TPTP Problem File: DAT181^1.p
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%------------------------------------------------------------------------------
% File : DAT181^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Data Structures
% Problem : Lazy lists II 762
% Version : [Bla16] axioms : Especial.
% English :
% Refs : [Fri04] Friedrich (2004), Lazy Lists II
% : [RB15] Reynolds & Blanchette (2015), A Decision Procedure for
% : [Bla16] Blanchette (2016), Email to Geoff Sutcliffe
% Source : [Bla16]
% Names : llist2__762.p [Bla16]
% Status : Theorem
% Rating : 1.00 v7.1.0
% Syntax : Number of formulae : 325 ( 135 unt; 56 typ; 0 def)
% Number of atoms : 659 ( 264 equ; 0 cnn)
% Maximal formula atoms : 7 ( 2 avg)
% Number of connectives : 4007 ( 102 ~; 26 |; 63 &;3532 @)
% ( 0 <=>; 284 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 7 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 158 ( 158 >; 0 *; 0 +; 0 <<)
% Number of symbols : 54 ( 53 usr; 3 con; 0-5 aty)
% Number of variables : 950 ( 17 ^; 851 !; 36 ?; 950 :)
% ( 46 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 14:49:03.131
%------------------------------------------------------------------------------
%----Could-be-implicit typings (6)
thf(ty_t_Coinductive__List_Ollist,type,
coinductive_llist: $tType > $tType ).
thf(ty_t_Extended__Nat_Oenat,type,
extended_enat: $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_a,type,
a: $tType ).
%----Explicit typings (50)
thf(sy_cl_Lattices_Obounded__lattice,type,
bounded_lattice:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Otop,type,
top:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Omonoid__add,type,
monoid_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Lattices_Obounded__lattice__top,type,
bounded_lattice_top:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_c_Coinductive__List_Ofinite__lprefix,type,
coindu328551480prefix:
!>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List_Ogen__lset,type,
coinductive_gen_lset:
!>[A: $tType] : ( ( set @ A ) > ( coinductive_llist @ A ) > ( set @ A ) ) ).
thf(sy_c_Coinductive__List_Olappend,type,
coinductive_lappend:
!>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
thf(sy_c_Coinductive__List_Oldistinct,type,
coindu351974385stinct:
!>[A: $tType] : ( ( coinductive_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List_Olfilter,type,
coinductive_lfilter:
!>[A: $tType] : ( ( A > $o ) > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
thf(sy_c_Coinductive__List_Olfinite,type,
coinductive_lfinite:
!>[A: $tType] : ( ( coinductive_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List_Ollast,type,
coinductive_llast:
!>[A: $tType] : ( ( coinductive_llist @ A ) > A ) ).
thf(sy_c_Coinductive__List_Ollcp,type,
coinductive_llcp:
!>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > extended_enat ) ).
thf(sy_c_Coinductive__List_Ollexord,type,
coinductive_llexord:
!>[A: $tType] : ( ( A > A > $o ) > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List_Ollist_OLCons,type,
coinductive_LCons:
!>[A: $tType] : ( A > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
thf(sy_c_Coinductive__List_Ollist_OLNil,type,
coinductive_LNil:
!>[A: $tType] : ( coinductive_llist @ A ) ).
thf(sy_c_Coinductive__List_Ollist_Olnull,type,
coinductive_lnull:
!>[A: $tType] : ( ( coinductive_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List_Ollist_Olset,type,
coinductive_lset:
!>[A: $tType] : ( ( coinductive_llist @ A ) > ( set @ A ) ) ).
thf(sy_c_Coinductive__List_Olmember,type,
coinductive_lmember:
!>[A: $tType] : ( A > ( coinductive_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List_Olstrict__prefix,type,
coindu1478340336prefix:
!>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List_Olsublist,type,
coinductive_lsublist:
!>[A: $tType] : ( ( coinductive_llist @ A ) > ( set @ nat ) > ( coinductive_llist @ A ) ) ).
thf(sy_c_Coinductive__List_Omonoid__add__class_Ollistsum,type,
coindu780009021istsum:
!>[A: $tType] : ( ( coinductive_llist @ A ) > A ) ).
thf(sy_c_Extended__Nat_OeSuc,type,
extended_eSuc: extended_enat > extended_enat ).
thf(sy_c_Groups_Ominus__class_Ominus,type,
minus_minus:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Groups_Oplus__class_Oplus,type,
plus_plus:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Groups_Ozero__class_Ozero,type,
zero_zero:
!>[A: $tType] : A ).
thf(sy_c_HOL_Oundefined,type,
undefined:
!>[A: $tType] : A ).
thf(sy_c_LList2__Mirabelle__hamjzmohle_Oalllsts,type,
lList2435255213lllsts:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( coinductive_llist @ A ) ) ) ).
thf(sy_c_LList2__Mirabelle__hamjzmohle_Oalllstsp,type,
lList21511617539llstsp:
!>[A: $tType] : ( ( A > $o ) > ( coinductive_llist @ A ) > $o ) ).
thf(sy_c_LList2__Mirabelle__hamjzmohle_Ofinlsts,type,
lList2236698231inlsts:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( coinductive_llist @ A ) ) ) ).
thf(sy_c_LList2__Mirabelle__hamjzmohle_Ofinlsts__rec,type,
lList21916056377ts_rec:
!>[B: $tType,A: $tType] : ( B > ( A > ( coinductive_llist @ A ) > B > B ) > ( coinductive_llist @ A ) > B ) ).
thf(sy_c_LList2__Mirabelle__hamjzmohle_Ofinlstsp,type,
lList2860480441nlstsp:
!>[A: $tType] : ( ( A > $o ) > ( coinductive_llist @ A ) > $o ) ).
thf(sy_c_LList2__Mirabelle__hamjzmohle_Ofpslsts,type,
lList22096119349pslsts:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( coinductive_llist @ A ) ) ) ).
thf(sy_c_LList2__Mirabelle__hamjzmohle_Oinflsts,type,
lList21612149805nflsts:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( coinductive_llist @ A ) ) ) ).
thf(sy_c_LList2__Mirabelle__hamjzmohle_Olbutlast,type,
lList2370560421utlast:
!>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
thf(sy_c_LList2__Mirabelle__hamjzmohle_Oldrop,type,
lList2508575361_ldrop:
!>[A: $tType] : ( ( coinductive_llist @ A ) > nat > ( coinductive_llist @ A ) ) ).
thf(sy_c_LList2__Mirabelle__hamjzmohle_Ollast,type,
lList2170638824_llast:
!>[A: $tType] : ( ( coinductive_llist @ A ) > A ) ).
thf(sy_c_LList2__Mirabelle__hamjzmohle_Ollength,type,
lList21232602520length:
!>[A: $tType] : ( ( coinductive_llist @ A ) > nat ) ).
thf(sy_c_LList2__Mirabelle__hamjzmohle_Olrev,type,
lList2281150353e_lrev:
!>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
thf(sy_c_LList2__Mirabelle__hamjzmohle_Oltake,type,
lList22119844313_ltake:
!>[A: $tType] : ( ( coinductive_llist @ A ) > nat > ( coinductive_llist @ A ) ) ).
thf(sy_c_LList2__Mirabelle__hamjzmohle_Oposlsts,type,
lList21148268032oslsts:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( coinductive_llist @ A ) ) ) ).
thf(sy_c_Lattices_Osup__class_Osup,type,
sup_sup:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Orderings_Otop__class_Otop,type,
top_top:
!>[A: $tType] : A ).
thf(sy_c_Pure_Otype,type,
type2:
!>[A: $tType] : ( itself @ A ) ).
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_A,type,
a2: set @ a ).
thf(sy_v_x,type,
x: a ).
thf(sy_v_xs,type,
xs: coinductive_llist @ a ).
%----Relevant facts (254)
thf(fact_0_fin,axiom,
member @ ( coinductive_llist @ a ) @ xs @ ( lList2236698231inlsts @ a @ a2 ) ).
% fin
thf(fact_1_LNil__is__lappend__conv,axiom,
! [A: $tType,S: coinductive_llist @ A,T: coinductive_llist @ A] :
( ( ( coinductive_LNil @ A )
= ( coinductive_lappend @ A @ S @ T ) )
= ( ( S
= ( coinductive_LNil @ A ) )
& ( T
= ( coinductive_LNil @ A ) ) ) ) ).
% LNil_is_lappend_conv
thf(fact_2_lappend__is__LNil__conv,axiom,
! [A: $tType,S: coinductive_llist @ A,T: coinductive_llist @ A] :
( ( ( coinductive_lappend @ A @ S @ T )
= ( coinductive_LNil @ A ) )
= ( ( S
= ( coinductive_LNil @ A ) )
& ( T
= ( coinductive_LNil @ A ) ) ) ) ).
% lappend_is_LNil_conv
thf(fact_3_lbutlast__LNil,axiom,
! [A: $tType] :
( ( lList2370560421utlast @ A @ ( coinductive_LNil @ A ) )
= ( coinductive_LNil @ A ) ) ).
% lbutlast_LNil
thf(fact_4_llistE,axiom,
! [A: $tType,Y: coinductive_llist @ A] :
( ( Y
!= ( coinductive_LNil @ A ) )
=> ~ ! [X21: A,X22: coinductive_llist @ A] :
( Y
!= ( coinductive_LCons @ A @ X21 @ X22 ) ) ) ).
% llistE
thf(fact_5_lbutlast__LCons,axiom,
! [A: $tType,R: coinductive_llist @ A,A2: set @ A,A3: A] :
( ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( ( R
= ( coinductive_LNil @ A ) )
=> ( ( lList2370560421utlast @ A @ ( coinductive_LCons @ A @ A3 @ R ) )
= ( coinductive_LNil @ A ) ) )
& ( ( R
!= ( coinductive_LNil @ A ) )
=> ( ( lList2370560421utlast @ A @ ( coinductive_LCons @ A @ A3 @ R ) )
= ( coinductive_LCons @ A @ A3 @ ( lList2370560421utlast @ A @ R ) ) ) ) ) ) ).
% lbutlast_LCons
thf(fact_6_lappend__code_I1_J,axiom,
! [A: $tType,Ys: coinductive_llist @ A] :
( ( coinductive_lappend @ A @ ( coinductive_LNil @ A ) @ Ys )
= Ys ) ).
% lappend_code(1)
thf(fact_7_lappend__LNil2,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( coinductive_lappend @ A @ Xs @ ( coinductive_LNil @ A ) )
= Xs ) ).
% lappend_LNil2
thf(fact_8_LNil__eq__lappend__iff,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( ( coinductive_LNil @ A )
= ( coinductive_lappend @ A @ Xs @ Ys ) )
= ( ( Xs
= ( coinductive_LNil @ A ) )
& ( Ys
= ( coinductive_LNil @ A ) ) ) ) ).
% LNil_eq_lappend_iff
thf(fact_9_lappend__eq__LNil__iff,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( ( coinductive_lappend @ A @ Xs @ Ys )
= ( coinductive_LNil @ A ) )
= ( ( Xs
= ( coinductive_LNil @ A ) )
& ( Ys
= ( coinductive_LNil @ A ) ) ) ) ).
% lappend_eq_LNil_iff
thf(fact_10_lappend__code_I2_J,axiom,
! [A: $tType,Xa: A,X: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_lappend @ A @ ( coinductive_LCons @ A @ Xa @ X ) @ Ys )
= ( coinductive_LCons @ A @ Xa @ ( coinductive_lappend @ A @ X @ Ys ) ) ) ).
% lappend_code(2)
thf(fact_11_same__lappend__eq,axiom,
! [A: $tType,R: coinductive_llist @ A,A2: set @ A,S: coinductive_llist @ A,T: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( ( coinductive_lappend @ A @ R @ S )
= ( coinductive_lappend @ A @ R @ T ) )
= ( S = T ) ) ) ).
% same_lappend_eq
thf(fact_12_lapp__fin__fin__iff,axiom,
! [A: $tType,R: coinductive_llist @ A,S: coinductive_llist @ A,A2: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ ( coinductive_lappend @ A @ R @ S ) @ ( lList2236698231inlsts @ A @ A2 ) )
= ( ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ A2 ) )
& ( member @ ( coinductive_llist @ A ) @ S @ ( lList2236698231inlsts @ A @ A2 ) ) ) ) ).
% lapp_fin_fin_iff
thf(fact_13_lrev__induct,axiom,
! [A: $tType,Xs: coinductive_llist @ A,A2: set @ A,P: ( coinductive_llist @ A ) > $o] :
( ( member @ ( coinductive_llist @ A ) @ Xs @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( P @ ( coinductive_LNil @ A ) )
=> ( ! [X2: A,Xs2: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ Xs2 @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( P @ Xs2 )
=> ( ( member @ A @ X2 @ A2 )
=> ( P @ ( coinductive_lappend @ A @ Xs2 @ ( coinductive_LCons @ A @ X2 @ ( coinductive_LNil @ A ) ) ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% lrev_induct
thf(fact_14_finlsts__rev__cases,axiom,
! [A: $tType,T: coinductive_llist @ A,A2: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ T @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( T
!= ( coinductive_LNil @ A ) )
=> ~ ! [A4: A,L: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ L @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( member @ A @ A4 @ A2 )
=> ( T
!= ( coinductive_lappend @ A @ L @ ( coinductive_LCons @ A @ A4 @ ( coinductive_LNil @ A ) ) ) ) ) ) ) ) ).
% finlsts_rev_cases
thf(fact_15_lappend__snocL1__conv__LCons2,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Y: A,Ys: coinductive_llist @ A] :
( ( coinductive_lappend @ A @ ( coinductive_lappend @ A @ Xs @ ( coinductive_LCons @ A @ Y @ ( coinductive_LNil @ A ) ) ) @ Ys )
= ( coinductive_lappend @ A @ Xs @ ( coinductive_LCons @ A @ Y @ Ys ) ) ) ).
% lappend_snocL1_conv_LCons2
thf(fact_16_llist_Oinject,axiom,
! [A: $tType,X212: A,X222: coinductive_llist @ A,Y21: A,Y22: coinductive_llist @ A] :
( ( ( coinductive_LCons @ A @ X212 @ X222 )
= ( coinductive_LCons @ A @ Y21 @ Y22 ) )
= ( ( X212 = Y21 )
& ( X222 = Y22 ) ) ) ).
% llist.inject
thf(fact_17_lappend__assoc,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A] :
( ( coinductive_lappend @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) @ Zs )
= ( coinductive_lappend @ A @ Xs @ ( coinductive_lappend @ A @ Ys @ Zs ) ) ) ).
% lappend_assoc
thf(fact_18_finlsts_OLCons__fin,axiom,
! [A: $tType,L2: coinductive_llist @ A,A2: set @ A,A3: A] :
( ( member @ ( coinductive_llist @ A ) @ L2 @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( member @ A @ A3 @ A2 )
=> ( member @ ( coinductive_llist @ A ) @ ( coinductive_LCons @ A @ A3 @ L2 ) @ ( lList2236698231inlsts @ A @ A2 ) ) ) ) ).
% finlsts.LCons_fin
thf(fact_19_finlsts_OLNil__fin,axiom,
! [A: $tType,A2: set @ A] : ( member @ ( coinductive_llist @ A ) @ ( coinductive_LNil @ A ) @ ( lList2236698231inlsts @ A @ A2 ) ) ).
% finlsts.LNil_fin
thf(fact_20_lapp__fin__fin__lemma,axiom,
! [A: $tType,R: coinductive_llist @ A,S: coinductive_llist @ A,A2: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ ( coinductive_lappend @ A @ R @ S ) @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ A2 ) ) ) ).
% lapp_fin_fin_lemma
thf(fact_21_lappfin__finT,axiom,
! [A: $tType,S: coinductive_llist @ A,A2: set @ A,T: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ S @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( member @ ( coinductive_llist @ A ) @ T @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( member @ ( coinductive_llist @ A ) @ ( coinductive_lappend @ A @ S @ T ) @ ( lList2236698231inlsts @ A @ A2 ) ) ) ) ).
% lappfin_finT
thf(fact_22_neq__LNil__conv,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( Xs
!= ( coinductive_LNil @ A ) )
= ( ? [X3: A,Xs3: coinductive_llist @ A] :
( Xs
= ( coinductive_LCons @ A @ X3 @ Xs3 ) ) ) ) ).
% neq_LNil_conv
thf(fact_23_llist_Odistinct_I1_J,axiom,
! [A: $tType,X212: A,X222: coinductive_llist @ A] :
( ( coinductive_LNil @ A )
!= ( coinductive_LCons @ A @ X212 @ X222 ) ) ).
% llist.distinct(1)
thf(fact_24_lappend__LNil__LNil,axiom,
! [A: $tType] :
( ( coinductive_lappend @ A @ ( coinductive_LNil @ A ) @ ( coinductive_LNil @ A ) )
= ( coinductive_LNil @ A ) ) ).
% lappend_LNil_LNil
thf(fact_25_finlsts_Oinducts,axiom,
! [A: $tType,X: coinductive_llist @ A,A2: set @ A,P: ( coinductive_llist @ A ) > $o] :
( ( member @ ( coinductive_llist @ A ) @ X @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( P @ ( coinductive_LNil @ A ) )
=> ( ! [L: coinductive_llist @ A,A4: A] :
( ( member @ ( coinductive_llist @ A ) @ L @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( P @ L )
=> ( ( member @ A @ A4 @ A2 )
=> ( P @ ( coinductive_LCons @ A @ A4 @ L ) ) ) ) )
=> ( P @ X ) ) ) ) ).
% finlsts.inducts
thf(fact_26_finlsts__induct,axiom,
! [A: $tType,X: coinductive_llist @ A,A2: set @ A,P: ( coinductive_llist @ A ) > $o] :
( ( member @ ( coinductive_llist @ A ) @ X @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ! [L: coinductive_llist @ A] :
( ( L
= ( coinductive_LNil @ A ) )
=> ( P @ L ) )
=> ( ! [A4: A,L: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ L @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( P @ L )
=> ( ( member @ A @ A4 @ A2 )
=> ( P @ ( coinductive_LCons @ A @ A4 @ L ) ) ) ) )
=> ( P @ X ) ) ) ) ).
% finlsts_induct
thf(fact_27_finlsts_Osimps,axiom,
! [A: $tType,A3: coinductive_llist @ A,A2: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ A3 @ ( lList2236698231inlsts @ A @ A2 ) )
= ( ( A3
= ( coinductive_LNil @ A ) )
| ? [L3: coinductive_llist @ A,A5: A] :
( ( A3
= ( coinductive_LCons @ A @ A5 @ L3 ) )
& ( member @ ( coinductive_llist @ A ) @ L3 @ ( lList2236698231inlsts @ A @ A2 ) )
& ( member @ A @ A5 @ A2 ) ) ) ) ).
% finlsts.simps
thf(fact_28_finlsts_Ocases,axiom,
! [A: $tType,A3: coinductive_llist @ A,A2: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ A3 @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( A3
!= ( coinductive_LNil @ A ) )
=> ~ ! [L: coinductive_llist @ A,A4: A] :
( ( A3
= ( coinductive_LCons @ A @ A4 @ L ) )
=> ( ( member @ ( coinductive_llist @ A ) @ L @ ( lList2236698231inlsts @ A @ A2 ) )
=> ~ ( member @ A @ A4 @ A2 ) ) ) ) ) ).
% finlsts.cases
thf(fact_29_llast__snoc,axiom,
! [A: $tType,Xs: coinductive_llist @ A,A2: set @ A,X: A] :
( ( member @ ( coinductive_llist @ A ) @ Xs @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( lList2170638824_llast @ A @ ( coinductive_lappend @ A @ Xs @ ( coinductive_LCons @ A @ X @ ( coinductive_LNil @ A ) ) ) )
= X ) ) ).
% llast_snoc
thf(fact_30_LList2__Mirabelle__hamjzmohle_Ollast__LCons,axiom,
! [B: $tType,R: coinductive_llist @ B,A2: set @ B,A3: B] :
( ( member @ ( coinductive_llist @ B ) @ R @ ( lList2236698231inlsts @ B @ A2 ) )
=> ( ( ( R
= ( coinductive_LNil @ B ) )
=> ( ( lList2170638824_llast @ B @ ( coinductive_LCons @ B @ A3 @ R ) )
= A3 ) )
& ( ( R
!= ( coinductive_LNil @ B ) )
=> ( ( lList2170638824_llast @ B @ ( coinductive_LCons @ B @ A3 @ R ) )
= ( lList2170638824_llast @ B @ R ) ) ) ) ) ).
% LList2_Mirabelle_hamjzmohle.llast_LCons
thf(fact_31_lrev__LCons,axiom,
! [A: $tType,R: coinductive_llist @ A,A2: set @ A,A3: A] :
( ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( lList2281150353e_lrev @ A @ ( coinductive_LCons @ A @ A3 @ R ) )
= ( coinductive_lappend @ A @ ( lList2281150353e_lrev @ A @ R ) @ ( coinductive_LCons @ A @ A3 @ ( coinductive_LNil @ A ) ) ) ) ) ).
% lrev_LCons
thf(fact_32_fpslsts__iff,axiom,
! [A: $tType,S: coinductive_llist @ A,A2: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ S @ ( lList22096119349pslsts @ A @ A2 ) )
= ( ( member @ ( coinductive_llist @ A ) @ S @ ( lList2236698231inlsts @ A @ A2 ) )
& ( S
!= ( coinductive_LNil @ A ) ) ) ) ).
% fpslsts_iff
thf(fact_33_Coinductive__List_Ofinite__lprefix__nitpick__simps_I3_J,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Y: A,Ys: coinductive_llist @ A] :
( ( coindu328551480prefix @ A @ Xs @ ( coinductive_LCons @ A @ Y @ Ys ) )
= ( ( Xs
= ( coinductive_LNil @ A ) )
| ? [Xs3: coinductive_llist @ A] :
( ( Xs
= ( coinductive_LCons @ A @ Y @ Xs3 ) )
& ( coindu328551480prefix @ A @ Xs3 @ Ys ) ) ) ) ).
% Coinductive_List.finite_lprefix_nitpick_simps(3)
thf(fact_34_lstrict__prefix__code_I2_J,axiom,
! [B: $tType,Y: B,Ys: coinductive_llist @ B] : ( coindu1478340336prefix @ B @ ( coinductive_LNil @ B ) @ ( coinductive_LCons @ B @ Y @ Ys ) ) ).
% lstrict_prefix_code(2)
thf(fact_35_lstrict__prefix__code_I3_J,axiom,
! [B: $tType,X: B,Xs: coinductive_llist @ B] :
~ ( coindu1478340336prefix @ B @ ( coinductive_LCons @ B @ X @ Xs ) @ ( coinductive_LNil @ B ) ) ).
% lstrict_prefix_code(3)
thf(fact_36_lbutlast__lapp__llast,axiom,
! [A: $tType,L2: coinductive_llist @ A,A2: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ L2 @ ( lList22096119349pslsts @ A @ A2 ) )
=> ( L2
= ( coinductive_lappend @ A @ ( lList2370560421utlast @ A @ L2 ) @ ( coinductive_LCons @ A @ ( lList2170638824_llast @ A @ L2 ) @ ( coinductive_LNil @ A ) ) ) ) ) ).
% lbutlast_lapp_llast
thf(fact_37_finlsts__rec__LCons__def,axiom,
! [B: $tType,A: $tType,F: ( coinductive_llist @ A ) > B,C: B,D: A > ( coinductive_llist @ A ) > B > B,R: coinductive_llist @ A,A2: set @ A,A3: A] :
( ( F
= ( lList21916056377ts_rec @ B @ A @ C @ D ) )
=> ( ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( F @ ( coinductive_LCons @ A @ A3 @ R ) )
= ( D @ A3 @ R @ ( F @ R ) ) ) ) ) ).
% finlsts_rec_LCons_def
thf(fact_38_finlsts__rec__LCons,axiom,
! [B: $tType,A: $tType,R: coinductive_llist @ A,A2: set @ A,C: B,D: A > ( coinductive_llist @ A ) > B > B,A3: A] :
( ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( lList21916056377ts_rec @ B @ A @ C @ D @ ( coinductive_LCons @ A @ A3 @ R ) )
= ( D @ A3 @ R @ ( lList21916056377ts_rec @ B @ A @ C @ D @ R ) ) ) ) ).
% finlsts_rec_LCons
thf(fact_39_lstrict__prefix__code_I4_J,axiom,
! [B: $tType,X: B,Xs: coinductive_llist @ B,Y: B,Ys: coinductive_llist @ B] :
( ( coindu1478340336prefix @ B @ ( coinductive_LCons @ B @ X @ Xs ) @ ( coinductive_LCons @ B @ Y @ Ys ) )
= ( ( X = Y )
& ( coindu1478340336prefix @ B @ Xs @ Ys ) ) ) ).
% lstrict_prefix_code(4)
thf(fact_40_lrev__LNil,axiom,
! [A: $tType] :
( ( lList2281150353e_lrev @ A @ ( coinductive_LNil @ A ) )
= ( coinductive_LNil @ A ) ) ).
% lrev_LNil
thf(fact_41_lstrict__prefix__code_I1_J,axiom,
! [A: $tType] :
~ ( coindu1478340336prefix @ A @ ( coinductive_LNil @ A ) @ ( coinductive_LNil @ A ) ) ).
% lstrict_prefix_code(1)
thf(fact_42_lrevT,axiom,
! [A: $tType,Xs: coinductive_llist @ A,A2: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ Xs @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( member @ ( coinductive_llist @ A ) @ ( lList2281150353e_lrev @ A @ Xs ) @ ( lList2236698231inlsts @ A @ A2 ) ) ) ).
% lrevT
thf(fact_43_llist__less__induct,axiom,
! [A: $tType,P: ( coinductive_llist @ A ) > $o,Xs: coinductive_llist @ A] :
( ! [Xs2: coinductive_llist @ A] :
( ! [Ys2: coinductive_llist @ A] :
( ( coindu1478340336prefix @ A @ Ys2 @ Xs2 )
=> ( P @ Ys2 ) )
=> ( P @ Xs2 ) )
=> ( P @ Xs ) ) ).
% llist_less_induct
thf(fact_44_finlsts__rec__LNil,axiom,
! [B: $tType,A: $tType,C: A,D: B > ( coinductive_llist @ B ) > A > A] :
( ( lList21916056377ts_rec @ A @ B @ C @ D @ ( coinductive_LNil @ B ) )
= C ) ).
% finlsts_rec_LNil
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,A2: set @ A] :
( ( collect @ A
@ ^ [X3: A] : ( member @ A @ X3 @ A2 ) )
= A2 ) ).
% 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_finlsts__rec__LNil__def,axiom,
! [A: $tType,B: $tType,F: ( coinductive_llist @ A ) > B,C: B,D: A > ( coinductive_llist @ A ) > B > B] :
( ( F
= ( lList21916056377ts_rec @ B @ A @ C @ D ) )
=> ( ( F @ ( coinductive_LNil @ A ) )
= C ) ) ).
% finlsts_rec_LNil_def
thf(fact_50_Coinductive__List_Ofinite__lprefix__nitpick__simps_I2_J,axiom,
! [A: $tType,Xs: coinductive_llist @ A] : ( coindu328551480prefix @ A @ ( coinductive_LNil @ A ) @ Xs ) ).
% Coinductive_List.finite_lprefix_nitpick_simps(2)
thf(fact_51_Coinductive__List_Ofinite__lprefix__nitpick__simps_I1_J,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( coindu328551480prefix @ A @ Xs @ ( coinductive_LNil @ A ) )
= ( Xs
= ( coinductive_LNil @ A ) ) ) ).
% Coinductive_List.finite_lprefix_nitpick_simps(1)
thf(fact_52_fps__induct,axiom,
! [A: $tType,L2: coinductive_llist @ A,A2: set @ A,P: ( coinductive_llist @ A ) > $o] :
( ( member @ ( coinductive_llist @ A ) @ L2 @ ( lList22096119349pslsts @ A @ A2 ) )
=> ( ! [A4: A] :
( ( member @ A @ A4 @ A2 )
=> ( P @ ( coinductive_LCons @ A @ A4 @ ( coinductive_LNil @ A ) ) ) )
=> ( ! [A4: A,L: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ L @ ( lList22096119349pslsts @ A @ A2 ) )
=> ( ( P @ L )
=> ( ( member @ A @ A4 @ A2 )
=> ( P @ ( coinductive_LCons @ A @ A4 @ L ) ) ) ) )
=> ( P @ L2 ) ) ) ) ).
% fps_induct
thf(fact_53_fpslsts__cases,axiom,
! [A: $tType,R: coinductive_llist @ A,A2: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ R @ ( lList22096119349pslsts @ A @ A2 ) )
=> ~ ! [A4: A,Rs: coinductive_llist @ A] :
( ( R
= ( coinductive_LCons @ A @ A4 @ Rs ) )
=> ( ( member @ A @ A4 @ A2 )
=> ~ ( member @ ( coinductive_llist @ A ) @ Rs @ ( lList2236698231inlsts @ A @ A2 ) ) ) ) ) ).
% fpslsts_cases
thf(fact_54_lrev__lappend,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ Xs @ ( lList2236698231inlsts @ A @ ( top_top @ ( set @ A ) ) ) )
=> ( ( member @ ( coinductive_llist @ A ) @ Ys @ ( lList2236698231inlsts @ A @ ( top_top @ ( set @ A ) ) ) )
=> ( ( lList2281150353e_lrev @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
= ( coinductive_lappend @ A @ ( lList2281150353e_lrev @ A @ Ys ) @ ( lList2281150353e_lrev @ A @ Xs ) ) ) ) ) ).
% lrev_lappend
thf(fact_55_LNil__is__lrev__conv,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ Xs @ ( lList2236698231inlsts @ A @ ( top_top @ ( set @ A ) ) ) )
=> ( ( ( coinductive_LNil @ A )
= ( lList2281150353e_lrev @ A @ Xs ) )
= ( Xs
= ( coinductive_LNil @ A ) ) ) ) ).
% LNil_is_lrev_conv
thf(fact_56_lrev__is__LNil__conv,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ Xs @ ( lList2236698231inlsts @ A @ ( top_top @ ( set @ A ) ) ) )
=> ( ( ( lList2281150353e_lrev @ A @ Xs )
= ( coinductive_LNil @ A ) )
= ( Xs
= ( coinductive_LNil @ A ) ) ) ) ).
% lrev_is_LNil_conv
thf(fact_57_llast__singleton,axiom,
! [A: $tType,X: A] :
( ( coinductive_llast @ A @ ( coinductive_LCons @ A @ X @ ( coinductive_LNil @ A ) ) )
= X ) ).
% llast_singleton
thf(fact_58_alllstsp_Ocases,axiom,
! [A: $tType,A2: A > $o,A3: coinductive_llist @ A] :
( ( lList21511617539llstsp @ A @ A2 @ A3 )
=> ( ( A3
!= ( coinductive_LNil @ A ) )
=> ~ ! [L: coinductive_llist @ A,A4: A] :
( ( A3
= ( coinductive_LCons @ A @ A4 @ L ) )
=> ( ( lList21511617539llstsp @ A @ A2 @ L )
=> ~ ( A2 @ A4 ) ) ) ) ) ).
% alllstsp.cases
thf(fact_59_alllstsp_Osimps,axiom,
! [A: $tType] :
( ( lList21511617539llstsp @ A )
= ( ^ [A6: A > $o,A5: coinductive_llist @ A] :
( ( A5
= ( coinductive_LNil @ A ) )
| ? [L3: coinductive_llist @ A,B2: A] :
( ( A5
= ( coinductive_LCons @ A @ B2 @ L3 ) )
& ( lList21511617539llstsp @ A @ A6 @ L3 )
& ( A6 @ B2 ) ) ) ) ) ).
% alllstsp.simps
thf(fact_60_finlstsp_Ocases,axiom,
! [A: $tType,A2: A > $o,A3: coinductive_llist @ A] :
( ( lList2860480441nlstsp @ A @ A2 @ A3 )
=> ( ( A3
!= ( coinductive_LNil @ A ) )
=> ~ ! [L: coinductive_llist @ A,A4: A] :
( ( A3
= ( coinductive_LCons @ A @ A4 @ L ) )
=> ( ( lList2860480441nlstsp @ A @ A2 @ L )
=> ~ ( A2 @ A4 ) ) ) ) ) ).
% finlstsp.cases
thf(fact_61_finlstsp_Osimps,axiom,
! [A: $tType] :
( ( lList2860480441nlstsp @ A )
= ( ^ [A6: A > $o,A5: coinductive_llist @ A] :
( ( A5
= ( coinductive_LNil @ A ) )
| ? [L3: coinductive_llist @ A,B2: A] :
( ( A5
= ( coinductive_LCons @ A @ B2 @ L3 ) )
& ( lList2860480441nlstsp @ A @ A6 @ L3 )
& ( A6 @ B2 ) ) ) ) ) ).
% finlstsp.simps
thf(fact_62_llast__LCons2,axiom,
! [A: $tType,X: A,Y: A,Xs: coinductive_llist @ A] :
( ( coinductive_llast @ A @ ( coinductive_LCons @ A @ X @ ( coinductive_LCons @ A @ Y @ Xs ) ) )
= ( coinductive_llast @ A @ ( coinductive_LCons @ A @ Y @ Xs ) ) ) ).
% llast_LCons2
thf(fact_63_lrev__lrev__ident,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ Xs @ ( lList2236698231inlsts @ A @ ( top_top @ ( set @ A ) ) ) )
=> ( ( lList2281150353e_lrev @ A @ ( lList2281150353e_lrev @ A @ Xs ) )
= Xs ) ) ).
% lrev_lrev_ident
thf(fact_64_lrev__is__lrev__conv,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ Xs @ ( lList2236698231inlsts @ A @ ( top_top @ ( set @ A ) ) ) )
=> ( ( member @ ( coinductive_llist @ A ) @ Ys @ ( lList2236698231inlsts @ A @ ( top_top @ ( set @ A ) ) ) )
=> ( ( ( lList2281150353e_lrev @ A @ Xs )
= ( lList2281150353e_lrev @ A @ Ys ) )
= ( Xs = Ys ) ) ) ) ).
% lrev_is_lrev_conv
thf(fact_65_finT__simp,axiom,
! [A: $tType,R: coinductive_llist @ A,A2: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% finT_simp
thf(fact_66_fin__finite,axiom,
! [A: $tType,R: coinductive_llist @ A,A2: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% fin_finite
thf(fact_67_finlstsp_OLCons__fin,axiom,
! [A: $tType,A2: A > $o,L2: coinductive_llist @ A,A3: A] :
( ( lList2860480441nlstsp @ A @ A2 @ L2 )
=> ( ( A2 @ A3 )
=> ( lList2860480441nlstsp @ A @ A2 @ ( coinductive_LCons @ A @ A3 @ L2 ) ) ) ) ).
% finlstsp.LCons_fin
thf(fact_68_alllstsp_OLCons__all,axiom,
! [A: $tType,A2: A > $o,L2: coinductive_llist @ A,A3: A] :
( ( lList21511617539llstsp @ A @ A2 @ L2 )
=> ( ( A2 @ A3 )
=> ( lList21511617539llstsp @ A @ A2 @ ( coinductive_LCons @ A @ A3 @ L2 ) ) ) ) ).
% alllstsp.LCons_all
thf(fact_69_finlstsp_OLNil__fin,axiom,
! [A: $tType,A2: A > $o] : ( lList2860480441nlstsp @ A @ A2 @ ( coinductive_LNil @ A ) ) ).
% finlstsp.LNil_fin
thf(fact_70_alllstsp_OLNil__all,axiom,
! [A: $tType,A2: A > $o] : ( lList21511617539llstsp @ A @ A2 @ ( coinductive_LNil @ A ) ) ).
% alllstsp.LNil_all
thf(fact_71_alllstsp_Ocoinduct,axiom,
! [A: $tType,X4: ( coinductive_llist @ A ) > $o,X: coinductive_llist @ A,A2: A > $o] :
( ( X4 @ X )
=> ( ! [X2: coinductive_llist @ A] :
( ( X4 @ X2 )
=> ( ( X2
= ( coinductive_LNil @ A ) )
| ? [L4: coinductive_llist @ A,A7: A] :
( ( X2
= ( coinductive_LCons @ A @ A7 @ L4 ) )
& ( ( X4 @ L4 )
| ( lList21511617539llstsp @ A @ A2 @ L4 ) )
& ( A2 @ A7 ) ) ) )
=> ( lList21511617539llstsp @ A @ A2 @ X ) ) ) ).
% alllstsp.coinduct
thf(fact_72_finlstsp_Oinducts,axiom,
! [A: $tType,A2: A > $o,X: coinductive_llist @ A,P: ( coinductive_llist @ A ) > $o] :
( ( lList2860480441nlstsp @ A @ A2 @ X )
=> ( ( P @ ( coinductive_LNil @ A ) )
=> ( ! [L: coinductive_llist @ A,A4: A] :
( ( lList2860480441nlstsp @ A @ A2 @ L )
=> ( ( P @ L )
=> ( ( A2 @ A4 )
=> ( P @ ( coinductive_LCons @ A @ A4 @ L ) ) ) ) )
=> ( P @ X ) ) ) ) ).
% finlstsp.inducts
thf(fact_73_UNIV__I,axiom,
! [A: $tType,X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ).
% UNIV_I
thf(fact_74_iso__tuple__UNIV__I,axiom,
! [A: $tType,X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ).
% iso_tuple_UNIV_I
thf(fact_75_top__apply,axiom,
! [C2: $tType,D2: $tType] :
( ( top @ C2 @ ( type2 @ C2 ) )
=> ( ( top_top @ ( D2 > C2 ) )
= ( ^ [X3: D2] : ( top_top @ C2 ) ) ) ) ).
% top_apply
thf(fact_76_poslsts__UNIV,axiom,
! [A: $tType,S: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ S @ ( lList21148268032oslsts @ A @ ( top_top @ ( set @ A ) ) ) )
= ( S
!= ( coinductive_LNil @ A ) ) ) ).
% poslsts_UNIV
thf(fact_77_ltake__fin,axiom,
! [A: $tType,R: coinductive_llist @ A,I: nat] : ( member @ ( coinductive_llist @ A ) @ ( lList22119844313_ltake @ A @ R @ I ) @ ( lList2236698231inlsts @ A @ ( top_top @ ( set @ A ) ) ) ) ).
% ltake_fin
thf(fact_78_ldrop__fin__iffT,axiom,
! [A: $tType,T: coinductive_llist @ A,I: nat] :
( ( member @ ( coinductive_llist @ A ) @ ( lList2508575361_ldrop @ A @ T @ I ) @ ( lList2236698231inlsts @ A @ ( top_top @ ( set @ A ) ) ) )
= ( member @ ( coinductive_llist @ A ) @ T @ ( lList2236698231inlsts @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% ldrop_fin_iffT
thf(fact_79_llast__lappend__LCons,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Y: A,Ys: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ Xs )
=> ( ( coinductive_llast @ A @ ( coinductive_lappend @ A @ Xs @ ( coinductive_LCons @ A @ Y @ Ys ) ) )
= ( coinductive_llast @ A @ ( coinductive_LCons @ A @ Y @ Ys ) ) ) ) ).
% llast_lappend_LCons
thf(fact_80_lfinite__code_I2_J,axiom,
! [B: $tType,X: B,Xs: coinductive_llist @ B] :
( ( coinductive_lfinite @ B @ ( coinductive_LCons @ B @ X @ Xs ) )
= ( coinductive_lfinite @ B @ Xs ) ) ).
% lfinite_code(2)
thf(fact_81_lfinite__LCons,axiom,
! [A: $tType,X: A,Xs: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ ( coinductive_LCons @ A @ X @ Xs ) )
= ( coinductive_lfinite @ A @ Xs ) ) ).
% lfinite_LCons
thf(fact_82_lfinite__code_I1_J,axiom,
! [A: $tType] : ( coinductive_lfinite @ A @ ( coinductive_LNil @ A ) ) ).
% lfinite_code(1)
thf(fact_83_lfinite__lappend,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
= ( ( coinductive_lfinite @ A @ Xs )
& ( coinductive_lfinite @ A @ Ys ) ) ) ).
% lfinite_lappend
thf(fact_84_LList2__Mirabelle__hamjzmohle_Oldrop__LNil,axiom,
! [A: $tType,I: nat] :
( ( lList2508575361_ldrop @ A @ ( coinductive_LNil @ A ) @ I )
= ( coinductive_LNil @ A ) ) ).
% LList2_Mirabelle_hamjzmohle.ldrop_LNil
thf(fact_85_LList2__Mirabelle__hamjzmohle_Oltake__LNil,axiom,
! [A: $tType,I: nat] :
( ( lList22119844313_ltake @ A @ ( coinductive_LNil @ A ) @ I )
= ( coinductive_LNil @ A ) ) ).
% LList2_Mirabelle_hamjzmohle.ltake_LNil
thf(fact_86_top__set__def,axiom,
! [A: $tType] :
( ( top_top @ ( set @ A ) )
= ( collect @ A @ ( top_top @ ( A > $o ) ) ) ) ).
% top_set_def
thf(fact_87_ltake__ldrop__id,axiom,
! [A: $tType,X: coinductive_llist @ A,I: nat] :
( ( coinductive_lappend @ A @ ( lList22119844313_ltake @ A @ X @ I ) @ ( lList2508575361_ldrop @ A @ X @ I ) )
= X ) ).
% ltake_ldrop_id
thf(fact_88_lfinite__LConsI,axiom,
! [A: $tType,Xs: coinductive_llist @ A,X: A] :
( ( coinductive_lfinite @ A @ Xs )
=> ( coinductive_lfinite @ A @ ( coinductive_LCons @ A @ X @ Xs ) ) ) ).
% lfinite_LConsI
thf(fact_89_lfinite__LNil,axiom,
! [A: $tType] : ( coinductive_lfinite @ A @ ( coinductive_LNil @ A ) ) ).
% lfinite_LNil
thf(fact_90_lappend__inf,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ~ ( coinductive_lfinite @ A @ Xs )
=> ( ( coinductive_lappend @ A @ Xs @ Ys )
= Xs ) ) ).
% lappend_inf
thf(fact_91_lstrict__prefix__lfinite1,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coindu1478340336prefix @ A @ Xs @ Ys )
=> ( coinductive_lfinite @ A @ Xs ) ) ).
% lstrict_prefix_lfinite1
thf(fact_92_drop__nonLNil,axiom,
! [A: $tType,T: coinductive_llist @ A,I: nat] :
( ( ( lList2508575361_ldrop @ A @ T @ I )
!= ( coinductive_LNil @ A ) )
=> ( T
!= ( coinductive_LNil @ A ) ) ) ).
% drop_nonLNil
thf(fact_93_ldrop__finT,axiom,
! [A: $tType,T: coinductive_llist @ A,A2: set @ A,I: nat] :
( ( member @ ( coinductive_llist @ A ) @ T @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( member @ ( coinductive_llist @ A ) @ ( lList2508575361_ldrop @ A @ T @ I ) @ ( lList2236698231inlsts @ A @ A2 ) ) ) ).
% ldrop_finT
thf(fact_94_lfinite_Ocases,axiom,
! [A: $tType,A3: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ A3 )
=> ( ( A3
!= ( coinductive_LNil @ A ) )
=> ~ ! [Xs2: coinductive_llist @ A] :
( ? [X2: A] :
( A3
= ( coinductive_LCons @ A @ X2 @ Xs2 ) )
=> ~ ( coinductive_lfinite @ A @ Xs2 ) ) ) ) ).
% lfinite.cases
thf(fact_95_lfinite_Osimps,axiom,
! [A: $tType] :
( ( coinductive_lfinite @ A )
= ( ^ [A5: coinductive_llist @ A] :
( ( A5
= ( coinductive_LNil @ A ) )
| ? [Xs4: coinductive_llist @ A,X3: A] :
( ( A5
= ( coinductive_LCons @ A @ X3 @ Xs4 ) )
& ( coinductive_lfinite @ A @ Xs4 ) ) ) ) ) ).
% lfinite.simps
thf(fact_96_lfinite_Oinducts,axiom,
! [A: $tType,X: coinductive_llist @ A,P: ( coinductive_llist @ A ) > $o] :
( ( coinductive_lfinite @ A @ X )
=> ( ( P @ ( coinductive_LNil @ A ) )
=> ( ! [Xs2: coinductive_llist @ A,X2: A] :
( ( coinductive_lfinite @ A @ Xs2 )
=> ( ( P @ Xs2 )
=> ( P @ ( coinductive_LCons @ A @ X2 @ Xs2 ) ) ) )
=> ( P @ X ) ) ) ) ).
% lfinite.inducts
thf(fact_97_lfinite__rev__induct,axiom,
! [A: $tType,Xs: coinductive_llist @ A,P: ( coinductive_llist @ A ) > $o] :
( ( coinductive_lfinite @ A @ Xs )
=> ( ( P @ ( coinductive_LNil @ A ) )
=> ( ! [X2: A,Xs2: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ Xs2 )
=> ( ( P @ Xs2 )
=> ( P @ ( coinductive_lappend @ A @ Xs2 @ ( coinductive_LCons @ A @ X2 @ ( coinductive_LNil @ A ) ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% lfinite_rev_induct
thf(fact_98_llimit__induct,axiom,
! [A: $tType,P: ( coinductive_llist @ A ) > $o,Xs: coinductive_llist @ A] :
( ( P @ ( coinductive_LNil @ A ) )
=> ( ! [X2: A,Xs2: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ Xs2 )
=> ( ( P @ Xs2 )
=> ( P @ ( coinductive_LCons @ A @ X2 @ Xs2 ) ) ) )
=> ( ( ! [Ys2: coinductive_llist @ A] :
( ( coindu1478340336prefix @ A @ Ys2 @ Xs )
=> ( P @ Ys2 ) )
=> ( P @ Xs ) )
=> ( P @ Xs ) ) ) ) ).
% llimit_induct
thf(fact_99_UNIV__witness,axiom,
! [A: $tType] :
? [X2: A] : ( member @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ).
% UNIV_witness
thf(fact_100_UNIV__eq__I,axiom,
! [A: $tType,A2: set @ A] :
( ! [X2: A] : ( member @ A @ X2 @ A2 )
=> ( ( top_top @ ( set @ A ) )
= A2 ) ) ).
% UNIV_eq_I
thf(fact_101_ltake__lappend__llength,axiom,
! [A: $tType,R: coinductive_llist @ A,A2: set @ A,S: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( lList22119844313_ltake @ A @ ( coinductive_lappend @ A @ R @ S ) @ ( lList21232602520length @ A @ R ) )
= R ) ) ).
% ltake_lappend_llength
thf(fact_102_llength__drop__take,axiom,
! [A: $tType,T: coinductive_llist @ A,I: nat] :
( ( ( lList2508575361_ldrop @ A @ T @ I )
!= ( coinductive_LNil @ A ) )
=> ( ( lList21232602520length @ A @ ( lList22119844313_ltake @ A @ T @ I ) )
= I ) ) ).
% llength_drop_take
thf(fact_103_lapp__suff__llength,axiom,
! [A: $tType,R: coinductive_llist @ A,A2: set @ A,S: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( lList2508575361_ldrop @ A @ ( coinductive_lappend @ A @ R @ S ) @ ( lList21232602520length @ A @ R ) )
= S ) ) ).
% lapp_suff_llength
thf(fact_104_poslsts__iff,axiom,
! [A: $tType,S: coinductive_llist @ A,A2: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ S @ ( lList21148268032oslsts @ A @ A2 ) )
= ( ( member @ ( coinductive_llist @ A ) @ S @ ( lList2435255213lllsts @ A @ A2 ) )
& ( S
!= ( coinductive_LNil @ A ) ) ) ) ).
% poslsts_iff
thf(fact_105_take__fin,axiom,
! [A: $tType,T: coinductive_llist @ A,A2: set @ A,I: nat] :
( ( member @ ( coinductive_llist @ A ) @ T @ ( lList2435255213lllsts @ A @ A2 ) )
=> ( member @ ( coinductive_llist @ A ) @ ( lList22119844313_ltake @ A @ T @ I ) @ ( lList2236698231inlsts @ A @ A2 ) ) ) ).
% take_fin
thf(fact_106_ldrop__inf__iffT,axiom,
! [A: $tType,T: coinductive_llist @ A,I: nat] :
( ( member @ ( coinductive_llist @ A ) @ ( lList2508575361_ldrop @ A @ T @ I ) @ ( lList21612149805nflsts @ A @ ( top_top @ ( set @ A ) ) ) )
= ( member @ ( coinductive_llist @ A ) @ T @ ( lList21612149805nflsts @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% ldrop_inf_iffT
thf(fact_107_top1I,axiom,
! [A: $tType,X: A] : ( top_top @ ( A > $o ) @ X ) ).
% top1I
thf(fact_108_alllsts__UNIV,axiom,
! [A: $tType,S: coinductive_llist @ A] : ( member @ ( coinductive_llist @ A ) @ S @ ( lList2435255213lllsts @ A @ ( top_top @ ( set @ A ) ) ) ) ).
% alllsts_UNIV
thf(fact_109_LConsE,axiom,
! [A: $tType,X: A,Xs: coinductive_llist @ A,A2: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ ( coinductive_LCons @ A @ X @ Xs ) @ ( lList2435255213lllsts @ A @ A2 ) )
= ( ( member @ A @ X @ A2 )
& ( member @ ( coinductive_llist @ A ) @ Xs @ ( lList2435255213lllsts @ A @ A2 ) ) ) ) ).
% LConsE
thf(fact_110_lapp__inf,axiom,
! [A: $tType,S: coinductive_llist @ A,A2: set @ A,T: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ S @ ( lList21612149805nflsts @ A @ A2 ) )
=> ( ( coinductive_lappend @ A @ S @ T )
= S ) ) ).
% lapp_inf
thf(fact_111_notfin__inf,axiom,
! [A: $tType,X: coinductive_llist @ A] :
( ( ~ ( member @ ( coinductive_llist @ A ) @ X @ ( lList2236698231inlsts @ A @ ( top_top @ ( set @ A ) ) ) ) )
= ( member @ ( coinductive_llist @ A ) @ X @ ( lList21612149805nflsts @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% notfin_inf
thf(fact_112_notinf__fin,axiom,
! [A: $tType,X: coinductive_llist @ A] :
( ( ~ ( member @ ( coinductive_llist @ A ) @ X @ ( lList21612149805nflsts @ A @ ( top_top @ ( set @ A ) ) ) ) )
= ( member @ ( coinductive_llist @ A ) @ X @ ( lList2236698231inlsts @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% notinf_fin
thf(fact_113_llength__take,axiom,
! [A: $tType,T: coinductive_llist @ A,A2: set @ A,I: nat] :
( ( member @ ( coinductive_llist @ A ) @ T @ ( lList21612149805nflsts @ A @ A2 ) )
=> ( ( lList21232602520length @ A @ ( lList22119844313_ltake @ A @ T @ I ) )
= I ) ) ).
% llength_take
thf(fact_114_inflstsI,axiom,
! [A: $tType,X: coinductive_llist @ A,A2: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ X @ ( lList2435255213lllsts @ A @ A2 ) )
=> ( ~ ( member @ ( coinductive_llist @ A ) @ X @ ( lList2236698231inlsts @ A @ ( top_top @ ( set @ A ) ) ) )
=> ( member @ ( coinductive_llist @ A ) @ X @ ( lList21612149805nflsts @ A @ A2 ) ) ) ) ).
% inflstsI
thf(fact_115_infsubsetall,axiom,
! [A: $tType,X: coinductive_llist @ A,A2: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ X @ ( lList21612149805nflsts @ A @ A2 ) )
=> ( member @ ( coinductive_llist @ A ) @ X @ ( lList2435255213lllsts @ A @ A2 ) ) ) ).
% infsubsetall
thf(fact_116_alllstsE,axiom,
! [A: $tType,X: coinductive_llist @ A,A2: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ X @ ( lList2435255213lllsts @ A @ A2 ) )
=> ( ~ ( member @ ( coinductive_llist @ A ) @ X @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( member @ ( coinductive_llist @ A ) @ X @ ( lList21612149805nflsts @ A @ A2 ) ) ) ) ).
% alllstsE
thf(fact_117_inflstsE,axiom,
! [A: $tType,X: coinductive_llist @ A,A2: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ X @ ( lList21612149805nflsts @ A @ A2 ) )
=> ~ ( ( member @ ( coinductive_llist @ A ) @ X @ ( lList2435255213lllsts @ A @ A2 ) )
=> ( member @ ( coinductive_llist @ A ) @ X @ ( lList2236698231inlsts @ A @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% inflstsE
thf(fact_118_lapp__allT__iff,axiom,
! [A: $tType,R: coinductive_llist @ A,S: coinductive_llist @ A,A2: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ ( coinductive_lappend @ A @ R @ S ) @ ( lList2435255213lllsts @ A @ A2 ) )
= ( ( ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ A2 ) )
& ( member @ ( coinductive_llist @ A ) @ S @ ( lList2435255213lllsts @ A @ A2 ) ) )
| ( member @ ( coinductive_llist @ A ) @ R @ ( lList21612149805nflsts @ A @ A2 ) ) ) ) ).
% lapp_allT_iff
thf(fact_119_infT__simp,axiom,
! [A: $tType,R: coinductive_llist @ A,A2: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ R @ ( lList21612149805nflsts @ A @ A2 ) )
=> ( member @ ( coinductive_llist @ A ) @ R @ ( lList21612149805nflsts @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% infT_simp
thf(fact_120_inflsts__cases,axiom,
! [A: $tType,S: coinductive_llist @ A,A2: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ S @ ( lList21612149805nflsts @ A @ A2 ) )
=> ~ ! [A4: A,L: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ L @ ( lList21612149805nflsts @ A @ A2 ) )
=> ( ( member @ A @ A4 @ A2 )
=> ( S
!= ( coinductive_LCons @ A @ A4 @ L ) ) ) ) ) ).
% inflsts_cases
thf(fact_121_inflstsI2,axiom,
! [A: $tType,A3: A,A2: set @ A,T: coinductive_llist @ A] :
( ( member @ A @ A3 @ A2 )
=> ( ( member @ ( coinductive_llist @ A ) @ T @ ( lList21612149805nflsts @ A @ A2 ) )
=> ( member @ ( coinductive_llist @ A ) @ ( coinductive_LCons @ A @ A3 @ T ) @ ( lList21612149805nflsts @ A @ A2 ) ) ) ) ).
% inflstsI2
thf(fact_122_alllsts_OLCons__all,axiom,
! [A: $tType,L2: coinductive_llist @ A,A2: set @ A,A3: A] :
( ( member @ ( coinductive_llist @ A ) @ L2 @ ( lList2435255213lllsts @ A @ A2 ) )
=> ( ( member @ A @ A3 @ A2 )
=> ( member @ ( coinductive_llist @ A ) @ ( coinductive_LCons @ A @ A3 @ L2 ) @ ( lList2435255213lllsts @ A @ A2 ) ) ) ) ).
% alllsts.LCons_all
thf(fact_123_alllsts_OLNil__all,axiom,
! [A: $tType,A2: set @ A] : ( member @ ( coinductive_llist @ A ) @ ( coinductive_LNil @ A ) @ ( lList2435255213lllsts @ A @ A2 ) ) ).
% alllsts.LNil_all
thf(fact_124_lapp__all__invT,axiom,
! [A: $tType,R: coinductive_llist @ A,S: coinductive_llist @ A,A2: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ ( coinductive_lappend @ A @ R @ S ) @ ( lList2435255213lllsts @ A @ A2 ) )
=> ( member @ ( coinductive_llist @ A ) @ R @ ( lList2435255213lllsts @ A @ A2 ) ) ) ).
% lapp_all_invT
thf(fact_125_lappT,axiom,
! [A: $tType,S: coinductive_llist @ A,A2: set @ A,T: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ S @ ( lList2435255213lllsts @ A @ A2 ) )
=> ( ( member @ ( coinductive_llist @ A ) @ T @ ( lList2435255213lllsts @ A @ A2 ) )
=> ( member @ ( coinductive_llist @ A ) @ ( coinductive_lappend @ A @ S @ T ) @ ( lList2435255213lllsts @ A @ A2 ) ) ) ) ).
% lappT
thf(fact_126_finite__lemma,axiom,
! [A: $tType,X: coinductive_llist @ A,A2: set @ A,B3: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ X @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( member @ ( coinductive_llist @ A ) @ X @ ( lList2435255213lllsts @ A @ B3 ) )
=> ( member @ ( coinductive_llist @ A ) @ X @ ( lList2236698231inlsts @ A @ B3 ) ) ) ) ).
% finite_lemma
thf(fact_127_finsubsetall,axiom,
! [A: $tType,X: coinductive_llist @ A,A2: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ X @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( member @ ( coinductive_llist @ A ) @ X @ ( lList2435255213lllsts @ A @ A2 ) ) ) ).
% finsubsetall
thf(fact_128_ldrop__infT,axiom,
! [A: $tType,T: coinductive_llist @ A,A2: set @ A,I: nat] :
( ( member @ ( coinductive_llist @ A ) @ T @ ( lList21612149805nflsts @ A @ A2 ) )
=> ( member @ ( coinductive_llist @ A ) @ ( lList2508575361_ldrop @ A @ T @ I ) @ ( lList21612149805nflsts @ A @ A2 ) ) ) ).
% ldrop_infT
thf(fact_129_ldropT,axiom,
! [A: $tType,T: coinductive_llist @ A,A2: set @ A,I: nat] :
( ( member @ ( coinductive_llist @ A ) @ T @ ( lList2435255213lllsts @ A @ A2 ) )
=> ( member @ ( coinductive_llist @ A ) @ ( lList2508575361_ldrop @ A @ T @ I ) @ ( lList2435255213lllsts @ A @ A2 ) ) ) ).
% ldropT
thf(fact_130_alllsts_Ocoinduct,axiom,
! [A: $tType,X4: ( coinductive_llist @ A ) > $o,X: coinductive_llist @ A,A2: set @ A] :
( ( X4 @ X )
=> ( ! [X2: coinductive_llist @ A] :
( ( X4 @ X2 )
=> ( ( X2
= ( coinductive_LNil @ A ) )
| ? [L4: coinductive_llist @ A,A7: A] :
( ( X2
= ( coinductive_LCons @ A @ A7 @ L4 ) )
& ( ( X4 @ L4 )
| ( member @ ( coinductive_llist @ A ) @ L4 @ ( lList2435255213lllsts @ A @ A2 ) ) )
& ( member @ A @ A7 @ A2 ) ) ) )
=> ( member @ ( coinductive_llist @ A ) @ X @ ( lList2435255213lllsts @ A @ A2 ) ) ) ) ).
% alllsts.coinduct
thf(fact_131_alllsts_Osimps,axiom,
! [A: $tType,A3: coinductive_llist @ A,A2: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ A3 @ ( lList2435255213lllsts @ A @ A2 ) )
= ( ( A3
= ( coinductive_LNil @ A ) )
| ? [L3: coinductive_llist @ A,A5: A] :
( ( A3
= ( coinductive_LCons @ A @ A5 @ L3 ) )
& ( member @ ( coinductive_llist @ A ) @ L3 @ ( lList2435255213lllsts @ A @ A2 ) )
& ( member @ A @ A5 @ A2 ) ) ) ) ).
% alllsts.simps
thf(fact_132_alllsts_Ocases,axiom,
! [A: $tType,A3: coinductive_llist @ A,A2: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ A3 @ ( lList2435255213lllsts @ A @ A2 ) )
=> ( ( A3
!= ( coinductive_LNil @ A ) )
=> ~ ! [L: coinductive_llist @ A,A4: A] :
( ( A3
= ( coinductive_LCons @ A @ A4 @ L ) )
=> ( ( member @ ( coinductive_llist @ A ) @ L @ ( lList2435255213lllsts @ A @ A2 ) )
=> ~ ( member @ A @ A4 @ A2 ) ) ) ) ) ).
% alllsts.cases
thf(fact_133_fin__inf__cases,axiom,
! [A: $tType,R: coinductive_llist @ A] :
( ~ ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ ( top_top @ ( set @ A ) ) ) )
=> ( member @ ( coinductive_llist @ A ) @ R @ ( lList21612149805nflsts @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% fin_inf_cases
thf(fact_134_lapp__fin__infT,axiom,
! [A: $tType,S: coinductive_llist @ A,A2: set @ A,T: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ S @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( member @ ( coinductive_llist @ A ) @ T @ ( lList21612149805nflsts @ A @ A2 ) )
=> ( member @ ( coinductive_llist @ A ) @ ( coinductive_lappend @ A @ S @ T ) @ ( lList21612149805nflsts @ A @ A2 ) ) ) ) ).
% lapp_fin_infT
thf(fact_135_lapp__inv2T,axiom,
! [A: $tType,R: coinductive_llist @ A,S: coinductive_llist @ A,A2: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ ( coinductive_lappend @ A @ R @ S ) @ ( lList21612149805nflsts @ A @ A2 ) )
=> ( ( ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ A2 ) )
& ( member @ ( coinductive_llist @ A ) @ S @ ( lList21612149805nflsts @ A @ A2 ) ) )
| ( member @ ( coinductive_llist @ A ) @ R @ ( lList21612149805nflsts @ A @ A2 ) ) ) ) ).
% lapp_inv2T
thf(fact_136_lapp__infT,axiom,
! [A: $tType,R: coinductive_llist @ A,S: coinductive_llist @ A,A2: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ ( coinductive_lappend @ A @ R @ S ) @ ( lList21612149805nflsts @ A @ A2 ) )
= ( ( ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ A2 ) )
& ( member @ ( coinductive_llist @ A ) @ S @ ( lList21612149805nflsts @ A @ A2 ) ) )
| ( member @ ( coinductive_llist @ A ) @ R @ ( lList21612149805nflsts @ A @ A2 ) ) ) ) ).
% lapp_infT
thf(fact_137_app__invT,axiom,
! [A: $tType,R: coinductive_llist @ A,A2: set @ A,S: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( member @ ( coinductive_llist @ A ) @ ( coinductive_lappend @ A @ R @ S ) @ ( lList21612149805nflsts @ A @ A2 ) )
=> ( member @ ( coinductive_llist @ A ) @ S @ ( lList21612149805nflsts @ A @ A2 ) ) ) ) ).
% app_invT
thf(fact_138_top__empty__eq,axiom,
! [A: $tType] :
( ( top_top @ ( A > $o ) )
= ( ^ [X3: A] : ( member @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ) ) ).
% top_empty_eq
thf(fact_139_top__conj_I2_J,axiom,
! [A: $tType,P: $o,X: A] :
( ( P
& ( top_top @ ( A > $o ) @ X ) )
= P ) ).
% top_conj(2)
thf(fact_140_top__conj_I1_J,axiom,
! [A: $tType,X: A,P: $o] :
( ( ( top_top @ ( A > $o ) @ X )
& P )
= P ) ).
% top_conj(1)
thf(fact_141_inflsts__def,axiom,
! [A: $tType] :
( ( lList21612149805nflsts @ A )
= ( ^ [A6: set @ A] : ( minus_minus @ ( set @ ( coinductive_llist @ A ) ) @ ( lList2435255213lllsts @ A @ A6 ) @ ( lList2236698231inlsts @ A @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% inflsts_def
thf(fact_142_LList2__Mirabelle__hamjzmohle_Ollength__LCons,axiom,
! [A: $tType,R: coinductive_llist @ A,A2: set @ A,A3: A] :
( ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( lList21232602520length @ A @ ( coinductive_LCons @ A @ A3 @ R ) )
= ( suc @ ( lList21232602520length @ A @ R ) ) ) ) ).
% LList2_Mirabelle_hamjzmohle.llength_LCons
thf(fact_143_DiffI,axiom,
! [A: $tType,C: A,A2: set @ A,B3: set @ A] :
( ( member @ A @ C @ A2 )
=> ( ~ ( member @ A @ C @ B3 )
=> ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A2 @ B3 ) ) ) ) ).
% DiffI
thf(fact_144_Diff__iff,axiom,
! [A: $tType,C: A,A2: set @ A,B3: set @ A] :
( ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A2 @ B3 ) )
= ( ( member @ A @ C @ A2 )
& ~ ( member @ A @ C @ B3 ) ) ) ).
% Diff_iff
thf(fact_145_Diff__idemp,axiom,
! [A: $tType,A2: set @ A,B3: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ B3 ) @ B3 )
= ( minus_minus @ ( set @ A ) @ A2 @ B3 ) ) ).
% Diff_idemp
thf(fact_146_DiffE,axiom,
! [A: $tType,C: A,A2: set @ A,B3: set @ A] :
( ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A2 @ B3 ) )
=> ~ ( ( member @ A @ C @ A2 )
=> ( member @ A @ C @ B3 ) ) ) ).
% DiffE
thf(fact_147_DiffD1,axiom,
! [A: $tType,C: A,A2: set @ A,B3: set @ A] :
( ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A2 @ B3 ) )
=> ( member @ A @ C @ A2 ) ) ).
% DiffD1
thf(fact_148_DiffD2,axiom,
! [A: $tType,C: A,A2: set @ A,B3: set @ A] :
( ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A2 @ B3 ) )
=> ~ ( member @ A @ C @ B3 ) ) ).
% DiffD2
thf(fact_149_ltake__LCons__Suc,axiom,
! [A: $tType,A3: A,L2: coinductive_llist @ A,I: nat] :
( ( lList22119844313_ltake @ A @ ( coinductive_LCons @ A @ A3 @ L2 ) @ ( suc @ I ) )
= ( coinductive_LCons @ A @ A3 @ ( lList22119844313_ltake @ A @ L2 @ I ) ) ) ).
% ltake_LCons_Suc
thf(fact_150_fin__Un__inf,axiom,
! [A: $tType,A2: set @ A] :
( ( sup_sup @ ( set @ ( coinductive_llist @ A ) ) @ ( lList2236698231inlsts @ A @ A2 ) @ ( lList21612149805nflsts @ A @ A2 ) )
= ( lList2435255213lllsts @ A @ A2 ) ) ).
% fin_Un_inf
thf(fact_151_LList2__Mirabelle__hamjzmohle_Ollength__LNil,axiom,
! [A: $tType] :
( ( lList21232602520length @ A @ ( coinductive_LNil @ A ) )
= ( zero_zero @ nat ) ) ).
% LList2_Mirabelle_hamjzmohle.llength_LNil
thf(fact_152_lstrict__prefix__lappend__conv,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coindu1478340336prefix @ A @ Xs @ ( coinductive_lappend @ A @ Xs @ Ys ) )
= ( ( coinductive_lfinite @ A @ Xs )
& ~ ( coinductive_lnull @ A @ Ys ) ) ) ).
% lstrict_prefix_lappend_conv
thf(fact_153_lmember__code_I1_J,axiom,
! [A: $tType,X: A] :
~ ( coinductive_lmember @ A @ X @ ( coinductive_LNil @ A ) ) ).
% lmember_code(1)
thf(fact_154_Un__iff,axiom,
! [A: $tType,C: A,A2: set @ A,B3: set @ A] :
( ( member @ A @ C @ ( sup_sup @ ( set @ A ) @ A2 @ B3 ) )
= ( ( member @ A @ C @ A2 )
| ( member @ A @ C @ B3 ) ) ) ).
% Un_iff
thf(fact_155_UnCI,axiom,
! [A: $tType,C: A,B3: set @ A,A2: set @ A] :
( ( ~ ( member @ A @ C @ B3 )
=> ( member @ A @ C @ A2 ) )
=> ( member @ A @ C @ ( sup_sup @ ( set @ A ) @ A2 @ B3 ) ) ) ).
% UnCI
thf(fact_156_Un__Diff__cancel,axiom,
! [A: $tType,A2: set @ A,B3: set @ A] :
( ( sup_sup @ ( set @ A ) @ A2 @ ( minus_minus @ ( set @ A ) @ B3 @ A2 ) )
= ( sup_sup @ ( set @ A ) @ A2 @ B3 ) ) ).
% Un_Diff_cancel
thf(fact_157_Un__Diff__cancel2,axiom,
! [A: $tType,B3: set @ A,A2: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ B3 @ A2 ) @ A2 )
= ( sup_sup @ ( set @ A ) @ B3 @ A2 ) ) ).
% Un_Diff_cancel2
thf(fact_158_lappend_Odisc__iff_I2_J,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( ~ ( coinductive_lnull @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) ) )
= ( ~ ( coinductive_lnull @ A @ Xs )
| ~ ( coinductive_lnull @ A @ Ys ) ) ) ).
% lappend.disc_iff(2)
thf(fact_159_lnull__lappend,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
= ( ( coinductive_lnull @ A @ Xs )
& ( coinductive_lnull @ A @ Ys ) ) ) ).
% lnull_lappend
thf(fact_160_Un__Diff,axiom,
! [A: $tType,A2: set @ A,B3: set @ A,C3: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A2 @ B3 ) @ C3 )
= ( sup_sup @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ C3 ) @ ( minus_minus @ ( set @ A ) @ B3 @ C3 ) ) ) ).
% Un_Diff
thf(fact_161_LList2__Mirabelle__hamjzmohle_Oldrop_Osimps_I1_J,axiom,
! [A: $tType,L2: coinductive_llist @ A] :
( ( lList2508575361_ldrop @ A @ L2 @ ( zero_zero @ nat ) )
= L2 ) ).
% LList2_Mirabelle_hamjzmohle.ldrop.simps(1)
thf(fact_162_Un__left__commute,axiom,
! [A: $tType,A2: set @ A,B3: set @ A,C3: set @ A] :
( ( sup_sup @ ( set @ A ) @ A2 @ ( sup_sup @ ( set @ A ) @ B3 @ C3 ) )
= ( sup_sup @ ( set @ A ) @ B3 @ ( sup_sup @ ( set @ A ) @ A2 @ C3 ) ) ) ).
% Un_left_commute
thf(fact_163_Un__left__absorb,axiom,
! [A: $tType,A2: set @ A,B3: set @ A] :
( ( sup_sup @ ( set @ A ) @ A2 @ ( sup_sup @ ( set @ A ) @ A2 @ B3 ) )
= ( sup_sup @ ( set @ A ) @ A2 @ B3 ) ) ).
% Un_left_absorb
thf(fact_164_Un__commute,axiom,
! [A: $tType] :
( ( sup_sup @ ( set @ A ) )
= ( ^ [A6: set @ A,B4: set @ A] : ( sup_sup @ ( set @ A ) @ B4 @ A6 ) ) ) ).
% Un_commute
thf(fact_165_Un__absorb,axiom,
! [A: $tType,A2: set @ A] :
( ( sup_sup @ ( set @ A ) @ A2 @ A2 )
= A2 ) ).
% Un_absorb
thf(fact_166_Un__assoc,axiom,
! [A: $tType,A2: set @ A,B3: set @ A,C3: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A2 @ B3 ) @ C3 )
= ( sup_sup @ ( set @ A ) @ A2 @ ( sup_sup @ ( set @ A ) @ B3 @ C3 ) ) ) ).
% Un_assoc
thf(fact_167_ball__Un,axiom,
! [A: $tType,A2: set @ A,B3: set @ A,P: A > $o] :
( ( ! [X3: A] :
( ( member @ A @ X3 @ ( sup_sup @ ( set @ A ) @ A2 @ B3 ) )
=> ( P @ X3 ) ) )
= ( ! [X3: A] :
( ( member @ A @ X3 @ A2 )
=> ( P @ X3 ) )
& ! [X3: A] :
( ( member @ A @ X3 @ B3 )
=> ( P @ X3 ) ) ) ) ).
% ball_Un
thf(fact_168_bex__Un,axiom,
! [A: $tType,A2: set @ A,B3: set @ A,P: A > $o] :
( ( ? [X3: A] :
( ( member @ A @ X3 @ ( sup_sup @ ( set @ A ) @ A2 @ B3 ) )
& ( P @ X3 ) ) )
= ( ? [X3: A] :
( ( member @ A @ X3 @ A2 )
& ( P @ X3 ) )
| ? [X3: A] :
( ( member @ A @ X3 @ B3 )
& ( P @ X3 ) ) ) ) ).
% bex_Un
thf(fact_169_UnI2,axiom,
! [A: $tType,C: A,B3: set @ A,A2: set @ A] :
( ( member @ A @ C @ B3 )
=> ( member @ A @ C @ ( sup_sup @ ( set @ A ) @ A2 @ B3 ) ) ) ).
% UnI2
thf(fact_170_UnI1,axiom,
! [A: $tType,C: A,A2: set @ A,B3: set @ A] :
( ( member @ A @ C @ A2 )
=> ( member @ A @ C @ ( sup_sup @ ( set @ A ) @ A2 @ B3 ) ) ) ).
% UnI1
thf(fact_171_UnE,axiom,
! [A: $tType,C: A,A2: set @ A,B3: set @ A] :
( ( member @ A @ C @ ( sup_sup @ ( set @ A ) @ A2 @ B3 ) )
=> ( ~ ( member @ A @ C @ A2 )
=> ( member @ A @ C @ B3 ) ) ) ).
% UnE
thf(fact_172_lappend_Oexhaust,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( ( coinductive_lnull @ A @ Xs )
=> ~ ( coinductive_lnull @ A @ Ys ) )
=> ( ~ ( coinductive_lnull @ A @ Xs )
| ~ ( coinductive_lnull @ A @ Ys ) ) ) ).
% lappend.exhaust
thf(fact_173_lzip_Oexhaust,axiom,
! [A: $tType,B: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B] :
( ~ ( ( coinductive_lnull @ A @ Xs )
| ( coinductive_lnull @ B @ Ys ) )
=> ~ ( ~ ( coinductive_lnull @ A @ Xs )
=> ( coinductive_lnull @ B @ Ys ) ) ) ).
% lzip.exhaust
thf(fact_174_lnull__imp__lfinite,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ Xs )
=> ( coinductive_lfinite @ A @ Xs ) ) ).
% lnull_imp_lfinite
thf(fact_175_lappend_Odisc_I2_J,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( ~ ( coinductive_lnull @ A @ Xs )
| ~ ( coinductive_lnull @ A @ Ys ) )
=> ~ ( coinductive_lnull @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) ) ) ).
% lappend.disc(2)
thf(fact_176_lappend_Odisc_I1_J,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ Xs )
=> ( ( coinductive_lnull @ A @ Ys )
=> ( coinductive_lnull @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) ) ) ) ).
% lappend.disc(1)
thf(fact_177_lappend__lnull1,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ Xs )
=> ( ( coinductive_lappend @ A @ Xs @ Ys )
= Ys ) ) ).
% lappend_lnull1
thf(fact_178_lappend__lnull2,axiom,
! [A: $tType,Ys: coinductive_llist @ A,Xs: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ Ys )
=> ( ( coinductive_lappend @ A @ Xs @ Ys )
= Xs ) ) ).
% lappend_lnull2
thf(fact_179_llist_Odisc_I1_J,axiom,
! [A: $tType] : ( coinductive_lnull @ A @ ( coinductive_LNil @ A ) ) ).
% llist.disc(1)
thf(fact_180_llist_OdiscI_I1_J,axiom,
! [A: $tType,Llist: coinductive_llist @ A] :
( ( Llist
= ( coinductive_LNil @ A ) )
=> ( coinductive_lnull @ A @ Llist ) ) ).
% llist.discI(1)
thf(fact_181_llist_Ocollapse_I1_J,axiom,
! [A: $tType,Llist: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ Llist )
=> ( Llist
= ( coinductive_LNil @ A ) ) ) ).
% llist.collapse(1)
thf(fact_182_lnull__def,axiom,
! [A: $tType] :
( ( coinductive_lnull @ A )
= ( ^ [Llist2: coinductive_llist @ A] :
( Llist2
= ( coinductive_LNil @ A ) ) ) ) ).
% lnull_def
thf(fact_183_llist_Odisc_I2_J,axiom,
! [A: $tType,X212: A,X222: coinductive_llist @ A] :
~ ( coinductive_lnull @ A @ ( coinductive_LCons @ A @ X212 @ X222 ) ) ).
% llist.disc(2)
thf(fact_184_llist_OdiscI_I2_J,axiom,
! [A: $tType,Llist: coinductive_llist @ A,X212: A,X222: coinductive_llist @ A] :
( ( Llist
= ( coinductive_LCons @ A @ X212 @ X222 ) )
=> ~ ( coinductive_lnull @ A @ Llist ) ) ).
% llist.discI(2)
thf(fact_185_not__lnull__conv,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( ~ ( coinductive_lnull @ A @ Xs ) )
= ( ? [X3: A,Xs3: coinductive_llist @ A] :
( Xs
= ( coinductive_LCons @ A @ X3 @ Xs3 ) ) ) ) ).
% not_lnull_conv
thf(fact_186_Un__UNIV__left,axiom,
! [A: $tType,B3: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( top_top @ ( set @ A ) ) @ B3 )
= ( top_top @ ( set @ A ) ) ) ).
% Un_UNIV_left
thf(fact_187_Un__UNIV__right,axiom,
! [A: $tType,A2: set @ A] :
( ( sup_sup @ ( set @ A ) @ A2 @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% Un_UNIV_right
thf(fact_188_lappend_Octr_I1_J,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ Xs )
=> ( ( coinductive_lnull @ A @ Ys )
=> ( ( coinductive_lappend @ A @ Xs @ Ys )
= ( coinductive_LNil @ A ) ) ) ) ).
% lappend.ctr(1)
thf(fact_189_Coinductive__List_Ollast__LCons,axiom,
! [A: $tType,Xs: coinductive_llist @ A,X: A] :
( ( ( coinductive_lnull @ A @ Xs )
=> ( ( coinductive_llast @ A @ ( coinductive_LCons @ A @ X @ Xs ) )
= X ) )
& ( ~ ( coinductive_lnull @ A @ Xs )
=> ( ( coinductive_llast @ A @ ( coinductive_LCons @ A @ X @ Xs ) )
= ( coinductive_llast @ A @ Xs ) ) ) ) ).
% Coinductive_List.llast_LCons
thf(fact_190_LList2__Mirabelle__hamjzmohle_Oltake_Osimps_I1_J,axiom,
! [A: $tType,L2: coinductive_llist @ A] :
( ( lList22119844313_ltake @ A @ L2 @ ( zero_zero @ nat ) )
= ( coinductive_LNil @ A ) ) ).
% LList2_Mirabelle_hamjzmohle.ltake.simps(1)
thf(fact_191_lmember__code_I2_J,axiom,
! [A: $tType,X: A,Y: A,Ys: coinductive_llist @ A] :
( ( coinductive_lmember @ A @ X @ ( coinductive_LCons @ A @ Y @ Ys ) )
= ( ( X = Y )
| ( coinductive_lmember @ A @ X @ Ys ) ) ) ).
% lmember_code(2)
thf(fact_192_sup__top__right,axiom,
! [A: $tType] :
( ( bounded_lattice_top @ A @ ( type2 @ A ) )
=> ! [X: A] :
( ( sup_sup @ A @ X @ ( top_top @ A ) )
= ( top_top @ A ) ) ) ).
% sup_top_right
thf(fact_193_sup__top__left,axiom,
! [A: $tType] :
( ( bounded_lattice_top @ A @ ( type2 @ A ) )
=> ! [X: A] :
( ( sup_sup @ A @ ( top_top @ A ) @ X )
= ( top_top @ A ) ) ) ).
% sup_top_left
thf(fact_194_lsublist__singleton,axiom,
! [A: $tType,A2: set @ nat,X: A] :
( ( ( member @ nat @ ( zero_zero @ nat ) @ A2 )
=> ( ( coinductive_lsublist @ A @ ( coinductive_LCons @ A @ X @ ( coinductive_LNil @ A ) ) @ A2 )
= ( coinductive_LCons @ A @ X @ ( coinductive_LNil @ A ) ) ) )
& ( ~ ( member @ nat @ ( zero_zero @ nat ) @ A2 )
=> ( ( coinductive_lsublist @ A @ ( coinductive_LCons @ A @ X @ ( coinductive_LNil @ A ) ) @ A2 )
= ( coinductive_LNil @ A ) ) ) ) ).
% lsublist_singleton
thf(fact_195_lsublist__LNil,axiom,
! [A: $tType,A2: set @ nat] :
( ( coinductive_lsublist @ A @ ( coinductive_LNil @ A ) @ A2 )
= ( coinductive_LNil @ A ) ) ).
% lsublist_LNil
thf(fact_196_llistsum__inf,axiom,
! [A: $tType] :
( ( monoid_add @ A @ ( type2 @ A ) )
=> ! [Xs: coinductive_llist @ A] :
( ~ ( coinductive_lfinite @ A @ Xs )
=> ( ( coindu780009021istsum @ A @ Xs )
= ( zero_zero @ A ) ) ) ) ).
% llistsum_inf
thf(fact_197_llistsum__LNil,axiom,
! [A: $tType] :
( ( monoid_add @ A @ ( type2 @ A ) )
=> ( ( coindu780009021istsum @ A @ ( coinductive_LNil @ A ) )
= ( zero_zero @ A ) ) ) ).
% llistsum_LNil
thf(fact_198_ltake_Oexhaust,axiom,
! [A: $tType,N: extended_enat,Xs: coinductive_llist @ A] :
( ~ ( ( N
= ( zero_zero @ extended_enat ) )
| ( coinductive_lnull @ A @ Xs ) )
=> ~ ( ( N
!= ( zero_zero @ extended_enat ) )
=> ( coinductive_lnull @ A @ Xs ) ) ) ).
% ltake.exhaust
thf(fact_199_llast__lappend,axiom,
! [A: $tType,Ys: coinductive_llist @ A,Xs: coinductive_llist @ A] :
( ( ( coinductive_lnull @ A @ Ys )
=> ( ( coinductive_llast @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
= ( coinductive_llast @ A @ Xs ) ) )
& ( ~ ( coinductive_lnull @ A @ Ys )
=> ( ( ( coinductive_lfinite @ A @ Xs )
=> ( ( coinductive_llast @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
= ( coinductive_llast @ A @ Ys ) ) )
& ( ~ ( coinductive_lfinite @ A @ Xs )
=> ( ( coinductive_llast @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
= ( undefined @ A ) ) ) ) ) ) ).
% llast_lappend
thf(fact_200_llistsum__LCons,axiom,
! [A: $tType] :
( ( monoid_add @ A @ ( type2 @ A ) )
=> ! [Xs: coinductive_llist @ A,X: A] :
( ( coinductive_lfinite @ A @ Xs )
=> ( ( coindu780009021istsum @ A @ ( coinductive_LCons @ A @ X @ Xs ) )
= ( plus_plus @ A @ X @ ( coindu780009021istsum @ A @ Xs ) ) ) ) ) ).
% llistsum_LCons
thf(fact_201_LList2__Mirabelle__hamjzmohle_Ollast__LNil,axiom,
! [B: $tType] :
( ( lList2170638824_llast @ B @ ( coinductive_LNil @ B ) )
= ( undefined @ B ) ) ).
% LList2_Mirabelle_hamjzmohle.llast_LNil
thf(fact_202_llast__linfinite,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ~ ( coinductive_lfinite @ A @ Xs )
=> ( ( coinductive_llast @ A @ Xs )
= ( undefined @ A ) ) ) ).
% llast_linfinite
thf(fact_203_ldrop__add,axiom,
! [A: $tType,T: coinductive_llist @ A,I: nat,K: nat] :
( ( lList2508575361_ldrop @ A @ T @ ( plus_plus @ nat @ I @ K ) )
= ( lList2508575361_ldrop @ A @ ( lList2508575361_ldrop @ A @ T @ I ) @ K ) ) ).
% ldrop_add
thf(fact_204_Coinductive__List_Ollast__LNil,axiom,
! [A: $tType] :
( ( coinductive_llast @ A @ ( coinductive_LNil @ A ) )
= ( undefined @ A ) ) ).
% Coinductive_List.llast_LNil
thf(fact_205_LList2__Mirabelle__hamjzmohle_Oltake__ldrop,axiom,
! [A: $tType,Xs: coinductive_llist @ A,M: nat,N: nat] :
( ( lList22119844313_ltake @ A @ ( lList2508575361_ldrop @ A @ Xs @ M ) @ N )
= ( lList2508575361_ldrop @ A @ ( lList22119844313_ltake @ A @ Xs @ ( plus_plus @ nat @ N @ M ) ) @ M ) ) ).
% LList2_Mirabelle_hamjzmohle.ltake_ldrop
thf(fact_206_llistsum__lappend,axiom,
! [A: $tType] :
( ( monoid_add @ A @ ( type2 @ A ) )
=> ! [Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ Xs )
=> ( ( coinductive_lfinite @ A @ Ys )
=> ( ( coindu780009021istsum @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
= ( plus_plus @ A @ ( coindu780009021istsum @ A @ Xs ) @ ( coindu780009021istsum @ A @ Ys ) ) ) ) ) ) ).
% llistsum_lappend
thf(fact_207_llcp__LNil2,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( coinductive_llcp @ A @ Xs @ ( coinductive_LNil @ A ) )
= ( zero_zero @ extended_enat ) ) ).
% llcp_LNil2
thf(fact_208_llcp__LNil1,axiom,
! [A: $tType,Ys: coinductive_llist @ A] :
( ( coinductive_llcp @ A @ ( coinductive_LNil @ A ) @ Ys )
= ( zero_zero @ extended_enat ) ) ).
% llcp_LNil1
thf(fact_209_llcp__commute,axiom,
! [A: $tType] :
( ( coinductive_llcp @ A )
= ( ^ [Xs4: coinductive_llist @ A,Ys3: coinductive_llist @ A] : ( coinductive_llcp @ A @ Ys3 @ Xs4 ) ) ) ).
% llcp_commute
thf(fact_210_llcp__LCons,axiom,
! [A: $tType,X: A,Y: A,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( ( X = Y )
=> ( ( coinductive_llcp @ A @ ( coinductive_LCons @ A @ X @ Xs ) @ ( coinductive_LCons @ A @ Y @ Ys ) )
= ( extended_eSuc @ ( coinductive_llcp @ A @ Xs @ Ys ) ) ) )
& ( ( X != Y )
=> ( ( coinductive_llcp @ A @ ( coinductive_LCons @ A @ X @ Xs ) @ ( coinductive_LCons @ A @ Y @ Ys ) )
= ( zero_zero @ extended_enat ) ) ) ) ).
% llcp_LCons
thf(fact_211_lset__lappend__lfinite,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ Xs )
=> ( ( coinductive_lset @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
= ( sup_sup @ ( set @ A ) @ ( coinductive_lset @ A @ Xs ) @ ( coinductive_lset @ A @ Ys ) ) ) ) ).
% lset_lappend_lfinite
thf(fact_212_lset__lmember,axiom,
! [A: $tType,X: A,Xs: coinductive_llist @ A] :
( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
= ( coinductive_lmember @ A @ X @ Xs ) ) ).
% lset_lmember
thf(fact_213_llist_Oset__induct,axiom,
! [A: $tType,X: A,A3: coinductive_llist @ A,P: A > ( coinductive_llist @ A ) > $o] :
( ( member @ A @ X @ ( coinductive_lset @ A @ A3 ) )
=> ( ! [Z1: A,Z2: coinductive_llist @ A] : ( P @ Z1 @ ( coinductive_LCons @ A @ Z1 @ Z2 ) )
=> ( ! [Z1: A,Z2: coinductive_llist @ A,Xa2: A] :
( ( member @ A @ Xa2 @ ( coinductive_lset @ A @ Z2 ) )
=> ( ( P @ Xa2 @ Z2 )
=> ( P @ Xa2 @ ( coinductive_LCons @ A @ Z1 @ Z2 ) ) ) )
=> ( P @ X @ A3 ) ) ) ) ).
% llist.set_induct
thf(fact_214_llist_Oset__cases,axiom,
! [A: $tType,E: A,A3: coinductive_llist @ A] :
( ( member @ A @ E @ ( coinductive_lset @ A @ A3 ) )
=> ( ! [Z2: coinductive_llist @ A] :
( A3
!= ( coinductive_LCons @ A @ E @ Z2 ) )
=> ~ ! [Z1: A,Z2: coinductive_llist @ A] :
( ( A3
= ( coinductive_LCons @ A @ Z1 @ Z2 ) )
=> ~ ( member @ A @ E @ ( coinductive_lset @ A @ Z2 ) ) ) ) ) ).
% llist.set_cases
thf(fact_215_lset__induct_H,axiom,
! [A: $tType,X: A,Xs: coinductive_llist @ A,P: ( coinductive_llist @ A ) > $o] :
( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
=> ( ! [Xs2: coinductive_llist @ A] : ( P @ ( coinductive_LCons @ A @ X @ Xs2 ) )
=> ( ! [X5: A,Xs2: coinductive_llist @ A] :
( ( member @ A @ X @ ( coinductive_lset @ A @ Xs2 ) )
=> ( ( P @ Xs2 )
=> ( P @ ( coinductive_LCons @ A @ X5 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% lset_induct'
thf(fact_216_lset__induct,axiom,
! [A: $tType,X: A,Xs: coinductive_llist @ A,P: ( coinductive_llist @ A ) > $o] :
( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
=> ( ! [Xs2: coinductive_llist @ A] : ( P @ ( coinductive_LCons @ A @ X @ Xs2 ) )
=> ( ! [X5: A,Xs2: coinductive_llist @ A] :
( ( member @ A @ X @ ( coinductive_lset @ A @ Xs2 ) )
=> ( ( X != X5 )
=> ( ( P @ Xs2 )
=> ( P @ ( coinductive_LCons @ A @ X5 @ Xs2 ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% lset_induct
thf(fact_217_lset__cases,axiom,
! [A: $tType,X: A,Xs: coinductive_llist @ A] :
( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
=> ( ! [Xs5: coinductive_llist @ A] :
( Xs
!= ( coinductive_LCons @ A @ X @ Xs5 ) )
=> ~ ! [X5: A,Xs5: coinductive_llist @ A] :
( ( Xs
= ( coinductive_LCons @ A @ X5 @ Xs5 ) )
=> ~ ( member @ A @ X @ ( coinductive_lset @ A @ Xs5 ) ) ) ) ) ).
% lset_cases
thf(fact_218_llist_Oset__intros_I1_J,axiom,
! [A: $tType,A1: A,A22: coinductive_llist @ A] : ( member @ A @ A1 @ ( coinductive_lset @ A @ ( coinductive_LCons @ A @ A1 @ A22 ) ) ) ).
% llist.set_intros(1)
thf(fact_219_llist_Oset__intros_I2_J,axiom,
! [A: $tType,X: A,A22: coinductive_llist @ A,A1: A] :
( ( member @ A @ X @ ( coinductive_lset @ A @ A22 ) )
=> ( member @ A @ X @ ( coinductive_lset @ A @ ( coinductive_LCons @ A @ A1 @ A22 ) ) ) ) ).
% llist.set_intros(2)
thf(fact_220_lset__intros_I1_J,axiom,
! [A: $tType,X: A,Xs: coinductive_llist @ A] : ( member @ A @ X @ ( coinductive_lset @ A @ ( coinductive_LCons @ A @ X @ Xs ) ) ) ).
% lset_intros(1)
thf(fact_221_lset__intros_I2_J,axiom,
! [A: $tType,X: A,Xs: coinductive_llist @ A,X6: A] :
( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
=> ( member @ A @ X @ ( coinductive_lset @ A @ ( coinductive_LCons @ A @ X6 @ Xs ) ) ) ) ).
% lset_intros(2)
thf(fact_222_in__lset__lappend__iff,axiom,
! [A: $tType,X: A,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( member @ A @ X @ ( coinductive_lset @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) ) )
= ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
| ( ( coinductive_lfinite @ A @ Xs )
& ( member @ A @ X @ ( coinductive_lset @ A @ Ys ) ) ) ) ) ).
% in_lset_lappend_iff
thf(fact_223_split__llist,axiom,
! [A: $tType,X: A,Xs: coinductive_llist @ A] :
( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
=> ? [Ys4: coinductive_llist @ A,Zs2: coinductive_llist @ A] :
( ( Xs
= ( coinductive_lappend @ A @ Ys4 @ ( coinductive_LCons @ A @ X @ Zs2 ) ) )
& ( coinductive_lfinite @ A @ Ys4 ) ) ) ).
% split_llist
thf(fact_224_split__llist__first,axiom,
! [A: $tType,X: A,Xs: coinductive_llist @ A] :
( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
=> ? [Ys4: coinductive_llist @ A,Zs2: coinductive_llist @ A] :
( ( Xs
= ( coinductive_lappend @ A @ Ys4 @ ( coinductive_LCons @ A @ X @ Zs2 ) ) )
& ( coinductive_lfinite @ A @ Ys4 )
& ~ ( member @ A @ X @ ( coinductive_lset @ A @ Ys4 ) ) ) ) ).
% split_llist_first
thf(fact_225_lset__lappend__conv,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( ( coinductive_lfinite @ A @ Xs )
=> ( ( coinductive_lset @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
= ( sup_sup @ ( set @ A ) @ ( coinductive_lset @ A @ Xs ) @ ( coinductive_lset @ A @ Ys ) ) ) )
& ( ~ ( coinductive_lfinite @ A @ Xs )
=> ( ( coinductive_lset @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
= ( coinductive_lset @ A @ Xs ) ) ) ) ).
% lset_lappend_conv
thf(fact_226_gen__lset__def,axiom,
! [A: $tType] :
( ( coinductive_gen_lset @ A )
= ( ^ [A6: set @ A,Xs4: coinductive_llist @ A] : ( sup_sup @ ( set @ A ) @ A6 @ ( coinductive_lset @ A @ Xs4 ) ) ) ) ).
% gen_lset_def
thf(fact_227_lfilter__eq__LConsD,axiom,
! [A: $tType,P: A > $o,Ys: coinductive_llist @ A,X: A,Xs: coinductive_llist @ A] :
( ( ( coinductive_lfilter @ A @ P @ Ys )
= ( coinductive_LCons @ A @ X @ Xs ) )
=> ? [Us: coinductive_llist @ A,Vs: coinductive_llist @ A] :
( ( Ys
= ( coinductive_lappend @ A @ Us @ ( coinductive_LCons @ A @ X @ Vs ) ) )
& ( coinductive_lfinite @ A @ Us )
& ! [X7: A] :
( ( member @ A @ X7 @ ( coinductive_lset @ A @ Us ) )
=> ~ ( P @ X7 ) )
& ( P @ X )
& ( Xs
= ( coinductive_lfilter @ A @ P @ Vs ) ) ) ) ).
% lfilter_eq_LConsD
thf(fact_228_lfilter__idem,axiom,
! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
( ( coinductive_lfilter @ A @ P @ ( coinductive_lfilter @ A @ P @ Xs ) )
= ( coinductive_lfilter @ A @ P @ Xs ) ) ).
% lfilter_idem
thf(fact_229_lfilter__LCons,axiom,
! [A: $tType,P: A > $o,X: A,Xs: coinductive_llist @ A] :
( ( ( P @ X )
=> ( ( coinductive_lfilter @ A @ P @ ( coinductive_LCons @ A @ X @ Xs ) )
= ( coinductive_LCons @ A @ X @ ( coinductive_lfilter @ A @ P @ Xs ) ) ) )
& ( ~ ( P @ X )
=> ( ( coinductive_lfilter @ A @ P @ ( coinductive_LCons @ A @ X @ Xs ) )
= ( coinductive_lfilter @ A @ P @ Xs ) ) ) ) ).
% lfilter_LCons
thf(fact_230_lfilter__LNil,axiom,
! [A: $tType,P: A > $o] :
( ( coinductive_lfilter @ A @ P @ ( coinductive_LNil @ A ) )
= ( coinductive_LNil @ A ) ) ).
% lfilter_LNil
thf(fact_231_diverge__lfilter__LNil,axiom,
! [A: $tType,Xs: coinductive_llist @ A,P: A > $o] :
( ! [X2: A] :
( ( member @ A @ X2 @ ( coinductive_lset @ A @ Xs ) )
=> ~ ( P @ X2 ) )
=> ( ( coinductive_lfilter @ A @ P @ Xs )
= ( coinductive_LNil @ A ) ) ) ).
% diverge_lfilter_LNil
thf(fact_232_lnull__lfilter,axiom,
! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ ( coinductive_lfilter @ A @ P @ Xs ) )
= ( ! [X3: A] :
( ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs ) )
=> ~ ( P @ X3 ) ) ) ) ).
% lnull_lfilter
thf(fact_233_lfilter__lappend__lfinite,axiom,
! [A: $tType,Xs: coinductive_llist @ A,P: A > $o,Ys: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ Xs )
=> ( ( coinductive_lfilter @ A @ P @ ( coinductive_lappend @ A @ Xs @ Ys ) )
= ( coinductive_lappend @ A @ ( coinductive_lfilter @ A @ P @ Xs ) @ ( coinductive_lfilter @ A @ P @ Ys ) ) ) ) ).
% lfilter_lappend_lfinite
thf(fact_234_lfilter__cong,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,P: A > $o,Q: A > $o] :
( ( Xs = Ys )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ ( coinductive_lset @ A @ Ys ) )
=> ( ( P @ X2 )
= ( Q @ X2 ) ) )
=> ( ( coinductive_lfilter @ A @ P @ Xs )
= ( coinductive_lfilter @ A @ Q @ Ys ) ) ) ) ).
% lfilter_cong
thf(fact_235_lfilter__id__conv,axiom,
! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
( ( ( coinductive_lfilter @ A @ P @ Xs )
= Xs )
= ( ! [X3: A] :
( ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs ) )
=> ( P @ X3 ) ) ) ) ).
% lfilter_id_conv
thf(fact_236_lfilter__empty__conv,axiom,
! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
( ( ( coinductive_lfilter @ A @ P @ Xs )
= ( coinductive_LNil @ A ) )
= ( ! [X3: A] :
( ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs ) )
=> ~ ( P @ X3 ) ) ) ) ).
% lfilter_empty_conv
thf(fact_237_lfilter__eq__lappend__lfiniteD,axiom,
! [A: $tType,P: A > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A] :
( ( ( coinductive_lfilter @ A @ P @ Xs )
= ( coinductive_lappend @ A @ Ys @ Zs ) )
=> ( ( coinductive_lfinite @ A @ Ys )
=> ? [Us: coinductive_llist @ A,Vs: coinductive_llist @ A] :
( ( Xs
= ( coinductive_lappend @ A @ Us @ Vs ) )
& ( coinductive_lfinite @ A @ Us )
& ( Ys
= ( coinductive_lfilter @ A @ P @ Us ) )
& ( Zs
= ( coinductive_lfilter @ A @ P @ Vs ) ) ) ) ) ).
% lfilter_eq_lappend_lfiniteD
thf(fact_238_lfinite__lfilterI,axiom,
! [A: $tType,Xs: coinductive_llist @ A,P: A > $o] :
( ( coinductive_lfinite @ A @ Xs )
=> ( coinductive_lfinite @ A @ ( coinductive_lfilter @ A @ P @ Xs ) ) ) ).
% lfinite_lfilterI
thf(fact_239_lfilter__LCons__seek,axiom,
! [A: $tType,P2: A > $o,X: A,L2: coinductive_llist @ A] :
( ~ ( P2 @ X )
=> ( ( coinductive_lfilter @ A @ P2 @ ( coinductive_LCons @ A @ X @ L2 ) )
= ( coinductive_lfilter @ A @ P2 @ L2 ) ) ) ).
% lfilter_LCons_seek
thf(fact_240_lfilter__LCons__found,axiom,
! [A: $tType,P: A > $o,X: A,Xs: coinductive_llist @ A] :
( ( P @ X )
=> ( ( coinductive_lfilter @ A @ P @ ( coinductive_LCons @ A @ X @ Xs ) )
= ( coinductive_LCons @ A @ X @ ( coinductive_lfilter @ A @ P @ Xs ) ) ) ) ).
% lfilter_LCons_found
thf(fact_241_gen__lset__code_I1_J,axiom,
! [A: $tType,A2: set @ A] :
( ( coinductive_gen_lset @ A @ A2 @ ( coinductive_LNil @ A ) )
= A2 ) ).
% gen_lset_code(1)
thf(fact_242_ldistinct_Ocases,axiom,
! [A: $tType,A3: coinductive_llist @ A] :
( ( coindu351974385stinct @ A @ A3 )
=> ( ( A3
!= ( coinductive_LNil @ A ) )
=> ~ ! [X2: A,Xs2: coinductive_llist @ A] :
( ( A3
= ( coinductive_LCons @ A @ X2 @ Xs2 ) )
=> ( ~ ( member @ A @ X2 @ ( coinductive_lset @ A @ Xs2 ) )
=> ~ ( coindu351974385stinct @ A @ Xs2 ) ) ) ) ) ).
% ldistinct.cases
thf(fact_243_ldistinct_Osimps,axiom,
! [A: $tType] :
( ( coindu351974385stinct @ A )
= ( ^ [A5: coinductive_llist @ A] :
( ( A5
= ( coinductive_LNil @ A ) )
| ? [X3: A,Xs4: coinductive_llist @ A] :
( ( A5
= ( coinductive_LCons @ A @ X3 @ Xs4 ) )
& ~ ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs4 ) )
& ( coindu351974385stinct @ A @ Xs4 ) ) ) ) ) ).
% ldistinct.simps
thf(fact_244_ldistinct__LNil__code,axiom,
! [A: $tType] : ( coindu351974385stinct @ A @ ( coinductive_LNil @ A ) ) ).
% ldistinct_LNil_code
thf(fact_245_ldistinct__LCons,axiom,
! [A: $tType,X: A,Xs: coinductive_llist @ A] :
( ( coindu351974385stinct @ A @ ( coinductive_LCons @ A @ X @ Xs ) )
= ( ~ ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
& ( coindu351974385stinct @ A @ Xs ) ) ) ).
% ldistinct_LCons
thf(fact_246_ldistinct__lfilterI,axiom,
! [A: $tType,Xs: coinductive_llist @ A,P: A > $o] :
( ( coindu351974385stinct @ A @ Xs )
=> ( coindu351974385stinct @ A @ ( coinductive_lfilter @ A @ P @ Xs ) ) ) ).
% ldistinct_lfilterI
thf(fact_247_ldistinct_OLNil,axiom,
! [A: $tType] : ( coindu351974385stinct @ A @ ( coinductive_LNil @ A ) ) ).
% ldistinct.LNil
thf(fact_248_ldistinct_OLCons,axiom,
! [A: $tType,X: A,Xs: coinductive_llist @ A] :
( ~ ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
=> ( ( coindu351974385stinct @ A @ Xs )
=> ( coindu351974385stinct @ A @ ( coinductive_LCons @ A @ X @ Xs ) ) ) ) ).
% ldistinct.LCons
thf(fact_249_ldistinct_Ocoinduct,axiom,
! [A: $tType,X4: ( coinductive_llist @ A ) > $o,X: coinductive_llist @ A] :
( ( X4 @ X )
=> ( ! [X2: coinductive_llist @ A] :
( ( X4 @ X2 )
=> ( ( X2
= ( coinductive_LNil @ A ) )
| ? [Xa3: A,Xs6: coinductive_llist @ A] :
( ( X2
= ( coinductive_LCons @ A @ Xa3 @ Xs6 ) )
& ~ ( member @ A @ Xa3 @ ( coinductive_lset @ A @ Xs6 ) )
& ( ( X4 @ Xs6 )
| ( coindu351974385stinct @ A @ Xs6 ) ) ) ) )
=> ( coindu351974385stinct @ A @ X ) ) ) ).
% ldistinct.coinduct
thf(fact_250_llexord__lappend__leftD,axiom,
! [A: $tType,R: A > A > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A] :
( ( coinductive_llexord @ A @ R @ ( coinductive_lappend @ A @ Xs @ Ys ) @ ( coinductive_lappend @ A @ Xs @ Zs ) )
=> ( ( coinductive_lfinite @ A @ Xs )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ ( coinductive_lset @ A @ Xs ) )
=> ~ ( R @ X2 @ X2 ) )
=> ( coinductive_llexord @ A @ R @ Ys @ Zs ) ) ) ) ).
% llexord_lappend_leftD
thf(fact_251_llexord__lappend__left,axiom,
! [A: $tType,Xs: coinductive_llist @ A,R: A > A > $o,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ Xs )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ ( coinductive_lset @ A @ Xs ) )
=> ~ ( R @ X2 @ X2 ) )
=> ( ( coinductive_llexord @ A @ R @ ( coinductive_lappend @ A @ Xs @ Ys ) @ ( coinductive_lappend @ A @ Xs @ Zs ) )
= ( coinductive_llexord @ A @ R @ Ys @ Zs ) ) ) ) ).
% llexord_lappend_left
thf(fact_252_llexord__refl,axiom,
! [A: $tType,R: A > A > $o,Xs: coinductive_llist @ A] : ( coinductive_llexord @ A @ R @ Xs @ Xs ) ).
% llexord_refl
thf(fact_253_llexord__LCons__LCons,axiom,
! [A: $tType,R: A > A > $o,X: A,Xs: coinductive_llist @ A,Y: A,Ys: coinductive_llist @ A] :
( ( coinductive_llexord @ A @ R @ ( coinductive_LCons @ A @ X @ Xs ) @ ( coinductive_LCons @ A @ Y @ Ys ) )
= ( ( ( X = Y )
& ( coinductive_llexord @ A @ R @ Xs @ Ys ) )
| ( R @ X @ Y ) ) ) ).
% llexord_LCons_LCons
%----Type constructors (14)
thf(tcon_Extended__Nat_Oenat___Lattices_Obounded__lattice,axiom,
bounded_lattice @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_HOL_Obool___Lattices_Obounded__lattice_1,axiom,
bounded_lattice @ $o @ ( type2 @ $o ) ).
thf(tcon_Set_Oset___Lattices_Obounded__lattice_2,axiom,
! [A8: $tType] : ( bounded_lattice @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_fun___Lattices_Obounded__lattice_3,axiom,
! [A8: $tType,A9: $tType] :
( ( bounded_lattice @ A9 @ ( type2 @ A9 ) )
=> ( bounded_lattice @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
thf(tcon_fun___Lattices_Obounded__lattice__top,axiom,
! [A8: $tType,A9: $tType] :
( ( bounded_lattice @ A9 @ ( type2 @ A9 ) )
=> ( bounded_lattice_top @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
thf(tcon_fun___Orderings_Otop,axiom,
! [A8: $tType,A9: $tType] :
( ( top @ A9 @ ( type2 @ A9 ) )
=> ( top @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
thf(tcon_Nat_Onat___Groups_Omonoid__add,axiom,
monoid_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Set_Oset___Lattices_Obounded__lattice__top_4,axiom,
! [A8: $tType] : ( bounded_lattice_top @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Otop_5,axiom,
! [A8: $tType] : ( top @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_HOL_Obool___Lattices_Obounded__lattice__top_6,axiom,
bounded_lattice_top @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Otop_7,axiom,
top @ $o @ ( type2 @ $o ) ).
thf(tcon_Extended__Nat_Oenat___Lattices_Obounded__lattice__top_8,axiom,
bounded_lattice_top @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Groups_Omonoid__add_9,axiom,
monoid_add @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Orderings_Otop_10,axiom,
top @ extended_enat @ ( type2 @ extended_enat ) ).
%----Conjectures (1)
thf(conj_0,conjecture,
( ( lList2370560421utlast @ a @ ( coinductive_lappend @ a @ xs @ ( coinductive_LCons @ a @ x @ ( coinductive_LNil @ a ) ) ) )
= xs ) ).
%------------------------------------------------------------------------------