TPTP Problem File: DAT118^1.p
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%------------------------------------------------------------------------------
% File : DAT118^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Data Structures
% Problem : Coinductive list 970
% 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__970.p [Bla16]
% Status : Theorem
% Rating : 1.00 v7.1.0
% Syntax : Number of formulae : 303 ( 110 unt; 38 typ; 0 def)
% Number of atoms : 705 ( 256 equ; 0 cnn)
% Maximal formula atoms : 8 ( 2 avg)
% Number of connectives : 4092 ( 98 ~; 22 |; 54 &;3599 @)
% ( 0 <=>; 319 =>; 0 <=; 0 <~>)
% Maximal formula depth : 28 ( 8 avg)
% Number of types : 2 ( 1 usr)
% Number of type conns : 212 ( 212 >; 0 *; 0 +; 0 <<)
% Number of symbols : 40 ( 37 usr; 3 con; 0-7 aty)
% Number of variables : 1130 ( 42 ^;1015 !; 32 ?;1130 :)
% ( 41 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 14:47:51.610
%------------------------------------------------------------------------------
%----Could-be-implicit typings (5)
thf(ty_t_Coinductive__List__Mirabelle__kmikjhschf_Ollist,type,
coindu1593790203_llist: $tType > $tType ).
thf(ty_t_Product__Type_Oprod,type,
product_prod: $tType > $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 (33)
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_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Lattices_Osemilattice__sup,type,
semilattice_sup:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ogen__lset,type,
coindu1928975208n_lset:
!>[A: $tType] : ( ( set @ A ) > ( coindu1593790203_llist @ A ) > ( set @ A ) ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_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_Olfinite,type,
coindu1213758845finite:
!>[A: $tType] : ( ( 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_Olmap,type,
coindu1062782156e_lmap:
!>[A: $tType,Aa: $tType] : ( ( A > Aa ) > ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ Aa ) ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollist_Olnull,type,
coindu335574135_lnull:
!>[A: $tType] : ( ( coindu1593790203_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollist_Olset,type,
coindu1112613586e_lset:
!>[A: $tType] : ( ( coindu1593790203_llist @ A ) > ( set @ A ) ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollist_Oltl,type,
coindu1047225960le_ltl:
!>[A: $tType] : ( ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Olmember,type,
coindu567634248member:
!>[A: $tType] : ( A > ( coindu1593790203_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Olprefix,type,
coindu1696667936prefix:
!>[A: $tType] : ( ( coindu1593790203_llist @ A ) > ( 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_If,type,
if:
!>[A: $tType] : ( $o > A > A > A ) ).
thf(sy_c_Lattices_Osup__class_Osup,type,
sup_sup:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Product__Type_OPair,type,
product_Pair:
!>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
thf(sy_c_Product__Type_Ointernal__case__prod,type,
produc2004651681e_prod:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
product_rec_prod:
!>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
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_xs,type,
xs: coindu1593790203_llist @ a ).
%----Relevant facts (254)
thf(fact_0_assms,axiom,
comple2143767107ompact @ ( coindu1593790203_llist @ a ) @ ( coindu526335360e_lSup @ a ) @ ( coindu1696667936prefix @ a ) @ xs ).
% assms
thf(fact_1_lfinite__lappend,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu1213758845finite @ A @ ( coindu268472904append @ A @ Xs @ Ys ) )
= ( ( coindu1213758845finite @ A @ Xs )
& ( coindu1213758845finite @ A @ Ys ) ) ) ).
% lfinite_lappend
thf(fact_2_lfinite__lmap,axiom,
! [A: $tType,B: $tType,F: B > A,Xs: coindu1593790203_llist @ B] :
( ( coindu1213758845finite @ A @ ( coindu1062782156e_lmap @ B @ A @ F @ Xs ) )
= ( coindu1213758845finite @ B @ Xs ) ) ).
% lfinite_lmap
thf(fact_3_lfinite__code_I1_J,axiom,
! [A: $tType] : ( coindu1213758845finite @ A @ ( coindu1598213697e_LNil @ A ) ) ).
% lfinite_code(1)
thf(fact_4_lfinite__ltl,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( ( coindu1213758845finite @ A @ ( coindu1047225960le_ltl @ A @ Xs ) )
= ( coindu1213758845finite @ A @ Xs ) ) ).
% lfinite_ltl
thf(fact_5_lfinite__code_I2_J,axiom,
! [B: $tType,X: B,Xs: coindu1593790203_llist @ B] :
( ( coindu1213758845finite @ B @ ( coindu1121789889_LCons @ B @ X @ Xs ) )
= ( coindu1213758845finite @ B @ Xs ) ) ).
% lfinite_code(2)
thf(fact_6_lfinite__LCons,axiom,
! [A: $tType,X: A,Xs: coindu1593790203_llist @ A] :
( ( coindu1213758845finite @ A @ ( coindu1121789889_LCons @ A @ X @ Xs ) )
= ( coindu1213758845finite @ A @ Xs ) ) ).
% lfinite_LCons
thf(fact_7_lappend__inf,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ~ ( coindu1213758845finite @ A @ Xs )
=> ( ( coindu268472904append @ A @ Xs @ Ys )
= Xs ) ) ).
% lappend_inf
thf(fact_8_lfinite__LNil,axiom,
! [A: $tType] : ( coindu1213758845finite @ A @ ( coindu1598213697e_LNil @ A ) ) ).
% lfinite_LNil
thf(fact_9_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_10_lfinite__lSupD,axiom,
! [A: $tType,A2: set @ ( coindu1593790203_llist @ A )] :
( ( coindu1213758845finite @ A @ ( coindu526335360e_lSup @ A @ A2 ) )
=> ! [X2: coindu1593790203_llist @ A] :
( ( member @ ( coindu1593790203_llist @ A ) @ X2 @ A2 )
=> ( coindu1213758845finite @ A @ X2 ) ) ) ).
% lfinite_lSupD
thf(fact_11_lnull__imp__lfinite,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ A @ Xs )
=> ( coindu1213758845finite @ A @ Xs ) ) ).
% lnull_imp_lfinite
thf(fact_12_lprefix__refl,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] : ( coindu1696667936prefix @ A @ Xs @ Xs ) ).
% lprefix_refl
thf(fact_13_llist_Oleq__refl,axiom,
! [A: $tType,X: coindu1593790203_llist @ A] : ( coindu1696667936prefix @ A @ X @ X ) ).
% llist.leq_refl
thf(fact_14_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_15_LCons__lprefix__LCons,axiom,
! [A: $tType,X: A,Xs: coindu1593790203_llist @ A,Y: A,Ys: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ ( coindu1121789889_LCons @ A @ X @ Xs ) @ ( coindu1121789889_LCons @ A @ Y @ Ys ) )
= ( ( X = Y )
& ( coindu1696667936prefix @ A @ Xs @ Ys ) ) ) ).
% LCons_lprefix_LCons
thf(fact_16_lprefix__code_I1_J,axiom,
! [A: $tType,Ys: coindu1593790203_llist @ A] : ( coindu1696667936prefix @ A @ ( coindu1598213697e_LNil @ A ) @ Ys ) ).
% lprefix_code(1)
thf(fact_17_lnull__lSup,axiom,
! [A: $tType,A2: set @ ( coindu1593790203_llist @ A )] :
( ( coindu335574135_lnull @ A @ ( coindu526335360e_lSup @ A @ A2 ) )
= ( ! [X3: coindu1593790203_llist @ A] :
( ( member @ ( coindu1593790203_llist @ A ) @ X3 @ A2 )
=> ( coindu335574135_lnull @ A @ X3 ) ) ) ) ).
% lnull_lSup
thf(fact_18_llist_Omap__disc__iff,axiom,
! [B: $tType,A: $tType,F: A > B,A3: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ B @ ( coindu1062782156e_lmap @ A @ B @ F @ A3 ) )
= ( coindu335574135_lnull @ A @ A3 ) ) ).
% llist.map_disc_iff
thf(fact_19_lappend_Odisc__iff_I2_J,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( ~ ( coindu335574135_lnull @ A @ ( coindu268472904append @ A @ Xs @ Ys ) ) )
= ( ~ ( coindu335574135_lnull @ A @ Xs )
| ~ ( coindu335574135_lnull @ A @ Ys ) ) ) ).
% lappend.disc_iff(2)
thf(fact_20_lnull__lappend,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ A @ ( coindu268472904append @ A @ Xs @ Ys ) )
= ( ( coindu335574135_lnull @ A @ Xs )
& ( coindu335574135_lnull @ A @ Ys ) ) ) ).
% lnull_lappend
thf(fact_21_lappend__code_I2_J,axiom,
! [A: $tType,Xa: A,X: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu268472904append @ A @ ( coindu1121789889_LCons @ A @ Xa @ X ) @ Ys )
= ( coindu1121789889_LCons @ A @ Xa @ ( coindu268472904append @ A @ X @ Ys ) ) ) ).
% lappend_code(2)
thf(fact_22_lappend__code_I1_J,axiom,
! [A: $tType,Ys: coindu1593790203_llist @ A] :
( ( coindu268472904append @ A @ ( coindu1598213697e_LNil @ A ) @ Ys )
= Ys ) ).
% lappend_code(1)
thf(fact_23_lappend__LNil2,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( ( coindu268472904append @ A @ Xs @ ( coindu1598213697e_LNil @ A ) )
= Xs ) ).
% lappend_LNil2
thf(fact_24_ltl__lmap,axiom,
! [A: $tType,B: $tType,F: B > A,Xs: coindu1593790203_llist @ B] :
( ( coindu1047225960le_ltl @ A @ ( coindu1062782156e_lmap @ B @ A @ F @ Xs ) )
= ( coindu1062782156e_lmap @ B @ A @ F @ ( coindu1047225960le_ltl @ B @ Xs ) ) ) ).
% ltl_lmap
thf(fact_25_lprefix__LNil,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ Xs @ ( coindu1598213697e_LNil @ A ) )
= ( coindu335574135_lnull @ A @ Xs ) ) ).
% lprefix_LNil
thf(fact_26_lprefix__lappend__same,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A,Zs: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ ( coindu268472904append @ A @ Xs @ Ys ) @ ( coindu268472904append @ A @ Xs @ Zs ) )
= ( ( coindu1213758845finite @ A @ Xs )
=> ( coindu1696667936prefix @ A @ Ys @ Zs ) ) ) ).
% lprefix_lappend_same
thf(fact_27_ltl__lappend,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( ( coindu335574135_lnull @ A @ Xs )
=> ( ( coindu1047225960le_ltl @ A @ ( coindu268472904append @ A @ Xs @ Ys ) )
= ( coindu1047225960le_ltl @ A @ Ys ) ) )
& ( ~ ( coindu335574135_lnull @ A @ Xs )
=> ( ( coindu1047225960le_ltl @ A @ ( coindu268472904append @ A @ Xs @ Ys ) )
= ( coindu268472904append @ A @ ( coindu1047225960le_ltl @ A @ Xs ) @ Ys ) ) ) ) ).
% ltl_lappend
thf(fact_28_compact__LCons__iff,axiom,
! [A: $tType,X: A,Xs: coindu1593790203_llist @ A] :
( ( comple2143767107ompact @ ( coindu1593790203_llist @ A ) @ ( coindu526335360e_lSup @ A ) @ ( coindu1696667936prefix @ A ) @ ( coindu1121789889_LCons @ A @ X @ Xs ) )
= ( comple2143767107ompact @ ( coindu1593790203_llist @ A ) @ ( coindu526335360e_lSup @ A ) @ ( coindu1696667936prefix @ A ) @ Xs ) ) ).
% compact_LCons_iff
thf(fact_29_lSup_Octr_I1_J,axiom,
! [A: $tType,A2: set @ ( coindu1593790203_llist @ A )] :
( ! [X4: coindu1593790203_llist @ A] :
( ( member @ ( coindu1593790203_llist @ A ) @ X4 @ A2 )
=> ( coindu335574135_lnull @ A @ X4 ) )
=> ( ( coindu526335360e_lSup @ A @ A2 )
= ( coindu1598213697e_LNil @ A ) ) ) ).
% lSup.ctr(1)
thf(fact_30_lSup_Odisc_I2_J,axiom,
! [A: $tType,A2: set @ ( coindu1593790203_llist @ A )] :
( ~ ! [X2: coindu1593790203_llist @ A] :
( ( member @ ( coindu1593790203_llist @ A ) @ X2 @ A2 )
=> ( coindu335574135_lnull @ A @ X2 ) )
=> ~ ( coindu335574135_lnull @ A @ ( coindu526335360e_lSup @ A @ A2 ) ) ) ).
% lSup.disc(2)
thf(fact_31_lSup_Odisc_I1_J,axiom,
! [A: $tType,A2: set @ ( coindu1593790203_llist @ A )] :
( ! [X4: coindu1593790203_llist @ A] :
( ( member @ ( coindu1593790203_llist @ A ) @ X4 @ A2 )
=> ( coindu335574135_lnull @ A @ X4 ) )
=> ( coindu335574135_lnull @ A @ ( coindu526335360e_lSup @ A @ A2 ) ) ) ).
% lSup.disc(1)
thf(fact_32_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_33_ltl__simps_I1_J,axiom,
! [A: $tType] :
( ( coindu1047225960le_ltl @ A @ ( coindu1598213697e_LNil @ A ) )
= ( coindu1598213697e_LNil @ A ) ) ).
% ltl_simps(1)
thf(fact_34_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_35_llist_Odisc_I1_J,axiom,
! [A: $tType] : ( coindu335574135_lnull @ A @ ( coindu1598213697e_LNil @ A ) ) ).
% llist.disc(1)
thf(fact_36_lappend_Octr_I1_J,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ A @ Xs )
=> ( ( coindu335574135_lnull @ A @ Ys )
=> ( ( coindu268472904append @ A @ Xs @ Ys )
= ( coindu1598213697e_LNil @ A ) ) ) ) ).
% lappend.ctr(1)
thf(fact_37_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_38_llist_OdiscI_I1_J,axiom,
! [A: $tType,Llist: coindu1593790203_llist @ A] :
( ( Llist
= ( coindu1598213697e_LNil @ A ) )
=> ( coindu335574135_lnull @ A @ Llist ) ) ).
% llist.discI(1)
thf(fact_39_llist_Osimps_I13_J,axiom,
! [B: $tType,A: $tType,F: A > B,X21: A,X22: coindu1593790203_llist @ A] :
( ( coindu1062782156e_lmap @ A @ B @ F @ ( coindu1121789889_LCons @ A @ X21 @ X22 ) )
= ( coindu1121789889_LCons @ B @ ( F @ X21 ) @ ( coindu1062782156e_lmap @ A @ B @ F @ X22 ) ) ) ).
% llist.simps(13)
thf(fact_40_llist_Osimps_I12_J,axiom,
! [A: $tType,B: $tType,F: A > B] :
( ( coindu1062782156e_lmap @ A @ B @ F @ ( coindu1598213697e_LNil @ A ) )
= ( coindu1598213697e_LNil @ B ) ) ).
% llist.simps(12)
thf(fact_41_lappend_Odisc_I2_J,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( ~ ( coindu335574135_lnull @ A @ Xs )
| ~ ( coindu335574135_lnull @ A @ Ys ) )
=> ~ ( coindu335574135_lnull @ A @ ( coindu268472904append @ A @ Xs @ Ys ) ) ) ).
% lappend.disc(2)
thf(fact_42_lappend_Odisc_I1_J,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ A @ Xs )
=> ( ( coindu335574135_lnull @ A @ Ys )
=> ( coindu335574135_lnull @ A @ ( coindu268472904append @ A @ Xs @ Ys ) ) ) ) ).
% lappend.disc(1)
thf(fact_43_lprefix__code_I2_J,axiom,
! [A: $tType,X: A,Xs: coindu1593790203_llist @ A] :
~ ( coindu1696667936prefix @ A @ ( coindu1121789889_LCons @ A @ X @ Xs ) @ ( coindu1598213697e_LNil @ A ) ) ).
% lprefix_code(2)
thf(fact_44_lSup_Odisc__iff_I2_J,axiom,
! [A: $tType,A2: set @ ( coindu1593790203_llist @ A )] :
( ( coindu335574135_lnull @ A @ ( coindu526335360e_lSup @ A @ A2 ) )
= ( ! [X3: coindu1593790203_llist @ A] :
( ( member @ ( coindu1593790203_llist @ A ) @ X3 @ A2 )
=> ( coindu335574135_lnull @ A @ X3 ) ) ) ) ).
% lSup.disc_iff(2)
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] :
( ! [X4: A] :
( ( P @ X4 )
= ( Q @ X4 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_48_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X4: A] :
( ( F @ X4 )
= ( G @ X4 ) )
=> ( F = G ) ) ).
% ext
thf(fact_49_llist_Omap__sel_I2_J,axiom,
! [B: $tType,A: $tType,A3: coindu1593790203_llist @ A,F: A > B] :
( ~ ( coindu335574135_lnull @ A @ A3 )
=> ( ( coindu1047225960le_ltl @ B @ ( coindu1062782156e_lmap @ A @ B @ F @ A3 ) )
= ( coindu1062782156e_lmap @ A @ B @ F @ ( coindu1047225960le_ltl @ A @ A3 ) ) ) ) ).
% llist.map_sel(2)
thf(fact_50_llist_Ocollapse_I1_J,axiom,
! [A: $tType,Llist: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ A @ Llist )
=> ( Llist
= ( coindu1598213697e_LNil @ A ) ) ) ).
% llist.collapse(1)
thf(fact_51_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_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_lappend__ltl,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ~ ( coindu335574135_lnull @ A @ Xs )
=> ( ( coindu268472904append @ A @ ( coindu1047225960le_ltl @ A @ Xs ) @ Ys )
= ( coindu1047225960le_ltl @ A @ ( coindu268472904append @ A @ Xs @ Ys ) ) ) ) ).
% lappend_ltl
thf(fact_54_LNil__eq__lmap,axiom,
! [A: $tType,B: $tType,F: B > A,Xs: coindu1593790203_llist @ B] :
( ( ( coindu1598213697e_LNil @ A )
= ( coindu1062782156e_lmap @ B @ A @ F @ Xs ) )
= ( Xs
= ( coindu1598213697e_LNil @ B ) ) ) ).
% LNil_eq_lmap
thf(fact_55_lmap__eq__LNil,axiom,
! [A: $tType,B: $tType,F: B > A,Xs: coindu1593790203_llist @ B] :
( ( ( coindu1062782156e_lmap @ B @ A @ F @ Xs )
= ( coindu1598213697e_LNil @ A ) )
= ( Xs
= ( coindu1598213697e_LNil @ B ) ) ) ).
% lmap_eq_LNil
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_lappend__assoc,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A,Zs: coindu1593790203_llist @ A] :
( ( coindu268472904append @ A @ ( coindu268472904append @ A @ Xs @ Ys ) @ Zs )
= ( coindu268472904append @ A @ Xs @ ( coindu268472904append @ A @ Ys @ Zs ) ) ) ).
% lappend_assoc
thf(fact_58_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_59_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_60_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,X4: A] :
( ( A1
= ( coindu1121789889_LCons @ A @ X4 @ Xs2 ) )
=> ( ( A22
= ( coindu1121789889_LCons @ A @ X4 @ Ys2 ) )
=> ~ ( coindu1696667936prefix @ A @ Xs2 @ Ys2 ) ) ) ) ) ).
% lprefix.cases
thf(fact_61_lprefix_Osimps,axiom,
! [A: $tType] :
( ( coindu1696667936prefix @ A )
= ( ^ [A12: coindu1593790203_llist @ A,A23: coindu1593790203_llist @ A] :
( ? [Xs3: coindu1593790203_llist @ A] :
( ( A12
= ( coindu1598213697e_LNil @ A ) )
& ( A23 = Xs3 ) )
| ? [Xs3: coindu1593790203_llist @ A,Ys3: coindu1593790203_llist @ A,X3: A] :
( ( A12
= ( coindu1121789889_LCons @ A @ X3 @ Xs3 ) )
& ( A23
= ( coindu1121789889_LCons @ A @ X3 @ Ys3 ) )
& ( coindu1696667936prefix @ A @ Xs3 @ Ys3 ) ) ) ) ) ).
% lprefix.simps
thf(fact_62_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_63_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_64_neq__LNil__conv,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( ( Xs
!= ( coindu1598213697e_LNil @ A ) )
= ( ? [X3: A,Xs4: coindu1593790203_llist @ A] :
( Xs
= ( coindu1121789889_LCons @ A @ X3 @ Xs4 ) ) ) ) ).
% neq_LNil_conv
thf(fact_65_compact__LConsD,axiom,
! [A: $tType,X: A,Xs: coindu1593790203_llist @ A] :
( ( comple2143767107ompact @ ( coindu1593790203_llist @ A ) @ ( coindu526335360e_lSup @ A ) @ ( coindu1696667936prefix @ A ) @ ( coindu1121789889_LCons @ A @ X @ Xs ) )
=> ( comple2143767107ompact @ ( coindu1593790203_llist @ A ) @ ( coindu526335360e_lSup @ A ) @ ( coindu1696667936prefix @ A ) @ Xs ) ) ).
% compact_LConsD
thf(fact_66_compact__LConsI,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,X: A] :
( ( comple2143767107ompact @ ( coindu1593790203_llist @ A ) @ ( coindu526335360e_lSup @ A ) @ ( coindu1696667936prefix @ A ) @ Xs )
=> ( comple2143767107ompact @ ( coindu1593790203_llist @ A ) @ ( coindu526335360e_lSup @ A ) @ ( coindu1696667936prefix @ A ) @ ( coindu1121789889_LCons @ A @ X @ Xs ) ) ) ).
% compact_LConsI
thf(fact_67_lappend__lnull1,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ A @ Xs )
=> ( ( coindu268472904append @ A @ Xs @ Ys )
= Ys ) ) ).
% lappend_lnull1
thf(fact_68_lappend__lnull2,axiom,
! [A: $tType,Ys: coindu1593790203_llist @ A,Xs: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ A @ Ys )
=> ( ( coindu268472904append @ A @ Xs @ Ys )
= Xs ) ) ).
% lappend_lnull2
thf(fact_69_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_70_not__lnull__conv,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( ( ~ ( coindu335574135_lnull @ A @ Xs ) )
= ( ? [X3: A,Xs4: coindu1593790203_llist @ A] :
( Xs
= ( coindu1121789889_LCons @ A @ X3 @ Xs4 ) ) ) ) ).
% not_lnull_conv
thf(fact_71_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_72_llist_Oleq__trans,axiom,
! [A: $tType,X: coindu1593790203_llist @ A,Y: coindu1593790203_llist @ A,Z: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ X @ Y )
=> ( ( coindu1696667936prefix @ A @ Y @ Z )
=> ( coindu1696667936prefix @ A @ X @ Z ) ) ) ).
% llist.leq_trans
thf(fact_73_lnull__def,axiom,
! [A: $tType] :
( ( coindu335574135_lnull @ A )
= ( ^ [Llist2: coindu1593790203_llist @ A] :
( Llist2
= ( coindu1598213697e_LNil @ A ) ) ) ) ).
% lnull_def
thf(fact_74_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_75_lprefix__lappend,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] : ( coindu1696667936prefix @ A @ Xs @ ( coindu268472904append @ A @ Xs @ Ys ) ) ).
% lprefix_lappend
thf(fact_76_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_77_Le__LCons,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A,X: A] :
( ( coindu1696667936prefix @ A @ Xs @ Ys )
=> ( coindu1696667936prefix @ A @ ( coindu1121789889_LCons @ A @ X @ Xs ) @ ( coindu1121789889_LCons @ A @ X @ Ys ) ) ) ).
% Le_LCons
thf(fact_78_lprefix_Ocoinduct,axiom,
! [A: $tType,X5: ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) > $o,X: coindu1593790203_llist @ A,Xa: coindu1593790203_llist @ A] :
( ( X5 @ X @ Xa )
=> ( ! [X4: coindu1593790203_llist @ A,Xa2: coindu1593790203_llist @ A] :
( ( X5 @ X4 @ Xa2 )
=> ( ? [Xs5: coindu1593790203_llist @ A] :
( ( X4
= ( coindu1598213697e_LNil @ A ) )
& ( Xa2 = Xs5 ) )
| ? [Xs5: coindu1593790203_llist @ A,Ys4: coindu1593790203_llist @ A,Xb: A] :
( ( X4
= ( coindu1121789889_LCons @ A @ Xb @ Xs5 ) )
& ( Xa2
= ( coindu1121789889_LCons @ A @ Xb @ Ys4 ) )
& ( ( X5 @ Xs5 @ Ys4 )
| ( coindu1696667936prefix @ A @ Xs5 @ Ys4 ) ) ) ) )
=> ( coindu1696667936prefix @ A @ X @ Xa ) ) ) ).
% lprefix.coinduct
thf(fact_79_lappend__LNil__LNil,axiom,
! [A: $tType] :
( ( coindu268472904append @ A @ ( coindu1598213697e_LNil @ A ) @ ( coindu1598213697e_LNil @ A ) )
= ( coindu1598213697e_LNil @ A ) ) ).
% lappend_LNil_LNil
thf(fact_80_llist_Oleq__antisym,axiom,
! [A: $tType,X: coindu1593790203_llist @ A,Y: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ X @ Y )
=> ( ( coindu1696667936prefix @ A @ Y @ X )
=> ( X = Y ) ) ) ).
% llist.leq_antisym
thf(fact_81_LCons__lprefix__conv,axiom,
! [A: $tType,X: A,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ ( coindu1121789889_LCons @ A @ X @ Xs ) @ Ys )
= ( ? [Ys5: coindu1593790203_llist @ A] :
( ( Ys
= ( coindu1121789889_LCons @ A @ X @ Ys5 ) )
& ( coindu1696667936prefix @ A @ Xs @ Ys5 ) ) ) ) ).
% LCons_lprefix_conv
thf(fact_82_lmap__eq__LCons__conv,axiom,
! [A: $tType,B: $tType,F: B > A,Xs: coindu1593790203_llist @ B,Y: A,Ys: coindu1593790203_llist @ A] :
( ( ( coindu1062782156e_lmap @ B @ A @ F @ Xs )
= ( coindu1121789889_LCons @ A @ Y @ Ys ) )
= ( ? [X3: B,Xs4: coindu1593790203_llist @ B] :
( ( Xs
= ( coindu1121789889_LCons @ B @ X3 @ Xs4 ) )
& ( Y
= ( F @ X3 ) )
& ( Ys
= ( coindu1062782156e_lmap @ B @ A @ F @ Xs4 ) ) ) ) ) ).
% lmap_eq_LCons_conv
thf(fact_83_lprefix__LCons__conv,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Y: A,Ys: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ Xs @ ( coindu1121789889_LCons @ A @ Y @ Ys ) )
= ( ( Xs
= ( coindu1598213697e_LNil @ A ) )
| ? [Xs4: coindu1593790203_llist @ A] :
( ( Xs
= ( coindu1121789889_LCons @ A @ Y @ Xs4 ) )
& ( coindu1696667936prefix @ A @ Xs4 @ Ys ) ) ) ) ).
% lprefix_LCons_conv
thf(fact_84_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_85_LNil__eq__lappend__iff,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( ( coindu1598213697e_LNil @ A )
= ( coindu268472904append @ A @ Xs @ Ys ) )
= ( ( Xs
= ( coindu1598213697e_LNil @ A ) )
& ( Ys
= ( coindu1598213697e_LNil @ A ) ) ) ) ).
% LNil_eq_lappend_iff
thf(fact_86_lappend__eq__LNil__iff,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( ( coindu268472904append @ A @ Xs @ Ys )
= ( coindu1598213697e_LNil @ A ) )
= ( ( Xs
= ( coindu1598213697e_LNil @ A ) )
& ( Ys
= ( coindu1598213697e_LNil @ A ) ) ) ) ).
% lappend_eq_LNil_iff
thf(fact_87_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_88_lmap__lappend__distrib,axiom,
! [A: $tType,B: $tType,F: B > A,Xs: coindu1593790203_llist @ B,Ys: coindu1593790203_llist @ B] :
( ( coindu1062782156e_lmap @ B @ A @ F @ ( coindu268472904append @ B @ Xs @ Ys ) )
= ( coindu268472904append @ A @ ( coindu1062782156e_lmap @ B @ A @ F @ Xs ) @ ( coindu1062782156e_lmap @ B @ A @ F @ Ys ) ) ) ).
% lmap_lappend_distrib
thf(fact_89_LNil__lprefix,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] : ( coindu1696667936prefix @ A @ ( coindu1598213697e_LNil @ A ) @ Xs ) ).
% LNil_lprefix
thf(fact_90_lprefix__lappend__sameI,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A,Zs: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ Xs @ Ys )
=> ( coindu1696667936prefix @ A @ ( coindu268472904append @ A @ Zs @ Xs ) @ ( coindu268472904append @ A @ Zs @ Ys ) ) ) ).
% lprefix_lappend_sameI
thf(fact_91_lappend__snocL1__conv__LCons2,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Y: A,Ys: coindu1593790203_llist @ A] :
( ( coindu268472904append @ A @ ( coindu268472904append @ A @ Xs @ ( coindu1121789889_LCons @ A @ Y @ ( coindu1598213697e_LNil @ A ) ) ) @ Ys )
= ( coindu268472904append @ A @ Xs @ ( coindu1121789889_LCons @ A @ Y @ Ys ) ) ) ).
% lappend_snocL1_conv_LCons2
thf(fact_92_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_93_lfinite_Oinducts,axiom,
! [A: $tType,X: coindu1593790203_llist @ A,P: ( coindu1593790203_llist @ A ) > $o] :
( ( coindu1213758845finite @ A @ X )
=> ( ( P @ ( coindu1598213697e_LNil @ A ) )
=> ( ! [Xs2: coindu1593790203_llist @ A,X4: A] :
( ( coindu1213758845finite @ A @ Xs2 )
=> ( ( P @ Xs2 )
=> ( P @ ( coindu1121789889_LCons @ A @ X4 @ Xs2 ) ) ) )
=> ( P @ X ) ) ) ) ).
% lfinite.inducts
thf(fact_94_lfinite_Osimps,axiom,
! [A: $tType] :
( ( coindu1213758845finite @ A )
= ( ^ [A4: coindu1593790203_llist @ A] :
( ( A4
= ( coindu1598213697e_LNil @ A ) )
| ? [Xs3: coindu1593790203_llist @ A,X3: A] :
( ( A4
= ( coindu1121789889_LCons @ A @ X3 @ Xs3 ) )
& ( coindu1213758845finite @ A @ Xs3 ) ) ) ) ) ).
% lfinite.simps
thf(fact_95_lfinite_Ocases,axiom,
! [A: $tType,A3: coindu1593790203_llist @ A] :
( ( coindu1213758845finite @ A @ A3 )
=> ( ( A3
!= ( coindu1598213697e_LNil @ A ) )
=> ~ ! [Xs2: coindu1593790203_llist @ A] :
( ? [X4: A] :
( A3
= ( coindu1121789889_LCons @ A @ X4 @ Xs2 ) )
=> ~ ( coindu1213758845finite @ A @ Xs2 ) ) ) ) ).
% lfinite.cases
thf(fact_96_lfinite__LConsI,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,X: A] :
( ( coindu1213758845finite @ A @ Xs )
=> ( coindu1213758845finite @ A @ ( coindu1121789889_LCons @ A @ X @ Xs ) ) ) ).
% lfinite_LConsI
thf(fact_97_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 ) )
=> ( ! [X4: A,Xs2: coindu1593790203_llist @ A] :
( ( coindu1213758845finite @ A @ Xs2 )
=> ( ( P @ Xs2 )
=> ( P @ ( coindu1121789889_LCons @ A @ X4 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% llist_induct
thf(fact_98_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_99_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_100_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_101_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_102_lmember__code_I1_J,axiom,
! [A: $tType,X: A] :
~ ( coindu567634248member @ A @ X @ ( coindu1598213697e_LNil @ A ) ) ).
% lmember_code(1)
thf(fact_103_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_104_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_105_lmember__code_I2_J,axiom,
! [A: $tType,X: A,Y: A,Ys: coindu1593790203_llist @ A] :
( ( coindu567634248member @ A @ X @ ( coindu1121789889_LCons @ A @ Y @ Ys ) )
= ( ( X = Y )
| ( coindu567634248member @ A @ X @ Ys ) ) ) ).
% lmember_code(2)
thf(fact_106_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_107_lprefixI,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A,X5: set @ ( product_prod @ ( coindu1593790203_llist @ A ) @ ( coindu1593790203_llist @ A ) )] :
( ( member @ ( product_prod @ ( coindu1593790203_llist @ A ) @ ( coindu1593790203_llist @ A ) ) @ ( product_Pair @ ( coindu1593790203_llist @ A ) @ ( coindu1593790203_llist @ A ) @ Xs @ Ys ) @ X5 )
=> ( ! [Xs2: coindu1593790203_llist @ A,Ys2: coindu1593790203_llist @ A] :
( ( member @ ( product_prod @ ( coindu1593790203_llist @ A ) @ ( coindu1593790203_llist @ A ) ) @ ( product_Pair @ ( coindu1593790203_llist @ A ) @ ( coindu1593790203_llist @ A ) @ Xs2 @ Ys2 ) @ X5 )
=> ( ( coindu335574135_lnull @ A @ Xs2 )
| ? [X2: A,Xs6: coindu1593790203_llist @ A,Ys6: coindu1593790203_llist @ A] :
( ( Xs2
= ( coindu1121789889_LCons @ A @ X2 @ Xs6 ) )
& ( Ys2
= ( coindu1121789889_LCons @ A @ X2 @ Ys6 ) )
& ( ( member @ ( product_prod @ ( coindu1593790203_llist @ A ) @ ( coindu1593790203_llist @ A ) ) @ ( product_Pair @ ( coindu1593790203_llist @ A ) @ ( coindu1593790203_llist @ A ) @ Xs6 @ Ys6 ) @ X5 )
| ( coindu1696667936prefix @ A @ Xs6 @ Ys6 ) ) ) ) )
=> ( coindu1696667936prefix @ A @ Xs @ Ys ) ) ) ).
% lprefixI
thf(fact_108_llist_Omap__sel_I1_J,axiom,
! [B: $tType,A: $tType,A3: coindu1593790203_llist @ A,F: A > B] :
( ~ ( coindu335574135_lnull @ A @ A3 )
=> ( ( coindu1046438764le_lhd @ B @ ( coindu1062782156e_lmap @ A @ B @ F @ A3 ) )
= ( F @ ( coindu1046438764le_lhd @ A @ A3 ) ) ) ) ).
% llist.map_sel(1)
thf(fact_109_lhd__lappend,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( ( coindu335574135_lnull @ A @ Xs )
=> ( ( coindu1046438764le_lhd @ A @ ( coindu268472904append @ A @ Xs @ Ys ) )
= ( coindu1046438764le_lhd @ A @ Ys ) ) )
& ( ~ ( coindu335574135_lnull @ A @ Xs )
=> ( ( coindu1046438764le_lhd @ A @ ( coindu268472904append @ A @ Xs @ Ys ) )
= ( coindu1046438764le_lhd @ A @ Xs ) ) ) ) ).
% lhd_lappend
thf(fact_110_lhd__LCons,axiom,
! [A: $tType,X21: A,X22: coindu1593790203_llist @ A] :
( ( coindu1046438764le_lhd @ A @ ( coindu1121789889_LCons @ A @ X21 @ X22 ) )
= X21 ) ).
% lhd_LCons
thf(fact_111_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_112_llist_Ocoinduct__strong,axiom,
! [A: $tType,R: ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) > $o,Llist: coindu1593790203_llist @ A,Llist3: coindu1593790203_llist @ A] :
( ( R @ Llist @ Llist3 )
=> ( ! [Llist4: coindu1593790203_llist @ A,Llist5: coindu1593790203_llist @ A] :
( ( R @ Llist4 @ Llist5 )
=> ( ( ( coindu335574135_lnull @ A @ Llist4 )
= ( coindu335574135_lnull @ A @ Llist5 ) )
& ( ~ ( coindu335574135_lnull @ A @ Llist4 )
=> ( ~ ( coindu335574135_lnull @ A @ Llist5 )
=> ( ( ( coindu1046438764le_lhd @ A @ Llist4 )
= ( coindu1046438764le_lhd @ A @ Llist5 ) )
& ( ( R @ ( coindu1047225960le_ltl @ A @ Llist4 ) @ ( coindu1047225960le_ltl @ A @ Llist5 ) )
| ( ( coindu1047225960le_ltl @ A @ Llist4 )
= ( coindu1047225960le_ltl @ A @ Llist5 ) ) ) ) ) ) ) )
=> ( Llist = Llist3 ) ) ) ).
% llist.coinduct_strong
thf(fact_113_llist_Ocoinduct,axiom,
! [A: $tType,R: ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) > $o,Llist: coindu1593790203_llist @ A,Llist3: coindu1593790203_llist @ A] :
( ( R @ Llist @ Llist3 )
=> ( ! [Llist4: coindu1593790203_llist @ A,Llist5: coindu1593790203_llist @ A] :
( ( R @ Llist4 @ Llist5 )
=> ( ( ( coindu335574135_lnull @ A @ Llist4 )
= ( coindu335574135_lnull @ A @ Llist5 ) )
& ( ~ ( coindu335574135_lnull @ A @ Llist4 )
=> ( ~ ( coindu335574135_lnull @ A @ Llist5 )
=> ( ( ( coindu1046438764le_lhd @ A @ Llist4 )
= ( coindu1046438764le_lhd @ A @ Llist5 ) )
& ( R @ ( coindu1047225960le_ltl @ A @ Llist4 ) @ ( coindu1047225960le_ltl @ A @ Llist5 ) ) ) ) ) ) )
=> ( Llist = Llist3 ) ) ) ).
% llist.coinduct
thf(fact_114_llist_Oexpand,axiom,
! [A: $tType,Llist: coindu1593790203_llist @ A,Llist3: coindu1593790203_llist @ A] :
( ( ( coindu335574135_lnull @ A @ Llist )
= ( coindu335574135_lnull @ A @ Llist3 ) )
=> ( ( ~ ( coindu335574135_lnull @ A @ Llist )
=> ( ~ ( coindu335574135_lnull @ A @ Llist3 )
=> ( ( ( coindu1046438764le_lhd @ A @ Llist )
= ( coindu1046438764le_lhd @ A @ Llist3 ) )
& ( ( coindu1047225960le_ltl @ A @ Llist )
= ( coindu1047225960le_ltl @ A @ Llist3 ) ) ) ) )
=> ( Llist = Llist3 ) ) ) ).
% llist.expand
thf(fact_115_old_Oprod_Oinject,axiom,
! [A: $tType,B: $tType,A3: A,B2: B,A5: A,B3: B] :
( ( ( product_Pair @ A @ B @ A3 @ B2 )
= ( product_Pair @ A @ B @ A5 @ B3 ) )
= ( ( A3 = A5 )
& ( B2 = B3 ) ) ) ).
% old.prod.inject
thf(fact_116_prod_Oinject,axiom,
! [A: $tType,B: $tType,X1: A,X23: B,Y1: A,Y2: B] :
( ( ( product_Pair @ A @ B @ X1 @ X23 )
= ( product_Pair @ A @ B @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X23 = Y2 ) ) ) ).
% prod.inject
thf(fact_117_llist_Osplit__sel__asm,axiom,
! [B: $tType,A: $tType,P: B > $o,F1: B,F2: A > ( coindu1593790203_llist @ A ) > B,Llist: coindu1593790203_llist @ A] :
( ( P @ ( coindu882539134_llist @ B @ A @ F1 @ F2 @ Llist ) )
= ( ~ ( ( ( Llist
= ( coindu1598213697e_LNil @ A ) )
& ~ ( P @ F1 ) )
| ( ( Llist
= ( coindu1121789889_LCons @ A @ ( coindu1046438764le_lhd @ A @ Llist ) @ ( coindu1047225960le_ltl @ A @ Llist ) ) )
& ~ ( P @ ( F2 @ ( coindu1046438764le_lhd @ A @ Llist ) @ ( coindu1047225960le_ltl @ A @ Llist ) ) ) ) ) ) ) ).
% llist.split_sel_asm
thf(fact_118_llist_Osplit__sel,axiom,
! [B: $tType,A: $tType,P: B > $o,F1: B,F2: A > ( coindu1593790203_llist @ A ) > B,Llist: coindu1593790203_llist @ A] :
( ( P @ ( coindu882539134_llist @ B @ A @ F1 @ F2 @ Llist ) )
= ( ( ( Llist
= ( coindu1598213697e_LNil @ A ) )
=> ( P @ F1 ) )
& ( ( Llist
= ( coindu1121789889_LCons @ A @ ( coindu1046438764le_lhd @ A @ Llist ) @ ( coindu1047225960le_ltl @ A @ Llist ) ) )
=> ( P @ ( F2 @ ( coindu1046438764le_lhd @ A @ Llist ) @ ( coindu1047225960le_ltl @ A @ Llist ) ) ) ) ) ) ).
% llist.split_sel
thf(fact_119_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_120_unfold__llist_Odisc__iff_I2_J,axiom,
! [B: $tType,A: $tType,P2: A > $o,G21: A > B,G22: A > A,A3: A] :
( ( ~ ( coindu335574135_lnull @ B @ ( coindu1599971794_llist @ A @ B @ P2 @ G21 @ G22 @ A3 ) ) )
= ( ~ ( P2 @ A3 ) ) ) ).
% unfold_llist.disc_iff(2)
thf(fact_121_unfold__llist_Odisc__iff_I1_J,axiom,
! [B: $tType,A: $tType,P2: A > $o,G21: A > B,G22: A > A,A3: A] :
( ( coindu335574135_lnull @ B @ ( coindu1599971794_llist @ A @ B @ P2 @ G21 @ G22 @ A3 ) )
= ( P2 @ A3 ) ) ).
% unfold_llist.disc_iff(1)
thf(fact_122_unfold__llist__eq__LCons,axiom,
! [A: $tType,B: $tType,IS_LNIL: B > $o,LHD: B > A,LTL: B > B,B2: B,X: A,Xs: coindu1593790203_llist @ A] :
( ( ( coindu1599971794_llist @ B @ A @ IS_LNIL @ LHD @ LTL @ B2 )
= ( coindu1121789889_LCons @ A @ X @ Xs ) )
= ( ~ ( IS_LNIL @ B2 )
& ( X
= ( LHD @ B2 ) )
& ( Xs
= ( coindu1599971794_llist @ B @ A @ IS_LNIL @ LHD @ LTL @ ( LTL @ B2 ) ) ) ) ) ).
% unfold_llist_eq_LCons
thf(fact_123_unfold__llist_Odisc_I2_J,axiom,
! [B: $tType,A: $tType,P2: A > $o,A3: A,G21: A > B,G22: A > A] :
( ~ ( P2 @ A3 )
=> ~ ( coindu335574135_lnull @ B @ ( coindu1599971794_llist @ A @ B @ P2 @ G21 @ G22 @ A3 ) ) ) ).
% unfold_llist.disc(2)
thf(fact_124_unfold__llist_Odisc_I1_J,axiom,
! [B: $tType,A: $tType,P2: A > $o,A3: A,G21: A > B,G22: A > A] :
( ( P2 @ A3 )
=> ( coindu335574135_lnull @ B @ ( coindu1599971794_llist @ A @ B @ P2 @ G21 @ G22 @ A3 ) ) ) ).
% unfold_llist.disc(1)
thf(fact_125_unfold__llist_Octr_I2_J,axiom,
! [B: $tType,A: $tType,P2: A > $o,A3: A,G21: A > B,G22: A > A] :
( ~ ( P2 @ A3 )
=> ( ( coindu1599971794_llist @ A @ B @ P2 @ G21 @ G22 @ A3 )
= ( coindu1121789889_LCons @ B @ ( G21 @ A3 ) @ ( coindu1599971794_llist @ A @ B @ P2 @ G21 @ G22 @ ( G22 @ A3 ) ) ) ) ) ).
% unfold_llist.ctr(2)
thf(fact_126_unfold__llist_Octr_I1_J,axiom,
! [A: $tType,B: $tType,P2: A > $o,A3: A,G21: A > B,G22: A > A] :
( ( P2 @ A3 )
=> ( ( coindu1599971794_llist @ A @ B @ P2 @ G21 @ G22 @ A3 )
= ( coindu1598213697e_LNil @ B ) ) ) ).
% unfold_llist.ctr(1)
thf(fact_127_unfold__llist_Osimps_I4_J,axiom,
! [B: $tType,A: $tType,P2: A > $o,A3: A,G21: A > B,G22: A > A] :
( ~ ( P2 @ A3 )
=> ( ( coindu1047225960le_ltl @ B @ ( coindu1599971794_llist @ A @ B @ P2 @ G21 @ G22 @ A3 ) )
= ( coindu1599971794_llist @ A @ B @ P2 @ G21 @ G22 @ ( G22 @ A3 ) ) ) ) ).
% unfold_llist.simps(4)
thf(fact_128_unfold__llist_Osimps_I3_J,axiom,
! [B: $tType,A: $tType,P2: A > $o,A3: A,G21: A > B,G22: A > A] :
( ~ ( P2 @ A3 )
=> ( ( coindu1046438764le_lhd @ B @ ( coindu1599971794_llist @ A @ B @ P2 @ G21 @ G22 @ A3 ) )
= ( G21 @ A3 ) ) ) ).
% unfold_llist.simps(3)
thf(fact_129_llist_Osimps_I5_J,axiom,
! [B: $tType,A: $tType,F1: B,F2: A > ( coindu1593790203_llist @ A ) > B,X21: A,X22: coindu1593790203_llist @ A] :
( ( coindu882539134_llist @ B @ A @ F1 @ F2 @ ( coindu1121789889_LCons @ A @ X21 @ X22 ) )
= ( F2 @ X21 @ X22 ) ) ).
% llist.simps(5)
thf(fact_130_llist_Osimps_I4_J,axiom,
! [A: $tType,B: $tType,F1: B,F2: A > ( coindu1593790203_llist @ A ) > B] :
( ( coindu882539134_llist @ B @ A @ F1 @ F2 @ ( coindu1598213697e_LNil @ A ) )
= F1 ) ).
% llist.simps(4)
thf(fact_131_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_132_ltl__unfold__llist,axiom,
! [A: $tType,B: $tType,IS_LNIL: B > $o,A3: B,LHD: B > A,LTL: B > B] :
( ( ( IS_LNIL @ A3 )
=> ( ( coindu1047225960le_ltl @ A @ ( coindu1599971794_llist @ B @ A @ IS_LNIL @ LHD @ LTL @ A3 ) )
= ( coindu1598213697e_LNil @ A ) ) )
& ( ~ ( IS_LNIL @ A3 )
=> ( ( coindu1047225960le_ltl @ A @ ( coindu1599971794_llist @ B @ A @ IS_LNIL @ LHD @ LTL @ A3 ) )
= ( coindu1599971794_llist @ B @ A @ IS_LNIL @ LHD @ LTL @ ( LTL @ A3 ) ) ) ) ) ).
% ltl_unfold_llist
thf(fact_133_surj__pair,axiom,
! [A: $tType,B: $tType,P2: product_prod @ A @ B] :
? [X4: A,Y3: B] :
( P2
= ( product_Pair @ A @ B @ X4 @ Y3 ) ) ).
% surj_pair
thf(fact_134_prod__cases,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P2: product_prod @ A @ B] :
( ! [A6: A,B4: B] : ( P @ ( product_Pair @ A @ B @ A6 @ B4 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_135_Pair__inject,axiom,
! [A: $tType,B: $tType,A3: A,B2: B,A5: A,B3: B] :
( ( ( product_Pair @ A @ B @ A3 @ B2 )
= ( product_Pair @ A @ B @ A5 @ B3 ) )
=> ~ ( ( A3 = A5 )
=> ( B2 != B3 ) ) ) ).
% Pair_inject
thf(fact_136_prod__cases3,axiom,
! [A: $tType,B: $tType,C: $tType,Y: product_prod @ A @ ( product_prod @ B @ C )] :
~ ! [A6: A,B4: B,C2: C] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ C ) @ A6 @ ( product_Pair @ B @ C @ B4 @ C2 ) ) ) ).
% prod_cases3
thf(fact_137_prod__cases4,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
~ ! [A6: A,B4: B,C2: C,D2: D] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B4 @ ( product_Pair @ C @ D @ C2 @ D2 ) ) ) ) ).
% prod_cases4
thf(fact_138_prod__cases5,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
~ ! [A6: A,B4: B,C2: C,D2: D,E2: E] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C2 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) ) ).
% prod_cases5
thf(fact_139_prod__cases6,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F3: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) )] :
~ ! [A6: A,B4: B,C2: C,D2: D,E2: E,F4: F3] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ F3 ) @ D2 @ ( product_Pair @ E @ F3 @ E2 @ F4 ) ) ) ) ) ) ).
% prod_cases6
thf(fact_140_prod__cases7,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F3: $tType,G2: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G2 ) ) ) ) )] :
~ ! [A6: A,B4: B,C2: C,D2: D,E2: E,F4: F3,G3: G2] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G2 ) ) ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G2 ) ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G2 ) ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G2 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F3 @ G2 ) @ E2 @ ( product_Pair @ F3 @ G2 @ F4 @ G3 ) ) ) ) ) ) ) ).
% prod_cases7
thf(fact_141_prod__induct3,axiom,
! [C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ C )] :
( ! [A6: A,B4: B,C2: C] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A6 @ ( product_Pair @ B @ C @ B4 @ C2 ) ) )
=> ( P @ X ) ) ).
% prod_induct3
thf(fact_142_prod__induct4,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
( ! [A6: A,B4: B,C2: C,D2: D] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B4 @ ( product_Pair @ C @ D @ C2 @ D2 ) ) ) )
=> ( P @ X ) ) ).
% prod_induct4
thf(fact_143_prod__induct5,axiom,
! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
( ! [A6: A,B4: B,C2: C,D2: D,E2: E] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C2 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) )
=> ( P @ X ) ) ).
% prod_induct5
thf(fact_144_prod__induct6,axiom,
! [F3: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) )] :
( ! [A6: A,B4: B,C2: C,D2: D,E2: E,F4: F3] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ F3 ) @ D2 @ ( product_Pair @ E @ F3 @ E2 @ F4 ) ) ) ) ) )
=> ( P @ X ) ) ).
% prod_induct6
thf(fact_145_prod__induct7,axiom,
! [G2: $tType,F3: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G2 ) ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G2 ) ) ) ) )] :
( ! [A6: A,B4: B,C2: C,D2: D,E2: E,F4: F3,G3: G2] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G2 ) ) ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G2 ) ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G2 ) ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G2 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F3 @ G2 ) @ E2 @ ( product_Pair @ F3 @ G2 @ F4 @ G3 ) ) ) ) ) ) )
=> ( P @ X ) ) ).
% prod_induct7
thf(fact_146_old_Oprod_Oexhaust,axiom,
! [A: $tType,B: $tType,Y: product_prod @ A @ B] :
~ ! [A6: A,B4: B] :
( Y
!= ( product_Pair @ A @ B @ A6 @ B4 ) ) ).
% old.prod.exhaust
thf(fact_147_old_Oprod_Oinducts,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
( ! [A6: A,B4: B] : ( P @ ( product_Pair @ A @ B @ A6 @ B4 ) )
=> ( P @ Prod ) ) ).
% old.prod.inducts
thf(fact_148_llist_Ocase__eq__if,axiom,
! [A: $tType,B: $tType] :
( ( coindu882539134_llist @ B @ A )
= ( ^ [F12: B,F22: A > ( coindu1593790203_llist @ A ) > B,Llist2: coindu1593790203_llist @ A] : ( if @ B @ ( coindu335574135_lnull @ A @ Llist2 ) @ F12 @ ( F22 @ ( coindu1046438764le_lhd @ A @ Llist2 ) @ ( coindu1047225960le_ltl @ A @ Llist2 ) ) ) ) ) ).
% llist.case_eq_if
thf(fact_149_old_Oprod_Orec,axiom,
! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A3: A,B2: B] :
( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A3 @ B2 ) )
= ( F1 @ A3 @ B2 ) ) ).
% old.prod.rec
thf(fact_150_internal__case__prod__conv,axiom,
! [B: $tType,A: $tType,C: $tType,C3: B > C > A,A3: B,B2: C] :
( ( produc2004651681e_prod @ B @ C @ A @ C3 @ ( product_Pair @ B @ C @ A3 @ B2 ) )
= ( C3 @ A3 @ B2 ) ) ).
% internal_case_prod_conv
thf(fact_151_lhd__lSup__eq,axiom,
! [A: $tType,Y4: set @ ( coindu1593790203_llist @ A ),Xs: coindu1593790203_llist @ A] :
( ( comple1193779247_chain @ ( coindu1593790203_llist @ A ) @ ( coindu1696667936prefix @ A ) @ Y4 )
=> ( ( member @ ( coindu1593790203_llist @ A ) @ Xs @ Y4 )
=> ( ~ ( coindu335574135_lnull @ A @ Xs )
=> ( ( coindu1046438764le_lhd @ A @ ( coindu526335360e_lSup @ A @ Y4 ) )
= ( coindu1046438764le_lhd @ A @ Xs ) ) ) ) ) ).
% lhd_lSup_eq
thf(fact_152_split__llist,axiom,
! [A: $tType,X: A,Xs: coindu1593790203_llist @ A] :
( ( member @ A @ X @ ( coindu1112613586e_lset @ A @ Xs ) )
=> ? [Ys2: coindu1593790203_llist @ A,Zs3: coindu1593790203_llist @ A] :
( ( Xs
= ( coindu268472904append @ A @ Ys2 @ ( coindu1121789889_LCons @ A @ X @ Zs3 ) ) )
& ( coindu1213758845finite @ A @ Ys2 ) ) ) ).
% split_llist
thf(fact_153_llist_Omap__cong,axiom,
! [B: $tType,A: $tType,X: coindu1593790203_llist @ A,Ya: coindu1593790203_llist @ A,F: A > B,G: A > B] :
( ( X = Ya )
=> ( ! [Z2: A] :
( ( member @ A @ Z2 @ ( coindu1112613586e_lset @ A @ Ya ) )
=> ( ( F @ Z2 )
= ( G @ Z2 ) ) )
=> ( ( coindu1062782156e_lmap @ A @ B @ F @ X )
= ( coindu1062782156e_lmap @ A @ B @ G @ Ya ) ) ) ) ).
% llist.map_cong
thf(fact_154_llist_Omap__cong0,axiom,
! [B: $tType,A: $tType,X: coindu1593790203_llist @ A,F: A > B,G: A > B] :
( ! [Z2: A] :
( ( member @ A @ Z2 @ ( coindu1112613586e_lset @ A @ X ) )
=> ( ( F @ Z2 )
= ( G @ Z2 ) ) )
=> ( ( coindu1062782156e_lmap @ A @ B @ F @ X )
= ( coindu1062782156e_lmap @ A @ B @ G @ X ) ) ) ).
% llist.map_cong0
thf(fact_155_llist_Oinj__map__strong,axiom,
! [B: $tType,A: $tType,X: coindu1593790203_llist @ A,Xa: coindu1593790203_llist @ A,F: A > B,Fa: A > B] :
( ! [Z2: A,Za: A] :
( ( member @ A @ Z2 @ ( coindu1112613586e_lset @ A @ X ) )
=> ( ( member @ A @ Za @ ( coindu1112613586e_lset @ A @ Xa ) )
=> ( ( ( F @ Z2 )
= ( Fa @ Za ) )
=> ( Z2 = Za ) ) ) )
=> ( ( ( coindu1062782156e_lmap @ A @ B @ F @ X )
= ( coindu1062782156e_lmap @ A @ B @ Fa @ Xa ) )
=> ( X = Xa ) ) ) ).
% llist.inj_map_strong
thf(fact_156_in__lset__ltlD,axiom,
! [A: $tType,X: A,Xs: coindu1593790203_llist @ A] :
( ( member @ A @ X @ ( coindu1112613586e_lset @ A @ ( coindu1047225960le_ltl @ A @ Xs ) ) )
=> ( member @ A @ X @ ( coindu1112613586e_lset @ A @ Xs ) ) ) ).
% in_lset_ltlD
thf(fact_157_lset__intros_I2_J,axiom,
! [A: $tType,X: A,Xs: coindu1593790203_llist @ A,X6: A] :
( ( member @ A @ X @ ( coindu1112613586e_lset @ A @ Xs ) )
=> ( member @ A @ X @ ( coindu1112613586e_lset @ A @ ( coindu1121789889_LCons @ A @ X6 @ Xs ) ) ) ) ).
% lset_intros(2)
thf(fact_158_lset__intros_I1_J,axiom,
! [A: $tType,X: A,Xs: coindu1593790203_llist @ A] : ( member @ A @ X @ ( coindu1112613586e_lset @ A @ ( coindu1121789889_LCons @ A @ X @ Xs ) ) ) ).
% lset_intros(1)
thf(fact_159_llist_Oset__intros_I2_J,axiom,
! [A: $tType,X: A,A22: coindu1593790203_llist @ A,A1: A] :
( ( member @ A @ X @ ( coindu1112613586e_lset @ A @ A22 ) )
=> ( member @ A @ X @ ( coindu1112613586e_lset @ A @ ( coindu1121789889_LCons @ A @ A1 @ A22 ) ) ) ) ).
% llist.set_intros(2)
thf(fact_160_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_161_lset__cases,axiom,
! [A: $tType,X: A,Xs: coindu1593790203_llist @ A] :
( ( member @ A @ X @ ( coindu1112613586e_lset @ A @ Xs ) )
=> ( ! [Xs7: coindu1593790203_llist @ A] :
( Xs
!= ( coindu1121789889_LCons @ A @ X @ Xs7 ) )
=> ~ ! [X7: A,Xs7: coindu1593790203_llist @ A] :
( ( Xs
= ( coindu1121789889_LCons @ A @ X7 @ Xs7 ) )
=> ~ ( member @ A @ X @ ( coindu1112613586e_lset @ A @ Xs7 ) ) ) ) ) ).
% lset_cases
thf(fact_162_lset__induct,axiom,
! [A: $tType,X: A,Xs: coindu1593790203_llist @ A,P: ( coindu1593790203_llist @ A ) > $o] :
( ( member @ A @ X @ ( coindu1112613586e_lset @ A @ Xs ) )
=> ( ! [Xs2: coindu1593790203_llist @ A] : ( P @ ( coindu1121789889_LCons @ A @ X @ Xs2 ) )
=> ( ! [X7: A,Xs2: coindu1593790203_llist @ A] :
( ( member @ A @ X @ ( coindu1112613586e_lset @ A @ Xs2 ) )
=> ( ( X != X7 )
=> ( ( P @ Xs2 )
=> ( P @ ( coindu1121789889_LCons @ A @ X7 @ Xs2 ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% lset_induct
thf(fact_163_lset__induct_H,axiom,
! [A: $tType,X: A,Xs: coindu1593790203_llist @ A,P: ( coindu1593790203_llist @ A ) > $o] :
( ( member @ A @ X @ ( coindu1112613586e_lset @ A @ Xs ) )
=> ( ! [Xs2: coindu1593790203_llist @ A] : ( P @ ( coindu1121789889_LCons @ A @ X @ Xs2 ) )
=> ( ! [X7: A,Xs2: coindu1593790203_llist @ A] :
( ( member @ A @ X @ ( coindu1112613586e_lset @ A @ Xs2 ) )
=> ( ( P @ Xs2 )
=> ( P @ ( coindu1121789889_LCons @ A @ X7 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% lset_induct'
thf(fact_164_llist_Oset__cases,axiom,
! [A: $tType,E3: A,A3: coindu1593790203_llist @ A] :
( ( member @ A @ E3 @ ( coindu1112613586e_lset @ A @ A3 ) )
=> ( ! [Z22: coindu1593790203_llist @ A] :
( A3
!= ( coindu1121789889_LCons @ A @ E3 @ Z22 ) )
=> ~ ! [Z1: A,Z22: coindu1593790203_llist @ A] :
( ( A3
= ( coindu1121789889_LCons @ A @ Z1 @ Z22 ) )
=> ~ ( member @ A @ E3 @ ( coindu1112613586e_lset @ A @ Z22 ) ) ) ) ) ).
% llist.set_cases
thf(fact_165_llist_Oset__induct,axiom,
! [A: $tType,X: A,A3: coindu1593790203_llist @ A,P: A > ( coindu1593790203_llist @ A ) > $o] :
( ( member @ A @ X @ ( coindu1112613586e_lset @ A @ A3 ) )
=> ( ! [Z1: A,Z22: coindu1593790203_llist @ A] : ( P @ Z1 @ ( coindu1121789889_LCons @ A @ Z1 @ Z22 ) )
=> ( ! [Z1: A,Z22: coindu1593790203_llist @ A,Xa2: A] :
( ( member @ A @ Xa2 @ ( coindu1112613586e_lset @ A @ Z22 ) )
=> ( ( P @ Xa2 @ Z22 )
=> ( P @ Xa2 @ ( coindu1121789889_LCons @ A @ Z1 @ Z22 ) ) ) )
=> ( P @ X @ A3 ) ) ) ) ).
% llist.set_induct
thf(fact_166_lset__lmember,axiom,
! [A: $tType,X: A,Xs: coindu1593790203_llist @ A] :
( ( member @ A @ X @ ( coindu1112613586e_lset @ A @ Xs ) )
= ( coindu567634248member @ A @ X @ Xs ) ) ).
% lset_lmember
thf(fact_167_chain__lprefix__lSup,axiom,
! [A: $tType,A2: set @ ( coindu1593790203_llist @ A ),Xs: coindu1593790203_llist @ A] :
( ( comple1193779247_chain @ ( coindu1593790203_llist @ A ) @ ( coindu1696667936prefix @ A ) @ A2 )
=> ( ( member @ ( coindu1593790203_llist @ A ) @ Xs @ A2 )
=> ( coindu1696667936prefix @ A @ Xs @ ( coindu526335360e_lSup @ A @ A2 ) ) ) ) ).
% chain_lprefix_lSup
thf(fact_168_chain__lSup__lprefix,axiom,
! [A: $tType,A2: set @ ( coindu1593790203_llist @ A ),Zs: coindu1593790203_llist @ A] :
( ( comple1193779247_chain @ ( coindu1593790203_llist @ A ) @ ( coindu1696667936prefix @ A ) @ A2 )
=> ( ! [Xs2: coindu1593790203_llist @ A] :
( ( member @ ( coindu1593790203_llist @ A ) @ Xs2 @ A2 )
=> ( coindu1696667936prefix @ A @ Xs2 @ Zs ) )
=> ( coindu1696667936prefix @ A @ ( coindu526335360e_lSup @ A @ A2 ) @ Zs ) ) ) ).
% chain_lSup_lprefix
thf(fact_169_llist_Olub__upper,axiom,
! [A: $tType,A2: set @ ( coindu1593790203_llist @ A ),X: coindu1593790203_llist @ A] :
( ( comple1193779247_chain @ ( coindu1593790203_llist @ A ) @ ( coindu1696667936prefix @ A ) @ A2 )
=> ( ( member @ ( coindu1593790203_llist @ A ) @ X @ A2 )
=> ( coindu1696667936prefix @ A @ X @ ( coindu526335360e_lSup @ A @ A2 ) ) ) ) ).
% llist.lub_upper
thf(fact_170_llist_Olub__least,axiom,
! [A: $tType,A2: set @ ( coindu1593790203_llist @ A ),Z: coindu1593790203_llist @ A] :
( ( comple1193779247_chain @ ( coindu1593790203_llist @ A ) @ ( coindu1696667936prefix @ A ) @ A2 )
=> ( ! [X4: coindu1593790203_llist @ A] :
( ( member @ ( coindu1593790203_llist @ A ) @ X4 @ A2 )
=> ( coindu1696667936prefix @ A @ X4 @ Z ) )
=> ( coindu1696667936prefix @ A @ ( coindu526335360e_lSup @ A @ A2 ) @ Z ) ) ) ).
% llist.lub_least
thf(fact_171_llist_Oset__sel_I2_J,axiom,
! [A: $tType,A3: coindu1593790203_llist @ A,X: A] :
( ~ ( coindu335574135_lnull @ A @ A3 )
=> ( ( member @ A @ X @ ( coindu1112613586e_lset @ A @ ( coindu1047225960le_ltl @ A @ A3 ) ) )
=> ( member @ A @ X @ ( coindu1112613586e_lset @ A @ A3 ) ) ) ) ).
% llist.set_sel(2)
thf(fact_172_llist_Oset__sel_I1_J,axiom,
! [A: $tType,A3: coindu1593790203_llist @ A] :
( ~ ( coindu335574135_lnull @ A @ A3 )
=> ( member @ A @ ( coindu1046438764le_lhd @ A @ A3 ) @ ( coindu1112613586e_lset @ A @ A3 ) ) ) ).
% llist.set_sel(1)
thf(fact_173_in__lset__lappend__iff,axiom,
! [A: $tType,X: A,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( member @ A @ X @ ( coindu1112613586e_lset @ A @ ( coindu268472904append @ A @ Xs @ Ys ) ) )
= ( ( member @ A @ X @ ( coindu1112613586e_lset @ A @ Xs ) )
| ( ( coindu1213758845finite @ A @ Xs )
& ( member @ A @ X @ ( coindu1112613586e_lset @ A @ Ys ) ) ) ) ) ).
% in_lset_lappend_iff
thf(fact_174_llist__set__induct,axiom,
! [A: $tType,X: A,Xs: coindu1593790203_llist @ A,P: A > ( coindu1593790203_llist @ A ) > $o] :
( ( member @ A @ X @ ( coindu1112613586e_lset @ A @ Xs ) )
=> ( ! [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 @ X @ Xs ) ) ) ) ).
% llist_set_induct
thf(fact_175_split__llist__first,axiom,
! [A: $tType,X: A,Xs: coindu1593790203_llist @ A] :
( ( member @ A @ X @ ( coindu1112613586e_lset @ A @ Xs ) )
=> ? [Ys2: coindu1593790203_llist @ A,Zs3: coindu1593790203_llist @ A] :
( ( Xs
= ( coindu268472904append @ A @ Ys2 @ ( coindu1121789889_LCons @ A @ X @ Zs3 ) ) )
& ( coindu1213758845finite @ A @ Ys2 )
& ~ ( member @ A @ X @ ( coindu1112613586e_lset @ A @ Ys2 ) ) ) ) ).
% split_llist_first
thf(fact_176_lset__lappend__lfinite,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu1213758845finite @ A @ Xs )
=> ( ( coindu1112613586e_lset @ A @ ( coindu268472904append @ A @ Xs @ Ys ) )
= ( sup_sup @ ( set @ A ) @ ( coindu1112613586e_lset @ A @ Xs ) @ ( coindu1112613586e_lset @ A @ Ys ) ) ) ) ).
% lset_lappend_lfinite
thf(fact_177_chain__equality,axiom,
! [A: $tType,A2: set @ A] :
( ( comple1193779247_chain @ A
@ ^ [Y5: A,Z3: A] : Y5 = Z3
@ A2 )
= ( ! [X3: A] :
( ( member @ A @ X3 @ A2 )
=> ! [Y6: A] :
( ( member @ A @ Y6 @ A2 )
=> ( X3 = Y6 ) ) ) ) ) ).
% chain_equality
thf(fact_178_chain__def,axiom,
! [A: $tType] :
( ( comple1193779247_chain @ A )
= ( ^ [Ord: A > A > $o,S: set @ A] :
! [X3: A] :
( ( member @ A @ X3 @ S )
=> ! [Y6: A] :
( ( member @ A @ Y6 @ S )
=> ( ( Ord @ X3 @ Y6 )
| ( Ord @ Y6 @ X3 ) ) ) ) ) ) ).
% chain_def
thf(fact_179_lset__lappend__conv,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( ( coindu1213758845finite @ A @ Xs )
=> ( ( coindu1112613586e_lset @ A @ ( coindu268472904append @ A @ Xs @ Ys ) )
= ( sup_sup @ ( set @ A ) @ ( coindu1112613586e_lset @ A @ Xs ) @ ( coindu1112613586e_lset @ A @ Ys ) ) ) )
& ( ~ ( coindu1213758845finite @ A @ Xs )
=> ( ( coindu1112613586e_lset @ A @ ( coindu268472904append @ A @ Xs @ Ys ) )
= ( coindu1112613586e_lset @ A @ Xs ) ) ) ) ).
% lset_lappend_conv
thf(fact_180_chainD,axiom,
! [A: $tType,Ord2: A > A > $o,S2: set @ A,X: A,Y: A] :
( ( comple1193779247_chain @ A @ Ord2 @ S2 )
=> ( ( member @ A @ X @ S2 )
=> ( ( member @ A @ Y @ S2 )
=> ( ( Ord2 @ X @ Y )
| ( Ord2 @ Y @ X ) ) ) ) ) ).
% chainD
thf(fact_181_chainE,axiom,
! [A: $tType,Ord2: A > A > $o,S2: set @ A,X: A,Y: A] :
( ( comple1193779247_chain @ A @ Ord2 @ S2 )
=> ( ( member @ A @ X @ S2 )
=> ( ( member @ A @ Y @ S2 )
=> ( ~ ( Ord2 @ X @ Y )
=> ( Ord2 @ Y @ X ) ) ) ) ) ).
% chainE
thf(fact_182_chainI,axiom,
! [A: $tType,S2: set @ A,Ord2: A > A > $o] :
( ! [X4: A,Y3: A] :
( ( member @ A @ X4 @ S2 )
=> ( ( member @ A @ Y3 @ S2 )
=> ( ( Ord2 @ X4 @ Y3 )
| ( Ord2 @ Y3 @ X4 ) ) ) )
=> ( comple1193779247_chain @ A @ Ord2 @ S2 ) ) ).
% chainI
thf(fact_183_UnCI,axiom,
! [A: $tType,C3: A,B5: set @ A,A2: set @ A] :
( ( ~ ( member @ A @ C3 @ B5 )
=> ( member @ A @ C3 @ A2 ) )
=> ( member @ A @ C3 @ ( sup_sup @ ( set @ A ) @ A2 @ B5 ) ) ) ).
% UnCI
thf(fact_184_Un__iff,axiom,
! [A: $tType,C3: A,A2: set @ A,B5: set @ A] :
( ( member @ A @ C3 @ ( sup_sup @ ( set @ A ) @ A2 @ B5 ) )
= ( ( member @ A @ C3 @ A2 )
| ( member @ A @ C3 @ B5 ) ) ) ).
% Un_iff
thf(fact_185_sup__apply,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_sup @ B @ ( type2 @ B ) )
=> ( ( sup_sup @ ( A > B ) )
= ( ^ [F5: A > B,G4: A > B,X3: A] : ( sup_sup @ B @ ( F5 @ X3 ) @ ( G4 @ X3 ) ) ) ) ) ).
% sup_apply
thf(fact_186_sup_Oright__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [A3: A,B2: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ A3 @ B2 ) @ B2 )
= ( sup_sup @ A @ A3 @ B2 ) ) ) ).
% sup.right_idem
thf(fact_187_sup__left__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( sup_sup @ A @ X @ ( sup_sup @ A @ X @ Y ) )
= ( sup_sup @ A @ X @ Y ) ) ) ).
% sup_left_idem
thf(fact_188_sup_Oleft__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [A3: A,B2: A] :
( ( sup_sup @ A @ A3 @ ( sup_sup @ A @ A3 @ B2 ) )
= ( sup_sup @ A @ A3 @ B2 ) ) ) ).
% sup.left_idem
thf(fact_189_sup__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [X: A] :
( ( sup_sup @ A @ X @ X )
= X ) ) ).
% sup_idem
thf(fact_190_sup_Oidem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [A3: A] :
( ( sup_sup @ A @ A3 @ A3 )
= A3 ) ) ).
% sup.idem
thf(fact_191_sup__left__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z: A] :
( ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z ) )
= ( sup_sup @ A @ Y @ ( sup_sup @ A @ X @ Z ) ) ) ) ).
% sup_left_commute
thf(fact_192_sup_Oleft__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [B2: A,A3: A,C3: A] :
( ( sup_sup @ A @ B2 @ ( sup_sup @ A @ A3 @ C3 ) )
= ( sup_sup @ A @ A3 @ ( sup_sup @ A @ B2 @ C3 ) ) ) ) ).
% sup.left_commute
thf(fact_193_sup__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ( ( sup_sup @ A )
= ( ^ [X3: A,Y6: A] : ( sup_sup @ A @ Y6 @ X3 ) ) ) ) ).
% sup_commute
thf(fact_194_sup_Ocommute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ( ( sup_sup @ A )
= ( ^ [A4: A,B6: A] : ( sup_sup @ A @ B6 @ A4 ) ) ) ) ).
% sup.commute
thf(fact_195_sup__assoc,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ X @ Y ) @ Z )
= ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z ) ) ) ) ).
% sup_assoc
thf(fact_196_sup_Oassoc,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [A3: A,B2: A,C3: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ A3 @ B2 ) @ C3 )
= ( sup_sup @ A @ A3 @ ( sup_sup @ A @ B2 @ C3 ) ) ) ) ).
% sup.assoc
thf(fact_197_sup__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_sup @ B @ ( type2 @ B ) )
=> ( ( sup_sup @ ( A > B ) )
= ( ^ [F5: A > B,G4: A > B,X3: A] : ( sup_sup @ B @ ( F5 @ X3 ) @ ( G4 @ X3 ) ) ) ) ) ).
% sup_fun_def
thf(fact_198_inf__sup__aci_I5_J,axiom,
! [A: $tType] :
( ( lattice @ A @ ( type2 @ A ) )
=> ( ( sup_sup @ A )
= ( ^ [X3: A,Y6: A] : ( sup_sup @ A @ Y6 @ X3 ) ) ) ) ).
% inf_sup_aci(5)
thf(fact_199_inf__sup__aci_I6_J,axiom,
! [A: $tType] :
( ( lattice @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ X @ Y ) @ Z )
= ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z ) ) ) ) ).
% inf_sup_aci(6)
thf(fact_200_inf__sup__aci_I7_J,axiom,
! [A: $tType] :
( ( lattice @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z: A] :
( ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z ) )
= ( sup_sup @ A @ Y @ ( sup_sup @ A @ X @ Z ) ) ) ) ).
% inf_sup_aci(7)
thf(fact_201_inf__sup__aci_I8_J,axiom,
! [A: $tType] :
( ( lattice @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( sup_sup @ A @ X @ ( sup_sup @ A @ X @ Y ) )
= ( sup_sup @ A @ X @ Y ) ) ) ).
% inf_sup_aci(8)
thf(fact_202_Un__left__commute,axiom,
! [A: $tType,A2: set @ A,B5: set @ A,C4: set @ A] :
( ( sup_sup @ ( set @ A ) @ A2 @ ( sup_sup @ ( set @ A ) @ B5 @ C4 ) )
= ( sup_sup @ ( set @ A ) @ B5 @ ( sup_sup @ ( set @ A ) @ A2 @ C4 ) ) ) ).
% Un_left_commute
thf(fact_203_Un__left__absorb,axiom,
! [A: $tType,A2: set @ A,B5: set @ A] :
( ( sup_sup @ ( set @ A ) @ A2 @ ( sup_sup @ ( set @ A ) @ A2 @ B5 ) )
= ( sup_sup @ ( set @ A ) @ A2 @ B5 ) ) ).
% Un_left_absorb
thf(fact_204_Un__commute,axiom,
! [A: $tType] :
( ( sup_sup @ ( set @ A ) )
= ( ^ [A7: set @ A,B7: set @ A] : ( sup_sup @ ( set @ A ) @ B7 @ A7 ) ) ) ).
% Un_commute
thf(fact_205_Un__absorb,axiom,
! [A: $tType,A2: set @ A] :
( ( sup_sup @ ( set @ A ) @ A2 @ A2 )
= A2 ) ).
% Un_absorb
thf(fact_206_Un__assoc,axiom,
! [A: $tType,A2: set @ A,B5: set @ A,C4: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A2 @ B5 ) @ C4 )
= ( sup_sup @ ( set @ A ) @ A2 @ ( sup_sup @ ( set @ A ) @ B5 @ C4 ) ) ) ).
% Un_assoc
thf(fact_207_ball__Un,axiom,
! [A: $tType,A2: set @ A,B5: set @ A,P: A > $o] :
( ( ! [X3: A] :
( ( member @ A @ X3 @ ( sup_sup @ ( set @ A ) @ A2 @ B5 ) )
=> ( P @ X3 ) ) )
= ( ! [X3: A] :
( ( member @ A @ X3 @ A2 )
=> ( P @ X3 ) )
& ! [X3: A] :
( ( member @ A @ X3 @ B5 )
=> ( P @ X3 ) ) ) ) ).
% ball_Un
thf(fact_208_bex__Un,axiom,
! [A: $tType,A2: set @ A,B5: set @ A,P: A > $o] :
( ( ? [X3: A] :
( ( member @ A @ X3 @ ( sup_sup @ ( set @ A ) @ A2 @ B5 ) )
& ( P @ X3 ) ) )
= ( ? [X3: A] :
( ( member @ A @ X3 @ A2 )
& ( P @ X3 ) )
| ? [X3: A] :
( ( member @ A @ X3 @ B5 )
& ( P @ X3 ) ) ) ) ).
% bex_Un
thf(fact_209_UnI2,axiom,
! [A: $tType,C3: A,B5: set @ A,A2: set @ A] :
( ( member @ A @ C3 @ B5 )
=> ( member @ A @ C3 @ ( sup_sup @ ( set @ A ) @ A2 @ B5 ) ) ) ).
% UnI2
thf(fact_210_UnI1,axiom,
! [A: $tType,C3: A,A2: set @ A,B5: set @ A] :
( ( member @ A @ C3 @ A2 )
=> ( member @ A @ C3 @ ( sup_sup @ ( set @ A ) @ A2 @ B5 ) ) ) ).
% UnI1
thf(fact_211_UnE,axiom,
! [A: $tType,C3: A,A2: set @ A,B5: set @ A] :
( ( member @ A @ C3 @ ( sup_sup @ ( set @ A ) @ A2 @ B5 ) )
=> ( ~ ( member @ A @ C3 @ A2 )
=> ( member @ A @ C3 @ B5 ) ) ) ).
% UnE
thf(fact_212_gen__lset__def,axiom,
! [A: $tType] :
( ( coindu1928975208n_lset @ A )
= ( ^ [A7: set @ A,Xs3: coindu1593790203_llist @ A] : ( sup_sup @ ( set @ A ) @ A7 @ ( coindu1112613586e_lset @ A @ Xs3 ) ) ) ) ).
% gen_lset_def
thf(fact_213_lset__lappend,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] : ( ord_less_eq @ ( set @ A ) @ ( coindu1112613586e_lset @ A @ ( coindu268472904append @ A @ Xs @ Ys ) ) @ ( sup_sup @ ( set @ A ) @ ( coindu1112613586e_lset @ A @ Xs ) @ ( coindu1112613586e_lset @ A @ Ys ) ) ) ).
% lset_lappend
thf(fact_214_le__sup__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ X @ Y ) @ Z )
= ( ( ord_less_eq @ A @ X @ Z )
& ( ord_less_eq @ A @ Y @ Z ) ) ) ) ).
% le_sup_iff
thf(fact_215_sup_Obounded__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [B2: A,C3: A,A3: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ B2 @ C3 ) @ A3 )
= ( ( ord_less_eq @ A @ B2 @ A3 )
& ( ord_less_eq @ A @ C3 @ A3 ) ) ) ) ).
% sup.bounded_iff
thf(fact_216_Un__subset__iff,axiom,
! [A: $tType,A2: set @ A,B5: set @ A,C4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A2 @ B5 ) @ C4 )
= ( ( ord_less_eq @ ( set @ A ) @ A2 @ C4 )
& ( ord_less_eq @ ( set @ A ) @ B5 @ C4 ) ) ) ).
% Un_subset_iff
thf(fact_217_inf__sup__ord_I4_J,axiom,
! [A: $tType] :
( ( lattice @ A @ ( type2 @ A ) )
=> ! [Y: A,X: A] : ( ord_less_eq @ A @ Y @ ( sup_sup @ A @ X @ Y ) ) ) ).
% inf_sup_ord(4)
thf(fact_218_inf__sup__ord_I3_J,axiom,
! [A: $tType] :
( ( lattice @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] : ( ord_less_eq @ A @ X @ ( sup_sup @ A @ X @ Y ) ) ) ).
% inf_sup_ord(3)
thf(fact_219_Un__mono,axiom,
! [A: $tType,A2: set @ A,C4: set @ A,B5: set @ A,D3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ C4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ D3 )
=> ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A2 @ B5 ) @ ( sup_sup @ ( set @ A ) @ C4 @ D3 ) ) ) ) ).
% Un_mono
thf(fact_220_Un__least,axiom,
! [A: $tType,A2: set @ A,C4: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ C4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ C4 )
=> ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A2 @ B5 ) @ C4 ) ) ) ).
% Un_least
thf(fact_221_Un__upper1,axiom,
! [A: $tType,A2: set @ A,B5: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ ( sup_sup @ ( set @ A ) @ A2 @ B5 ) ) ).
% Un_upper1
thf(fact_222_Un__upper2,axiom,
! [A: $tType,B5: set @ A,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ B5 @ ( sup_sup @ ( set @ A ) @ A2 @ B5 ) ) ).
% Un_upper2
thf(fact_223_Un__absorb1,axiom,
! [A: $tType,A2: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B5 )
=> ( ( sup_sup @ ( set @ A ) @ A2 @ B5 )
= B5 ) ) ).
% Un_absorb1
thf(fact_224_Un__absorb2,axiom,
! [A: $tType,B5: set @ A,A2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B5 @ A2 )
=> ( ( sup_sup @ ( set @ A ) @ A2 @ B5 )
= A2 ) ) ).
% Un_absorb2
thf(fact_225_subset__Un__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A7: set @ A,B7: set @ A] :
( ( sup_sup @ ( set @ A ) @ A7 @ B7 )
= B7 ) ) ) ).
% subset_Un_eq
thf(fact_226_le__supE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [A3: A,B2: A,X: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ A3 @ B2 ) @ X )
=> ~ ( ( ord_less_eq @ A @ A3 @ X )
=> ~ ( ord_less_eq @ A @ B2 @ X ) ) ) ) ).
% le_supE
thf(fact_227_le__supI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [A3: A,X: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ X )
=> ( ( ord_less_eq @ A @ B2 @ X )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ A3 @ B2 ) @ X ) ) ) ) ).
% le_supI
thf(fact_228_sup__ge1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] : ( ord_less_eq @ A @ X @ ( sup_sup @ A @ X @ Y ) ) ) ).
% sup_ge1
thf(fact_229_sup__ge2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [Y: A,X: A] : ( ord_less_eq @ A @ Y @ ( sup_sup @ A @ X @ Y ) ) ) ).
% sup_ge2
thf(fact_230_le__supI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [X: A,A3: A,B2: A] :
( ( ord_less_eq @ A @ X @ A3 )
=> ( ord_less_eq @ A @ X @ ( sup_sup @ A @ A3 @ B2 ) ) ) ) ).
% le_supI1
thf(fact_231_le__supI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [X: A,B2: A,A3: A] :
( ( ord_less_eq @ A @ X @ B2 )
=> ( ord_less_eq @ A @ X @ ( sup_sup @ A @ A3 @ B2 ) ) ) ) ).
% le_supI2
thf(fact_232_sup_Omono,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [C3: A,A3: A,D4: A,B2: A] :
( ( ord_less_eq @ A @ C3 @ A3 )
=> ( ( ord_less_eq @ A @ D4 @ B2 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ C3 @ D4 ) @ ( sup_sup @ A @ A3 @ B2 ) ) ) ) ) ).
% sup.mono
thf(fact_233_sup__mono,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [A3: A,C3: A,B2: A,D4: A] :
( ( ord_less_eq @ A @ A3 @ C3 )
=> ( ( ord_less_eq @ A @ B2 @ D4 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ A3 @ B2 ) @ ( sup_sup @ A @ C3 @ D4 ) ) ) ) ) ).
% sup_mono
thf(fact_234_sup__least,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [Y: A,X: A,Z: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( ord_less_eq @ A @ Z @ X )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ Y @ Z ) @ X ) ) ) ) ).
% sup_least
thf(fact_235_le__iff__sup,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ( ( ord_less_eq @ A )
= ( ^ [X3: A,Y6: A] :
( ( sup_sup @ A @ X3 @ Y6 )
= Y6 ) ) ) ) ).
% le_iff_sup
thf(fact_236_sup_OorderE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [B2: A,A3: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( A3
= ( sup_sup @ A @ A3 @ B2 ) ) ) ) ).
% sup.orderE
thf(fact_237_sup_OorderI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [A3: A,B2: A] :
( ( A3
= ( sup_sup @ A @ A3 @ B2 ) )
=> ( ord_less_eq @ A @ B2 @ A3 ) ) ) ).
% sup.orderI
thf(fact_238_sup__unique,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [F: A > A > A,X: A,Y: A] :
( ! [X4: A,Y3: A] : ( ord_less_eq @ A @ X4 @ ( F @ X4 @ Y3 ) )
=> ( ! [X4: A,Y3: A] : ( ord_less_eq @ A @ Y3 @ ( F @ X4 @ Y3 ) )
=> ( ! [X4: A,Y3: A,Z2: A] :
( ( ord_less_eq @ A @ Y3 @ X4 )
=> ( ( ord_less_eq @ A @ Z2 @ X4 )
=> ( ord_less_eq @ A @ ( F @ Y3 @ Z2 ) @ X4 ) ) )
=> ( ( sup_sup @ A @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ) ).
% sup_unique
thf(fact_239_sup_Oabsorb1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [B2: A,A3: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ( sup_sup @ A @ A3 @ B2 )
= A3 ) ) ) ).
% sup.absorb1
thf(fact_240_sup_Oabsorb2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( sup_sup @ A @ A3 @ B2 )
= B2 ) ) ) ).
% sup.absorb2
thf(fact_241_sup__absorb1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( sup_sup @ A @ X @ Y )
= X ) ) ) ).
% sup_absorb1
thf(fact_242_sup__absorb2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( sup_sup @ A @ X @ Y )
= Y ) ) ) ).
% sup_absorb2
thf(fact_243_sup_OboundedE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [B2: A,C3: A,A3: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ B2 @ C3 ) @ A3 )
=> ~ ( ( ord_less_eq @ A @ B2 @ A3 )
=> ~ ( ord_less_eq @ A @ C3 @ A3 ) ) ) ) ).
% sup.boundedE
thf(fact_244_sup_OboundedI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [B2: A,A3: A,C3: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ( ord_less_eq @ A @ C3 @ A3 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ B2 @ C3 ) @ A3 ) ) ) ) ).
% sup.boundedI
thf(fact_245_sup_Oorder__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ( ( ord_less_eq @ A )
= ( ^ [B6: A,A4: A] :
( A4
= ( sup_sup @ A @ A4 @ B6 ) ) ) ) ) ).
% sup.order_iff
thf(fact_246_sup_Ocobounded1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [A3: A,B2: A] : ( ord_less_eq @ A @ A3 @ ( sup_sup @ A @ A3 @ B2 ) ) ) ).
% sup.cobounded1
thf(fact_247_sup_Ocobounded2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [B2: A,A3: A] : ( ord_less_eq @ A @ B2 @ ( sup_sup @ A @ A3 @ B2 ) ) ) ).
% sup.cobounded2
thf(fact_248_sup_Oabsorb__iff1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ( ( ord_less_eq @ A )
= ( ^ [B6: A,A4: A] :
( ( sup_sup @ A @ A4 @ B6 )
= A4 ) ) ) ) ).
% sup.absorb_iff1
thf(fact_249_sup_Oabsorb__iff2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ( ( ord_less_eq @ A )
= ( ^ [A4: A,B6: A] :
( ( sup_sup @ A @ A4 @ B6 )
= B6 ) ) ) ) ).
% sup.absorb_iff2
thf(fact_250_sup_OcoboundedI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [C3: A,A3: A,B2: A] :
( ( ord_less_eq @ A @ C3 @ A3 )
=> ( ord_less_eq @ A @ C3 @ ( sup_sup @ A @ A3 @ B2 ) ) ) ) ).
% sup.coboundedI1
thf(fact_251_sup_OcoboundedI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [C3: A,B2: A,A3: A] :
( ( ord_less_eq @ A @ C3 @ B2 )
=> ( ord_less_eq @ A @ C3 @ ( sup_sup @ A @ A3 @ B2 ) ) ) ) ).
% sup.coboundedI2
thf(fact_252_wlog__linorder__le,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > A > $o,B2: A,A3: A] :
( ! [A6: A,B4: A] :
( ( ord_less_eq @ A @ A6 @ B4 )
=> ( P @ A6 @ B4 ) )
=> ( ( ( P @ B2 @ A3 )
=> ( P @ A3 @ B2 ) )
=> ( P @ A3 @ B2 ) ) ) ) ).
% wlog_linorder_le
thf(fact_253_chain__subset,axiom,
! [A: $tType,Ord2: A > A > $o,A2: set @ A,B5: set @ A] :
( ( comple1193779247_chain @ A @ Ord2 @ A2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ A2 )
=> ( comple1193779247_chain @ A @ Ord2 @ B5 ) ) ) ).
% chain_subset
%----Type constructors (7)
thf(tcon_fun___Lattices_Osemilattice__sup,axiom,
! [A8: $tType,A9: $tType] :
( ( semilattice_sup @ A9 @ ( type2 @ A9 ) )
=> ( semilattice_sup @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
thf(tcon_fun___Lattices_Olattice,axiom,
! [A8: $tType,A9: $tType] :
( ( lattice @ A9 @ ( type2 @ A9 ) )
=> ( lattice @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
thf(tcon_Set_Oset___Lattices_Osemilattice__sup_1,axiom,
! [A8: $tType] : ( semilattice_sup @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Lattices_Olattice_2,axiom,
! [A8: $tType] : ( lattice @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_3,axiom,
semilattice_sup @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
linorder @ $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,X: A,Y: A] :
( ( if @ A @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X: A,Y: A] :
( ( if @ A @ $true @ X @ Y )
= X ) ).
%----Conjectures (1)
thf(conj_0,conjecture,
coindu1213758845finite @ a @ xs ).
%------------------------------------------------------------------------------