TPTP Problem File: DAT134^1.p
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%------------------------------------------------------------------------------
% File : DAT134^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Data Structures
% Problem : Coinductive list 4309
% 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__4309.p [Bla16]
% Status : Theorem
% Rating : 1.00 v7.1.0
% Syntax : Number of formulae : 311 ( 134 unt; 46 typ; 0 def)
% Number of atoms : 707 ( 274 equ; 3 cnn)
% Maximal formula atoms : 8 ( 2 avg)
% Number of connectives : 4181 ( 136 ~; 33 |; 83 &;3582 @)
% ( 0 <=>; 347 =>; 0 <=; 0 <~>)
% Maximal formula depth : 22 ( 9 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 420 ( 420 >; 0 *; 0 +; 0 <<)
% Number of symbols : 47 ( 43 usr; 5 con; 0-8 aty)
% Number of variables : 1209 ( 43 ^;1070 !; 54 ?;1209 :)
% ( 42 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 15:06:08.277
%------------------------------------------------------------------------------
%----Could-be-implicit typings (6)
thf(ty_t_Coinductive__List__Mirabelle__kmikjhschf_Ollist,type,
coindu1593790203_llist: $tType > $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_t_Nat_Onat,type,
nat: $tType ).
thf(ty_t_itself,type,
itself: $tType > $tType ).
thf(ty_tf_b,type,
b: $tType ).
thf(ty_tf_a,type,
a: $tType ).
%----Explicit typings (40)
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Opreorder,type,
preorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
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_Ogen__lset,type,
coindu1928975208n_lset:
!>[A: $tType] : ( ( set @ A ) > ( coindu1593790203_llist @ A ) > ( set @ 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_Oldistinct,type,
coindu6345450stinct:
!>[A: $tType] : ( ( coindu1593790203_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_OldropWhile,type,
coindu438612276pWhile:
!>[A: $tType] : ( ( A > $o ) > ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Oldropn,type,
coindu531130065ldropn:
!>[A: $tType] : ( nat > ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Olfilter,type,
coindu1889960678filter:
!>[A: $tType] : ( ( A > $o ) > ( 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_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_Ocase__llist,type,
coindu882539134_llist:
!>[B: $tType,A: $tType] : ( B > ( A > ( coindu1593790203_llist @ A ) > B ) > ( coindu1593790203_llist @ A ) > B ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollist_Ocorec__llist,type,
coindu1244876290_llist:
!>[C: $tType,A: $tType] : ( ( C > $o ) > ( C > A ) > ( C > $o ) > ( C > ( coindu1593790203_llist @ A ) ) > ( C > C ) > C > ( 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_Ollist__all2,type,
coindu987187967t_all2:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ B ) > $o ) ).
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_Oltl,type,
coindu1047225960le_ltl:
!>[A: $tType] : ( ( coindu1593790203_llist @ 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_Coinductive__List__Mirabelle__kmikjhschf_Oord__class_Olsorted,type,
coindu48241764sorted:
!>[A: $tType] : ( ( coindu1593790203_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ounfold__llist,type,
coindu1599971794_llist:
!>[A: $tType,B: $tType] : ( ( A > $o ) > ( A > B ) > ( A > A ) > A > ( coindu1593790203_llist @ B ) ) ).
thf(sy_c_Fun_Ocomp,type,
comp:
!>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( A > B ) > A > C ) ).
thf(sy_c_If,type,
if:
!>[A: $tType] : ( $o > A > A > A ) ).
thf(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Pure_Otype,type,
type2:
!>[A: $tType] : ( itself @ A ) ).
thf(sy_c_Relation_Oinv__imagep,type,
inv_imagep:
!>[B: $tType,A: $tType] : ( ( B > B > $o ) > ( A > B ) > A > A > $o ) ).
thf(sy_c_Relation_Otransp,type,
transp:
!>[A: $tType] : ( ( A > A > $o ) > $o ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_P,type,
p: a > b > $o ).
thf(sy_v_Q1,type,
q1: a > $o ).
thf(sy_v_Q2,type,
q2: b > $o ).
thf(sy_v_xs,type,
xs: coindu1593790203_llist @ a ).
thf(sy_v_ys,type,
ys: coindu1593790203_llist @ b ).
%----Relevant facts (254)
thf(fact_0_lfilter__idem,axiom,
! [A: $tType,P: A > $o,Xs: coindu1593790203_llist @ A] :
( ( coindu1889960678filter @ A @ P @ ( coindu1889960678filter @ A @ P @ Xs ) )
= ( coindu1889960678filter @ A @ P @ Xs ) ) ).
% lfilter_idem
thf(fact_1__C_K_C,axiom,
coindu987187967t_all2 @ a @ b @ p @ xs @ ys ).
% "*"
thf(fact_2_Q,axiom,
! [X: a,Y: b] :
( ( p @ X @ Y )
=> ( ( q1 @ X )
= ( q2 @ Y ) ) ) ).
% Q
thf(fact_3_llist__all2__mono,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ B,P2: A > B > $o] :
( ( coindu987187967t_all2 @ A @ B @ P @ Xs @ Ys )
=> ( ! [X2: A,Y2: B] :
( ( P @ X2 @ Y2 )
=> ( P2 @ X2 @ Y2 ) )
=> ( coindu987187967t_all2 @ A @ B @ P2 @ Xs @ Ys ) ) ) ).
% llist_all2_mono
thf(fact_4_llist_Orel__eq,axiom,
! [A: $tType] :
( ( coindu987187967t_all2 @ A @ A
@ ^ [Y3: A,Z: A] : Y3 = Z )
= ( ^ [Y3: coindu1593790203_llist @ A,Z: coindu1593790203_llist @ A] : Y3 = Z ) ) ).
% llist.rel_eq
thf(fact_5_llist_Orel__refl,axiom,
! [B: $tType,Ra: B > B > $o,X: coindu1593790203_llist @ B] :
( ! [X2: B] : ( Ra @ X2 @ X2 )
=> ( coindu987187967t_all2 @ B @ B @ Ra @ X @ X ) ) ).
% llist.rel_refl
thf(fact_6_lfilter__LNil,axiom,
! [A: $tType,P: A > $o] :
( ( coindu1889960678filter @ A @ P @ ( coindu1598213697e_LNil @ A ) )
= ( coindu1598213697e_LNil @ A ) ) ).
% lfilter_LNil
thf(fact_7_llist__all2__LNil1,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Xs: coindu1593790203_llist @ B] :
( ( coindu987187967t_all2 @ A @ B @ P @ ( coindu1598213697e_LNil @ A ) @ Xs )
= ( Xs
= ( coindu1598213697e_LNil @ B ) ) ) ).
% llist_all2_LNil1
thf(fact_8_llist__all2__LNil2,axiom,
! [B: $tType,A: $tType,P: A > B > $o,Xs: coindu1593790203_llist @ A] :
( ( coindu987187967t_all2 @ A @ B @ P @ Xs @ ( coindu1598213697e_LNil @ B ) )
= ( Xs
= ( coindu1598213697e_LNil @ A ) ) ) ).
% llist_all2_LNil2
thf(fact_9_lfilter__LCons,axiom,
! [A: $tType,P: A > $o,X: A,Xs: coindu1593790203_llist @ A] :
( ( ( P @ X )
=> ( ( coindu1889960678filter @ A @ P @ ( coindu1121789889_LCons @ A @ X @ Xs ) )
= ( coindu1121789889_LCons @ A @ X @ ( coindu1889960678filter @ A @ P @ Xs ) ) ) )
& ( ~ ( P @ X )
=> ( ( coindu1889960678filter @ A @ P @ ( coindu1121789889_LCons @ A @ X @ Xs ) )
= ( coindu1889960678filter @ A @ P @ Xs ) ) ) ) ).
% lfilter_LCons
thf(fact_10_ldistinct__lfilterI,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,P: A > $o] :
( ( coindu6345450stinct @ A @ Xs )
=> ( coindu6345450stinct @ A @ ( coindu1889960678filter @ A @ P @ Xs ) ) ) ).
% ldistinct_lfilterI
thf(fact_11_llist__all2__LCons__LCons,axiom,
! [A: $tType,B: $tType,R: A > B > $o,X21: A,X22: coindu1593790203_llist @ A,Y21: B,Y22: coindu1593790203_llist @ B] :
( ( coindu987187967t_all2 @ A @ B @ R @ ( coindu1121789889_LCons @ A @ X21 @ X22 ) @ ( coindu1121789889_LCons @ B @ Y21 @ Y22 ) )
= ( ( R @ X21 @ Y21 )
& ( coindu987187967t_all2 @ A @ B @ R @ X22 @ Y22 ) ) ) ).
% llist_all2_LCons_LCons
thf(fact_12_llist__all2__same,axiom,
! [A: $tType,P: A > A > $o,Xs: coindu1593790203_llist @ A] :
( ( coindu987187967t_all2 @ A @ A @ P @ Xs @ Xs )
= ( ! [X3: A] :
( ( member @ A @ X3 @ ( coindu1112613586e_lset @ A @ Xs ) )
=> ( P @ X3 @ X3 ) ) ) ) ).
% llist_all2_same
thf(fact_13_llist__all2__trans,axiom,
! [A: $tType,P: A > A > $o,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A,Zs: coindu1593790203_llist @ A] :
( ( coindu987187967t_all2 @ A @ A @ P @ Xs @ Ys )
=> ( ( coindu987187967t_all2 @ A @ A @ P @ Ys @ Zs )
=> ( ( transp @ A @ P )
=> ( coindu987187967t_all2 @ A @ A @ P @ Xs @ Zs ) ) ) ) ).
% llist_all2_trans
thf(fact_14_llist__all2__ldropWhileI,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ B,Q1: A > $o,Q2: B > $o] :
( ( coindu987187967t_all2 @ A @ B @ P @ Xs @ Ys )
=> ( ! [X2: A,Y2: B] :
( ( P @ X2 @ Y2 )
=> ( ( Q1 @ X2 )
= ( Q2 @ Y2 ) ) )
=> ( coindu987187967t_all2 @ A @ B @ P @ ( coindu438612276pWhile @ A @ Q1 @ Xs ) @ ( coindu438612276pWhile @ B @ Q2 @ Ys ) ) ) ) ).
% llist_all2_ldropWhileI
thf(fact_15_llist__all2__ldropnI,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ B,N: nat] :
( ( coindu987187967t_all2 @ A @ B @ P @ Xs @ Ys )
=> ( coindu987187967t_all2 @ A @ B @ P @ ( coindu531130065ldropn @ A @ N @ Xs ) @ ( coindu531130065ldropn @ B @ N @ Ys ) ) ) ).
% llist_all2_ldropnI
thf(fact_16_lfilter__LCons__seek,axiom,
! [A: $tType,P3: A > $o,X: A,L: coindu1593790203_llist @ A] :
( ~ ( P3 @ X )
=> ( ( coindu1889960678filter @ A @ P3 @ ( coindu1121789889_LCons @ A @ X @ L ) )
= ( coindu1889960678filter @ A @ P3 @ L ) ) ) ).
% lfilter_LCons_seek
thf(fact_17_lfilter__LCons__found,axiom,
! [A: $tType,P: A > $o,X: A,Xs: coindu1593790203_llist @ A] :
( ( P @ X )
=> ( ( coindu1889960678filter @ A @ P @ ( coindu1121789889_LCons @ A @ X @ Xs ) )
= ( coindu1121789889_LCons @ A @ X @ ( coindu1889960678filter @ A @ P @ Xs ) ) ) ) ).
% lfilter_LCons_found
thf(fact_18_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_19_ldropn__LNil,axiom,
! [A: $tType,N: nat] :
( ( coindu531130065ldropn @ A @ N @ ( coindu1598213697e_LNil @ A ) )
= ( coindu1598213697e_LNil @ A ) ) ).
% ldropn_LNil
thf(fact_20_ldistinct__LNil__code,axiom,
! [A: $tType] : ( coindu6345450stinct @ A @ ( coindu1598213697e_LNil @ A ) ) ).
% ldistinct_LNil_code
thf(fact_21_ldropWhile__LCons,axiom,
! [A: $tType,P: A > $o,X: A,Xs: coindu1593790203_llist @ A] :
( ( ( P @ X )
=> ( ( coindu438612276pWhile @ A @ P @ ( coindu1121789889_LCons @ A @ X @ Xs ) )
= ( coindu438612276pWhile @ A @ P @ Xs ) ) )
& ( ~ ( P @ X )
=> ( ( coindu438612276pWhile @ A @ P @ ( coindu1121789889_LCons @ A @ X @ Xs ) )
= ( coindu1121789889_LCons @ A @ X @ Xs ) ) ) ) ).
% ldropWhile_LCons
thf(fact_22_ldropWhile__LNil,axiom,
! [A: $tType,P: A > $o] :
( ( coindu438612276pWhile @ A @ P @ ( coindu1598213697e_LNil @ A ) )
= ( coindu1598213697e_LNil @ A ) ) ).
% ldropWhile_LNil
thf(fact_23_diverge__lfilter__LNil,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,P: A > $o] :
( ! [X2: A] :
( ( member @ A @ X2 @ ( coindu1112613586e_lset @ A @ Xs ) )
=> ~ ( P @ X2 ) )
=> ( ( coindu1889960678filter @ A @ P @ Xs )
= ( coindu1598213697e_LNil @ A ) ) ) ).
% diverge_lfilter_LNil
thf(fact_24_ldistinct__LCons,axiom,
! [A: $tType,X: A,Xs: coindu1593790203_llist @ A] :
( ( coindu6345450stinct @ A @ ( coindu1121789889_LCons @ A @ X @ Xs ) )
= ( ~ ( member @ A @ X @ ( coindu1112613586e_lset @ A @ Xs ) )
& ( coindu6345450stinct @ A @ Xs ) ) ) ).
% ldistinct_LCons
thf(fact_25_lset__intros_I2_J,axiom,
! [A: $tType,X: A,Xs: coindu1593790203_llist @ A,X4: A] :
( ( member @ A @ X @ ( coindu1112613586e_lset @ A @ Xs ) )
=> ( member @ A @ X @ ( coindu1112613586e_lset @ A @ ( coindu1121789889_LCons @ A @ X4 @ Xs ) ) ) ) ).
% lset_intros(2)
thf(fact_26_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_27_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_28_llist_Oset__intros_I2_J,axiom,
! [A: $tType,X: A,A2: coindu1593790203_llist @ A,A1: A] :
( ( member @ A @ X @ ( coindu1112613586e_lset @ A @ A2 ) )
=> ( member @ A @ X @ ( coindu1112613586e_lset @ A @ ( coindu1121789889_LCons @ A @ A1 @ A2 ) ) ) ) ).
% llist.set_intros(2)
thf(fact_29_llist_Oset__intros_I1_J,axiom,
! [A: $tType,A1: A,A2: coindu1593790203_llist @ A] : ( member @ A @ A1 @ ( coindu1112613586e_lset @ A @ ( coindu1121789889_LCons @ A @ A1 @ A2 ) ) ) ).
% llist.set_intros(1)
thf(fact_30_lset__cases,axiom,
! [A: $tType,X: A,Xs: coindu1593790203_llist @ A] :
( ( member @ A @ X @ ( coindu1112613586e_lset @ A @ Xs ) )
=> ( ! [Xs2: coindu1593790203_llist @ A] :
( Xs
!= ( coindu1121789889_LCons @ A @ X @ Xs2 ) )
=> ~ ! [X5: A,Xs2: coindu1593790203_llist @ A] :
( ( Xs
= ( coindu1121789889_LCons @ A @ X5 @ Xs2 ) )
=> ~ ( member @ A @ X @ ( coindu1112613586e_lset @ A @ Xs2 ) ) ) ) ) ).
% lset_cases
thf(fact_31_lset__induct,axiom,
! [A: $tType,X: A,Xs: coindu1593790203_llist @ A,P: ( coindu1593790203_llist @ A ) > $o] :
( ( member @ A @ X @ ( coindu1112613586e_lset @ A @ Xs ) )
=> ( ! [Xs3: coindu1593790203_llist @ A] : ( P @ ( coindu1121789889_LCons @ A @ X @ Xs3 ) )
=> ( ! [X5: A,Xs3: coindu1593790203_llist @ A] :
( ( member @ A @ X @ ( coindu1112613586e_lset @ A @ Xs3 ) )
=> ( ( X != X5 )
=> ( ( P @ Xs3 )
=> ( P @ ( coindu1121789889_LCons @ A @ X5 @ Xs3 ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% lset_induct
thf(fact_32_lset__induct_H,axiom,
! [A: $tType,X: A,Xs: coindu1593790203_llist @ A,P: ( coindu1593790203_llist @ A ) > $o] :
( ( member @ A @ X @ ( coindu1112613586e_lset @ A @ Xs ) )
=> ( ! [Xs3: coindu1593790203_llist @ A] : ( P @ ( coindu1121789889_LCons @ A @ X @ Xs3 ) )
=> ( ! [X5: A,Xs3: coindu1593790203_llist @ A] :
( ( member @ A @ X @ ( coindu1112613586e_lset @ A @ Xs3 ) )
=> ( ( P @ Xs3 )
=> ( P @ ( coindu1121789889_LCons @ A @ X5 @ Xs3 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% lset_induct'
thf(fact_33_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_34_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_35_ldistinct_OLNil,axiom,
! [A: $tType] : ( coindu6345450stinct @ A @ ( coindu1598213697e_LNil @ A ) ) ).
% ldistinct.LNil
thf(fact_36_in__lset__ldropnD,axiom,
! [A: $tType,X: A,N: nat,Xs: coindu1593790203_llist @ A] :
( ( member @ A @ X @ ( coindu1112613586e_lset @ A @ ( coindu531130065ldropn @ A @ N @ Xs ) ) )
=> ( member @ A @ X @ ( coindu1112613586e_lset @ A @ Xs ) ) ) ).
% in_lset_ldropnD
thf(fact_37_ldistinct_OLCons,axiom,
! [A: $tType,X: A,Xs: coindu1593790203_llist @ A] :
( ~ ( member @ A @ X @ ( coindu1112613586e_lset @ A @ Xs ) )
=> ( ( coindu6345450stinct @ A @ Xs )
=> ( coindu6345450stinct @ A @ ( coindu1121789889_LCons @ A @ X @ Xs ) ) ) ) ).
% ldistinct.LCons
thf(fact_38_ldistinct_Ocases,axiom,
! [A: $tType,A3: coindu1593790203_llist @ A] :
( ( coindu6345450stinct @ A @ A3 )
=> ( ( A3
!= ( coindu1598213697e_LNil @ A ) )
=> ~ ! [X2: A,Xs3: coindu1593790203_llist @ A] :
( ( A3
= ( coindu1121789889_LCons @ A @ X2 @ Xs3 ) )
=> ( ~ ( member @ A @ X2 @ ( coindu1112613586e_lset @ A @ Xs3 ) )
=> ~ ( coindu6345450stinct @ A @ Xs3 ) ) ) ) ) ).
% ldistinct.cases
thf(fact_39_ldistinct_Osimps,axiom,
! [A: $tType] :
( ( coindu6345450stinct @ A )
= ( ^ [A4: coindu1593790203_llist @ A] :
( ( A4
= ( coindu1598213697e_LNil @ A ) )
| ? [X3: A,Xs5: coindu1593790203_llist @ A] :
( ( A4
= ( coindu1121789889_LCons @ A @ X3 @ Xs5 ) )
& ~ ( member @ A @ X3 @ ( coindu1112613586e_lset @ A @ Xs5 ) )
& ( coindu6345450stinct @ A @ Xs5 ) ) ) ) ) ).
% ldistinct.simps
thf(fact_40_ldropWhile__cong,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A,P: A > $o,Q: A > $o] :
( ( Xs = Ys )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ ( coindu1112613586e_lset @ A @ Ys ) )
=> ( ( P @ X2 )
= ( Q @ X2 ) ) )
=> ( ( coindu438612276pWhile @ A @ P @ Xs )
= ( coindu438612276pWhile @ A @ Q @ Ys ) ) ) ) ).
% ldropWhile_cong
thf(fact_41_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_42_ldistinct__ldropn,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,N: nat] :
( ( coindu6345450stinct @ A @ Xs )
=> ( coindu6345450stinct @ A @ ( coindu531130065ldropn @ A @ N @ Xs ) ) ) ).
% ldistinct_ldropn
thf(fact_43_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,Xa: A] :
( ( member @ A @ Xa @ ( coindu1112613586e_lset @ A @ Z2 ) )
=> ( ( P @ Xa @ Z2 )
=> ( P @ Xa @ ( coindu1121789889_LCons @ A @ Z1 @ Z2 ) ) ) )
=> ( P @ X @ A3 ) ) ) ) ).
% llist.set_induct
thf(fact_44_ldistinct_Ocoinduct,axiom,
! [A: $tType,X6: ( coindu1593790203_llist @ A ) > $o,X: coindu1593790203_llist @ A] :
( ( X6 @ X )
=> ( ! [X2: coindu1593790203_llist @ A] :
( ( X6 @ X2 )
=> ( ( X2
= ( coindu1598213697e_LNil @ A ) )
| ? [Xa2: A,Xs6: coindu1593790203_llist @ A] :
( ( X2
= ( coindu1121789889_LCons @ A @ Xa2 @ Xs6 ) )
& ~ ( member @ A @ Xa2 @ ( coindu1112613586e_lset @ A @ Xs6 ) )
& ( ( X6 @ Xs6 )
| ( coindu6345450stinct @ A @ Xs6 ) ) ) ) )
=> ( coindu6345450stinct @ A @ X ) ) ) ).
% ldistinct.coinduct
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,A5: set @ A] :
( ( collect @ A
@ ^ [X3: A] : ( member @ A @ X3 @ A5 ) )
= A5 ) ).
% 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_in__lset__ldropWhileD,axiom,
! [A: $tType,X: A,P: A > $o,Xs: coindu1593790203_llist @ A] :
( ( member @ A @ X @ ( coindu1112613586e_lset @ A @ ( coindu438612276pWhile @ A @ P @ Xs ) ) )
=> ( member @ A @ X @ ( coindu1112613586e_lset @ A @ Xs ) ) ) ).
% in_lset_ldropWhileD
thf(fact_50_ldropWhile__eq__LNil__iff,axiom,
! [A: $tType,P: A > $o,Xs: coindu1593790203_llist @ A] :
( ( ( coindu438612276pWhile @ A @ P @ Xs )
= ( coindu1598213697e_LNil @ A ) )
= ( ! [X3: A] :
( ( member @ A @ X3 @ ( coindu1112613586e_lset @ A @ Xs ) )
=> ( P @ X3 ) ) ) ) ).
% ldropWhile_eq_LNil_iff
thf(fact_51_llist__all2__LNil__LCons,axiom,
! [A: $tType,B: $tType,R: A > B > $o,Y21: B,Y22: coindu1593790203_llist @ B] :
~ ( coindu987187967t_all2 @ A @ B @ R @ ( coindu1598213697e_LNil @ A ) @ ( coindu1121789889_LCons @ B @ Y21 @ Y22 ) ) ).
% llist_all2_LNil_LCons
thf(fact_52_llist__all2__LCons__LNil,axiom,
! [A: $tType,B: $tType,R: A > B > $o,Y21: A,Y22: coindu1593790203_llist @ A] :
~ ( coindu987187967t_all2 @ A @ B @ R @ ( coindu1121789889_LCons @ A @ Y21 @ Y22 ) @ ( coindu1598213697e_LNil @ B ) ) ).
% llist_all2_LCons_LNil
thf(fact_53_llist__all2__cases,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ B] :
( ( coindu987187967t_all2 @ A @ B @ P @ Xs @ Ys )
=> ( ( ( Xs
= ( coindu1598213697e_LNil @ A ) )
=> ( Ys
!= ( coindu1598213697e_LNil @ B ) ) )
=> ~ ! [X2: A,Xs2: coindu1593790203_llist @ A] :
( ( Xs
= ( coindu1121789889_LCons @ A @ X2 @ Xs2 ) )
=> ! [Y2: B,Ys2: coindu1593790203_llist @ B] :
( ( Ys
= ( coindu1121789889_LCons @ B @ Y2 @ Ys2 ) )
=> ( ( P @ X2 @ Y2 )
=> ~ ( coindu987187967t_all2 @ A @ B @ P @ Xs2 @ Ys2 ) ) ) ) ) ) ).
% llist_all2_cases
thf(fact_54_llist_Orel__cases,axiom,
! [A: $tType,B: $tType,R: A > B > $o,A3: coindu1593790203_llist @ A,B2: coindu1593790203_llist @ B] :
( ( coindu987187967t_all2 @ A @ B @ R @ A3 @ B2 )
=> ( ( ( A3
= ( coindu1598213697e_LNil @ A ) )
=> ( B2
!= ( coindu1598213697e_LNil @ B ) ) )
=> ~ ! [X1: A,X23: coindu1593790203_llist @ A] :
( ( A3
= ( coindu1121789889_LCons @ A @ X1 @ X23 ) )
=> ! [Y1: B,Y23: coindu1593790203_llist @ B] :
( ( B2
= ( coindu1121789889_LCons @ B @ Y1 @ Y23 ) )
=> ( ( R @ X1 @ Y1 )
=> ~ ( coindu987187967t_all2 @ A @ B @ R @ X23 @ Y23 ) ) ) ) ) ) ).
% llist.rel_cases
thf(fact_55_lfilter__empty__conv,axiom,
! [A: $tType,P: A > $o,Xs: coindu1593790203_llist @ A] :
( ( ( coindu1889960678filter @ A @ P @ Xs )
= ( coindu1598213697e_LNil @ A ) )
= ( ! [X3: A] :
( ( member @ A @ X3 @ ( coindu1112613586e_lset @ A @ Xs ) )
=> ~ ( P @ X3 ) ) ) ) ).
% lfilter_empty_conv
thf(fact_56_llist_Orel__transp,axiom,
! [A: $tType,R: A > A > $o] :
( ( transp @ A @ R )
=> ( transp @ ( coindu1593790203_llist @ A ) @ ( coindu987187967t_all2 @ A @ A @ R ) ) ) ).
% llist.rel_transp
thf(fact_57_llist_Orel__refl__strong,axiom,
! [A: $tType,X: coindu1593790203_llist @ A,Ra: A > A > $o] :
( ! [Z3: A] :
( ( member @ A @ Z3 @ ( coindu1112613586e_lset @ A @ X ) )
=> ( Ra @ Z3 @ Z3 ) )
=> ( coindu987187967t_all2 @ A @ A @ Ra @ X @ X ) ) ).
% llist.rel_refl_strong
thf(fact_58_llist_Orel__mono__strong,axiom,
! [A: $tType,B: $tType,R: A > B > $o,X: coindu1593790203_llist @ A,Y: coindu1593790203_llist @ B,Ra: A > B > $o] :
( ( coindu987187967t_all2 @ A @ B @ R @ X @ Y )
=> ( ! [Z3: A,Yb: B] :
( ( member @ A @ Z3 @ ( coindu1112613586e_lset @ A @ X ) )
=> ( ( member @ B @ Yb @ ( coindu1112613586e_lset @ B @ Y ) )
=> ( ( R @ Z3 @ Yb )
=> ( Ra @ Z3 @ Yb ) ) ) )
=> ( coindu987187967t_all2 @ A @ B @ Ra @ X @ Y ) ) ) ).
% llist.rel_mono_strong
thf(fact_59_llist_Orel__cong,axiom,
! [A: $tType,B: $tType,X: coindu1593790203_llist @ A,Ya: coindu1593790203_llist @ A,Y: coindu1593790203_llist @ B,Xa3: coindu1593790203_llist @ B,R: A > B > $o,Ra: A > B > $o] :
( ( X = Ya )
=> ( ( Y = Xa3 )
=> ( ! [Z3: A,Yb: B] :
( ( member @ A @ Z3 @ ( coindu1112613586e_lset @ A @ Ya ) )
=> ( ( member @ B @ Yb @ ( coindu1112613586e_lset @ B @ Xa3 ) )
=> ( ( R @ Z3 @ Yb )
= ( Ra @ Z3 @ Yb ) ) ) )
=> ( ( coindu987187967t_all2 @ A @ B @ R @ X @ Y )
= ( coindu987187967t_all2 @ A @ B @ Ra @ Ya @ Xa3 ) ) ) ) ) ).
% llist.rel_cong
thf(fact_60_llist__all2__lsetD2,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ B,Y: B] :
( ( coindu987187967t_all2 @ A @ B @ P @ Xs @ Ys )
=> ( ( member @ B @ Y @ ( coindu1112613586e_lset @ B @ Ys ) )
=> ? [X2: A] :
( ( member @ A @ X2 @ ( coindu1112613586e_lset @ A @ Xs ) )
& ( P @ X2 @ Y ) ) ) ) ).
% llist_all2_lsetD2
thf(fact_61_llist__all2__lsetD1,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ B,X: A] :
( ( coindu987187967t_all2 @ A @ B @ P @ Xs @ Ys )
=> ( ( member @ A @ X @ ( coindu1112613586e_lset @ A @ Xs ) )
=> ? [X2: B] :
( ( member @ B @ X2 @ ( coindu1112613586e_lset @ B @ Ys ) )
& ( P @ X @ X2 ) ) ) ) ).
% llist_all2_lsetD1
thf(fact_62_llist__all2__reflI,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,P: A > A > $o] :
( ! [X2: A] :
( ( member @ A @ X2 @ ( coindu1112613586e_lset @ A @ Xs ) )
=> ( P @ X2 @ X2 ) )
=> ( coindu987187967t_all2 @ A @ A @ P @ Xs @ Xs ) ) ).
% llist_all2_reflI
thf(fact_63_llist__all2__LCons2,axiom,
! [B: $tType,A: $tType,P: A > B > $o,Xs: coindu1593790203_llist @ A,Y: B,Ys: coindu1593790203_llist @ B] :
( ( coindu987187967t_all2 @ A @ B @ P @ Xs @ ( coindu1121789889_LCons @ B @ Y @ Ys ) )
= ( ? [X3: A,Xs4: coindu1593790203_llist @ A] :
( ( Xs
= ( coindu1121789889_LCons @ A @ X3 @ Xs4 ) )
& ( P @ X3 @ Y )
& ( coindu987187967t_all2 @ A @ B @ P @ Xs4 @ Ys ) ) ) ) ).
% llist_all2_LCons2
thf(fact_64_llist__all2__LCons1,axiom,
! [A: $tType,B: $tType,P: A > B > $o,X: A,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ B] :
( ( coindu987187967t_all2 @ A @ B @ P @ ( coindu1121789889_LCons @ A @ X @ Xs ) @ Ys )
= ( ? [Y4: B,Ys3: coindu1593790203_llist @ B] :
( ( Ys
= ( coindu1121789889_LCons @ B @ Y4 @ Ys3 ) )
& ( P @ X @ Y4 )
& ( coindu987187967t_all2 @ A @ B @ P @ Xs @ Ys3 ) ) ) ) ).
% llist_all2_LCons1
thf(fact_65_llist_Orel__intros_I2_J,axiom,
! [A: $tType,B: $tType,R: A > B > $o,X21: A,Y21: B,X22: coindu1593790203_llist @ A,Y22: coindu1593790203_llist @ B] :
( ( R @ X21 @ Y21 )
=> ( ( coindu987187967t_all2 @ A @ B @ R @ X22 @ Y22 )
=> ( coindu987187967t_all2 @ A @ B @ R @ ( coindu1121789889_LCons @ A @ X21 @ X22 ) @ ( coindu1121789889_LCons @ B @ Y21 @ Y22 ) ) ) ) ).
% llist.rel_intros(2)
thf(fact_66_lfilter__cong,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A,P: A > $o,Q: A > $o] :
( ( Xs = Ys )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ ( coindu1112613586e_lset @ A @ Ys ) )
=> ( ( P @ X2 )
= ( Q @ X2 ) ) )
=> ( ( coindu1889960678filter @ A @ P @ Xs )
= ( coindu1889960678filter @ A @ Q @ Ys ) ) ) ) ).
% lfilter_cong
thf(fact_67_llist__all2__LNil__LNil,axiom,
! [A: $tType,B: $tType,R: A > B > $o] : ( coindu987187967t_all2 @ A @ B @ R @ ( coindu1598213697e_LNil @ A ) @ ( coindu1598213697e_LNil @ B ) ) ).
% llist_all2_LNil_LNil
thf(fact_68_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_69_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_70_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_71_llast__singleton,axiom,
! [A: $tType,X: A] :
( ( coindu2000965700_llast @ A @ ( coindu1121789889_LCons @ A @ X @ ( coindu1598213697e_LNil @ A ) ) )
= X ) ).
% llast_singleton
thf(fact_72_transpD,axiom,
! [A: $tType,R2: A > A > $o,X: A,Y: A,Z4: A] :
( ( transp @ A @ R2 )
=> ( ( R2 @ X @ Y )
=> ( ( R2 @ Y @ Z4 )
=> ( R2 @ X @ Z4 ) ) ) ) ).
% transpD
thf(fact_73_transpE,axiom,
! [A: $tType,R2: A > A > $o,X: A,Y: A,Z4: A] :
( ( transp @ A @ R2 )
=> ( ( R2 @ X @ Y )
=> ( ( R2 @ Y @ Z4 )
=> ( R2 @ X @ Z4 ) ) ) ) ).
% transpE
thf(fact_74_transpI,axiom,
! [A: $tType,R2: A > A > $o] :
( ! [X2: A,Y2: A,Z3: A] :
( ( R2 @ X2 @ Y2 )
=> ( ( R2 @ Y2 @ Z3 )
=> ( R2 @ X2 @ Z3 ) ) )
=> ( transp @ A @ R2 ) ) ).
% transpI
thf(fact_75_transp__def,axiom,
! [A: $tType] :
( ( transp @ A )
= ( ^ [R3: A > A > $o] :
! [X3: A,Y4: A,Z5: A] :
( ( R3 @ X3 @ Y4 )
=> ( ( R3 @ Y4 @ Z5 )
=> ( R3 @ X3 @ Z5 ) ) ) ) ) ).
% transp_def
thf(fact_76_DEADID_Orel__transp,axiom,
! [A: $tType] :
( transp @ A
@ ^ [Y3: A,Z: A] : Y3 = Z ) ).
% DEADID.rel_transp
thf(fact_77_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_78_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_79_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_80_lstrict__prefix__code_I1_J,axiom,
! [A: $tType] :
~ ( coindu574146665prefix @ A @ ( coindu1598213697e_LNil @ A ) @ ( coindu1598213697e_LNil @ A ) ) ).
% lstrict_prefix_code(1)
thf(fact_81_llist__less__induct,axiom,
! [A: $tType,P: ( coindu1593790203_llist @ A ) > $o,Xs: coindu1593790203_llist @ A] :
( ! [Xs3: coindu1593790203_llist @ A] :
( ! [Ys4: coindu1593790203_llist @ A] :
( ( coindu574146665prefix @ A @ Ys4 @ Xs3 )
=> ( P @ Ys4 ) )
=> ( P @ Xs3 ) )
=> ( P @ Xs ) ) ).
% llist_less_induct
thf(fact_82_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_83_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_84_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_85_lmember__code_I1_J,axiom,
! [A: $tType,X: A] :
~ ( coindu567634248member @ A @ X @ ( coindu1598213697e_LNil @ A ) ) ).
% lmember_code(1)
thf(fact_86_llimit__induct,axiom,
! [A: $tType,P: ( coindu1593790203_llist @ A ) > $o,Xs: coindu1593790203_llist @ A] :
( ( P @ ( coindu1598213697e_LNil @ A ) )
=> ( ! [X2: A,Xs3: coindu1593790203_llist @ A] :
( ( coindu1213758845finite @ A @ Xs3 )
=> ( ( P @ Xs3 )
=> ( P @ ( coindu1121789889_LCons @ A @ X2 @ Xs3 ) ) ) )
=> ( ( ! [Ys4: coindu1593790203_llist @ A] :
( ( coindu574146665prefix @ A @ Ys4 @ Xs )
=> ( P @ Ys4 ) )
=> ( P @ Xs ) )
=> ( P @ Xs ) ) ) ) ).
% llimit_induct
thf(fact_87_lfilter__eq__LCons,axiom,
! [A: $tType,P: A > $o,Xs: coindu1593790203_llist @ A,X: A,Xs7: coindu1593790203_llist @ A] :
( ( ( coindu1889960678filter @ A @ P @ Xs )
= ( coindu1121789889_LCons @ A @ X @ Xs7 ) )
=> ? [Xs8: coindu1593790203_llist @ A] :
( ( Xs7
= ( coindu1889960678filter @ A @ P @ Xs8 ) )
& ( ( coindu438612276pWhile @ A @ ( comp @ $o @ $o @ A @ (~) @ P ) @ Xs )
= ( coindu1121789889_LCons @ A @ X @ Xs8 ) ) ) ) ).
% lfilter_eq_LCons
thf(fact_88_in__inv__imagep,axiom,
! [B: $tType,A: $tType] :
( ( inv_imagep @ A @ B )
= ( ^ [R3: A > A > $o,F2: B > A,X3: B,Y4: B] : ( R3 @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) ) ) ).
% in_inv_imagep
thf(fact_89_lsorted__code_I2_J,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [X: A] : ( coindu48241764sorted @ A @ ( coindu1121789889_LCons @ A @ X @ ( coindu1598213697e_LNil @ A ) ) ) ) ).
% lsorted_code(2)
thf(fact_90_gen__lset__code_I1_J,axiom,
! [A: $tType,A5: set @ A] :
( ( coindu1928975208n_lset @ A @ A5 @ ( coindu1598213697e_LNil @ A ) )
= A5 ) ).
% gen_lset_code(1)
thf(fact_91_llist_Ocorec__code,axiom,
! [A: $tType,C: $tType] :
( ( coindu1244876290_llist @ C @ A )
= ( ^ [P4: C > $o,G21: C > A,Q22: C > $o,G221: C > ( coindu1593790203_llist @ A ),G222: C > C,A4: C] : ( if @ ( coindu1593790203_llist @ A ) @ ( P4 @ A4 ) @ ( coindu1598213697e_LNil @ A ) @ ( coindu1121789889_LCons @ A @ ( G21 @ A4 ) @ ( if @ ( coindu1593790203_llist @ A ) @ ( Q22 @ A4 ) @ ( G221 @ A4 ) @ ( coindu1244876290_llist @ C @ A @ P4 @ G21 @ Q22 @ G221 @ G222 @ ( G222 @ A4 ) ) ) ) ) ) ) ).
% llist.corec_code
thf(fact_92_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_93_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_94_lfinite__code_I1_J,axiom,
! [A: $tType] : ( coindu1213758845finite @ A @ ( coindu1598213697e_LNil @ A ) ) ).
% lfinite_code(1)
thf(fact_95_lsorted__code_I1_J,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ( coindu48241764sorted @ A @ ( coindu1598213697e_LNil @ A ) ) ) ).
% lsorted_code(1)
thf(fact_96_lfinite__ldropn,axiom,
! [A: $tType,N: nat,Xs: coindu1593790203_llist @ A] :
( ( coindu1213758845finite @ A @ ( coindu531130065ldropn @ A @ N @ Xs ) )
= ( coindu1213758845finite @ A @ Xs ) ) ).
% lfinite_ldropn
thf(fact_97_type__copy__map__cong0,axiom,
! [B: $tType,D: $tType,E2: $tType,A: $tType,C: $tType,M: B > A,G: C > B,X: C,N2: D > A,H: C > D,F: A > E2] :
( ( ( M @ ( G @ X ) )
= ( N2 @ ( H @ X ) ) )
=> ( ( comp @ B @ E2 @ C @ ( comp @ A @ E2 @ B @ F @ M ) @ G @ X )
= ( comp @ D @ E2 @ C @ ( comp @ A @ E2 @ D @ F @ N2 ) @ H @ X ) ) ) ).
% type_copy_map_cong0
thf(fact_98_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_99_lfinite__LNil,axiom,
! [A: $tType] : ( coindu1213758845finite @ A @ ( coindu1598213697e_LNil @ A ) ) ).
% lfinite_LNil
thf(fact_100_llist__all2__lfiniteD,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ B] :
( ( coindu987187967t_all2 @ A @ B @ P @ Xs @ Ys )
=> ( ( coindu1213758845finite @ A @ Xs )
= ( coindu1213758845finite @ B @ Ys ) ) ) ).
% llist_all2_lfiniteD
thf(fact_101_lfinite__lfilterI,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,P: A > $o] :
( ( coindu1213758845finite @ A @ Xs )
=> ( coindu1213758845finite @ A @ ( coindu1889960678filter @ A @ P @ Xs ) ) ) ).
% lfinite_lfilterI
thf(fact_102_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_103_LNil,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ( coindu48241764sorted @ A @ ( coindu1598213697e_LNil @ A ) ) ) ).
% LNil
thf(fact_104_lsorted__ldropn,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [Xs: coindu1593790203_llist @ A,N: nat] :
( ( coindu48241764sorted @ A @ Xs )
=> ( coindu48241764sorted @ A @ ( coindu531130065ldropn @ A @ N @ Xs ) ) ) ) ).
% lsorted_ldropn
thf(fact_105_llist_Ocorec_I2_J,axiom,
! [A: $tType,C: $tType,P3: C > $o,A3: C,G212: C > A,Q222: C > $o,G2212: C > ( coindu1593790203_llist @ A ),G2222: C > C] :
( ~ ( P3 @ A3 )
=> ( ( coindu1244876290_llist @ C @ A @ P3 @ G212 @ Q222 @ G2212 @ G2222 @ A3 )
= ( coindu1121789889_LCons @ A @ ( G212 @ A3 ) @ ( if @ ( coindu1593790203_llist @ A ) @ ( Q222 @ A3 ) @ ( G2212 @ A3 ) @ ( coindu1244876290_llist @ C @ A @ P3 @ G212 @ Q222 @ G2212 @ G2222 @ ( G2222 @ A3 ) ) ) ) ) ) ).
% llist.corec(2)
thf(fact_106_llist_Ocorec_I1_J,axiom,
! [C: $tType,A: $tType,P3: C > $o,A3: C,G212: C > A,Q222: C > $o,G2212: C > ( coindu1593790203_llist @ A ),G2222: C > C] :
( ( P3 @ A3 )
=> ( ( coindu1244876290_llist @ C @ A @ P3 @ G212 @ Q222 @ G2212 @ G2222 @ A3 )
= ( coindu1598213697e_LNil @ A ) ) ) ).
% llist.corec(1)
thf(fact_107_lfinite_Ocases,axiom,
! [A: $tType,A3: coindu1593790203_llist @ A] :
( ( coindu1213758845finite @ A @ A3 )
=> ( ( A3
!= ( coindu1598213697e_LNil @ A ) )
=> ~ ! [Xs3: coindu1593790203_llist @ A] :
( ? [X2: A] :
( A3
= ( coindu1121789889_LCons @ A @ X2 @ Xs3 ) )
=> ~ ( coindu1213758845finite @ A @ Xs3 ) ) ) ) ).
% lfinite.cases
thf(fact_108_lfinite_Osimps,axiom,
! [A: $tType] :
( ( coindu1213758845finite @ A )
= ( ^ [A4: coindu1593790203_llist @ A] :
( ( A4
= ( coindu1598213697e_LNil @ A ) )
| ? [Xs5: coindu1593790203_llist @ A,X3: A] :
( ( A4
= ( coindu1121789889_LCons @ A @ X3 @ Xs5 ) )
& ( coindu1213758845finite @ A @ Xs5 ) ) ) ) ) ).
% lfinite.simps
thf(fact_109_lfinite_Oinducts,axiom,
! [A: $tType,X: coindu1593790203_llist @ A,P: ( coindu1593790203_llist @ A ) > $o] :
( ( coindu1213758845finite @ A @ X )
=> ( ( P @ ( coindu1598213697e_LNil @ A ) )
=> ( ! [Xs3: coindu1593790203_llist @ A,X2: A] :
( ( coindu1213758845finite @ A @ Xs3 )
=> ( ( P @ Xs3 )
=> ( P @ ( coindu1121789889_LCons @ A @ X2 @ Xs3 ) ) ) )
=> ( P @ X ) ) ) ) ).
% lfinite.inducts
thf(fact_110_lfinite__ldropWhile,axiom,
! [A: $tType,P: A > $o,Xs: coindu1593790203_llist @ A] :
( ( coindu1213758845finite @ A @ ( coindu438612276pWhile @ A @ P @ Xs ) )
= ( ? [X3: A] :
( ( member @ A @ X3 @ ( coindu1112613586e_lset @ A @ Xs ) )
& ~ ( P @ X3 ) )
=> ( coindu1213758845finite @ A @ Xs ) ) ) ).
% lfinite_ldropWhile
thf(fact_111_Singleton,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [X: A] : ( coindu48241764sorted @ A @ ( coindu1121789889_LCons @ A @ X @ ( coindu1598213697e_LNil @ A ) ) ) ) ).
% Singleton
thf(fact_112_lsorted__ldistinct__lset__unique,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu48241764sorted @ A @ Xs )
=> ( ( coindu6345450stinct @ A @ Xs )
=> ( ( coindu48241764sorted @ A @ Ys )
=> ( ( coindu6345450stinct @ A @ Ys )
=> ( ( ( coindu1112613586e_lset @ A @ Xs )
= ( coindu1112613586e_lset @ A @ Ys ) )
=> ( Xs = Ys ) ) ) ) ) ) ) ).
% lsorted_ldistinct_lset_unique
thf(fact_113_comp__apply,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comp @ B @ A @ C )
= ( ^ [F2: B > A,G2: C > B,X3: C] : ( F2 @ ( G2 @ X3 ) ) ) ) ).
% comp_apply
thf(fact_114_lhd__lfilter,axiom,
! [A: $tType,P: A > $o,Xs: coindu1593790203_llist @ A] :
( ( coindu1046438764le_lhd @ A @ ( coindu1889960678filter @ A @ P @ Xs ) )
= ( coindu1046438764le_lhd @ A @ ( coindu438612276pWhile @ A @ ( comp @ $o @ $o @ A @ (~) @ P ) @ Xs ) ) ) ).
% lhd_lfilter
thf(fact_115_lfilter__eq__LConsD,axiom,
! [A: $tType,P: A > $o,Ys: coindu1593790203_llist @ A,X: A,Xs: coindu1593790203_llist @ A] :
( ( ( coindu1889960678filter @ A @ P @ Ys )
= ( coindu1121789889_LCons @ A @ X @ Xs ) )
=> ? [Us: coindu1593790203_llist @ A,Vs: coindu1593790203_llist @ A] :
( ( Ys
= ( coindu268472904append @ A @ Us @ ( coindu1121789889_LCons @ A @ X @ Vs ) ) )
& ( coindu1213758845finite @ A @ Us )
& ! [X7: A] :
( ( member @ A @ X7 @ ( coindu1112613586e_lset @ A @ Us ) )
=> ~ ( P @ X7 ) )
& ( P @ X )
& ( Xs
= ( coindu1889960678filter @ A @ P @ Vs ) ) ) ) ).
% lfilter_eq_LConsD
thf(fact_116_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_117_ltl__lfilter,axiom,
! [A: $tType,P: A > $o,Xs: coindu1593790203_llist @ A] :
( ( coindu1047225960le_ltl @ A @ ( coindu1889960678filter @ A @ P @ Xs ) )
= ( coindu1889960678filter @ A @ P @ ( coindu1047225960le_ltl @ A @ ( coindu438612276pWhile @ A @ ( comp @ $o @ $o @ A @ (~) @ P ) @ Xs ) ) ) ) ).
% ltl_lfilter
thf(fact_118_lappend__code_I2_J,axiom,
! [A: $tType,Xa3: A,X: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu268472904append @ A @ ( coindu1121789889_LCons @ A @ Xa3 @ X ) @ Ys )
= ( coindu1121789889_LCons @ A @ Xa3 @ ( coindu268472904append @ A @ X @ Ys ) ) ) ).
% lappend_code(2)
thf(fact_119_lappend__LNil2,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( ( coindu268472904append @ A @ Xs @ ( coindu1598213697e_LNil @ A ) )
= Xs ) ).
% lappend_LNil2
thf(fact_120_lappend__code_I1_J,axiom,
! [A: $tType,Ys: coindu1593790203_llist @ A] :
( ( coindu268472904append @ A @ ( coindu1598213697e_LNil @ A ) @ Ys )
= Ys ) ).
% lappend_code(1)
thf(fact_121_lfinite__ltl,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( ( coindu1213758845finite @ A @ ( coindu1047225960le_ltl @ A @ Xs ) )
= ( coindu1213758845finite @ A @ Xs ) ) ).
% lfinite_ltl
thf(fact_122_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_123_lfilter__lappend__lfinite,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,P: A > $o,Ys: coindu1593790203_llist @ A] :
( ( coindu1213758845finite @ A @ Xs )
=> ( ( coindu1889960678filter @ A @ P @ ( coindu268472904append @ A @ Xs @ Ys ) )
= ( coindu268472904append @ A @ ( coindu1889960678filter @ A @ P @ Xs ) @ ( coindu1889960678filter @ A @ P @ Ys ) ) ) ) ).
% lfilter_lappend_lfinite
thf(fact_124_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_125_eq__LConsD,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Y: A,Ys: coindu1593790203_llist @ A] :
( ( Xs
= ( coindu1121789889_LCons @ A @ Y @ Ys ) )
=> ( ( Xs
!= ( coindu1598213697e_LNil @ A ) )
& ( ( coindu1046438764le_lhd @ A @ Xs )
= Y )
& ( ( coindu1047225960le_ltl @ A @ Xs )
= Ys ) ) ) ).
% eq_LConsD
thf(fact_126_llist_Oexhaust__sel,axiom,
! [A: $tType,Llist: coindu1593790203_llist @ A] :
( ( Llist
!= ( coindu1598213697e_LNil @ A ) )
=> ( Llist
= ( coindu1121789889_LCons @ A @ ( coindu1046438764le_lhd @ A @ Llist ) @ ( coindu1047225960le_ltl @ A @ Llist ) ) ) ) ).
% llist.exhaust_sel
thf(fact_127_lhd__LCons,axiom,
! [A: $tType,X21: A,X22: coindu1593790203_llist @ A] :
( ( coindu1046438764le_lhd @ A @ ( coindu1121789889_LCons @ A @ X21 @ X22 ) )
= X21 ) ).
% lhd_LCons
thf(fact_128_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_129_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_130_lappend__LNil__LNil,axiom,
! [A: $tType] :
( ( coindu268472904append @ A @ ( coindu1598213697e_LNil @ A ) @ ( coindu1598213697e_LNil @ A ) )
= ( coindu1598213697e_LNil @ A ) ) ).
% lappend_LNil_LNil
thf(fact_131_ltl__simps_I2_J,axiom,
! [A: $tType,X21: A,X22: coindu1593790203_llist @ A] :
( ( coindu1047225960le_ltl @ A @ ( coindu1121789889_LCons @ A @ X21 @ X22 ) )
= X22 ) ).
% ltl_simps(2)
thf(fact_132_ltl__simps_I1_J,axiom,
! [A: $tType] :
( ( coindu1047225960le_ltl @ A @ ( coindu1598213697e_LNil @ A ) )
= ( coindu1598213697e_LNil @ A ) ) ).
% ltl_simps(1)
thf(fact_133_in__lset__ltlD,axiom,
! [A: $tType,X: A,Xs: coindu1593790203_llist @ A] :
( ( member @ A @ X @ ( coindu1112613586e_lset @ A @ ( coindu1047225960le_ltl @ A @ Xs ) ) )
=> ( member @ A @ X @ ( coindu1112613586e_lset @ A @ Xs ) ) ) ).
% in_lset_ltlD
thf(fact_134_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_135_llist__all2__ltlI,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ B] :
( ( coindu987187967t_all2 @ A @ B @ P @ Xs @ Ys )
=> ( coindu987187967t_all2 @ A @ B @ P @ ( coindu1047225960le_ltl @ A @ Xs ) @ ( coindu1047225960le_ltl @ B @ Ys ) ) ) ).
% llist_all2_ltlI
thf(fact_136_ltl__ldropn,axiom,
! [A: $tType,N: nat,Xs: coindu1593790203_llist @ A] :
( ( coindu1047225960le_ltl @ A @ ( coindu531130065ldropn @ A @ N @ Xs ) )
= ( coindu531130065ldropn @ A @ N @ ( coindu1047225960le_ltl @ A @ Xs ) ) ) ).
% ltl_ldropn
thf(fact_137_ldistinct__ltlI,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( ( coindu6345450stinct @ A @ Xs )
=> ( coindu6345450stinct @ A @ ( coindu1047225960le_ltl @ A @ Xs ) ) ) ).
% ldistinct_ltlI
thf(fact_138_lsorted__ltlI,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [Xs: coindu1593790203_llist @ A] :
( ( coindu48241764sorted @ A @ Xs )
=> ( coindu48241764sorted @ A @ ( coindu1047225960le_ltl @ A @ Xs ) ) ) ) ).
% lsorted_ltlI
thf(fact_139_llist_Ocorec__sel_I1_J,axiom,
! [A: $tType,C: $tType,P3: C > $o,A3: C,G212: C > A,Q222: C > $o,G2212: C > ( coindu1593790203_llist @ A ),G2222: C > C] :
( ~ ( P3 @ A3 )
=> ( ( coindu1046438764le_lhd @ A @ ( coindu1244876290_llist @ C @ A @ P3 @ G212 @ Q222 @ G2212 @ G2222 @ A3 ) )
= ( G212 @ A3 ) ) ) ).
% llist.corec_sel(1)
thf(fact_140_llist_Ocorec__sel_I2_J,axiom,
! [A: $tType,C: $tType,P3: C > $o,A3: C,Q222: C > $o,G212: C > A,G2212: C > ( coindu1593790203_llist @ A ),G2222: C > C] :
( ~ ( P3 @ A3 )
=> ( ( ( Q222 @ A3 )
=> ( ( coindu1047225960le_ltl @ A @ ( coindu1244876290_llist @ C @ A @ P3 @ G212 @ Q222 @ G2212 @ G2222 @ A3 ) )
= ( G2212 @ A3 ) ) )
& ( ~ ( Q222 @ A3 )
=> ( ( coindu1047225960le_ltl @ A @ ( coindu1244876290_llist @ C @ A @ P3 @ G212 @ Q222 @ G2212 @ G2222 @ A3 ) )
= ( coindu1244876290_llist @ C @ A @ P3 @ G212 @ Q222 @ G2212 @ G2222 @ ( G2222 @ A3 ) ) ) ) ) ) ).
% llist.corec_sel(2)
thf(fact_141_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_142_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_143_llist__all2__lappendI,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ B,Xs7: coindu1593790203_llist @ A,Ys5: coindu1593790203_llist @ B] :
( ( coindu987187967t_all2 @ A @ B @ P @ Xs @ Ys )
=> ( ( ( coindu1213758845finite @ A @ Xs )
=> ( ( coindu1213758845finite @ B @ Ys )
=> ( coindu987187967t_all2 @ A @ B @ P @ Xs7 @ Ys5 ) ) )
=> ( coindu987187967t_all2 @ A @ B @ P @ ( coindu268472904append @ A @ Xs @ Xs7 ) @ ( coindu268472904append @ B @ Ys @ Ys5 ) ) ) ) ).
% llist_all2_lappendI
thf(fact_144_lfilter__eq__lappend__lfiniteD,axiom,
! [A: $tType,P: A > $o,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A,Zs: coindu1593790203_llist @ A] :
( ( ( coindu1889960678filter @ A @ P @ Xs )
= ( coindu268472904append @ A @ Ys @ Zs ) )
=> ( ( coindu1213758845finite @ A @ Ys )
=> ? [Us: coindu1593790203_llist @ A,Vs: coindu1593790203_llist @ A] :
( ( Xs
= ( coindu268472904append @ A @ Us @ Vs ) )
& ( coindu1213758845finite @ A @ Us )
& ( Ys
= ( coindu1889960678filter @ A @ P @ Us ) )
& ( Zs
= ( coindu1889960678filter @ A @ P @ Vs ) ) ) ) ) ).
% lfilter_eq_lappend_lfiniteD
thf(fact_145_lhd__ldropWhile__in__lset,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,P: A > $o] :
( ? [X7: A] :
( ( member @ A @ X7 @ ( coindu1112613586e_lset @ A @ Xs ) )
& ~ ( P @ X7 ) )
=> ( member @ A @ ( coindu1046438764le_lhd @ A @ ( coindu438612276pWhile @ A @ P @ Xs ) ) @ ( coindu1112613586e_lset @ A @ Xs ) ) ) ).
% lhd_ldropWhile_in_lset
thf(fact_146_lhd__ldropWhile,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,P: A > $o] :
( ? [X7: A] :
( ( member @ A @ X7 @ ( coindu1112613586e_lset @ A @ Xs ) )
& ~ ( P @ X7 ) )
=> ~ ( P @ ( coindu1046438764le_lhd @ A @ ( coindu438612276pWhile @ A @ P @ Xs ) ) ) ) ).
% lhd_ldropWhile
thf(fact_147_lfinite__rev__induct,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,P: ( coindu1593790203_llist @ A ) > $o] :
( ( coindu1213758845finite @ A @ Xs )
=> ( ( P @ ( coindu1598213697e_LNil @ A ) )
=> ( ! [X2: A,Xs3: coindu1593790203_llist @ A] :
( ( coindu1213758845finite @ A @ Xs3 )
=> ( ( P @ Xs3 )
=> ( P @ ( coindu268472904append @ A @ Xs3 @ ( coindu1121789889_LCons @ A @ X2 @ ( coindu1598213697e_LNil @ A ) ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% lfinite_rev_induct
thf(fact_148_split__llist,axiom,
! [A: $tType,X: A,Xs: coindu1593790203_llist @ A] :
( ( member @ A @ X @ ( coindu1112613586e_lset @ A @ Xs ) )
=> ? [Ys6: coindu1593790203_llist @ A,Zs2: coindu1593790203_llist @ A] :
( ( Xs
= ( coindu268472904append @ A @ Ys6 @ ( coindu1121789889_LCons @ A @ X @ Zs2 ) ) )
& ( coindu1213758845finite @ A @ Ys6 ) ) ) ).
% split_llist
thf(fact_149_split__llist__first,axiom,
! [A: $tType,X: A,Xs: coindu1593790203_llist @ A] :
( ( member @ A @ X @ ( coindu1112613586e_lset @ A @ Xs ) )
=> ? [Ys6: coindu1593790203_llist @ A,Zs2: coindu1593790203_llist @ A] :
( ( Xs
= ( coindu268472904append @ A @ Ys6 @ ( coindu1121789889_LCons @ A @ X @ Zs2 ) ) )
& ( coindu1213758845finite @ A @ Ys6 )
& ~ ( member @ A @ X @ ( coindu1112613586e_lset @ A @ Ys6 ) ) ) ) ).
% split_llist_first
thf(fact_150_comp__eq__dest__lhs,axiom,
! [C: $tType,B: $tType,A: $tType,A3: C > B,B2: A > C,C2: A > B,V: A] :
( ( ( comp @ C @ B @ A @ A3 @ B2 )
= C2 )
=> ( ( A3 @ ( B2 @ V ) )
= ( C2 @ V ) ) ) ).
% comp_eq_dest_lhs
thf(fact_151_comp__eq__elim,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,A3: C > B,B2: A > C,C2: D > B,D2: A > D] :
( ( ( comp @ C @ B @ A @ A3 @ B2 )
= ( comp @ D @ B @ A @ C2 @ D2 ) )
=> ! [V2: A] :
( ( A3 @ ( B2 @ V2 ) )
= ( C2 @ ( D2 @ V2 ) ) ) ) ).
% comp_eq_elim
thf(fact_152_comp__eq__dest,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,A3: C > B,B2: A > C,C2: D > B,D2: A > D,V: A] :
( ( ( comp @ C @ B @ A @ A3 @ B2 )
= ( comp @ D @ B @ A @ C2 @ D2 ) )
=> ( ( A3 @ ( B2 @ V ) )
= ( C2 @ ( D2 @ V ) ) ) ) ).
% comp_eq_dest
thf(fact_153_comp__assoc,axiom,
! [B: $tType,D: $tType,C: $tType,A: $tType,F: D > B,G: C > D,H: A > C] :
( ( comp @ C @ B @ A @ ( comp @ D @ B @ C @ F @ G ) @ H )
= ( comp @ D @ B @ A @ F @ ( comp @ C @ D @ A @ G @ H ) ) ) ).
% comp_assoc
thf(fact_154_comp__def,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comp @ B @ C @ A )
= ( ^ [F2: B > C,G2: A > B,X3: A] : ( F2 @ ( G2 @ X3 ) ) ) ) ).
% comp_def
thf(fact_155_llist_Osplit__sel,axiom,
! [B: $tType,A: $tType,P: B > $o,F1: B,F22: A > ( coindu1593790203_llist @ A ) > B,Llist: coindu1593790203_llist @ A] :
( ( P @ ( coindu882539134_llist @ B @ A @ F1 @ F22 @ Llist ) )
= ( ( ( Llist
= ( coindu1598213697e_LNil @ A ) )
=> ( P @ F1 ) )
& ( ( Llist
= ( coindu1121789889_LCons @ A @ ( coindu1046438764le_lhd @ A @ Llist ) @ ( coindu1047225960le_ltl @ A @ Llist ) ) )
=> ( P @ ( F22 @ ( coindu1046438764le_lhd @ A @ Llist ) @ ( coindu1047225960le_ltl @ A @ Llist ) ) ) ) ) ) ).
% llist.split_sel
thf(fact_156_llist_Osplit__sel__asm,axiom,
! [B: $tType,A: $tType,P: B > $o,F1: B,F22: A > ( coindu1593790203_llist @ A ) > B,Llist: coindu1593790203_llist @ A] :
( ( P @ ( coindu882539134_llist @ B @ A @ F1 @ F22 @ Llist ) )
= ( ~ ( ( ( Llist
= ( coindu1598213697e_LNil @ A ) )
& ~ ( P @ F1 ) )
| ( ( Llist
= ( coindu1121789889_LCons @ A @ ( coindu1046438764le_lhd @ A @ Llist ) @ ( coindu1047225960le_ltl @ A @ Llist ) ) )
& ~ ( P @ ( F22 @ ( coindu1046438764le_lhd @ A @ Llist ) @ ( coindu1047225960le_ltl @ A @ Llist ) ) ) ) ) ) ) ).
% llist.split_sel_asm
thf(fact_157_llexord__conv,axiom,
! [A: $tType] :
( ( coindu300403952lexord @ A )
= ( ^ [R3: A > A > $o,Xs5: coindu1593790203_llist @ A,Ys7: coindu1593790203_llist @ A] :
( ( Xs5 = Ys7 )
| ? [Zs3: coindu1593790203_llist @ A,Xs4: coindu1593790203_llist @ A,Y4: A,Ys3: coindu1593790203_llist @ A] :
( ( coindu1213758845finite @ A @ Zs3 )
& ( Xs5
= ( coindu268472904append @ A @ Zs3 @ Xs4 ) )
& ( Ys7
= ( coindu268472904append @ A @ Zs3 @ ( coindu1121789889_LCons @ A @ Y4 @ Ys3 ) ) )
& ( ( Xs4
= ( coindu1598213697e_LNil @ A ) )
| ( R3 @ ( coindu1046438764le_lhd @ A @ Xs4 ) @ Y4 ) ) ) ) ) ) ).
% llexord_conv
thf(fact_158_ldistinct__lhdD,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( ( coindu6345450stinct @ A @ Xs )
=> ( ~ ( coindu335574135_lnull @ A @ Xs )
=> ~ ( member @ A @ ( coindu1046438764le_lhd @ A @ Xs ) @ ( coindu1112613586e_lset @ A @ ( coindu1047225960le_ltl @ A @ Xs ) ) ) ) ) ).
% ldistinct_lhdD
thf(fact_159_llexord__refl,axiom,
! [A: $tType,R2: A > A > $o,Xs: coindu1593790203_llist @ A] : ( coindu300403952lexord @ A @ R2 @ Xs @ Xs ) ).
% llexord_refl
thf(fact_160_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_161_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_162_llexord__LCons__LCons,axiom,
! [A: $tType,R2: A > A > $o,X: A,Xs: coindu1593790203_llist @ A,Y: A,Ys: coindu1593790203_llist @ A] :
( ( coindu300403952lexord @ A @ R2 @ ( coindu1121789889_LCons @ A @ X @ Xs ) @ ( coindu1121789889_LCons @ A @ Y @ Ys ) )
= ( ( ( X = Y )
& ( coindu300403952lexord @ A @ R2 @ Xs @ Ys ) )
| ( R2 @ X @ Y ) ) ) ).
% llexord_LCons_LCons
thf(fact_163_llexord__LNil__right,axiom,
! [A: $tType,Ys: coindu1593790203_llist @ A,R2: A > A > $o,Xs: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ A @ Ys )
=> ( ( coindu300403952lexord @ A @ R2 @ Xs @ Ys )
= ( coindu335574135_lnull @ A @ Xs ) ) ) ).
% llexord_LNil_right
thf(fact_164_llexord__code_I1_J,axiom,
! [A: $tType,R2: A > A > $o,Ys: coindu1593790203_llist @ A] : ( coindu300403952lexord @ A @ R2 @ ( coindu1598213697e_LNil @ A ) @ Ys ) ).
% llexord_code(1)
thf(fact_165_llist_Ocorec__disc__iff_I2_J,axiom,
! [A: $tType,C: $tType,P3: C > $o,G212: C > A,Q222: C > $o,G2212: C > ( coindu1593790203_llist @ A ),G2222: C > C,A3: C] :
( ( ~ ( coindu335574135_lnull @ A @ ( coindu1244876290_llist @ C @ A @ P3 @ G212 @ Q222 @ G2212 @ G2222 @ A3 ) ) )
= ( ~ ( P3 @ A3 ) ) ) ).
% llist.corec_disc_iff(2)
thf(fact_166_llist_Ocorec__disc__iff_I1_J,axiom,
! [A: $tType,C: $tType,P3: C > $o,G212: C > A,Q222: C > $o,G2212: C > ( coindu1593790203_llist @ A ),G2222: C > C,A3: C] :
( ( coindu335574135_lnull @ A @ ( coindu1244876290_llist @ C @ A @ P3 @ G212 @ Q222 @ G2212 @ G2222 @ A3 ) )
= ( P3 @ A3 ) ) ).
% llist.corec_disc_iff(1)
thf(fact_167_ltl__lappend,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( ( coindu335574135_lnull @ A @ Xs )
=> ( ( coindu1047225960le_ltl @ A @ ( coindu268472904append @ A @ Xs @ Ys ) )
= ( coindu1047225960le_ltl @ A @ Ys ) ) )
& ( ~ ( coindu335574135_lnull @ A @ Xs )
=> ( ( coindu1047225960le_ltl @ A @ ( coindu268472904append @ A @ Xs @ Ys ) )
= ( coindu268472904append @ A @ ( coindu1047225960le_ltl @ A @ Xs ) @ Ys ) ) ) ) ).
% ltl_lappend
thf(fact_168_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_169_lnull__lfilter,axiom,
! [A: $tType,P: A > $o,Xs: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ A @ ( coindu1889960678filter @ A @ P @ Xs ) )
= ( ! [X3: A] :
( ( member @ A @ X3 @ ( coindu1112613586e_lset @ A @ Xs ) )
=> ~ ( P @ X3 ) ) ) ) ).
% lnull_lfilter
thf(fact_170_lnull__ldropWhile,axiom,
! [A: $tType,P: A > $o,Xs: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ A @ ( coindu438612276pWhile @ A @ P @ Xs ) )
= ( ! [X3: A] :
( ( member @ A @ X3 @ ( coindu1112613586e_lset @ A @ Xs ) )
=> ( P @ X3 ) ) ) ) ).
% lnull_ldropWhile
thf(fact_171_lhd__LCons__ltl,axiom,
! [A: $tType,Llist: coindu1593790203_llist @ A] :
( ~ ( coindu335574135_lnull @ A @ Llist )
=> ( ( coindu1121789889_LCons @ A @ ( coindu1046438764le_lhd @ A @ Llist ) @ ( coindu1047225960le_ltl @ A @ Llist ) )
= Llist ) ) ).
% lhd_LCons_ltl
thf(fact_172_llexord__lappend__leftI,axiom,
! [A: $tType,R2: A > A > $o,Ys: coindu1593790203_llist @ A,Zs: coindu1593790203_llist @ A,Xs: coindu1593790203_llist @ A] :
( ( coindu300403952lexord @ A @ R2 @ Ys @ Zs )
=> ( coindu300403952lexord @ A @ R2 @ ( coindu268472904append @ A @ Xs @ Ys ) @ ( coindu268472904append @ A @ Xs @ Zs ) ) ) ).
% llexord_lappend_leftI
thf(fact_173_llexord__append__right,axiom,
! [A: $tType,R2: A > A > $o,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] : ( coindu300403952lexord @ A @ R2 @ Xs @ ( coindu268472904append @ A @ Xs @ Ys ) ) ).
% llexord_append_right
thf(fact_174_ltakeWhile_Oexhaust,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,P: A > $o] :
( ~ ( ( coindu335574135_lnull @ A @ Xs )
| ~ ( P @ ( coindu1046438764le_lhd @ A @ Xs ) ) )
=> ~ ( ~ ( coindu335574135_lnull @ A @ Xs )
=> ~ ( P @ ( coindu1046438764le_lhd @ A @ Xs ) ) ) ) ).
% ltakeWhile.exhaust
thf(fact_175_llist_Ocase__eq__if,axiom,
! [A: $tType,B: $tType] :
( ( coindu882539134_llist @ B @ A )
= ( ^ [F12: B,F23: A > ( coindu1593790203_llist @ A ) > B,Llist2: coindu1593790203_llist @ A] : ( if @ B @ ( coindu335574135_lnull @ A @ Llist2 ) @ F12 @ ( F23 @ ( coindu1046438764le_lhd @ A @ Llist2 ) @ ( coindu1047225960le_ltl @ A @ Llist2 ) ) ) ) ) ).
% llist.case_eq_if
thf(fact_176_llexord__coinduct,axiom,
! [A: $tType,X6: ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) > $o,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A,R2: A > A > $o] :
( ( X6 @ Xs @ Ys )
=> ( ! [Xs3: coindu1593790203_llist @ A,Ys6: coindu1593790203_llist @ A] :
( ( X6 @ Xs3 @ Ys6 )
=> ( ~ ( coindu335574135_lnull @ A @ Xs3 )
=> ( ~ ( coindu335574135_lnull @ A @ Ys6 )
& ( ~ ( coindu335574135_lnull @ A @ Ys6 )
=> ( ( R2 @ ( coindu1046438764le_lhd @ A @ Xs3 ) @ ( coindu1046438764le_lhd @ A @ Ys6 ) )
| ( ( ( coindu1046438764le_lhd @ A @ Xs3 )
= ( coindu1046438764le_lhd @ A @ Ys6 ) )
& ( ( X6 @ ( coindu1047225960le_ltl @ A @ Xs3 ) @ ( coindu1047225960le_ltl @ A @ Ys6 ) )
| ( coindu300403952lexord @ A @ R2 @ ( coindu1047225960le_ltl @ A @ Xs3 ) @ ( coindu1047225960le_ltl @ A @ Ys6 ) ) ) ) ) ) ) ) )
=> ( coindu300403952lexord @ A @ R2 @ Xs @ Ys ) ) ) ).
% llexord_coinduct
thf(fact_177_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_178_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_179_lnull__ltlI,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ A @ Xs )
=> ( coindu335574135_lnull @ A @ ( coindu1047225960le_ltl @ A @ Xs ) ) ) ).
% lnull_ltlI
thf(fact_180_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_181_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_182_not__lnull__conv,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( ( ~ ( coindu335574135_lnull @ A @ Xs ) )
= ( ? [X3: A,Xs4: coindu1593790203_llist @ A] :
( Xs
= ( coindu1121789889_LCons @ A @ X3 @ Xs4 ) ) ) ) ).
% not_lnull_conv
thf(fact_183_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_184_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_185_lnull__def,axiom,
! [A: $tType] :
( ( coindu335574135_lnull @ A )
= ( ^ [Llist2: coindu1593790203_llist @ A] :
( Llist2
= ( coindu1598213697e_LNil @ A ) ) ) ) ).
% lnull_def
thf(fact_186_llist_Ocollapse_I1_J,axiom,
! [A: $tType,Llist: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ A @ Llist )
=> ( Llist
= ( coindu1598213697e_LNil @ A ) ) ) ).
% llist.collapse(1)
thf(fact_187_llist_OdiscI_I1_J,axiom,
! [A: $tType,Llist: coindu1593790203_llist @ A] :
( ( Llist
= ( coindu1598213697e_LNil @ A ) )
=> ( coindu335574135_lnull @ A @ Llist ) ) ).
% llist.discI(1)
thf(fact_188_llist_Odisc_I1_J,axiom,
! [A: $tType] : ( coindu335574135_lnull @ A @ ( coindu1598213697e_LNil @ A ) ) ).
% llist.disc(1)
thf(fact_189_llist__all2__lnullD,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ B] :
( ( coindu987187967t_all2 @ A @ B @ P @ Xs @ Ys )
=> ( ( coindu335574135_lnull @ A @ Xs )
= ( coindu335574135_lnull @ B @ Ys ) ) ) ).
% llist_all2_lnullD
thf(fact_190_lnull__imp__lfinite,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ A @ Xs )
=> ( coindu1213758845finite @ A @ Xs ) ) ).
% lnull_imp_lfinite
thf(fact_191_llexord__LCons__less,axiom,
! [A: $tType,R2: A > A > $o,X: A,Y: A,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( R2 @ X @ Y )
=> ( coindu300403952lexord @ A @ R2 @ ( coindu1121789889_LCons @ A @ X @ Xs ) @ ( coindu1121789889_LCons @ A @ Y @ Ys ) ) ) ).
% llexord_LCons_less
thf(fact_192_llexord__LCons__eq,axiom,
! [A: $tType,R2: A > A > $o,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A,X: A] :
( ( coindu300403952lexord @ A @ R2 @ Xs @ Ys )
=> ( coindu300403952lexord @ A @ R2 @ ( coindu1121789889_LCons @ A @ X @ Xs ) @ ( coindu1121789889_LCons @ A @ X @ Ys ) ) ) ).
% llexord_LCons_eq
thf(fact_193_llexord__LCons__left,axiom,
! [A: $tType,R2: A > A > $o,X: A,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu300403952lexord @ A @ R2 @ ( coindu1121789889_LCons @ A @ X @ Xs ) @ Ys )
= ( ? [Y4: A,Ys3: coindu1593790203_llist @ A] :
( ( Ys
= ( coindu1121789889_LCons @ A @ Y4 @ Ys3 ) )
& ( ( ( X = Y4 )
& ( coindu300403952lexord @ A @ R2 @ Xs @ Ys3 ) )
| ( R2 @ X @ Y4 ) ) ) ) ) ).
% llexord_LCons_left
thf(fact_194_llexord__code_I3_J,axiom,
! [A: $tType,R2: A > A > $o,X: A,Xs: coindu1593790203_llist @ A,Y: A,Ys: coindu1593790203_llist @ A] :
( ( coindu300403952lexord @ A @ R2 @ ( coindu1121789889_LCons @ A @ X @ Xs ) @ ( coindu1121789889_LCons @ A @ Y @ Ys ) )
= ( ( R2 @ X @ Y )
| ( ( X = Y )
& ( coindu300403952lexord @ A @ R2 @ Xs @ Ys ) ) ) ) ).
% llexord_code(3)
thf(fact_195_llexord__LNil,axiom,
! [A: $tType,R2: A > A > $o,Ys: coindu1593790203_llist @ A] : ( coindu300403952lexord @ A @ R2 @ ( coindu1598213697e_LNil @ A ) @ Ys ) ).
% llexord_LNil
thf(fact_196_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_197_llexord__trans,axiom,
! [A: $tType,R2: A > A > $o,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A,Zs: coindu1593790203_llist @ A] :
( ( coindu300403952lexord @ A @ R2 @ Xs @ Ys )
=> ( ( coindu300403952lexord @ A @ R2 @ Ys @ Zs )
=> ( ! [A6: A,B3: A,C3: A] :
( ( R2 @ A6 @ B3 )
=> ( ( R2 @ B3 @ C3 )
=> ( R2 @ A6 @ C3 ) ) )
=> ( coindu300403952lexord @ A @ R2 @ Xs @ Zs ) ) ) ) ).
% llexord_trans
thf(fact_198_lnull__llexord,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,R2: A > A > $o,Ys: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ A @ Xs )
=> ( coindu300403952lexord @ A @ R2 @ Xs @ Ys ) ) ).
% lnull_llexord
thf(fact_199_llexord__linear,axiom,
! [A: $tType,R2: A > A > $o,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ! [X2: A,Y2: A] :
( ( R2 @ X2 @ Y2 )
| ( X2 = Y2 )
| ( R2 @ Y2 @ X2 ) )
=> ( ( coindu300403952lexord @ A @ R2 @ Xs @ Ys )
| ( coindu300403952lexord @ A @ R2 @ Ys @ Xs ) ) ) ).
% llexord_linear
thf(fact_200_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_201_llexord__antisym,axiom,
! [A: $tType,R2: A > A > $o,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu300403952lexord @ A @ R2 @ Xs @ Ys )
=> ( ( coindu300403952lexord @ A @ R2 @ Ys @ Xs )
=> ( ! [A6: A,B3: A] :
( ( R2 @ A6 @ B3 )
=> ~ ( R2 @ B3 @ A6 ) )
=> ( Xs = Ys ) ) ) ) ).
% llexord_antisym
thf(fact_202_llist_Ocorec__disc_I2_J,axiom,
! [A: $tType,C: $tType,P3: C > $o,A3: C,G212: C > A,Q222: C > $o,G2212: C > ( coindu1593790203_llist @ A ),G2222: C > C] :
( ~ ( P3 @ A3 )
=> ~ ( coindu335574135_lnull @ A @ ( coindu1244876290_llist @ C @ A @ P3 @ G212 @ Q222 @ G2212 @ G2222 @ A3 ) ) ) ).
% llist.corec_disc(2)
thf(fact_203_llist_Ocorec__disc_I1_J,axiom,
! [A: $tType,C: $tType,P3: C > $o,A3: C,G212: C > A,Q222: C > $o,G2212: C > ( coindu1593790203_llist @ A ),G2222: C > C] :
( ( P3 @ A3 )
=> ( coindu335574135_lnull @ A @ ( coindu1244876290_llist @ C @ A @ P3 @ G212 @ Q222 @ G2212 @ G2222 @ A3 ) ) ) ).
% llist.corec_disc(1)
thf(fact_204_llist_Osimps_I5_J,axiom,
! [B: $tType,A: $tType,F1: B,F22: A > ( coindu1593790203_llist @ A ) > B,X21: A,X22: coindu1593790203_llist @ A] :
( ( coindu882539134_llist @ B @ A @ F1 @ F22 @ ( coindu1121789889_LCons @ A @ X21 @ X22 ) )
= ( F22 @ X21 @ X22 ) ) ).
% llist.simps(5)
thf(fact_205_llist_Osimps_I4_J,axiom,
! [A: $tType,B: $tType,F1: B,F22: A > ( coindu1593790203_llist @ A ) > B] :
( ( coindu882539134_llist @ B @ A @ F1 @ F22 @ ( coindu1598213697e_LNil @ A ) )
= F1 ) ).
% llist.simps(4)
thf(fact_206_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_207_llist_Oset__sel_I2_J,axiom,
! [A: $tType,A3: coindu1593790203_llist @ A,X: A] :
( ~ ( coindu335574135_lnull @ A @ A3 )
=> ( ( member @ A @ X @ ( coindu1112613586e_lset @ A @ ( coindu1047225960le_ltl @ A @ A3 ) ) )
=> ( member @ A @ X @ ( coindu1112613586e_lset @ A @ A3 ) ) ) ) ).
% llist.set_sel(2)
thf(fact_208_llist_Oset__sel_I1_J,axiom,
! [A: $tType,A3: coindu1593790203_llist @ A] :
( ~ ( coindu335574135_lnull @ A @ A3 )
=> ( member @ A @ ( coindu1046438764le_lhd @ A @ A3 ) @ ( coindu1112613586e_lset @ A @ A3 ) ) ) ).
% llist.set_sel(1)
thf(fact_209_llist_Oexpand,axiom,
! [A: $tType,Llist: coindu1593790203_llist @ A,Llist3: coindu1593790203_llist @ A] :
( ( ( coindu335574135_lnull @ A @ Llist )
= ( coindu335574135_lnull @ A @ Llist3 ) )
=> ( ( ~ ( coindu335574135_lnull @ A @ Llist )
=> ( ~ ( coindu335574135_lnull @ A @ Llist3 )
=> ( ( ( coindu1046438764le_lhd @ A @ Llist )
= ( coindu1046438764le_lhd @ A @ Llist3 ) )
& ( ( coindu1047225960le_ltl @ A @ Llist )
= ( coindu1047225960le_ltl @ A @ Llist3 ) ) ) ) )
=> ( Llist = Llist3 ) ) ) ).
% llist.expand
thf(fact_210_llist_Ocoinduct,axiom,
! [A: $tType,R: ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) > $o,Llist: coindu1593790203_llist @ A,Llist3: coindu1593790203_llist @ A] :
( ( R @ Llist @ Llist3 )
=> ( ! [Llist4: coindu1593790203_llist @ A,Llist5: coindu1593790203_llist @ A] :
( ( R @ Llist4 @ Llist5 )
=> ( ( ( coindu335574135_lnull @ A @ Llist4 )
= ( coindu335574135_lnull @ A @ Llist5 ) )
& ( ~ ( coindu335574135_lnull @ A @ Llist4 )
=> ( ~ ( coindu335574135_lnull @ A @ Llist5 )
=> ( ( ( coindu1046438764le_lhd @ A @ Llist4 )
= ( coindu1046438764le_lhd @ A @ Llist5 ) )
& ( R @ ( coindu1047225960le_ltl @ A @ Llist4 ) @ ( coindu1047225960le_ltl @ A @ Llist5 ) ) ) ) ) ) )
=> ( Llist = Llist3 ) ) ) ).
% llist.coinduct
thf(fact_211_llist_Ocoinduct__strong,axiom,
! [A: $tType,R: ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) > $o,Llist: coindu1593790203_llist @ A,Llist3: coindu1593790203_llist @ A] :
( ( R @ Llist @ Llist3 )
=> ( ! [Llist4: coindu1593790203_llist @ A,Llist5: coindu1593790203_llist @ A] :
( ( R @ Llist4 @ Llist5 )
=> ( ( ( coindu335574135_lnull @ A @ Llist4 )
= ( coindu335574135_lnull @ A @ Llist5 ) )
& ( ~ ( coindu335574135_lnull @ A @ Llist4 )
=> ( ~ ( coindu335574135_lnull @ A @ Llist5 )
=> ( ( ( coindu1046438764le_lhd @ A @ Llist4 )
= ( coindu1046438764le_lhd @ A @ Llist5 ) )
& ( ( R @ ( coindu1047225960le_ltl @ A @ Llist4 ) @ ( coindu1047225960le_ltl @ A @ Llist5 ) )
| ( ( coindu1047225960le_ltl @ A @ Llist4 )
= ( coindu1047225960le_ltl @ A @ Llist5 ) ) ) ) ) ) ) )
=> ( Llist = Llist3 ) ) ) ).
% llist.coinduct_strong
thf(fact_212_lappend__ltl,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ~ ( coindu335574135_lnull @ A @ Xs )
=> ( ( coindu268472904append @ A @ ( coindu1047225960le_ltl @ A @ Xs ) @ Ys )
= ( coindu1047225960le_ltl @ A @ ( coindu268472904append @ A @ Xs @ Ys ) ) ) ) ).
% lappend_ltl
thf(fact_213_lfinite__induct,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,P: ( coindu1593790203_llist @ A ) > $o] :
( ( coindu1213758845finite @ A @ Xs )
=> ( ! [Xs3: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ A @ Xs3 )
=> ( P @ Xs3 ) )
=> ( ! [Xs3: coindu1593790203_llist @ A] :
( ( coindu1213758845finite @ A @ Xs3 )
=> ( ~ ( coindu335574135_lnull @ A @ Xs3 )
=> ( ( P @ ( coindu1047225960le_ltl @ A @ Xs3 ) )
=> ( P @ Xs3 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% lfinite_induct
thf(fact_214_llist__all2__lhdD,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ B] :
( ( coindu987187967t_all2 @ A @ B @ P @ Xs @ Ys )
=> ( ~ ( coindu335574135_lnull @ A @ Xs )
=> ( P @ ( coindu1046438764le_lhd @ A @ Xs ) @ ( coindu1046438764le_lhd @ B @ Ys ) ) ) ) ).
% llist_all2_lhdD
thf(fact_215_llist__all2__lhdD2,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ B] :
( ( coindu987187967t_all2 @ A @ B @ P @ Xs @ Ys )
=> ( ~ ( coindu335574135_lnull @ B @ Ys )
=> ( P @ ( coindu1046438764le_lhd @ A @ Xs ) @ ( coindu1046438764le_lhd @ B @ Ys ) ) ) ) ).
% llist_all2_lhdD2
thf(fact_216_ldropn__lnull,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,N: nat] :
( ( coindu335574135_lnull @ A @ Xs )
=> ( ( coindu531130065ldropn @ A @ N @ Xs )
= ( coindu1598213697e_LNil @ A ) ) ) ).
% ldropn_lnull
thf(fact_217_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_218_llexord__code_I2_J,axiom,
! [A: $tType,R2: A > A > $o,X: A,Xs: coindu1593790203_llist @ A] :
~ ( coindu300403952lexord @ A @ R2 @ ( coindu1121789889_LCons @ A @ X @ Xs ) @ ( coindu1598213697e_LNil @ A ) ) ).
% llexord_code(2)
thf(fact_219_llexord_Ocases,axiom,
! [A: $tType,R2: A > A > $o,A1: coindu1593790203_llist @ A,A2: coindu1593790203_llist @ A] :
( ( coindu300403952lexord @ A @ R2 @ A1 @ A2 )
=> ( ! [Xs3: coindu1593790203_llist @ A,Ys6: coindu1593790203_llist @ A,X2: A] :
( ( A1
= ( coindu1121789889_LCons @ A @ X2 @ Xs3 ) )
=> ( ( A2
= ( coindu1121789889_LCons @ A @ X2 @ Ys6 ) )
=> ~ ( coindu300403952lexord @ A @ R2 @ Xs3 @ Ys6 ) ) )
=> ( ! [X2: A] :
( ? [Xs3: coindu1593790203_llist @ A] :
( A1
= ( coindu1121789889_LCons @ A @ X2 @ Xs3 ) )
=> ! [Y2: A] :
( ? [Ys6: coindu1593790203_llist @ A] :
( A2
= ( coindu1121789889_LCons @ A @ Y2 @ Ys6 ) )
=> ~ ( R2 @ X2 @ Y2 ) ) )
=> ~ ( ( A1
= ( coindu1598213697e_LNil @ A ) )
=> ! [Ys6: coindu1593790203_llist @ A] : A2 != Ys6 ) ) ) ) ).
% llexord.cases
thf(fact_220_llexord_Osimps,axiom,
! [A: $tType] :
( ( coindu300403952lexord @ A )
= ( ^ [R3: A > A > $o,A12: coindu1593790203_llist @ A,A22: coindu1593790203_llist @ A] :
( ? [Xs5: coindu1593790203_llist @ A,Ys7: coindu1593790203_llist @ A,X3: A] :
( ( A12
= ( coindu1121789889_LCons @ A @ X3 @ Xs5 ) )
& ( A22
= ( coindu1121789889_LCons @ A @ X3 @ Ys7 ) )
& ( coindu300403952lexord @ A @ R3 @ Xs5 @ Ys7 ) )
| ? [X3: A,Y4: A,Xs5: coindu1593790203_llist @ A,Ys7: coindu1593790203_llist @ A] :
( ( A12
= ( coindu1121789889_LCons @ A @ X3 @ Xs5 ) )
& ( A22
= ( coindu1121789889_LCons @ A @ Y4 @ Ys7 ) )
& ( R3 @ X3 @ Y4 ) )
| ? [Ys7: coindu1593790203_llist @ A] :
( ( A12
= ( coindu1598213697e_LNil @ A ) )
& ( A22 = Ys7 ) ) ) ) ) ).
% llexord.simps
thf(fact_221_llexord_Ocoinduct,axiom,
! [A: $tType,X6: ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) > $o,X: coindu1593790203_llist @ A,Xa3: coindu1593790203_llist @ A,R2: A > A > $o] :
( ( X6 @ X @ Xa3 )
=> ( ! [X2: coindu1593790203_llist @ A,Xa: coindu1593790203_llist @ A] :
( ( X6 @ X2 @ Xa )
=> ( ? [Xs6: coindu1593790203_llist @ A,Ys4: coindu1593790203_llist @ A,Xb: A] :
( ( X2
= ( coindu1121789889_LCons @ A @ Xb @ Xs6 ) )
& ( Xa
= ( coindu1121789889_LCons @ A @ Xb @ Ys4 ) )
& ( ( X6 @ Xs6 @ Ys4 )
| ( coindu300403952lexord @ A @ R2 @ Xs6 @ Ys4 ) ) )
| ? [Xb: A,Y5: A,Xs6: coindu1593790203_llist @ A,Ys4: coindu1593790203_llist @ A] :
( ( X2
= ( coindu1121789889_LCons @ A @ Xb @ Xs6 ) )
& ( Xa
= ( coindu1121789889_LCons @ A @ Y5 @ Ys4 ) )
& ( R2 @ Xb @ Y5 ) )
| ? [Ys4: coindu1593790203_llist @ A] :
( ( X2
= ( coindu1598213697e_LNil @ A ) )
& ( Xa = Ys4 ) ) ) )
=> ( coindu300403952lexord @ A @ R2 @ X @ Xa3 ) ) ) ).
% llexord.coinduct
thf(fact_222_llist__set__induct,axiom,
! [A: $tType,X: A,Xs: coindu1593790203_llist @ A,P: A > ( coindu1593790203_llist @ A ) > $o] :
( ( member @ A @ X @ ( coindu1112613586e_lset @ A @ Xs ) )
=> ( ! [Xs3: coindu1593790203_llist @ A] :
( ~ ( coindu335574135_lnull @ A @ Xs3 )
=> ( P @ ( coindu1046438764le_lhd @ A @ Xs3 ) @ Xs3 ) )
=> ( ! [Xs3: coindu1593790203_llist @ A,Y2: A] :
( ~ ( coindu335574135_lnull @ A @ Xs3 )
=> ( ( member @ A @ Y2 @ ( coindu1112613586e_lset @ A @ ( coindu1047225960le_ltl @ A @ Xs3 ) ) )
=> ( ( P @ Y2 @ ( coindu1047225960le_ltl @ A @ Xs3 ) )
=> ( P @ Y2 @ Xs3 ) ) ) )
=> ( P @ X @ Xs ) ) ) ) ).
% llist_set_induct
thf(fact_223_llist__all2__coinduct,axiom,
! [A: $tType,B: $tType,X6: ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ B ) > $o,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ B,P: A > B > $o] :
( ( X6 @ Xs @ Ys )
=> ( ! [Xs3: coindu1593790203_llist @ A,Ys6: coindu1593790203_llist @ B] :
( ( X6 @ Xs3 @ Ys6 )
=> ( ( coindu335574135_lnull @ A @ Xs3 )
= ( coindu335574135_lnull @ B @ Ys6 ) ) )
=> ( ! [Xs3: coindu1593790203_llist @ A,Ys6: coindu1593790203_llist @ B] :
( ( X6 @ Xs3 @ Ys6 )
=> ( ~ ( coindu335574135_lnull @ A @ Xs3 )
=> ( ~ ( coindu335574135_lnull @ B @ Ys6 )
=> ( ( P @ ( coindu1046438764le_lhd @ A @ Xs3 ) @ ( coindu1046438764le_lhd @ B @ Ys6 ) )
& ( ( X6 @ ( coindu1047225960le_ltl @ A @ Xs3 ) @ ( coindu1047225960le_ltl @ B @ Ys6 ) )
| ( coindu987187967t_all2 @ A @ B @ P @ ( coindu1047225960le_ltl @ A @ Xs3 ) @ ( coindu1047225960le_ltl @ B @ Ys6 ) ) ) ) ) ) )
=> ( coindu987187967t_all2 @ A @ B @ P @ Xs @ Ys ) ) ) ) ).
% llist_all2_coinduct
thf(fact_224_llist_Orel__coinduct,axiom,
! [A: $tType,B: $tType,P: ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ B ) > $o,X: coindu1593790203_llist @ A,Y: coindu1593790203_llist @ B,R: A > B > $o] :
( ( P @ X @ Y )
=> ( ! [Llist4: coindu1593790203_llist @ A,Llist5: coindu1593790203_llist @ B] :
( ( P @ Llist4 @ Llist5 )
=> ( ( ( coindu335574135_lnull @ A @ Llist4 )
= ( coindu335574135_lnull @ B @ Llist5 ) )
& ( ~ ( coindu335574135_lnull @ A @ Llist4 )
=> ( ~ ( coindu335574135_lnull @ B @ Llist5 )
=> ( ( R @ ( coindu1046438764le_lhd @ A @ Llist4 ) @ ( coindu1046438764le_lhd @ B @ Llist5 ) )
& ( P @ ( coindu1047225960le_ltl @ A @ Llist4 ) @ ( coindu1047225960le_ltl @ B @ Llist5 ) ) ) ) ) ) )
=> ( coindu987187967t_all2 @ A @ B @ R @ X @ Y ) ) ) ).
% llist.rel_coinduct
thf(fact_225_llist__all2__expand,axiom,
! [A: $tType,B: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ B,P: A > B > $o] :
( ( ( coindu335574135_lnull @ A @ Xs )
= ( coindu335574135_lnull @ B @ Ys ) )
=> ( ( ~ ( coindu335574135_lnull @ A @ Xs )
=> ( ~ ( coindu335574135_lnull @ B @ Ys )
=> ( ( P @ ( coindu1046438764le_lhd @ A @ Xs ) @ ( coindu1046438764le_lhd @ B @ Ys ) )
& ( coindu987187967t_all2 @ A @ B @ P @ ( coindu1047225960le_ltl @ A @ Xs ) @ ( coindu1047225960le_ltl @ B @ Ys ) ) ) ) )
=> ( coindu987187967t_all2 @ A @ B @ P @ Xs @ Ys ) ) ) ).
% llist_all2_expand
thf(fact_226_llist_Orel__sel,axiom,
! [B: $tType,A: $tType] :
( ( coindu987187967t_all2 @ A @ B )
= ( ^ [R4: A > B > $o,A4: coindu1593790203_llist @ A,B4: coindu1593790203_llist @ B] :
( ( ( coindu335574135_lnull @ A @ A4 )
= ( coindu335574135_lnull @ B @ B4 ) )
& ( ~ ( coindu335574135_lnull @ A @ A4 )
=> ( ~ ( coindu335574135_lnull @ B @ B4 )
=> ( ( R4 @ ( coindu1046438764le_lhd @ A @ A4 ) @ ( coindu1046438764le_lhd @ B @ B4 ) )
& ( coindu987187967t_all2 @ A @ B @ R4 @ ( coindu1047225960le_ltl @ A @ A4 ) @ ( coindu1047225960le_ltl @ B @ B4 ) ) ) ) ) ) ) ) ).
% llist.rel_sel
thf(fact_227_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_228_llexord__lappend__left,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,R2: A > A > $o,Ys: coindu1593790203_llist @ A,Zs: coindu1593790203_llist @ A] :
( ( coindu1213758845finite @ A @ Xs )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ ( coindu1112613586e_lset @ A @ Xs ) )
=> ~ ( R2 @ X2 @ X2 ) )
=> ( ( coindu300403952lexord @ A @ R2 @ ( coindu268472904append @ A @ Xs @ Ys ) @ ( coindu268472904append @ A @ Xs @ Zs ) )
= ( coindu300403952lexord @ A @ R2 @ Ys @ Zs ) ) ) ) ).
% llexord_lappend_left
thf(fact_229_llexord__lappend__leftD,axiom,
! [A: $tType,R2: A > A > $o,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A,Zs: coindu1593790203_llist @ A] :
( ( coindu300403952lexord @ A @ R2 @ ( coindu268472904append @ A @ Xs @ Ys ) @ ( coindu268472904append @ A @ Xs @ Zs ) )
=> ( ( coindu1213758845finite @ A @ Xs )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ ( coindu1112613586e_lset @ A @ Xs ) )
=> ~ ( R2 @ X2 @ X2 ) )
=> ( coindu300403952lexord @ A @ R2 @ Ys @ Zs ) ) ) ) ).
% llexord_lappend_leftD
thf(fact_230_ldistinct__coinduct,axiom,
! [A: $tType,X6: ( coindu1593790203_llist @ A ) > $o,Xs: coindu1593790203_llist @ A] :
( ( X6 @ Xs )
=> ( ! [Xs3: coindu1593790203_llist @ A] :
( ( X6 @ Xs3 )
=> ( ~ ( coindu335574135_lnull @ A @ Xs3 )
=> ( ~ ( member @ A @ ( coindu1046438764le_lhd @ A @ Xs3 ) @ ( coindu1112613586e_lset @ A @ ( coindu1047225960le_ltl @ A @ Xs3 ) ) )
& ( ( X6 @ ( coindu1047225960le_ltl @ A @ Xs3 ) )
| ( coindu6345450stinct @ A @ ( coindu1047225960le_ltl @ A @ Xs3 ) ) ) ) ) )
=> ( coindu6345450stinct @ A @ Xs ) ) ) ).
% ldistinct_coinduct
thf(fact_231_lsortedD,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [Xs: coindu1593790203_llist @ A,Y: A] :
( ( coindu48241764sorted @ A @ Xs )
=> ( ~ ( coindu335574135_lnull @ A @ Xs )
=> ( ( member @ A @ Y @ ( coindu1112613586e_lset @ A @ ( coindu1047225960le_ltl @ A @ Xs ) ) )
=> ( ord_less_eq @ A @ ( coindu1046438764le_lhd @ A @ Xs ) @ Y ) ) ) ) ) ).
% lsortedD
thf(fact_232_lsorted__coinduct,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X6: ( coindu1593790203_llist @ A ) > $o,Xs: coindu1593790203_llist @ A] :
( ( X6 @ Xs )
=> ( ! [Xs3: coindu1593790203_llist @ A] :
( ( X6 @ Xs3 )
=> ( ~ ( coindu335574135_lnull @ A @ Xs3 )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ ( coindu1112613586e_lset @ A @ ( coindu1047225960le_ltl @ A @ Xs3 ) ) )
=> ( ord_less_eq @ A @ ( coindu1046438764le_lhd @ A @ Xs3 ) @ X2 ) )
& ( ( X6 @ ( coindu1047225960le_ltl @ A @ Xs3 ) )
| ( coindu48241764sorted @ A @ ( coindu1047225960le_ltl @ A @ Xs3 ) ) ) ) ) )
=> ( coindu48241764sorted @ A @ Xs ) ) ) ) ).
% lsorted_coinduct
thf(fact_233_unfold__llist__id,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( ( coindu1599971794_llist @ ( coindu1593790203_llist @ A ) @ A @ ( coindu335574135_lnull @ A ) @ ( coindu1046438764le_lhd @ A ) @ ( coindu1047225960le_ltl @ A ) @ Xs )
= Xs ) ).
% unfold_llist_id
thf(fact_234_unfold__llist__eq__LCons,axiom,
! [A: $tType,B: $tType,IS_LNIL: B > $o,LHD: B > A,LTL: B > B,B2: B,X: A,Xs: coindu1593790203_llist @ A] :
( ( ( coindu1599971794_llist @ B @ A @ IS_LNIL @ LHD @ LTL @ B2 )
= ( coindu1121789889_LCons @ A @ X @ Xs ) )
= ( ~ ( IS_LNIL @ B2 )
& ( X
= ( LHD @ B2 ) )
& ( Xs
= ( coindu1599971794_llist @ B @ A @ IS_LNIL @ LHD @ LTL @ ( LTL @ B2 ) ) ) ) ) ).
% unfold_llist_eq_LCons
thf(fact_235_unfold__llist_Odisc__iff_I1_J,axiom,
! [B: $tType,A: $tType,P3: A > $o,G212: A > B,G22: A > A,A3: A] :
( ( coindu335574135_lnull @ B @ ( coindu1599971794_llist @ A @ B @ P3 @ G212 @ G22 @ A3 ) )
= ( P3 @ A3 ) ) ).
% unfold_llist.disc_iff(1)
thf(fact_236_unfold__llist_Odisc__iff_I2_J,axiom,
! [B: $tType,A: $tType,P3: A > $o,G212: A > B,G22: A > A,A3: A] :
( ( ~ ( coindu335574135_lnull @ B @ ( coindu1599971794_llist @ A @ B @ P3 @ G212 @ G22 @ A3 ) ) )
= ( ~ ( P3 @ A3 ) ) ) ).
% unfold_llist.disc_iff(2)
thf(fact_237_lsorted__LCons__LCons,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Xs: coindu1593790203_llist @ A] :
( ( coindu48241764sorted @ A @ ( coindu1121789889_LCons @ A @ X @ ( coindu1121789889_LCons @ A @ Y @ Xs ) ) )
= ( ( ord_less_eq @ A @ X @ Y )
& ( coindu48241764sorted @ A @ ( coindu1121789889_LCons @ A @ Y @ Xs ) ) ) ) ) ).
% lsorted_LCons_LCons
thf(fact_238_unfold__llist_Odisc_I2_J,axiom,
! [B: $tType,A: $tType,P3: A > $o,A3: A,G212: A > B,G22: A > A] :
( ~ ( P3 @ A3 )
=> ~ ( coindu335574135_lnull @ B @ ( coindu1599971794_llist @ A @ B @ P3 @ G212 @ G22 @ A3 ) ) ) ).
% unfold_llist.disc(2)
thf(fact_239_unfold__llist_Odisc_I1_J,axiom,
! [B: $tType,A: $tType,P3: A > $o,A3: A,G212: A > B,G22: A > A] :
( ( P3 @ A3 )
=> ( coindu335574135_lnull @ B @ ( coindu1599971794_llist @ A @ B @ P3 @ G212 @ G22 @ A3 ) ) ) ).
% unfold_llist.disc(1)
thf(fact_240_unfold__llist_Octr_I2_J,axiom,
! [B: $tType,A: $tType,P3: A > $o,A3: A,G212: A > B,G22: A > A] :
( ~ ( P3 @ A3 )
=> ( ( coindu1599971794_llist @ A @ B @ P3 @ G212 @ G22 @ A3 )
= ( coindu1121789889_LCons @ B @ ( G212 @ A3 ) @ ( coindu1599971794_llist @ A @ B @ P3 @ G212 @ G22 @ ( G22 @ A3 ) ) ) ) ) ).
% unfold_llist.ctr(2)
thf(fact_241_unfold__llist_Octr_I1_J,axiom,
! [A: $tType,B: $tType,P3: A > $o,A3: A,G212: A > B,G22: A > A] :
( ( P3 @ A3 )
=> ( ( coindu1599971794_llist @ A @ B @ P3 @ G212 @ G22 @ A3 )
= ( coindu1598213697e_LNil @ B ) ) ) ).
% unfold_llist.ctr(1)
thf(fact_242_unfold__llist_Osimps_I4_J,axiom,
! [B: $tType,A: $tType,P3: A > $o,A3: A,G212: A > B,G22: A > A] :
( ~ ( P3 @ A3 )
=> ( ( coindu1047225960le_ltl @ B @ ( coindu1599971794_llist @ A @ B @ P3 @ G212 @ G22 @ A3 ) )
= ( coindu1599971794_llist @ A @ B @ P3 @ G212 @ G22 @ ( G22 @ A3 ) ) ) ) ).
% unfold_llist.simps(4)
thf(fact_243_unfold__llist_Osimps_I3_J,axiom,
! [B: $tType,A: $tType,P3: A > $o,A3: A,G212: A > B,G22: A > A] :
( ~ ( P3 @ A3 )
=> ( ( coindu1046438764le_lhd @ B @ ( coindu1599971794_llist @ A @ B @ P3 @ G212 @ G22 @ A3 ) )
= ( G212 @ A3 ) ) ) ).
% unfold_llist.simps(3)
thf(fact_244_wlog__linorder__le,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > A > $o,B2: A,A3: A] :
( ! [A6: A,B3: A] :
( ( ord_less_eq @ A @ A6 @ B3 )
=> ( P @ A6 @ B3 ) )
=> ( ( ( P @ B2 @ A3 )
=> ( P @ A3 @ B2 ) )
=> ( P @ A3 @ B2 ) ) ) ) ).
% wlog_linorder_le
thf(fact_245_llist_Orel__mono,axiom,
! [B: $tType,A: $tType,R: A > B > $o,Ra: A > B > $o] :
( ( ord_less_eq @ ( A > B > $o ) @ R @ Ra )
=> ( ord_less_eq @ ( ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ B ) > $o ) @ ( coindu987187967t_all2 @ A @ B @ R ) @ ( coindu987187967t_all2 @ A @ B @ Ra ) ) ) ).
% llist.rel_mono
thf(fact_246_lset__ltl,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] : ( ord_less_eq @ ( set @ A ) @ ( coindu1112613586e_lset @ A @ ( coindu1047225960le_ltl @ A @ Xs ) ) @ ( coindu1112613586e_lset @ A @ Xs ) ) ).
% lset_ltl
thf(fact_247_lset__lappend1,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] : ( ord_less_eq @ ( set @ A ) @ ( coindu1112613586e_lset @ A @ Xs ) @ ( coindu1112613586e_lset @ A @ ( coindu268472904append @ A @ Xs @ Ys ) ) ) ).
% lset_lappend1
thf(fact_248_LCons__LCons,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Xs: coindu1593790203_llist @ A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( coindu48241764sorted @ A @ ( coindu1121789889_LCons @ A @ Y @ Xs ) )
=> ( coindu48241764sorted @ A @ ( coindu1121789889_LCons @ A @ X @ ( coindu1121789889_LCons @ A @ Y @ Xs ) ) ) ) ) ) ).
% LCons_LCons
thf(fact_249_lset__ldropn__subset,axiom,
! [A: $tType,N: nat,Xs: coindu1593790203_llist @ A] : ( ord_less_eq @ ( set @ A ) @ ( coindu1112613586e_lset @ A @ ( coindu531130065ldropn @ A @ N @ Xs ) ) @ ( coindu1112613586e_lset @ A @ Xs ) ) ).
% lset_ldropn_subset
thf(fact_250_lset__ldropWhile__subset,axiom,
! [A: $tType,P: A > $o,Xs: coindu1593790203_llist @ A] : ( ord_less_eq @ ( set @ A ) @ ( coindu1112613586e_lset @ A @ ( coindu438612276pWhile @ A @ P @ Xs ) ) @ ( coindu1112613586e_lset @ A @ Xs ) ) ).
% lset_ldropWhile_subset
thf(fact_251_unfold__llist_Ocode,axiom,
! [B: $tType,A: $tType] :
( ( coindu1599971794_llist @ A @ B )
= ( ^ [P4: A > $o,G21: A > B,G223: A > A,A4: A] : ( if @ ( coindu1593790203_llist @ B ) @ ( P4 @ A4 ) @ ( coindu1598213697e_LNil @ B ) @ ( coindu1121789889_LCons @ B @ ( G21 @ A4 ) @ ( coindu1599971794_llist @ A @ B @ P4 @ G21 @ G223 @ ( G223 @ A4 ) ) ) ) ) ) ).
% unfold_llist.code
thf(fact_252_ltl__unfold__llist,axiom,
! [A: $tType,B: $tType,IS_LNIL: B > $o,A3: B,LHD: B > A,LTL: B > B] :
( ( ( IS_LNIL @ A3 )
=> ( ( coindu1047225960le_ltl @ A @ ( coindu1599971794_llist @ B @ A @ IS_LNIL @ LHD @ LTL @ A3 ) )
= ( coindu1598213697e_LNil @ A ) ) )
& ( ~ ( IS_LNIL @ A3 )
=> ( ( coindu1047225960le_ltl @ A @ ( coindu1599971794_llist @ B @ A @ IS_LNIL @ LHD @ LTL @ A3 ) )
= ( coindu1599971794_llist @ B @ A @ IS_LNIL @ LHD @ LTL @ ( LTL @ A3 ) ) ) ) ) ).
% ltl_unfold_llist
thf(fact_253_unfold__llist__ltl__unroll,axiom,
! [A: $tType,B: $tType,IS_LNIL: B > $o,LHD: B > A,LTL: B > B,B2: B] :
( ( coindu1599971794_llist @ B @ A @ IS_LNIL @ LHD @ LTL @ ( LTL @ B2 ) )
= ( coindu1599971794_llist @ B @ A @ ( comp @ B @ $o @ B @ IS_LNIL @ LTL ) @ ( comp @ B @ A @ B @ LHD @ LTL ) @ LTL @ B2 ) ) ).
% unfold_llist_ltl_unroll
%----Type constructors (7)
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A7: $tType,A8: $tType] :
( ( preorder @ A8 @ ( type2 @ A8 ) )
=> ( preorder @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A7: $tType,A8: $tType] :
( ( ord @ A8 @ ( type2 @ A8 ) )
=> ( ord @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_1,axiom,
! [A7: $tType] : ( preorder @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_2,axiom,
! [A7: $tType] : ( ord @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_HOL_Obool___Orderings_Opreorder_3,axiom,
preorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
linorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oord_4,axiom,
ord @ $o @ ( type2 @ $o ) ).
%----Helper facts (3)
thf(help_If_3_1_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_T,axiom,
! [A: $tType,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,
coindu987187967t_all2 @ a @ b @ p @ ( coindu1889960678filter @ a @ q1 @ xs ) @ ( coindu1889960678filter @ b @ q2 @ ys ) ).
%------------------------------------------------------------------------------