TPTP Problem File: DAT132^1.p
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%------------------------------------------------------------------------------
% File : DAT132^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Data Structures
% Problem : Coinductive list 3961
% Version : [Bla16] axioms : Especial.
% English :
% Refs : [Loc10] Lochbihler (2010), Coinductive
% : [RB15] Reynolds & Blanchette (2015), A Decision Procedure for
% : [Bla16] Blanchette (2016), Email to Geoff Sutcliffe
% Source : [Bla16]
% Names : coinductive_list__3961.p [Bla16]
% Status : Theorem
% Rating : 0.33 v8.1.0, 0.25 v7.5.0, 0.33 v7.3.0, 0.00 v7.1.0
% Syntax : Number of formulae : 317 ( 165 unt; 58 typ; 0 def)
% Number of atoms : 615 ( 452 equ; 0 cnn)
% Maximal formula atoms : 11 ( 2 avg)
% Number of connectives : 3591 ( 112 ~; 34 |; 71 &;3158 @)
% ( 0 <=>; 216 =>; 0 <=; 0 <~>)
% Maximal formula depth : 22 ( 6 avg)
% Number of types : 2 ( 1 usr)
% Number of type conns : 198 ( 198 >; 0 *; 0 +; 0 <<)
% Number of symbols : 60 ( 57 usr; 11 con; 0-5 aty)
% Number of variables : 1005 ( 17 ^; 881 !; 58 ?;1005 :)
% ( 49 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 15:04:06.870
%------------------------------------------------------------------------------
%----Could-be-implicit typings (5)
thf(ty_t_Coinductive__List__Mirabelle__kmikjhschf_Ollist,type,
coindu1593790203_llist: $tType > $tType ).
thf(ty_t_List_Olist,type,
list: $tType > $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_t_itself,type,
itself: $tType > $tType ).
thf(ty_tf_a,type,
a: $tType ).
%----Explicit typings (53)
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ofinite__lprefix,type,
coindu1571841457prefix:
!>[A: $tType] : ( ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Olappend,type,
coindu268472904append:
!>[A: $tType] : ( ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Olfinite,type,
coindu1213758845finite:
!>[A: $tType] : ( ( coindu1593790203_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Olist__of,type,
coindu1736001939ist_of:
!>[A: $tType] : ( ( coindu1593790203_llist @ A ) > ( list @ A ) ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Olist__of__aux,type,
coindu1542816657of_aux:
!>[A: $tType] : ( ( list @ A ) > ( coindu1593790203_llist @ A ) > ( list @ A ) ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollast,type,
coindu2000965700_llast:
!>[A: $tType] : ( ( coindu1593790203_llist @ A ) > A ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollexord,type,
coindu300403952lexord:
!>[A: $tType] : ( ( A > A > $o ) > ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollist_OLCons,type,
coindu1121789889_LCons:
!>[A: $tType] : ( A > ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollist_OLNil,type,
coindu1598213697e_LNil:
!>[A: $tType] : ( coindu1593790203_llist @ A ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollist_Olhd,type,
coindu1046438764le_lhd:
!>[A: $tType] : ( ( coindu1593790203_llist @ A ) > A ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollist_Olnull,type,
coindu335574135_lnull:
!>[A: $tType] : ( ( coindu1593790203_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollist_Olset,type,
coindu1112613586e_lset:
!>[A: $tType] : ( ( coindu1593790203_llist @ A ) > ( set @ A ) ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollist__of,type,
coindu1079202755ist_of:
!>[A: $tType] : ( ( list @ A ) > ( coindu1593790203_llist @ A ) ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Olmember,type,
coindu567634248member:
!>[A: $tType] : ( A > ( coindu1593790203_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Olstrict__prefix,type,
coindu574146665prefix:
!>[A: $tType] : ( ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) > $o ) ).
thf(sy_c_HOL_Oundefined,type,
undefined:
!>[A: $tType] : A ).
thf(sy_c_If,type,
if:
!>[A: $tType] : ( $o > A > A > A ) ).
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_Obutlast,type,
butlast:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
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_Omap,type,
map:
!>[A: $tType,Aa: $tType] : ( ( A > Aa ) > ( list @ A ) > ( list @ Aa ) ) ).
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_Omap__tailrec,type,
map_tailrec:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( list @ A ) > ( list @ B ) ) ).
thf(sy_c_List_Omap__tailrec__rev,type,
map_tailrec_rev:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( list @ A ) > ( list @ B ) > ( list @ B ) ) ).
thf(sy_c_List_Omaps,type,
maps:
!>[A: $tType,B: $tType] : ( ( A > ( list @ B ) ) > ( list @ A ) > ( list @ B ) ) ).
thf(sy_c_List_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_Orev,type,
rev:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Orotate1,type,
rotate1:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Osublists,type,
sublists:
!>[A: $tType] : ( ( list @ A ) > ( list @ ( list @ A ) ) ) ).
thf(sy_c_Pure_Otype,type,
type:
!>[A: $tType] : ( itself @ A ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_Sublist_Oprefixes,type,
prefixes:
!>[A: $tType] : ( ( list @ A ) > ( list @ ( list @ A ) ) ) ).
thf(sy_c_Sublist_Oprefixes__rel,type,
prefixes_rel:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > $o ) ).
thf(sy_c_Wellfounded_Oaccp,type,
accp:
!>[A: $tType] : ( ( A > A > $o ) > A > $o ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_XS_H____,type,
xs: list @ a ).
thf(sy_v_r,type,
r: a > a > $o ).
thf(sy_v_thesis____,type,
thesis: $o ).
thf(sy_v_xs,type,
xs2: list @ a ).
thf(sy_v_xs_H____,type,
xs3: coindu1593790203_llist @ a ).
thf(sy_v_y____,type,
y: a ).
thf(sy_v_ys,type,
ys: list @ a ).
thf(sy_v_ys_H____,type,
ys2: coindu1593790203_llist @ a ).
thf(sy_v_zs_H____,type,
zs: list @ a ).
thf(sy_v_zs____,type,
zs2: coindu1593790203_llist @ a ).
%----Relevant facts (254)
thf(fact_0_llist__of__inject,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( coindu1079202755ist_of @ A @ Xs )
= ( coindu1079202755ist_of @ A @ Ys ) )
= ( Xs = Ys ) ) ).
% llist_of_inject
thf(fact_1__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062zs_H_O_Azs_A_061_Allist__of_Azs_H_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [Zs: list @ a] :
( zs2
!= ( coindu1079202755ist_of @ a @ Zs ) ) ).
% \<open>\<And>thesis. (\<And>zs'. zs = llist_of zs' \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_2__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062XS_H_O_Axs_H_A_061_Allist__of_AXS_H_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [XS: list @ a] :
( xs3
!= ( coindu1079202755ist_of @ a @ XS ) ) ).
% \<open>\<And>thesis. (\<And>XS'. xs' = llist_of XS' \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_3__092_060open_062lfinite_Ays_H_092_060close_062,axiom,
coindu1213758845finite @ a @ ys2 ).
% \<open>lfinite ys'\<close>
thf(fact_4_list__of__llist__of,axiom,
! [A: $tType,Xs: list @ A] :
( ( coindu1736001939ist_of @ A @ ( coindu1079202755ist_of @ A @ Xs ) )
= Xs ) ).
% list_of_llist_of
thf(fact_5__092_060open_062zs_A_061_Allist__of_Azs_H_092_060close_062,axiom,
( zs2
= ( coindu1079202755ist_of @ a @ zs ) ) ).
% \<open>zs = llist_of zs'\<close>
thf(fact_6__092_060open_062llist__of_Ays_A_061_Alappend_Azs_A_ILCons_Ay_Ays_H_J_092_060close_062,axiom,
( ( coindu1079202755ist_of @ a @ ys )
= ( coindu268472904append @ a @ zs2 @ ( coindu1121789889_LCons @ a @ y @ ys2 ) ) ) ).
% \<open>llist_of ys = lappend zs (LCons y ys')\<close>
thf(fact_7__092_060open_062xs_H_A_061_Allist__of_AXS_H_092_060close_062,axiom,
( xs3
= ( coindu1079202755ist_of @ a @ xs ) ) ).
% \<open>xs' = llist_of XS'\<close>
thf(fact_8__092_060open_062lfinite_A_Illist__of_Ays_J_092_060close_062,axiom,
coindu1213758845finite @ a @ ( coindu1079202755ist_of @ a @ ys ) ).
% \<open>lfinite (llist_of ys)\<close>
thf(fact_9__092_060open_062lfinite_A_Illist__of_Axs_J_092_060close_062,axiom,
coindu1213758845finite @ a @ ( coindu1079202755ist_of @ a @ xs2 ) ).
% \<open>lfinite (llist_of xs)\<close>
thf(fact_10__092_060open_062llexord_Ar_A_Illist__of_Axs_J_A_Illist__of_Ays_J_092_060close_062,axiom,
coindu300403952lexord @ a @ r @ ( coindu1079202755ist_of @ a @ xs2 ) @ ( coindu1079202755ist_of @ a @ ys ) ).
% \<open>llexord r (llist_of xs) (llist_of ys)\<close>
thf(fact_11_lfinite__llist__of,axiom,
! [A: $tType,Xs: list @ A] : ( coindu1213758845finite @ A @ ( coindu1079202755ist_of @ A @ Xs ) ) ).
% lfinite_llist_of
thf(fact_12_llast__llist__of,axiom,
! [A: $tType,Xs: list @ A] :
( ( coindu2000965700_llast @ A @ ( coindu1079202755ist_of @ A @ Xs ) )
= ( last @ A @ Xs ) ) ).
% llast_llist_of
thf(fact_13__092_060open_062lfinite_Azs_092_060close_062,axiom,
coindu1213758845finite @ a @ zs2 ).
% \<open>lfinite zs\<close>
thf(fact_14__092_060open_062lfinite_Axs_H_092_060close_062,axiom,
coindu1213758845finite @ a @ xs3 ).
% \<open>lfinite xs'\<close>
thf(fact_15_llist_Oinject,axiom,
! [A: $tType,X21: A,X22: coindu1593790203_llist @ A,Y21: A,Y22: coindu1593790203_llist @ A] :
( ( ( coindu1121789889_LCons @ A @ X21 @ X22 )
= ( coindu1121789889_LCons @ A @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% llist.inject
thf(fact_16_llexord__refl,axiom,
! [A: $tType,R: A > A > $o,Xs: coindu1593790203_llist @ A] : ( coindu300403952lexord @ A @ R @ Xs @ Xs ) ).
% llexord_refl
thf(fact_17__092_060open_062llist__of_Axs_A_061_Alappend_Azs_Axs_H_092_060close_062,axiom,
( ( coindu1079202755ist_of @ a @ xs2 )
= ( coindu268472904append @ a @ zs2 @ xs3 ) ) ).
% \<open>llist_of xs = lappend zs xs'\<close>
thf(fact_18_lappend__code_I2_J,axiom,
! [A: $tType,Xa: A,X: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu268472904append @ A @ ( coindu1121789889_LCons @ A @ Xa @ X ) @ Ys )
= ( coindu1121789889_LCons @ A @ Xa @ ( coindu268472904append @ A @ X @ Ys ) ) ) ).
% lappend_code(2)
thf(fact_19_lfinite__code_I2_J,axiom,
! [B: $tType,X: B,Xs: coindu1593790203_llist @ B] :
( ( coindu1213758845finite @ B @ ( coindu1121789889_LCons @ B @ X @ Xs ) )
= ( coindu1213758845finite @ B @ Xs ) ) ).
% lfinite_code(2)
thf(fact_20_lfinite__LCons,axiom,
! [A: $tType,X: A,Xs: coindu1593790203_llist @ A] :
( ( coindu1213758845finite @ A @ ( coindu1121789889_LCons @ A @ X @ Xs ) )
= ( coindu1213758845finite @ A @ Xs ) ) ).
% lfinite_LCons
thf(fact_21_lfinite__lappend,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu1213758845finite @ A @ ( coindu268472904append @ A @ Xs @ Ys ) )
= ( ( coindu1213758845finite @ A @ Xs )
& ( coindu1213758845finite @ A @ Ys ) ) ) ).
% lfinite_lappend
thf(fact_22_llexord__LCons__LCons,axiom,
! [A: $tType,R: A > A > $o,X: A,Xs: coindu1593790203_llist @ A,Y: A,Ys: coindu1593790203_llist @ A] :
( ( coindu300403952lexord @ A @ R @ ( coindu1121789889_LCons @ A @ X @ Xs ) @ ( coindu1121789889_LCons @ A @ Y @ Ys ) )
= ( ( ( X = Y )
& ( coindu300403952lexord @ A @ R @ Xs @ Ys ) )
| ( R @ X @ Y ) ) ) ).
% llexord_LCons_LCons
thf(fact_23_llast__LCons2,axiom,
! [A: $tType,X: A,Y: A,Xs: coindu1593790203_llist @ A] :
( ( coindu2000965700_llast @ A @ ( coindu1121789889_LCons @ A @ X @ ( coindu1121789889_LCons @ A @ Y @ Xs ) ) )
= ( coindu2000965700_llast @ A @ ( coindu1121789889_LCons @ A @ Y @ Xs ) ) ) ).
% llast_LCons2
thf(fact_24_llist__of__list__of,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( ( coindu1213758845finite @ A @ Xs )
=> ( ( coindu1079202755ist_of @ A @ ( coindu1736001939ist_of @ A @ Xs ) )
= Xs ) ) ).
% llist_of_list_of
thf(fact_25_llast__lappend__LCons,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Y: A,Ys: coindu1593790203_llist @ A] :
( ( coindu1213758845finite @ A @ Xs )
=> ( ( coindu2000965700_llast @ A @ ( coindu268472904append @ A @ Xs @ ( coindu1121789889_LCons @ A @ Y @ Ys ) ) )
= ( coindu2000965700_llast @ A @ ( coindu1121789889_LCons @ A @ Y @ Ys ) ) ) ) ).
% llast_lappend_LCons
thf(fact_26_llexord__code_I3_J,axiom,
! [A: $tType,R: A > A > $o,X: A,Xs: coindu1593790203_llist @ A,Y: A,Ys: coindu1593790203_llist @ A] :
( ( coindu300403952lexord @ A @ R @ ( coindu1121789889_LCons @ A @ X @ Xs ) @ ( coindu1121789889_LCons @ A @ Y @ Ys ) )
= ( ( R @ X @ Y )
| ( ( X = Y )
& ( coindu300403952lexord @ A @ R @ Xs @ Ys ) ) ) ) ).
% llexord_code(3)
thf(fact_27_lappend__inf,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ~ ( coindu1213758845finite @ A @ Xs )
=> ( ( coindu268472904append @ A @ Xs @ Ys )
= Xs ) ) ).
% lappend_inf
thf(fact_28_lappend__assoc,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A,Zs2: coindu1593790203_llist @ A] :
( ( coindu268472904append @ A @ ( coindu268472904append @ A @ Xs @ Ys ) @ Zs2 )
= ( coindu268472904append @ A @ Xs @ ( coindu268472904append @ A @ Ys @ Zs2 ) ) ) ).
% lappend_assoc
thf(fact_29_llexord__trans,axiom,
! [A: $tType,R: A > A > $o,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A,Zs2: coindu1593790203_llist @ A] :
( ( coindu300403952lexord @ A @ R @ Xs @ Ys )
=> ( ( coindu300403952lexord @ A @ R @ Ys @ Zs2 )
=> ( ! [A2: A,B2: A,C: A] :
( ( R @ A2 @ B2 )
=> ( ( R @ B2 @ C )
=> ( R @ A2 @ C ) ) )
=> ( coindu300403952lexord @ A @ R @ Xs @ Zs2 ) ) ) ) ).
% llexord_trans
thf(fact_30_llexord__linear,axiom,
! [A: $tType,R: A > A > $o,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ! [X2: A,Y2: A] :
( ( R @ X2 @ Y2 )
| ( X2 = Y2 )
| ( R @ Y2 @ X2 ) )
=> ( ( coindu300403952lexord @ A @ R @ Xs @ Ys )
| ( coindu300403952lexord @ A @ R @ Ys @ Xs ) ) ) ).
% llexord_linear
thf(fact_31_llexord__antisym,axiom,
! [A: $tType,R: A > A > $o,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu300403952lexord @ A @ R @ Xs @ Ys )
=> ( ( coindu300403952lexord @ A @ R @ Ys @ Xs )
=> ( ! [A2: A,B2: A] :
( ( R @ A2 @ B2 )
=> ~ ( R @ B2 @ A2 ) )
=> ( Xs = Ys ) ) ) ) ).
% llexord_antisym
thf(fact_32_llexord__LCons__left,axiom,
! [A: $tType,R: A > A > $o,X: A,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu300403952lexord @ A @ R @ ( coindu1121789889_LCons @ A @ X @ Xs ) @ Ys )
= ( ? [Y3: A,Ys2: coindu1593790203_llist @ A] :
( ( Ys
= ( coindu1121789889_LCons @ A @ Y3 @ Ys2 ) )
& ( ( ( X = Y3 )
& ( coindu300403952lexord @ A @ R @ Xs @ Ys2 ) )
| ( R @ X @ Y3 ) ) ) ) ) ).
% llexord_LCons_left
thf(fact_33_llexord__append__right,axiom,
! [A: $tType,R: A > A > $o,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] : ( coindu300403952lexord @ A @ R @ Xs @ ( coindu268472904append @ A @ Xs @ Ys ) ) ).
% llexord_append_right
thf(fact_34_llexord__lappend__leftI,axiom,
! [A: $tType,R: A > A > $o,Ys: coindu1593790203_llist @ A,Zs2: coindu1593790203_llist @ A,Xs: coindu1593790203_llist @ A] :
( ( coindu300403952lexord @ A @ R @ Ys @ Zs2 )
=> ( coindu300403952lexord @ A @ R @ ( coindu268472904append @ A @ Xs @ Ys ) @ ( coindu268472904append @ A @ Xs @ Zs2 ) ) ) ).
% llexord_lappend_leftI
thf(fact_35_lfinite__LConsI,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,X: A] :
( ( coindu1213758845finite @ A @ Xs )
=> ( coindu1213758845finite @ A @ ( coindu1121789889_LCons @ A @ X @ Xs ) ) ) ).
% lfinite_LConsI
thf(fact_36_llexord__LCons__eq,axiom,
! [A: $tType,R: A > A > $o,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A,X: A] :
( ( coindu300403952lexord @ A @ R @ Xs @ Ys )
=> ( coindu300403952lexord @ A @ R @ ( coindu1121789889_LCons @ A @ X @ Xs ) @ ( coindu1121789889_LCons @ A @ X @ Ys ) ) ) ).
% llexord_LCons_eq
thf(fact_37_llexord__LCons__less,axiom,
! [A: $tType,R: A > A > $o,X: A,Y: A,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( R @ X @ Y )
=> ( coindu300403952lexord @ A @ R @ ( coindu1121789889_LCons @ A @ X @ Xs ) @ ( coindu1121789889_LCons @ A @ Y @ Ys ) ) ) ).
% llexord_LCons_less
thf(fact_38__092_060open_062xs_A_061_Azs_H_A_064_AXS_H_092_060close_062,axiom,
( xs2
= ( append @ a @ zs @ xs ) ) ).
% \<open>xs = zs' @ XS'\<close>
thf(fact_39__092_060open_062xs_H_A_061_ALNil_A_092_060or_062_Ar_A_Ilhd_Axs_H_J_Ay_092_060close_062,axiom,
( ( xs3
= ( coindu1598213697e_LNil @ a ) )
| ( r @ ( coindu1046438764le_lhd @ a @ xs3 ) @ y ) ) ).
% \<open>xs' = LNil \<or> r (lhd xs') y\<close>
thf(fact_40_list__of__LCons,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,X: A] :
( ( coindu1213758845finite @ A @ Xs )
=> ( ( coindu1736001939ist_of @ A @ ( coindu1121789889_LCons @ A @ X @ Xs ) )
= ( cons @ A @ X @ ( coindu1736001939ist_of @ A @ Xs ) ) ) ) ).
% list_of_LCons
thf(fact_41_lmember__code_I2_J,axiom,
! [A: $tType,X: A,Y: A,Ys: coindu1593790203_llist @ A] :
( ( coindu567634248member @ A @ X @ ( coindu1121789889_LCons @ A @ Y @ Ys ) )
= ( ( X = Y )
| ( coindu567634248member @ A @ X @ Ys ) ) ) ).
% lmember_code(2)
thf(fact_42_list__of__lappend,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu1213758845finite @ A @ Xs )
=> ( ( coindu1213758845finite @ A @ Ys )
=> ( ( coindu1736001939ist_of @ A @ ( coindu268472904append @ A @ Xs @ Ys ) )
= ( append @ A @ ( coindu1736001939ist_of @ A @ Xs ) @ ( coindu1736001939ist_of @ A @ Ys ) ) ) ) ) ).
% list_of_lappend
thf(fact_43_lfinite__rev__induct,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,P: ( coindu1593790203_llist @ A ) > $o] :
( ( coindu1213758845finite @ A @ Xs )
=> ( ( P @ ( coindu1598213697e_LNil @ A ) )
=> ( ! [X2: A,Xs2: coindu1593790203_llist @ A] :
( ( coindu1213758845finite @ A @ Xs2 )
=> ( ( P @ Xs2 )
=> ( P @ ( coindu268472904append @ A @ Xs2 @ ( coindu1121789889_LCons @ A @ X2 @ ( coindu1598213697e_LNil @ A ) ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% lfinite_rev_induct
thf(fact_44_llexord__lappend__left,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,R: A > A > $o,Ys: coindu1593790203_llist @ A,Zs2: coindu1593790203_llist @ A] :
( ( coindu1213758845finite @ A @ Xs )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ ( coindu1112613586e_lset @ A @ Xs ) )
=> ~ ( R @ X2 @ X2 ) )
=> ( ( coindu300403952lexord @ A @ R @ ( coindu268472904append @ A @ Xs @ Ys ) @ ( coindu268472904append @ A @ Xs @ Zs2 ) )
= ( coindu300403952lexord @ A @ R @ Ys @ Zs2 ) ) ) ) ).
% llexord_lappend_left
thf(fact_45_mem__Collect__eq,axiom,
! [A: $tType,A3: A,P: A > $o] :
( ( member @ A @ A3 @ ( collect @ A @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: $tType,A4: set @ A] :
( ( collect @ A
@ ^ [X3: A] : ( member @ A @ X3 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X2: A] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_48_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X2: A] :
( ( F @ X2 )
= ( G @ X2 ) )
=> ( F = G ) ) ).
% ext
thf(fact_49_llexord__lappend__leftD,axiom,
! [A: $tType,R: A > A > $o,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A,Zs2: coindu1593790203_llist @ A] :
( ( coindu300403952lexord @ A @ R @ ( coindu268472904append @ A @ Xs @ Ys ) @ ( coindu268472904append @ A @ Xs @ Zs2 ) )
=> ( ( coindu1213758845finite @ A @ Xs )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ ( coindu1112613586e_lset @ A @ Xs ) )
=> ~ ( R @ X2 @ X2 ) )
=> ( coindu300403952lexord @ A @ R @ Ys @ Zs2 ) ) ) ) ).
% llexord_lappend_leftD
thf(fact_50_split__llist,axiom,
! [A: $tType,X: A,Xs: coindu1593790203_llist @ A] :
( ( member @ A @ X @ ( coindu1112613586e_lset @ A @ Xs ) )
=> ? [Ys3: coindu1593790203_llist @ A,Zs3: coindu1593790203_llist @ A] :
( ( Xs
= ( coindu268472904append @ A @ Ys3 @ ( coindu1121789889_LCons @ A @ X @ Zs3 ) ) )
& ( coindu1213758845finite @ A @ Ys3 ) ) ) ).
% split_llist
thf(fact_51_split__llist__first,axiom,
! [A: $tType,X: A,Xs: coindu1593790203_llist @ A] :
( ( member @ A @ X @ ( coindu1112613586e_lset @ A @ Xs ) )
=> ? [Ys3: coindu1593790203_llist @ A,Zs3: coindu1593790203_llist @ A] :
( ( Xs
= ( coindu268472904append @ A @ Ys3 @ ( coindu1121789889_LCons @ A @ X @ Zs3 ) ) )
& ( coindu1213758845finite @ A @ Ys3 )
& ~ ( member @ A @ X @ ( coindu1112613586e_lset @ A @ Ys3 ) ) ) ) ).
% split_llist_first
thf(fact_52_llast__singleton,axiom,
! [A: $tType,X: A] :
( ( coindu2000965700_llast @ A @ ( coindu1121789889_LCons @ A @ X @ ( coindu1598213697e_LNil @ A ) ) )
= X ) ).
% llast_singleton
thf(fact_53_lappend__LNil2,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( ( coindu268472904append @ A @ Xs @ ( coindu1598213697e_LNil @ A ) )
= Xs ) ).
% lappend_LNil2
thf(fact_54_lappend__code_I1_J,axiom,
! [A: $tType,Ys: coindu1593790203_llist @ A] :
( ( coindu268472904append @ A @ ( coindu1598213697e_LNil @ A ) @ Ys )
= Ys ) ).
% lappend_code(1)
thf(fact_55_lfinite__code_I1_J,axiom,
! [A: $tType] : ( coindu1213758845finite @ A @ ( coindu1598213697e_LNil @ A ) ) ).
% lfinite_code(1)
thf(fact_56_llexord__code_I1_J,axiom,
! [A: $tType,R: A > A > $o,Ys: coindu1593790203_llist @ A] : ( coindu300403952lexord @ A @ R @ ( coindu1598213697e_LNil @ A ) @ Ys ) ).
% llexord_code(1)
thf(fact_57_lset__lmember,axiom,
! [A: $tType,X: A,Xs: coindu1593790203_llist @ A] :
( ( member @ A @ X @ ( coindu1112613586e_lset @ A @ Xs ) )
= ( coindu567634248member @ A @ X @ Xs ) ) ).
% lset_lmember
thf(fact_58_lmember__code_I1_J,axiom,
! [A: $tType,X: A] :
~ ( coindu567634248member @ A @ X @ ( coindu1598213697e_LNil @ A ) ) ).
% lmember_code(1)
thf(fact_59_lhd__LCons,axiom,
! [A: $tType,X21: A,X22: coindu1593790203_llist @ A] :
( ( coindu1046438764le_lhd @ A @ ( coindu1121789889_LCons @ A @ X21 @ X22 ) )
= X21 ) ).
% lhd_LCons
thf(fact_60_llist_Oset__induct,axiom,
! [A: $tType,X: A,A3: coindu1593790203_llist @ A,P: A > ( coindu1593790203_llist @ A ) > $o] :
( ( member @ A @ X @ ( coindu1112613586e_lset @ A @ A3 ) )
=> ( ! [Z1: A,Z2: coindu1593790203_llist @ A] : ( P @ Z1 @ ( coindu1121789889_LCons @ A @ Z1 @ Z2 ) )
=> ( ! [Z1: A,Z2: coindu1593790203_llist @ A,Xa2: A] :
( ( member @ A @ Xa2 @ ( coindu1112613586e_lset @ A @ Z2 ) )
=> ( ( P @ Xa2 @ Z2 )
=> ( P @ Xa2 @ ( coindu1121789889_LCons @ A @ Z1 @ Z2 ) ) ) )
=> ( P @ X @ A3 ) ) ) ) ).
% llist.set_induct
thf(fact_61_llist_Oset__cases,axiom,
! [A: $tType,E: A,A3: coindu1593790203_llist @ A] :
( ( member @ A @ E @ ( coindu1112613586e_lset @ A @ A3 ) )
=> ( ! [Z2: coindu1593790203_llist @ A] :
( A3
!= ( coindu1121789889_LCons @ A @ E @ Z2 ) )
=> ~ ! [Z1: A,Z2: coindu1593790203_llist @ A] :
( ( A3
= ( coindu1121789889_LCons @ A @ Z1 @ Z2 ) )
=> ~ ( member @ A @ E @ ( coindu1112613586e_lset @ A @ Z2 ) ) ) ) ) ).
% llist.set_cases
thf(fact_62_lset__induct_H,axiom,
! [A: $tType,X: A,Xs: coindu1593790203_llist @ A,P: ( coindu1593790203_llist @ A ) > $o] :
( ( member @ A @ X @ ( coindu1112613586e_lset @ A @ Xs ) )
=> ( ! [Xs2: coindu1593790203_llist @ A] : ( P @ ( coindu1121789889_LCons @ A @ X @ Xs2 ) )
=> ( ! [X4: A,Xs2: coindu1593790203_llist @ A] :
( ( member @ A @ X @ ( coindu1112613586e_lset @ A @ Xs2 ) )
=> ( ( P @ Xs2 )
=> ( P @ ( coindu1121789889_LCons @ A @ X4 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% lset_induct'
thf(fact_63_lset__induct,axiom,
! [A: $tType,X: A,Xs: coindu1593790203_llist @ A,P: ( coindu1593790203_llist @ A ) > $o] :
( ( member @ A @ X @ ( coindu1112613586e_lset @ A @ Xs ) )
=> ( ! [Xs2: coindu1593790203_llist @ A] : ( P @ ( coindu1121789889_LCons @ A @ X @ Xs2 ) )
=> ( ! [X4: A,Xs2: coindu1593790203_llist @ A] :
( ( member @ A @ X @ ( coindu1112613586e_lset @ A @ Xs2 ) )
=> ( ( X != X4 )
=> ( ( P @ Xs2 )
=> ( P @ ( coindu1121789889_LCons @ A @ X4 @ Xs2 ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% lset_induct
thf(fact_64_lset__cases,axiom,
! [A: $tType,X: A,Xs: coindu1593790203_llist @ A] :
( ( member @ A @ X @ ( coindu1112613586e_lset @ A @ Xs ) )
=> ( ! [Xs3: coindu1593790203_llist @ A] :
( Xs
!= ( coindu1121789889_LCons @ A @ X @ Xs3 ) )
=> ~ ! [X4: A,Xs3: coindu1593790203_llist @ A] :
( ( Xs
= ( coindu1121789889_LCons @ A @ X4 @ Xs3 ) )
=> ~ ( member @ A @ X @ ( coindu1112613586e_lset @ A @ Xs3 ) ) ) ) ) ).
% lset_cases
thf(fact_65_llist_Oset__intros_I1_J,axiom,
! [A: $tType,A1: A,A22: coindu1593790203_llist @ A] : ( member @ A @ A1 @ ( coindu1112613586e_lset @ A @ ( coindu1121789889_LCons @ A @ A1 @ A22 ) ) ) ).
% llist.set_intros(1)
thf(fact_66_llist_Oset__intros_I2_J,axiom,
! [A: $tType,X: A,A22: coindu1593790203_llist @ A,A1: A] :
( ( member @ A @ X @ ( coindu1112613586e_lset @ A @ A22 ) )
=> ( member @ A @ X @ ( coindu1112613586e_lset @ A @ ( coindu1121789889_LCons @ A @ A1 @ A22 ) ) ) ) ).
% llist.set_intros(2)
thf(fact_67_lset__intros_I1_J,axiom,
! [A: $tType,X: A,Xs: coindu1593790203_llist @ A] : ( member @ A @ X @ ( coindu1112613586e_lset @ A @ ( coindu1121789889_LCons @ A @ X @ Xs ) ) ) ).
% lset_intros(1)
thf(fact_68_lset__intros_I2_J,axiom,
! [A: $tType,X: A,Xs: coindu1593790203_llist @ A,X5: A] :
( ( member @ A @ X @ ( coindu1112613586e_lset @ A @ Xs ) )
=> ( member @ A @ X @ ( coindu1112613586e_lset @ A @ ( coindu1121789889_LCons @ A @ X5 @ Xs ) ) ) ) ).
% lset_intros(2)
thf(fact_69_neq__LNil__conv,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( ( Xs
!= ( coindu1598213697e_LNil @ A ) )
= ( ? [X3: A,Xs4: coindu1593790203_llist @ A] :
( Xs
= ( coindu1121789889_LCons @ A @ X3 @ Xs4 ) ) ) ) ).
% neq_LNil_conv
thf(fact_70_llist_Oexhaust,axiom,
! [A: $tType,Y: coindu1593790203_llist @ A] :
( ( Y
!= ( coindu1598213697e_LNil @ A ) )
=> ~ ! [X212: A,X222: coindu1593790203_llist @ A] :
( Y
!= ( coindu1121789889_LCons @ A @ X212 @ X222 ) ) ) ).
% llist.exhaust
thf(fact_71_llist_Odistinct_I1_J,axiom,
! [A: $tType,X21: A,X22: coindu1593790203_llist @ A] :
( ( coindu1598213697e_LNil @ A )
!= ( coindu1121789889_LCons @ A @ X21 @ X22 ) ) ).
% llist.distinct(1)
thf(fact_72_lappend__eq__LNil__iff,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( ( coindu268472904append @ A @ Xs @ Ys )
= ( coindu1598213697e_LNil @ A ) )
= ( ( Xs
= ( coindu1598213697e_LNil @ A ) )
& ( Ys
= ( coindu1598213697e_LNil @ A ) ) ) ) ).
% lappend_eq_LNil_iff
thf(fact_73_LNil__eq__lappend__iff,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( ( coindu1598213697e_LNil @ A )
= ( coindu268472904append @ A @ Xs @ Ys ) )
= ( ( Xs
= ( coindu1598213697e_LNil @ A ) )
& ( Ys
= ( coindu1598213697e_LNil @ A ) ) ) ) ).
% LNil_eq_lappend_iff
thf(fact_74_lappend__LNil__LNil,axiom,
! [A: $tType] :
( ( coindu268472904append @ A @ ( coindu1598213697e_LNil @ A ) @ ( coindu1598213697e_LNil @ A ) )
= ( coindu1598213697e_LNil @ A ) ) ).
% lappend_LNil_LNil
thf(fact_75_lfinite__LNil,axiom,
! [A: $tType] : ( coindu1213758845finite @ A @ ( coindu1598213697e_LNil @ A ) ) ).
% lfinite_LNil
thf(fact_76_llexord__LNil,axiom,
! [A: $tType,R: A > A > $o,Ys: coindu1593790203_llist @ A] : ( coindu300403952lexord @ A @ R @ ( coindu1598213697e_LNil @ A ) @ Ys ) ).
% llexord_LNil
thf(fact_77_llexord__conv,axiom,
! [A: $tType] :
( ( coindu300403952lexord @ A )
= ( ^ [R2: A > A > $o,Xs5: coindu1593790203_llist @ A,Ys4: coindu1593790203_llist @ A] :
( ( Xs5 = Ys4 )
| ? [Zs4: coindu1593790203_llist @ A,Xs4: coindu1593790203_llist @ A,Y3: A,Ys2: coindu1593790203_llist @ A] :
( ( coindu1213758845finite @ A @ Zs4 )
& ( Xs5
= ( coindu268472904append @ A @ Zs4 @ Xs4 ) )
& ( Ys4
= ( coindu268472904append @ A @ Zs4 @ ( coindu1121789889_LCons @ A @ Y3 @ Ys2 ) ) )
& ( ( Xs4
= ( coindu1598213697e_LNil @ A ) )
| ( R2 @ ( coindu1046438764le_lhd @ A @ Xs4 ) @ Y3 ) ) ) ) ) ) ).
% llexord_conv
thf(fact_78_in__lset__lappend__iff,axiom,
! [A: $tType,X: A,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( member @ A @ X @ ( coindu1112613586e_lset @ A @ ( coindu268472904append @ A @ Xs @ Ys ) ) )
= ( ( member @ A @ X @ ( coindu1112613586e_lset @ A @ Xs ) )
| ( ( coindu1213758845finite @ A @ Xs )
& ( member @ A @ X @ ( coindu1112613586e_lset @ A @ Ys ) ) ) ) ) ).
% in_lset_lappend_iff
thf(fact_79_lappend__snocL1__conv__LCons2,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Y: A,Ys: coindu1593790203_llist @ A] :
( ( coindu268472904append @ A @ ( coindu268472904append @ A @ Xs @ ( coindu1121789889_LCons @ A @ Y @ ( coindu1598213697e_LNil @ A ) ) ) @ Ys )
= ( coindu268472904append @ A @ Xs @ ( coindu1121789889_LCons @ A @ Y @ Ys ) ) ) ).
% lappend_snocL1_conv_LCons2
thf(fact_80_lfinite_Oinducts,axiom,
! [A: $tType,X: coindu1593790203_llist @ A,P: ( coindu1593790203_llist @ A ) > $o] :
( ( coindu1213758845finite @ A @ X )
=> ( ( P @ ( coindu1598213697e_LNil @ A ) )
=> ( ! [Xs2: coindu1593790203_llist @ A,X2: A] :
( ( coindu1213758845finite @ A @ Xs2 )
=> ( ( P @ Xs2 )
=> ( P @ ( coindu1121789889_LCons @ A @ X2 @ Xs2 ) ) ) )
=> ( P @ X ) ) ) ) ).
% lfinite.inducts
thf(fact_81_lfinite_Osimps,axiom,
! [A: $tType] :
( ( coindu1213758845finite @ A )
= ( ^ [A5: coindu1593790203_llist @ A] :
( ( A5
= ( coindu1598213697e_LNil @ A ) )
| ? [Xs5: coindu1593790203_llist @ A,X3: A] :
( ( A5
= ( coindu1121789889_LCons @ A @ X3 @ Xs5 ) )
& ( coindu1213758845finite @ A @ Xs5 ) ) ) ) ) ).
% lfinite.simps
thf(fact_82_lfinite_Ocases,axiom,
! [A: $tType,A3: coindu1593790203_llist @ A] :
( ( coindu1213758845finite @ A @ A3 )
=> ( ( A3
!= ( coindu1598213697e_LNil @ A ) )
=> ~ ! [Xs2: coindu1593790203_llist @ A] :
( ? [X2: A] :
( A3
= ( coindu1121789889_LCons @ A @ X2 @ Xs2 ) )
=> ~ ( coindu1213758845finite @ A @ Xs2 ) ) ) ) ).
% lfinite.cases
thf(fact_83_lappend__llist__of__llist__of,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( coindu268472904append @ A @ ( coindu1079202755ist_of @ A @ Xs ) @ ( coindu1079202755ist_of @ A @ Ys ) )
= ( coindu1079202755ist_of @ A @ ( append @ A @ Xs @ Ys ) ) ) ).
% lappend_llist_of_llist_of
thf(fact_84_llist__of_Osimps_I2_J,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( coindu1079202755ist_of @ A @ ( cons @ A @ X @ Xs ) )
= ( coindu1121789889_LCons @ A @ X @ ( coindu1079202755ist_of @ A @ Xs ) ) ) ).
% llist_of.simps(2)
thf(fact_85_llist__of__eq__LCons__conv,axiom,
! [A: $tType,Xs: list @ A,Y: A,Ys: coindu1593790203_llist @ A] :
( ( ( coindu1079202755ist_of @ A @ Xs )
= ( coindu1121789889_LCons @ A @ Y @ Ys ) )
= ( ? [Xs4: list @ A] :
( ( Xs
= ( cons @ A @ Y @ Xs4 ) )
& ( Ys
= ( coindu1079202755ist_of @ A @ Xs4 ) ) ) ) ) ).
% llist_of_eq_LCons_conv
thf(fact_86_llexord_Ocoinduct,axiom,
! [A: $tType,X6: ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) > $o,X: coindu1593790203_llist @ A,Xa: coindu1593790203_llist @ A,R: A > A > $o] :
( ( X6 @ X @ Xa )
=> ( ! [X2: coindu1593790203_llist @ A,Xa2: coindu1593790203_llist @ A] :
( ( X6 @ X2 @ Xa2 )
=> ( ? [Xs6: coindu1593790203_llist @ A,Ys5: coindu1593790203_llist @ A,Xb: A] :
( ( X2
= ( coindu1121789889_LCons @ A @ Xb @ Xs6 ) )
& ( Xa2
= ( coindu1121789889_LCons @ A @ Xb @ Ys5 ) )
& ( ( X6 @ Xs6 @ Ys5 )
| ( coindu300403952lexord @ A @ R @ Xs6 @ Ys5 ) ) )
| ? [Xb: A,Y4: A,Xs6: coindu1593790203_llist @ A,Ys5: coindu1593790203_llist @ A] :
( ( X2
= ( coindu1121789889_LCons @ A @ Xb @ Xs6 ) )
& ( Xa2
= ( coindu1121789889_LCons @ A @ Y4 @ Ys5 ) )
& ( R @ Xb @ Y4 ) )
| ? [Ys5: coindu1593790203_llist @ A] :
( ( X2
= ( coindu1598213697e_LNil @ A ) )
& ( Xa2 = Ys5 ) ) ) )
=> ( coindu300403952lexord @ A @ R @ X @ Xa ) ) ) ).
% llexord.coinduct
thf(fact_87_llexord_Osimps,axiom,
! [A: $tType] :
( ( coindu300403952lexord @ A )
= ( ^ [R2: A > A > $o,A12: coindu1593790203_llist @ A,A23: coindu1593790203_llist @ A] :
( ? [Xs5: coindu1593790203_llist @ A,Ys4: coindu1593790203_llist @ A,X3: A] :
( ( A12
= ( coindu1121789889_LCons @ A @ X3 @ Xs5 ) )
& ( A23
= ( coindu1121789889_LCons @ A @ X3 @ Ys4 ) )
& ( coindu300403952lexord @ A @ R2 @ Xs5 @ Ys4 ) )
| ? [X3: A,Y3: A,Xs5: coindu1593790203_llist @ A,Ys4: coindu1593790203_llist @ A] :
( ( A12
= ( coindu1121789889_LCons @ A @ X3 @ Xs5 ) )
& ( A23
= ( coindu1121789889_LCons @ A @ Y3 @ Ys4 ) )
& ( R2 @ X3 @ Y3 ) )
| ? [Ys4: coindu1593790203_llist @ A] :
( ( A12
= ( coindu1598213697e_LNil @ A ) )
& ( A23 = Ys4 ) ) ) ) ) ).
% llexord.simps
thf(fact_88_llexord_Ocases,axiom,
! [A: $tType,R: A > A > $o,A1: coindu1593790203_llist @ A,A22: coindu1593790203_llist @ A] :
( ( coindu300403952lexord @ A @ R @ A1 @ A22 )
=> ( ! [Xs2: coindu1593790203_llist @ A,Ys3: coindu1593790203_llist @ A,X2: A] :
( ( A1
= ( coindu1121789889_LCons @ A @ X2 @ Xs2 ) )
=> ( ( A22
= ( coindu1121789889_LCons @ A @ X2 @ Ys3 ) )
=> ~ ( coindu300403952lexord @ A @ R @ Xs2 @ Ys3 ) ) )
=> ( ! [X2: A] :
( ? [Xs2: coindu1593790203_llist @ A] :
( A1
= ( coindu1121789889_LCons @ A @ X2 @ Xs2 ) )
=> ! [Y2: A] :
( ? [Ys3: coindu1593790203_llist @ A] :
( A22
= ( coindu1121789889_LCons @ A @ Y2 @ Ys3 ) )
=> ~ ( R @ X2 @ Y2 ) ) )
=> ~ ( ( A1
= ( coindu1598213697e_LNil @ A ) )
=> ! [Ys3: coindu1593790203_llist @ A] : A22 != Ys3 ) ) ) ) ).
% llexord.cases
thf(fact_89_llexord__code_I2_J,axiom,
! [A: $tType,R: A > A > $o,X: A,Xs: coindu1593790203_llist @ A] :
~ ( coindu300403952lexord @ A @ R @ ( coindu1121789889_LCons @ A @ X @ Xs ) @ ( coindu1598213697e_LNil @ A ) ) ).
% llexord_code(2)
thf(fact_90_same__append__eq,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs2: list @ A] :
( ( ( append @ A @ Xs @ Ys )
= ( append @ A @ Xs @ Zs2 ) )
= ( Ys = Zs2 ) ) ).
% same_append_eq
thf(fact_91_append__same__eq,axiom,
! [A: $tType,Ys: list @ A,Xs: list @ A,Zs2: list @ A] :
( ( ( append @ A @ Ys @ Xs )
= ( append @ A @ Zs2 @ Xs ) )
= ( Ys = Zs2 ) ) ).
% append_same_eq
thf(fact_92_append__assoc,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs2: list @ A] :
( ( append @ A @ ( append @ A @ Xs @ Ys ) @ Zs2 )
= ( append @ A @ Xs @ ( append @ A @ Ys @ Zs2 ) ) ) ).
% append_assoc
thf(fact_93_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_94_append__Cons,axiom,
! [A: $tType,X: A,Xs: list @ A,Ys: list @ A] :
( ( append @ A @ ( cons @ A @ X @ Xs ) @ Ys )
= ( cons @ A @ X @ ( append @ A @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_95_Cons__eq__appendI,axiom,
! [A: $tType,X: A,Xs1: list @ A,Ys: list @ A,Xs: list @ A,Zs2: list @ A] :
( ( ( cons @ A @ X @ Xs1 )
= Ys )
=> ( ( Xs
= ( append @ A @ Xs1 @ Zs2 ) )
=> ( ( cons @ A @ X @ Xs )
= ( append @ A @ Ys @ Zs2 ) ) ) ) ).
% Cons_eq_appendI
thf(fact_96_lappend__llist__of__LCons,axiom,
! [A: $tType,Xs: list @ A,Y: A,Ys: coindu1593790203_llist @ A] :
( ( coindu268472904append @ A @ ( coindu1079202755ist_of @ A @ Xs ) @ ( coindu1121789889_LCons @ A @ Y @ Ys ) )
= ( coindu268472904append @ A @ ( coindu1079202755ist_of @ A @ ( append @ A @ Xs @ ( cons @ A @ Y @ ( nil @ A ) ) ) ) @ Ys ) ) ).
% lappend_llist_of_LCons
thf(fact_97_list__of__aux__code_I2_J,axiom,
! [A: $tType,Xs: list @ A,Y: A,Ys: coindu1593790203_llist @ A] :
( ( coindu1542816657of_aux @ A @ Xs @ ( coindu1121789889_LCons @ A @ Y @ Ys ) )
= ( coindu1542816657of_aux @ A @ ( cons @ A @ Y @ Xs ) @ Ys ) ) ).
% list_of_aux_code(2)
thf(fact_98_append__Nil2,axiom,
! [A: $tType,Xs: list @ A] :
( ( append @ A @ Xs @ ( nil @ A ) )
= Xs ) ).
% append_Nil2
thf(fact_99_append__self__conv,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( append @ A @ Xs @ Ys )
= Xs )
= ( Ys
= ( nil @ A ) ) ) ).
% append_self_conv
thf(fact_100_self__append__conv,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( Xs
= ( append @ A @ Xs @ Ys ) )
= ( Ys
= ( nil @ A ) ) ) ).
% self_append_conv
thf(fact_101_append__self__conv2,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( append @ A @ Xs @ Ys )
= Ys )
= ( Xs
= ( nil @ A ) ) ) ).
% append_self_conv2
thf(fact_102_self__append__conv2,axiom,
! [A: $tType,Ys: list @ A,Xs: list @ A] :
( ( Ys
= ( append @ A @ Xs @ Ys ) )
= ( Xs
= ( nil @ A ) ) ) ).
% self_append_conv2
thf(fact_103_Nil__is__append__conv,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( nil @ A )
= ( append @ A @ Xs @ Ys ) )
= ( ( Xs
= ( nil @ A ) )
& ( Ys
= ( nil @ A ) ) ) ) ).
% Nil_is_append_conv
thf(fact_104_append__is__Nil__conv,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( append @ A @ Xs @ Ys )
= ( nil @ A ) )
= ( ( Xs
= ( nil @ A ) )
& ( Ys
= ( nil @ A ) ) ) ) ).
% append_is_Nil_conv
thf(fact_105_append1__eq__conv,axiom,
! [A: $tType,Xs: list @ A,X: A,Ys: list @ A,Y: A] :
( ( ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) )
= ( append @ A @ Ys @ ( cons @ A @ Y @ ( nil @ A ) ) ) )
= ( ( Xs = Ys )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_106_last__appendR,axiom,
! [A: $tType,Ys: list @ A,Xs: list @ A] :
( ( Ys
!= ( nil @ A ) )
=> ( ( last @ A @ ( append @ A @ Xs @ Ys ) )
= ( last @ A @ Ys ) ) ) ).
% last_appendR
thf(fact_107_last__appendL,axiom,
! [A: $tType,Ys: list @ A,Xs: list @ A] :
( ( Ys
= ( nil @ A ) )
=> ( ( last @ A @ ( append @ A @ Xs @ Ys ) )
= ( last @ A @ Xs ) ) ) ).
% last_appendL
thf(fact_108_list__of__LNil,axiom,
! [A: $tType] :
( ( coindu1736001939ist_of @ A @ ( coindu1598213697e_LNil @ A ) )
= ( nil @ A ) ) ).
% list_of_LNil
thf(fact_109_last__snoc,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( last @ A @ ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) ) )
= X ) ).
% last_snoc
thf(fact_110_map__tailrec__rev_Oinduct,axiom,
! [A: $tType,B: $tType,P: ( A > B ) > ( list @ A ) > ( list @ B ) > $o,A0: A > B,A1: list @ A,A22: list @ B] :
( ! [F2: A > B,X1: list @ B] : ( P @ F2 @ ( nil @ A ) @ X1 )
=> ( ! [F2: A > B,A2: A,As: list @ A,Bs: list @ B] :
( ( P @ F2 @ As @ ( cons @ B @ ( F2 @ A2 ) @ Bs ) )
=> ( P @ F2 @ ( cons @ A @ A2 @ As ) @ Bs ) )
=> ( P @ A0 @ A1 @ A22 ) ) ) ).
% map_tailrec_rev.induct
thf(fact_111_list__nonempty__induct,axiom,
! [A: $tType,Xs: list @ A,P: ( list @ A ) > $o] :
( ( Xs
!= ( nil @ A ) )
=> ( ! [X2: A] : ( P @ ( cons @ A @ X2 @ ( nil @ A ) ) )
=> ( ! [X2: A,Xs2: list @ A] :
( ( Xs2
!= ( nil @ A ) )
=> ( ( P @ Xs2 )
=> ( P @ ( cons @ A @ X2 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_112_remdups__adj_Oinduct,axiom,
! [A: $tType,P: ( list @ A ) > $o,A0: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [X2: A] : ( P @ ( cons @ A @ X2 @ ( nil @ A ) ) )
=> ( ! [X2: A,Y2: A,Xs2: list @ A] :
( ( ( X2 = Y2 )
=> ( P @ ( cons @ A @ X2 @ Xs2 ) ) )
=> ( ( ( X2 != Y2 )
=> ( P @ ( cons @ A @ Y2 @ Xs2 ) ) )
=> ( P @ ( cons @ A @ X2 @ ( cons @ A @ Y2 @ Xs2 ) ) ) ) )
=> ( P @ A0 ) ) ) ) ).
% remdups_adj.induct
thf(fact_113_remdups__adj_Ocases,axiom,
! [A: $tType,X: list @ A] :
( ( X
!= ( nil @ A ) )
=> ( ! [X2: A] :
( X
!= ( cons @ A @ X2 @ ( nil @ A ) ) )
=> ~ ! [X2: A,Y2: A,Xs2: list @ A] :
( X
!= ( cons @ A @ X2 @ ( cons @ A @ Y2 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_114_append__Nil,axiom,
! [A: $tType,Ys: list @ A] :
( ( append @ A @ ( nil @ A ) @ Ys )
= Ys ) ).
% append_Nil
thf(fact_115_transpose_Ocases,axiom,
! [A: $tType,X: list @ ( list @ A )] :
( ( X
!= ( nil @ ( list @ A ) ) )
=> ( ! [Xss: list @ ( list @ A )] :
( X
!= ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss ) )
=> ~ ! [X2: A,Xs2: list @ A,Xss: list @ ( list @ A )] :
( X
!= ( cons @ ( list @ A ) @ ( cons @ A @ X2 @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_116_eq__Nil__appendI,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( Xs = Ys )
=> ( Xs
= ( append @ A @ ( nil @ A ) @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_117_splice_Oinduct,axiom,
! [A: $tType,P: ( list @ A ) > ( list @ A ) > $o,A0: list @ A,A1: list @ A] :
( ! [X1: list @ A] : ( P @ ( nil @ A ) @ X1 )
=> ( ! [V: A,Va: list @ A] : ( P @ ( cons @ A @ V @ Va ) @ ( nil @ A ) )
=> ( ! [X2: A,Xs2: list @ A,Y2: A,Ys3: list @ A] :
( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons @ A @ X2 @ Xs2 ) @ ( cons @ A @ Y2 @ Ys3 ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% splice.induct
thf(fact_118_list__induct2_H,axiom,
! [A: $tType,B: $tType,P: ( list @ A ) > ( list @ B ) > $o,Xs: list @ A,Ys: list @ B] :
( ( P @ ( nil @ A ) @ ( nil @ B ) )
=> ( ! [X2: A,Xs2: list @ A] : ( P @ ( cons @ A @ X2 @ Xs2 ) @ ( nil @ B ) )
=> ( ! [Y2: B,Ys3: list @ B] : ( P @ ( nil @ A ) @ ( cons @ B @ Y2 @ Ys3 ) )
=> ( ! [X2: A,Xs2: list @ A,Y2: B,Ys3: list @ B] :
( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons @ A @ X2 @ Xs2 ) @ ( cons @ B @ Y2 @ Ys3 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_119_neq__Nil__conv,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
= ( ? [Y3: A,Ys4: list @ A] :
( Xs
= ( cons @ A @ Y3 @ Ys4 ) ) ) ) ).
% neq_Nil_conv
thf(fact_120_list_Oinducts,axiom,
! [A: $tType,P: ( list @ A ) > $o,List: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [X1: A,X23: list @ A] :
( ( P @ X23 )
=> ( P @ ( cons @ A @ X1 @ X23 ) ) )
=> ( P @ List ) ) ) ).
% list.inducts
thf(fact_121_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_122_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_123_list_Odistinct_I1_J,axiom,
! [A: $tType,X21: A,X22: list @ A] :
( ( nil @ A )
!= ( cons @ A @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_124_list__of__code,axiom,
! [A: $tType] :
( ( coindu1736001939ist_of @ A )
= ( coindu1542816657of_aux @ A @ ( nil @ A ) ) ) ).
% list_of_code
thf(fact_125_rev__induct,axiom,
! [A: $tType,P: ( list @ A ) > $o,Xs: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [X2: A,Xs2: list @ A] :
( ( P @ Xs2 )
=> ( P @ ( append @ A @ Xs2 @ ( cons @ A @ X2 @ ( nil @ A ) ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_126_rev__exhaust,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ~ ! [Ys3: list @ A,Y2: A] :
( Xs
!= ( append @ A @ Ys3 @ ( cons @ A @ Y2 @ ( nil @ A ) ) ) ) ) ).
% rev_exhaust
thf(fact_127_Cons__eq__append__conv,axiom,
! [A: $tType,X: A,Xs: list @ A,Ys: list @ A,Zs2: list @ A] :
( ( ( cons @ A @ X @ Xs )
= ( append @ A @ Ys @ Zs2 ) )
= ( ( ( Ys
= ( nil @ A ) )
& ( ( cons @ A @ X @ Xs )
= Zs2 ) )
| ? [Ys2: list @ A] :
( ( ( cons @ A @ X @ Ys2 )
= Ys )
& ( Xs
= ( append @ A @ Ys2 @ Zs2 ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_128_append__eq__Cons__conv,axiom,
! [A: $tType,Ys: list @ A,Zs2: list @ A,X: A,Xs: list @ A] :
( ( ( append @ A @ Ys @ Zs2 )
= ( cons @ A @ X @ Xs ) )
= ( ( ( Ys
= ( nil @ A ) )
& ( Zs2
= ( cons @ A @ X @ Xs ) ) )
| ? [Ys2: list @ A] :
( ( Ys
= ( cons @ A @ X @ Ys2 ) )
& ( ( append @ A @ Ys2 @ Zs2 )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_129_rev__nonempty__induct,axiom,
! [A: $tType,Xs: list @ A,P: ( list @ A ) > $o] :
( ( Xs
!= ( nil @ A ) )
=> ( ! [X2: A] : ( P @ ( cons @ A @ X2 @ ( nil @ A ) ) )
=> ( ! [X2: A,Xs2: list @ A] :
( ( Xs2
!= ( nil @ A ) )
=> ( ( P @ Xs2 )
=> ( P @ ( append @ A @ Xs2 @ ( cons @ A @ X2 @ ( nil @ A ) ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_130_last__ConsR,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( last @ A @ ( cons @ A @ X @ Xs ) )
= ( last @ A @ Xs ) ) ) ).
% last_ConsR
thf(fact_131_last__ConsL,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( Xs
= ( nil @ A ) )
=> ( ( last @ A @ ( cons @ A @ X @ Xs ) )
= X ) ) ).
% last_ConsL
thf(fact_132_last_Osimps,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( ( Xs
= ( nil @ A ) )
=> ( ( last @ A @ ( cons @ A @ X @ Xs ) )
= X ) )
& ( ( Xs
!= ( nil @ A ) )
=> ( ( last @ A @ ( cons @ A @ X @ Xs ) )
= ( last @ A @ Xs ) ) ) ) ).
% last.simps
thf(fact_133_longest__common__suffix,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
? [Ss: list @ A,Xs3: list @ A,Ys6: list @ A] :
( ( Xs
= ( append @ A @ Xs3 @ Ss ) )
& ( Ys
= ( append @ A @ Ys6 @ Ss ) )
& ( ( Xs3
= ( nil @ A ) )
| ( Ys6
= ( nil @ A ) )
| ( ( last @ A @ Xs3 )
!= ( last @ A @ Ys6 ) ) ) ) ).
% longest_common_suffix
thf(fact_134_last__append,axiom,
! [A: $tType,Ys: list @ A,Xs: list @ A] :
( ( ( Ys
= ( nil @ A ) )
=> ( ( last @ A @ ( append @ A @ Xs @ Ys ) )
= ( last @ A @ Xs ) ) )
& ( ( Ys
!= ( nil @ A ) )
=> ( ( last @ A @ ( append @ A @ Xs @ Ys ) )
= ( last @ A @ Ys ) ) ) ) ).
% last_append
thf(fact_135_llist__of__eq__LNil__conv,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( coindu1079202755ist_of @ A @ Xs )
= ( coindu1598213697e_LNil @ A ) )
= ( Xs
= ( nil @ A ) ) ) ).
% llist_of_eq_LNil_conv
thf(fact_136_llist__of_Osimps_I1_J,axiom,
! [A: $tType] :
( ( coindu1079202755ist_of @ A @ ( nil @ A ) )
= ( coindu1598213697e_LNil @ A ) ) ).
% llist_of.simps(1)
thf(fact_137_not__Cons__self2,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( cons @ A @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_138_append__eq__appendI,axiom,
! [A: $tType,Xs: list @ A,Xs1: list @ A,Zs2: list @ A,Ys: list @ A,Us: list @ A] :
( ( ( append @ A @ Xs @ Xs1 )
= Zs2 )
=> ( ( Ys
= ( append @ A @ Xs1 @ Us ) )
=> ( ( append @ A @ Xs @ Ys )
= ( append @ A @ Zs2 @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_139_append__eq__append__conv2,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs2: list @ A,Ts: list @ A] :
( ( ( append @ A @ Xs @ Ys )
= ( append @ A @ Zs2 @ Ts ) )
= ( ? [Us2: list @ A] :
( ( ( Xs
= ( append @ A @ Zs2 @ Us2 ) )
& ( ( append @ A @ Us2 @ Ys )
= Ts ) )
| ( ( ( append @ A @ Xs @ Us2 )
= Zs2 )
& ( Ys
= ( append @ A @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_140_longest__common__prefix_Oinduct,axiom,
! [A: $tType,P: ( list @ A ) > ( list @ A ) > $o,A0: list @ A,A1: list @ A] :
( ! [X2: A,Xs2: list @ A,Y2: A,Ys3: list @ A] :
( ( ( X2 = Y2 )
=> ( P @ Xs2 @ Ys3 ) )
=> ( P @ ( cons @ A @ X2 @ Xs2 ) @ ( cons @ A @ Y2 @ Ys3 ) ) )
=> ( ! [X1: list @ A] : ( P @ ( nil @ A ) @ X1 )
=> ( ! [Uu: list @ A] : ( P @ Uu @ ( nil @ A ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% longest_common_prefix.induct
thf(fact_141_prefixes_Oinduct,axiom,
! [A: $tType,P: ( list @ A ) > $o,A0: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [X2: A,Xs2: list @ A] :
( ( P @ Xs2 )
=> ( P @ ( cons @ A @ X2 @ Xs2 ) ) )
=> ( P @ A0 ) ) ) ).
% prefixes.induct
thf(fact_142_prefixes_Ocases,axiom,
! [A: $tType,X: list @ A] :
( ( X
!= ( nil @ A ) )
=> ~ ! [X2: A,Xs2: list @ A] :
( X
!= ( cons @ A @ X2 @ Xs2 ) ) ) ).
% prefixes.cases
thf(fact_143_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_144_bind__simps_I2_J,axiom,
! [A: $tType,B: $tType,X: B,Xs: list @ B,F: B > ( list @ A )] :
( ( bind @ B @ A @ ( cons @ B @ X @ Xs ) @ F )
= ( append @ A @ ( F @ X ) @ ( bind @ B @ A @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_145_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_146_butlast__snoc,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( butlast @ A @ ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) ) )
= Xs ) ).
% butlast_snoc
thf(fact_147_butlast_Osimps_I1_J,axiom,
! [A: $tType] :
( ( butlast @ A @ ( nil @ A ) )
= ( nil @ A ) ) ).
% butlast.simps(1)
thf(fact_148_butlast_Osimps_I2_J,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( ( Xs
= ( nil @ A ) )
=> ( ( butlast @ A @ ( cons @ A @ X @ Xs ) )
= ( nil @ A ) ) )
& ( ( Xs
!= ( nil @ A ) )
=> ( ( butlast @ A @ ( cons @ A @ X @ Xs ) )
= ( cons @ A @ X @ ( butlast @ A @ Xs ) ) ) ) ) ).
% butlast.simps(2)
thf(fact_149_butlast__append,axiom,
! [A: $tType,Ys: list @ A,Xs: list @ A] :
( ( ( Ys
= ( nil @ A ) )
=> ( ( butlast @ A @ ( append @ A @ Xs @ Ys ) )
= ( butlast @ A @ Xs ) ) )
& ( ( Ys
!= ( nil @ A ) )
=> ( ( butlast @ A @ ( append @ A @ Xs @ Ys ) )
= ( append @ A @ Xs @ ( butlast @ A @ Ys ) ) ) ) ) ).
% butlast_append
thf(fact_150_snoc__eq__iff__butlast,axiom,
! [A: $tType,Xs: list @ A,X: A,Ys: list @ A] :
( ( ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) )
= Ys )
= ( ( Ys
!= ( nil @ A ) )
& ( ( butlast @ A @ Ys )
= Xs )
& ( ( last @ A @ Ys )
= X ) ) ) ).
% snoc_eq_iff_butlast
thf(fact_151_prefixes__snoc,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( prefixes @ A @ ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) ) )
= ( append @ ( list @ A ) @ ( prefixes @ A @ Xs ) @ ( cons @ ( list @ A ) @ ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) ) @ ( nil @ ( list @ A ) ) ) ) ) ).
% prefixes_snoc
thf(fact_152_prefixes__eq__Snoc,axiom,
! [A: $tType,Ys: list @ A,Xs: list @ ( list @ A ),X: list @ A] :
( ( ( prefixes @ A @ Ys )
= ( append @ ( list @ A ) @ Xs @ ( cons @ ( list @ A ) @ X @ ( nil @ ( list @ A ) ) ) ) )
= ( ( ( ( Ys
= ( nil @ A ) )
& ( Xs
= ( nil @ ( list @ A ) ) ) )
| ? [Z: A,Zs4: list @ A] :
( ( Ys
= ( append @ A @ Zs4 @ ( cons @ A @ Z @ ( nil @ A ) ) ) )
& ( Xs
= ( prefixes @ A @ Zs4 ) ) ) )
& ( X = Ys ) ) ) ).
% prefixes_eq_Snoc
thf(fact_153_sublists_Osimps_I1_J,axiom,
! [A: $tType] :
( ( sublists @ A @ ( nil @ A ) )
= ( cons @ ( list @ A ) @ ( nil @ A ) @ ( nil @ ( list @ A ) ) ) ) ).
% sublists.simps(1)
thf(fact_154_prefixes_Osimps_I1_J,axiom,
! [A: $tType] :
( ( prefixes @ A @ ( nil @ A ) )
= ( cons @ ( list @ A ) @ ( nil @ A ) @ ( nil @ ( list @ A ) ) ) ) ).
% prefixes.simps(1)
thf(fact_155_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_156_llimit__induct,axiom,
! [A: $tType,P: ( coindu1593790203_llist @ A ) > $o,Xs: coindu1593790203_llist @ A] :
( ( P @ ( coindu1598213697e_LNil @ A ) )
=> ( ! [X2: A,Xs2: coindu1593790203_llist @ A] :
( ( coindu1213758845finite @ A @ Xs2 )
=> ( ( P @ Xs2 )
=> ( P @ ( coindu1121789889_LCons @ A @ X2 @ Xs2 ) ) ) )
=> ( ( ! [Ys5: coindu1593790203_llist @ A] :
( ( coindu574146665prefix @ A @ Ys5 @ Xs )
=> ( P @ Ys5 ) )
=> ( P @ Xs ) )
=> ( P @ Xs ) ) ) ) ).
% llimit_induct
thf(fact_157_prefixes_Oelims,axiom,
! [A: $tType,X: list @ A,Y: list @ ( list @ A )] :
( ( ( prefixes @ A @ X )
= Y )
=> ( ( ( X
= ( nil @ A ) )
=> ( Y
!= ( cons @ ( list @ A ) @ ( nil @ A ) @ ( nil @ ( list @ A ) ) ) ) )
=> ~ ! [X2: A,Xs2: list @ A] :
( ( X
= ( cons @ A @ X2 @ Xs2 ) )
=> ( Y
!= ( cons @ ( list @ A ) @ ( nil @ A ) @ ( map @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X2 ) @ ( prefixes @ A @ Xs2 ) ) ) ) ) ) ) ).
% prefixes.elims
thf(fact_158_Nil__is__map__conv,axiom,
! [A: $tType,B: $tType,F: B > A,Xs: list @ B] :
( ( ( nil @ A )
= ( map @ B @ A @ F @ Xs ) )
= ( Xs
= ( nil @ B ) ) ) ).
% Nil_is_map_conv
thf(fact_159_map__is__Nil__conv,axiom,
! [A: $tType,B: $tType,F: B > A,Xs: list @ B] :
( ( ( map @ B @ A @ F @ Xs )
= ( nil @ A ) )
= ( Xs
= ( nil @ B ) ) ) ).
% map_is_Nil_conv
thf(fact_160_list_Omap__disc__iff,axiom,
! [B: $tType,A: $tType,F: A > B,A3: list @ A] :
( ( ( map @ A @ B @ F @ A3 )
= ( nil @ B ) )
= ( A3
= ( nil @ A ) ) ) ).
% list.map_disc_iff
thf(fact_161_map__append,axiom,
! [A: $tType,B: $tType,F: B > A,Xs: list @ B,Ys: list @ B] :
( ( map @ B @ A @ F @ ( append @ B @ Xs @ Ys ) )
= ( append @ A @ ( map @ B @ A @ F @ Xs ) @ ( map @ B @ A @ F @ Ys ) ) ) ).
% map_append
thf(fact_162_lstrict__prefix__code_I4_J,axiom,
! [B: $tType,X: B,Xs: coindu1593790203_llist @ B,Y: B,Ys: coindu1593790203_llist @ B] :
( ( coindu574146665prefix @ B @ ( coindu1121789889_LCons @ B @ X @ Xs ) @ ( coindu1121789889_LCons @ B @ Y @ Ys ) )
= ( ( X = Y )
& ( coindu574146665prefix @ B @ Xs @ Ys ) ) ) ).
% lstrict_prefix_code(4)
thf(fact_163_lstrict__prefix__code_I1_J,axiom,
! [A: $tType] :
~ ( coindu574146665prefix @ A @ ( coindu1598213697e_LNil @ A ) @ ( coindu1598213697e_LNil @ A ) ) ).
% lstrict_prefix_code(1)
thf(fact_164_lstrict__prefix__code_I2_J,axiom,
! [B: $tType,Y: B,Ys: coindu1593790203_llist @ B] : ( coindu574146665prefix @ B @ ( coindu1598213697e_LNil @ B ) @ ( coindu1121789889_LCons @ B @ Y @ Ys ) ) ).
% lstrict_prefix_code(2)
thf(fact_165_lstrict__prefix__code_I3_J,axiom,
! [B: $tType,X: B,Xs: coindu1593790203_llist @ B] :
~ ( coindu574146665prefix @ B @ ( coindu1121789889_LCons @ B @ X @ Xs ) @ ( coindu1598213697e_LNil @ B ) ) ).
% lstrict_prefix_code(3)
thf(fact_166_llist__less__induct,axiom,
! [A: $tType,P: ( coindu1593790203_llist @ A ) > $o,Xs: coindu1593790203_llist @ A] :
( ! [Xs2: coindu1593790203_llist @ A] :
( ! [Ys5: coindu1593790203_llist @ A] :
( ( coindu574146665prefix @ A @ Ys5 @ Xs2 )
=> ( P @ Ys5 ) )
=> ( P @ Xs2 ) )
=> ( P @ Xs ) ) ).
% llist_less_induct
thf(fact_167_map__butlast,axiom,
! [A: $tType,B: $tType,F: B > A,Xs: list @ B] :
( ( map @ B @ A @ F @ ( butlast @ B @ Xs ) )
= ( butlast @ A @ ( map @ B @ A @ F @ Xs ) ) ) ).
% map_butlast
thf(fact_168_list_Osimps_I8_J,axiom,
! [A: $tType,B: $tType,F: A > B] :
( ( map @ A @ B @ F @ ( nil @ A ) )
= ( nil @ B ) ) ).
% list.simps(8)
thf(fact_169_list_Osimps_I9_J,axiom,
! [B: $tType,A: $tType,F: A > B,X21: A,X22: list @ A] :
( ( map @ A @ B @ F @ ( cons @ A @ X21 @ X22 ) )
= ( cons @ B @ ( F @ X21 ) @ ( map @ A @ B @ F @ X22 ) ) ) ).
% list.simps(9)
thf(fact_170_Cons__eq__map__D,axiom,
! [A: $tType,B: $tType,X: A,Xs: list @ A,F: B > A,Ys: list @ B] :
( ( ( cons @ A @ X @ Xs )
= ( map @ B @ A @ F @ Ys ) )
=> ? [Z3: B,Zs3: list @ B] :
( ( Ys
= ( cons @ B @ Z3 @ Zs3 ) )
& ( X
= ( F @ Z3 ) )
& ( Xs
= ( map @ B @ A @ F @ Zs3 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_171_map__eq__Cons__D,axiom,
! [B: $tType,A: $tType,F: B > A,Xs: list @ B,Y: A,Ys: list @ A] :
( ( ( map @ B @ A @ F @ Xs )
= ( cons @ A @ Y @ Ys ) )
=> ? [Z3: B,Zs3: list @ B] :
( ( Xs
= ( cons @ B @ Z3 @ Zs3 ) )
& ( ( F @ Z3 )
= Y )
& ( ( map @ B @ A @ F @ Zs3 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_172_Cons__eq__map__conv,axiom,
! [A: $tType,B: $tType,X: A,Xs: list @ A,F: B > A,Ys: list @ B] :
( ( ( cons @ A @ X @ Xs )
= ( map @ B @ A @ F @ Ys ) )
= ( ? [Z: B,Zs4: list @ B] :
( ( Ys
= ( cons @ B @ Z @ Zs4 ) )
& ( X
= ( F @ Z ) )
& ( Xs
= ( map @ B @ A @ F @ Zs4 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_173_map__eq__Cons__conv,axiom,
! [A: $tType,B: $tType,F: B > A,Xs: list @ B,Y: A,Ys: list @ A] :
( ( ( map @ B @ A @ F @ Xs )
= ( cons @ A @ Y @ Ys ) )
= ( ? [Z: B,Zs4: list @ B] :
( ( Xs
= ( cons @ B @ Z @ Zs4 ) )
& ( ( F @ Z )
= Y )
& ( ( map @ B @ A @ F @ Zs4 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_174_lstrict__prefix__lfinite1,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu574146665prefix @ A @ Xs @ Ys )
=> ( coindu1213758845finite @ A @ Xs ) ) ).
% lstrict_prefix_lfinite1
thf(fact_175_last__map,axiom,
! [B: $tType,A: $tType,Xs: list @ A,F: A > B] :
( ( Xs
!= ( nil @ A ) )
=> ( ( last @ B @ ( map @ A @ B @ F @ Xs ) )
= ( F @ ( last @ A @ Xs ) ) ) ) ).
% last_map
thf(fact_176_prefixes_Osimps_I2_J,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( prefixes @ A @ ( cons @ A @ X @ Xs ) )
= ( cons @ ( list @ A ) @ ( nil @ A ) @ ( map @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X ) @ ( prefixes @ A @ Xs ) ) ) ) ).
% prefixes.simps(2)
thf(fact_177_prefixes_Opelims,axiom,
! [A: $tType,X: list @ A,Y: list @ ( list @ A )] :
( ( ( prefixes @ A @ X )
= Y )
=> ( ( accp @ ( list @ A ) @ ( prefixes_rel @ A ) @ X )
=> ( ( ( X
= ( nil @ A ) )
=> ( ( Y
= ( cons @ ( list @ A ) @ ( nil @ A ) @ ( nil @ ( list @ A ) ) ) )
=> ~ ( accp @ ( list @ A ) @ ( prefixes_rel @ A ) @ ( nil @ A ) ) ) )
=> ~ ! [X2: A,Xs2: list @ A] :
( ( X
= ( cons @ A @ X2 @ Xs2 ) )
=> ( ( Y
= ( cons @ ( list @ A ) @ ( nil @ A ) @ ( map @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X2 ) @ ( prefixes @ A @ Xs2 ) ) ) )
=> ~ ( accp @ ( list @ A ) @ ( prefixes_rel @ A ) @ ( cons @ A @ X2 @ Xs2 ) ) ) ) ) ) ) ).
% prefixes.pelims
thf(fact_178_Coinductive__List__Mirabelle__kmikjhschf_Ofinite__lprefix__nitpick__simps_I3_J,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Y: A,Ys: coindu1593790203_llist @ A] :
( ( coindu1571841457prefix @ A @ Xs @ ( coindu1121789889_LCons @ A @ Y @ Ys ) )
= ( ( Xs
= ( coindu1598213697e_LNil @ A ) )
| ? [Xs4: coindu1593790203_llist @ A] :
( ( Xs
= ( coindu1121789889_LCons @ A @ Y @ Xs4 ) )
& ( coindu1571841457prefix @ A @ Xs4 @ Ys ) ) ) ) ).
% Coinductive_List_Mirabelle_kmikjhschf.finite_lprefix_nitpick_simps(3)
thf(fact_179_maps__simps_I1_J,axiom,
! [A: $tType,B: $tType,F: B > ( list @ A ),X: B,Xs: list @ B] :
( ( maps @ B @ A @ F @ ( cons @ B @ X @ Xs ) )
= ( append @ A @ ( F @ X ) @ ( maps @ B @ A @ F @ Xs ) ) ) ).
% maps_simps(1)
thf(fact_180_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_181_Coinductive__List__Mirabelle__kmikjhschf_Ofinite__lprefix__nitpick__simps_I2_J,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] : ( coindu1571841457prefix @ A @ ( coindu1598213697e_LNil @ A ) @ Xs ) ).
% Coinductive_List_Mirabelle_kmikjhschf.finite_lprefix_nitpick_simps(2)
thf(fact_182_Coinductive__List__Mirabelle__kmikjhschf_Ofinite__lprefix__nitpick__simps_I1_J,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( ( coindu1571841457prefix @ A @ Xs @ ( coindu1598213697e_LNil @ A ) )
= ( Xs
= ( coindu1598213697e_LNil @ A ) ) ) ).
% Coinductive_List_Mirabelle_kmikjhschf.finite_lprefix_nitpick_simps(1)
thf(fact_183_insert__Nil,axiom,
! [A: $tType,X: A] :
( ( insert @ A @ X @ ( nil @ A ) )
= ( cons @ A @ X @ ( nil @ A ) ) ) ).
% insert_Nil
thf(fact_184_rotate1_Osimps_I2_J,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( rotate1 @ A @ ( cons @ A @ X @ Xs ) )
= ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) ) ) ).
% rotate1.simps(2)
thf(fact_185_map__eq__map__tailrec,axiom,
! [B: $tType,A: $tType] :
( ( map @ A @ B )
= ( map_tailrec @ A @ B ) ) ).
% map_eq_map_tailrec
thf(fact_186_rotate1__is__Nil__conv,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( rotate1 @ A @ Xs )
= ( nil @ A ) )
= ( Xs
= ( nil @ A ) ) ) ).
% rotate1_is_Nil_conv
thf(fact_187_rotate1_Osimps_I1_J,axiom,
! [A: $tType] :
( ( rotate1 @ A @ ( nil @ A ) )
= ( nil @ A ) ) ).
% rotate1.simps(1)
thf(fact_188_hd__list__of,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( ( coindu1213758845finite @ A @ Xs )
=> ( ( hd @ A @ ( coindu1736001939ist_of @ A @ Xs ) )
= ( coindu1046438764le_lhd @ A @ Xs ) ) ) ).
% hd_list_of
thf(fact_189_list__ex1__simps_I1_J,axiom,
! [A: $tType,P: A > $o] :
~ ( list_ex1 @ A @ P @ ( nil @ A ) ) ).
% list_ex1_simps(1)
thf(fact_190_list__of__LCons__conv,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,X: A] :
( ( ( coindu1213758845finite @ A @ Xs )
=> ( ( coindu1736001939ist_of @ A @ ( coindu1121789889_LCons @ A @ X @ Xs ) )
= ( cons @ A @ X @ ( coindu1736001939ist_of @ A @ Xs ) ) ) )
& ( ~ ( coindu1213758845finite @ A @ Xs )
=> ( ( coindu1736001939ist_of @ A @ ( coindu1121789889_LCons @ A @ X @ Xs ) )
= ( undefined @ ( list @ A ) ) ) ) ) ).
% list_of_LCons_conv
thf(fact_191_hd__append2,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( hd @ A @ ( append @ A @ Xs @ Ys ) )
= ( hd @ A @ Xs ) ) ) ).
% hd_append2
thf(fact_192_lhd__llist__of,axiom,
! [A: $tType,Xs: list @ A] :
( ( coindu1046438764le_lhd @ A @ ( coindu1079202755ist_of @ A @ Xs ) )
= ( hd @ A @ Xs ) ) ).
% lhd_llist_of
thf(fact_193_list_Osel_I1_J,axiom,
! [A: $tType,X21: A,X22: list @ A] :
( ( hd @ A @ ( cons @ A @ X21 @ X22 ) )
= X21 ) ).
% list.sel(1)
thf(fact_194_list_Omap__sel_I1_J,axiom,
! [B: $tType,A: $tType,A3: list @ A,F: A > B] :
( ( A3
!= ( nil @ A ) )
=> ( ( hd @ B @ ( map @ A @ B @ F @ A3 ) )
= ( F @ ( hd @ A @ A3 ) ) ) ) ).
% list.map_sel(1)
thf(fact_195_hd__map,axiom,
! [B: $tType,A: $tType,Xs: list @ A,F: A > B] :
( ( Xs
!= ( nil @ A ) )
=> ( ( hd @ B @ ( map @ A @ B @ F @ Xs ) )
= ( F @ ( hd @ A @ Xs ) ) ) ) ).
% hd_map
thf(fact_196_longest__common__prefix,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
? [Ps: list @ A,Xs3: list @ A,Ys6: list @ A] :
( ( Xs
= ( append @ A @ Ps @ Xs3 ) )
& ( Ys
= ( append @ A @ Ps @ Ys6 ) )
& ( ( Xs3
= ( nil @ A ) )
| ( Ys6
= ( nil @ A ) )
| ( ( hd @ A @ Xs3 )
!= ( hd @ A @ Ys6 ) ) ) ) ).
% longest_common_prefix
thf(fact_197_hd__append,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( Xs
= ( nil @ A ) )
=> ( ( hd @ A @ ( append @ A @ Xs @ Ys ) )
= ( hd @ A @ Ys ) ) )
& ( ( Xs
!= ( nil @ A ) )
=> ( ( hd @ A @ ( append @ A @ Xs @ Ys ) )
= ( hd @ A @ Xs ) ) ) ) ).
% hd_append
thf(fact_198_llast__LNil,axiom,
! [A: $tType] :
( ( coindu2000965700_llast @ A @ ( coindu1598213697e_LNil @ A ) )
= ( undefined @ A ) ) ).
% llast_LNil
thf(fact_199_llast__linfinite,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( ~ ( coindu1213758845finite @ A @ Xs )
=> ( ( coindu2000965700_llast @ A @ Xs )
= ( undefined @ A ) ) ) ).
% llast_linfinite
thf(fact_200_list__of__aux__def,axiom,
! [A: $tType] :
( ( coindu1542816657of_aux @ A )
= ( ^ [Xs5: list @ A,Ys4: coindu1593790203_llist @ A] : ( if @ ( list @ A ) @ ( coindu1213758845finite @ A @ Ys4 ) @ ( append @ A @ ( rev @ A @ Xs5 ) @ ( coindu1736001939ist_of @ A @ Ys4 ) ) @ ( undefined @ ( list @ A ) ) ) ) ) ).
% list_of_aux_def
thf(fact_201_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_202_rev__rev__ident,axiom,
! [A: $tType,Xs: list @ A] :
( ( rev @ A @ ( rev @ A @ Xs ) )
= Xs ) ).
% rev_rev_ident
thf(fact_203_rev__is__rev__conv,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( rev @ A @ Xs )
= ( rev @ A @ Ys ) )
= ( Xs = Ys ) ) ).
% rev_is_rev_conv
thf(fact_204_Nil__is__rev__conv,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( nil @ A )
= ( rev @ A @ Xs ) )
= ( Xs
= ( nil @ A ) ) ) ).
% Nil_is_rev_conv
thf(fact_205_rev__is__Nil__conv,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( rev @ A @ Xs )
= ( nil @ A ) )
= ( Xs
= ( nil @ A ) ) ) ).
% rev_is_Nil_conv
thf(fact_206_rev__append,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( rev @ A @ ( append @ A @ Xs @ Ys ) )
= ( append @ A @ ( rev @ A @ Ys ) @ ( rev @ A @ Xs ) ) ) ).
% rev_append
thf(fact_207_rev__singleton__conv,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( ( rev @ A @ Xs )
= ( cons @ A @ X @ ( nil @ A ) ) )
= ( Xs
= ( cons @ A @ X @ ( nil @ A ) ) ) ) ).
% rev_singleton_conv
thf(fact_208_singleton__rev__conv,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( ( cons @ A @ X @ ( nil @ A ) )
= ( rev @ A @ Xs ) )
= ( Xs
= ( cons @ A @ X @ ( nil @ A ) ) ) ) ).
% singleton_rev_conv
thf(fact_209_tl__append2,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( tl @ A @ ( append @ A @ Xs @ Ys ) )
= ( append @ A @ ( tl @ A @ Xs ) @ Ys ) ) ) ).
% tl_append2
thf(fact_210_rev__eq__Cons__iff,axiom,
! [A: $tType,Xs: list @ A,Y: A,Ys: list @ A] :
( ( ( rev @ A @ Xs )
= ( cons @ A @ Y @ Ys ) )
= ( Xs
= ( append @ A @ ( rev @ A @ Ys ) @ ( cons @ A @ Y @ ( nil @ A ) ) ) ) ) ).
% rev_eq_Cons_iff
thf(fact_211_list_Ocollapse,axiom,
! [A: $tType,List: list @ A] :
( ( List
!= ( nil @ A ) )
=> ( ( cons @ A @ ( hd @ A @ List ) @ ( tl @ A @ List ) )
= List ) ) ).
% list.collapse
thf(fact_212_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_213_map__tl,axiom,
! [A: $tType,B: $tType,F: B > A,Xs: list @ B] :
( ( map @ B @ A @ F @ ( tl @ B @ Xs ) )
= ( tl @ A @ ( map @ B @ A @ F @ Xs ) ) ) ).
% map_tl
thf(fact_214_rev__map,axiom,
! [A: $tType,B: $tType,F: B > A,Xs: list @ B] :
( ( rev @ A @ ( map @ B @ A @ F @ Xs ) )
= ( map @ B @ A @ F @ ( rev @ B @ Xs ) ) ) ).
% rev_map
thf(fact_215_rev__swap,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( rev @ A @ Xs )
= Ys )
= ( Xs
= ( rev @ A @ Ys ) ) ) ).
% rev_swap
thf(fact_216_list_Osel_I2_J,axiom,
! [A: $tType] :
( ( tl @ A @ ( nil @ A ) )
= ( nil @ A ) ) ).
% list.sel(2)
thf(fact_217_list_Osel_I3_J,axiom,
! [A: $tType,X21: A,X22: list @ A] :
( ( tl @ A @ ( cons @ A @ X21 @ X22 ) )
= X22 ) ).
% list.sel(3)
thf(fact_218_butlast__tl,axiom,
! [A: $tType,Xs: list @ A] :
( ( butlast @ A @ ( tl @ A @ Xs ) )
= ( tl @ A @ ( butlast @ A @ Xs ) ) ) ).
% butlast_tl
thf(fact_219_rev_Osimps_I1_J,axiom,
! [A: $tType] :
( ( rev @ A @ ( nil @ A ) )
= ( nil @ A ) ) ).
% rev.simps(1)
thf(fact_220_list_Omap__sel_I2_J,axiom,
! [B: $tType,A: $tType,A3: list @ A,F: A > B] :
( ( A3
!= ( nil @ A ) )
=> ( ( tl @ B @ ( map @ A @ B @ F @ A3 ) )
= ( map @ A @ B @ F @ ( tl @ A @ A3 ) ) ) ) ).
% list.map_sel(2)
thf(fact_221_tl__Nil,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( tl @ A @ Xs )
= ( nil @ A ) )
= ( ( Xs
= ( nil @ A ) )
| ? [X3: A] :
( Xs
= ( cons @ A @ X3 @ ( nil @ A ) ) ) ) ) ).
% tl_Nil
thf(fact_222_Nil__tl,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( nil @ A )
= ( tl @ A @ Xs ) )
= ( ( Xs
= ( nil @ A ) )
| ? [X3: A] :
( Xs
= ( cons @ A @ X3 @ ( nil @ A ) ) ) ) ) ).
% Nil_tl
thf(fact_223_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_224_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_225_list__of__aux__code_I1_J,axiom,
! [A: $tType,Xs: list @ A] :
( ( coindu1542816657of_aux @ A @ Xs @ ( coindu1598213697e_LNil @ A ) )
= ( rev @ A @ Xs ) ) ).
% list_of_aux_code(1)
thf(fact_226_rev_Osimps_I2_J,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( rev @ A @ ( cons @ A @ X @ Xs ) )
= ( append @ A @ ( rev @ A @ Xs ) @ ( cons @ A @ X @ ( nil @ A ) ) ) ) ).
% rev.simps(2)
thf(fact_227_hd__rev,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( hd @ A @ ( rev @ A @ Xs ) )
= ( last @ A @ Xs ) ) ) ).
% hd_rev
thf(fact_228_last__rev,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( last @ A @ ( rev @ A @ Xs ) )
= ( hd @ A @ Xs ) ) ) ).
% last_rev
thf(fact_229_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_230_map__tailrec__def,axiom,
! [B: $tType,A: $tType] :
( ( map_tailrec @ A @ B )
= ( ^ [F3: A > B,As2: list @ A] : ( rev @ B @ ( map_tailrec_rev @ A @ B @ F3 @ As2 @ ( nil @ B ) ) ) ) ) ).
% map_tailrec_def
thf(fact_231_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_232_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_233_map__tailrec__rev_Osimps_I2_J,axiom,
! [A: $tType,B: $tType,F: A > B,A3: A,As3: list @ A,Bs2: list @ B] :
( ( map_tailrec_rev @ A @ B @ F @ ( cons @ A @ A3 @ As3 ) @ Bs2 )
= ( map_tailrec_rev @ A @ B @ F @ As3 @ ( cons @ B @ ( F @ A3 ) @ Bs2 ) ) ) ).
% map_tailrec_rev.simps(2)
thf(fact_234_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_235_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_236_map__tailrec__rev_Oelims,axiom,
! [A: $tType,B: $tType,X: A > B,Xa: list @ A,Xb2: list @ B,Y: list @ B] :
( ( ( map_tailrec_rev @ A @ B @ X @ Xa @ Xb2 )
= Y )
=> ( ( ( Xa
= ( nil @ A ) )
=> ( Y != Xb2 ) )
=> ~ ! [A2: A,As: list @ A] :
( ( Xa
= ( cons @ A @ A2 @ As ) )
=> ( Y
!= ( map_tailrec_rev @ A @ B @ X @ As @ ( cons @ B @ ( X @ A2 ) @ Xb2 ) ) ) ) ) ) ).
% map_tailrec_rev.elims
thf(fact_237_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_238_map__tailrec__rev,axiom,
! [A: $tType,B: $tType] :
( ( map_tailrec_rev @ B @ A )
= ( ^ [F3: B > A,As2: list @ B] : ( append @ A @ ( rev @ A @ ( map @ B @ A @ F3 @ As2 ) ) ) ) ) ).
% map_tailrec_rev
thf(fact_239_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_240_remdups__adj__append,axiom,
! [A: $tType,Xs_1: list @ A,X: A,Xs_2: list @ A] :
( ( remdups_adj @ A @ ( append @ A @ Xs_1 @ ( cons @ A @ X @ Xs_2 ) ) )
= ( append @ A @ ( remdups_adj @ A @ ( append @ A @ Xs_1 @ ( cons @ A @ X @ ( nil @ A ) ) ) ) @ ( tl @ A @ ( remdups_adj @ A @ ( cons @ A @ X @ Xs_2 ) ) ) ) ) ).
% remdups_adj_append
thf(fact_241_llast__lappend,axiom,
! [A: $tType,Ys: coindu1593790203_llist @ A,Xs: coindu1593790203_llist @ A] :
( ( ( coindu335574135_lnull @ A @ Ys )
=> ( ( coindu2000965700_llast @ A @ ( coindu268472904append @ A @ Xs @ Ys ) )
= ( coindu2000965700_llast @ A @ Xs ) ) )
& ( ~ ( coindu335574135_lnull @ A @ Ys )
=> ( ( ( coindu1213758845finite @ A @ Xs )
=> ( ( coindu2000965700_llast @ A @ ( coindu268472904append @ A @ Xs @ Ys ) )
= ( coindu2000965700_llast @ A @ Ys ) ) )
& ( ~ ( coindu1213758845finite @ A @ Xs )
=> ( ( coindu2000965700_llast @ A @ ( coindu268472904append @ A @ Xs @ Ys ) )
= ( undefined @ A ) ) ) ) ) ) ).
% llast_lappend
thf(fact_242_lappend_Odisc__iff_I2_J,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( ~ ( coindu335574135_lnull @ A @ ( coindu268472904append @ A @ Xs @ Ys ) ) )
= ( ~ ( coindu335574135_lnull @ A @ Xs )
| ~ ( coindu335574135_lnull @ A @ Ys ) ) ) ).
% lappend.disc_iff(2)
thf(fact_243_lnull__lappend,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ A @ ( coindu268472904append @ A @ Xs @ Ys ) )
= ( ( coindu335574135_lnull @ A @ Xs )
& ( coindu335574135_lnull @ A @ Ys ) ) ) ).
% lnull_lappend
thf(fact_244_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_245_llexord__LNil__right,axiom,
! [A: $tType,Ys: coindu1593790203_llist @ A,R: A > A > $o,Xs: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ A @ Ys )
=> ( ( coindu300403952lexord @ A @ R @ Xs @ Ys )
= ( coindu335574135_lnull @ A @ Xs ) ) ) ).
% llexord_LNil_right
thf(fact_246_remdups__adj__rev,axiom,
! [A: $tType,Xs: list @ A] :
( ( remdups_adj @ A @ ( rev @ A @ Xs ) )
= ( rev @ A @ ( remdups_adj @ A @ Xs ) ) ) ).
% remdups_adj_rev
thf(fact_247_hd__remdups__adj,axiom,
! [A: $tType,Xs: list @ A] :
( ( hd @ A @ ( remdups_adj @ A @ Xs ) )
= ( hd @ A @ Xs ) ) ).
% hd_remdups_adj
thf(fact_248_lnull__llist__of,axiom,
! [A: $tType,Xs: list @ A] :
( ( coindu335574135_lnull @ A @ ( coindu1079202755ist_of @ A @ Xs ) )
= ( Xs
= ( nil @ A ) ) ) ).
% lnull_llist_of
thf(fact_249_lhd__lappend,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( ( coindu335574135_lnull @ A @ Xs )
=> ( ( coindu1046438764le_lhd @ A @ ( coindu268472904append @ A @ Xs @ Ys ) )
= ( coindu1046438764le_lhd @ A @ Ys ) ) )
& ( ~ ( coindu335574135_lnull @ A @ Xs )
=> ( ( coindu1046438764le_lhd @ A @ ( coindu268472904append @ A @ Xs @ Ys ) )
= ( coindu1046438764le_lhd @ A @ Xs ) ) ) ) ).
% lhd_lappend
thf(fact_250_remdups__adj__Cons__alt,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( cons @ A @ X @ ( tl @ A @ ( remdups_adj @ A @ ( cons @ A @ X @ Xs ) ) ) )
= ( remdups_adj @ A @ ( cons @ A @ X @ Xs ) ) ) ).
% remdups_adj_Cons_alt
thf(fact_251_llast__LCons,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,X: A] :
( ( ( coindu335574135_lnull @ A @ Xs )
=> ( ( coindu2000965700_llast @ A @ ( coindu1121789889_LCons @ A @ X @ Xs ) )
= X ) )
& ( ~ ( coindu335574135_lnull @ A @ Xs )
=> ( ( coindu2000965700_llast @ A @ ( coindu1121789889_LCons @ A @ X @ Xs ) )
= ( coindu2000965700_llast @ A @ Xs ) ) ) ) ).
% llast_LCons
thf(fact_252_lappend_Oexhaust,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( ( coindu335574135_lnull @ A @ Xs )
=> ~ ( coindu335574135_lnull @ A @ Ys ) )
=> ( ~ ( coindu335574135_lnull @ A @ Xs )
| ~ ( coindu335574135_lnull @ A @ Ys ) ) ) ).
% lappend.exhaust
thf(fact_253_lzip_Oexhaust,axiom,
! [A: $tType,B: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ B] :
( ~ ( ( coindu335574135_lnull @ A @ Xs )
| ( coindu335574135_lnull @ B @ Ys ) )
=> ~ ( ~ ( coindu335574135_lnull @ A @ Xs )
=> ( coindu335574135_lnull @ B @ Ys ) ) ) ).
% lzip.exhaust
%----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,X: A,Y: A] :
( ( if @ A @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X: A,Y: A] :
( ( if @ A @ $true @ X @ Y )
= X ) ).
%----Conjectures (2)
thf(conj_0,hypothesis,
! [YS: list @ a] :
( ( ys2
= ( coindu1079202755ist_of @ a @ YS ) )
=> thesis ) ).
thf(conj_1,conjecture,
thesis ).
%------------------------------------------------------------------------------