TPTP Problem File: DAT130^1.p
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%------------------------------------------------------------------------------
% File : DAT130^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Data Structures
% Problem : Coinductive list 3543
% 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__3543.p [Bla16]
% Status : Theorem
% Rating : 0.00 v7.2.0, 0.25 v7.1.0
% Syntax : Number of formulae : 308 ( 119 unt; 44 typ; 0 def)
% Number of atoms : 751 ( 251 equ; 0 cnn)
% Maximal formula atoms : 8 ( 2 avg)
% Number of connectives : 3848 ( 137 ~; 41 |; 76 &;3272 @)
% ( 0 <=>; 322 =>; 0 <=; 0 <~>)
% Maximal formula depth : 22 ( 8 avg)
% Number of types : 2 ( 1 usr)
% Number of type conns : 271 ( 271 >; 0 *; 0 +; 0 <<)
% Number of symbols : 46 ( 43 usr; 4 con; 0-7 aty)
% Number of variables : 955 ( 47 ^; 810 !; 58 ?; 955 :)
% ( 40 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 15:02:13.700
%------------------------------------------------------------------------------
%----Could-be-implicit typings (4)
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_itself,type,
itself: $tType > $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_Lattices_Olattice,type,
lattice:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Lattices_Osemilattice__inf,type,
semilattice_inf:
!>[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_OlSup,type,
coindu526335360e_lSup:
!>[A: $tType] : ( ( set @ ( coindu1593790203_llist @ A ) ) > ( coindu1593790203_llist @ A ) ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Olappend,type,
coindu268472904append:
!>[A: $tType] : ( ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Oldistinct,type,
coindu6345450stinct:
!>[A: $tType] : ( ( coindu1593790203_llist @ A ) > $o ) ).
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_Olhd,type,
coindu1046438764le_lhd:
!>[A: $tType] : ( ( coindu1593790203_llist @ A ) > A ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollist_Olnull,type,
coindu335574135_lnull:
!>[A: $tType] : ( ( coindu1593790203_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollist_Olset,type,
coindu1112613586e_lset:
!>[A: $tType] : ( ( coindu1593790203_llist @ A ) > ( set @ A ) ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollist_Oltl,type,
coindu1047225960le_ltl:
!>[A: $tType] : ( ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Olprefix,type,
coindu1696667936prefix:
!>[A: $tType] : ( ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Olstrict__prefix,type,
coindu574146665prefix:
!>[A: $tType] : ( ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_OltakeWhile,type,
coindu721411036eWhile:
!>[A: $tType] : ( ( A > $o ) > ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Oord_Olsorted,type,
coindu543932918sorted:
!>[A: $tType] : ( ( A > A > $o ) > ( 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_Complete__Partial__Order2_Ocompact,type,
comple2143767107ompact:
!>[A: $tType] : ( ( ( set @ A ) > A ) > ( A > A > $o ) > A > $o ) ).
thf(sy_c_Complete__Partial__Order2_Ocont,type,
comple1412571634l_cont:
!>[A: $tType,B: $tType] : ( ( ( set @ A ) > A ) > ( A > A > $o ) > ( ( set @ B ) > B ) > ( B > B > $o ) > ( A > B ) > $o ) ).
thf(sy_c_Complete__Partial__Order_Occpo_Oadmissible,type,
comple1986059218ssible:
!>[A: $tType] : ( ( ( set @ A ) > A ) > ( A > A > $o ) > ( A > $o ) > $o ) ).
thf(sy_c_Complete__Partial__Order_Ochain,type,
comple1193779247_chain:
!>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > $o ) ).
thf(sy_c_Finite__Set_Ofinite,type,
finite_finite:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_If,type,
if:
!>[A: $tType] : ( $o > A > A > A ) ).
thf(sy_c_Lattices_Oinf__class_Oinf,type,
inf_inf:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Orderings_Oord_Omax,type,
max:
!>[A: $tType] : ( ( A > A > $o ) > A > A > A ) ).
thf(sy_c_Orderings_Oord_Omin,type,
min:
!>[A: $tType] : ( ( A > A > $o ) > A > A > 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_Set__Interval_Oord_OatLeast,type,
set_atLeast:
!>[A: $tType] : ( ( A > A > $o ) > A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord_OatLeastAtMost,type,
set_atLeastAtMost:
!>[A: $tType] : ( ( A > A > $o ) > A > A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord_OatMost,type,
set_atMost:
!>[A: $tType] : ( ( A > A > $o ) > A > ( set @ A ) ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_Ya____,type,
ya: set @ ( coindu1593790203_llist @ a ) ).
thf(sy_v_less__eq,type,
less_eq: a > a > $o ).
thf(sy_v_xs____,type,
xs: coindu1593790203_llist @ a ).
%----Relevant facts (254)
thf(fact_0__092_060open_062xs_A_092_060in_062_AY_092_060close_062,axiom,
member @ ( coindu1593790203_llist @ a ) @ xs @ ya ).
% \<open>xs \<in> Y\<close>
thf(fact_1__092_060open_062lhd_Axs_A_092_060le_062_Alhd_A_Iltl_Axs_J_092_060close_062,axiom,
less_eq @ ( coindu1046438764le_lhd @ a @ xs ) @ ( coindu1046438764le_lhd @ a @ ( coindu1047225960le_ltl @ a @ xs ) ) ).
% \<open>lhd xs \<le> lhd (ltl xs)\<close>
thf(fact_2_local_Oord__eq__le__trans,axiom,
! [A2: a,B2: a,C: a] :
( ( A2 = B2 )
=> ( ( less_eq @ B2 @ C )
=> ( less_eq @ A2 @ C ) ) ) ).
% local.ord_eq_le_trans
thf(fact_3_local_Oord__le__eq__trans,axiom,
! [A2: a,B2: a,C: a] :
( ( less_eq @ A2 @ B2 )
=> ( ( B2 = C )
=> ( less_eq @ A2 @ C ) ) ) ).
% local.ord_le_eq_trans
thf(fact_4__092_060open_062lhd_A_IlSup_AY_J_A_061_Alhd_Axs_092_060close_062,axiom,
( ( coindu1046438764le_lhd @ a @ ( coindu526335360e_lSup @ a @ ya ) )
= ( coindu1046438764le_lhd @ a @ xs ) ) ).
% \<open>lhd (lSup Y) = lhd xs\<close>
thf(fact_5__092_060open_062lhd_A_Iltl_A_IlSup_AY_J_J_A_061_Alhd_A_Iltl_Axs_J_092_060close_062,axiom,
( ( coindu1046438764le_lhd @ a @ ( coindu1047225960le_ltl @ a @ ( coindu526335360e_lSup @ a @ ya ) ) )
= ( coindu1046438764le_lhd @ a @ ( coindu1047225960le_ltl @ a @ xs ) ) ) ).
% \<open>lhd (ltl (lSup Y)) = lhd (ltl xs)\<close>
thf(fact_6_local_Omax__def,axiom,
! [A2: a,B2: a] :
( ( ( less_eq @ A2 @ B2 )
=> ( ( max @ a @ less_eq @ A2 @ B2 )
= B2 ) )
& ( ~ ( less_eq @ A2 @ B2 )
=> ( ( max @ a @ less_eq @ A2 @ B2 )
= A2 ) ) ) ).
% local.max_def
thf(fact_7_local_Omin__def,axiom,
! [A2: a,B2: a] :
( ( ( less_eq @ A2 @ B2 )
=> ( ( min @ a @ less_eq @ A2 @ B2 )
= A2 ) )
& ( ~ ( less_eq @ A2 @ B2 )
=> ( ( min @ a @ less_eq @ A2 @ B2 )
= B2 ) ) ) ).
% local.min_def
thf(fact_8_lsorted_I2_J,axiom,
! [X: coindu1593790203_llist @ a] :
( ( member @ ( coindu1593790203_llist @ a ) @ X @ ya )
=> ( coindu543932918sorted @ a @ less_eq @ X ) ) ).
% lsorted(2)
thf(fact_9_lsorted_I3_J,axiom,
~ ( coindu335574135_lnull @ a @ ( coindu526335360e_lSup @ a @ ya ) ) ).
% lsorted(3)
thf(fact_10_lsorted_I4_J,axiom,
~ ( coindu335574135_lnull @ a @ ( coindu1047225960le_ltl @ a @ ( coindu526335360e_lSup @ a @ ya ) ) ) ).
% lsorted(4)
thf(fact_11__092_060open_062_092_060not_062_Alnull_A_Iltl_Axs_J_092_060close_062,axiom,
~ ( coindu335574135_lnull @ a @ ( coindu1047225960le_ltl @ a @ xs ) ) ).
% \<open>\<not> lnull (ltl xs)\<close>
thf(fact_12_sorted,axiom,
! [Xs: coindu1593790203_llist @ a] :
( ( member @ ( coindu1593790203_llist @ a ) @ Xs @ ya )
=> ( coindu543932918sorted @ a @ less_eq @ Xs ) ) ).
% sorted
thf(fact_13_lsorted__ltlI,axiom,
! [Xs: coindu1593790203_llist @ a] :
( ( coindu543932918sorted @ a @ less_eq @ Xs )
=> ( coindu543932918sorted @ a @ less_eq @ ( coindu1047225960le_ltl @ a @ Xs ) ) ) ).
% lsorted_ltlI
thf(fact_14__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062xs_O_A_092_060lbrakk_062xs_A_092_060in_062_AY_059_A_092_060not_062_Alnull_Axs_059_A_092_060not_062_Alnull_A_Iltl_Axs_J_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [Xs2: coindu1593790203_llist @ a] :
( ( member @ ( coindu1593790203_llist @ a ) @ Xs2 @ ya )
=> ( ~ ( coindu335574135_lnull @ a @ Xs2 )
=> ( coindu335574135_lnull @ a @ ( coindu1047225960le_ltl @ a @ Xs2 ) ) ) ) ).
% \<open>\<And>thesis. (\<And>xs. \<lbrakk>xs \<in> Y; \<not> lnull xs; \<not> lnull (ltl xs)\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_15_local_OatLeastAtMost__iff,axiom,
! [I: a,L: a,U: a] :
( ( member @ a @ I @ ( set_atLeastAtMost @ a @ less_eq @ L @ U ) )
= ( ( less_eq @ L @ I )
& ( less_eq @ I @ U ) ) ) ).
% local.atLeastAtMost_iff
thf(fact_16__092_060open_062_092_060not_062_Alnull_Axs_092_060close_062,axiom,
~ ( coindu335574135_lnull @ a @ xs ) ).
% \<open>\<not> lnull xs\<close>
thf(fact_17_local_OatLeast__iff,axiom,
! [I: a,K: a] :
( ( member @ a @ I @ ( set_atLeast @ a @ less_eq @ K ) )
= ( less_eq @ K @ I ) ) ).
% local.atLeast_iff
thf(fact_18_local_OatMost__iff,axiom,
! [I: a,K: a] :
( ( member @ a @ I @ ( set_atMost @ a @ less_eq @ K ) )
= ( less_eq @ I @ K ) ) ).
% local.atMost_iff
thf(fact_19_lsorted__coinduct_H,axiom,
! [X2: ( coindu1593790203_llist @ a ) > $o,Xs: coindu1593790203_llist @ a] :
( ( X2 @ Xs )
=> ( ! [Xs2: coindu1593790203_llist @ a] :
( ( X2 @ Xs2 )
=> ( ~ ( coindu335574135_lnull @ a @ Xs2 )
=> ( ~ ( coindu335574135_lnull @ a @ ( coindu1047225960le_ltl @ a @ Xs2 ) )
=> ( ( less_eq @ ( coindu1046438764le_lhd @ a @ Xs2 ) @ ( coindu1046438764le_lhd @ a @ ( coindu1047225960le_ltl @ a @ Xs2 ) ) )
& ( ( X2 @ ( coindu1047225960le_ltl @ a @ Xs2 ) )
| ( coindu543932918sorted @ a @ less_eq @ ( coindu1047225960le_ltl @ a @ Xs2 ) ) ) ) ) ) )
=> ( coindu543932918sorted @ a @ less_eq @ Xs ) ) ) ).
% lsorted_coinduct'
thf(fact_20_lsorted__lhdD,axiom,
! [Xs: coindu1593790203_llist @ a] :
( ( coindu543932918sorted @ a @ less_eq @ Xs )
=> ( ~ ( coindu335574135_lnull @ a @ Xs )
=> ( ~ ( coindu335574135_lnull @ a @ ( coindu1047225960le_ltl @ a @ Xs ) )
=> ( less_eq @ ( coindu1046438764le_lhd @ a @ Xs ) @ ( coindu1046438764le_lhd @ a @ ( coindu1047225960le_ltl @ a @ Xs ) ) ) ) ) ) ).
% lsorted_lhdD
thf(fact_21_chain,axiom,
comple1193779247_chain @ ( coindu1593790203_llist @ a ) @ ( coindu1696667936prefix @ a ) @ ya ).
% chain
thf(fact_22_llist_Oexpand,axiom,
! [A: $tType,Llist: coindu1593790203_llist @ A,Llist2: coindu1593790203_llist @ A] :
( ( ( coindu335574135_lnull @ A @ Llist )
= ( coindu335574135_lnull @ A @ Llist2 ) )
=> ( ( ~ ( coindu335574135_lnull @ A @ Llist )
=> ( ~ ( coindu335574135_lnull @ A @ Llist2 )
=> ( ( ( coindu1046438764le_lhd @ A @ Llist )
= ( coindu1046438764le_lhd @ A @ Llist2 ) )
& ( ( coindu1047225960le_ltl @ A @ Llist )
= ( coindu1047225960le_ltl @ A @ Llist2 ) ) ) ) )
=> ( Llist = Llist2 ) ) ) ).
% llist.expand
thf(fact_23_lprefix__refl,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] : ( coindu1696667936prefix @ A @ Xs @ Xs ) ).
% lprefix_refl
thf(fact_24_llist_Oleq__refl,axiom,
! [A: $tType,X3: coindu1593790203_llist @ A] : ( coindu1696667936prefix @ A @ X3 @ X3 ) ).
% llist.leq_refl
thf(fact_25_lnull__lSup,axiom,
! [A: $tType,A3: set @ ( coindu1593790203_llist @ A )] :
( ( coindu335574135_lnull @ A @ ( coindu526335360e_lSup @ A @ A3 ) )
= ( ! [X4: coindu1593790203_llist @ A] :
( ( member @ ( coindu1593790203_llist @ A ) @ X4 @ A3 )
=> ( coindu335574135_lnull @ A @ X4 ) ) ) ) ).
% lnull_lSup
thf(fact_26_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_27_lnull__lprefix,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ A @ Xs )
=> ( coindu1696667936prefix @ A @ Xs @ Ys ) ) ).
% lnull_lprefix
thf(fact_28_lprefix__lnull,axiom,
! [A: $tType,Ys: coindu1593790203_llist @ A,Xs: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ A @ Ys )
=> ( ( coindu1696667936prefix @ A @ Xs @ Ys )
= ( coindu335574135_lnull @ A @ Xs ) ) ) ).
% lprefix_lnull
thf(fact_29_lprefix__trans,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A,Zs: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ Xs @ Ys )
=> ( ( coindu1696667936prefix @ A @ Ys @ Zs )
=> ( coindu1696667936prefix @ A @ Xs @ Zs ) ) ) ).
% lprefix_trans
thf(fact_30_lprefix__lnullD,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ Xs @ Ys )
=> ( ( coindu335574135_lnull @ A @ Ys )
=> ( coindu335574135_lnull @ A @ Xs ) ) ) ).
% lprefix_lnullD
thf(fact_31_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_32_llist_Oleq__trans,axiom,
! [A: $tType,X3: coindu1593790203_llist @ A,Y: coindu1593790203_llist @ A,Z: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ X3 @ Y )
=> ( ( coindu1696667936prefix @ A @ Y @ Z )
=> ( coindu1696667936prefix @ A @ X3 @ Z ) ) ) ).
% llist.leq_trans
thf(fact_33_lprefix__antisym,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ Xs @ Ys )
=> ( ( coindu1696667936prefix @ A @ Ys @ Xs )
=> ( Xs = Ys ) ) ) ).
% lprefix_antisym
thf(fact_34_llist_Oleq__antisym,axiom,
! [A: $tType,X3: coindu1593790203_llist @ A,Y: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ X3 @ Y )
=> ( ( coindu1696667936prefix @ A @ Y @ X3 )
=> ( X3 = Y ) ) ) ).
% llist.leq_antisym
thf(fact_35_lprefix__not__lnullD,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ Xs @ Ys )
=> ( ~ ( coindu335574135_lnull @ A @ Xs )
=> ~ ( coindu335574135_lnull @ A @ Ys ) ) ) ).
% lprefix_not_lnullD
thf(fact_36_lprefix__down__linear,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Zs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ Xs @ Zs )
=> ( ( coindu1696667936prefix @ A @ Ys @ Zs )
=> ( ( coindu1696667936prefix @ A @ Xs @ Ys )
| ( coindu1696667936prefix @ A @ Ys @ Xs ) ) ) ) ).
% lprefix_down_linear
thf(fact_37_chain__lprefix__lSup,axiom,
! [A: $tType,A3: set @ ( coindu1593790203_llist @ A ),Xs: coindu1593790203_llist @ A] :
( ( comple1193779247_chain @ ( coindu1593790203_llist @ A ) @ ( coindu1696667936prefix @ A ) @ A3 )
=> ( ( member @ ( coindu1593790203_llist @ A ) @ Xs @ A3 )
=> ( coindu1696667936prefix @ A @ Xs @ ( coindu526335360e_lSup @ A @ A3 ) ) ) ) ).
% chain_lprefix_lSup
thf(fact_38_chain__lSup__lprefix,axiom,
! [A: $tType,A3: set @ ( coindu1593790203_llist @ A ),Zs: coindu1593790203_llist @ A] :
( ( comple1193779247_chain @ ( coindu1593790203_llist @ A ) @ ( coindu1696667936prefix @ A ) @ A3 )
=> ( ! [Xs2: coindu1593790203_llist @ A] :
( ( member @ ( coindu1593790203_llist @ A ) @ Xs2 @ A3 )
=> ( coindu1696667936prefix @ A @ Xs2 @ Zs ) )
=> ( coindu1696667936prefix @ A @ ( coindu526335360e_lSup @ A @ A3 ) @ Zs ) ) ) ).
% chain_lSup_lprefix
thf(fact_39_llist_Olub__upper,axiom,
! [A: $tType,A3: set @ ( coindu1593790203_llist @ A ),X3: coindu1593790203_llist @ A] :
( ( comple1193779247_chain @ ( coindu1593790203_llist @ A ) @ ( coindu1696667936prefix @ A ) @ A3 )
=> ( ( member @ ( coindu1593790203_llist @ A ) @ X3 @ A3 )
=> ( coindu1696667936prefix @ A @ X3 @ ( coindu526335360e_lSup @ A @ A3 ) ) ) ) ).
% llist.lub_upper
thf(fact_40_llist_Olub__least,axiom,
! [A: $tType,A3: set @ ( coindu1593790203_llist @ A ),Z: coindu1593790203_llist @ A] :
( ( comple1193779247_chain @ ( coindu1593790203_llist @ A ) @ ( coindu1696667936prefix @ A ) @ A3 )
=> ( ! [X5: coindu1593790203_llist @ A] :
( ( member @ ( coindu1593790203_llist @ A ) @ X5 @ A3 )
=> ( coindu1696667936prefix @ A @ X5 @ Z ) )
=> ( coindu1696667936prefix @ A @ ( coindu526335360e_lSup @ A @ A3 ) @ Z ) ) ) ).
% llist.lub_least
thf(fact_41_lprefix__lhdD,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ Xs @ Ys )
=> ( ~ ( coindu335574135_lnull @ A @ Xs )
=> ( ( coindu1046438764le_lhd @ A @ Xs )
= ( coindu1046438764le_lhd @ A @ Ys ) ) ) ) ).
% lprefix_lhdD
thf(fact_42_lhd__lSup__eq,axiom,
! [A: $tType,Y2: set @ ( coindu1593790203_llist @ A ),Xs: coindu1593790203_llist @ A] :
( ( comple1193779247_chain @ ( coindu1593790203_llist @ A ) @ ( coindu1696667936prefix @ A ) @ Y2 )
=> ( ( member @ ( coindu1593790203_llist @ A ) @ Xs @ Y2 )
=> ( ~ ( coindu335574135_lnull @ A @ Xs )
=> ( ( coindu1046438764le_lhd @ A @ ( coindu526335360e_lSup @ A @ Y2 ) )
= ( coindu1046438764le_lhd @ A @ Xs ) ) ) ) ) ).
% lhd_lSup_eq
thf(fact_43_mem__Collect__eq,axiom,
! [A: $tType,A2: A,P: A > $o] :
( ( member @ A @ A2 @ ( collect @ A @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_44_Collect__mem__eq,axiom,
! [A: $tType,A3: set @ A] :
( ( collect @ A
@ ^ [X4: A] : ( member @ A @ X4 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_45_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X5: A] :
( ( P @ X5 )
= ( Q @ X5 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_46_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X5: A] :
( ( F @ X5 )
= ( G @ X5 ) )
=> ( F = G ) ) ).
% ext
thf(fact_47_lprefix__coinduct,axiom,
! [A: $tType,P: ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) > $o,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( P @ Xs @ Ys )
=> ( ! [Xs2: coindu1593790203_llist @ A,Ys2: coindu1593790203_llist @ A] :
( ( P @ Xs2 @ Ys2 )
=> ( ( ( coindu335574135_lnull @ A @ Ys2 )
=> ( coindu335574135_lnull @ A @ Xs2 ) )
& ( ~ ( coindu335574135_lnull @ A @ Xs2 )
=> ( ~ ( coindu335574135_lnull @ A @ Ys2 )
=> ( ( ( coindu1046438764le_lhd @ A @ Xs2 )
= ( coindu1046438764le_lhd @ A @ Ys2 ) )
& ( ( P @ ( coindu1047225960le_ltl @ A @ Xs2 ) @ ( coindu1047225960le_ltl @ A @ Ys2 ) )
| ( coindu1696667936prefix @ A @ ( coindu1047225960le_ltl @ A @ Xs2 ) @ ( coindu1047225960le_ltl @ A @ Ys2 ) ) ) ) ) ) ) )
=> ( coindu1696667936prefix @ A @ Xs @ Ys ) ) ) ).
% lprefix_coinduct
thf(fact_48_lprefix__expand,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( ~ ( coindu335574135_lnull @ A @ Xs )
=> ( ~ ( coindu335574135_lnull @ A @ Ys )
& ( ( coindu1046438764le_lhd @ A @ Xs )
= ( coindu1046438764le_lhd @ A @ Ys ) )
& ( coindu1696667936prefix @ A @ ( coindu1047225960le_ltl @ A @ Xs ) @ ( coindu1047225960le_ltl @ A @ Ys ) ) ) )
=> ( coindu1696667936prefix @ A @ Xs @ Ys ) ) ).
% lprefix_expand
thf(fact_49_lSup_Odisc__iff_I2_J,axiom,
! [A: $tType,A3: set @ ( coindu1593790203_llist @ A )] :
( ( coindu335574135_lnull @ A @ ( coindu526335360e_lSup @ A @ A3 ) )
= ( ! [X4: coindu1593790203_llist @ A] :
( ( member @ ( coindu1593790203_llist @ A ) @ X4 @ A3 )
=> ( coindu335574135_lnull @ A @ X4 ) ) ) ) ).
% lSup.disc_iff(2)
thf(fact_50_lSup_Odisc_I1_J,axiom,
! [A: $tType,A3: set @ ( coindu1593790203_llist @ A )] :
( ! [X5: coindu1593790203_llist @ A] :
( ( member @ ( coindu1593790203_llist @ A ) @ X5 @ A3 )
=> ( coindu335574135_lnull @ A @ X5 ) )
=> ( coindu335574135_lnull @ A @ ( coindu526335360e_lSup @ A @ A3 ) ) ) ).
% lSup.disc(1)
thf(fact_51_lSup_Odisc_I2_J,axiom,
! [A: $tType,A3: set @ ( coindu1593790203_llist @ A )] :
( ~ ! [X: coindu1593790203_llist @ A] :
( ( member @ ( coindu1593790203_llist @ A ) @ X @ A3 )
=> ( coindu335574135_lnull @ A @ X ) )
=> ~ ( coindu335574135_lnull @ A @ ( coindu526335360e_lSup @ A @ A3 ) ) ) ).
% lSup.disc(2)
thf(fact_52_lnull__ltlI,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ A @ Xs )
=> ( coindu335574135_lnull @ A @ ( coindu1047225960le_ltl @ A @ Xs ) ) ) ).
% lnull_ltlI
thf(fact_53_ord_Olsorted__coinduct_H,axiom,
! [A: $tType,X2: ( coindu1593790203_llist @ A ) > $o,Xs: coindu1593790203_llist @ A,Less_eq: A > A > $o] :
( ( X2 @ Xs )
=> ( ! [Xs2: coindu1593790203_llist @ A] :
( ( X2 @ Xs2 )
=> ( ~ ( coindu335574135_lnull @ A @ Xs2 )
=> ( ~ ( coindu335574135_lnull @ A @ ( coindu1047225960le_ltl @ A @ Xs2 ) )
=> ( ( Less_eq @ ( coindu1046438764le_lhd @ A @ Xs2 ) @ ( coindu1046438764le_lhd @ A @ ( coindu1047225960le_ltl @ A @ Xs2 ) ) )
& ( ( X2 @ ( coindu1047225960le_ltl @ A @ Xs2 ) )
| ( coindu543932918sorted @ A @ Less_eq @ ( coindu1047225960le_ltl @ A @ Xs2 ) ) ) ) ) ) )
=> ( coindu543932918sorted @ A @ Less_eq @ Xs ) ) ) ).
% ord.lsorted_coinduct'
thf(fact_54_ord_Olsorted__lhdD,axiom,
! [A: $tType,Less_eq: A > A > $o,Xs: coindu1593790203_llist @ A] :
( ( coindu543932918sorted @ A @ Less_eq @ Xs )
=> ( ~ ( coindu335574135_lnull @ A @ Xs )
=> ( ~ ( coindu335574135_lnull @ A @ ( coindu1047225960le_ltl @ A @ Xs ) )
=> ( Less_eq @ ( coindu1046438764le_lhd @ A @ Xs ) @ ( coindu1046438764le_lhd @ A @ ( coindu1047225960le_ltl @ A @ Xs ) ) ) ) ) ) ).
% ord.lsorted_lhdD
thf(fact_55_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_56_lprefix__ltlI,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ Xs @ Ys )
=> ( coindu1696667936prefix @ A @ ( coindu1047225960le_ltl @ A @ Xs ) @ ( coindu1047225960le_ltl @ A @ Ys ) ) ) ).
% lprefix_ltlI
thf(fact_57_ord_Olsorted__ltlI,axiom,
! [A: $tType,Less_eq: A > A > $o,Xs: coindu1593790203_llist @ A] :
( ( coindu543932918sorted @ A @ Less_eq @ Xs )
=> ( coindu543932918sorted @ A @ Less_eq @ ( coindu1047225960le_ltl @ A @ Xs ) ) ) ).
% ord.lsorted_ltlI
thf(fact_58_llist_Ocoinduct__strong,axiom,
! [A: $tType,R: ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) > $o,Llist: coindu1593790203_llist @ A,Llist2: coindu1593790203_llist @ A] :
( ( R @ Llist @ Llist2 )
=> ( ! [Llist3: coindu1593790203_llist @ A,Llist4: coindu1593790203_llist @ A] :
( ( R @ Llist3 @ Llist4 )
=> ( ( ( coindu335574135_lnull @ A @ Llist3 )
= ( coindu335574135_lnull @ A @ Llist4 ) )
& ( ~ ( coindu335574135_lnull @ A @ Llist3 )
=> ( ~ ( coindu335574135_lnull @ A @ Llist4 )
=> ( ( ( coindu1046438764le_lhd @ A @ Llist3 )
= ( coindu1046438764le_lhd @ A @ Llist4 ) )
& ( ( R @ ( coindu1047225960le_ltl @ A @ Llist3 ) @ ( coindu1047225960le_ltl @ A @ Llist4 ) )
| ( ( coindu1047225960le_ltl @ A @ Llist3 )
= ( coindu1047225960le_ltl @ A @ Llist4 ) ) ) ) ) ) ) )
=> ( Llist = Llist2 ) ) ) ).
% llist.coinduct_strong
thf(fact_59_llist_Ocoinduct,axiom,
! [A: $tType,R: ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) > $o,Llist: coindu1593790203_llist @ A,Llist2: coindu1593790203_llist @ A] :
( ( R @ Llist @ Llist2 )
=> ( ! [Llist3: coindu1593790203_llist @ A,Llist4: coindu1593790203_llist @ A] :
( ( R @ Llist3 @ Llist4 )
=> ( ( ( coindu335574135_lnull @ A @ Llist3 )
= ( coindu335574135_lnull @ A @ Llist4 ) )
& ( ~ ( coindu335574135_lnull @ A @ Llist3 )
=> ( ~ ( coindu335574135_lnull @ A @ Llist4 )
=> ( ( ( coindu1046438764le_lhd @ A @ Llist3 )
= ( coindu1046438764le_lhd @ A @ Llist4 ) )
& ( R @ ( coindu1047225960le_ltl @ A @ Llist3 ) @ ( coindu1047225960le_ltl @ A @ Llist4 ) ) ) ) ) ) )
=> ( Llist = Llist2 ) ) ) ).
% llist.coinduct
thf(fact_60_local_OatLeastAtMost__def,axiom,
! [L: a,U: a] :
( ( set_atLeastAtMost @ a @ less_eq @ L @ U )
= ( inf_inf @ ( set @ a ) @ ( set_atLeast @ a @ less_eq @ L ) @ ( set_atMost @ a @ less_eq @ U ) ) ) ).
% local.atLeastAtMost_def
thf(fact_61_lsorted__LCons_H,axiom,
! [X3: a,Xs: coindu1593790203_llist @ a] :
( ( coindu543932918sorted @ a @ less_eq @ ( coindu1121789889_LCons @ a @ X3 @ Xs ) )
= ( ~ ( coindu335574135_lnull @ a @ Xs )
=> ( ( less_eq @ X3 @ ( coindu1046438764le_lhd @ a @ Xs ) )
& ( coindu543932918sorted @ a @ less_eq @ Xs ) ) ) ) ).
% lsorted_LCons'
thf(fact_62_LNil,axiom,
coindu543932918sorted @ a @ less_eq @ ( coindu1598213697e_LNil @ a ) ).
% LNil
thf(fact_63_LCons__LCons,axiom,
! [X3: a,Y: a,Xs: coindu1593790203_llist @ a] :
( ( less_eq @ X3 @ Y )
=> ( ( coindu543932918sorted @ a @ less_eq @ ( coindu1121789889_LCons @ a @ Y @ Xs ) )
=> ( coindu543932918sorted @ a @ less_eq @ ( coindu1121789889_LCons @ a @ X3 @ ( coindu1121789889_LCons @ a @ Y @ Xs ) ) ) ) ) ).
% LCons_LCons
thf(fact_64_ord_Omax__def,axiom,
! [A: $tType] :
( ( max @ A )
= ( ^ [Less_eq2: A > A > $o,A4: A,B3: A] : ( if @ A @ ( Less_eq2 @ A4 @ B3 ) @ B3 @ A4 ) ) ) ).
% ord.max_def
thf(fact_65_ord_Omin__def,axiom,
! [A: $tType] :
( ( min @ A )
= ( ^ [Less_eq2: A > A > $o,A4: A,B3: A] : ( if @ A @ ( Less_eq2 @ A4 @ B3 ) @ A4 @ B3 ) ) ) ).
% ord.min_def
thf(fact_66_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_67_llist_Ocase__eq__if,axiom,
! [A: $tType,B: $tType] :
( ( coindu882539134_llist @ B @ A )
= ( ^ [F1: B,F2: A > ( coindu1593790203_llist @ A ) > B,Llist5: coindu1593790203_llist @ A] : ( if @ B @ ( coindu335574135_lnull @ A @ Llist5 ) @ F1 @ ( F2 @ ( coindu1046438764le_lhd @ A @ Llist5 ) @ ( coindu1047225960le_ltl @ A @ Llist5 ) ) ) ) ) ).
% llist.case_eq_if
thf(fact_68_ord_OatMost__iff,axiom,
! [A: $tType,I: A,Less_eq: A > A > $o,K: A] :
( ( member @ A @ I @ ( set_atMost @ A @ Less_eq @ K ) )
= ( Less_eq @ I @ K ) ) ).
% ord.atMost_iff
thf(fact_69_ord_OatLeast__iff,axiom,
! [A: $tType,I: A,Less_eq: A > A > $o,K: A] :
( ( member @ A @ I @ ( set_atLeast @ A @ Less_eq @ K ) )
= ( Less_eq @ K @ I ) ) ).
% ord.atLeast_iff
thf(fact_70_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_71_Singleton,axiom,
! [X3: a] : ( coindu543932918sorted @ a @ less_eq @ ( coindu1121789889_LCons @ a @ X3 @ ( coindu1598213697e_LNil @ a ) ) ) ).
% Singleton
thf(fact_72_lsorted_Ocases,axiom,
! [A2: coindu1593790203_llist @ a] :
( ( coindu543932918sorted @ a @ less_eq @ A2 )
=> ( ( A2
!= ( coindu1598213697e_LNil @ a ) )
=> ( ! [X5: a] :
( A2
!= ( coindu1121789889_LCons @ a @ X5 @ ( coindu1598213697e_LNil @ a ) ) )
=> ~ ! [X5: a,Y3: a,Xs2: coindu1593790203_llist @ a] :
( ( A2
= ( coindu1121789889_LCons @ a @ X5 @ ( coindu1121789889_LCons @ a @ Y3 @ Xs2 ) ) )
=> ( ( less_eq @ X5 @ Y3 )
=> ~ ( coindu543932918sorted @ a @ less_eq @ ( coindu1121789889_LCons @ a @ Y3 @ Xs2 ) ) ) ) ) ) ) ).
% lsorted.cases
thf(fact_73_lsorted_Ocoinduct,axiom,
! [X2: ( coindu1593790203_llist @ a ) > $o,X3: coindu1593790203_llist @ a] :
( ( X2 @ X3 )
=> ( ! [X5: coindu1593790203_llist @ a] :
( ( X2 @ X5 )
=> ( ( X5
= ( coindu1598213697e_LNil @ a ) )
| ? [Xa: a] :
( X5
= ( coindu1121789889_LCons @ a @ Xa @ ( coindu1598213697e_LNil @ a ) ) )
| ? [Xa: a,Y4: a,Xs3: coindu1593790203_llist @ a] :
( ( X5
= ( coindu1121789889_LCons @ a @ Xa @ ( coindu1121789889_LCons @ a @ Y4 @ Xs3 ) ) )
& ( less_eq @ Xa @ Y4 )
& ( ( X2 @ ( coindu1121789889_LCons @ a @ Y4 @ Xs3 ) )
| ( coindu543932918sorted @ a @ less_eq @ ( coindu1121789889_LCons @ a @ Y4 @ Xs3 ) ) ) ) ) )
=> ( coindu543932918sorted @ a @ less_eq @ X3 ) ) ) ).
% lsorted.coinduct
thf(fact_74_lsorted_Osimps,axiom,
! [A2: coindu1593790203_llist @ a] :
( ( coindu543932918sorted @ a @ less_eq @ A2 )
= ( ( A2
= ( coindu1598213697e_LNil @ a ) )
| ? [X4: a] :
( A2
= ( coindu1121789889_LCons @ a @ X4 @ ( coindu1598213697e_LNil @ a ) ) )
| ? [X4: a,Y5: a,Xs4: coindu1593790203_llist @ a] :
( ( A2
= ( coindu1121789889_LCons @ a @ X4 @ ( coindu1121789889_LCons @ a @ Y5 @ Xs4 ) ) )
& ( less_eq @ X4 @ Y5 )
& ( coindu543932918sorted @ a @ less_eq @ ( coindu1121789889_LCons @ a @ Y5 @ Xs4 ) ) ) ) ) ).
% lsorted.simps
thf(fact_75_LCons__lprefix__LCons,axiom,
! [A: $tType,X3: A,Xs: coindu1593790203_llist @ A,Y: A,Ys: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ ( coindu1121789889_LCons @ A @ X3 @ Xs ) @ ( coindu1121789889_LCons @ A @ Y @ Ys ) )
= ( ( X3 = Y )
& ( coindu1696667936prefix @ A @ Xs @ Ys ) ) ) ).
% LCons_lprefix_LCons
thf(fact_76_lprefix__code_I1_J,axiom,
! [A: $tType,Ys: coindu1593790203_llist @ A] : ( coindu1696667936prefix @ A @ ( coindu1598213697e_LNil @ A ) @ Ys ) ).
% lprefix_code(1)
thf(fact_77_unfold__llist__eq__LCons,axiom,
! [A: $tType,B: $tType,IS_LNIL: B > $o,LHD: B > A,LTL: B > B,B2: B,X3: A,Xs: coindu1593790203_llist @ A] :
( ( ( coindu1599971794_llist @ B @ A @ IS_LNIL @ LHD @ LTL @ B2 )
= ( coindu1121789889_LCons @ A @ X3 @ Xs ) )
= ( ~ ( IS_LNIL @ B2 )
& ( X3
= ( LHD @ B2 ) )
& ( Xs
= ( coindu1599971794_llist @ B @ A @ IS_LNIL @ LHD @ LTL @ ( LTL @ B2 ) ) ) ) ) ).
% unfold_llist_eq_LCons
thf(fact_78_unfold__llist_Odisc__iff_I1_J,axiom,
! [B: $tType,A: $tType,P2: A > $o,G21: A > B,G22: A > A,A2: A] :
( ( coindu335574135_lnull @ B @ ( coindu1599971794_llist @ A @ B @ P2 @ G21 @ G22 @ A2 ) )
= ( P2 @ A2 ) ) ).
% unfold_llist.disc_iff(1)
thf(fact_79_unfold__llist_Odisc__iff_I2_J,axiom,
! [B: $tType,A: $tType,P2: A > $o,G21: A > B,G22: A > A,A2: A] :
( ( ~ ( coindu335574135_lnull @ B @ ( coindu1599971794_llist @ A @ B @ P2 @ G21 @ G22 @ A2 ) ) )
= ( ~ ( P2 @ A2 ) ) ) ).
% unfold_llist.disc_iff(2)
thf(fact_80_lprefix__LNil,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ Xs @ ( coindu1598213697e_LNil @ A ) )
= ( coindu335574135_lnull @ A @ Xs ) ) ).
% lprefix_LNil
thf(fact_81_lsorted__LCons__LCons,axiom,
! [X3: a,Y: a,Xs: coindu1593790203_llist @ a] :
( ( coindu543932918sorted @ a @ less_eq @ ( coindu1121789889_LCons @ a @ X3 @ ( coindu1121789889_LCons @ a @ Y @ Xs ) ) )
= ( ( less_eq @ X3 @ Y )
& ( coindu543932918sorted @ a @ less_eq @ ( coindu1121789889_LCons @ a @ Y @ Xs ) ) ) ) ).
% lsorted_LCons_LCons
thf(fact_82_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_83_lsorted__code_I1_J,axiom,
coindu543932918sorted @ a @ less_eq @ ( coindu1598213697e_LNil @ a ) ).
% lsorted_code(1)
thf(fact_84_lsorted__code_I2_J,axiom,
! [X3: a] : ( coindu543932918sorted @ a @ less_eq @ ( coindu1121789889_LCons @ a @ X3 @ ( coindu1598213697e_LNil @ a ) ) ) ).
% lsorted_code(2)
thf(fact_85_lprefix__code_I2_J,axiom,
! [A: $tType,X3: A,Xs: coindu1593790203_llist @ A] :
~ ( coindu1696667936prefix @ A @ ( coindu1121789889_LCons @ A @ X3 @ Xs ) @ ( coindu1598213697e_LNil @ A ) ) ).
% lprefix_code(2)
thf(fact_86_lprefix_Ocases,axiom,
! [A: $tType,A1: coindu1593790203_llist @ A,A22: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ A1 @ A22 )
=> ( ( ( A1
= ( coindu1598213697e_LNil @ A ) )
=> ! [Xs2: coindu1593790203_llist @ A] : A22 != Xs2 )
=> ~ ! [Xs2: coindu1593790203_llist @ A,Ys2: coindu1593790203_llist @ A,X5: A] :
( ( A1
= ( coindu1121789889_LCons @ A @ X5 @ Xs2 ) )
=> ( ( A22
= ( coindu1121789889_LCons @ A @ X5 @ Ys2 ) )
=> ~ ( coindu1696667936prefix @ A @ Xs2 @ Ys2 ) ) ) ) ) ).
% lprefix.cases
thf(fact_87_lprefix_Osimps,axiom,
! [A: $tType] :
( ( coindu1696667936prefix @ A )
= ( ^ [A12: coindu1593790203_llist @ A,A23: coindu1593790203_llist @ A] :
( ? [Xs4: coindu1593790203_llist @ A] :
( ( A12
= ( coindu1598213697e_LNil @ A ) )
& ( A23 = Xs4 ) )
| ? [Xs4: coindu1593790203_llist @ A,Ys3: coindu1593790203_llist @ A,X4: A] :
( ( A12
= ( coindu1121789889_LCons @ A @ X4 @ Xs4 ) )
& ( A23
= ( coindu1121789889_LCons @ A @ X4 @ Ys3 ) )
& ( coindu1696667936prefix @ A @ Xs4 @ Ys3 ) ) ) ) ) ).
% lprefix.simps
thf(fact_88_lprefix_Ocoinduct,axiom,
! [A: $tType,X2: ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) > $o,X3: coindu1593790203_llist @ A,Xa2: coindu1593790203_llist @ A] :
( ( X2 @ X3 @ Xa2 )
=> ( ! [X5: coindu1593790203_llist @ A,Xa3: coindu1593790203_llist @ A] :
( ( X2 @ X5 @ Xa3 )
=> ( ? [Xs3: coindu1593790203_llist @ A] :
( ( X5
= ( coindu1598213697e_LNil @ A ) )
& ( Xa3 = Xs3 ) )
| ? [Xs3: coindu1593790203_llist @ A,Ys4: coindu1593790203_llist @ A,Xb: A] :
( ( X5
= ( coindu1121789889_LCons @ A @ Xb @ Xs3 ) )
& ( Xa3
= ( coindu1121789889_LCons @ A @ Xb @ Ys4 ) )
& ( ( X2 @ Xs3 @ Ys4 )
| ( coindu1696667936prefix @ A @ Xs3 @ Ys4 ) ) ) ) )
=> ( coindu1696667936prefix @ A @ X3 @ Xa2 ) ) ) ).
% lprefix.coinduct
thf(fact_89_lprefix__LCons__conv,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Y: A,Ys: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ Xs @ ( coindu1121789889_LCons @ A @ Y @ Ys ) )
= ( ( Xs
= ( coindu1598213697e_LNil @ A ) )
| ? [Xs5: coindu1593790203_llist @ A] :
( ( Xs
= ( coindu1121789889_LCons @ A @ Y @ Xs5 ) )
& ( coindu1696667936prefix @ A @ Xs5 @ Ys ) ) ) ) ).
% lprefix_LCons_conv
thf(fact_90_unfold__llist_Ocode,axiom,
! [B: $tType,A: $tType] :
( ( coindu1599971794_llist @ A @ B )
= ( ^ [P3: A > $o,G212: A > B,G222: A > A,A4: A] : ( if @ ( coindu1593790203_llist @ B ) @ ( P3 @ A4 ) @ ( coindu1598213697e_LNil @ B ) @ ( coindu1121789889_LCons @ B @ ( G212 @ A4 ) @ ( coindu1599971794_llist @ A @ B @ P3 @ G212 @ G222 @ ( G222 @ A4 ) ) ) ) ) ) ).
% unfold_llist.code
thf(fact_91_neq__LNil__conv,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( ( Xs
!= ( coindu1598213697e_LNil @ A ) )
= ( ? [X4: A,Xs5: coindu1593790203_llist @ A] :
( Xs
= ( coindu1121789889_LCons @ A @ X4 @ Xs5 ) ) ) ) ).
% neq_LNil_conv
thf(fact_92_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_93_unfold__llist_Octr_I1_J,axiom,
! [A: $tType,B: $tType,P2: A > $o,A2: A,G21: A > B,G22: A > A] :
( ( P2 @ A2 )
=> ( ( coindu1599971794_llist @ A @ B @ P2 @ G21 @ G22 @ A2 )
= ( coindu1598213697e_LNil @ B ) ) ) ).
% unfold_llist.ctr(1)
thf(fact_94_unfold__llist_Octr_I2_J,axiom,
! [B: $tType,A: $tType,P2: A > $o,A2: A,G21: A > B,G22: A > A] :
( ~ ( P2 @ A2 )
=> ( ( coindu1599971794_llist @ A @ B @ P2 @ G21 @ G22 @ A2 )
= ( coindu1121789889_LCons @ B @ ( G21 @ A2 ) @ ( coindu1599971794_llist @ A @ B @ P2 @ G21 @ G22 @ ( G22 @ A2 ) ) ) ) ) ).
% unfold_llist.ctr(2)
thf(fact_95_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_96_llist_Osimps_I4_J,axiom,
! [A: $tType,B: $tType,F12: B,F22: A > ( coindu1593790203_llist @ A ) > B] :
( ( coindu882539134_llist @ B @ A @ F12 @ F22 @ ( coindu1598213697e_LNil @ A ) )
= F12 ) ).
% llist.simps(4)
thf(fact_97_llist_Osimps_I5_J,axiom,
! [B: $tType,A: $tType,F12: B,F22: A > ( coindu1593790203_llist @ A ) > B,X21: A,X22: coindu1593790203_llist @ A] :
( ( coindu882539134_llist @ B @ A @ F12 @ F22 @ ( coindu1121789889_LCons @ A @ X21 @ X22 ) )
= ( F22 @ X21 @ X22 ) ) ).
% llist.simps(5)
thf(fact_98_ord_OSingleton,axiom,
! [A: $tType,Less_eq: A > A > $o,X3: A] : ( coindu543932918sorted @ A @ Less_eq @ ( coindu1121789889_LCons @ A @ X3 @ ( coindu1598213697e_LNil @ A ) ) ) ).
% ord.Singleton
thf(fact_99_ord_Olsorted_Ocoinduct,axiom,
! [A: $tType,X2: ( coindu1593790203_llist @ A ) > $o,X3: coindu1593790203_llist @ A,Less_eq: A > A > $o] :
( ( X2 @ X3 )
=> ( ! [X5: coindu1593790203_llist @ A] :
( ( X2 @ X5 )
=> ( ( X5
= ( coindu1598213697e_LNil @ A ) )
| ? [Xa: A] :
( X5
= ( coindu1121789889_LCons @ A @ Xa @ ( coindu1598213697e_LNil @ A ) ) )
| ? [Xa: A,Y4: A,Xs3: coindu1593790203_llist @ A] :
( ( X5
= ( coindu1121789889_LCons @ A @ Xa @ ( coindu1121789889_LCons @ A @ Y4 @ Xs3 ) ) )
& ( Less_eq @ Xa @ Y4 )
& ( ( X2 @ ( coindu1121789889_LCons @ A @ Y4 @ Xs3 ) )
| ( coindu543932918sorted @ A @ Less_eq @ ( coindu1121789889_LCons @ A @ Y4 @ Xs3 ) ) ) ) ) )
=> ( coindu543932918sorted @ A @ Less_eq @ X3 ) ) ) ).
% ord.lsorted.coinduct
thf(fact_100_ord_Olsorted_Osimps,axiom,
! [A: $tType] :
( ( coindu543932918sorted @ A )
= ( ^ [Less_eq2: A > A > $o,A4: coindu1593790203_llist @ A] :
( ( A4
= ( coindu1598213697e_LNil @ A ) )
| ? [X4: A] :
( A4
= ( coindu1121789889_LCons @ A @ X4 @ ( coindu1598213697e_LNil @ A ) ) )
| ? [X4: A,Y5: A,Xs4: coindu1593790203_llist @ A] :
( ( A4
= ( coindu1121789889_LCons @ A @ X4 @ ( coindu1121789889_LCons @ A @ Y5 @ Xs4 ) ) )
& ( Less_eq2 @ X4 @ Y5 )
& ( coindu543932918sorted @ A @ Less_eq2 @ ( coindu1121789889_LCons @ A @ Y5 @ Xs4 ) ) ) ) ) ) ).
% ord.lsorted.simps
thf(fact_101_ord_Olsorted_Ocases,axiom,
! [A: $tType,Less_eq: A > A > $o,A2: coindu1593790203_llist @ A] :
( ( coindu543932918sorted @ A @ Less_eq @ A2 )
=> ( ( A2
!= ( coindu1598213697e_LNil @ A ) )
=> ( ! [X5: A] :
( A2
!= ( coindu1121789889_LCons @ A @ X5 @ ( coindu1598213697e_LNil @ A ) ) )
=> ~ ! [X5: A,Y3: A,Xs2: coindu1593790203_llist @ A] :
( ( A2
= ( coindu1121789889_LCons @ A @ X5 @ ( coindu1121789889_LCons @ A @ Y3 @ Xs2 ) ) )
=> ( ( Less_eq @ X5 @ Y3 )
=> ~ ( coindu543932918sorted @ A @ Less_eq @ ( coindu1121789889_LCons @ A @ Y3 @ Xs2 ) ) ) ) ) ) ) ).
% ord.lsorted.cases
thf(fact_102_ord_Olsorted__code_I2_J,axiom,
! [A: $tType,Less_eq: A > A > $o,X3: A] : ( coindu543932918sorted @ A @ Less_eq @ ( coindu1121789889_LCons @ A @ X3 @ ( coindu1598213697e_LNil @ A ) ) ) ).
% ord.lsorted_code(2)
thf(fact_103_ltl__unfold__llist,axiom,
! [A: $tType,B: $tType,IS_LNIL: B > $o,A2: B,LHD: B > A,LTL: B > B] :
( ( ( IS_LNIL @ A2 )
=> ( ( coindu1047225960le_ltl @ A @ ( coindu1599971794_llist @ B @ A @ IS_LNIL @ LHD @ LTL @ A2 ) )
= ( coindu1598213697e_LNil @ A ) ) )
& ( ~ ( IS_LNIL @ A2 )
=> ( ( coindu1047225960le_ltl @ A @ ( coindu1599971794_llist @ B @ A @ IS_LNIL @ LHD @ LTL @ A2 ) )
= ( coindu1599971794_llist @ B @ A @ IS_LNIL @ LHD @ LTL @ ( LTL @ A2 ) ) ) ) ) ).
% ltl_unfold_llist
thf(fact_104_llist_Osplit__sel,axiom,
! [B: $tType,A: $tType,P: B > $o,F12: B,F22: A > ( coindu1593790203_llist @ A ) > B,Llist: coindu1593790203_llist @ A] :
( ( P @ ( coindu882539134_llist @ B @ A @ F12 @ F22 @ Llist ) )
= ( ( ( Llist
= ( coindu1598213697e_LNil @ A ) )
=> ( P @ F12 ) )
& ( ( 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_105_llist_Osplit__sel__asm,axiom,
! [B: $tType,A: $tType,P: B > $o,F12: B,F22: A > ( coindu1593790203_llist @ A ) > B,Llist: coindu1593790203_llist @ A] :
( ( P @ ( coindu882539134_llist @ B @ A @ F12 @ F22 @ Llist ) )
= ( ~ ( ( ( Llist
= ( coindu1598213697e_LNil @ A ) )
& ~ ( P @ F12 ) )
| ( ( 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_106_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_107_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_108_LCons__lprefix__conv,axiom,
! [A: $tType,X3: A,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ ( coindu1121789889_LCons @ A @ X3 @ Xs ) @ Ys )
= ( ? [Ys5: coindu1593790203_llist @ A] :
( ( Ys
= ( coindu1121789889_LCons @ A @ X3 @ Ys5 ) )
& ( coindu1696667936prefix @ A @ Xs @ Ys5 ) ) ) ) ).
% LCons_lprefix_conv
thf(fact_109_Le__LCons,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A,X3: A] :
( ( coindu1696667936prefix @ A @ Xs @ Ys )
=> ( coindu1696667936prefix @ A @ ( coindu1121789889_LCons @ A @ X3 @ Xs ) @ ( coindu1121789889_LCons @ A @ X3 @ Ys ) ) ) ).
% Le_LCons
thf(fact_110_not__lnull__conv,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( ( ~ ( coindu335574135_lnull @ A @ Xs ) )
= ( ? [X4: A,Xs5: coindu1593790203_llist @ A] :
( Xs
= ( coindu1121789889_LCons @ A @ X4 @ Xs5 ) ) ) ) ).
% not_lnull_conv
thf(fact_111_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_112_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_113_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_114_LNil__lprefix,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] : ( coindu1696667936prefix @ A @ ( coindu1598213697e_LNil @ A ) @ Xs ) ).
% LNil_lprefix
thf(fact_115_lhd__LCons,axiom,
! [A: $tType,X21: A,X22: coindu1593790203_llist @ A] :
( ( coindu1046438764le_lhd @ A @ ( coindu1121789889_LCons @ A @ X21 @ X22 ) )
= X21 ) ).
% lhd_LCons
thf(fact_116_lnull__def,axiom,
! [A: $tType] :
( ( coindu335574135_lnull @ A )
= ( ^ [Llist5: coindu1593790203_llist @ A] :
( Llist5
= ( coindu1598213697e_LNil @ A ) ) ) ) ).
% lnull_def
thf(fact_117_llist_Ocollapse_I1_J,axiom,
! [A: $tType,Llist: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ A @ Llist )
=> ( Llist
= ( coindu1598213697e_LNil @ A ) ) ) ).
% llist.collapse(1)
thf(fact_118_llist_OdiscI_I1_J,axiom,
! [A: $tType,Llist: coindu1593790203_llist @ A] :
( ( Llist
= ( coindu1598213697e_LNil @ A ) )
=> ( coindu335574135_lnull @ A @ Llist ) ) ).
% llist.discI(1)
thf(fact_119_llist_Odisc_I1_J,axiom,
! [A: $tType] : ( coindu335574135_lnull @ A @ ( coindu1598213697e_LNil @ A ) ) ).
% llist.disc(1)
thf(fact_120_ltl__simps_I1_J,axiom,
! [A: $tType] :
( ( coindu1047225960le_ltl @ A @ ( coindu1598213697e_LNil @ A ) )
= ( coindu1598213697e_LNil @ A ) ) ).
% ltl_simps(1)
thf(fact_121_ord_Olsorted__LCons__LCons,axiom,
! [A: $tType,Less_eq: A > A > $o,X3: A,Y: A,Xs: coindu1593790203_llist @ A] :
( ( coindu543932918sorted @ A @ Less_eq @ ( coindu1121789889_LCons @ A @ X3 @ ( coindu1121789889_LCons @ A @ Y @ Xs ) ) )
= ( ( Less_eq @ X3 @ Y )
& ( coindu543932918sorted @ A @ Less_eq @ ( coindu1121789889_LCons @ A @ Y @ Xs ) ) ) ) ).
% ord.lsorted_LCons_LCons
thf(fact_122_ord_OLCons__LCons,axiom,
! [A: $tType,Less_eq: A > A > $o,X3: A,Y: A,Xs: coindu1593790203_llist @ A] :
( ( Less_eq @ X3 @ Y )
=> ( ( coindu543932918sorted @ A @ Less_eq @ ( coindu1121789889_LCons @ A @ Y @ Xs ) )
=> ( coindu543932918sorted @ A @ Less_eq @ ( coindu1121789889_LCons @ A @ X3 @ ( coindu1121789889_LCons @ A @ Y @ Xs ) ) ) ) ) ).
% ord.LCons_LCons
thf(fact_123_ord_OLNil,axiom,
! [A: $tType,Less_eq: A > A > $o] : ( coindu543932918sorted @ A @ Less_eq @ ( coindu1598213697e_LNil @ A ) ) ).
% ord.LNil
thf(fact_124_ord_Olsorted__code_I1_J,axiom,
! [A: $tType,Less_eq: A > A > $o] : ( coindu543932918sorted @ A @ Less_eq @ ( coindu1598213697e_LNil @ A ) ) ).
% ord.lsorted_code(1)
thf(fact_125_unfold__llist_Odisc_I1_J,axiom,
! [B: $tType,A: $tType,P2: A > $o,A2: A,G21: A > B,G22: A > A] :
( ( P2 @ A2 )
=> ( coindu335574135_lnull @ B @ ( coindu1599971794_llist @ A @ B @ P2 @ G21 @ G22 @ A2 ) ) ) ).
% unfold_llist.disc(1)
thf(fact_126_unfold__llist_Odisc_I2_J,axiom,
! [B: $tType,A: $tType,P2: A > $o,A2: A,G21: A > B,G22: A > A] :
( ~ ( P2 @ A2 )
=> ~ ( coindu335574135_lnull @ B @ ( coindu1599971794_llist @ A @ B @ P2 @ G21 @ G22 @ A2 ) ) ) ).
% unfold_llist.disc(2)
thf(fact_127_unfold__llist_Osimps_I4_J,axiom,
! [B: $tType,A: $tType,P2: A > $o,A2: A,G21: A > B,G22: A > A] :
( ~ ( P2 @ A2 )
=> ( ( coindu1047225960le_ltl @ B @ ( coindu1599971794_llist @ A @ B @ P2 @ G21 @ G22 @ A2 ) )
= ( coindu1599971794_llist @ A @ B @ P2 @ G21 @ G22 @ ( G22 @ A2 ) ) ) ) ).
% unfold_llist.simps(4)
thf(fact_128_unfold__llist_Osimps_I3_J,axiom,
! [B: $tType,A: $tType,P2: A > $o,A2: A,G21: A > B,G22: A > A] :
( ~ ( P2 @ A2 )
=> ( ( coindu1046438764le_lhd @ B @ ( coindu1599971794_llist @ A @ B @ P2 @ G21 @ G22 @ A2 ) )
= ( G21 @ A2 ) ) ) ).
% unfold_llist.simps(3)
thf(fact_129_lSup_Octr_I1_J,axiom,
! [A: $tType,A3: set @ ( coindu1593790203_llist @ A )] :
( ! [X5: coindu1593790203_llist @ A] :
( ( member @ ( coindu1593790203_llist @ A ) @ X5 @ A3 )
=> ( coindu335574135_lnull @ A @ X5 ) )
=> ( ( coindu526335360e_lSup @ A @ A3 )
= ( coindu1598213697e_LNil @ A ) ) ) ).
% lSup.ctr(1)
thf(fact_130_ord_OatLeastAtMost__def,axiom,
! [A: $tType] :
( ( set_atLeastAtMost @ A )
= ( ^ [Less_eq2: A > A > $o,L2: A,U2: A] : ( inf_inf @ ( set @ A ) @ ( set_atLeast @ A @ Less_eq2 @ L2 ) @ ( set_atMost @ A @ Less_eq2 @ U2 ) ) ) ) ).
% ord.atLeastAtMost_def
thf(fact_131_ord_Olsorted__LCons_H,axiom,
! [A: $tType,Less_eq: A > A > $o,X3: A,Xs: coindu1593790203_llist @ A] :
( ( coindu543932918sorted @ A @ Less_eq @ ( coindu1121789889_LCons @ A @ X3 @ Xs ) )
= ( ~ ( coindu335574135_lnull @ A @ Xs )
=> ( ( Less_eq @ X3 @ ( coindu1046438764le_lhd @ A @ Xs ) )
& ( coindu543932918sorted @ A @ Less_eq @ Xs ) ) ) ) ).
% ord.lsorted_LCons'
thf(fact_132_ord_OatLeastAtMost__iff,axiom,
! [A: $tType,I: A,Less_eq: A > A > $o,L: A,U: A] :
( ( member @ A @ I @ ( set_atLeastAtMost @ A @ Less_eq @ L @ U ) )
= ( ( Less_eq @ L @ I )
& ( Less_eq @ I @ U ) ) ) ).
% ord.atLeastAtMost_iff
thf(fact_133_Int__iff,axiom,
! [A: $tType,C: A,A3: set @ A,B4: set @ A] :
( ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A3 @ B4 ) )
= ( ( member @ A @ C @ A3 )
& ( member @ A @ C @ B4 ) ) ) ).
% Int_iff
thf(fact_134_IntI,axiom,
! [A: $tType,C: A,A3: set @ A,B4: set @ A] :
( ( member @ A @ C @ A3 )
=> ( ( member @ A @ C @ B4 )
=> ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A3 @ B4 ) ) ) ) ).
% IntI
thf(fact_135_inf__right__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A @ ( type2 @ A ) )
=> ! [X3: A,Y: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ X3 @ Y ) @ Y )
= ( inf_inf @ A @ X3 @ Y ) ) ) ).
% inf_right_idem
thf(fact_136_inf_Oright__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ A2 @ B2 ) @ B2 )
= ( inf_inf @ A @ A2 @ B2 ) ) ) ).
% inf.right_idem
thf(fact_137_inf__left__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A @ ( type2 @ A ) )
=> ! [X3: A,Y: A] :
( ( inf_inf @ A @ X3 @ ( inf_inf @ A @ X3 @ Y ) )
= ( inf_inf @ A @ X3 @ Y ) ) ) ).
% inf_left_idem
thf(fact_138_inf_Oleft__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( inf_inf @ A @ A2 @ ( inf_inf @ A @ A2 @ B2 ) )
= ( inf_inf @ A @ A2 @ B2 ) ) ) ).
% inf.left_idem
thf(fact_139_inf__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A @ ( type2 @ A ) )
=> ! [X3: A] :
( ( inf_inf @ A @ X3 @ X3 )
= X3 ) ) ).
% inf_idem
thf(fact_140_inf__apply,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_inf @ B @ ( type2 @ B ) )
=> ( ( inf_inf @ ( A > B ) )
= ( ^ [F3: A > B,G2: A > B,X4: A] : ( inf_inf @ B @ ( F3 @ X4 ) @ ( G2 @ X4 ) ) ) ) ) ).
% inf_apply
thf(fact_141_inf_Oidem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( inf_inf @ A @ A2 @ A2 )
= A2 ) ) ).
% inf.idem
thf(fact_142_inf__sup__aci_I4_J,axiom,
! [A: $tType] :
( ( lattice @ A @ ( type2 @ A ) )
=> ! [X3: A,Y: A] :
( ( inf_inf @ A @ X3 @ ( inf_inf @ A @ X3 @ Y ) )
= ( inf_inf @ A @ X3 @ Y ) ) ) ).
% inf_sup_aci(4)
thf(fact_143_inf__sup__aci_I3_J,axiom,
! [A: $tType] :
( ( lattice @ A @ ( type2 @ A ) )
=> ! [X3: A,Y: A,Z: A] :
( ( inf_inf @ A @ X3 @ ( inf_inf @ A @ Y @ Z ) )
= ( inf_inf @ A @ Y @ ( inf_inf @ A @ X3 @ Z ) ) ) ) ).
% inf_sup_aci(3)
thf(fact_144_inf__sup__aci_I2_J,axiom,
! [A: $tType] :
( ( lattice @ A @ ( type2 @ A ) )
=> ! [X3: A,Y: A,Z: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ X3 @ Y ) @ Z )
= ( inf_inf @ A @ X3 @ ( inf_inf @ A @ Y @ Z ) ) ) ) ).
% inf_sup_aci(2)
thf(fact_145_inf__sup__aci_I1_J,axiom,
! [A: $tType] :
( ( lattice @ A @ ( type2 @ A ) )
=> ( ( inf_inf @ A )
= ( ^ [X4: A,Y5: A] : ( inf_inf @ A @ Y5 @ X4 ) ) ) ) ).
% inf_sup_aci(1)
thf(fact_146_inf__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_inf @ B @ ( type2 @ B ) )
=> ( ( inf_inf @ ( A > B ) )
= ( ^ [F3: A > B,G2: A > B,X4: A] : ( inf_inf @ B @ ( F3 @ X4 ) @ ( G2 @ X4 ) ) ) ) ) ).
% inf_fun_def
thf(fact_147_inf_Oassoc,axiom,
! [A: $tType] :
( ( semilattice_inf @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ A2 @ B2 ) @ C )
= ( inf_inf @ A @ A2 @ ( inf_inf @ A @ B2 @ C ) ) ) ) ).
% inf.assoc
thf(fact_148_inf__assoc,axiom,
! [A: $tType] :
( ( semilattice_inf @ A @ ( type2 @ A ) )
=> ! [X3: A,Y: A,Z: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ X3 @ Y ) @ Z )
= ( inf_inf @ A @ X3 @ ( inf_inf @ A @ Y @ Z ) ) ) ) ).
% inf_assoc
thf(fact_149_inf_Ocommute,axiom,
! [A: $tType] :
( ( semilattice_inf @ A @ ( type2 @ A ) )
=> ( ( inf_inf @ A )
= ( ^ [A4: A,B3: A] : ( inf_inf @ A @ B3 @ A4 ) ) ) ) ).
% inf.commute
thf(fact_150_inf__commute,axiom,
! [A: $tType] :
( ( semilattice_inf @ A @ ( type2 @ A ) )
=> ( ( inf_inf @ A )
= ( ^ [X4: A,Y5: A] : ( inf_inf @ A @ Y5 @ X4 ) ) ) ) ).
% inf_commute
thf(fact_151_inf_Oleft__commute,axiom,
! [A: $tType] :
( ( semilattice_inf @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A,C: A] :
( ( inf_inf @ A @ B2 @ ( inf_inf @ A @ A2 @ C ) )
= ( inf_inf @ A @ A2 @ ( inf_inf @ A @ B2 @ C ) ) ) ) ).
% inf.left_commute
thf(fact_152_inf__left__commute,axiom,
! [A: $tType] :
( ( semilattice_inf @ A @ ( type2 @ A ) )
=> ! [X3: A,Y: A,Z: A] :
( ( inf_inf @ A @ X3 @ ( inf_inf @ A @ Y @ Z ) )
= ( inf_inf @ A @ Y @ ( inf_inf @ A @ X3 @ Z ) ) ) ) ).
% inf_left_commute
thf(fact_153_IntE,axiom,
! [A: $tType,C: A,A3: set @ A,B4: set @ A] :
( ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A3 @ B4 ) )
=> ~ ( ( member @ A @ C @ A3 )
=> ~ ( member @ A @ C @ B4 ) ) ) ).
% IntE
thf(fact_154_IntD1,axiom,
! [A: $tType,C: A,A3: set @ A,B4: set @ A] :
( ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A3 @ B4 ) )
=> ( member @ A @ C @ A3 ) ) ).
% IntD1
thf(fact_155_IntD2,axiom,
! [A: $tType,C: A,A3: set @ A,B4: set @ A] :
( ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A3 @ B4 ) )
=> ( member @ A @ C @ B4 ) ) ).
% IntD2
thf(fact_156_Int__assoc,axiom,
! [A: $tType,A3: set @ A,B4: set @ A,C2: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ B4 ) @ C2 )
= ( inf_inf @ ( set @ A ) @ A3 @ ( inf_inf @ ( set @ A ) @ B4 @ C2 ) ) ) ).
% Int_assoc
thf(fact_157_Int__absorb,axiom,
! [A: $tType,A3: set @ A] :
( ( inf_inf @ ( set @ A ) @ A3 @ A3 )
= A3 ) ).
% Int_absorb
thf(fact_158_Int__commute,axiom,
! [A: $tType] :
( ( inf_inf @ ( set @ A ) )
= ( ^ [A5: set @ A,B5: set @ A] : ( inf_inf @ ( set @ A ) @ B5 @ A5 ) ) ) ).
% Int_commute
thf(fact_159_Int__left__absorb,axiom,
! [A: $tType,A3: set @ A,B4: set @ A] :
( ( inf_inf @ ( set @ A ) @ A3 @ ( inf_inf @ ( set @ A ) @ A3 @ B4 ) )
= ( inf_inf @ ( set @ A ) @ A3 @ B4 ) ) ).
% Int_left_absorb
thf(fact_160_Int__left__commute,axiom,
! [A: $tType,A3: set @ A,B4: set @ A,C2: set @ A] :
( ( inf_inf @ ( set @ A ) @ A3 @ ( inf_inf @ ( set @ A ) @ B4 @ C2 ) )
= ( inf_inf @ ( set @ A ) @ B4 @ ( inf_inf @ ( set @ A ) @ A3 @ C2 ) ) ) ).
% Int_left_commute
thf(fact_161_compact__LCons__iff,axiom,
! [A: $tType,X3: A,Xs: coindu1593790203_llist @ A] :
( ( comple2143767107ompact @ ( coindu1593790203_llist @ A ) @ ( coindu526335360e_lSup @ A ) @ ( coindu1696667936prefix @ A ) @ ( coindu1121789889_LCons @ A @ X3 @ Xs ) )
= ( comple2143767107ompact @ ( coindu1593790203_llist @ A ) @ ( coindu526335360e_lSup @ A ) @ ( coindu1696667936prefix @ A ) @ Xs ) ) ).
% compact_LCons_iff
thf(fact_162_ltakeWhile_Ocode,axiom,
! [A: $tType] :
( ( coindu721411036eWhile @ A )
= ( ^ [P4: A > $o,Xs4: coindu1593790203_llist @ A] :
( if @ ( coindu1593790203_llist @ A )
@ ( ( coindu335574135_lnull @ A @ Xs4 )
| ~ ( P4 @ ( coindu1046438764le_lhd @ A @ Xs4 ) ) )
@ ( coindu1598213697e_LNil @ A )
@ ( coindu1121789889_LCons @ A @ ( coindu1046438764le_lhd @ A @ Xs4 ) @ ( coindu721411036eWhile @ A @ P4 @ ( coindu1047225960le_ltl @ A @ Xs4 ) ) ) ) ) ) ).
% ltakeWhile.code
thf(fact_163_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 ) )
| ? [Xs5: coindu1593790203_llist @ A] :
( ( Xs
= ( coindu1121789889_LCons @ A @ Y @ Xs5 ) )
& ( coindu1571841457prefix @ A @ Xs5 @ Ys ) ) ) ) ).
% Coinductive_List_Mirabelle_kmikjhschf.finite_lprefix_nitpick_simps(3)
thf(fact_164_lstrict__prefix__code_I3_J,axiom,
! [B: $tType,X3: B,Xs: coindu1593790203_llist @ B] :
~ ( coindu574146665prefix @ B @ ( coindu1121789889_LCons @ B @ X3 @ Xs ) @ ( coindu1598213697e_LNil @ B ) ) ).
% lstrict_prefix_code(3)
thf(fact_165_ltakeWhile__LNil,axiom,
! [A: $tType,P: A > $o] :
( ( coindu721411036eWhile @ A @ P @ ( coindu1598213697e_LNil @ A ) )
= ( coindu1598213697e_LNil @ A ) ) ).
% ltakeWhile_LNil
thf(fact_166_lstrict__prefix__code_I4_J,axiom,
! [B: $tType,X3: B,Xs: coindu1593790203_llist @ B,Y: B,Ys: coindu1593790203_llist @ B] :
( ( coindu574146665prefix @ B @ ( coindu1121789889_LCons @ B @ X3 @ Xs ) @ ( coindu1121789889_LCons @ B @ Y @ Ys ) )
= ( ( X3 = Y )
& ( coindu574146665prefix @ B @ Xs @ Ys ) ) ) ).
% lstrict_prefix_code(4)
thf(fact_167_lstrict__prefix__code_I1_J,axiom,
! [A: $tType] :
~ ( coindu574146665prefix @ A @ ( coindu1598213697e_LNil @ A ) @ ( coindu1598213697e_LNil @ A ) ) ).
% lstrict_prefix_code(1)
thf(fact_168_ltakeWhile__LCons,axiom,
! [A: $tType,P: A > $o,X3: A,Xs: coindu1593790203_llist @ A] :
( ( ( P @ X3 )
=> ( ( coindu721411036eWhile @ A @ P @ ( coindu1121789889_LCons @ A @ X3 @ Xs ) )
= ( coindu1121789889_LCons @ A @ X3 @ ( coindu721411036eWhile @ A @ P @ Xs ) ) ) )
& ( ~ ( P @ X3 )
=> ( ( coindu721411036eWhile @ A @ P @ ( coindu1121789889_LCons @ A @ X3 @ Xs ) )
= ( coindu1598213697e_LNil @ A ) ) ) ) ).
% ltakeWhile_LCons
thf(fact_169_ltakeWhile_Odisc__iff_I2_J,axiom,
! [A: $tType,P: A > $o,Xs: coindu1593790203_llist @ A] :
( ( ~ ( coindu335574135_lnull @ A @ ( coindu721411036eWhile @ A @ P @ Xs ) ) )
= ( ~ ( coindu335574135_lnull @ A @ Xs )
& ( P @ ( coindu1046438764le_lhd @ A @ Xs ) ) ) ) ).
% ltakeWhile.disc_iff(2)
thf(fact_170_ltakeWhile_Odisc__iff_I1_J,axiom,
! [A: $tType,P: A > $o,Xs: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ A @ ( coindu721411036eWhile @ A @ P @ Xs ) )
= ( ( coindu335574135_lnull @ A @ Xs )
| ~ ( P @ ( coindu1046438764le_lhd @ A @ Xs ) ) ) ) ).
% ltakeWhile.disc_iff(1)
thf(fact_171_lnull__ltakeWhile,axiom,
! [A: $tType,P: A > $o,Xs: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ A @ ( coindu721411036eWhile @ A @ P @ Xs ) )
= ( ~ ( coindu335574135_lnull @ A @ Xs )
=> ~ ( P @ ( coindu1046438764le_lhd @ A @ Xs ) ) ) ) ).
% lnull_ltakeWhile
thf(fact_172_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_173_lprefix__ltakeWhile,axiom,
! [A: $tType,P: A > $o,Xs: coindu1593790203_llist @ A] : ( coindu1696667936prefix @ A @ ( coindu721411036eWhile @ A @ P @ Xs ) @ Xs ) ).
% lprefix_ltakeWhile
thf(fact_174_llist__less__induct,axiom,
! [A: $tType,P: ( coindu1593790203_llist @ A ) > $o,Xs: coindu1593790203_llist @ A] :
( ! [Xs2: coindu1593790203_llist @ A] :
( ! [Ys4: coindu1593790203_llist @ A] :
( ( coindu574146665prefix @ A @ Ys4 @ Xs2 )
=> ( P @ Ys4 ) )
=> ( P @ Xs2 ) )
=> ( P @ Xs ) ) ).
% llist_less_induct
thf(fact_175_lstrict__prefix__def,axiom,
! [A: $tType] :
( ( coindu574146665prefix @ A )
= ( ^ [Xs4: coindu1593790203_llist @ A,Ys3: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ Xs4 @ Ys3 )
& ( Xs4 != Ys3 ) ) ) ) ).
% lstrict_prefix_def
thf(fact_176_ltakeWhile_Odisc_I2_J,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,P: A > $o] :
( ~ ( coindu335574135_lnull @ A @ Xs )
=> ( ( P @ ( coindu1046438764le_lhd @ A @ Xs ) )
=> ~ ( coindu335574135_lnull @ A @ ( coindu721411036eWhile @ A @ P @ Xs ) ) ) ) ).
% ltakeWhile.disc(2)
thf(fact_177_ltakeWhile_Odisc_I1_J,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,P: A > $o] :
( ( ( coindu335574135_lnull @ A @ Xs )
| ~ ( P @ ( coindu1046438764le_lhd @ A @ Xs ) ) )
=> ( coindu335574135_lnull @ A @ ( coindu721411036eWhile @ A @ P @ Xs ) ) ) ).
% ltakeWhile.disc(1)
thf(fact_178_lhd__ltakeWhile,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,P: A > $o] :
( ~ ( coindu335574135_lnull @ A @ Xs )
=> ( ( P @ ( coindu1046438764le_lhd @ A @ Xs ) )
=> ( ( coindu1046438764le_lhd @ A @ ( coindu721411036eWhile @ A @ P @ Xs ) )
= ( coindu1046438764le_lhd @ A @ Xs ) ) ) ) ).
% lhd_ltakeWhile
thf(fact_179_ltakeWhile__eq__LNil__iff,axiom,
! [A: $tType,P: A > $o,Xs: coindu1593790203_llist @ A] :
( ( ( coindu721411036eWhile @ A @ P @ Xs )
= ( coindu1598213697e_LNil @ A ) )
= ( ( Xs
!= ( coindu1598213697e_LNil @ A ) )
=> ~ ( P @ ( coindu1046438764le_lhd @ A @ Xs ) ) ) ) ).
% ltakeWhile_eq_LNil_iff
thf(fact_180_Coinductive__List__Mirabelle__kmikjhschf_Ofinite__lprefix__def,axiom,
! [A: $tType] :
( ( coindu1571841457prefix @ A )
= ( coindu1696667936prefix @ A ) ) ).
% Coinductive_List_Mirabelle_kmikjhschf.finite_lprefix_def
thf(fact_181_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_182_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_183_ltakeWhile_Octr_I1_J,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,P: A > $o] :
( ( ( coindu335574135_lnull @ A @ Xs )
| ~ ( P @ ( coindu1046438764le_lhd @ A @ Xs ) ) )
=> ( ( coindu721411036eWhile @ A @ P @ Xs )
= ( coindu1598213697e_LNil @ A ) ) ) ).
% ltakeWhile.ctr(1)
thf(fact_184_ltakeWhile_Osimps_I4_J,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,P: A > $o] :
( ~ ( coindu335574135_lnull @ A @ Xs )
=> ( ( P @ ( coindu1046438764le_lhd @ A @ Xs ) )
=> ( ( coindu1047225960le_ltl @ A @ ( coindu721411036eWhile @ A @ P @ Xs ) )
= ( coindu721411036eWhile @ A @ P @ ( coindu1047225960le_ltl @ A @ Xs ) ) ) ) ) ).
% ltakeWhile.simps(4)
thf(fact_185_ltl__ltakeWhile,axiom,
! [A: $tType,P: A > $o,Xs: coindu1593790203_llist @ A] :
( ( ( P @ ( coindu1046438764le_lhd @ A @ Xs ) )
=> ( ( coindu1047225960le_ltl @ A @ ( coindu721411036eWhile @ A @ P @ Xs ) )
= ( coindu721411036eWhile @ A @ P @ ( coindu1047225960le_ltl @ A @ Xs ) ) ) )
& ( ~ ( P @ ( coindu1046438764le_lhd @ A @ Xs ) )
=> ( ( coindu1047225960le_ltl @ A @ ( coindu721411036eWhile @ A @ P @ Xs ) )
= ( coindu1598213697e_LNil @ A ) ) ) ) ).
% ltl_ltakeWhile
thf(fact_186_ltakeWhile_Octr_I2_J,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,P: A > $o] :
( ~ ( coindu335574135_lnull @ A @ Xs )
=> ( ( P @ ( coindu1046438764le_lhd @ A @ Xs ) )
=> ( ( coindu721411036eWhile @ A @ P @ Xs )
= ( coindu1121789889_LCons @ A @ ( coindu1046438764le_lhd @ A @ Xs ) @ ( coindu721411036eWhile @ A @ P @ ( coindu1047225960le_ltl @ A @ Xs ) ) ) ) ) ) ).
% ltakeWhile.ctr(2)
thf(fact_187_compact__LConsI,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,X3: A] :
( ( comple2143767107ompact @ ( coindu1593790203_llist @ A ) @ ( coindu526335360e_lSup @ A ) @ ( coindu1696667936prefix @ A ) @ Xs )
=> ( comple2143767107ompact @ ( coindu1593790203_llist @ A ) @ ( coindu526335360e_lSup @ A ) @ ( coindu1696667936prefix @ A ) @ ( coindu1121789889_LCons @ A @ X3 @ Xs ) ) ) ).
% compact_LConsI
thf(fact_188_compact__LConsD,axiom,
! [A: $tType,X3: A,Xs: coindu1593790203_llist @ A] :
( ( comple2143767107ompact @ ( coindu1593790203_llist @ A ) @ ( coindu526335360e_lSup @ A ) @ ( coindu1696667936prefix @ A ) @ ( coindu1121789889_LCons @ A @ X3 @ Xs ) )
=> ( comple2143767107ompact @ ( coindu1593790203_llist @ A ) @ ( coindu526335360e_lSup @ A ) @ ( coindu1696667936prefix @ A ) @ Xs ) ) ).
% compact_LConsD
thf(fact_189_ldistinct__lSup,axiom,
! [A: $tType,Y2: set @ ( coindu1593790203_llist @ A )] :
( ( comple1193779247_chain @ ( coindu1593790203_llist @ A ) @ ( coindu1696667936prefix @ A ) @ Y2 )
=> ( ! [X5: coindu1593790203_llist @ A] :
( ( member @ ( coindu1593790203_llist @ A ) @ X5 @ Y2 )
=> ( coindu6345450stinct @ A @ X5 ) )
=> ( coindu6345450stinct @ A @ ( coindu526335360e_lSup @ A @ Y2 ) ) ) ) ).
% ldistinct_lSup
thf(fact_190_llast__singleton,axiom,
! [A: $tType,X3: A] :
( ( coindu2000965700_llast @ A @ ( coindu1121789889_LCons @ A @ X3 @ ( coindu1598213697e_LNil @ A ) ) )
= X3 ) ).
% llast_singleton
thf(fact_191_cont__ltl,axiom,
! [A: $tType] : ( comple1412571634l_cont @ ( coindu1593790203_llist @ A ) @ ( coindu1593790203_llist @ A ) @ ( coindu526335360e_lSup @ A ) @ ( coindu1696667936prefix @ A ) @ ( coindu526335360e_lSup @ A ) @ ( coindu1696667936prefix @ A ) @ ( coindu1047225960le_ltl @ A ) ) ).
% cont_ltl
thf(fact_192_ldistinct__LNil__code,axiom,
! [A: $tType] : ( coindu6345450stinct @ A @ ( coindu1598213697e_LNil @ A ) ) ).
% ldistinct_LNil_code
thf(fact_193_llast__LCons2,axiom,
! [A: $tType,X3: A,Y: A,Xs: coindu1593790203_llist @ A] :
( ( coindu2000965700_llast @ A @ ( coindu1121789889_LCons @ A @ X3 @ ( coindu1121789889_LCons @ A @ Y @ Xs ) ) )
= ( coindu2000965700_llast @ A @ ( coindu1121789889_LCons @ A @ Y @ Xs ) ) ) ).
% llast_LCons2
thf(fact_194_ldistinct__ltlI,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( ( coindu6345450stinct @ A @ Xs )
=> ( coindu6345450stinct @ A @ ( coindu1047225960le_ltl @ A @ Xs ) ) ) ).
% ldistinct_ltlI
thf(fact_195_ldistinct_OLNil,axiom,
! [A: $tType] : ( coindu6345450stinct @ A @ ( coindu1598213697e_LNil @ A ) ) ).
% ldistinct.LNil
thf(fact_196_ldistinct__lprefix,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu6345450stinct @ A @ Xs )
=> ( ( coindu1696667936prefix @ A @ Ys @ Xs )
=> ( coindu6345450stinct @ A @ Ys ) ) ) ).
% ldistinct_lprefix
thf(fact_197_llast__LCons,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,X3: A] :
( ( ( coindu335574135_lnull @ A @ Xs )
=> ( ( coindu2000965700_llast @ A @ ( coindu1121789889_LCons @ A @ X3 @ Xs ) )
= X3 ) )
& ( ~ ( coindu335574135_lnull @ A @ Xs )
=> ( ( coindu2000965700_llast @ A @ ( coindu1121789889_LCons @ A @ X3 @ Xs ) )
= ( coindu2000965700_llast @ A @ Xs ) ) ) ) ).
% llast_LCons
thf(fact_198_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_199_ldistinct__coinduct,axiom,
! [A: $tType,X2: ( coindu1593790203_llist @ A ) > $o,Xs: coindu1593790203_llist @ A] :
( ( X2 @ Xs )
=> ( ! [Xs2: coindu1593790203_llist @ A] :
( ( X2 @ Xs2 )
=> ( ~ ( coindu335574135_lnull @ A @ Xs2 )
=> ( ~ ( member @ A @ ( coindu1046438764le_lhd @ A @ Xs2 ) @ ( coindu1112613586e_lset @ A @ ( coindu1047225960le_ltl @ A @ Xs2 ) ) )
& ( ( X2 @ ( coindu1047225960le_ltl @ A @ Xs2 ) )
| ( coindu6345450stinct @ A @ ( coindu1047225960le_ltl @ A @ Xs2 ) ) ) ) ) )
=> ( coindu6345450stinct @ A @ Xs ) ) ) ).
% ldistinct_coinduct
thf(fact_200_ldistinct__LCons,axiom,
! [A: $tType,X3: A,Xs: coindu1593790203_llist @ A] :
( ( coindu6345450stinct @ A @ ( coindu1121789889_LCons @ A @ X3 @ Xs ) )
= ( ~ ( member @ A @ X3 @ ( coindu1112613586e_lset @ A @ Xs ) )
& ( coindu6345450stinct @ A @ Xs ) ) ) ).
% ldistinct_LCons
thf(fact_201_ldistinct_OLCons,axiom,
! [A: $tType,X3: A,Xs: coindu1593790203_llist @ A] :
( ~ ( member @ A @ X3 @ ( coindu1112613586e_lset @ A @ Xs ) )
=> ( ( coindu6345450stinct @ A @ Xs )
=> ( coindu6345450stinct @ A @ ( coindu1121789889_LCons @ A @ X3 @ Xs ) ) ) ) ).
% ldistinct.LCons
thf(fact_202_llist_Oset__sel_I2_J,axiom,
! [A: $tType,A2: coindu1593790203_llist @ A,X3: A] :
( ~ ( coindu335574135_lnull @ A @ A2 )
=> ( ( member @ A @ X3 @ ( coindu1112613586e_lset @ A @ ( coindu1047225960le_ltl @ A @ A2 ) ) )
=> ( member @ A @ X3 @ ( coindu1112613586e_lset @ A @ A2 ) ) ) ) ).
% llist.set_sel(2)
thf(fact_203_llist_Oset__sel_I1_J,axiom,
! [A: $tType,A2: coindu1593790203_llist @ A] :
( ~ ( coindu335574135_lnull @ A @ A2 )
=> ( member @ A @ ( coindu1046438764le_lhd @ A @ A2 ) @ ( coindu1112613586e_lset @ A @ A2 ) ) ) ).
% llist.set_sel(1)
thf(fact_204_lset__intros_I2_J,axiom,
! [A: $tType,X3: A,Xs: coindu1593790203_llist @ A,X6: A] :
( ( member @ A @ X3 @ ( coindu1112613586e_lset @ A @ Xs ) )
=> ( member @ A @ X3 @ ( coindu1112613586e_lset @ A @ ( coindu1121789889_LCons @ A @ X6 @ Xs ) ) ) ) ).
% lset_intros(2)
thf(fact_205_lset__intros_I1_J,axiom,
! [A: $tType,X3: A,Xs: coindu1593790203_llist @ A] : ( member @ A @ X3 @ ( coindu1112613586e_lset @ A @ ( coindu1121789889_LCons @ A @ X3 @ Xs ) ) ) ).
% lset_intros(1)
thf(fact_206_llist_Oset__intros_I2_J,axiom,
! [A: $tType,X3: A,A22: coindu1593790203_llist @ A,A1: A] :
( ( member @ A @ X3 @ ( coindu1112613586e_lset @ A @ A22 ) )
=> ( member @ A @ X3 @ ( coindu1112613586e_lset @ A @ ( coindu1121789889_LCons @ A @ A1 @ A22 ) ) ) ) ).
% llist.set_intros(2)
thf(fact_207_llist_Oset__intros_I1_J,axiom,
! [A: $tType,A1: A,A22: coindu1593790203_llist @ A] : ( member @ A @ A1 @ ( coindu1112613586e_lset @ A @ ( coindu1121789889_LCons @ A @ A1 @ A22 ) ) ) ).
% llist.set_intros(1)
thf(fact_208_lset__cases,axiom,
! [A: $tType,X3: A,Xs: coindu1593790203_llist @ A] :
( ( member @ A @ X3 @ ( coindu1112613586e_lset @ A @ Xs ) )
=> ( ! [Xs6: coindu1593790203_llist @ A] :
( Xs
!= ( coindu1121789889_LCons @ A @ X3 @ Xs6 ) )
=> ~ ! [X7: A,Xs6: coindu1593790203_llist @ A] :
( ( Xs
= ( coindu1121789889_LCons @ A @ X7 @ Xs6 ) )
=> ~ ( member @ A @ X3 @ ( coindu1112613586e_lset @ A @ Xs6 ) ) ) ) ) ).
% lset_cases
thf(fact_209_lset__induct,axiom,
! [A: $tType,X3: A,Xs: coindu1593790203_llist @ A,P: ( coindu1593790203_llist @ A ) > $o] :
( ( member @ A @ X3 @ ( coindu1112613586e_lset @ A @ Xs ) )
=> ( ! [Xs2: coindu1593790203_llist @ A] : ( P @ ( coindu1121789889_LCons @ A @ X3 @ Xs2 ) )
=> ( ! [X7: A,Xs2: coindu1593790203_llist @ A] :
( ( member @ A @ X3 @ ( coindu1112613586e_lset @ A @ Xs2 ) )
=> ( ( X3 != X7 )
=> ( ( P @ Xs2 )
=> ( P @ ( coindu1121789889_LCons @ A @ X7 @ Xs2 ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% lset_induct
thf(fact_210_lset__induct_H,axiom,
! [A: $tType,X3: A,Xs: coindu1593790203_llist @ A,P: ( coindu1593790203_llist @ A ) > $o] :
( ( member @ A @ X3 @ ( coindu1112613586e_lset @ A @ Xs ) )
=> ( ! [Xs2: coindu1593790203_llist @ A] : ( P @ ( coindu1121789889_LCons @ A @ X3 @ Xs2 ) )
=> ( ! [X7: A,Xs2: coindu1593790203_llist @ A] :
( ( member @ A @ X3 @ ( coindu1112613586e_lset @ A @ Xs2 ) )
=> ( ( P @ Xs2 )
=> ( P @ ( coindu1121789889_LCons @ A @ X7 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% lset_induct'
thf(fact_211_llist_Oset__cases,axiom,
! [A: $tType,E: A,A2: coindu1593790203_llist @ A] :
( ( member @ A @ E @ ( coindu1112613586e_lset @ A @ A2 ) )
=> ( ! [Z2: coindu1593790203_llist @ A] :
( A2
!= ( coindu1121789889_LCons @ A @ E @ Z2 ) )
=> ~ ! [Z1: A,Z2: coindu1593790203_llist @ A] :
( ( A2
= ( coindu1121789889_LCons @ A @ Z1 @ Z2 ) )
=> ~ ( member @ A @ E @ ( coindu1112613586e_lset @ A @ Z2 ) ) ) ) ) ).
% llist.set_cases
thf(fact_212_llist_Oset__induct,axiom,
! [A: $tType,X3: A,A2: coindu1593790203_llist @ A,P: A > ( coindu1593790203_llist @ A ) > $o] :
( ( member @ A @ X3 @ ( coindu1112613586e_lset @ A @ A2 ) )
=> ( ! [Z1: A,Z2: coindu1593790203_llist @ A] : ( P @ Z1 @ ( coindu1121789889_LCons @ A @ Z1 @ Z2 ) )
=> ( ! [Z1: A,Z2: coindu1593790203_llist @ A,Xa3: A] :
( ( member @ A @ Xa3 @ ( coindu1112613586e_lset @ A @ Z2 ) )
=> ( ( P @ Xa3 @ Z2 )
=> ( P @ Xa3 @ ( coindu1121789889_LCons @ A @ Z1 @ Z2 ) ) ) )
=> ( P @ X3 @ A2 ) ) ) ) ).
% llist.set_induct
thf(fact_213_in__lset__ltlD,axiom,
! [A: $tType,X3: A,Xs: coindu1593790203_llist @ A] :
( ( member @ A @ X3 @ ( coindu1112613586e_lset @ A @ ( coindu1047225960le_ltl @ A @ Xs ) ) )
=> ( member @ A @ X3 @ ( coindu1112613586e_lset @ A @ Xs ) ) ) ).
% in_lset_ltlD
thf(fact_214_ltakeWhile__all,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,P: A > $o] :
( ! [X5: A] :
( ( member @ A @ X5 @ ( coindu1112613586e_lset @ A @ Xs ) )
=> ( P @ X5 ) )
=> ( ( coindu721411036eWhile @ A @ P @ Xs )
= Xs ) ) ).
% ltakeWhile_all
thf(fact_215_ltakeWhile__cong,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A,P: A > $o,Q: A > $o] :
( ( Xs = Ys )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ ( coindu1112613586e_lset @ A @ Ys ) )
=> ( ( P @ X5 )
= ( Q @ X5 ) ) )
=> ( ( coindu721411036eWhile @ A @ P @ Xs )
= ( coindu721411036eWhile @ A @ Q @ Ys ) ) ) ) ).
% ltakeWhile_cong
thf(fact_216_lset__ltakeWhileD,axiom,
! [A: $tType,X3: A,P: A > $o,Xs: coindu1593790203_llist @ A] :
( ( member @ A @ X3 @ ( coindu1112613586e_lset @ A @ ( coindu721411036eWhile @ A @ P @ Xs ) ) )
=> ( ( member @ A @ X3 @ ( coindu1112613586e_lset @ A @ Xs ) )
& ( P @ X3 ) ) ) ).
% lset_ltakeWhileD
thf(fact_217_llist__set__induct,axiom,
! [A: $tType,X3: A,Xs: coindu1593790203_llist @ A,P: A > ( coindu1593790203_llist @ A ) > $o] :
( ( member @ A @ X3 @ ( coindu1112613586e_lset @ A @ Xs ) )
=> ( ! [Xs2: coindu1593790203_llist @ A] :
( ~ ( coindu335574135_lnull @ A @ Xs2 )
=> ( P @ ( coindu1046438764le_lhd @ A @ Xs2 ) @ Xs2 ) )
=> ( ! [Xs2: coindu1593790203_llist @ A,Y3: A] :
( ~ ( coindu335574135_lnull @ A @ Xs2 )
=> ( ( member @ A @ Y3 @ ( coindu1112613586e_lset @ A @ ( coindu1047225960le_ltl @ A @ Xs2 ) ) )
=> ( ( P @ Y3 @ ( coindu1047225960le_ltl @ A @ Xs2 ) )
=> ( P @ Y3 @ Xs2 ) ) ) )
=> ( P @ X3 @ Xs ) ) ) ) ).
% llist_set_induct
thf(fact_218_ldistinct_Ocases,axiom,
! [A: $tType,A2: coindu1593790203_llist @ A] :
( ( coindu6345450stinct @ A @ A2 )
=> ( ( A2
!= ( coindu1598213697e_LNil @ A ) )
=> ~ ! [X5: A,Xs2: coindu1593790203_llist @ A] :
( ( A2
= ( coindu1121789889_LCons @ A @ X5 @ Xs2 ) )
=> ( ~ ( member @ A @ X5 @ ( coindu1112613586e_lset @ A @ Xs2 ) )
=> ~ ( coindu6345450stinct @ A @ Xs2 ) ) ) ) ) ).
% ldistinct.cases
thf(fact_219_ldistinct_Osimps,axiom,
! [A: $tType] :
( ( coindu6345450stinct @ A )
= ( ^ [A4: coindu1593790203_llist @ A] :
( ( A4
= ( coindu1598213697e_LNil @ A ) )
| ? [X4: A,Xs4: coindu1593790203_llist @ A] :
( ( A4
= ( coindu1121789889_LCons @ A @ X4 @ Xs4 ) )
& ~ ( member @ A @ X4 @ ( coindu1112613586e_lset @ A @ Xs4 ) )
& ( coindu6345450stinct @ A @ Xs4 ) ) ) ) ) ).
% ldistinct.simps
thf(fact_220_ldistinct_Ocoinduct,axiom,
! [A: $tType,X2: ( coindu1593790203_llist @ A ) > $o,X3: coindu1593790203_llist @ A] :
( ( X2 @ X3 )
=> ( ! [X5: coindu1593790203_llist @ A] :
( ( X2 @ X5 )
=> ( ( X5
= ( coindu1598213697e_LNil @ A ) )
| ? [Xa: A,Xs3: coindu1593790203_llist @ A] :
( ( X5
= ( coindu1121789889_LCons @ A @ Xa @ Xs3 ) )
& ~ ( member @ A @ Xa @ ( coindu1112613586e_lset @ A @ Xs3 ) )
& ( ( X2 @ Xs3 )
| ( coindu6345450stinct @ A @ Xs3 ) ) ) ) )
=> ( coindu6345450stinct @ A @ X3 ) ) ) ).
% ldistinct.coinduct
thf(fact_221_compact__eq__lfinite,axiom,
! [A: $tType] :
( ( comple2143767107ompact @ ( coindu1593790203_llist @ A ) @ ( coindu526335360e_lSup @ A ) @ ( coindu1696667936prefix @ A ) )
= ( coindu1213758845finite @ A ) ) ).
% compact_eq_lfinite
thf(fact_222_llexord_Ocoinduct,axiom,
! [A: $tType,X2: ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) > $o,X3: coindu1593790203_llist @ A,Xa2: coindu1593790203_llist @ A,R2: A > A > $o] :
( ( X2 @ X3 @ Xa2 )
=> ( ! [X5: coindu1593790203_llist @ A,Xa3: coindu1593790203_llist @ A] :
( ( X2 @ X5 @ Xa3 )
=> ( ? [Xs3: coindu1593790203_llist @ A,Ys4: coindu1593790203_llist @ A,Xb: A] :
( ( X5
= ( coindu1121789889_LCons @ A @ Xb @ Xs3 ) )
& ( Xa3
= ( coindu1121789889_LCons @ A @ Xb @ Ys4 ) )
& ( ( X2 @ Xs3 @ Ys4 )
| ( coindu300403952lexord @ A @ R2 @ Xs3 @ Ys4 ) ) )
| ? [Xb: A,Y4: A,Xs3: coindu1593790203_llist @ A,Ys4: coindu1593790203_llist @ A] :
( ( X5
= ( coindu1121789889_LCons @ A @ Xb @ Xs3 ) )
& ( Xa3
= ( coindu1121789889_LCons @ A @ Y4 @ Ys4 ) )
& ( R2 @ Xb @ Y4 ) )
| ? [Ys4: coindu1593790203_llist @ A] :
( ( X5
= ( coindu1598213697e_LNil @ A ) )
& ( Xa3 = Ys4 ) ) ) )
=> ( coindu300403952lexord @ A @ R2 @ X3 @ Xa2 ) ) ) ).
% llexord.coinduct
thf(fact_223_lfinite__LCons,axiom,
! [A: $tType,X3: A,Xs: coindu1593790203_llist @ A] :
( ( coindu1213758845finite @ A @ ( coindu1121789889_LCons @ A @ X3 @ Xs ) )
= ( coindu1213758845finite @ A @ Xs ) ) ).
% lfinite_LCons
thf(fact_224_lfinite__code_I2_J,axiom,
! [B: $tType,X3: B,Xs: coindu1593790203_llist @ B] :
( ( coindu1213758845finite @ B @ ( coindu1121789889_LCons @ B @ X3 @ Xs ) )
= ( coindu1213758845finite @ B @ Xs ) ) ).
% lfinite_code(2)
thf(fact_225_lfinite__code_I1_J,axiom,
! [A: $tType] : ( coindu1213758845finite @ A @ ( coindu1598213697e_LNil @ A ) ) ).
% lfinite_code(1)
thf(fact_226_lfinite__ltl,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( ( coindu1213758845finite @ A @ ( coindu1047225960le_ltl @ A @ Xs ) )
= ( coindu1213758845finite @ A @ Xs ) ) ).
% lfinite_ltl
thf(fact_227_lfinite__ltakeWhile,axiom,
! [A: $tType,P: A > $o,Xs: coindu1593790203_llist @ A] :
( ( coindu1213758845finite @ A @ ( coindu721411036eWhile @ A @ P @ Xs ) )
= ( ( coindu1213758845finite @ A @ Xs )
| ? [X4: A] :
( ( member @ A @ X4 @ ( coindu1112613586e_lset @ A @ Xs ) )
& ~ ( P @ X4 ) ) ) ) ).
% lfinite_ltakeWhile
thf(fact_228_Coinductive__List__Mirabelle__kmikjhschf_Olprefix__nitpick__simps,axiom,
! [A: $tType] :
( ( coindu1696667936prefix @ A )
= ( ^ [Xs4: coindu1593790203_llist @ A,Ys3: coindu1593790203_llist @ A] :
( ( ( coindu1213758845finite @ A @ Xs4 )
=> ( coindu1571841457prefix @ A @ Xs4 @ Ys3 ) )
& ( ~ ( coindu1213758845finite @ A @ Xs4 )
=> ( Xs4 = Ys3 ) ) ) ) ) ).
% Coinductive_List_Mirabelle_kmikjhschf.lprefix_nitpick_simps
thf(fact_229_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_230_lfinite__lSupD,axiom,
! [A: $tType,A3: set @ ( coindu1593790203_llist @ A )] :
( ( coindu1213758845finite @ A @ ( coindu526335360e_lSup @ A @ A3 ) )
=> ! [X: coindu1593790203_llist @ A] :
( ( member @ ( coindu1593790203_llist @ A ) @ X @ A3 )
=> ( coindu1213758845finite @ A @ X ) ) ) ).
% lfinite_lSupD
thf(fact_231_lfinite__LNil,axiom,
! [A: $tType] : ( coindu1213758845finite @ A @ ( coindu1598213697e_LNil @ A ) ) ).
% lfinite_LNil
thf(fact_232_lnull__imp__lfinite,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ A @ Xs )
=> ( coindu1213758845finite @ A @ Xs ) ) ).
% lnull_imp_lfinite
thf(fact_233_lfinite__LConsI,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,X3: A] :
( ( coindu1213758845finite @ A @ Xs )
=> ( coindu1213758845finite @ A @ ( coindu1121789889_LCons @ A @ X3 @ Xs ) ) ) ).
% lfinite_LConsI
thf(fact_234_not__lfinite__lprefix__conv__eq,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ~ ( coindu1213758845finite @ A @ Xs )
=> ( ( coindu1696667936prefix @ A @ Xs @ Ys )
= ( Xs = Ys ) ) ) ).
% not_lfinite_lprefix_conv_eq
thf(fact_235_lprefix__lfiniteD,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ Xs @ Ys )
=> ( ( coindu1213758845finite @ A @ Ys )
=> ( coindu1213758845finite @ A @ Xs ) ) ) ).
% lprefix_lfiniteD
thf(fact_236_llexord__LNil,axiom,
! [A: $tType,R2: A > A > $o,Ys: coindu1593790203_llist @ A] : ( coindu300403952lexord @ A @ R2 @ ( coindu1598213697e_LNil @ A ) @ Ys ) ).
% llexord_LNil
thf(fact_237_llexord__LCons__less,axiom,
! [A: $tType,R2: A > A > $o,X3: A,Y: A,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( R2 @ X3 @ Y )
=> ( coindu300403952lexord @ A @ R2 @ ( coindu1121789889_LCons @ A @ X3 @ Xs ) @ ( coindu1121789889_LCons @ A @ Y @ Ys ) ) ) ).
% llexord_LCons_less
thf(fact_238_llexord__LCons__eq,axiom,
! [A: $tType,R2: A > A > $o,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A,X3: A] :
( ( coindu300403952lexord @ A @ R2 @ Xs @ Ys )
=> ( coindu300403952lexord @ A @ R2 @ ( coindu1121789889_LCons @ A @ X3 @ Xs ) @ ( coindu1121789889_LCons @ A @ X3 @ Ys ) ) ) ).
% llexord_LCons_eq
thf(fact_239_lfinite__induct,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,P: ( coindu1593790203_llist @ A ) > $o] :
( ( coindu1213758845finite @ A @ Xs )
=> ( ! [Xs2: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ A @ Xs2 )
=> ( P @ Xs2 ) )
=> ( ! [Xs2: coindu1593790203_llist @ A] :
( ( coindu1213758845finite @ A @ Xs2 )
=> ( ~ ( coindu335574135_lnull @ A @ Xs2 )
=> ( ( P @ ( coindu1047225960le_ltl @ A @ Xs2 ) )
=> ( P @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% lfinite_induct
thf(fact_240_lfinite_Oinducts,axiom,
! [A: $tType,X3: coindu1593790203_llist @ A,P: ( coindu1593790203_llist @ A ) > $o] :
( ( coindu1213758845finite @ A @ X3 )
=> ( ( P @ ( coindu1598213697e_LNil @ A ) )
=> ( ! [Xs2: coindu1593790203_llist @ A,X5: A] :
( ( coindu1213758845finite @ A @ Xs2 )
=> ( ( P @ Xs2 )
=> ( P @ ( coindu1121789889_LCons @ A @ X5 @ Xs2 ) ) ) )
=> ( P @ X3 ) ) ) ) ).
% lfinite.inducts
thf(fact_241_lfinite_Osimps,axiom,
! [A: $tType] :
( ( coindu1213758845finite @ A )
= ( ^ [A4: coindu1593790203_llist @ A] :
( ( A4
= ( coindu1598213697e_LNil @ A ) )
| ? [Xs4: coindu1593790203_llist @ A,X4: A] :
( ( A4
= ( coindu1121789889_LCons @ A @ X4 @ Xs4 ) )
& ( coindu1213758845finite @ A @ Xs4 ) ) ) ) ) ).
% lfinite.simps
thf(fact_242_lfinite_Ocases,axiom,
! [A: $tType,A2: coindu1593790203_llist @ A] :
( ( coindu1213758845finite @ A @ A2 )
=> ( ( A2
!= ( coindu1598213697e_LNil @ A ) )
=> ~ ! [Xs2: coindu1593790203_llist @ A] :
( ? [X5: A] :
( A2
= ( coindu1121789889_LCons @ A @ X5 @ Xs2 ) )
=> ~ ( coindu1213758845finite @ A @ Xs2 ) ) ) ) ).
% lfinite.cases
thf(fact_243_llimit__induct,axiom,
! [A: $tType,P: ( coindu1593790203_llist @ A ) > $o,Xs: coindu1593790203_llist @ A] :
( ( P @ ( coindu1598213697e_LNil @ A ) )
=> ( ! [X5: A,Xs2: coindu1593790203_llist @ A] :
( ( coindu1213758845finite @ A @ Xs2 )
=> ( ( P @ Xs2 )
=> ( P @ ( coindu1121789889_LCons @ A @ X5 @ Xs2 ) ) ) )
=> ( ( ! [Ys4: coindu1593790203_llist @ A] :
( ( coindu574146665prefix @ A @ Ys4 @ Xs )
=> ( P @ Ys4 ) )
=> ( P @ Xs ) )
=> ( P @ Xs ) ) ) ) ).
% llimit_induct
thf(fact_244_compact__lfiniteI,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( ( coindu1213758845finite @ A @ Xs )
=> ( comple2143767107ompact @ ( coindu1593790203_llist @ A ) @ ( coindu526335360e_lSup @ A ) @ ( coindu1696667936prefix @ A ) @ Xs ) ) ).
% compact_lfiniteI
thf(fact_245_compact__lfiniteD,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( ( comple2143767107ompact @ ( coindu1593790203_llist @ A ) @ ( coindu526335360e_lSup @ A ) @ ( coindu1696667936prefix @ A ) @ Xs )
=> ( coindu1213758845finite @ A @ Xs ) ) ).
% compact_lfiniteD
thf(fact_246_llexord_Ocases,axiom,
! [A: $tType,R2: A > A > $o,A1: coindu1593790203_llist @ A,A22: coindu1593790203_llist @ A] :
( ( coindu300403952lexord @ A @ R2 @ A1 @ A22 )
=> ( ! [Xs2: coindu1593790203_llist @ A,Ys2: coindu1593790203_llist @ A,X5: A] :
( ( A1
= ( coindu1121789889_LCons @ A @ X5 @ Xs2 ) )
=> ( ( A22
= ( coindu1121789889_LCons @ A @ X5 @ Ys2 ) )
=> ~ ( coindu300403952lexord @ A @ R2 @ Xs2 @ Ys2 ) ) )
=> ( ! [X5: A] :
( ? [Xs2: coindu1593790203_llist @ A] :
( A1
= ( coindu1121789889_LCons @ A @ X5 @ Xs2 ) )
=> ! [Y3: A] :
( ? [Ys2: coindu1593790203_llist @ A] :
( A22
= ( coindu1121789889_LCons @ A @ Y3 @ Ys2 ) )
=> ~ ( R2 @ X5 @ Y3 ) ) )
=> ~ ( ( A1
= ( coindu1598213697e_LNil @ A ) )
=> ! [Ys2: coindu1593790203_llist @ A] : A22 != Ys2 ) ) ) ) ).
% llexord.cases
thf(fact_247_llexord_Osimps,axiom,
! [A: $tType] :
( ( coindu300403952lexord @ A )
= ( ^ [R3: A > A > $o,A12: coindu1593790203_llist @ A,A23: coindu1593790203_llist @ A] :
( ? [Xs4: coindu1593790203_llist @ A,Ys3: coindu1593790203_llist @ A,X4: A] :
( ( A12
= ( coindu1121789889_LCons @ A @ X4 @ Xs4 ) )
& ( A23
= ( coindu1121789889_LCons @ A @ X4 @ Ys3 ) )
& ( coindu300403952lexord @ A @ R3 @ Xs4 @ Ys3 ) )
| ? [X4: A,Y5: A,Xs4: coindu1593790203_llist @ A,Ys3: coindu1593790203_llist @ A] :
( ( A12
= ( coindu1121789889_LCons @ A @ X4 @ Xs4 ) )
& ( A23
= ( coindu1121789889_LCons @ A @ Y5 @ Ys3 ) )
& ( R3 @ X4 @ Y5 ) )
| ? [Ys3: coindu1593790203_llist @ A] :
( ( A12
= ( coindu1598213697e_LNil @ A ) )
& ( A23 = Ys3 ) ) ) ) ) ).
% llexord.simps
thf(fact_248_llist__induct,axiom,
! [A: $tType,P: ( coindu1593790203_llist @ A ) > $o,Xs: coindu1593790203_llist @ A] :
( ( comple1986059218ssible @ ( coindu1593790203_llist @ A ) @ ( coindu526335360e_lSup @ A ) @ ( coindu1696667936prefix @ A ) @ P )
=> ( ( P @ ( coindu1598213697e_LNil @ A ) )
=> ( ! [X5: A,Xs2: coindu1593790203_llist @ A] :
( ( coindu1213758845finite @ A @ Xs2 )
=> ( ( P @ Xs2 )
=> ( P @ ( coindu1121789889_LCons @ A @ X5 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% llist_induct
thf(fact_249_lfinite__lSup__chain,axiom,
! [A: $tType,A3: set @ ( coindu1593790203_llist @ A )] :
( ( comple1193779247_chain @ ( coindu1593790203_llist @ A ) @ ( coindu1696667936prefix @ A ) @ A3 )
=> ( ( coindu1213758845finite @ A @ ( coindu526335360e_lSup @ A @ A3 ) )
= ( ( finite_finite @ ( coindu1593790203_llist @ A ) @ A3 )
& ! [X4: coindu1593790203_llist @ A] :
( ( member @ ( coindu1593790203_llist @ A ) @ X4 @ A3 )
=> ( coindu1213758845finite @ A @ X4 ) ) ) ) ) ).
% lfinite_lSup_chain
thf(fact_250_lfinite__imp__finite__lset,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( ( coindu1213758845finite @ A @ Xs )
=> ( finite_finite @ A @ ( coindu1112613586e_lset @ A @ Xs ) ) ) ).
% lfinite_imp_finite_lset
thf(fact_251_llist__gen__induct,axiom,
! [A: $tType,P: ( coindu1593790203_llist @ A ) > $o,Xs: coindu1593790203_llist @ A] :
( ( comple1986059218ssible @ ( coindu1593790203_llist @ A ) @ ( coindu526335360e_lSup @ A ) @ ( coindu1696667936prefix @ A ) @ P )
=> ( ? [Zs2: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ Zs2 @ Xs )
& ( coindu1213758845finite @ A @ Zs2 )
& ! [Ys2: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ Zs2 @ Ys2 )
=> ( ( coindu1696667936prefix @ A @ Ys2 @ Xs )
=> ( ( coindu1213758845finite @ A @ Ys2 )
=> ( P @ Ys2 ) ) ) ) )
=> ( P @ Xs ) ) ) ).
% llist_gen_induct
thf(fact_252_finite__Int,axiom,
! [A: $tType,F4: set @ A,G3: set @ A] :
( ( ( finite_finite @ A @ F4 )
| ( finite_finite @ A @ G3 ) )
=> ( finite_finite @ A @ ( inf_inf @ ( set @ A ) @ F4 @ G3 ) ) ) ).
% finite_Int
thf(fact_253_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
%----Type constructors (6)
thf(tcon_fun___Lattices_Osemilattice__inf,axiom,
! [A6: $tType,A7: $tType] :
( ( semilattice_inf @ A7 @ ( type2 @ A7 ) )
=> ( semilattice_inf @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
thf(tcon_fun___Lattices_Olattice,axiom,
! [A6: $tType,A7: $tType] :
( ( lattice @ A7 @ ( type2 @ A7 ) )
=> ( lattice @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
thf(tcon_Set_Oset___Lattices_Osemilattice__inf_1,axiom,
! [A6: $tType] : ( semilattice_inf @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
thf(tcon_Set_Oset___Lattices_Olattice_2,axiom,
! [A6: $tType] : ( lattice @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
thf(tcon_HOL_Obool___Lattices_Osemilattice__inf_3,axiom,
semilattice_inf @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Lattices_Olattice_4,axiom,
lattice @ $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,X3: A,Y: A] :
( ( if @ A @ $false @ X3 @ Y )
= Y ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X3: A,Y: A] :
( ( if @ A @ $true @ X3 @ Y )
= X3 ) ).
%----Conjectures (1)
thf(conj_0,conjecture,
less_eq @ ( coindu1046438764le_lhd @ a @ ( coindu526335360e_lSup @ a @ ya ) ) @ ( coindu1046438764le_lhd @ a @ ( coindu1047225960le_ltl @ a @ ( coindu526335360e_lSup @ a @ ya ) ) ) ).
%------------------------------------------------------------------------------