TPTP Problem File: DAT138^1.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : DAT138^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Data Structures
% Problem : Coinductive list 5031
% 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__5031.p [Bla16]
% Status : Theorem
% Rating : 0.00 v7.2.0, 0.25 v7.1.0
% Syntax : Number of formulae : 306 ( 164 unt; 49 typ; 0 def)
% Number of atoms : 623 ( 427 equ; 0 cnn)
% Maximal formula atoms : 11 ( 2 avg)
% Number of connectives : 3522 ( 117 ~; 24 |; 64 &;3088 @)
% ( 0 <=>; 229 =>; 0 <=; 0 <~>)
% Maximal formula depth : 34 ( 7 avg)
% Number of types : 3 ( 2 usr)
% Number of type conns : 181 ( 181 >; 0 *; 0 +; 0 <<)
% Number of symbols : 50 ( 47 usr; 3 con; 0-5 aty)
% Number of variables : 1028 ( 23 ^; 907 !; 51 ?;1028 :)
% ( 47 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 15:10:27.734
%------------------------------------------------------------------------------
%----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_Nat_Onat,type,
nat: $tType ).
thf(ty_t_itself,type,
itself: $tType > $tType ).
thf(ty_tf_a,type,
a: $tType ).
%----Explicit typings (44)
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_Oinf__llist,type,
coindu288502823_llist:
!>[A: $tType] : ( ( nat > A ) > ( coindu1593790203_llist @ A ) ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Oiterates,type,
coindu1529956631erates:
!>[A: $tType] : ( ( A > A ) > A > ( coindu1593790203_llist @ A ) ) ).
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_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_Olnull,type,
coindu335574135_lnull:
!>[A: $tType] : ( ( coindu1593790203_llist @ A ) > $o ) ).
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_Olnth,type,
coindu749330388e_lnth:
!>[A: $tType] : ( ( coindu1593790203_llist @ A ) > nat > A ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Olprefix,type,
coindu1696667936prefix:
!>[A: $tType] : ( ( coindu1593790203_llist @ 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_Omap,type,
map:
!>[A: $tType,Aa: $tType] : ( ( A > Aa ) > ( list @ A ) > ( list @ Aa ) ) ).
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_Onth,type,
nth:
!>[A: $tType] : ( ( list @ A ) > nat > A ) ).
thf(sy_c_List_Oord_Olexordp__eq,type,
lexordp_eq:
!>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Oproduct__lists,type,
product_lists:
!>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ ( list @ A ) ) ) ).
thf(sy_c_List_Orev,type,
rev:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Orotate1,type,
rotate1:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Osublists,type,
sublists:
!>[A: $tType] : ( ( list @ A ) > ( list @ ( list @ A ) ) ) ).
thf(sy_c_Nat_Osize__class_Osize,type,
size_size:
!>[A: $tType] : ( A > nat ) ).
thf(sy_c_Pure_Otype,type,
type:
!>[A: $tType] : ( itself @ 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_Sublist_Ostrict__suffix,type,
strict_suffix:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > $o ) ).
thf(sy_c_Wellfounded_Oaccp,type,
accp:
!>[A: $tType] : ( ( A > A > $o ) > A > $o ) ).
thf(sy_v_f,type,
f: nat > a ).
thf(sy_v_xs,type,
xs: coindu1593790203_llist @ a ).
%----Relevant facts (253)
thf(fact_0_inf__llist__inj,axiom,
! [A: $tType,F: nat > A,G: nat > A] :
( ( ( coindu288502823_llist @ A @ F )
= ( coindu288502823_llist @ A @ G ) )
= ( F = G ) ) ).
% inf_llist_inj
thf(fact_1_lappend__assoc,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A,Zs: coindu1593790203_llist @ A] :
( ( coindu268472904append @ A @ ( coindu268472904append @ A @ Xs @ Ys ) @ Zs )
= ( coindu268472904append @ A @ Xs @ ( coindu268472904append @ A @ Ys @ Zs ) ) ) ).
% lappend_assoc
thf(fact_2_inf__llist__lprefix,axiom,
! [A: $tType,F: nat > A,Xs: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ ( coindu288502823_llist @ A @ F ) @ Xs )
= ( Xs
= ( coindu288502823_llist @ A @ F ) ) ) ).
% inf_llist_lprefix
thf(fact_3_lnth__inf__llist,axiom,
! [A: $tType,F: nat > A,N: nat] :
( ( coindu749330388e_lnth @ A @ ( coindu288502823_llist @ A @ F ) @ N )
= ( F @ N ) ) ).
% lnth_inf_llist
thf(fact_4_lappend__code_I1_J,axiom,
! [A: $tType,Ys: coindu1593790203_llist @ A] :
( ( coindu268472904append @ A @ ( coindu1598213697e_LNil @ A ) @ Ys )
= Ys ) ).
% lappend_code(1)
thf(fact_5_lappend__LNil2,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( ( coindu268472904append @ A @ Xs @ ( coindu1598213697e_LNil @ A ) )
= Xs ) ).
% lappend_LNil2
thf(fact_6_inf__llist__neq__llist__of_I1_J,axiom,
! [A: $tType,Xs: list @ A,F: nat > A] :
( ( coindu1079202755ist_of @ A @ Xs )
!= ( coindu288502823_llist @ A @ F ) ) ).
% inf_llist_neq_llist_of(1)
thf(fact_7_llist__exhaust,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( ! [Ys2: list @ A] :
( Xs
!= ( coindu1079202755ist_of @ A @ Ys2 ) )
=> ~ ! [F2: nat > A] :
( Xs
!= ( coindu288502823_llist @ A @ F2 ) ) ) ).
% llist_exhaust
thf(fact_8_lappend__iterates,axiom,
! [A: $tType,F: A > A,X: A,Xs: coindu1593790203_llist @ A] :
( ( coindu268472904append @ A @ ( coindu1529956631erates @ A @ F @ X ) @ Xs )
= ( coindu1529956631erates @ A @ F @ X ) ) ).
% lappend_iterates
thf(fact_9_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_10_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_11_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_12_inf__llist__neq__LNil,axiom,
! [A: $tType,F: nat > A] :
( ( coindu288502823_llist @ A @ F )
!= ( coindu1598213697e_LNil @ A ) ) ).
% inf_llist_neq_LNil
thf(fact_13_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_14_lprefix__refl,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] : ( coindu1696667936prefix @ A @ Xs @ Xs ) ).
% lprefix_refl
thf(fact_15_llist_Oleq__refl,axiom,
! [A: $tType,X: coindu1593790203_llist @ A] : ( coindu1696667936prefix @ A @ X @ X ) ).
% llist.leq_refl
thf(fact_16_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_17_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_18_LCons__lprefix__LCons,axiom,
! [A: $tType,X: A,Xs: coindu1593790203_llist @ A,Y: A,Ys: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ ( coindu1121789889_LCons @ A @ X @ Xs ) @ ( coindu1121789889_LCons @ A @ Y @ Ys ) )
= ( ( X = Y )
& ( coindu1696667936prefix @ A @ Xs @ Ys ) ) ) ).
% LCons_lprefix_LCons
thf(fact_19_lprefix__code_I1_J,axiom,
! [A: $tType,Ys: coindu1593790203_llist @ A] : ( coindu1696667936prefix @ A @ ( coindu1598213697e_LNil @ A ) @ Ys ) ).
% lprefix_code(1)
thf(fact_20_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_21_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_22_lfinite__code_I1_J,axiom,
! [A: $tType] : ( coindu1213758845finite @ A @ ( coindu1598213697e_LNil @ A ) ) ).
% lfinite_code(1)
thf(fact_23_lprefix__LNil,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ Xs @ ( coindu1598213697e_LNil @ A ) )
= ( coindu335574135_lnull @ A @ Xs ) ) ).
% lprefix_LNil
thf(fact_24_lprefix__lappend__same,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A,Zs: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ ( coindu268472904append @ A @ Xs @ Ys ) @ ( coindu268472904append @ A @ Xs @ Zs ) )
= ( ( coindu1213758845finite @ A @ Xs )
=> ( coindu1696667936prefix @ A @ Ys @ Zs ) ) ) ).
% lprefix_lappend_same
thf(fact_25_inf__llist__lnth,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( ~ ( coindu1213758845finite @ A @ Xs )
=> ( ( coindu288502823_llist @ A @ ( coindu749330388e_lnth @ A @ Xs ) )
= Xs ) ) ).
% inf_llist_lnth
thf(fact_26_llist_Odisc_I2_J,axiom,
! [A: $tType,X21: A,X22: coindu1593790203_llist @ A] :
~ ( coindu335574135_lnull @ A @ ( coindu1121789889_LCons @ A @ X21 @ X22 ) ) ).
% llist.disc(2)
thf(fact_27_llist_Odisc_I1_J,axiom,
! [A: $tType] : ( coindu335574135_lnull @ A @ ( coindu1598213697e_LNil @ A ) ) ).
% llist.disc(1)
thf(fact_28_llist_OdiscI_I2_J,axiom,
! [A: $tType,Llist: coindu1593790203_llist @ A,X21: A,X22: coindu1593790203_llist @ A] :
( ( Llist
= ( coindu1121789889_LCons @ A @ X21 @ X22 ) )
=> ~ ( coindu335574135_lnull @ A @ Llist ) ) ).
% llist.discI(2)
thf(fact_29_llist_OdiscI_I1_J,axiom,
! [A: $tType,Llist: coindu1593790203_llist @ A] :
( ( Llist
= ( coindu1598213697e_LNil @ A ) )
=> ( coindu335574135_lnull @ A @ Llist ) ) ).
% llist.discI(1)
thf(fact_30_lprefix__code_I2_J,axiom,
! [A: $tType,X: A,Xs: coindu1593790203_llist @ A] :
~ ( coindu1696667936prefix @ A @ ( coindu1121789889_LCons @ A @ X @ Xs ) @ ( coindu1598213697e_LNil @ A ) ) ).
% lprefix_code(2)
thf(fact_31_llist_Ocollapse_I1_J,axiom,
! [A: $tType,Llist: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ A @ Llist )
=> ( Llist
= ( coindu1598213697e_LNil @ A ) ) ) ).
% llist.collapse(1)
thf(fact_32_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_33_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
thf(fact_34_iterates_Ocode,axiom,
! [A: $tType] :
( ( coindu1529956631erates @ A )
= ( ^ [F3: A > A,X2: A] : ( coindu1121789889_LCons @ A @ X2 @ ( coindu1529956631erates @ A @ F3 @ ( F3 @ X2 ) ) ) ) ) ).
% iterates.code
thf(fact_35_lfinite_Ocases,axiom,
! [A: $tType,A2: coindu1593790203_llist @ A] :
( ( coindu1213758845finite @ A @ A2 )
=> ( ( A2
!= ( coindu1598213697e_LNil @ A ) )
=> ~ ! [Xs2: coindu1593790203_llist @ A] :
( ? [X3: A] :
( A2
= ( coindu1121789889_LCons @ A @ X3 @ Xs2 ) )
=> ~ ( coindu1213758845finite @ A @ Xs2 ) ) ) ) ).
% lfinite.cases
thf(fact_36_lfinite_Osimps,axiom,
! [A: $tType] :
( ( coindu1213758845finite @ A )
= ( ^ [A3: coindu1593790203_llist @ A] :
( ( A3
= ( coindu1598213697e_LNil @ A ) )
| ? [Xs3: coindu1593790203_llist @ A,X2: A] :
( ( A3
= ( coindu1121789889_LCons @ A @ X2 @ Xs3 ) )
& ( coindu1213758845finite @ A @ Xs3 ) ) ) ) ) ).
% lfinite.simps
thf(fact_37_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_38_lnull__lprefix,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ A @ Xs )
=> ( coindu1696667936prefix @ A @ Xs @ Ys ) ) ).
% lnull_lprefix
thf(fact_39_lprefix_Ocases,axiom,
! [A: $tType,A1: coindu1593790203_llist @ A,A22: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ A1 @ A22 )
=> ( ( ( A1
= ( coindu1598213697e_LNil @ A ) )
=> ! [Xs2: coindu1593790203_llist @ A] : A22 != Xs2 )
=> ~ ! [Xs2: coindu1593790203_llist @ A,Ys2: coindu1593790203_llist @ A,X3: A] :
( ( A1
= ( coindu1121789889_LCons @ A @ X3 @ Xs2 ) )
=> ( ( A22
= ( coindu1121789889_LCons @ A @ X3 @ Ys2 ) )
=> ~ ( coindu1696667936prefix @ A @ Xs2 @ Ys2 ) ) ) ) ) ).
% lprefix.cases
thf(fact_40_lprefix_Osimps,axiom,
! [A: $tType] :
( ( coindu1696667936prefix @ A )
= ( ^ [A12: coindu1593790203_llist @ A,A23: coindu1593790203_llist @ A] :
( ? [Xs3: coindu1593790203_llist @ A] :
( ( A12
= ( coindu1598213697e_LNil @ A ) )
& ( A23 = Xs3 ) )
| ? [Xs3: coindu1593790203_llist @ A,Ys3: coindu1593790203_llist @ A,X2: A] :
( ( A12
= ( coindu1121789889_LCons @ A @ X2 @ Xs3 ) )
& ( A23
= ( coindu1121789889_LCons @ A @ X2 @ Ys3 ) )
& ( coindu1696667936prefix @ A @ Xs3 @ Ys3 ) ) ) ) ) ).
% lprefix.simps
thf(fact_41_lprefix__lnull,axiom,
! [A: $tType,Ys: coindu1593790203_llist @ A,Xs: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ A @ Ys )
=> ( ( coindu1696667936prefix @ A @ Xs @ Ys )
= ( coindu335574135_lnull @ A @ Xs ) ) ) ).
% lprefix_lnull
thf(fact_42_lprefix__trans,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A,Zs: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ Xs @ Ys )
=> ( ( coindu1696667936prefix @ A @ Ys @ Zs )
=> ( coindu1696667936prefix @ A @ Xs @ Zs ) ) ) ).
% lprefix_trans
thf(fact_43_neq__LNil__conv,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( ( Xs
!= ( coindu1598213697e_LNil @ A ) )
= ( ? [X2: A,Xs4: coindu1593790203_llist @ A] :
( Xs
= ( coindu1121789889_LCons @ A @ X2 @ Xs4 ) ) ) ) ).
% neq_LNil_conv
thf(fact_44_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X3: A] :
( ( F @ X3 )
= ( G @ X3 ) )
=> ( F = G ) ) ).
% ext
thf(fact_45_lprefix__lnullD,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ Xs @ Ys )
=> ( ( coindu335574135_lnull @ A @ Ys )
=> ( coindu335574135_lnull @ A @ Xs ) ) ) ).
% lprefix_lnullD
thf(fact_46_not__lnull__conv,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( ( ~ ( coindu335574135_lnull @ A @ Xs ) )
= ( ? [X2: A,Xs4: coindu1593790203_llist @ A] :
( Xs
= ( coindu1121789889_LCons @ A @ X2 @ Xs4 ) ) ) ) ).
% not_lnull_conv
thf(fact_47_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_48_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,X3: A] :
( ( coindu1213758845finite @ A @ Xs2 )
=> ( ( P @ Xs2 )
=> ( P @ ( coindu1121789889_LCons @ A @ X3 @ Xs2 ) ) ) )
=> ( P @ X ) ) ) ) ).
% lfinite.inducts
thf(fact_49_llist_Oleq__trans,axiom,
! [A: $tType,X: coindu1593790203_llist @ A,Y: coindu1593790203_llist @ A,Z: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ X @ Y )
=> ( ( coindu1696667936prefix @ A @ Y @ Z )
=> ( coindu1696667936prefix @ A @ X @ Z ) ) ) ).
% llist.leq_trans
thf(fact_50_lnull__def,axiom,
! [A: $tType] :
( ( coindu335574135_lnull @ A )
= ( ^ [Llist2: coindu1593790203_llist @ A] :
( Llist2
= ( coindu1598213697e_LNil @ A ) ) ) ) ).
% lnull_def
thf(fact_51_lprefix__antisym,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ Xs @ Ys )
=> ( ( coindu1696667936prefix @ A @ Ys @ Xs )
=> ( Xs = Ys ) ) ) ).
% lprefix_antisym
thf(fact_52_lfinite__iterates,axiom,
! [A: $tType,F: A > A,X: A] :
~ ( coindu1213758845finite @ A @ ( coindu1529956631erates @ A @ F @ X ) ) ).
% lfinite_iterates
thf(fact_53_lfinite__llist__of,axiom,
! [A: $tType,Xs: list @ A] : ( coindu1213758845finite @ A @ ( coindu1079202755ist_of @ A @ Xs ) ) ).
% lfinite_llist_of
thf(fact_54_Le__LCons,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A,X: A] :
( ( coindu1696667936prefix @ A @ Xs @ Ys )
=> ( coindu1696667936prefix @ A @ ( coindu1121789889_LCons @ A @ X @ Xs ) @ ( coindu1121789889_LCons @ A @ X @ Ys ) ) ) ).
% Le_LCons
thf(fact_55_lprefix_Ocoinduct,axiom,
! [A: $tType,X4: ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) > $o,X: coindu1593790203_llist @ A,Xa: coindu1593790203_llist @ A] :
( ( X4 @ X @ Xa )
=> ( ! [X3: coindu1593790203_llist @ A,Xa2: coindu1593790203_llist @ A] :
( ( X4 @ X3 @ Xa2 )
=> ( ? [Xs5: coindu1593790203_llist @ A] :
( ( X3
= ( coindu1598213697e_LNil @ A ) )
& ( Xa2 = Xs5 ) )
| ? [Xs5: coindu1593790203_llist @ A,Ys4: coindu1593790203_llist @ A,Xb: A] :
( ( X3
= ( coindu1121789889_LCons @ A @ Xb @ Xs5 ) )
& ( Xa2
= ( coindu1121789889_LCons @ A @ Xb @ Ys4 ) )
& ( ( X4 @ Xs5 @ Ys4 )
| ( coindu1696667936prefix @ A @ Xs5 @ Ys4 ) ) ) ) )
=> ( coindu1696667936prefix @ A @ X @ Xa ) ) ) ).
% lprefix.coinduct
thf(fact_56_lprefix__lfiniteD,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ Xs @ Ys )
=> ( ( coindu1213758845finite @ A @ Ys )
=> ( coindu1213758845finite @ A @ Xs ) ) ) ).
% lprefix_lfiniteD
thf(fact_57_iterates_Odisc__iff,axiom,
! [A: $tType,F: A > A,X: A] :
~ ( coindu335574135_lnull @ A @ ( coindu1529956631erates @ A @ F @ X ) ) ).
% iterates.disc_iff
thf(fact_58_llist_Oleq__antisym,axiom,
! [A: $tType,X: coindu1593790203_llist @ A,Y: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ X @ Y )
=> ( ( coindu1696667936prefix @ A @ Y @ X )
=> ( X = Y ) ) ) ).
% llist.leq_antisym
thf(fact_59_lnull__imp__lfinite,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ A @ Xs )
=> ( coindu1213758845finite @ A @ Xs ) ) ).
% lnull_imp_lfinite
thf(fact_60_LCons__lprefix__conv,axiom,
! [A: $tType,X: A,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ ( coindu1121789889_LCons @ A @ X @ Xs ) @ Ys )
= ( ? [Ys5: coindu1593790203_llist @ A] :
( ( Ys
= ( coindu1121789889_LCons @ A @ X @ Ys5 ) )
& ( coindu1696667936prefix @ A @ Xs @ Ys5 ) ) ) ) ).
% LCons_lprefix_conv
thf(fact_61_lfinite__rev__induct,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,P: ( coindu1593790203_llist @ A ) > $o] :
( ( coindu1213758845finite @ A @ Xs )
=> ( ( P @ ( coindu1598213697e_LNil @ A ) )
=> ( ! [X3: A,Xs2: coindu1593790203_llist @ A] :
( ( coindu1213758845finite @ A @ Xs2 )
=> ( ( P @ Xs2 )
=> ( P @ ( coindu268472904append @ A @ Xs2 @ ( coindu1121789889_LCons @ A @ X3 @ ( coindu1598213697e_LNil @ A ) ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% lfinite_rev_induct
thf(fact_62_lprefix__LCons__conv,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Y: A,Ys: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ Xs @ ( coindu1121789889_LCons @ A @ Y @ Ys ) )
= ( ( Xs
= ( coindu1598213697e_LNil @ A ) )
| ? [Xs4: coindu1593790203_llist @ A] :
( ( Xs
= ( coindu1121789889_LCons @ A @ Y @ Xs4 ) )
& ( coindu1696667936prefix @ A @ Xs4 @ Ys ) ) ) ) ).
% lprefix_LCons_conv
thf(fact_63_lprefix__not__lnullD,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ Xs @ Ys )
=> ( ~ ( coindu335574135_lnull @ A @ Xs )
=> ~ ( coindu335574135_lnull @ A @ Ys ) ) ) ).
% lprefix_not_lnullD
thf(fact_64_lprefix__down__linear,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Zs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ Xs @ Zs )
=> ( ( coindu1696667936prefix @ A @ Ys @ Zs )
=> ( ( coindu1696667936prefix @ A @ Xs @ Ys )
| ( coindu1696667936prefix @ A @ Ys @ Xs ) ) ) ) ).
% lprefix_down_linear
thf(fact_65_lfinite__LNil,axiom,
! [A: $tType] : ( coindu1213758845finite @ A @ ( coindu1598213697e_LNil @ A ) ) ).
% lfinite_LNil
thf(fact_66_LNil__lprefix,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] : ( coindu1696667936prefix @ A @ ( coindu1598213697e_LNil @ A ) @ Xs ) ).
% LNil_lprefix
thf(fact_67_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_68_not__lfinite__lprefix__conv__eq,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ~ ( coindu1213758845finite @ A @ Xs )
=> ( ( coindu1696667936prefix @ A @ Xs @ Ys )
= ( Xs = Ys ) ) ) ).
% not_lfinite_lprefix_conv_eq
thf(fact_69_lappend_Octr_I1_J,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ A @ Xs )
=> ( ( coindu335574135_lnull @ A @ Ys )
=> ( ( coindu268472904append @ A @ Xs @ Ys )
= ( coindu1598213697e_LNil @ A ) ) ) ) ).
% lappend.ctr(1)
thf(fact_70_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_71_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_72_lappend__lnull2,axiom,
! [A: $tType,Ys: coindu1593790203_llist @ A,Xs: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ A @ Ys )
=> ( ( coindu268472904append @ A @ Xs @ Ys )
= Xs ) ) ).
% lappend_lnull2
thf(fact_73_lappend__lnull1,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ A @ Xs )
=> ( ( coindu268472904append @ A @ Xs @ Ys )
= Ys ) ) ).
% lappend_lnull1
thf(fact_74_lappend_Odisc_I1_J,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ A @ Xs )
=> ( ( coindu335574135_lnull @ A @ Ys )
=> ( coindu335574135_lnull @ A @ ( coindu268472904append @ A @ Xs @ Ys ) ) ) ) ).
% lappend.disc(1)
thf(fact_75_lappend_Odisc_I2_J,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( ~ ( coindu335574135_lnull @ A @ Xs )
| ~ ( coindu335574135_lnull @ A @ Ys ) )
=> ~ ( coindu335574135_lnull @ A @ ( coindu268472904append @ A @ Xs @ Ys ) ) ) ).
% lappend.disc(2)
thf(fact_76_lprefix__lappend__sameI,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A,Zs: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ Xs @ Ys )
=> ( coindu1696667936prefix @ A @ ( coindu268472904append @ A @ Zs @ Xs ) @ ( coindu268472904append @ A @ Zs @ Ys ) ) ) ).
% lprefix_lappend_sameI
thf(fact_77_lprefix__conv__lappend,axiom,
! [A: $tType] :
( ( coindu1696667936prefix @ A )
= ( ^ [Xs3: coindu1593790203_llist @ A,Ys3: coindu1593790203_llist @ A] :
? [Zs2: coindu1593790203_llist @ A] :
( Ys3
= ( coindu268472904append @ A @ Xs3 @ Zs2 ) ) ) ) ).
% lprefix_conv_lappend
thf(fact_78_lprefix__lappendD,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A,Zs: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ Xs @ ( coindu268472904append @ A @ Ys @ Zs ) )
=> ( ( coindu1696667936prefix @ A @ Xs @ Ys )
| ( coindu1696667936prefix @ A @ Ys @ Xs ) ) ) ).
% lprefix_lappendD
thf(fact_79_lappend__lprefixE,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A,Zs: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ ( coindu268472904append @ A @ Xs @ Ys ) @ Zs )
=> ~ ! [Zs3: coindu1593790203_llist @ A] :
( Zs
!= ( coindu268472904append @ A @ Xs @ Zs3 ) ) ) ).
% lappend_lprefixE
thf(fact_80_lprefix__lappend,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] : ( coindu1696667936prefix @ A @ Xs @ ( coindu268472904append @ A @ Xs @ Ys ) ) ).
% lprefix_lappend
thf(fact_81_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_82_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_83_lappend__LNil__LNil,axiom,
! [A: $tType] :
( ( coindu268472904append @ A @ ( coindu1598213697e_LNil @ A ) @ ( coindu1598213697e_LNil @ A ) )
= ( coindu1598213697e_LNil @ A ) ) ).
% lappend_LNil_LNil
thf(fact_84_lfinite__inf__llist,axiom,
! [A: $tType,F: nat > A] :
~ ( coindu1213758845finite @ A @ ( coindu288502823_llist @ A @ F ) ) ).
% lfinite_inf_llist
thf(fact_85_lnull__inf__llist,axiom,
! [A: $tType,F: nat > A] :
~ ( coindu335574135_lnull @ A @ ( coindu288502823_llist @ A @ F ) ) ).
% lnull_inf_llist
thf(fact_86_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_87_lstrict__prefix__lappend__conv,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu574146665prefix @ A @ Xs @ ( coindu268472904append @ A @ Xs @ Ys ) )
= ( ( coindu1213758845finite @ A @ Xs )
& ~ ( coindu335574135_lnull @ A @ Ys ) ) ) ).
% lstrict_prefix_lappend_conv
thf(fact_88_llimit__induct,axiom,
! [A: $tType,P: ( coindu1593790203_llist @ A ) > $o,Xs: coindu1593790203_llist @ A] :
( ( P @ ( coindu1598213697e_LNil @ A ) )
=> ( ! [X3: A,Xs2: coindu1593790203_llist @ A] :
( ( coindu1213758845finite @ A @ Xs2 )
=> ( ( P @ Xs2 )
=> ( P @ ( coindu1121789889_LCons @ A @ X3 @ Xs2 ) ) ) )
=> ( ( ! [Ys4: coindu1593790203_llist @ A] :
( ( coindu574146665prefix @ A @ Ys4 @ Xs )
=> ( P @ Ys4 ) )
=> ( P @ Xs ) )
=> ( P @ Xs ) ) ) ) ).
% llimit_induct
thf(fact_89_Coinductive__List__Mirabelle__kmikjhschf_Olprefix__nitpick__simps,axiom,
! [A: $tType] :
( ( coindu1696667936prefix @ A )
= ( ^ [Xs3: coindu1593790203_llist @ A,Ys3: coindu1593790203_llist @ A] :
( ( ( coindu1213758845finite @ A @ Xs3 )
=> ( coindu1571841457prefix @ A @ Xs3 @ Ys3 ) )
& ( ~ ( coindu1213758845finite @ A @ Xs3 )
=> ( Xs3 = Ys3 ) ) ) ) ) ).
% Coinductive_List_Mirabelle_kmikjhschf.lprefix_nitpick_simps
thf(fact_90_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_91_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_92_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_93_llast__singleton,axiom,
! [A: $tType,X: A] :
( ( coindu2000965700_llast @ A @ ( coindu1121789889_LCons @ A @ X @ ( coindu1598213697e_LNil @ A ) ) )
= X ) ).
% llast_singleton
thf(fact_94_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_95_lmember__code_I1_J,axiom,
! [A: $tType,X: A] :
~ ( coindu567634248member @ A @ X @ ( coindu1598213697e_LNil @ A ) ) ).
% lmember_code(1)
thf(fact_96_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_97_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_98_list__of__llist__of,axiom,
! [A: $tType,Xs: list @ A] :
( ( coindu1736001939ist_of @ A @ ( coindu1079202755ist_of @ A @ Xs ) )
= Xs ) ).
% list_of_llist_of
thf(fact_99_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_100_lstrict__prefix__code_I1_J,axiom,
! [A: $tType] :
~ ( coindu574146665prefix @ A @ ( coindu1598213697e_LNil @ A ) @ ( coindu1598213697e_LNil @ A ) ) ).
% lstrict_prefix_code(1)
thf(fact_101_llist__less__induct,axiom,
! [A: $tType,P: ( coindu1593790203_llist @ A ) > $o,Xs: coindu1593790203_llist @ A] :
( ! [Xs2: coindu1593790203_llist @ A] :
( ! [Ys4: coindu1593790203_llist @ A] :
( ( coindu574146665prefix @ A @ Ys4 @ Xs2 )
=> ( P @ Ys4 ) )
=> ( P @ Xs2 ) )
=> ( P @ Xs ) ) ).
% llist_less_induct
thf(fact_102_lstrict__prefix__def,axiom,
! [A: $tType] :
( ( coindu574146665prefix @ A )
= ( ^ [Xs3: coindu1593790203_llist @ A,Ys3: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ Xs3 @ Ys3 )
& ( Xs3 != Ys3 ) ) ) ) ).
% lstrict_prefix_def
thf(fact_103_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_104_Coinductive__List__Mirabelle__kmikjhschf_Ofinite__lprefix__def,axiom,
! [A: $tType] :
( ( coindu1571841457prefix @ A )
= ( coindu1696667936prefix @ A ) ) ).
% Coinductive_List_Mirabelle_kmikjhschf.finite_lprefix_def
thf(fact_105_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_106_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_107_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_108_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_109_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_110_nth__list__of,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( ( coindu1213758845finite @ A @ Xs )
=> ( ( nth @ A @ ( coindu1736001939ist_of @ A @ Xs ) )
= ( coindu749330388e_lnth @ A @ Xs ) ) ) ).
% nth_list_of
thf(fact_111_llast__llist__of,axiom,
! [A: $tType,Xs: list @ A] :
( ( coindu2000965700_llast @ A @ ( coindu1079202755ist_of @ A @ Xs ) )
= ( last @ A @ Xs ) ) ).
% llast_llist_of
thf(fact_112_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_113_lnth__llist__of,axiom,
! [A: $tType,Xs: list @ A] :
( ( coindu749330388e_lnth @ A @ ( coindu1079202755ist_of @ A @ Xs ) )
= ( nth @ A @ Xs ) ) ).
% lnth_llist_of
thf(fact_114_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_115_lnth__LNil,axiom,
! [A: $tType,N: nat] :
( ( coindu749330388e_lnth @ A @ ( coindu1598213697e_LNil @ A ) @ N )
= ( undefined @ ( nat > A ) @ N ) ) ).
% lnth_LNil
thf(fact_116_lprefix__llist__ofI,axiom,
! [A: $tType,Ys: list @ A,Xs: list @ A] :
( ? [Zs4: list @ A] :
( Ys
= ( append @ A @ Xs @ Zs4 ) )
=> ( coindu1696667936prefix @ A @ ( coindu1079202755ist_of @ A @ Xs ) @ ( coindu1079202755ist_of @ A @ Ys ) ) ) ).
% lprefix_llist_ofI
thf(fact_117_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_118_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_119_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_120_llast__LNil,axiom,
! [A: $tType] :
( ( coindu2000965700_llast @ A @ ( coindu1598213697e_LNil @ A ) )
= ( undefined @ A ) ) ).
% llast_LNil
thf(fact_121_llast__linfinite,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( ~ ( coindu1213758845finite @ A @ Xs )
=> ( ( coindu2000965700_llast @ A @ Xs )
= ( undefined @ A ) ) ) ).
% llast_linfinite
thf(fact_122_append__assoc,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( append @ A @ ( append @ A @ Xs @ Ys ) @ Zs )
= ( append @ A @ Xs @ ( append @ A @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_123_append__same__eq,axiom,
! [A: $tType,Ys: list @ A,Xs: list @ A,Zs: list @ A] :
( ( ( append @ A @ Ys @ Xs )
= ( append @ A @ Zs @ Xs ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_124_same__append__eq,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( ( append @ A @ Xs @ Ys )
= ( append @ A @ Xs @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_125_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_126_not__Cons__self2,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( cons @ A @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_127_append__eq__append__conv2,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A,Ts: list @ A] :
( ( ( append @ A @ Xs @ Ys )
= ( append @ A @ Zs @ Ts ) )
= ( ? [Us: list @ A] :
( ( ( Xs
= ( append @ A @ Zs @ Us ) )
& ( ( append @ A @ Us @ Ys )
= Ts ) )
| ( ( ( append @ A @ Xs @ Us )
= Zs )
& ( Ys
= ( append @ A @ Us @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_128_append__eq__appendI,axiom,
! [A: $tType,Xs: list @ A,Xs1: list @ A,Zs: list @ A,Ys: list @ A,Us2: list @ A] :
( ( ( append @ A @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append @ A @ Xs1 @ Us2 ) )
=> ( ( append @ A @ Xs @ Ys )
= ( append @ A @ Zs @ Us2 ) ) ) ) ).
% append_eq_appendI
thf(fact_129_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_130_Cons__eq__appendI,axiom,
! [A: $tType,X: A,Xs1: list @ A,Ys: list @ A,Xs: list @ A,Zs: list @ A] :
( ( ( cons @ A @ X @ Xs1 )
= Ys )
=> ( ( Xs
= ( append @ A @ Xs1 @ Zs ) )
=> ( ( cons @ A @ X @ Xs )
= ( append @ A @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_131_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_132_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_133_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_134_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_135_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_136_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_137_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_138_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_139_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_140_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_141_append__Nil2,axiom,
! [A: $tType,Xs: list @ A] :
( ( append @ A @ Xs @ ( nil @ A ) )
= Xs ) ).
% append_Nil2
thf(fact_142_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_143_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_144_lnull__llist__of,axiom,
! [A: $tType,Xs: list @ A] :
( ( coindu335574135_lnull @ A @ ( coindu1079202755ist_of @ A @ Xs ) )
= ( Xs
= ( nil @ A ) ) ) ).
% lnull_llist_of
thf(fact_145_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_146_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_147_list__of__LNil,axiom,
! [A: $tType] :
( ( coindu1736001939ist_of @ A @ ( coindu1598213697e_LNil @ A ) )
= ( nil @ A ) ) ).
% list_of_LNil
thf(fact_148_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_149_list_Odistinct_I1_J,axiom,
! [A: $tType,X21: A,X22: list @ A] :
( ( nil @ A )
!= ( cons @ A @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_150_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_151_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_152_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_153_neq__Nil__conv,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
= ( ? [Y2: A,Ys3: list @ A] :
( Xs
= ( cons @ A @ Y2 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_154_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 ) )
=> ( ! [X3: A,Xs2: list @ A] : ( P @ ( cons @ A @ X3 @ Xs2 ) @ ( nil @ B ) )
=> ( ! [Y3: B,Ys2: list @ B] : ( P @ ( nil @ A ) @ ( cons @ B @ Y3 @ Ys2 ) )
=> ( ! [X3: A,Xs2: list @ A,Y3: B,Ys2: list @ B] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons @ A @ X3 @ Xs2 ) @ ( cons @ B @ Y3 @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_155_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 ) )
=> ( ! [X3: A,Xs2: list @ A,Y3: A,Ys2: list @ A] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons @ A @ X3 @ Xs2 ) @ ( cons @ A @ Y3 @ Ys2 ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% splice.induct
thf(fact_156_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_157_transpose_Ocases,axiom,
! [A: $tType,X: list @ ( list @ A )] :
( ( X
!= ( nil @ ( list @ A ) ) )
=> ( ! [Xss: list @ ( list @ A )] :
( X
!= ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss ) )
=> ~ ! [X3: A,Xs2: list @ A,Xss: list @ ( list @ A )] :
( X
!= ( cons @ ( list @ A ) @ ( cons @ A @ X3 @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_158_append__Nil,axiom,
! [A: $tType,Ys: list @ A] :
( ( append @ A @ ( nil @ A ) @ Ys )
= Ys ) ).
% append_Nil
thf(fact_159_remdups__adj_Ocases,axiom,
! [A: $tType,X: list @ A] :
( ( X
!= ( nil @ A ) )
=> ( ! [X3: A] :
( X
!= ( cons @ A @ X3 @ ( nil @ A ) ) )
=> ~ ! [X3: A,Y3: A,Xs2: list @ A] :
( X
!= ( cons @ A @ X3 @ ( cons @ A @ Y3 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_160_remdups__adj_Oinduct,axiom,
! [A: $tType,P: ( list @ A ) > $o,A0: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [X3: A] : ( P @ ( cons @ A @ X3 @ ( nil @ A ) ) )
=> ( ! [X3: A,Y3: A,Xs2: list @ A] :
( ( ( X3 = Y3 )
=> ( P @ ( cons @ A @ X3 @ Xs2 ) ) )
=> ( ( ( X3 != Y3 )
=> ( P @ ( cons @ A @ Y3 @ Xs2 ) ) )
=> ( P @ ( cons @ A @ X3 @ ( cons @ A @ Y3 @ Xs2 ) ) ) ) )
=> ( P @ A0 ) ) ) ) ).
% remdups_adj.induct
thf(fact_161_list__nonempty__induct,axiom,
! [A: $tType,Xs: list @ A,P: ( list @ A ) > $o] :
( ( Xs
!= ( nil @ A ) )
=> ( ! [X3: A] : ( P @ ( cons @ A @ X3 @ ( nil @ A ) ) )
=> ( ! [X3: A,Xs2: list @ A] :
( ( Xs2
!= ( nil @ A ) )
=> ( ( P @ Xs2 )
=> ( P @ ( cons @ A @ X3 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_162_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,A4: A,As: list @ A,Bs: list @ B] :
( ( P @ F2 @ As @ ( cons @ B @ ( F2 @ A4 ) @ Bs ) )
=> ( P @ F2 @ ( cons @ A @ A4 @ As ) @ Bs ) )
=> ( P @ A0 @ A1 @ A22 ) ) ) ).
% map_tailrec_rev.induct
thf(fact_163_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_164_list__of__code,axiom,
! [A: $tType] :
( ( coindu1736001939ist_of @ A )
= ( coindu1542816657of_aux @ A @ ( nil @ A ) ) ) ).
% list_of_code
thf(fact_165_rev__induct,axiom,
! [A: $tType,P: ( list @ A ) > $o,Xs: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [X3: A,Xs2: list @ A] :
( ( P @ Xs2 )
=> ( P @ ( append @ A @ Xs2 @ ( cons @ A @ X3 @ ( nil @ A ) ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_166_rev__exhaust,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ~ ! [Ys2: list @ A,Y3: A] :
( Xs
!= ( append @ A @ Ys2 @ ( cons @ A @ Y3 @ ( nil @ A ) ) ) ) ) ).
% rev_exhaust
thf(fact_167_Cons__eq__append__conv,axiom,
! [A: $tType,X: A,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( ( cons @ A @ X @ Xs )
= ( append @ A @ Ys @ Zs ) )
= ( ( ( Ys
= ( nil @ A ) )
& ( ( cons @ A @ X @ Xs )
= Zs ) )
| ? [Ys5: list @ A] :
( ( ( cons @ A @ X @ Ys5 )
= Ys )
& ( Xs
= ( append @ A @ Ys5 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_168_append__eq__Cons__conv,axiom,
! [A: $tType,Ys: list @ A,Zs: list @ A,X: A,Xs: list @ A] :
( ( ( append @ A @ Ys @ Zs )
= ( cons @ A @ X @ Xs ) )
= ( ( ( Ys
= ( nil @ A ) )
& ( Zs
= ( cons @ A @ X @ Xs ) ) )
| ? [Ys5: list @ A] :
( ( Ys
= ( cons @ A @ X @ Ys5 ) )
& ( ( append @ A @ Ys5 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_169_rev__nonempty__induct,axiom,
! [A: $tType,Xs: list @ A,P: ( list @ A ) > $o] :
( ( Xs
!= ( nil @ A ) )
=> ( ! [X3: A] : ( P @ ( cons @ A @ X3 @ ( nil @ A ) ) )
=> ( ! [X3: A,Xs2: list @ A] :
( ( Xs2
!= ( nil @ A ) )
=> ( ( P @ Xs2 )
=> ( P @ ( append @ A @ Xs2 @ ( cons @ A @ X3 @ ( nil @ A ) ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_170_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_171_last__ConsL,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( Xs
= ( nil @ A ) )
=> ( ( last @ A @ ( cons @ A @ X @ Xs ) )
= X ) ) ).
% last_ConsL
thf(fact_172_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_173_llist__of_Osimps_I1_J,axiom,
! [A: $tType] :
( ( coindu1079202755ist_of @ A @ ( nil @ A ) )
= ( coindu1598213697e_LNil @ A ) ) ).
% llist_of.simps(1)
thf(fact_174_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_175_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_176_longest__common__suffix,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
? [Ss: list @ A,Xs6: list @ A,Ys6: list @ A] :
( ( Xs
= ( append @ A @ Xs6 @ Ss ) )
& ( Ys
= ( append @ A @ Ys6 @ Ss ) )
& ( ( Xs6
= ( nil @ A ) )
| ( Ys6
= ( nil @ A ) )
| ( ( last @ A @ Xs6 )
!= ( last @ A @ Ys6 ) ) ) ) ).
% longest_common_suffix
thf(fact_177_longest__common__prefix_Oinduct,axiom,
! [A: $tType,P: ( list @ A ) > ( list @ A ) > $o,A0: list @ A,A1: list @ A] :
( ! [X3: A,Xs2: list @ A,Y3: A,Ys2: list @ A] :
( ( ( X3 = Y3 )
=> ( P @ Xs2 @ Ys2 ) )
=> ( P @ ( cons @ A @ X3 @ Xs2 ) @ ( cons @ A @ Y3 @ Ys2 ) ) )
=> ( ! [X1: list @ A] : ( P @ ( nil @ A ) @ X1 )
=> ( ! [Uu: list @ A] : ( P @ Uu @ ( nil @ A ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% longest_common_prefix.induct
thf(fact_178_prefixes_Oinduct,axiom,
! [A: $tType,P: ( list @ A ) > $o,A0: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [X3: A,Xs2: list @ A] :
( ( P @ Xs2 )
=> ( P @ ( cons @ A @ X3 @ Xs2 ) ) )
=> ( P @ A0 ) ) ) ).
% prefixes.induct
thf(fact_179_prefixes_Ocases,axiom,
! [A: $tType,X: list @ A] :
( ( X
!= ( nil @ A ) )
=> ~ ! [X3: A,Xs2: list @ A] :
( X
!= ( cons @ A @ X3 @ Xs2 ) ) ) ).
% prefixes.cases
thf(fact_180_list__of__aux__def,axiom,
! [A: $tType] :
( ( coindu1542816657of_aux @ A )
= ( ^ [Xs3: list @ A,Ys3: coindu1593790203_llist @ A] : ( if @ ( list @ A ) @ ( coindu1213758845finite @ A @ Ys3 ) @ ( append @ A @ ( rev @ A @ Xs3 ) @ ( coindu1736001939ist_of @ A @ Ys3 ) ) @ ( undefined @ ( list @ A ) ) ) ) ) ).
% list_of_aux_def
thf(fact_181_rev__rev__ident,axiom,
! [A: $tType,Xs: list @ A] :
( ( rev @ A @ ( rev @ A @ Xs ) )
= Xs ) ).
% rev_rev_ident
thf(fact_182_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_183_rev__is__Nil__conv,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( rev @ A @ Xs )
= ( nil @ A ) )
= ( Xs
= ( nil @ A ) ) ) ).
% rev_is_Nil_conv
thf(fact_184_Nil__is__rev__conv,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( nil @ A )
= ( rev @ A @ Xs ) )
= ( Xs
= ( nil @ A ) ) ) ).
% Nil_is_rev_conv
thf(fact_185_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_186_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_187_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_188_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_189_rev_Osimps_I1_J,axiom,
! [A: $tType] :
( ( rev @ A @ ( nil @ A ) )
= ( nil @ A ) ) ).
% rev.simps(1)
thf(fact_190_rev__swap,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( rev @ A @ Xs )
= Ys )
= ( Xs
= ( rev @ A @ Ys ) ) ) ).
% rev_swap
thf(fact_191_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_192_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_193_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_194_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 ) ) ) )
| ? [Z2: A,Zs2: list @ A] :
( ( Ys
= ( append @ A @ Zs2 @ ( cons @ A @ Z2 @ ( nil @ A ) ) ) )
& ( Xs
= ( prefixes @ A @ Zs2 ) ) ) )
& ( X = Ys ) ) ) ).
% prefixes_eq_Snoc
thf(fact_195_sublists_Osimps_I1_J,axiom,
! [A: $tType] :
( ( sublists @ A @ ( nil @ A ) )
= ( cons @ ( list @ A ) @ ( nil @ A ) @ ( nil @ ( list @ A ) ) ) ) ).
% sublists.simps(1)
thf(fact_196_prefixes_Osimps_I1_J,axiom,
! [A: $tType] :
( ( prefixes @ A @ ( nil @ A ) )
= ( cons @ ( list @ A ) @ ( nil @ A ) @ ( nil @ ( list @ A ) ) ) ) ).
% prefixes.simps(1)
thf(fact_197_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_198_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_199_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_200_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_201_butlast_Osimps_I1_J,axiom,
! [A: $tType] :
( ( butlast @ A @ ( nil @ A ) )
= ( nil @ A ) ) ).
% butlast.simps(1)
thf(fact_202_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_203_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_204_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 ) ) ) ) )
=> ~ ! [X3: A,Xs2: list @ A] :
( ( X
= ( cons @ A @ X3 @ Xs2 ) )
=> ( Y
!= ( cons @ ( list @ A ) @ ( nil @ A ) @ ( map @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 ) @ ( prefixes @ A @ Xs2 ) ) ) ) ) ) ) ).
% prefixes.elims
thf(fact_205_insert__Nil,axiom,
! [A: $tType,X: A] :
( ( insert @ A @ X @ ( nil @ A ) )
= ( cons @ A @ X @ ( nil @ A ) ) ) ).
% insert_Nil
thf(fact_206_list_Omap__disc__iff,axiom,
! [B: $tType,A: $tType,F: A > B,A2: list @ A] :
( ( ( map @ A @ B @ F @ A2 )
= ( nil @ B ) )
= ( A2
= ( nil @ A ) ) ) ).
% list.map_disc_iff
thf(fact_207_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_208_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_209_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_210_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_211_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_212_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_213_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 ) )
= ( ? [Z2: B,Zs2: list @ B] :
( ( Xs
= ( cons @ B @ Z2 @ Zs2 ) )
& ( ( F @ Z2 )
= Y )
& ( ( map @ B @ A @ F @ Zs2 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_214_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 ) )
= ( ? [Z2: B,Zs2: list @ B] :
( ( Ys
= ( cons @ B @ Z2 @ Zs2 ) )
& ( X
= ( F @ Z2 ) )
& ( Xs
= ( map @ B @ A @ F @ Zs2 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_215_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,Zs5: list @ B] :
( ( Xs
= ( cons @ B @ Z3 @ Zs5 ) )
& ( ( F @ Z3 )
= Y )
& ( ( map @ B @ A @ F @ Zs5 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_216_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,Zs5: list @ B] :
( ( Ys
= ( cons @ B @ Z3 @ Zs5 ) )
& ( X
= ( F @ Z3 ) )
& ( Xs
= ( map @ B @ A @ F @ Zs5 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_217_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_218_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_219_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_220_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_221_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 ) ) ) )
=> ~ ! [X3: A,Xs2: list @ A] :
( ( X
= ( cons @ A @ X3 @ Xs2 ) )
=> ( ( Y
= ( cons @ ( list @ A ) @ ( nil @ A ) @ ( map @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 ) @ ( prefixes @ A @ Xs2 ) ) ) )
=> ~ ( accp @ ( list @ A ) @ ( prefixes_rel @ A ) @ ( cons @ A @ X3 @ Xs2 ) ) ) ) ) ) ) ).
% prefixes.pelims
thf(fact_222_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_223_map__tailrec__rev_Osimps_I2_J,axiom,
! [A: $tType,B: $tType,F: A > B,A2: A,As3: list @ A,Bs2: list @ B] :
( ( map_tailrec_rev @ A @ B @ F @ ( cons @ A @ A2 @ As3 ) @ Bs2 )
= ( map_tailrec_rev @ A @ B @ F @ As3 @ ( cons @ B @ ( F @ A2 ) @ Bs2 ) ) ) ).
% map_tailrec_rev.simps(2)
thf(fact_224_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 ) )
=> ~ ! [A4: A,As: list @ A] :
( ( Xa
= ( cons @ A @ A4 @ As ) )
=> ( Y
!= ( map_tailrec_rev @ A @ B @ X @ As @ ( cons @ B @ ( X @ A4 ) @ Xb2 ) ) ) ) ) ) ).
% map_tailrec_rev.elims
thf(fact_225_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_226_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_227_rotate1__is__Nil__conv,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( rotate1 @ A @ Xs )
= ( nil @ A ) )
= ( Xs
= ( nil @ A ) ) ) ).
% rotate1_is_Nil_conv
thf(fact_228_rotate1_Osimps_I1_J,axiom,
! [A: $tType] :
( ( rotate1 @ A @ ( nil @ A ) )
= ( nil @ A ) ) ).
% rotate1.simps(1)
thf(fact_229_map__eq__map__tailrec,axiom,
! [B: $tType,A: $tType] :
( ( map @ A @ B )
= ( map_tailrec @ A @ B ) ) ).
% map_eq_map_tailrec
thf(fact_230_list__ex1__simps_I1_J,axiom,
! [A: $tType,P: A > $o] :
~ ( list_ex1 @ A @ P @ ( nil @ A ) ) ).
% list_ex1_simps(1)
thf(fact_231_strict__suffix__def,axiom,
! [A: $tType] :
( ( strict_suffix @ A )
= ( ^ [Xs3: list @ A,Ys3: list @ A] :
? [Us: list @ A] :
( ( Ys3
= ( append @ A @ Us @ Xs3 ) )
& ( Us
!= ( nil @ A ) ) ) ) ) ).
% strict_suffix_def
thf(fact_232_ord_Olexordp__eq__simps_I3_J,axiom,
! [A: $tType,Less: A > A > $o,X: A,Xs: list @ A] :
~ ( lexordp_eq @ A @ Less @ ( cons @ A @ X @ Xs ) @ ( nil @ A ) ) ).
% ord.lexordp_eq_simps(3)
thf(fact_233_nth__append__length,axiom,
! [A: $tType,Xs: list @ A,X: A,Ys: list @ A] :
( ( nth @ A @ ( append @ A @ Xs @ ( cons @ A @ X @ Ys ) ) @ ( size_size @ ( list @ A ) @ Xs ) )
= X ) ).
% nth_append_length
thf(fact_234_length__map,axiom,
! [A: $tType,B: $tType,F: B > A,Xs: list @ B] :
( ( size_size @ ( list @ A ) @ ( map @ B @ A @ F @ Xs ) )
= ( size_size @ ( list @ B ) @ Xs ) ) ).
% length_map
thf(fact_235_append__eq__append__conv,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Us2: list @ A,Vs: list @ A] :
( ( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ Ys ) )
| ( ( size_size @ ( list @ A ) @ Us2 )
= ( size_size @ ( list @ A ) @ Vs ) ) )
=> ( ( ( append @ A @ Xs @ Us2 )
= ( append @ A @ Ys @ Vs ) )
= ( ( Xs = Ys )
& ( Us2 = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_236_length__rev,axiom,
! [A: $tType,Xs: list @ A] :
( ( size_size @ ( list @ A ) @ ( rev @ A @ Xs ) )
= ( size_size @ ( list @ A ) @ Xs ) ) ).
% length_rev
thf(fact_237_ord_Olexordp__eq__simps_I4_J,axiom,
! [A: $tType,Less: A > A > $o,X: A,Xs: list @ A,Y: A,Ys: list @ A] :
( ( lexordp_eq @ A @ Less @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys ) )
= ( ( Less @ X @ Y )
| ( ~ ( Less @ Y @ X )
& ( lexordp_eq @ A @ Less @ Xs @ Ys ) ) ) ) ).
% ord.lexordp_eq_simps(4)
thf(fact_238_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_239_ord_Olexordp__eq__simps_I1_J,axiom,
! [A: $tType,Less: A > A > $o,Ys: list @ A] : ( lexordp_eq @ A @ Less @ ( nil @ A ) @ Ys ) ).
% ord.lexordp_eq_simps(1)
thf(fact_240_length__rotate1,axiom,
! [A: $tType,Xs: list @ A] :
( ( size_size @ ( list @ A ) @ ( rotate1 @ A @ Xs ) )
= ( size_size @ ( list @ A ) @ Xs ) ) ).
% length_rotate1
thf(fact_241_map__eq__imp__length__eq,axiom,
! [A: $tType,B: $tType,C: $tType,F: B > A,Xs: list @ B,G: C > A,Ys: list @ C] :
( ( ( map @ B @ A @ F @ Xs )
= ( map @ C @ A @ G @ Ys ) )
=> ( ( size_size @ ( list @ B ) @ Xs )
= ( size_size @ ( list @ C ) @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_242_ord_Olexordp__eq_Oinducts,axiom,
! [A: $tType,Less: A > A > $o,X12: list @ A,X24: list @ A,P: ( list @ A ) > ( list @ A ) > $o] :
( ( lexordp_eq @ A @ Less @ X12 @ X24 )
=> ( ! [X1: list @ A] : ( P @ ( nil @ A ) @ X1 )
=> ( ! [X3: A,Y3: A,Xs2: list @ A,Ys2: list @ A] :
( ( Less @ X3 @ Y3 )
=> ( P @ ( cons @ A @ X3 @ Xs2 ) @ ( cons @ A @ Y3 @ Ys2 ) ) )
=> ( ! [X3: A,Y3: A,Xs2: list @ A,Ys2: list @ A] :
( ~ ( Less @ X3 @ Y3 )
=> ( ~ ( Less @ Y3 @ X3 )
=> ( ( lexordp_eq @ A @ Less @ Xs2 @ Ys2 )
=> ( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons @ A @ X3 @ Xs2 ) @ ( cons @ A @ Y3 @ Ys2 ) ) ) ) ) )
=> ( P @ X12 @ X24 ) ) ) ) ) ).
% ord.lexordp_eq.inducts
thf(fact_243_ord_Olexordp__eq_Osimps,axiom,
! [A: $tType] :
( ( lexordp_eq @ A )
= ( ^ [Less2: A > A > $o,A12: list @ A,A23: list @ A] :
( ? [Ys3: list @ A] :
( ( A12
= ( nil @ A ) )
& ( A23 = Ys3 ) )
| ? [X2: A,Y2: A,Xs3: list @ A,Ys3: list @ A] :
( ( A12
= ( cons @ A @ X2 @ Xs3 ) )
& ( A23
= ( cons @ A @ Y2 @ Ys3 ) )
& ( Less2 @ X2 @ Y2 ) )
| ? [X2: A,Y2: A,Xs3: list @ A,Ys3: list @ A] :
( ( A12
= ( cons @ A @ X2 @ Xs3 ) )
& ( A23
= ( cons @ A @ Y2 @ Ys3 ) )
& ~ ( Less2 @ X2 @ Y2 )
& ~ ( Less2 @ Y2 @ X2 )
& ( lexordp_eq @ A @ Less2 @ Xs3 @ Ys3 ) ) ) ) ) ).
% ord.lexordp_eq.simps
thf(fact_244_ord_Olexordp__eq_Ocases,axiom,
! [A: $tType,Less: A > A > $o,A1: list @ A,A22: list @ A] :
( ( lexordp_eq @ A @ Less @ A1 @ A22 )
=> ( ( ( A1
= ( nil @ A ) )
=> ! [Ys2: list @ A] : A22 != Ys2 )
=> ( ! [X3: A] :
( ? [Xs2: list @ A] :
( A1
= ( cons @ A @ X3 @ Xs2 ) )
=> ! [Y3: A] :
( ? [Ys2: list @ A] :
( A22
= ( cons @ A @ Y3 @ Ys2 ) )
=> ~ ( Less @ X3 @ Y3 ) ) )
=> ~ ! [X3: A,Y3: A,Xs2: list @ A] :
( ( A1
= ( cons @ A @ X3 @ Xs2 ) )
=> ! [Ys2: list @ A] :
( ( A22
= ( cons @ A @ Y3 @ Ys2 ) )
=> ( ~ ( Less @ X3 @ Y3 )
=> ( ~ ( Less @ Y3 @ X3 )
=> ~ ( lexordp_eq @ A @ Less @ Xs2 @ Ys2 ) ) ) ) ) ) ) ) ).
% ord.lexordp_eq.cases
thf(fact_245_ord_Olexordp__eq_OCons,axiom,
! [A: $tType,Less: A > A > $o,X: A,Y: A,Xs: list @ A,Ys: list @ A] :
( ( Less @ X @ Y )
=> ( lexordp_eq @ A @ Less @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys ) ) ) ).
% ord.lexordp_eq.Cons
thf(fact_246_ord_Olexordp__eq_OCons__eq,axiom,
! [A: $tType,Less: A > A > $o,X: A,Y: A,Xs: list @ A,Ys: list @ A] :
( ~ ( Less @ X @ Y )
=> ( ~ ( Less @ Y @ X )
=> ( ( lexordp_eq @ A @ Less @ Xs @ Ys )
=> ( lexordp_eq @ A @ Less @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys ) ) ) ) ) ).
% ord.lexordp_eq.Cons_eq
thf(fact_247_ord_Olexordp__eq_ONil,axiom,
! [A: $tType,Less: A > A > $o,Ys: list @ A] : ( lexordp_eq @ A @ Less @ ( nil @ A ) @ Ys ) ).
% ord.lexordp_eq.Nil
thf(fact_248_list__induct4,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,Xs: list @ A,Ys: list @ B,Zs: list @ C,Ws: list @ D,P: ( list @ A ) > ( list @ B ) > ( list @ C ) > ( list @ D ) > $o] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( ( size_size @ ( list @ B ) @ Ys )
= ( size_size @ ( list @ C ) @ Zs ) )
=> ( ( ( size_size @ ( list @ C ) @ Zs )
= ( size_size @ ( list @ D ) @ Ws ) )
=> ( ( P @ ( nil @ A ) @ ( nil @ B ) @ ( nil @ C ) @ ( nil @ D ) )
=> ( ! [X3: A,Xs2: list @ A,Y3: B,Ys2: list @ B,Z3: C,Zs5: list @ C,W: D,Ws2: list @ D] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ B ) @ Ys2 ) )
=> ( ( ( size_size @ ( list @ B ) @ Ys2 )
= ( size_size @ ( list @ C ) @ Zs5 ) )
=> ( ( ( size_size @ ( list @ C ) @ Zs5 )
= ( size_size @ ( list @ D ) @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs5 @ Ws2 )
=> ( P @ ( cons @ A @ X3 @ Xs2 ) @ ( cons @ B @ Y3 @ Ys2 ) @ ( cons @ C @ Z3 @ Zs5 ) @ ( cons @ D @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_249_list__induct3,axiom,
! [B: $tType,A: $tType,C: $tType,Xs: list @ A,Ys: list @ B,Zs: list @ C,P: ( list @ A ) > ( list @ B ) > ( list @ C ) > $o] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( ( size_size @ ( list @ B ) @ Ys )
= ( size_size @ ( list @ C ) @ Zs ) )
=> ( ( P @ ( nil @ A ) @ ( nil @ B ) @ ( nil @ C ) )
=> ( ! [X3: A,Xs2: list @ A,Y3: B,Ys2: list @ B,Z3: C,Zs5: list @ C] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ B ) @ Ys2 ) )
=> ( ( ( size_size @ ( list @ B ) @ Ys2 )
= ( size_size @ ( list @ C ) @ Zs5 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs5 )
=> ( P @ ( cons @ A @ X3 @ Xs2 ) @ ( cons @ B @ Y3 @ Ys2 ) @ ( cons @ C @ Z3 @ Zs5 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_250_list__induct2,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,P: ( list @ A ) > ( list @ B ) > $o] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( P @ ( nil @ A ) @ ( nil @ B ) )
=> ( ! [X3: A,Xs2: list @ A,Y3: B,Ys2: list @ B] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ B ) @ Ys2 ) )
=> ( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons @ A @ X3 @ Xs2 ) @ ( cons @ B @ Y3 @ Ys2 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_251_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_252_neq__if__length__neq,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs )
!= ( size_size @ ( list @ A ) @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
%----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 (1)
thf(conj_0,conjecture,
( ( coindu268472904append @ a @ ( coindu288502823_llist @ a @ f ) @ xs )
= ( coindu288502823_llist @ a @ f ) ) ).
%------------------------------------------------------------------------------