ) @ X7 )
= Y3 ) )
& ( ( X3 != X7 )
=> ( ( permutation_of_list @ A @ ( cons @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ Xs2 ) @ X7 )
= ( permutation_of_list @ A @ Xs2 @ X7 ) ) ) ) ).
% permutation_of_list_Cons
thf(fact_227_part__code_I1_J,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,Pivot: A] :
( ( linorder_part @ B @ A @ F3 @ Pivot @ ( nil @ B ) )
= ( product_Pair @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) ) @ ( nil @ B ) @ ( product_Pair @ ( list @ B ) @ ( list @ B ) @ ( nil @ B ) @ ( nil @ B ) ) ) ) ) ).
% part_code(1)
thf(fact_228_not__in__set__insert,axiom,
! [A: $tType,X3: A,Xs2: list @ A] :
( ~ ( member2 @ A @ X3 @ ( set2 @ A @ Xs2 ) )
=> ( ( insert @ A @ X3 @ Xs2 )
= ( cons @ A @ X3 @ Xs2 ) ) ) ).
% not_in_set_insert
thf(fact_229_in__set__insert,axiom,
! [A: $tType,X3: A,Xs2: list @ A] :
( ( member2 @ A @ X3 @ ( set2 @ A @ Xs2 ) )
=> ( ( insert @ A @ X3 @ Xs2 )
= Xs2 ) ) ).
% in_set_insert
thf(fact_230_insert__Nil,axiom,
! [A: $tType,X3: A] :
( ( insert @ A @ X3 @ ( nil @ A ) )
= ( cons @ A @ X3 @ ( nil @ A ) ) ) ).
% insert_Nil
thf(fact_231_List_Oinsert__def,axiom,
! [A: $tType] :
( ( insert @ A )
= ( ^ [X: A,Xs: list @ A] : ( if @ ( list @ A ) @ ( member2 @ A @ X @ ( set2 @ A @ Xs ) ) @ Xs @ ( cons @ A @ X @ Xs ) ) ) ) ).
% List.insert_def
thf(fact_232_concat__eq__append__conv,axiom,
! [A: $tType,Xss2: list @ ( list @ A ),Ys2: list @ A,Zs: list @ A] :
( ( ( concat @ A @ Xss2 )
= ( append @ A @ Ys2 @ Zs ) )
= ( ( ( Xss2
= ( nil @ ( list @ A ) ) )
=> ( ( Ys2
= ( nil @ A ) )
& ( Zs
= ( nil @ A ) ) ) )
& ( ( Xss2
!= ( nil @ ( list @ A ) ) )
=> ? [Xss1: list @ ( list @ A ),Xs: list @ A,Xs6: list @ A,Xss22: list @ ( list @ A )] :
( ( Xss2
= ( append @ ( list @ A ) @ Xss1 @ ( cons @ ( list @ A ) @ ( append @ A @ Xs @ Xs6 ) @ Xss22 ) ) )
& ( Ys2
= ( append @ A @ ( concat @ A @ Xss1 ) @ Xs ) )
& ( Zs
= ( append @ A @ Xs6 @ ( concat @ A @ Xss22 ) ) ) ) ) ) ) ).
% concat_eq_append_conv
thf(fact_233_inverse__permutation__of__list_Opelims,axiom,
! [A: $tType,X3: list @ ( product_prod @ A @ A ),Xa2: A,Y3: A] :
( ( ( invers965609412f_list @ A @ X3 @ Xa2 )
= Y3 )
=> ( ( accp @ ( product_prod @ ( list @ ( product_prod @ A @ A ) ) @ A ) @ ( invers1507583541st_rel @ A ) @ ( product_Pair @ ( list @ ( product_prod @ A @ A ) ) @ A @ X3 @ Xa2 ) )
=> ( ( ( X3
= ( nil @ ( product_prod @ A @ A ) ) )
=> ( ( Y3 = Xa2 )
=> ~ ( accp @ ( product_prod @ ( list @ ( product_prod @ A @ A ) ) @ A ) @ ( invers1507583541st_rel @ A ) @ ( product_Pair @ ( list @ ( product_prod @ A @ A ) ) @ A @ ( nil @ ( product_prod @ A @ A ) ) @ Xa2 ) ) ) )
=> ~ ! [Y4: A,X6: A,Xs4: list @ ( product_prod @ A @ A )] :
( ( X3
= ( cons @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ X6 ) @ Xs4 ) )
=> ( ( ( ( Xa2 = X6 )
=> ( Y3 = Y4 ) )
& ( ( Xa2 != X6 )
=> ( Y3
= ( invers965609412f_list @ A @ Xs4 @ Xa2 ) ) ) )
=> ~ ( accp @ ( product_prod @ ( list @ ( product_prod @ A @ A ) ) @ A ) @ ( invers1507583541st_rel @ A ) @ ( product_Pair @ ( list @ ( product_prod @ A @ A ) ) @ A @ ( cons @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ X6 ) @ Xs4 ) @ Xa2 ) ) ) ) ) ) ) ).
% inverse_permutation_of_list.pelims
thf(fact_234_Nil__eq__concat__conv,axiom,
! [A: $tType,Xss2: list @ ( list @ A )] :
( ( ( nil @ A )
= ( concat @ A @ Xss2 ) )
= ( ! [X: list @ A] :
( ( member2 @ ( list @ A ) @ X @ ( set2 @ ( list @ A ) @ Xss2 ) )
=> ( X
= ( nil @ A ) ) ) ) ) ).
% Nil_eq_concat_conv
thf(fact_235_concat__eq__Nil__conv,axiom,
! [A: $tType,Xss2: list @ ( list @ A )] :
( ( ( concat @ A @ Xss2 )
= ( nil @ A ) )
= ( ! [X: list @ A] :
( ( member2 @ ( list @ A ) @ X @ ( set2 @ ( list @ A ) @ Xss2 ) )
=> ( X
= ( nil @ A ) ) ) ) ) ).
% concat_eq_Nil_conv
thf(fact_236_concat__append,axiom,
! [A: $tType,Xs2: list @ ( list @ A ),Ys2: list @ ( list @ A )] :
( ( concat @ A @ ( append @ ( list @ A ) @ Xs2 @ Ys2 ) )
= ( append @ A @ ( concat @ A @ Xs2 ) @ ( concat @ A @ Ys2 ) ) ) ).
% concat_append
thf(fact_237_concat_Osimps_I2_J,axiom,
! [A: $tType,X3: list @ A,Xs2: list @ ( list @ A )] :
( ( concat @ A @ ( cons @ ( list @ A ) @ X3 @ Xs2 ) )
= ( append @ A @ X3 @ ( concat @ A @ Xs2 ) ) ) ).
% concat.simps(2)
thf(fact_238_concat_Osimps_I1_J,axiom,
! [A: $tType] :
( ( concat @ A @ ( nil @ ( list @ A ) ) )
= ( nil @ A ) ) ).
% concat.simps(1)
thf(fact_239_concat__eq__appendD,axiom,
! [A: $tType,Xss2: list @ ( list @ A ),Ys2: list @ A,Zs: list @ A] :
( ( ( concat @ A @ Xss2 )
= ( append @ A @ Ys2 @ Zs ) )
=> ( ( Xss2
!= ( nil @ ( list @ A ) ) )
=> ? [Xss12: list @ ( list @ A ),Xs4: list @ A,Xs3: list @ A,Xss23: list @ ( list @ A )] :
( ( Xss2
= ( append @ ( list @ A ) @ Xss12 @ ( cons @ ( list @ A ) @ ( append @ A @ Xs4 @ Xs3 ) @ Xss23 ) ) )
& ( Ys2
= ( append @ A @ ( concat @ A @ Xss12 ) @ Xs4 ) )
& ( Zs
= ( append @ A @ Xs3 @ ( concat @ A @ Xss23 ) ) ) ) ) ) ).
% concat_eq_appendD
thf(fact_240_length__n__lists__elem,axiom,
! [A: $tType,Ys2: list @ A,N: nat,Xs2: list @ A] :
( ( member2 @ ( list @ A ) @ Ys2 @ ( set2 @ ( list @ A ) @ ( n_lists @ A @ N @ Xs2 ) ) )
=> ( ( size_size @ ( list @ A ) @ Ys2 )
= N ) ) ).
% length_n_lists_elem
thf(fact_241_listrel_Oinducts,axiom,
! [A: $tType,B: $tType,X1: list @ A,X2: list @ B,R: set @ ( product_prod @ A @ B ),P2: ( list @ A ) > ( list @ B ) > $o] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ X1 @ X2 ) @ ( listrel @ A @ B @ R ) )
=> ( ( P2 @ ( nil @ A ) @ ( nil @ B ) )
=> ( ! [X4: A,Y4: B,Xs4: list @ A,Ys4: list @ B] :
( ( member2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ R )
=> ( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs4 @ Ys4 ) @ ( listrel @ A @ B @ R ) )
=> ( ( P2 @ Xs4 @ Ys4 )
=> ( P2 @ ( cons @ A @ X4 @ Xs4 ) @ ( cons @ B @ Y4 @ Ys4 ) ) ) ) )
=> ( P2 @ X1 @ X2 ) ) ) ) ).
% listrel.inducts
thf(fact_242_listrel__eq__len,axiom,
! [A: $tType,B: $tType,Xs2: list @ A,Ys2: list @ B,R: set @ ( product_prod @ A @ B )] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs2 @ Ys2 ) @ ( listrel @ A @ B @ R ) )
=> ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ B ) @ Ys2 ) ) ) ).
% listrel_eq_len
thf(fact_243_listrel_ONil,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B )] : ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ ( nil @ B ) ) @ ( listrel @ A @ B @ R ) ) ).
% listrel.Nil
thf(fact_244_listrel__Nil1,axiom,
! [A: $tType,B: $tType,Xs2: list @ B,R: set @ ( product_prod @ A @ B )] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ Xs2 ) @ ( listrel @ A @ B @ R ) )
=> ( Xs2
= ( nil @ B ) ) ) ).
% listrel_Nil1
thf(fact_245_listrel__Nil2,axiom,
! [B: $tType,A: $tType,Xs2: list @ A,R: set @ ( product_prod @ A @ B )] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs2 @ ( nil @ B ) ) @ ( listrel @ A @ B @ R ) )
=> ( Xs2
= ( nil @ A ) ) ) ).
% listrel_Nil2
thf(fact_246_listrel_OCons,axiom,
! [B: $tType,A: $tType,X3: A,Y3: B,R: set @ ( product_prod @ A @ B ),Xs2: list @ A,Ys2: list @ B] :
( ( member2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R )
=> ( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs2 @ Ys2 ) @ ( listrel @ A @ B @ R ) )
=> ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ X3 @ Xs2 ) @ ( cons @ B @ Y3 @ Ys2 ) ) @ ( listrel @ A @ B @ R ) ) ) ) ).
% listrel.Cons
thf(fact_247_listrel__Cons1,axiom,
! [B: $tType,A: $tType,Y3: A,Ys2: list @ A,Xs2: list @ B,R: set @ ( product_prod @ A @ B )] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ Y3 @ Ys2 ) @ Xs2 ) @ ( listrel @ A @ B @ R ) )
=> ~ ! [Y4: B,Ys4: list @ B] :
( ( Xs2
= ( cons @ B @ Y4 @ Ys4 ) )
=> ( ( member2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Y3 @ Y4 ) @ R )
=> ~ ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Ys2 @ Ys4 ) @ ( listrel @ A @ B @ R ) ) ) ) ) ).
% listrel_Cons1
thf(fact_248_listrel__Cons2,axiom,
! [B: $tType,A: $tType,Xs2: list @ A,Y3: B,Ys2: list @ B,R: set @ ( product_prod @ A @ B )] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs2 @ ( cons @ B @ Y3 @ Ys2 ) ) @ ( listrel @ A @ B @ R ) )
=> ~ ! [X4: A,Xs4: list @ A] :
( ( Xs2
= ( cons @ A @ X4 @ Xs4 ) )
=> ( ( member2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y3 ) @ R )
=> ~ ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs4 @ Ys2 ) @ ( listrel @ A @ B @ R ) ) ) ) ) ).
% listrel_Cons2
thf(fact_249_listrel_Ocases,axiom,
! [B: $tType,A: $tType,A1: list @ A,A22: list @ B,R: set @ ( product_prod @ A @ B )] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ A1 @ A22 ) @ ( listrel @ A @ B @ R ) )
=> ( ( ( A1
= ( nil @ A ) )
=> ( A22
!= ( nil @ B ) ) )
=> ~ ! [X4: A,Y4: B,Xs4: list @ A] :
( ( A1
= ( cons @ A @ X4 @ Xs4 ) )
=> ! [Ys4: list @ B] :
( ( A22
= ( cons @ B @ Y4 @ Ys4 ) )
=> ( ( member2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ R )
=> ~ ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs4 @ Ys4 ) @ ( listrel @ A @ B @ R ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_250_listrel_Osimps,axiom,
! [B: $tType,A: $tType,A1: list @ A,A22: list @ B,R: set @ ( product_prod @ A @ B )] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ A1 @ A22 ) @ ( listrel @ A @ B @ R ) )
= ( ( ( A1
= ( nil @ A ) )
& ( A22
= ( nil @ B ) ) )
| ? [X: A,Y: B,Xs: list @ A,Ys: list @ B] :
( ( A1
= ( cons @ A @ X @ Xs ) )
& ( A22
= ( cons @ B @ Y @ Ys ) )
& ( member2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ R )
& ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs @ Ys ) @ ( listrel @ A @ B @ R ) ) ) ) ) ).
% listrel.simps
thf(fact_251_n__lists__Nil,axiom,
! [A: $tType,N: nat] :
( ( ( N
= ( zero_zero @ nat ) )
=> ( ( n_lists @ A @ N @ ( nil @ A ) )
= ( cons @ ( list @ A ) @ ( nil @ A ) @ ( nil @ ( list @ A ) ) ) ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( ( n_lists @ A @ N @ ( nil @ A ) )
= ( nil @ ( list @ A ) ) ) ) ) ).
% n_lists_Nil
thf(fact_252_n__lists_Osimps_I1_J,axiom,
! [A: $tType,Xs2: list @ A] :
( ( n_lists @ A @ ( zero_zero @ nat ) @ Xs2 )
= ( cons @ ( list @ A ) @ ( nil @ A ) @ ( nil @ ( list @ A ) ) ) ) ).
% n_lists.simps(1)
thf(fact_253_length__0__conv,axiom,
! [A: $tType,Xs2: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( zero_zero @ nat ) )
= ( Xs2
= ( nil @ A ) ) ) ).
% length_0_conv
thf(fact_254_length__greater__0__conv,axiom,
! [A: $tType,Xs2: list @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs2 ) )
= ( Xs2
!= ( nil @ A ) ) ) ).
% length_greater_0_conv
% Type constructors (8)
thf(tcon_fun___Orderings_Oord,axiom,
! [A8: $tType,A9: $tType] :
( ( ord @ A9 )
=> ( ord @ ( A8 > A9 ) ) ) ).
thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
linorder @ nat ).
thf(tcon_Nat_Onat___Orderings_Oord_1,axiom,
ord @ nat ).
thf(tcon_Set_Oset___Orderings_Oord_2,axiom,
! [A8: $tType] : ( ord @ ( set @ A8 ) ) ).
thf(tcon_HOL_Obool___Orderings_Olinorder_3,axiom,
linorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Oord_4,axiom,
ord @ $o ).
thf(tcon_Product__Type_Ounit___Orderings_Olinorder_5,axiom,
linorder @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oord_6,axiom,
ord @ product_unit ).
% Helper facts (3)
thf(help_If_3_1_T,axiom,
! [P2: $o] :
( ( P2 = $true )
| ( P2 = $false ) ) ).
thf(help_If_2_1_T,axiom,
! [A: $tType,X3: A,Y3: A] :
( ( if @ A @ $false @ X3 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X3: A,Y3: A] :
( ( if @ A @ $true @ X3 @ Y3 )
= X3 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
! [Xs4: list @ ( product_prod @ a @ a ),X4: a,Y4: a,Ys4: list @ ( product_prod @ a @ a )] :
( es
!= ( append @ ( product_prod @ a @ a ) @ Xs4 @ ( cons @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ X4 @ Y4 ) @ ( cons @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ Y4 @ X4 ) @ Ys4 ) ) ) ) ).
%------------------------------------------------------------------------------