TPTP Problem File: ITP100^2.p
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%------------------------------------------------------------------------------
% File : ITP100^2 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer ListInf problem prob_52__5408414_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 : ListInf/prob_52__5408414_1 [Des21]
% Status : Theorem
% Rating : 0.00 v7.5.0
% Syntax : Number of formulae : 327 ( 132 unt; 54 typ; 0 def)
% Number of atoms : 768 ( 467 equ; 0 cnn)
% Maximal formula atoms : 13 ( 2 avg)
% Number of connectives : 4717 ( 163 ~; 42 |; 91 &;4026 @)
% ( 0 <=>; 395 =>; 0 <=; 0 <~>)
% Maximal formula depth : 34 ( 8 avg)
% Number of types : 3 ( 2 usr)
% Number of type conns : 316 ( 316 >; 0 *; 0 +; 0 <<)
% Number of symbols : 55 ( 52 usr; 4 con; 0-6 aty)
% Number of variables : 1255 ( 27 ^;1100 !; 71 ?;1255 :)
% ( 57 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 16:15:32.855
%------------------------------------------------------------------------------
% Could-be-implicit typings (4)
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 (50)
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder,type,
order:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Opreorder,type,
preorder:
!>[A: $tType] : $o ).
thf(sy_c_BNF__Greatest__Fixpoint_OShift,type,
bNF_Greatest_Shift:
!>[A: $tType] : ( ( set @ ( list @ A ) ) > A > ( set @ ( list @ A ) ) ) ).
thf(sy_c_BNF__Greatest__Fixpoint_OSucc,type,
bNF_Greatest_Succ:
!>[A: $tType] : ( ( set @ ( list @ A ) ) > ( list @ A ) > ( set @ A ) ) ).
thf(sy_c_BNF__Greatest__Fixpoint_Oshift,type,
bNF_Greatest_shift:
!>[A: $tType,B: $tType] : ( ( ( list @ A ) > B ) > A > ( list @ A ) > B ) ).
thf(sy_c_If,type,
if:
!>[A: $tType] : ( $o > A > A > A ) ).
thf(sy_c_List2_Olist__asc,type,
list_asc:
!>[A: $tType] : ( ( list @ A ) > $o ) ).
thf(sy_c_List2_Olist__ord,type,
list_ord:
!>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > $o ) ).
thf(sy_c_List2_Olist__strict__asc,type,
list_strict_asc:
!>[A: $tType] : ( ( list @ A ) > $o ) ).
thf(sy_c_List2_Omap2,type,
map2:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( list @ A ) > ( list @ B ) > ( list @ C ) ) ).
thf(sy_c_ListInf__Mirabelle__nfmaokebij_Oi__append,type,
listIn521021761append:
!>[A: $tType] : ( ( list @ A ) > ( nat > A ) > nat > A ) ).
thf(sy_c_List_Oappend,type,
append:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Oarg__min__list,type,
arg_min_list:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( list @ A ) > A ) ).
thf(sy_c_List_Obind,type,
bind:
!>[A: $tType,B: $tType] : ( ( list @ A ) > ( A > ( list @ B ) ) > ( list @ B ) ) ).
thf(sy_c_List_Obutlast,type,
butlast:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Oconcat,type,
concat:
!>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ A ) ) ).
thf(sy_c_List_Odistinct__adj,type,
distinct_adj:
!>[A: $tType] : ( ( list @ A ) > $o ) ).
thf(sy_c_List_Ofoldr,type,
foldr:
!>[A: $tType,B: $tType] : ( ( A > B > B ) > ( list @ A ) > B > B ) ).
thf(sy_c_List_Oinsert,type,
insert:
!>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Olast,type,
last:
!>[A: $tType] : ( ( list @ A ) > A ) ).
thf(sy_c_List_Olist_OCons,type,
cons:
!>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Olist_ONil,type,
nil:
!>[A: $tType] : ( list @ A ) ).
thf(sy_c_List_Olist_Ocase__list,type,
case_list:
!>[B: $tType,A: $tType] : ( B > ( A > ( list @ A ) > B ) > ( list @ A ) > B ) ).
thf(sy_c_List_Olist_Ohd,type,
hd:
!>[A: $tType] : ( ( list @ A ) > A ) ).
thf(sy_c_List_Olist_Otl,type,
tl:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Olist__ex1,type,
list_ex1:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Olistrelp,type,
listrelp:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( list @ A ) > ( list @ B ) > $o ) ).
thf(sy_c_List_Omap__tailrec__rev,type,
map_tailrec_rev:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( list @ A ) > ( list @ B ) > ( list @ B ) ) ).
thf(sy_c_List_Omaps,type,
maps:
!>[A: $tType,B: $tType] : ( ( A > ( list @ B ) ) > ( list @ A ) > ( list @ B ) ) ).
thf(sy_c_List_Omember,type,
member:
!>[A: $tType] : ( ( list @ A ) > A > $o ) ).
thf(sy_c_List_Oord_Olexordp__eq,type,
lexordp_eq:
!>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Oord__class_Olexordp,type,
ord_lexordp:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Oord__class_Olexordp__eq,type,
ord_lexordp_eq:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Oproduct__lists,type,
product_lists:
!>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ ( list @ A ) ) ) ).
thf(sy_c_List_Oremdups__adj,type,
remdups_adj:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Orotate1,type,
rotate1:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Oshuffles,type,
shuffles:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( set @ ( list @ A ) ) ) ).
thf(sy_c_List_Osplice,type,
splice:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Osubseqs,type,
subseqs:
!>[A: $tType] : ( ( list @ A ) > ( list @ ( list @ A ) ) ) ).
thf(sy_c_List_Osuccessively,type,
successively:
!>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > $o ) ).
thf(sy_c_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_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_member,type,
member2:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_f,type,
f: nat > a ).
thf(sy_v_x,type,
x: a ).
thf(sy_v_xs,type,
xs: list @ a ).
% Relevant facts (255)
thf(fact_0_i__append__Nil,axiom,
! [A: $tType,F: nat > A] :
( ( listIn521021761append @ A @ ( nil @ A ) @ F )
= F ) ).
% i_append_Nil
thf(fact_1_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_2_list_Odistinct_I1_J,axiom,
! [A: $tType,X21: A,X22: list @ A] :
( ( nil @ A )
!= ( cons @ A @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_3_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_4_list_Oexhaust,axiom,
! [A: $tType,Y: list @ A] :
( ( Y
!= ( nil @ A ) )
=> ~ ! [X212: A,X222: list @ A] :
( Y
!= ( cons @ A @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_5_list_Oinducts,axiom,
! [A: $tType,P: ( list @ A ) > $o,List: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [X1: A,X2: list @ A] :
( ( P @ X2 )
=> ( P @ ( cons @ A @ X1 @ X2 ) ) )
=> ( P @ List ) ) ) ).
% list.inducts
thf(fact_6_neq__Nil__conv,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
= ( ? [Y2: A,Ys: list @ A] :
( Xs
= ( cons @ A @ Y2 @ Ys ) ) ) ) ).
% neq_Nil_conv
thf(fact_7_list__induct2_H,axiom,
! [A: $tType,B: $tType,P: ( list @ A ) > ( list @ B ) > $o,Xs: list @ A,Ys2: list @ B] :
( ( P @ ( nil @ A ) @ ( nil @ B ) )
=> ( ! [X: A,Xs2: list @ A] : ( P @ ( cons @ A @ X @ Xs2 ) @ ( nil @ B ) )
=> ( ! [Y3: B,Ys3: list @ B] : ( P @ ( nil @ A ) @ ( cons @ B @ Y3 @ Ys3 ) )
=> ( ! [X: A,Xs2: list @ A,Y3: B,Ys3: list @ B] :
( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons @ A @ X @ Xs2 ) @ ( cons @ B @ Y3 @ Ys3 ) ) )
=> ( P @ Xs @ Ys2 ) ) ) ) ) ).
% list_induct2'
thf(fact_8_splice_Oinduct,axiom,
! [A: $tType,P: ( list @ A ) > ( list @ A ) > $o,A0: list @ A,A1: list @ A] :
( ! [X_1: list @ A] : ( P @ ( nil @ A ) @ X_1 )
=> ( ! [X: A,Xs2: list @ A,Ys3: list @ A] :
( ( P @ Ys3 @ Xs2 )
=> ( P @ ( cons @ A @ X @ Xs2 ) @ Ys3 ) )
=> ( P @ A0 @ A1 ) ) ) ).
% splice.induct
thf(fact_9_induct__list012,axiom,
! [A: $tType,P: ( list @ A ) > $o,Xs: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [X: A] : ( P @ ( cons @ A @ X @ ( nil @ A ) ) )
=> ( ! [X: A,Y3: A,Zs: list @ A] :
( ( P @ Zs )
=> ( ( P @ ( cons @ A @ Y3 @ Zs ) )
=> ( P @ ( cons @ A @ X @ ( cons @ A @ Y3 @ Zs ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% induct_list012
thf(fact_10_min__list_Ocases,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X3: list @ A] :
( ! [X: A,Xs2: list @ A] :
( X3
!= ( cons @ A @ X @ Xs2 ) )
=> ( X3
= ( nil @ A ) ) ) ) ).
% min_list.cases
thf(fact_11_min__list_Oinduct,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [P: ( list @ A ) > $o,A0: list @ A] :
( ! [X: A,Xs2: list @ A] :
( ! [X213: A,X223: list @ A] :
( ( Xs2
= ( cons @ A @ X213 @ X223 ) )
=> ( P @ Xs2 ) )
=> ( P @ ( cons @ A @ X @ Xs2 ) ) )
=> ( ( P @ ( nil @ A ) )
=> ( P @ A0 ) ) ) ) ).
% min_list.induct
thf(fact_12_shuffles_Oinduct,axiom,
! [A: $tType,P: ( list @ A ) > ( list @ A ) > $o,A0: list @ A,A1: list @ A] :
( ! [X_1: list @ A] : ( P @ ( nil @ A ) @ X_1 )
=> ( ! [Xs2: list @ A] : ( P @ Xs2 @ ( nil @ A ) )
=> ( ! [X: A,Xs2: list @ A,Y3: A,Ys3: list @ A] :
( ( P @ Xs2 @ ( cons @ A @ Y3 @ Ys3 ) )
=> ( ( P @ ( cons @ A @ X @ Xs2 ) @ Ys3 )
=> ( P @ ( cons @ A @ X @ Xs2 ) @ ( cons @ A @ Y3 @ Ys3 ) ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% shuffles.induct
thf(fact_13_not__Cons__self2,axiom,
! [A: $tType,X3: A,Xs: list @ A] :
( ( cons @ A @ X3 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_14_strict__sorted_Oinduct,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: ( list @ A ) > $o,A0: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [X: A,Ys3: list @ A] :
( ( P @ Ys3 )
=> ( P @ ( cons @ A @ X @ Ys3 ) ) )
=> ( P @ A0 ) ) ) ) ).
% strict_sorted.induct
thf(fact_15_strict__sorted_Ocases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: list @ A] :
( ( X3
!= ( nil @ A ) )
=> ~ ! [X: A,Ys3: list @ A] :
( X3
!= ( cons @ A @ X @ Ys3 ) ) ) ) ).
% strict_sorted.cases
thf(fact_16_map__tailrec__rev_Oinduct,axiom,
! [A: $tType,B: $tType,P: ( A > B ) > ( list @ A ) > ( list @ B ) > $o,A0: A > B,A1: list @ A,A2: list @ B] :
( ! [F2: A > B,X_1: list @ B] : ( P @ F2 @ ( nil @ A ) @ X_1 )
=> ( ! [F2: A > B,A3: A,As: list @ A,Bs: list @ B] :
( ( P @ F2 @ As @ ( cons @ B @ ( F2 @ A3 ) @ Bs ) )
=> ( P @ F2 @ ( cons @ A @ A3 @ As ) @ Bs ) )
=> ( P @ A0 @ A1 @ A2 ) ) ) ).
% map_tailrec_rev.induct
thf(fact_17_list__nonempty__induct,axiom,
! [A: $tType,Xs: list @ A,P: ( list @ A ) > $o] :
( ( Xs
!= ( nil @ A ) )
=> ( ! [X: A] : ( P @ ( cons @ A @ X @ ( nil @ A ) ) )
=> ( ! [X: A,Xs2: list @ A] :
( ( Xs2
!= ( nil @ A ) )
=> ( ( P @ Xs2 )
=> ( P @ ( cons @ A @ X @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_18_successively_Oinduct,axiom,
! [A: $tType,P: ( A > A > $o ) > ( list @ A ) > $o,A0: A > A > $o,A1: list @ A] :
( ! [P2: A > A > $o] : ( P @ P2 @ ( nil @ A ) )
=> ( ! [P2: A > A > $o,X: A] : ( P @ P2 @ ( cons @ A @ X @ ( nil @ A ) ) )
=> ( ! [P2: A > A > $o,X: A,Y3: A,Xs2: list @ A] :
( ( P @ P2 @ ( cons @ A @ Y3 @ Xs2 ) )
=> ( P @ P2 @ ( cons @ A @ X @ ( cons @ A @ Y3 @ Xs2 ) ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% successively.induct
thf(fact_19_arg__min__list_Oinduct,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [P: ( A > B ) > ( list @ A ) > $o,A0: A > B,A1: list @ A] :
( ! [F2: A > B,X: A] : ( P @ F2 @ ( cons @ A @ X @ ( nil @ A ) ) )
=> ( ! [F2: A > B,X: A,Y3: A,Zs: list @ A] :
( ( P @ F2 @ ( cons @ A @ Y3 @ Zs ) )
=> ( P @ F2 @ ( cons @ A @ X @ ( cons @ A @ Y3 @ Zs ) ) ) )
=> ( ! [A3: A > B] : ( P @ A3 @ ( nil @ A ) )
=> ( P @ A0 @ A1 ) ) ) ) ) ).
% arg_min_list.induct
thf(fact_20_remdups__adj_Oinduct,axiom,
! [A: $tType,P: ( list @ A ) > $o,A0: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [X: A] : ( P @ ( cons @ A @ X @ ( nil @ A ) ) )
=> ( ! [X: A,Y3: A,Xs2: list @ A] :
( ( ( X = Y3 )
=> ( P @ ( cons @ A @ X @ Xs2 ) ) )
=> ( ( ( X != Y3 )
=> ( P @ ( cons @ A @ Y3 @ Xs2 ) ) )
=> ( P @ ( cons @ A @ X @ ( cons @ A @ Y3 @ Xs2 ) ) ) ) )
=> ( P @ A0 ) ) ) ) ).
% remdups_adj.induct
thf(fact_21_sorted__wrt_Oinduct,axiom,
! [A: $tType,P: ( A > A > $o ) > ( list @ A ) > $o,A0: A > A > $o,A1: list @ A] :
( ! [P2: A > A > $o] : ( P @ P2 @ ( nil @ A ) )
=> ( ! [P2: A > A > $o,X: A,Ys3: list @ A] :
( ( P @ P2 @ Ys3 )
=> ( P @ P2 @ ( cons @ A @ X @ Ys3 ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% sorted_wrt.induct
thf(fact_22_remdups__adj_Ocases,axiom,
! [A: $tType,X3: list @ A] :
( ( X3
!= ( nil @ A ) )
=> ( ! [X: A] :
( X3
!= ( cons @ A @ X @ ( nil @ A ) ) )
=> ~ ! [X: A,Y3: A,Xs2: list @ A] :
( X3
!= ( cons @ A @ X @ ( cons @ A @ Y3 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_23_transpose_Ocases,axiom,
! [A: $tType,X3: list @ ( list @ A )] :
( ( X3
!= ( nil @ ( list @ A ) ) )
=> ( ! [Xss: list @ ( list @ A )] :
( X3
!= ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss ) )
=> ~ ! [X: A,Xs2: list @ A,Xss: list @ ( list @ A )] :
( X3
!= ( cons @ ( list @ A ) @ ( cons @ A @ X @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_24_list__ord_Oinduct,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [P: ( A > A > $o ) > ( list @ A ) > $o,A0: A > A > $o,A1: list @ A] :
( ! [Ord: A > A > $o,X1: A,X2: A,Xs2: list @ A] :
( ( P @ Ord @ ( cons @ A @ X2 @ Xs2 ) )
=> ( P @ Ord @ ( cons @ A @ X1 @ ( cons @ A @ X2 @ Xs2 ) ) ) )
=> ( ! [Ord: A > A > $o] : ( P @ Ord @ ( nil @ A ) )
=> ( ! [Ord: A > A > $o,V: A] : ( P @ Ord @ ( cons @ A @ V @ ( nil @ A ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ) ).
% list_ord.induct
thf(fact_25_insert__Nil,axiom,
! [A: $tType,X3: A] :
( ( insert @ A @ X3 @ ( nil @ A ) )
= ( cons @ A @ X3 @ ( nil @ A ) ) ) ).
% insert_Nil
thf(fact_26_list__ex1__simps_I1_J,axiom,
! [A: $tType,P: A > $o] :
~ ( list_ex1 @ A @ P @ ( nil @ A ) ) ).
% list_ex1_simps(1)
thf(fact_27_map__tailrec__rev_Oelims,axiom,
! [A: $tType,B: $tType,X3: A > B,Xa: list @ A,Xb: list @ B,Y: list @ B] :
( ( ( map_tailrec_rev @ A @ B @ X3 @ Xa @ Xb )
= Y )
=> ( ( ( Xa
= ( nil @ A ) )
=> ( Y != Xb ) )
=> ~ ! [A3: A,As: list @ A] :
( ( Xa
= ( cons @ A @ A3 @ As ) )
=> ( Y
!= ( map_tailrec_rev @ A @ B @ X3 @ As @ ( cons @ B @ ( X3 @ A3 ) @ Xb ) ) ) ) ) ) ).
% map_tailrec_rev.elims
thf(fact_28_arg__min__list_Osimps_I1_J,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [F: A > B,X3: A] :
( ( arg_min_list @ A @ B @ F @ ( cons @ A @ X3 @ ( nil @ A ) ) )
= X3 ) ) ).
% arg_min_list.simps(1)
thf(fact_29_ord_Olexordp__eq__simps_I3_J,axiom,
! [A: $tType,Less: A > A > $o,X3: A,Xs: list @ A] :
~ ( lexordp_eq @ A @ Less @ ( cons @ A @ X3 @ Xs ) @ ( nil @ A ) ) ).
% ord.lexordp_eq_simps(3)
thf(fact_30_bind__simps_I1_J,axiom,
! [B: $tType,A: $tType,F: B > ( list @ A )] :
( ( bind @ B @ A @ ( nil @ B ) @ F )
= ( nil @ A ) ) ).
% bind_simps(1)
thf(fact_31_listrelp_Oinducts,axiom,
! [A: $tType,B: $tType,R: A > B > $o,X12: list @ A,X23: list @ B,P: ( list @ A ) > ( list @ B ) > $o] :
( ( listrelp @ A @ B @ R @ X12 @ X23 )
=> ( ( P @ ( nil @ A ) @ ( nil @ B ) )
=> ( ! [X: A,Y3: B,Xs2: list @ A,Ys3: list @ B] :
( ( R @ X @ Y3 )
=> ( ( listrelp @ A @ B @ R @ Xs2 @ Ys3 )
=> ( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons @ A @ X @ Xs2 ) @ ( cons @ B @ Y3 @ Ys3 ) ) ) ) )
=> ( P @ X12 @ X23 ) ) ) ) ).
% listrelp.inducts
thf(fact_32_listrelp_Osimps,axiom,
! [B: $tType,A: $tType] :
( ( listrelp @ A @ B )
= ( ^ [R2: A > B > $o,A12: list @ A,A22: list @ B] :
( ( ( A12
= ( nil @ A ) )
& ( A22
= ( nil @ B ) ) )
| ? [X4: A,Y2: B,Xs3: list @ A,Ys: list @ B] :
( ( A12
= ( cons @ A @ X4 @ Xs3 ) )
& ( A22
= ( cons @ B @ Y2 @ Ys ) )
& ( R2 @ X4 @ Y2 )
& ( listrelp @ A @ B @ R2 @ Xs3 @ Ys ) ) ) ) ) ).
% listrelp.simps
thf(fact_33_listrelp_Ocases,axiom,
! [A: $tType,B: $tType,R: A > B > $o,A1: list @ A,A2: list @ B] :
( ( listrelp @ A @ B @ R @ A1 @ A2 )
=> ( ( ( A1
= ( nil @ A ) )
=> ( A2
!= ( nil @ B ) ) )
=> ~ ! [X: A,Y3: B,Xs2: list @ A] :
( ( A1
= ( cons @ A @ X @ Xs2 ) )
=> ! [Ys3: list @ B] :
( ( A2
= ( cons @ B @ Y3 @ Ys3 ) )
=> ( ( R @ X @ Y3 )
=> ~ ( listrelp @ A @ B @ R @ Xs2 @ Ys3 ) ) ) ) ) ) ).
% listrelp.cases
thf(fact_34_lexordp__eq__simps_I3_J,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X3: A,Xs: list @ A] :
~ ( ord_lexordp_eq @ A @ ( cons @ A @ X3 @ Xs ) @ ( nil @ A ) ) ) ).
% lexordp_eq_simps(3)
thf(fact_35_lexordp__eq__simps_I2_J,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [Xs: list @ A] :
( ( ord_lexordp_eq @ A @ Xs @ ( nil @ A ) )
= ( Xs
= ( nil @ A ) ) ) ) ).
% lexordp_eq_simps(2)
thf(fact_36_lexordp__eq__simps_I1_J,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [Ys2: list @ A] : ( ord_lexordp_eq @ A @ ( nil @ A ) @ Ys2 ) ) ).
% lexordp_eq_simps(1)
thf(fact_37_ord_Olexordp__eq__simps_I4_J,axiom,
! [A: $tType,Less: A > A > $o,X3: A,Xs: list @ A,Y: A,Ys2: list @ A] :
( ( lexordp_eq @ A @ Less @ ( cons @ A @ X3 @ Xs ) @ ( cons @ A @ Y @ Ys2 ) )
= ( ( Less @ X3 @ Y )
| ( ~ ( Less @ Y @ X3 )
& ( lexordp_eq @ A @ Less @ Xs @ Ys2 ) ) ) ) ).
% ord.lexordp_eq_simps(4)
thf(fact_38_ord_Olexordp__eq__simps_I2_J,axiom,
! [A: $tType,Less: A > A > $o,Xs: list @ A] :
( ( lexordp_eq @ A @ Less @ Xs @ ( nil @ A ) )
= ( Xs
= ( nil @ A ) ) ) ).
% ord.lexordp_eq_simps(2)
thf(fact_39_ord_Olexordp__eq__simps_I1_J,axiom,
! [A: $tType,Less: A > A > $o,Ys2: list @ A] : ( lexordp_eq @ A @ Less @ ( nil @ A ) @ Ys2 ) ).
% ord.lexordp_eq_simps(1)
thf(fact_40_ord_Olexordp__eq_Ocong,axiom,
! [A: $tType] :
( ( lexordp_eq @ A )
= ( lexordp_eq @ A ) ) ).
% ord.lexordp_eq.cong
thf(fact_41_ord_Olexordp__eq__refl,axiom,
! [A: $tType,Less: A > A > $o,Xs: list @ A] : ( lexordp_eq @ A @ Less @ Xs @ Xs ) ).
% ord.lexordp_eq_refl
thf(fact_42_lexordp__eq__refl,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [Xs: list @ A] : ( ord_lexordp_eq @ A @ Xs @ Xs ) ) ).
% lexordp_eq_refl
thf(fact_43_lexordp__eq__trans,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,Ys2: list @ A,Zs2: list @ A] :
( ( ord_lexordp_eq @ A @ Xs @ Ys2 )
=> ( ( ord_lexordp_eq @ A @ Ys2 @ Zs2 )
=> ( ord_lexordp_eq @ A @ Xs @ Zs2 ) ) ) ) ).
% lexordp_eq_trans
thf(fact_44_mem__Collect__eq,axiom,
! [A: $tType,A4: A,P: A > $o] :
( ( member2 @ A @ A4 @ ( collect @ A @ P ) )
= ( P @ A4 ) ) ).
% mem_Collect_eq
thf(fact_45_Collect__mem__eq,axiom,
! [A: $tType,A5: set @ A] :
( ( collect @ A
@ ^ [X4: A] : ( member2 @ A @ X4 @ A5 ) )
= A5 ) ).
% Collect_mem_eq
thf(fact_46_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X: A] :
( ( P @ X )
= ( Q @ X ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_47_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X: A] :
( ( F @ X )
= ( G @ X ) )
=> ( F = G ) ) ).
% ext
thf(fact_48_lexordp__eq__linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,Ys2: list @ A] :
( ( ord_lexordp_eq @ A @ Xs @ Ys2 )
| ( ord_lexordp_eq @ A @ Ys2 @ Xs ) ) ) ).
% lexordp_eq_linear
thf(fact_49_lexordp__eq__antisym,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,Ys2: list @ A] :
( ( ord_lexordp_eq @ A @ Xs @ Ys2 )
=> ( ( ord_lexordp_eq @ A @ Ys2 @ Xs )
=> ( Xs = Ys2 ) ) ) ) ).
% lexordp_eq_antisym
thf(fact_50_lexordp__eq_ONil,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [Ys2: list @ A] : ( ord_lexordp_eq @ A @ ( nil @ A ) @ Ys2 ) ) ).
% lexordp_eq.Nil
thf(fact_51_ord_Olexordp__eq_OCons,axiom,
! [A: $tType,Less: A > A > $o,X3: A,Y: A,Xs: list @ A,Ys2: list @ A] :
( ( Less @ X3 @ Y )
=> ( lexordp_eq @ A @ Less @ ( cons @ A @ X3 @ Xs ) @ ( cons @ A @ Y @ Ys2 ) ) ) ).
% ord.lexordp_eq.Cons
thf(fact_52_ord_Olexordp__eq_OCons__eq,axiom,
! [A: $tType,Less: A > A > $o,X3: A,Y: A,Xs: list @ A,Ys2: list @ A] :
( ~ ( Less @ X3 @ Y )
=> ( ~ ( Less @ Y @ X3 )
=> ( ( lexordp_eq @ A @ Less @ Xs @ Ys2 )
=> ( lexordp_eq @ A @ Less @ ( cons @ A @ X3 @ Xs ) @ ( cons @ A @ Y @ Ys2 ) ) ) ) ) ).
% ord.lexordp_eq.Cons_eq
thf(fact_53_ord_Olexordp__eq_ONil,axiom,
! [A: $tType,Less: A > A > $o,Ys2: list @ A] : ( lexordp_eq @ A @ Less @ ( nil @ A ) @ Ys2 ) ).
% ord.lexordp_eq.Nil
thf(fact_54_listrelp_OCons,axiom,
! [A: $tType,B: $tType,R: A > B > $o,X3: A,Y: B,Xs: list @ A,Ys2: list @ B] :
( ( R @ X3 @ Y )
=> ( ( listrelp @ A @ B @ R @ Xs @ Ys2 )
=> ( listrelp @ A @ B @ R @ ( cons @ A @ X3 @ Xs ) @ ( cons @ B @ Y @ Ys2 ) ) ) ) ).
% listrelp.Cons
thf(fact_55_listrelp_ONil,axiom,
! [A: $tType,B: $tType,R: A > B > $o] : ( listrelp @ A @ B @ R @ ( nil @ A ) @ ( nil @ B ) ) ).
% listrelp.Nil
thf(fact_56_map__tailrec__rev_Osimps_I2_J,axiom,
! [A: $tType,B: $tType,F: A > B,A4: A,As2: list @ A,Bs2: list @ B] :
( ( map_tailrec_rev @ A @ B @ F @ ( cons @ A @ A4 @ As2 ) @ Bs2 )
= ( map_tailrec_rev @ A @ B @ F @ As2 @ ( cons @ B @ ( F @ A4 ) @ Bs2 ) ) ) ).
% map_tailrec_rev.simps(2)
thf(fact_57_map__tailrec__rev_Osimps_I1_J,axiom,
! [A: $tType,B: $tType,F: A > B,Bs2: list @ B] :
( ( map_tailrec_rev @ A @ B @ F @ ( nil @ A ) @ Bs2 )
= Bs2 ) ).
% map_tailrec_rev.simps(1)
thf(fact_58_ord_Olexordp__eq_Ocases,axiom,
! [A: $tType,Less: A > A > $o,A1: list @ A,A2: list @ A] :
( ( lexordp_eq @ A @ Less @ A1 @ A2 )
=> ( ( A1
!= ( nil @ A ) )
=> ( ! [X: A] :
( ? [Xs2: list @ A] :
( A1
= ( cons @ A @ X @ Xs2 ) )
=> ! [Y3: A] :
( ? [Ys3: list @ A] :
( A2
= ( cons @ A @ Y3 @ Ys3 ) )
=> ~ ( Less @ X @ Y3 ) ) )
=> ~ ! [X: A,Y3: A,Xs2: list @ A] :
( ( A1
= ( cons @ A @ X @ Xs2 ) )
=> ! [Ys3: list @ A] :
( ( A2
= ( cons @ A @ Y3 @ Ys3 ) )
=> ( ~ ( Less @ X @ Y3 )
=> ( ~ ( Less @ Y3 @ X )
=> ~ ( lexordp_eq @ A @ Less @ Xs2 @ Ys3 ) ) ) ) ) ) ) ) ).
% ord.lexordp_eq.cases
thf(fact_59_ord_Olexordp__eq_Osimps,axiom,
! [A: $tType] :
( ( lexordp_eq @ A )
= ( ^ [Less2: A > A > $o,A12: list @ A,A22: list @ A] :
( ? [Ys: list @ A] :
( ( A12
= ( nil @ A ) )
& ( A22 = Ys ) )
| ? [X4: A,Y2: A,Xs3: list @ A,Ys: list @ A] :
( ( A12
= ( cons @ A @ X4 @ Xs3 ) )
& ( A22
= ( cons @ A @ Y2 @ Ys ) )
& ( Less2 @ X4 @ Y2 ) )
| ? [X4: A,Y2: A,Xs3: list @ A,Ys: list @ A] :
( ( A12
= ( cons @ A @ X4 @ Xs3 ) )
& ( A22
= ( cons @ A @ Y2 @ Ys ) )
& ~ ( Less2 @ X4 @ Y2 )
& ~ ( Less2 @ Y2 @ X4 )
& ( lexordp_eq @ A @ Less2 @ Xs3 @ Ys ) ) ) ) ) ).
% ord.lexordp_eq.simps
thf(fact_60_ord_Olexordp__eq_Oinducts,axiom,
! [A: $tType,Less: A > A > $o,X12: list @ A,X23: list @ A,P: ( list @ A ) > ( list @ A ) > $o] :
( ( lexordp_eq @ A @ Less @ X12 @ X23 )
=> ( ! [X_1: list @ A] : ( P @ ( nil @ A ) @ X_1 )
=> ( ! [X: A,Y3: A,Xs2: list @ A,Ys3: list @ A] :
( ( Less @ X @ Y3 )
=> ( P @ ( cons @ A @ X @ Xs2 ) @ ( cons @ A @ Y3 @ Ys3 ) ) )
=> ( ! [X: A,Y3: A,Xs2: list @ A,Ys3: list @ A] :
( ~ ( Less @ X @ Y3 )
=> ( ~ ( Less @ Y3 @ X )
=> ( ( lexordp_eq @ A @ Less @ Xs2 @ Ys3 )
=> ( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons @ A @ X @ Xs2 ) @ ( cons @ A @ Y3 @ Ys3 ) ) ) ) ) )
=> ( P @ X12 @ X23 ) ) ) ) ) ).
% ord.lexordp_eq.inducts
thf(fact_61_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_62_subseqs_Osimps_I1_J,axiom,
! [A: $tType] :
( ( subseqs @ A @ ( nil @ A ) )
= ( cons @ ( list @ A ) @ ( nil @ A ) @ ( nil @ ( list @ A ) ) ) ) ).
% subseqs.simps(1)
thf(fact_63_lexordp__eq_Ocases,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A1: list @ A,A2: list @ A] :
( ( ord_lexordp_eq @ A @ A1 @ A2 )
=> ( ( A1
!= ( nil @ A ) )
=> ( ! [X: A] :
( ? [Xs2: list @ A] :
( A1
= ( cons @ A @ X @ Xs2 ) )
=> ! [Y3: A] :
( ? [Ys3: list @ A] :
( A2
= ( cons @ A @ Y3 @ Ys3 ) )
=> ~ ( ord_less @ A @ X @ Y3 ) ) )
=> ~ ! [X: A,Y3: A,Xs2: list @ A] :
( ( A1
= ( cons @ A @ X @ Xs2 ) )
=> ! [Ys3: list @ A] :
( ( A2
= ( cons @ A @ Y3 @ Ys3 ) )
=> ( ~ ( ord_less @ A @ X @ Y3 )
=> ( ~ ( ord_less @ A @ Y3 @ X )
=> ~ ( ord_lexordp_eq @ A @ Xs2 @ Ys3 ) ) ) ) ) ) ) ) ) ).
% lexordp_eq.cases
thf(fact_64_lexordp__eq_Osimps,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_lexordp_eq @ A )
= ( ^ [A12: list @ A,A22: list @ A] :
( ? [Ys: list @ A] :
( ( A12
= ( nil @ A ) )
& ( A22 = Ys ) )
| ? [X4: A,Y2: A,Xs3: list @ A,Ys: list @ A] :
( ( A12
= ( cons @ A @ X4 @ Xs3 ) )
& ( A22
= ( cons @ A @ Y2 @ Ys ) )
& ( ord_less @ A @ X4 @ Y2 ) )
| ? [X4: A,Y2: A,Xs3: list @ A,Ys: list @ A] :
( ( A12
= ( cons @ A @ X4 @ Xs3 ) )
& ( A22
= ( cons @ A @ Y2 @ Ys ) )
& ~ ( ord_less @ A @ X4 @ Y2 )
& ~ ( ord_less @ A @ Y2 @ X4 )
& ( ord_lexordp_eq @ A @ Xs3 @ Ys ) ) ) ) ) ) ).
% lexordp_eq.simps
thf(fact_65_lexordp__eq_Oinducts,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X12: list @ A,X23: list @ A,P: ( list @ A ) > ( list @ A ) > $o] :
( ( ord_lexordp_eq @ A @ X12 @ X23 )
=> ( ! [X_1: list @ A] : ( P @ ( nil @ A ) @ X_1 )
=> ( ! [X: A,Y3: A,Xs2: list @ A,Ys3: list @ A] :
( ( ord_less @ A @ X @ Y3 )
=> ( P @ ( cons @ A @ X @ Xs2 ) @ ( cons @ A @ Y3 @ Ys3 ) ) )
=> ( ! [X: A,Y3: A,Xs2: list @ A,Ys3: list @ A] :
( ~ ( ord_less @ A @ X @ Y3 )
=> ( ~ ( ord_less @ A @ Y3 @ X )
=> ( ( ord_lexordp_eq @ A @ Xs2 @ Ys3 )
=> ( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons @ A @ X @ Xs2 ) @ ( cons @ A @ Y3 @ Ys3 ) ) ) ) ) )
=> ( P @ X12 @ X23 ) ) ) ) ) ) ).
% lexordp_eq.inducts
thf(fact_66_member__rec_I2_J,axiom,
! [A: $tType,Y: A] :
~ ( member @ A @ ( nil @ A ) @ Y ) ).
% member_rec(2)
thf(fact_67_member__rec_I1_J,axiom,
! [A: $tType,X3: A,Xs: list @ A,Y: A] :
( ( member @ A @ ( cons @ A @ X3 @ Xs ) @ Y )
= ( ( X3 = Y )
| ( member @ A @ Xs @ Y ) ) ) ).
% member_rec(1)
thf(fact_68_splice_Oelims,axiom,
! [A: $tType,X3: list @ A,Xa: list @ A,Y: list @ A] :
( ( ( splice @ A @ X3 @ Xa )
= Y )
=> ( ( ( X3
= ( nil @ A ) )
=> ( Y != Xa ) )
=> ~ ! [X: A,Xs2: list @ A] :
( ( X3
= ( cons @ A @ X @ Xs2 ) )
=> ( Y
!= ( cons @ A @ X @ ( splice @ A @ Xa @ Xs2 ) ) ) ) ) ) ).
% splice.elims
thf(fact_69_bind__simps_I2_J,axiom,
! [A: $tType,B: $tType,X3: B,Xs: list @ B,F: B > ( list @ A )] :
( ( bind @ B @ A @ ( cons @ B @ X3 @ Xs ) @ F )
= ( append @ A @ ( F @ X3 ) @ ( bind @ B @ A @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_70_same__append__eq,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A,Zs2: list @ A] :
( ( ( append @ A @ Xs @ Ys2 )
= ( append @ A @ Xs @ Zs2 ) )
= ( Ys2 = Zs2 ) ) ).
% same_append_eq
thf(fact_71_append__same__eq,axiom,
! [A: $tType,Ys2: list @ A,Xs: list @ A,Zs2: list @ A] :
( ( ( append @ A @ Ys2 @ Xs )
= ( append @ A @ Zs2 @ Xs ) )
= ( Ys2 = Zs2 ) ) ).
% append_same_eq
thf(fact_72_append__assoc,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A,Zs2: list @ A] :
( ( append @ A @ ( append @ A @ Xs @ Ys2 ) @ Zs2 )
= ( append @ A @ Xs @ ( append @ A @ Ys2 @ Zs2 ) ) ) ).
% append_assoc
thf(fact_73_append_Oassoc,axiom,
! [A: $tType,A4: list @ A,B2: list @ A,C2: list @ A] :
( ( append @ A @ ( append @ A @ A4 @ B2 ) @ C2 )
= ( append @ A @ A4 @ ( append @ A @ B2 @ C2 ) ) ) ).
% append.assoc
thf(fact_74_append_Oright__neutral,axiom,
! [A: $tType,A4: list @ A] :
( ( append @ A @ A4 @ ( nil @ A ) )
= A4 ) ).
% append.right_neutral
thf(fact_75_append__is__Nil__conv,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A] :
( ( ( append @ A @ Xs @ Ys2 )
= ( nil @ A ) )
= ( ( Xs
= ( nil @ A ) )
& ( Ys2
= ( nil @ A ) ) ) ) ).
% append_is_Nil_conv
thf(fact_76_Nil__is__append__conv,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A] :
( ( ( nil @ A )
= ( append @ A @ Xs @ Ys2 ) )
= ( ( Xs
= ( nil @ A ) )
& ( Ys2
= ( nil @ A ) ) ) ) ).
% Nil_is_append_conv
thf(fact_77_self__append__conv2,axiom,
! [A: $tType,Ys2: list @ A,Xs: list @ A] :
( ( Ys2
= ( append @ A @ Xs @ Ys2 ) )
= ( Xs
= ( nil @ A ) ) ) ).
% self_append_conv2
thf(fact_78_append__self__conv2,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A] :
( ( ( append @ A @ Xs @ Ys2 )
= Ys2 )
= ( Xs
= ( nil @ A ) ) ) ).
% append_self_conv2
thf(fact_79_self__append__conv,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A] :
( ( Xs
= ( append @ A @ Xs @ Ys2 ) )
= ( Ys2
= ( nil @ A ) ) ) ).
% self_append_conv
thf(fact_80_append__self__conv,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A] :
( ( ( append @ A @ Xs @ Ys2 )
= Xs )
= ( Ys2
= ( nil @ A ) ) ) ).
% append_self_conv
thf(fact_81_append__Nil2,axiom,
! [A: $tType,Xs: list @ A] :
( ( append @ A @ Xs @ ( nil @ A ) )
= Xs ) ).
% append_Nil2
thf(fact_82_split__Nil__iff,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A] :
( ( ( splice @ A @ Xs @ Ys2 )
= ( nil @ A ) )
= ( ( Xs
= ( nil @ A ) )
& ( Ys2
= ( nil @ A ) ) ) ) ).
% split_Nil_iff
thf(fact_83_splice__Nil2,axiom,
! [A: $tType,Xs: list @ A] :
( ( splice @ A @ Xs @ ( nil @ A ) )
= Xs ) ).
% splice_Nil2
thf(fact_84_i__append__assoc,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A,F: nat > A] :
( ( listIn521021761append @ A @ Xs @ ( listIn521021761append @ A @ Ys2 @ F ) )
= ( listIn521021761append @ A @ ( append @ A @ Xs @ Ys2 ) @ F ) ) ).
% i_append_assoc
thf(fact_85_append1__eq__conv,axiom,
! [A: $tType,Xs: list @ A,X3: A,Ys2: list @ A,Y: A] :
( ( ( append @ A @ Xs @ ( cons @ A @ X3 @ ( nil @ A ) ) )
= ( append @ A @ Ys2 @ ( cons @ A @ Y @ ( nil @ A ) ) ) )
= ( ( Xs = Ys2 )
& ( X3 = Y ) ) ) ).
% append1_eq_conv
thf(fact_86_lexordp__eq__simps_I4_J,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X3: A,Xs: list @ A,Y: A,Ys2: list @ A] :
( ( ord_lexordp_eq @ A @ ( cons @ A @ X3 @ Xs ) @ ( cons @ A @ Y @ Ys2 ) )
= ( ( ord_less @ A @ X3 @ Y )
| ( ~ ( ord_less @ A @ Y @ X3 )
& ( ord_lexordp_eq @ A @ Xs @ Ys2 ) ) ) ) ) ).
% lexordp_eq_simps(4)
thf(fact_87_append__eq__append__conv2,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A,Zs2: list @ A,Ts: list @ A] :
( ( ( append @ A @ Xs @ Ys2 )
= ( append @ A @ Zs2 @ Ts ) )
= ( ? [Us: list @ A] :
( ( ( Xs
= ( append @ A @ Zs2 @ Us ) )
& ( ( append @ A @ Us @ Ys2 )
= Ts ) )
| ( ( ( append @ A @ Xs @ Us )
= Zs2 )
& ( Ys2
= ( append @ A @ Us @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_88_append__eq__appendI,axiom,
! [A: $tType,Xs: list @ A,Xs1: list @ A,Zs2: list @ A,Ys2: list @ A,Us2: list @ A] :
( ( ( append @ A @ Xs @ Xs1 )
= Zs2 )
=> ( ( Ys2
= ( append @ A @ Xs1 @ Us2 ) )
=> ( ( append @ A @ Xs @ Ys2 )
= ( append @ A @ Zs2 @ Us2 ) ) ) ) ).
% append_eq_appendI
thf(fact_89_append__Cons,axiom,
! [A: $tType,X3: A,Xs: list @ A,Ys2: list @ A] :
( ( append @ A @ ( cons @ A @ X3 @ Xs ) @ Ys2 )
= ( cons @ A @ X3 @ ( append @ A @ Xs @ Ys2 ) ) ) ).
% append_Cons
thf(fact_90_Cons__eq__appendI,axiom,
! [A: $tType,X3: A,Xs1: list @ A,Ys2: list @ A,Xs: list @ A,Zs2: list @ A] :
( ( ( cons @ A @ X3 @ Xs1 )
= Ys2 )
=> ( ( Xs
= ( append @ A @ Xs1 @ Zs2 ) )
=> ( ( cons @ A @ X3 @ Xs )
= ( append @ A @ Ys2 @ Zs2 ) ) ) ) ).
% Cons_eq_appendI
thf(fact_91_append_Oleft__neutral,axiom,
! [A: $tType,A4: list @ A] :
( ( append @ A @ ( nil @ A ) @ A4 )
= A4 ) ).
% append.left_neutral
thf(fact_92_append__Nil,axiom,
! [A: $tType,Ys2: list @ A] :
( ( append @ A @ ( nil @ A ) @ Ys2 )
= Ys2 ) ).
% append_Nil
thf(fact_93_eq__Nil__appendI,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A] :
( ( Xs = Ys2 )
=> ( Xs
= ( append @ A @ ( nil @ A ) @ Ys2 ) ) ) ).
% eq_Nil_appendI
thf(fact_94_lexordp__eq__pref,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [U: list @ A,V2: list @ A] : ( ord_lexordp_eq @ A @ U @ ( append @ A @ U @ V2 ) ) ) ).
% lexordp_eq_pref
thf(fact_95_ord_Olexordp__eq__pref,axiom,
! [A: $tType,Less: A > A > $o,U: list @ A,V2: list @ A] : ( lexordp_eq @ A @ Less @ U @ ( append @ A @ U @ V2 ) ) ).
% ord.lexordp_eq_pref
thf(fact_96_splice_Osimps_I2_J,axiom,
! [A: $tType,X3: A,Xs: list @ A,Ys2: list @ A] :
( ( splice @ A @ ( cons @ A @ X3 @ Xs ) @ Ys2 )
= ( cons @ A @ X3 @ ( splice @ A @ Ys2 @ Xs ) ) ) ).
% splice.simps(2)
thf(fact_97_splice_Osimps_I1_J,axiom,
! [A: $tType,Ys2: list @ A] :
( ( splice @ A @ ( nil @ A ) @ Ys2 )
= Ys2 ) ).
% splice.simps(1)
thf(fact_98_rev__nonempty__induct,axiom,
! [A: $tType,Xs: list @ A,P: ( list @ A ) > $o] :
( ( Xs
!= ( nil @ A ) )
=> ( ! [X: A] : ( P @ ( cons @ A @ X @ ( nil @ A ) ) )
=> ( ! [X: A,Xs2: list @ A] :
( ( Xs2
!= ( nil @ A ) )
=> ( ( P @ Xs2 )
=> ( P @ ( append @ A @ Xs2 @ ( cons @ A @ X @ ( nil @ A ) ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_99_append__eq__Cons__conv,axiom,
! [A: $tType,Ys2: list @ A,Zs2: list @ A,X3: A,Xs: list @ A] :
( ( ( append @ A @ Ys2 @ Zs2 )
= ( cons @ A @ X3 @ Xs ) )
= ( ( ( Ys2
= ( nil @ A ) )
& ( Zs2
= ( cons @ A @ X3 @ Xs ) ) )
| ? [Ys4: list @ A] :
( ( Ys2
= ( cons @ A @ X3 @ Ys4 ) )
& ( ( append @ A @ Ys4 @ Zs2 )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_100_Cons__eq__append__conv,axiom,
! [A: $tType,X3: A,Xs: list @ A,Ys2: list @ A,Zs2: list @ A] :
( ( ( cons @ A @ X3 @ Xs )
= ( append @ A @ Ys2 @ Zs2 ) )
= ( ( ( Ys2
= ( nil @ A ) )
& ( ( cons @ A @ X3 @ Xs )
= Zs2 ) )
| ? [Ys4: list @ A] :
( ( ( cons @ A @ X3 @ Ys4 )
= Ys2 )
& ( Xs
= ( append @ A @ Ys4 @ Zs2 ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_101_rev__exhaust,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ~ ! [Ys3: list @ A,Y3: A] :
( Xs
!= ( append @ A @ Ys3 @ ( cons @ A @ Y3 @ ( nil @ A ) ) ) ) ) ).
% rev_exhaust
thf(fact_102_rev__induct,axiom,
! [A: $tType,P: ( list @ A ) > $o,Xs: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [X: A,Xs2: list @ A] :
( ( P @ Xs2 )
=> ( P @ ( append @ A @ Xs2 @ ( cons @ A @ X @ ( nil @ A ) ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_103_append__eq__Cons,axiom,
! [A: $tType,X3: A,Xs: list @ A] :
( ( append @ A @ ( cons @ A @ X3 @ ( nil @ A ) ) @ Xs )
= ( cons @ A @ X3 @ Xs ) ) ).
% append_eq_Cons
thf(fact_104_lexordp__eq_OCons,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X3: A,Y: A,Xs: list @ A,Ys2: list @ A] :
( ( ord_less @ A @ X3 @ Y )
=> ( ord_lexordp_eq @ A @ ( cons @ A @ X3 @ Xs ) @ ( cons @ A @ Y @ Ys2 ) ) ) ) ).
% lexordp_eq.Cons
thf(fact_105_lexordp__eq_OCons__eq,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X3: A,Y: A,Xs: list @ A,Ys2: list @ A] :
( ~ ( ord_less @ A @ X3 @ Y )
=> ( ~ ( ord_less @ A @ Y @ X3 )
=> ( ( ord_lexordp_eq @ A @ Xs @ Ys2 )
=> ( ord_lexordp_eq @ A @ ( cons @ A @ X3 @ Xs ) @ ( cons @ A @ Y @ Ys2 ) ) ) ) ) ) ).
% lexordp_eq.Cons_eq
thf(fact_106_maps__simps_I1_J,axiom,
! [A: $tType,B: $tType,F: B > ( list @ A ),X3: B,Xs: list @ B] :
( ( maps @ B @ A @ F @ ( cons @ B @ X3 @ Xs ) )
= ( append @ A @ ( F @ X3 ) @ ( maps @ B @ A @ F @ Xs ) ) ) ).
% maps_simps(1)
thf(fact_107_concat__eq__append__conv,axiom,
! [A: $tType,Xss2: list @ ( list @ A ),Ys2: list @ A,Zs2: list @ A] :
( ( ( concat @ A @ Xss2 )
= ( append @ A @ Ys2 @ Zs2 ) )
= ( ( ( Xss2
= ( nil @ ( list @ A ) ) )
=> ( ( Ys2
= ( nil @ A ) )
& ( Zs2
= ( nil @ A ) ) ) )
& ( ( Xss2
!= ( nil @ ( list @ A ) ) )
=> ? [Xss1: list @ ( list @ A ),Xs3: list @ A,Xs4: list @ A,Xss22: list @ ( list @ A )] :
( ( Xss2
= ( append @ ( list @ A ) @ Xss1 @ ( cons @ ( list @ A ) @ ( append @ A @ Xs3 @ Xs4 ) @ Xss22 ) ) )
& ( Ys2
= ( append @ A @ ( concat @ A @ Xss1 ) @ Xs3 ) )
& ( Zs2
= ( append @ A @ Xs4 @ ( concat @ A @ Xss22 ) ) ) ) ) ) ) ).
% concat_eq_append_conv
thf(fact_108_rotate1_Osimps_I2_J,axiom,
! [A: $tType,X3: A,Xs: list @ A] :
( ( rotate1 @ A @ ( cons @ A @ X3 @ Xs ) )
= ( append @ A @ Xs @ ( cons @ A @ X3 @ ( nil @ A ) ) ) ) ).
% rotate1.simps(2)
thf(fact_109_butlast__snoc,axiom,
! [A: $tType,Xs: list @ A,X3: A] :
( ( butlast @ A @ ( append @ A @ Xs @ ( cons @ A @ X3 @ ( nil @ A ) ) ) )
= Xs ) ).
% butlast_snoc
thf(fact_110_last__snoc,axiom,
! [A: $tType,Xs: list @ A,X3: A] :
( ( last @ A @ ( append @ A @ Xs @ ( cons @ A @ X3 @ ( nil @ A ) ) ) )
= X3 ) ).
% last_snoc
thf(fact_111_SuccI,axiom,
! [A: $tType,Kl: list @ A,K: A,Kl2: set @ ( list @ A )] :
( ( member2 @ ( list @ A ) @ ( append @ A @ Kl @ ( cons @ A @ K @ ( nil @ A ) ) ) @ Kl2 )
=> ( member2 @ A @ K @ ( bNF_Greatest_Succ @ A @ Kl2 @ Kl ) ) ) ).
% SuccI
thf(fact_112_SuccD,axiom,
! [A: $tType,K: A,Kl2: set @ ( list @ A ),Kl: list @ A] :
( ( member2 @ A @ K @ ( bNF_Greatest_Succ @ A @ Kl2 @ Kl ) )
=> ( member2 @ ( list @ A ) @ ( append @ A @ Kl @ ( cons @ A @ K @ ( nil @ A ) ) ) @ Kl2 ) ) ).
% SuccD
thf(fact_113_rotate1__is__Nil__conv,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( rotate1 @ A @ Xs )
= ( nil @ A ) )
= ( Xs
= ( nil @ A ) ) ) ).
% rotate1_is_Nil_conv
thf(fact_114_last__appendL,axiom,
! [A: $tType,Ys2: list @ A,Xs: list @ A] :
( ( Ys2
= ( nil @ A ) )
=> ( ( last @ A @ ( append @ A @ Xs @ Ys2 ) )
= ( last @ A @ Xs ) ) ) ).
% last_appendL
thf(fact_115_last__appendR,axiom,
! [A: $tType,Ys2: list @ A,Xs: list @ A] :
( ( Ys2
!= ( nil @ A ) )
=> ( ( last @ A @ ( append @ A @ Xs @ Ys2 ) )
= ( last @ A @ Ys2 ) ) ) ).
% last_appendR
thf(fact_116_concat__append,axiom,
! [A: $tType,Xs: list @ ( list @ A ),Ys2: list @ ( list @ A )] :
( ( concat @ A @ ( append @ ( list @ A ) @ Xs @ Ys2 ) )
= ( append @ A @ ( concat @ A @ Xs ) @ ( concat @ A @ Ys2 ) ) ) ).
% concat_append
thf(fact_117_append__butlast__last__id,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( append @ A @ ( butlast @ A @ Xs ) @ ( cons @ A @ ( last @ A @ Xs ) @ ( nil @ A ) ) )
= Xs ) ) ).
% append_butlast_last_id
thf(fact_118_butlast_Osimps_I1_J,axiom,
! [A: $tType] :
( ( butlast @ A @ ( nil @ A ) )
= ( nil @ A ) ) ).
% butlast.simps(1)
thf(fact_119_snoc__eq__iff__butlast,axiom,
! [A: $tType,Xs: list @ A,X3: A,Ys2: list @ A] :
( ( ( append @ A @ Xs @ ( cons @ A @ X3 @ ( nil @ A ) ) )
= Ys2 )
= ( ( Ys2
!= ( nil @ A ) )
& ( ( butlast @ A @ Ys2 )
= Xs )
& ( ( last @ A @ Ys2 )
= X3 ) ) ) ).
% snoc_eq_iff_butlast
thf(fact_120_rotate1_Osimps_I1_J,axiom,
! [A: $tType] :
( ( rotate1 @ A @ ( nil @ A ) )
= ( nil @ A ) ) ).
% rotate1.simps(1)
thf(fact_121_last_Osimps,axiom,
! [A: $tType,Xs: list @ A,X3: A] :
( ( ( Xs
= ( nil @ A ) )
=> ( ( last @ A @ ( cons @ A @ X3 @ Xs ) )
= X3 ) )
& ( ( Xs
!= ( nil @ A ) )
=> ( ( last @ A @ ( cons @ A @ X3 @ Xs ) )
= ( last @ A @ Xs ) ) ) ) ).
% last.simps
thf(fact_122_last__ConsL,axiom,
! [A: $tType,Xs: list @ A,X3: A] :
( ( Xs
= ( nil @ A ) )
=> ( ( last @ A @ ( cons @ A @ X3 @ Xs ) )
= X3 ) ) ).
% last_ConsL
thf(fact_123_last__ConsR,axiom,
! [A: $tType,Xs: list @ A,X3: A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( last @ A @ ( cons @ A @ X3 @ Xs ) )
= ( last @ A @ Xs ) ) ) ).
% last_ConsR
thf(fact_124_last__append,axiom,
! [A: $tType,Ys2: list @ A,Xs: list @ A] :
( ( ( Ys2
= ( nil @ A ) )
=> ( ( last @ A @ ( append @ A @ Xs @ Ys2 ) )
= ( last @ A @ Xs ) ) )
& ( ( Ys2
!= ( nil @ A ) )
=> ( ( last @ A @ ( append @ A @ Xs @ Ys2 ) )
= ( last @ A @ Ys2 ) ) ) ) ).
% last_append
thf(fact_125_longest__common__suffix,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A] :
? [Ss: list @ A,Xs5: list @ A,Ys5: list @ A] :
( ( Xs
= ( append @ A @ Xs5 @ Ss ) )
& ( Ys2
= ( append @ A @ Ys5 @ Ss ) )
& ( ( Xs5
= ( nil @ A ) )
| ( Ys5
= ( nil @ A ) )
| ( ( last @ A @ Xs5 )
!= ( last @ A @ Ys5 ) ) ) ) ).
% longest_common_suffix
thf(fact_126_butlast_Osimps_I2_J,axiom,
! [A: $tType,Xs: list @ A,X3: A] :
( ( ( Xs
= ( nil @ A ) )
=> ( ( butlast @ A @ ( cons @ A @ X3 @ Xs ) )
= ( nil @ A ) ) )
& ( ( Xs
!= ( nil @ A ) )
=> ( ( butlast @ A @ ( cons @ A @ X3 @ Xs ) )
= ( cons @ A @ X3 @ ( butlast @ A @ Xs ) ) ) ) ) ).
% butlast.simps(2)
thf(fact_127_concat_Osimps_I1_J,axiom,
! [A: $tType] :
( ( concat @ A @ ( nil @ ( list @ A ) ) )
= ( nil @ A ) ) ).
% concat.simps(1)
thf(fact_128_concat_Osimps_I2_J,axiom,
! [A: $tType,X3: list @ A,Xs: list @ ( list @ A )] :
( ( concat @ A @ ( cons @ ( list @ A ) @ X3 @ Xs ) )
= ( append @ A @ X3 @ ( concat @ A @ Xs ) ) ) ).
% concat.simps(2)
thf(fact_129_butlast__append,axiom,
! [A: $tType,Ys2: list @ A,Xs: list @ A] :
( ( ( Ys2
= ( nil @ A ) )
=> ( ( butlast @ A @ ( append @ A @ Xs @ Ys2 ) )
= ( butlast @ A @ Xs ) ) )
& ( ( Ys2
!= ( nil @ A ) )
=> ( ( butlast @ A @ ( append @ A @ Xs @ Ys2 ) )
= ( append @ A @ Xs @ ( butlast @ A @ Ys2 ) ) ) ) ) ).
% butlast_append
thf(fact_130_maps__simps_I2_J,axiom,
! [B: $tType,A: $tType,F: B > ( list @ A )] :
( ( maps @ B @ A @ F @ ( nil @ B ) )
= ( nil @ A ) ) ).
% maps_simps(2)
thf(fact_131_concat__eq__appendD,axiom,
! [A: $tType,Xss2: list @ ( list @ A ),Ys2: list @ A,Zs2: list @ A] :
( ( ( concat @ A @ Xss2 )
= ( append @ A @ Ys2 @ Zs2 ) )
=> ( ( Xss2
!= ( nil @ ( list @ A ) ) )
=> ? [Xss12: list @ ( list @ A ),Xs2: list @ A,Xs5: list @ A,Xss23: list @ ( list @ A )] :
( ( Xss2
= ( append @ ( list @ A ) @ Xss12 @ ( cons @ ( list @ A ) @ ( append @ A @ Xs2 @ Xs5 ) @ Xss23 ) ) )
& ( Ys2
= ( append @ A @ ( concat @ A @ Xss12 ) @ Xs2 ) )
& ( Zs2
= ( append @ A @ Xs5 @ ( concat @ A @ Xss23 ) ) ) ) ) ) ).
% concat_eq_appendD
thf(fact_132_shift__def,axiom,
! [B: $tType,A: $tType] :
( ( bNF_Greatest_shift @ A @ B )
= ( ^ [Lab: ( list @ A ) > B,K2: A,Kl3: list @ A] : ( Lab @ ( cons @ A @ K2 @ Kl3 ) ) ) ) ).
% shift_def
thf(fact_133_empty__Shift,axiom,
! [A: $tType,Kl2: set @ ( list @ A ),K: A] :
( ( member2 @ ( list @ A ) @ ( nil @ A ) @ Kl2 )
=> ( ( member2 @ A @ K @ ( bNF_Greatest_Succ @ A @ Kl2 @ ( nil @ A ) ) )
=> ( member2 @ ( list @ A ) @ ( nil @ A ) @ ( bNF_Greatest_Shift @ A @ Kl2 @ K ) ) ) ) ).
% empty_Shift
thf(fact_134_Succ__Shift,axiom,
! [A: $tType,Kl2: set @ ( list @ A ),K: A,Kl: list @ A] :
( ( bNF_Greatest_Succ @ A @ ( bNF_Greatest_Shift @ A @ Kl2 @ K ) @ Kl )
= ( bNF_Greatest_Succ @ A @ Kl2 @ ( cons @ A @ K @ Kl ) ) ) ).
% Succ_Shift
thf(fact_135_list__ord__snoc,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [Ord2: A > A > $o,Xs: list @ A,X3: A] :
( ( list_ord @ A @ Ord2 @ ( append @ A @ Xs @ ( cons @ A @ X3 @ ( nil @ A ) ) ) )
= ( ( Xs
= ( nil @ A ) )
| ( ( Ord2 @ ( last @ A @ Xs ) @ X3 )
& ( list_ord @ A @ Ord2 @ Xs ) ) ) ) ) ).
% list_ord_snoc
thf(fact_136_concat__conv__foldr,axiom,
! [A: $tType] :
( ( concat @ A )
= ( ^ [Xss3: list @ ( list @ A )] : ( foldr @ ( list @ A ) @ ( list @ A ) @ ( append @ A ) @ Xss3 @ ( nil @ A ) ) ) ) ).
% concat_conv_foldr
thf(fact_137_foldr__append,axiom,
! [B: $tType,A: $tType,F: B > A > A,Xs: list @ B,Ys2: list @ B,A4: A] :
( ( foldr @ B @ A @ F @ ( append @ B @ Xs @ Ys2 ) @ A4 )
= ( foldr @ B @ A @ F @ Xs @ ( foldr @ B @ A @ F @ Ys2 @ A4 ) ) ) ).
% foldr_append
thf(fact_138_list__ord_Oelims_I3_J,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X3: A > A > $o,Xa: list @ A] :
( ~ ( list_ord @ A @ X3 @ Xa )
=> ~ ! [X1: A,X2: A,Xs2: list @ A] :
( ( Xa
= ( cons @ A @ X1 @ ( cons @ A @ X2 @ Xs2 ) ) )
=> ( ( X3 @ X1 @ X2 )
& ( list_ord @ A @ X3 @ ( cons @ A @ X2 @ Xs2 ) ) ) ) ) ) ).
% list_ord.elims(3)
thf(fact_139_list__ord_Osimps_I1_J,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [Ord2: A > A > $o,X12: A,X23: A,Xs: list @ A] :
( ( list_ord @ A @ Ord2 @ ( cons @ A @ X12 @ ( cons @ A @ X23 @ Xs ) ) )
= ( ( Ord2 @ X12 @ X23 )
& ( list_ord @ A @ Ord2 @ ( cons @ A @ X23 @ Xs ) ) ) ) ) ).
% list_ord.simps(1)
thf(fact_140_list__ord_Osimps_I2_J,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [Ord2: A > A > $o] : ( list_ord @ A @ Ord2 @ ( nil @ A ) ) ) ).
% list_ord.simps(2)
thf(fact_141_list__ord__Nil,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [Ord2: A > A > $o] : ( list_ord @ A @ Ord2 @ ( nil @ A ) ) ) ).
% list_ord_Nil
thf(fact_142_list__ord__imp,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [Ord2: A > A > $o,Ord3: A > A > $o,Xs: list @ A] :
( ! [X: A,Y3: A] :
( ( Ord2 @ X @ Y3 )
=> ( Ord3 @ X @ Y3 ) )
=> ( ( list_ord @ A @ Ord2 @ Xs )
=> ( list_ord @ A @ Ord3 @ Xs ) ) ) ) ).
% list_ord_imp
thf(fact_143_list__ord__one,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [Ord2: A > A > $o,X3: A] : ( list_ord @ A @ Ord2 @ ( cons @ A @ X3 @ ( nil @ A ) ) ) ) ).
% list_ord_one
thf(fact_144_list__ord_Osimps_I3_J,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [Ord2: A > A > $o,V2: A] : ( list_ord @ A @ Ord2 @ ( cons @ A @ V2 @ ( nil @ A ) ) ) ) ).
% list_ord.simps(3)
thf(fact_145_list__ord_Oelims_I1_J,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X3: A > A > $o,Xa: list @ A,Y: $o] :
( ( ( list_ord @ A @ X3 @ Xa )
= Y )
=> ( ! [X1: A,X2: A,Xs2: list @ A] :
( ( Xa
= ( cons @ A @ X1 @ ( cons @ A @ X2 @ Xs2 ) ) )
=> ( Y
= ( ~ ( ( X3 @ X1 @ X2 )
& ( list_ord @ A @ X3 @ ( cons @ A @ X2 @ Xs2 ) ) ) ) ) )
=> ( ( ( Xa
= ( nil @ A ) )
=> ~ Y )
=> ~ ( ? [V: A] :
( Xa
= ( cons @ A @ V @ ( nil @ A ) ) )
=> ~ Y ) ) ) ) ) ).
% list_ord.elims(1)
thf(fact_146_list__ord_Oelims_I2_J,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X3: A > A > $o,Xa: list @ A] :
( ( list_ord @ A @ X3 @ Xa )
=> ( ! [X1: A,X2: A,Xs2: list @ A] :
( ( Xa
= ( cons @ A @ X1 @ ( cons @ A @ X2 @ Xs2 ) ) )
=> ~ ( ( X3 @ X1 @ X2 )
& ( list_ord @ A @ X3 @ ( cons @ A @ X2 @ Xs2 ) ) ) )
=> ( ( Xa
!= ( nil @ A ) )
=> ~ ! [V: A] :
( Xa
!= ( cons @ A @ V @ ( nil @ A ) ) ) ) ) ) ) ).
% list_ord.elims(2)
thf(fact_147_ShiftD,axiom,
! [A: $tType,Kl: list @ A,Kl2: set @ ( list @ A ),K: A] :
( ( member2 @ ( list @ A ) @ Kl @ ( bNF_Greatest_Shift @ A @ Kl2 @ K ) )
=> ( member2 @ ( list @ A ) @ ( cons @ A @ K @ Kl ) @ Kl2 ) ) ).
% ShiftD
thf(fact_148_list__strict__asc__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( list_strict_asc @ A )
= ( list_ord @ A @ ( ord_less @ A ) ) ) ) ).
% list_strict_asc_def
thf(fact_149_list__ord__append,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [Ord2: A > A > $o,Xs: list @ A,Ys2: list @ A] :
( ( list_ord @ A @ Ord2 @ ( append @ A @ Xs @ Ys2 ) )
= ( ( list_ord @ A @ Ord2 @ Xs )
& ( ( Ys2
= ( nil @ A ) )
| ( ( list_ord @ A @ Ord2 @ Ys2 )
& ( ( Xs
= ( nil @ A ) )
| ( Ord2 @ ( last @ A @ Xs ) @ ( hd @ A @ Ys2 ) ) ) ) ) ) ) ) ).
% list_ord_append
thf(fact_150_lexordp__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_lexordp @ A )
= ( ^ [Xs3: list @ A,Ys: list @ A] :
( ? [X4: A,Vs: list @ A] :
( Ys
= ( append @ A @ Xs3 @ ( cons @ A @ X4 @ Vs ) ) )
| ? [Us: list @ A,A6: A,B3: A,Vs: list @ A,Ws: list @ A] :
( ( ord_less @ A @ A6 @ B3 )
& ( Xs3
= ( append @ A @ Us @ ( cons @ A @ A6 @ Vs ) ) )
& ( Ys
= ( append @ A @ Us @ ( cons @ A @ B3 @ Ws ) ) ) ) ) ) ) ) ).
% lexordp_iff
thf(fact_151_lexordp__append__left__rightI,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X3: A,Y: A,Us2: list @ A,Xs: list @ A,Ys2: list @ A] :
( ( ord_less @ A @ X3 @ Y )
=> ( ord_lexordp @ A @ ( append @ A @ Us2 @ ( cons @ A @ X3 @ Xs ) ) @ ( append @ A @ Us2 @ ( cons @ A @ Y @ Ys2 ) ) ) ) ) ).
% lexordp_append_left_rightI
thf(fact_152_lexordp__simps_I2_J,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [Xs: list @ A] :
~ ( ord_lexordp @ A @ Xs @ ( nil @ A ) ) ) ).
% lexordp_simps(2)
thf(fact_153_lexordp__simps_I1_J,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [Ys2: list @ A] :
( ( ord_lexordp @ A @ ( nil @ A ) @ Ys2 )
= ( Ys2
!= ( nil @ A ) ) ) ) ).
% lexordp_simps(1)
thf(fact_154_lexordp__simps_I3_J,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X3: A,Xs: list @ A,Y: A,Ys2: list @ A] :
( ( ord_lexordp @ A @ ( cons @ A @ X3 @ Xs ) @ ( cons @ A @ Y @ Ys2 ) )
= ( ( ord_less @ A @ X3 @ Y )
| ( ~ ( ord_less @ A @ Y @ X3 )
& ( ord_lexordp @ A @ Xs @ Ys2 ) ) ) ) ) ).
% lexordp_simps(3)
thf(fact_155_hd__append2,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( hd @ A @ ( append @ A @ Xs @ Ys2 ) )
= ( hd @ A @ Xs ) ) ) ).
% hd_append2
thf(fact_156_lexordp__append__leftI,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [Us2: list @ A,Vs2: list @ A,Xs: list @ A] :
( ( ord_lexordp @ A @ Us2 @ Vs2 )
=> ( ord_lexordp @ A @ ( append @ A @ Xs @ Us2 ) @ ( append @ A @ Xs @ Vs2 ) ) ) ) ).
% lexordp_append_leftI
thf(fact_157_lexordp__antisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Xs: list @ A,Ys2: list @ A] :
( ( ord_lexordp @ A @ Xs @ Ys2 )
=> ~ ( ord_lexordp @ A @ Ys2 @ Xs ) ) ) ).
% lexordp_antisym
thf(fact_158_lexordp__trans,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,Ys2: list @ A,Zs2: list @ A] :
( ( ord_lexordp @ A @ Xs @ Ys2 )
=> ( ( ord_lexordp @ A @ Ys2 @ Zs2 )
=> ( ord_lexordp @ A @ Xs @ Zs2 ) ) ) ) ).
% lexordp_trans
thf(fact_159_lexordp__linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,Ys2: list @ A] :
( ( ord_lexordp @ A @ Xs @ Ys2 )
| ( Xs = Ys2 )
| ( ord_lexordp @ A @ Ys2 @ Xs ) ) ) ).
% lexordp_linear
thf(fact_160_lexordp__irreflexive_H,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Xs: list @ A] :
~ ( ord_lexordp @ A @ Xs @ Xs ) ) ).
% lexordp_irreflexive'
thf(fact_161_lexordp__irreflexive,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [Xs: list @ A] :
( ! [X: A] :
~ ( ord_less @ A @ X @ X )
=> ~ ( ord_lexordp @ A @ Xs @ Xs ) ) ) ).
% lexordp_irreflexive
thf(fact_162_list_Osel_I1_J,axiom,
! [A: $tType,X21: A,X22: list @ A] :
( ( hd @ A @ ( cons @ A @ X21 @ X22 ) )
= X21 ) ).
% list.sel(1)
thf(fact_163_hd__concat,axiom,
! [A: $tType,Xs: list @ ( list @ A )] :
( ( Xs
!= ( nil @ ( list @ A ) ) )
=> ( ( ( hd @ ( list @ A ) @ Xs )
!= ( nil @ A ) )
=> ( ( hd @ A @ ( concat @ A @ Xs ) )
= ( hd @ A @ ( hd @ ( list @ A ) @ Xs ) ) ) ) ) ).
% hd_concat
thf(fact_164_lexordp__into__lexordp__eq,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [Xs: list @ A,Ys2: list @ A] :
( ( ord_lexordp @ A @ Xs @ Ys2 )
=> ( ord_lexordp_eq @ A @ Xs @ Ys2 ) ) ) ).
% lexordp_into_lexordp_eq
thf(fact_165_lexordp__eq__conv__lexord,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_lexordp_eq @ A )
= ( ^ [Xs3: list @ A,Ys: list @ A] :
( ( Xs3 = Ys )
| ( ord_lexordp @ A @ Xs3 @ Ys ) ) ) ) ) ).
% lexordp_eq_conv_lexord
thf(fact_166_lexordp__conv__lexordp__eq,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_lexordp @ A )
= ( ^ [Xs3: list @ A,Ys: list @ A] :
( ( ord_lexordp_eq @ A @ Xs3 @ Ys )
& ~ ( ord_lexordp_eq @ A @ Ys @ Xs3 ) ) ) ) ) ).
% lexordp_conv_lexordp_eq
thf(fact_167_hd__append,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A] :
( ( ( Xs
= ( nil @ A ) )
=> ( ( hd @ A @ ( append @ A @ Xs @ Ys2 ) )
= ( hd @ A @ Ys2 ) ) )
& ( ( Xs
!= ( nil @ A ) )
=> ( ( hd @ A @ ( append @ A @ Xs @ Ys2 ) )
= ( hd @ A @ Xs ) ) ) ) ).
% hd_append
thf(fact_168_longest__common__prefix,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A] :
? [Ps: list @ A,Xs5: list @ A,Ys5: list @ A] :
( ( Xs
= ( append @ A @ Ps @ Xs5 ) )
& ( Ys2
= ( append @ A @ Ps @ Ys5 ) )
& ( ( Xs5
= ( nil @ A ) )
| ( Ys5
= ( nil @ A ) )
| ( ( hd @ A @ Xs5 )
!= ( hd @ A @ Ys5 ) ) ) ) ).
% longest_common_prefix
thf(fact_169_list__ord__Cons__imp,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [Ord2: A > A > $o,Xs: list @ A,X3: A] :
( ( list_ord @ A @ Ord2 @ Xs )
=> ( ( Ord2 @ X3 @ ( hd @ A @ Xs ) )
=> ( list_ord @ A @ Ord2 @ ( cons @ A @ X3 @ Xs ) ) ) ) ) ).
% list_ord_Cons_imp
thf(fact_170_lexordp_OCons,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X3: A,Y: A,Xs: list @ A,Ys2: list @ A] :
( ( ord_less @ A @ X3 @ Y )
=> ( ord_lexordp @ A @ ( cons @ A @ X3 @ Xs ) @ ( cons @ A @ Y @ Ys2 ) ) ) ) ).
% lexordp.Cons
thf(fact_171_lexordp_OCons__eq,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X3: A,Y: A,Xs: list @ A,Ys2: list @ A] :
( ~ ( ord_less @ A @ X3 @ Y )
=> ( ~ ( ord_less @ A @ Y @ X3 )
=> ( ( ord_lexordp @ A @ Xs @ Ys2 )
=> ( ord_lexordp @ A @ ( cons @ A @ X3 @ Xs ) @ ( cons @ A @ Y @ Ys2 ) ) ) ) ) ) ).
% lexordp.Cons_eq
thf(fact_172_lexordp__append__leftD,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [Xs: list @ A,Us2: list @ A,Vs2: list @ A] :
( ( ord_lexordp @ A @ ( append @ A @ Xs @ Us2 ) @ ( append @ A @ Xs @ Vs2 ) )
=> ( ! [A3: A] :
~ ( ord_less @ A @ A3 @ A3 )
=> ( ord_lexordp @ A @ Us2 @ Vs2 ) ) ) ) ).
% lexordp_append_leftD
thf(fact_173_lexordp_ONil,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [Y: A,Ys2: list @ A] : ( ord_lexordp @ A @ ( nil @ A ) @ ( cons @ A @ Y @ Ys2 ) ) ) ).
% lexordp.Nil
thf(fact_174_lexordp__append__rightI,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [Ys2: list @ A,Xs: list @ A] :
( ( Ys2
!= ( nil @ A ) )
=> ( ord_lexordp @ A @ Xs @ ( append @ A @ Xs @ Ys2 ) ) ) ) ).
% lexordp_append_rightI
thf(fact_175_list__ord__Cons,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [Ord2: A > A > $o,X3: A,Xs: list @ A] :
( ( list_ord @ A @ Ord2 @ ( cons @ A @ X3 @ Xs ) )
= ( ( Xs
= ( nil @ A ) )
| ( ( Ord2 @ X3 @ ( hd @ A @ Xs ) )
& ( list_ord @ A @ Ord2 @ Xs ) ) ) ) ) ).
% list_ord_Cons
thf(fact_176_lexordp_Ocases,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A1: list @ A,A2: list @ A] :
( ( ord_lexordp @ A @ A1 @ A2 )
=> ( ( ( A1
= ( nil @ A ) )
=> ! [Y3: A,Ys3: list @ A] :
( A2
!= ( cons @ A @ Y3 @ Ys3 ) ) )
=> ( ! [X: A] :
( ? [Xs2: list @ A] :
( A1
= ( cons @ A @ X @ Xs2 ) )
=> ! [Y3: A] :
( ? [Ys3: list @ A] :
( A2
= ( cons @ A @ Y3 @ Ys3 ) )
=> ~ ( ord_less @ A @ X @ Y3 ) ) )
=> ~ ! [X: A,Y3: A,Xs2: list @ A] :
( ( A1
= ( cons @ A @ X @ Xs2 ) )
=> ! [Ys3: list @ A] :
( ( A2
= ( cons @ A @ Y3 @ Ys3 ) )
=> ( ~ ( ord_less @ A @ X @ Y3 )
=> ( ~ ( ord_less @ A @ Y3 @ X )
=> ~ ( ord_lexordp @ A @ Xs2 @ Ys3 ) ) ) ) ) ) ) ) ) ).
% lexordp.cases
thf(fact_177_lexordp_Osimps,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_lexordp @ A )
= ( ^ [A12: list @ A,A22: list @ A] :
( ? [Y2: A,Ys: list @ A] :
( ( A12
= ( nil @ A ) )
& ( A22
= ( cons @ A @ Y2 @ Ys ) ) )
| ? [X4: A,Y2: A,Xs3: list @ A,Ys: list @ A] :
( ( A12
= ( cons @ A @ X4 @ Xs3 ) )
& ( A22
= ( cons @ A @ Y2 @ Ys ) )
& ( ord_less @ A @ X4 @ Y2 ) )
| ? [X4: A,Y2: A,Xs3: list @ A,Ys: list @ A] :
( ( A12
= ( cons @ A @ X4 @ Xs3 ) )
& ( A22
= ( cons @ A @ Y2 @ Ys ) )
& ~ ( ord_less @ A @ X4 @ Y2 )
& ~ ( ord_less @ A @ Y2 @ X4 )
& ( ord_lexordp @ A @ Xs3 @ Ys ) ) ) ) ) ) ).
% lexordp.simps
thf(fact_178_lexordp_Oinducts,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X12: list @ A,X23: list @ A,P: ( list @ A ) > ( list @ A ) > $o] :
( ( ord_lexordp @ A @ X12 @ X23 )
=> ( ! [Y3: A,Ys3: list @ A] : ( P @ ( nil @ A ) @ ( cons @ A @ Y3 @ Ys3 ) )
=> ( ! [X: A,Y3: A,Xs2: list @ A,Ys3: list @ A] :
( ( ord_less @ A @ X @ Y3 )
=> ( P @ ( cons @ A @ X @ Xs2 ) @ ( cons @ A @ Y3 @ Ys3 ) ) )
=> ( ! [X: A,Y3: A,Xs2: list @ A,Ys3: list @ A] :
( ~ ( ord_less @ A @ X @ Y3 )
=> ( ~ ( ord_less @ A @ Y3 @ X )
=> ( ( ord_lexordp @ A @ Xs2 @ Ys3 )
=> ( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons @ A @ X @ Xs2 ) @ ( cons @ A @ Y3 @ Ys3 ) ) ) ) ) )
=> ( P @ X12 @ X23 ) ) ) ) ) ) ).
% lexordp.inducts
thf(fact_179_lexordp__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,Ys2: list @ A] :
( ( ord_lexordp @ A @ Xs @ Ys2 )
=> ( ( ( Xs
= ( nil @ A ) )
=> ! [Y3: A,Ys5: list @ A] :
( Ys2
!= ( cons @ A @ Y3 @ Ys5 ) ) )
=> ( ! [X: A] :
( ? [Xs5: list @ A] :
( Xs
= ( cons @ A @ X @ Xs5 ) )
=> ! [Y3: A] :
( ? [Ys5: list @ A] :
( Ys2
= ( cons @ A @ Y3 @ Ys5 ) )
=> ~ ( ord_less @ A @ X @ Y3 ) ) )
=> ~ ! [X: A,Xs5: list @ A] :
( ( Xs
= ( cons @ A @ X @ Xs5 ) )
=> ! [Ys5: list @ A] :
( ( Ys2
= ( cons @ A @ X @ Ys5 ) )
=> ~ ( ord_lexordp @ A @ Xs5 @ Ys5 ) ) ) ) ) ) ) ).
% lexordp_cases
thf(fact_180_lexordp__induct,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,Ys2: list @ A,P: ( list @ A ) > ( list @ A ) > $o] :
( ( ord_lexordp @ A @ Xs @ Ys2 )
=> ( ! [Y3: A,Ys3: list @ A] : ( P @ ( nil @ A ) @ ( cons @ A @ Y3 @ Ys3 ) )
=> ( ! [X: A,Xs2: list @ A,Y3: A,Ys3: list @ A] :
( ( ord_less @ A @ X @ Y3 )
=> ( P @ ( cons @ A @ X @ Xs2 ) @ ( cons @ A @ Y3 @ Ys3 ) ) )
=> ( ! [X: A,Xs2: list @ A,Ys3: list @ A] :
( ( ord_lexordp @ A @ Xs2 @ Ys3 )
=> ( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons @ A @ X @ Xs2 ) @ ( cons @ A @ X @ Ys3 ) ) ) )
=> ( P @ Xs @ Ys2 ) ) ) ) ) ) ).
% lexordp_induct
thf(fact_181_distinct__adj__append__iff,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A] :
( ( distinct_adj @ A @ ( append @ A @ Xs @ Ys2 ) )
= ( ( distinct_adj @ A @ Xs )
& ( distinct_adj @ A @ Ys2 )
& ( ( Xs
= ( nil @ A ) )
| ( Ys2
= ( nil @ A ) )
| ( ( last @ A @ Xs )
!= ( hd @ A @ Ys2 ) ) ) ) ) ).
% distinct_adj_append_iff
thf(fact_182_rotate1__hd__tl,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( rotate1 @ A @ Xs )
= ( append @ A @ ( tl @ A @ Xs ) @ ( cons @ A @ ( hd @ A @ Xs ) @ ( nil @ A ) ) ) ) ) ).
% rotate1_hd_tl
thf(fact_183_successively__append__iff,axiom,
! [A: $tType,P: A > A > $o,Xs: list @ A,Ys2: list @ A] :
( ( successively @ A @ P @ ( append @ A @ Xs @ Ys2 ) )
= ( ( successively @ A @ P @ Xs )
& ( successively @ A @ P @ Ys2 )
& ( ( Xs
= ( nil @ A ) )
| ( Ys2
= ( nil @ A ) )
| ( P @ ( last @ A @ Xs ) @ ( hd @ A @ Ys2 ) ) ) ) ) ).
% successively_append_iff
thf(fact_184_distinct__adj__Cons__Cons,axiom,
! [B: $tType,X3: B,Y: B,Xs: list @ B] :
( ( distinct_adj @ B @ ( cons @ B @ X3 @ ( cons @ B @ Y @ Xs ) ) )
= ( ( X3 != Y )
& ( distinct_adj @ B @ ( cons @ B @ Y @ Xs ) ) ) ) ).
% distinct_adj_Cons_Cons
thf(fact_185_tl__append2,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( tl @ A @ ( append @ A @ Xs @ Ys2 ) )
= ( append @ A @ ( tl @ A @ Xs ) @ Ys2 ) ) ) ).
% tl_append2
thf(fact_186_hd__Cons__tl,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( cons @ A @ ( hd @ A @ Xs ) @ ( tl @ A @ Xs ) )
= Xs ) ) ).
% hd_Cons_tl
thf(fact_187_list_Ocollapse,axiom,
! [A: $tType,List: list @ A] :
( ( List
!= ( nil @ A ) )
=> ( ( cons @ A @ ( hd @ A @ List ) @ ( tl @ A @ List ) )
= List ) ) ).
% list.collapse
thf(fact_188_list_Osel_I3_J,axiom,
! [A: $tType,X21: A,X22: list @ A] :
( ( tl @ A @ ( cons @ A @ X21 @ X22 ) )
= X22 ) ).
% list.sel(3)
thf(fact_189_list_Osel_I2_J,axiom,
! [A: $tType] :
( ( tl @ A @ ( nil @ A ) )
= ( nil @ A ) ) ).
% list.sel(2)
thf(fact_190_successively_Oelims_I3_J,axiom,
! [A: $tType,X3: A > A > $o,Xa: list @ A] :
( ~ ( successively @ A @ X3 @ Xa )
=> ~ ! [X: A,Y3: A,Xs2: list @ A] :
( ( Xa
= ( cons @ A @ X @ ( cons @ A @ Y3 @ Xs2 ) ) )
=> ( ( X3 @ X @ Y3 )
& ( successively @ A @ X3 @ ( cons @ A @ Y3 @ Xs2 ) ) ) ) ) ).
% successively.elims(3)
thf(fact_191_successively_Osimps_I3_J,axiom,
! [A: $tType,P: A > A > $o,X3: A,Y: A,Xs: list @ A] :
( ( successively @ A @ P @ ( cons @ A @ X3 @ ( cons @ A @ Y @ Xs ) ) )
= ( ( P @ X3 @ Y )
& ( successively @ A @ P @ ( cons @ A @ Y @ Xs ) ) ) ) ).
% successively.simps(3)
thf(fact_192_successively_Osimps_I1_J,axiom,
! [A: $tType,P: A > A > $o] : ( successively @ A @ P @ ( nil @ A ) ) ).
% successively.simps(1)
thf(fact_193_distinct__adj__appendD1,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A] :
( ( distinct_adj @ A @ ( append @ A @ Xs @ Ys2 ) )
=> ( distinct_adj @ A @ Xs ) ) ).
% distinct_adj_appendD1
thf(fact_194_distinct__adj__appendD2,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A] :
( ( distinct_adj @ A @ ( append @ A @ Xs @ Ys2 ) )
=> ( distinct_adj @ A @ Ys2 ) ) ).
% distinct_adj_appendD2
thf(fact_195_distinct__adj__Nil,axiom,
! [A: $tType] : ( distinct_adj @ A @ ( nil @ A ) ) ).
% distinct_adj_Nil
thf(fact_196_distinct__adj__ConsD,axiom,
! [A: $tType,X3: A,Xs: list @ A] :
( ( distinct_adj @ A @ ( cons @ A @ X3 @ Xs ) )
=> ( distinct_adj @ A @ Xs ) ) ).
% distinct_adj_ConsD
thf(fact_197_butlast__tl,axiom,
! [A: $tType,Xs: list @ A] :
( ( butlast @ A @ ( tl @ A @ Xs ) )
= ( tl @ A @ ( butlast @ A @ Xs ) ) ) ).
% butlast_tl
thf(fact_198_tl__Nil,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( tl @ A @ Xs )
= ( nil @ A ) )
= ( ( Xs
= ( nil @ A ) )
| ? [X4: A] :
( Xs
= ( cons @ A @ X4 @ ( nil @ A ) ) ) ) ) ).
% tl_Nil
thf(fact_199_Nil__tl,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( nil @ A )
= ( tl @ A @ Xs ) )
= ( ( Xs
= ( nil @ A ) )
| ? [X4: A] :
( Xs
= ( cons @ A @ X4 @ ( nil @ A ) ) ) ) ) ).
% Nil_tl
thf(fact_200_list_Oexpand,axiom,
! [A: $tType,List: list @ A,List2: list @ A] :
( ( ( List
= ( nil @ A ) )
= ( List2
= ( nil @ A ) ) )
=> ( ( ( List
!= ( nil @ A ) )
=> ( ( List2
!= ( nil @ A ) )
=> ( ( ( hd @ A @ List )
= ( hd @ A @ List2 ) )
& ( ( tl @ A @ List )
= ( tl @ A @ List2 ) ) ) ) )
=> ( List = List2 ) ) ) ).
% list.expand
thf(fact_201_last__tl,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( Xs
= ( nil @ A ) )
| ( ( tl @ A @ Xs )
!= ( nil @ A ) ) )
=> ( ( last @ A @ ( tl @ A @ Xs ) )
= ( last @ A @ Xs ) ) ) ).
% last_tl
thf(fact_202_successively_Osimps_I2_J,axiom,
! [A: $tType,P: A > A > $o,X3: A] : ( successively @ A @ P @ ( cons @ A @ X3 @ ( nil @ A ) ) ) ).
% successively.simps(2)
thf(fact_203_successively_Oelims_I1_J,axiom,
! [A: $tType,X3: A > A > $o,Xa: list @ A,Y: $o] :
( ( ( successively @ A @ X3 @ Xa )
= Y )
=> ( ( ( Xa
= ( nil @ A ) )
=> ~ Y )
=> ( ( ? [X: A] :
( Xa
= ( cons @ A @ X @ ( nil @ A ) ) )
=> ~ Y )
=> ~ ! [X: A,Y3: A,Xs2: list @ A] :
( ( Xa
= ( cons @ A @ X @ ( cons @ A @ Y3 @ Xs2 ) ) )
=> ( Y
= ( ~ ( ( X3 @ X @ Y3 )
& ( successively @ A @ X3 @ ( cons @ A @ Y3 @ Xs2 ) ) ) ) ) ) ) ) ) ).
% successively.elims(1)
thf(fact_204_successively_Oelims_I2_J,axiom,
! [A: $tType,X3: A > A > $o,Xa: list @ A] :
( ( successively @ A @ X3 @ Xa )
=> ( ( Xa
!= ( nil @ A ) )
=> ( ! [X: A] :
( Xa
!= ( cons @ A @ X @ ( nil @ A ) ) )
=> ~ ! [X: A,Y3: A,Xs2: list @ A] :
( ( Xa
= ( cons @ A @ X @ ( cons @ A @ Y3 @ Xs2 ) ) )
=> ~ ( ( X3 @ X @ Y3 )
& ( successively @ A @ X3 @ ( cons @ A @ Y3 @ Xs2 ) ) ) ) ) ) ) ).
% successively.elims(2)
thf(fact_205_distinct__adj__singleton,axiom,
! [B: $tType,X3: B] : ( distinct_adj @ B @ ( cons @ B @ X3 @ ( nil @ B ) ) ) ).
% distinct_adj_singleton
thf(fact_206_list_Oexhaust__sel,axiom,
! [A: $tType,List: list @ A] :
( ( List
!= ( nil @ A ) )
=> ( List
= ( cons @ A @ ( hd @ A @ List ) @ ( tl @ A @ List ) ) ) ) ).
% list.exhaust_sel
thf(fact_207_successively__Cons,axiom,
! [A: $tType,P: A > A > $o,X3: A,Xs: list @ A] :
( ( successively @ A @ P @ ( cons @ A @ X3 @ Xs ) )
= ( ( Xs
= ( nil @ A ) )
| ( ( P @ X3 @ ( hd @ A @ Xs ) )
& ( successively @ A @ P @ Xs ) ) ) ) ).
% successively_Cons
thf(fact_208_distinct__adj__Cons,axiom,
! [A: $tType,X3: A,Xs: list @ A] :
( ( distinct_adj @ A @ ( cons @ A @ X3 @ Xs ) )
= ( ( Xs
= ( nil @ A ) )
| ( ( X3
!= ( hd @ A @ Xs ) )
& ( distinct_adj @ A @ Xs ) ) ) ) ).
% distinct_adj_Cons
thf(fact_209_list_Osplit__sel,axiom,
! [B: $tType,A: $tType,P: B > $o,F1: B,F22: A > ( list @ A ) > B,List: list @ A] :
( ( P @ ( case_list @ B @ A @ F1 @ F22 @ List ) )
= ( ( ( List
= ( nil @ A ) )
=> ( P @ F1 ) )
& ( ( List
= ( cons @ A @ ( hd @ A @ List ) @ ( tl @ A @ List ) ) )
=> ( P @ ( F22 @ ( hd @ A @ List ) @ ( tl @ A @ List ) ) ) ) ) ) ).
% list.split_sel
thf(fact_210_list_Osplit__sel__asm,axiom,
! [B: $tType,A: $tType,P: B > $o,F1: B,F22: A > ( list @ A ) > B,List: list @ A] :
( ( P @ ( case_list @ B @ A @ F1 @ F22 @ List ) )
= ( ~ ( ( ( List
= ( nil @ A ) )
& ~ ( P @ F1 ) )
| ( ( List
= ( cons @ A @ ( hd @ A @ List ) @ ( tl @ A @ List ) ) )
& ~ ( P @ ( F22 @ ( hd @ A @ List ) @ ( tl @ A @ List ) ) ) ) ) ) ) ).
% list.split_sel_asm
thf(fact_211_list_Ocase__eq__if,axiom,
! [A: $tType,B: $tType] :
( ( case_list @ B @ A )
= ( ^ [F12: B,F23: A > ( list @ A ) > B,List3: list @ A] :
( if @ B
@ ( List3
= ( nil @ A ) )
@ F12
@ ( F23 @ ( hd @ A @ List3 ) @ ( tl @ A @ List3 ) ) ) ) ) ).
% list.case_eq_if
thf(fact_212_list_Osimps_I5_J,axiom,
! [B: $tType,A: $tType,F1: B,F22: A > ( list @ A ) > B,X21: A,X22: list @ A] :
( ( case_list @ B @ A @ F1 @ F22 @ ( cons @ A @ X21 @ X22 ) )
= ( F22 @ X21 @ X22 ) ) ).
% list.simps(5)
thf(fact_213_list_Osimps_I4_J,axiom,
! [A: $tType,B: $tType,F1: B,F22: A > ( list @ A ) > B] :
( ( case_list @ B @ A @ F1 @ F22 @ ( nil @ A ) )
= F1 ) ).
% list.simps(4)
thf(fact_214_Cons__in__shuffles__iff,axiom,
! [A: $tType,Z: A,Zs2: list @ A,Xs: list @ A,Ys2: list @ A] :
( ( member2 @ ( list @ A ) @ ( cons @ A @ Z @ Zs2 ) @ ( shuffles @ A @ Xs @ Ys2 ) )
= ( ( ( Xs
!= ( nil @ A ) )
& ( ( hd @ A @ Xs )
= Z )
& ( member2 @ ( list @ A ) @ Zs2 @ ( shuffles @ A @ ( tl @ A @ Xs ) @ Ys2 ) ) )
| ( ( Ys2
!= ( nil @ A ) )
& ( ( hd @ A @ Ys2 )
= Z )
& ( member2 @ ( list @ A ) @ Zs2 @ ( shuffles @ A @ Xs @ ( tl @ A @ Ys2 ) ) ) ) ) ) ).
% Cons_in_shuffles_iff
thf(fact_215_map2__Cons__not__empty,axiom,
! [A: $tType,B: $tType,C: $tType,Xs: list @ A,F: A > C > B,Y: C,Ys2: list @ C] :
( ( Xs
!= ( nil @ A ) )
=> ( ( map2 @ A @ C @ B @ F @ Xs @ ( cons @ C @ Y @ Ys2 ) )
= ( cons @ B @ ( F @ ( hd @ A @ Xs ) @ Y ) @ ( map2 @ A @ C @ B @ F @ ( tl @ A @ Xs ) @ Ys2 ) ) ) ) ).
% map2_Cons_not_empty
thf(fact_216_Nil__in__shuffles,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A] :
( ( member2 @ ( list @ A ) @ ( nil @ A ) @ ( shuffles @ A @ Xs @ Ys2 ) )
= ( ( Xs
= ( nil @ A ) )
& ( Ys2
= ( nil @ A ) ) ) ) ).
% Nil_in_shuffles
thf(fact_217_splice__in__shuffles,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A] : ( member2 @ ( list @ A ) @ ( splice @ A @ Xs @ Ys2 ) @ ( shuffles @ A @ Xs @ Ys2 ) ) ).
% splice_in_shuffles
thf(fact_218_shufflesE,axiom,
! [A: $tType,Zs2: list @ A,Xs: list @ A,Ys2: list @ A] :
( ( member2 @ ( list @ A ) @ Zs2 @ ( shuffles @ A @ Xs @ Ys2 ) )
=> ( ( ( Zs2 = Xs )
=> ( Ys2
!= ( nil @ A ) ) )
=> ( ( ( Zs2 = Ys2 )
=> ( Xs
!= ( nil @ A ) ) )
=> ( ! [X: A,Xs5: list @ A] :
( ( Xs
= ( cons @ A @ X @ Xs5 ) )
=> ! [Z2: A,Zs3: list @ A] :
( ( Zs2
= ( cons @ A @ Z2 @ Zs3 ) )
=> ( ( X = Z2 )
=> ~ ( member2 @ ( list @ A ) @ Zs3 @ ( shuffles @ A @ Xs5 @ Ys2 ) ) ) ) )
=> ~ ! [Y3: A,Ys5: list @ A] :
( ( Ys2
= ( cons @ A @ Y3 @ Ys5 ) )
=> ! [Z2: A,Zs3: list @ A] :
( ( Zs2
= ( cons @ A @ Z2 @ Zs3 ) )
=> ( ( Y3 = Z2 )
=> ~ ( member2 @ ( list @ A ) @ Zs3 @ ( shuffles @ A @ Xs @ Ys5 ) ) ) ) ) ) ) ) ) ).
% shufflesE
thf(fact_219_map2__Cons__Cons,axiom,
! [B: $tType,A: $tType,C: $tType,F: B > C > A,X3: B,Xs: list @ B,Y: C,Ys2: list @ C] :
( ( map2 @ B @ C @ A @ F @ ( cons @ B @ X3 @ Xs ) @ ( cons @ C @ Y @ Ys2 ) )
= ( cons @ A @ ( F @ X3 @ Y ) @ ( map2 @ B @ C @ A @ F @ Xs @ Ys2 ) ) ) ).
% map2_Cons_Cons
thf(fact_220_map2_Osimps_I1_J,axiom,
! [B: $tType,A: $tType,C: $tType,F: A > B > C,Ys2: list @ B] :
( ( map2 @ A @ B @ C @ F @ ( nil @ A ) @ Ys2 )
= ( nil @ C ) ) ).
% map2.simps(1)
thf(fact_221_map2__Nil,axiom,
! [C: $tType,B: $tType,A: $tType,F: B > C > A,Ys2: list @ C] :
( ( map2 @ B @ C @ A @ F @ ( nil @ B ) @ Ys2 )
= ( nil @ A ) ) ).
% map2_Nil
thf(fact_222_map2__empty__conv,axiom,
! [A: $tType,C: $tType,B: $tType,F: B > C > A,Xs: list @ B,Ys2: list @ C] :
( ( ( map2 @ B @ C @ A @ F @ Xs @ Ys2 )
= ( nil @ A ) )
= ( Xs
= ( nil @ B ) ) ) ).
% map2_empty_conv
thf(fact_223_map2__not__empty__conv,axiom,
! [A: $tType,C: $tType,B: $tType,F: B > C > A,Xs: list @ B,Ys2: list @ C] :
( ( ( map2 @ B @ C @ A @ F @ Xs @ Ys2 )
!= ( nil @ A ) )
= ( Xs
!= ( nil @ B ) ) ) ).
% map2_not_empty_conv
thf(fact_224_Cons__in__shuffles__leftI,axiom,
! [A: $tType,Zs2: list @ A,Xs: list @ A,Ys2: list @ A,Z: A] :
( ( member2 @ ( list @ A ) @ Zs2 @ ( shuffles @ A @ Xs @ Ys2 ) )
=> ( member2 @ ( list @ A ) @ ( cons @ A @ Z @ Zs2 ) @ ( shuffles @ A @ ( cons @ A @ Z @ Xs ) @ Ys2 ) ) ) ).
% Cons_in_shuffles_leftI
thf(fact_225_Cons__in__shuffles__rightI,axiom,
! [A: $tType,Zs2: list @ A,Xs: list @ A,Ys2: list @ A,Z: A] :
( ( member2 @ ( list @ A ) @ Zs2 @ ( shuffles @ A @ Xs @ Ys2 ) )
=> ( member2 @ ( list @ A ) @ ( cons @ A @ Z @ Zs2 ) @ ( shuffles @ A @ Xs @ ( cons @ A @ Z @ Ys2 ) ) ) ) ).
% Cons_in_shuffles_rightI
thf(fact_226_Nil__in__shufflesI,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A] :
( ( Xs
= ( nil @ A ) )
=> ( ( Ys2
= ( nil @ A ) )
=> ( member2 @ ( list @ A ) @ ( nil @ A ) @ ( shuffles @ A @ Xs @ Ys2 ) ) ) ) ).
% Nil_in_shufflesI
thf(fact_227_shuffles__commutes,axiom,
! [A: $tType] :
( ( shuffles @ A )
= ( ^ [Xs3: list @ A,Ys: list @ A] : ( shuffles @ A @ Ys @ Xs3 ) ) ) ).
% shuffles_commutes
thf(fact_228_map2_Osimps_I2_J,axiom,
! [A: $tType,C: $tType,B: $tType,F: A > B > C,X3: A,Xs: list @ A,Ys2: list @ B] :
( ( map2 @ A @ B @ C @ F @ ( cons @ A @ X3 @ Xs ) @ Ys2 )
= ( cons @ C @ ( F @ X3 @ ( hd @ B @ Ys2 ) ) @ ( map2 @ A @ B @ C @ F @ Xs @ ( tl @ B @ Ys2 ) ) ) ) ).
% map2.simps(2)
thf(fact_229_map2__Cons__if,axiom,
! [B: $tType,A: $tType,C: $tType,Xs: list @ B,F: B > C > A,Y: C,Ys2: list @ C] :
( ( ( Xs
= ( nil @ B ) )
=> ( ( map2 @ B @ C @ A @ F @ Xs @ ( cons @ C @ Y @ Ys2 ) )
= ( nil @ A ) ) )
& ( ( Xs
!= ( nil @ B ) )
=> ( ( map2 @ B @ C @ A @ F @ Xs @ ( cons @ C @ Y @ Ys2 ) )
= ( cons @ A @ ( F @ ( hd @ B @ Xs ) @ Y ) @ ( map2 @ B @ C @ A @ F @ ( tl @ B @ Xs ) @ Ys2 ) ) ) ) ) ).
% map2_Cons_if
thf(fact_230_remdups__adj__append_H,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A] :
( ( ( Xs
= ( nil @ A ) )
| ( Ys2
= ( nil @ A ) )
| ( ( last @ A @ Xs )
!= ( hd @ A @ Ys2 ) ) )
=> ( ( remdups_adj @ A @ ( append @ A @ Xs @ Ys2 ) )
= ( append @ A @ ( remdups_adj @ A @ Xs ) @ ( remdups_adj @ A @ Ys2 ) ) ) ) ).
% remdups_adj_append'
thf(fact_231_remdups__adj__append,axiom,
! [A: $tType,Xs_1: list @ A,X3: A,Xs_2: list @ A] :
( ( remdups_adj @ A @ ( append @ A @ Xs_1 @ ( cons @ A @ X3 @ Xs_2 ) ) )
= ( append @ A @ ( remdups_adj @ A @ ( append @ A @ Xs_1 @ ( cons @ A @ X3 @ ( nil @ A ) ) ) ) @ ( tl @ A @ ( remdups_adj @ A @ ( cons @ A @ X3 @ Xs_2 ) ) ) ) ) ).
% remdups_adj_append
thf(fact_232_remdups__adj__Nil__iff,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( remdups_adj @ A @ Xs )
= ( nil @ A ) )
= ( Xs
= ( nil @ A ) ) ) ).
% remdups_adj_Nil_iff
thf(fact_233_hd__remdups__adj,axiom,
! [A: $tType,Xs: list @ A] :
( ( hd @ A @ ( remdups_adj @ A @ Xs ) )
= ( hd @ A @ Xs ) ) ).
% hd_remdups_adj
thf(fact_234_last__remdups__adj,axiom,
! [A: $tType,Xs: list @ A] :
( ( last @ A @ ( remdups_adj @ A @ Xs ) )
= ( last @ A @ Xs ) ) ).
% last_remdups_adj
thf(fact_235_remdups__adj__Cons__alt,axiom,
! [A: $tType,X3: A,Xs: list @ A] :
( ( cons @ A @ X3 @ ( tl @ A @ ( remdups_adj @ A @ ( cons @ A @ X3 @ Xs ) ) ) )
= ( remdups_adj @ A @ ( cons @ A @ X3 @ Xs ) ) ) ).
% remdups_adj_Cons_alt
thf(fact_236_successively__remdups__adjI,axiom,
! [A: $tType,P: A > A > $o,Xs: list @ A] :
( ( successively @ A @ P @ Xs )
=> ( successively @ A @ P @ ( remdups_adj @ A @ Xs ) ) ) ).
% successively_remdups_adjI
thf(fact_237_distinct__adj__remdups__adj,axiom,
! [A: $tType,Xs: list @ A] : ( distinct_adj @ A @ ( remdups_adj @ A @ Xs ) ) ).
% distinct_adj_remdups_adj
thf(fact_238_distinct__adj__altdef,axiom,
! [A: $tType] :
( ( distinct_adj @ A )
= ( ^ [Xs3: list @ A] :
( ( remdups_adj @ A @ Xs3 )
= Xs3 ) ) ) ).
% distinct_adj_altdef
thf(fact_239_remdups__adj_Oelims,axiom,
! [A: $tType,X3: list @ A,Y: list @ A] :
( ( ( remdups_adj @ A @ X3 )
= Y )
=> ( ( ( X3
= ( nil @ A ) )
=> ( Y
!= ( nil @ A ) ) )
=> ( ! [X: A] :
( ( X3
= ( cons @ A @ X @ ( nil @ A ) ) )
=> ( Y
!= ( cons @ A @ X @ ( nil @ A ) ) ) )
=> ~ ! [X: A,Y3: A,Xs2: list @ A] :
( ( X3
= ( cons @ A @ X @ ( cons @ A @ Y3 @ Xs2 ) ) )
=> ~ ( ( ( X = Y3 )
=> ( Y
= ( remdups_adj @ A @ ( cons @ A @ X @ Xs2 ) ) ) )
& ( ( X != Y3 )
=> ( Y
= ( cons @ A @ X @ ( remdups_adj @ A @ ( cons @ A @ Y3 @ Xs2 ) ) ) ) ) ) ) ) ) ) ).
% remdups_adj.elims
thf(fact_240_remdups__adj_Osimps_I2_J,axiom,
! [A: $tType,X3: A] :
( ( remdups_adj @ A @ ( cons @ A @ X3 @ ( nil @ A ) ) )
= ( cons @ A @ X3 @ ( nil @ A ) ) ) ).
% remdups_adj.simps(2)
thf(fact_241_remdups__adj_Osimps_I1_J,axiom,
! [A: $tType] :
( ( remdups_adj @ A @ ( nil @ A ) )
= ( nil @ A ) ) ).
% remdups_adj.simps(1)
thf(fact_242_remdups__adj_Osimps_I3_J,axiom,
! [A: $tType,X3: A,Y: A,Xs: list @ A] :
( ( ( X3 = Y )
=> ( ( remdups_adj @ A @ ( cons @ A @ X3 @ ( cons @ A @ Y @ Xs ) ) )
= ( remdups_adj @ A @ ( cons @ A @ X3 @ Xs ) ) ) )
& ( ( X3 != Y )
=> ( ( remdups_adj @ A @ ( cons @ A @ X3 @ ( cons @ A @ Y @ Xs ) ) )
= ( cons @ A @ X3 @ ( remdups_adj @ A @ ( cons @ A @ Y @ Xs ) ) ) ) ) ) ).
% remdups_adj.simps(3)
thf(fact_243_remdups__adj__append__two,axiom,
! [A: $tType,Xs: list @ A,X3: A,Y: A] :
( ( remdups_adj @ A @ ( append @ A @ Xs @ ( cons @ A @ X3 @ ( cons @ A @ Y @ ( nil @ A ) ) ) ) )
= ( append @ A @ ( remdups_adj @ A @ ( append @ A @ Xs @ ( cons @ A @ X3 @ ( nil @ A ) ) ) ) @ ( if @ ( list @ A ) @ ( X3 = Y ) @ ( nil @ A ) @ ( cons @ A @ Y @ ( nil @ A ) ) ) ) ) ).
% remdups_adj_append_two
thf(fact_244_map2__snoc__snoc,axiom,
! [B: $tType,A: $tType,C: $tType,Xs: list @ A,Ys2: list @ B,F: A > B > C,X3: A,Y: B] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys2 ) )
=> ( ( map2 @ A @ B @ C @ F @ ( append @ A @ Xs @ ( cons @ A @ X3 @ ( nil @ A ) ) ) @ ( append @ B @ Ys2 @ ( cons @ B @ Y @ ( nil @ B ) ) ) )
= ( append @ C @ ( map2 @ A @ B @ C @ F @ Xs @ Ys2 ) @ ( cons @ C @ ( F @ X3 @ Y ) @ ( nil @ C ) ) ) ) ) ).
% map2_snoc_snoc
thf(fact_245_list__strict__asc__imp__list__asc,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [Xs: list @ A] :
( ( list_strict_asc @ A @ Xs )
=> ( list_asc @ A @ Xs ) ) ) ).
% list_strict_asc_imp_list_asc
thf(fact_246_append__eq__append__conv,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A,Us2: list @ A,Vs2: list @ A] :
( ( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ Ys2 ) )
| ( ( size_size @ ( list @ A ) @ Us2 )
= ( size_size @ ( list @ A ) @ Vs2 ) ) )
=> ( ( ( append @ A @ Xs @ Us2 )
= ( append @ A @ Ys2 @ Vs2 ) )
= ( ( Xs = Ys2 )
& ( Us2 = Vs2 ) ) ) ) ).
% append_eq_append_conv
thf(fact_247_length__rotate1,axiom,
! [A: $tType,Xs: list @ A] :
( ( size_size @ ( list @ A ) @ ( rotate1 @ A @ Xs ) )
= ( size_size @ ( list @ A ) @ Xs ) ) ).
% length_rotate1
thf(fact_248_length__greater__imp__not__empty,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( Xs
!= ( nil @ A ) ) ) ).
% length_greater_imp_not_empty
thf(fact_249_neq__if__length__neq,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs )
!= ( size_size @ ( list @ A ) @ Ys2 ) )
=> ( Xs != Ys2 ) ) ).
% neq_if_length_neq
thf(fact_250_Ex__list__of__length,axiom,
! [A: $tType,N: nat] :
? [Xs2: list @ A] :
( ( size_size @ ( list @ A ) @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_251_length__induct,axiom,
! [A: $tType,P: ( list @ A ) > $o,Xs: list @ A] :
( ! [Xs2: list @ A] :
( ! [Ys6: list @ A] :
( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Ys6 ) @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( P @ Ys6 ) )
=> ( P @ Xs2 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_252_list__induct2,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys2: list @ B,P: ( list @ A ) > ( list @ B ) > $o] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys2 ) )
=> ( ( P @ ( nil @ A ) @ ( nil @ B ) )
=> ( ! [X: A,Xs2: list @ A,Y3: B,Ys3: list @ B] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ B ) @ Ys3 ) )
=> ( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons @ A @ X @ Xs2 ) @ ( cons @ B @ Y3 @ Ys3 ) ) ) )
=> ( P @ Xs @ Ys2 ) ) ) ) ).
% list_induct2
thf(fact_253_list__induct3,axiom,
! [B: $tType,A: $tType,C: $tType,Xs: list @ A,Ys2: list @ B,Zs2: list @ C,P: ( list @ A ) > ( list @ B ) > ( list @ C ) > $o] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys2 ) )
=> ( ( ( size_size @ ( list @ B ) @ Ys2 )
= ( size_size @ ( list @ C ) @ Zs2 ) )
=> ( ( P @ ( nil @ A ) @ ( nil @ B ) @ ( nil @ C ) )
=> ( ! [X: A,Xs2: list @ A,Y3: B,Ys3: list @ B,Z2: C,Zs: list @ C] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ B ) @ Ys3 ) )
=> ( ( ( size_size @ ( list @ B ) @ Ys3 )
= ( size_size @ ( list @ C ) @ Zs ) )
=> ( ( P @ Xs2 @ Ys3 @ Zs )
=> ( P @ ( cons @ A @ X @ Xs2 ) @ ( cons @ B @ Y3 @ Ys3 ) @ ( cons @ C @ Z2 @ Zs ) ) ) ) )
=> ( P @ Xs @ Ys2 @ Zs2 ) ) ) ) ) ).
% list_induct3
thf(fact_254_list__induct4,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,Xs: list @ A,Ys2: list @ B,Zs2: list @ C,Ws2: list @ D,P: ( list @ A ) > ( list @ B ) > ( list @ C ) > ( list @ D ) > $o] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys2 ) )
=> ( ( ( size_size @ ( list @ B ) @ Ys2 )
= ( size_size @ ( list @ C ) @ Zs2 ) )
=> ( ( ( size_size @ ( list @ C ) @ Zs2 )
= ( size_size @ ( list @ D ) @ Ws2 ) )
=> ( ( P @ ( nil @ A ) @ ( nil @ B ) @ ( nil @ C ) @ ( nil @ D ) )
=> ( ! [X: A,Xs2: list @ A,Y3: B,Ys3: list @ B,Z2: C,Zs: list @ C,W: D,Ws3: list @ D] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ B ) @ Ys3 ) )
=> ( ( ( size_size @ ( list @ B ) @ Ys3 )
= ( size_size @ ( list @ C ) @ Zs ) )
=> ( ( ( size_size @ ( list @ C ) @ Zs )
= ( size_size @ ( list @ D ) @ Ws3 ) )
=> ( ( P @ Xs2 @ Ys3 @ Zs @ Ws3 )
=> ( P @ ( cons @ A @ X @ Xs2 ) @ ( cons @ B @ Y3 @ Ys3 ) @ ( cons @ C @ Z2 @ Zs ) @ ( cons @ D @ W @ Ws3 ) ) ) ) ) )
=> ( P @ Xs @ Ys2 @ Zs2 @ Ws2 ) ) ) ) ) ) ).
% list_induct4
% Type constructors (14)
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A7: $tType,A8: $tType] :
( ( preorder @ A8 )
=> ( preorder @ ( A7 > A8 ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A7: $tType,A8: $tType] :
( ( order @ A8 )
=> ( order @ ( A7 > A8 ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A7: $tType,A8: $tType] :
( ( ord @ A8 )
=> ( ord @ ( A7 > A8 ) ) ) ).
thf(tcon_Nat_Onat___Orderings_Opreorder_1,axiom,
preorder @ nat ).
thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
linorder @ nat ).
thf(tcon_Nat_Onat___Orderings_Oorder_2,axiom,
order @ nat ).
thf(tcon_Nat_Onat___Orderings_Oord_3,axiom,
ord @ nat ).
thf(tcon_Set_Oset___Orderings_Opreorder_4,axiom,
! [A7: $tType] : ( preorder @ ( set @ A7 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_5,axiom,
! [A7: $tType] : ( order @ ( set @ A7 ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_6,axiom,
! [A7: $tType] : ( ord @ ( set @ A7 ) ) ).
thf(tcon_HOL_Obool___Orderings_Opreorder_7,axiom,
preorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Olinorder_8,axiom,
linorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder_9,axiom,
order @ $o ).
thf(tcon_HOL_Obool___Orderings_Oord_10,axiom,
ord @ $o ).
% Helper facts (3)
thf(help_If_3_1_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_T,axiom,
! [A: $tType,X3: A,Y: A] :
( ( if @ A @ $false @ X3 @ Y )
= Y ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X3: A,Y: A] :
( ( if @ A @ $true @ X3 @ Y )
= X3 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( listIn521021761append @ a @ ( cons @ a @ x @ xs ) @ f )
= ( listIn521021761append @ a @ ( cons @ a @ x @ ( nil @ a ) ) @ ( listIn521021761append @ a @ xs @ f ) ) ) ).
%------------------------------------------------------------------------------