TPTP Problem File: DAT117^1.p
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%------------------------------------------------------------------------------
% File : DAT117^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Data Structures
% Problem : Coinductive list 761
% 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__761.p [Bla16]
% Status : Theorem
% Rating : 0.33 v8.1.0, 0.25 v7.5.0, 0.00 v7.2.0, 0.25 v7.1.0
% Syntax : Number of formulae : 311 ( 123 unt; 38 typ; 0 def)
% Number of atoms : 721 ( 267 equ; 0 cnn)
% Maximal formula atoms : 9 ( 2 avg)
% Number of connectives : 3839 ( 106 ~; 20 |; 62 &;3324 @)
% ( 0 <=>; 327 =>; 0 <=; 0 <~>)
% Maximal formula depth : 24 ( 8 avg)
% Number of types : 2 ( 1 usr)
% Number of type conns : 270 ( 270 >; 0 *; 0 +; 0 <<)
% Number of symbols : 40 ( 37 usr; 3 con; 0-8 aty)
% Number of variables : 1101 ( 56 ^; 973 !; 34 ?;1101 :)
% ( 38 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 14:46:28.138
%------------------------------------------------------------------------------
%----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_Lattices_Obounded__lattice,type,
bounded_lattice:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Lattices_Olattice,type,
lattice:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Lattices_Osemilattice__sup,type,
semilattice_sup:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Lattices_Obounded__lattice__bot,type,
bounded_lattice_bot:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Lattices_Obounded__semilattice__sup__bot,type,
bounde1808546759up_bot:
!>[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_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_Ocorec__llist,type,
coindu1244876290_llist:
!>[C: $tType,A: $tType] : ( ( C > $o ) > ( C > A ) > ( C > $o ) > ( C > ( coindu1593790203_llist @ A ) ) > ( C > C ) > C > ( coindu1593790203_llist @ A ) ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollist_Olhd,type,
coindu1046438764le_lhd:
!>[A: $tType] : ( ( coindu1593790203_llist @ A ) > A ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollist_Ollist__all2,type,
coindu987187967t_all2:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ B ) > $o ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollist_Olnull,type,
coindu335574135_lnull:
!>[A: $tType] : ( ( coindu1593790203_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollist_Olset,type,
coindu1112613586e_lset:
!>[A: $tType] : ( ( coindu1593790203_llist @ A ) > ( set @ A ) ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollist_Oltl,type,
coindu1047225960le_ltl:
!>[A: $tType] : ( ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollist_Opred__llist,type,
coindu44415597_llist:
!>[A: $tType] : ( ( A > $o ) > ( coindu1593790203_llist @ A ) > $o ) ).
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_Olsetp,type,
coindu1055736764_lsetp:
!>[A: $tType] : ( ( coindu1593790203_llist @ A ) > 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__Order_Omonotone,type,
comple1396247847notone:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( B > B > $o ) > ( A > B ) > $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_Obot__class_Obot,type,
bot_bot:
!>[A: $tType] : 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_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_x,type,
x: a ).
%----Relevant facts (254)
thf(fact_0_lprefix__refl,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] : ( coindu1696667936prefix @ A @ Xs @ Xs ) ).
% lprefix_refl
thf(fact_1_llist_Oleq__refl,axiom,
! [A: $tType,X: coindu1593790203_llist @ A] : ( coindu1696667936prefix @ A @ X @ X ) ).
% llist.leq_refl
thf(fact_2_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_3_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_4_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_5_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_6_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_7_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_8_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_9_LCons__lprefix__conv,axiom,
! [A: $tType,X: A,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ ( coindu1121789889_LCons @ A @ X @ Xs ) @ Ys )
= ( ? [Ys2: coindu1593790203_llist @ A] :
( ( Ys
= ( coindu1121789889_LCons @ A @ X @ Ys2 ) )
& ( coindu1696667936prefix @ A @ Xs @ Ys2 ) ) ) ) ).
% LCons_lprefix_conv
thf(fact_10_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_11_llist_Omonotone__if__bot,axiom,
! [B: $tType,A: $tType,Bound: coindu1593790203_llist @ A,G: ( coindu1593790203_llist @ A ) > B,Bot: B,F: ( coindu1593790203_llist @ A ) > B,Ord: B > B > $o] :
( ! [X2: coindu1593790203_llist @ A] :
( ( ( coindu1696667936prefix @ A @ X2 @ Bound )
=> ( ( G @ X2 )
= Bot ) )
& ( ~ ( coindu1696667936prefix @ A @ X2 @ Bound )
=> ( ( G @ X2 )
= ( F @ X2 ) ) ) )
=> ( ! [X2: coindu1593790203_llist @ A,Y2: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ X2 @ Y2 )
=> ( ~ ( coindu1696667936prefix @ A @ X2 @ Bound )
=> ( Ord @ ( F @ X2 ) @ ( F @ Y2 ) ) ) )
=> ( ! [X2: coindu1593790203_llist @ A] :
( ~ ( coindu1696667936prefix @ A @ X2 @ Bound )
=> ( Ord @ Bot @ ( F @ X2 ) ) )
=> ( ( Ord @ Bot @ Bot )
=> ( comple1396247847notone @ ( coindu1593790203_llist @ A ) @ B @ ( coindu1696667936prefix @ A ) @ Ord @ G ) ) ) ) ) ).
% llist.monotone_if_bot
thf(fact_12_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_13_monotoneD,axiom,
! [B: $tType,A: $tType,Orda: A > A > $o,Ordb: B > B > $o,F: A > B,X: A,Y: A] :
( ( comple1396247847notone @ A @ B @ Orda @ Ordb @ F )
=> ( ( Orda @ X @ Y )
=> ( Ordb @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% monotoneD
thf(fact_14_monotoneI,axiom,
! [B: $tType,A: $tType,Orda: A > A > $o,Ordb: B > B > $o,F: A > B] :
( ! [X2: A,Y2: A] :
( ( Orda @ X2 @ Y2 )
=> ( Ordb @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( comple1396247847notone @ A @ B @ Orda @ Ordb @ F ) ) ).
% monotoneI
thf(fact_15_monotone__def,axiom,
! [B: $tType,A: $tType] :
( ( comple1396247847notone @ A @ B )
= ( ^ [Orda2: A > A > $o,Ordb2: B > B > $o,F2: A > B] :
! [X3: A,Y3: A] :
( ( Orda2 @ X3 @ Y3 )
=> ( Ordb2 @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) ) ) ) ).
% monotone_def
thf(fact_16_llist_Opred__inject_I2_J,axiom,
! [A: $tType,P: A > $o,A2: A,Aa: coindu1593790203_llist @ A] :
( ( coindu44415597_llist @ A @ P @ ( coindu1121789889_LCons @ A @ A2 @ Aa ) )
= ( ( P @ A2 )
& ( coindu44415597_llist @ A @ P @ Aa ) ) ) ).
% llist.pred_inject(2)
thf(fact_17_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_18_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,Ys3: coindu1593790203_llist @ A,X2: A] :
( ( A1
= ( coindu1121789889_LCons @ A @ X2 @ Xs2 ) )
=> ( ( A22
= ( coindu1121789889_LCons @ A @ X2 @ Ys3 ) )
=> ~ ( coindu1696667936prefix @ A @ Xs2 @ Ys3 ) ) ) ) ) ).
% lprefix.cases
thf(fact_19_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,Ys4: coindu1593790203_llist @ A,X3: A] :
( ( A12
= ( coindu1121789889_LCons @ A @ X3 @ Xs3 ) )
& ( A23
= ( coindu1121789889_LCons @ A @ X3 @ Ys4 ) )
& ( coindu1696667936prefix @ A @ Xs3 @ Ys4 ) ) ) ) ) ).
% lprefix.simps
thf(fact_20_lprefix_Ocoinduct,axiom,
! [A: $tType,X4: ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) > $o,X: coindu1593790203_llist @ A,Xa: coindu1593790203_llist @ A] :
( ( X4 @ X @ Xa )
=> ( ! [X2: coindu1593790203_llist @ A,Xa2: coindu1593790203_llist @ A] :
( ( X4 @ X2 @ Xa2 )
=> ( ? [Xs4: coindu1593790203_llist @ A] :
( ( X2
= ( coindu1598213697e_LNil @ A ) )
& ( Xa2 = Xs4 ) )
| ? [Xs4: coindu1593790203_llist @ A,Ys5: coindu1593790203_llist @ A,Xb: A] :
( ( X2
= ( coindu1121789889_LCons @ A @ Xb @ Xs4 ) )
& ( Xa2
= ( coindu1121789889_LCons @ A @ Xb @ Ys5 ) )
& ( ( X4 @ Xs4 @ Ys5 )
| ( coindu1696667936prefix @ A @ Xs4 @ Ys5 ) ) ) ) )
=> ( coindu1696667936prefix @ A @ X @ Xa ) ) ) ).
% lprefix.coinduct
thf(fact_21_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_22_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_23_lsetp_Ointros_I2_J,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,X: A,X5: A] :
( ( coindu1055736764_lsetp @ A @ Xs @ X )
=> ( coindu1055736764_lsetp @ A @ ( coindu1121789889_LCons @ A @ X5 @ Xs ) @ X ) ) ).
% lsetp.intros(2)
thf(fact_24_lprefix__code_I1_J,axiom,
! [A: $tType,Ys: coindu1593790203_llist @ A] : ( coindu1696667936prefix @ A @ ( coindu1598213697e_LNil @ A ) @ Ys ) ).
% lprefix_code(1)
thf(fact_25_lmember__code_I1_J,axiom,
! [A: $tType,X: A] :
~ ( coindu567634248member @ A @ X @ ( coindu1598213697e_LNil @ A ) ) ).
% lmember_code(1)
thf(fact_26_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_27_llist_Opred__inject_I1_J,axiom,
! [A: $tType,P: A > $o] : ( coindu44415597_llist @ A @ P @ ( coindu1598213697e_LNil @ A ) ) ).
% llist.pred_inject(1)
thf(fact_28_unfold__llist_Ocode,axiom,
! [B: $tType,A: $tType] :
( ( coindu1599971794_llist @ A @ B )
= ( ^ [P3: A > $o,G212: A > B,G222: A > A,A3: A] : ( if @ ( coindu1593790203_llist @ B ) @ ( P3 @ A3 ) @ ( coindu1598213697e_LNil @ B ) @ ( coindu1121789889_LCons @ B @ ( G212 @ A3 ) @ ( coindu1599971794_llist @ A @ B @ P3 @ G212 @ G222 @ ( G222 @ A3 ) ) ) ) ) ) ).
% unfold_llist.code
thf(fact_29_LNil__lprefix,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] : ( coindu1696667936prefix @ A @ ( coindu1598213697e_LNil @ A ) @ Xs ) ).
% LNil_lprefix
thf(fact_30_neq__LNil__conv,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( ( Xs
!= ( coindu1598213697e_LNil @ A ) )
= ( ? [X3: A,Xs5: coindu1593790203_llist @ A] :
( Xs
= ( coindu1121789889_LCons @ A @ X3 @ Xs5 ) ) ) ) ).
% neq_LNil_conv
thf(fact_31_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_32_llist_Odistinct_I1_J,axiom,
! [A: $tType,X21: A,X22: coindu1593790203_llist @ A] :
( ( coindu1598213697e_LNil @ A )
!= ( coindu1121789889_LCons @ A @ X21 @ X22 ) ) ).
% llist.distinct(1)
thf(fact_33_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_34_lsetp_Oinducts,axiom,
! [A: $tType,X1: coindu1593790203_llist @ A,X23: A,P: ( coindu1593790203_llist @ A ) > A > $o] :
( ( coindu1055736764_lsetp @ A @ X1 @ X23 )
=> ( ! [X2: A,Xs2: coindu1593790203_llist @ A] : ( P @ ( coindu1121789889_LCons @ A @ X2 @ Xs2 ) @ X2 )
=> ( ! [Xs2: coindu1593790203_llist @ A,X2: A,X6: A] :
( ( coindu1055736764_lsetp @ A @ Xs2 @ X2 )
=> ( ( P @ Xs2 @ X2 )
=> ( P @ ( coindu1121789889_LCons @ A @ X6 @ Xs2 ) @ X2 ) ) )
=> ( P @ X1 @ X23 ) ) ) ) ).
% lsetp.inducts
thf(fact_35_lsetp_Osimps,axiom,
! [A: $tType] :
( ( coindu1055736764_lsetp @ A )
= ( ^ [A12: coindu1593790203_llist @ A,A23: A] :
( ? [X3: A,Xs3: coindu1593790203_llist @ A] :
( ( A12
= ( coindu1121789889_LCons @ A @ X3 @ Xs3 ) )
& ( A23 = X3 ) )
| ? [Xs3: coindu1593790203_llist @ A,X3: A,X7: A] :
( ( A12
= ( coindu1121789889_LCons @ A @ X7 @ Xs3 ) )
& ( A23 = X3 )
& ( coindu1055736764_lsetp @ A @ Xs3 @ X3 ) ) ) ) ) ).
% lsetp.simps
thf(fact_36_lsetp_Ocases,axiom,
! [A: $tType,A1: coindu1593790203_llist @ A,A22: A] :
( ( coindu1055736764_lsetp @ A @ A1 @ A22 )
=> ( ! [X2: A] :
( ? [Xs2: coindu1593790203_llist @ A] :
( A1
= ( coindu1121789889_LCons @ A @ X2 @ Xs2 ) )
=> ( A22 != X2 ) )
=> ~ ! [Xs2: coindu1593790203_llist @ A] :
( ? [X6: A] :
( A1
= ( coindu1121789889_LCons @ A @ X6 @ Xs2 ) )
=> ~ ( coindu1055736764_lsetp @ A @ Xs2 @ A22 ) ) ) ) ).
% lsetp.cases
thf(fact_37_lsetp_Ointros_I1_J,axiom,
! [A: $tType,X: A,Xs: coindu1593790203_llist @ A] : ( coindu1055736764_lsetp @ A @ ( coindu1121789889_LCons @ A @ X @ Xs ) @ X ) ).
% lsetp.intros(1)
thf(fact_38_gen__lset__code_I1_J,axiom,
! [A: $tType,A4: set @ A] :
( ( coindu1928975208n_lset @ A @ A4 @ ( coindu1598213697e_LNil @ A ) )
= A4 ) ).
% gen_lset_code(1)
thf(fact_39_llist_Ocorec__code,axiom,
! [A: $tType,C: $tType] :
( ( coindu1244876290_llist @ C @ A )
= ( ^ [P3: C > $o,G212: C > A,Q22: C > $o,G221: C > ( coindu1593790203_llist @ A ),G2222: C > C,A3: C] : ( if @ ( coindu1593790203_llist @ A ) @ ( P3 @ A3 ) @ ( coindu1598213697e_LNil @ A ) @ ( coindu1121789889_LCons @ A @ ( G212 @ A3 ) @ ( if @ ( coindu1593790203_llist @ A ) @ ( Q22 @ A3 ) @ ( G221 @ A3 ) @ ( coindu1244876290_llist @ C @ A @ P3 @ G212 @ Q22 @ G221 @ G2222 @ ( G2222 @ A3 ) ) ) ) ) ) ) ).
% llist.corec_code
thf(fact_40_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,X2: A] :
( ( coindu1213758845finite @ A @ Xs2 )
=> ( ( P @ Xs2 )
=> ( P @ ( coindu1121789889_LCons @ A @ X2 @ Xs2 ) ) ) )
=> ( P @ X ) ) ) ) ).
% lfinite.inducts
thf(fact_41_lfinite_Osimps,axiom,
! [A: $tType] :
( ( coindu1213758845finite @ A )
= ( ^ [A3: coindu1593790203_llist @ A] :
( ( A3
= ( coindu1598213697e_LNil @ A ) )
| ? [Xs3: coindu1593790203_llist @ A,X3: A] :
( ( A3
= ( coindu1121789889_LCons @ A @ X3 @ Xs3 ) )
& ( coindu1213758845finite @ A @ Xs3 ) ) ) ) ) ).
% lfinite.simps
thf(fact_42_lfinite_Ocases,axiom,
! [A: $tType,A2: coindu1593790203_llist @ A] :
( ( coindu1213758845finite @ A @ A2 )
=> ( ( A2
!= ( coindu1598213697e_LNil @ A ) )
=> ~ ! [Xs2: coindu1593790203_llist @ A] :
( ? [X2: A] :
( A2
= ( coindu1121789889_LCons @ A @ X2 @ Xs2 ) )
=> ~ ( coindu1213758845finite @ A @ Xs2 ) ) ) ) ).
% lfinite.cases
thf(fact_43_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_44_lprefix__LNil,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ Xs @ ( coindu1598213697e_LNil @ A ) )
= ( coindu335574135_lnull @ A @ Xs ) ) ).
% lprefix_LNil
thf(fact_45_mem__Collect__eq,axiom,
! [A: $tType,A2: A,P: A > $o] :
( ( member @ A @ A2 @ ( collect @ A @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: $tType,A4: set @ A] :
( ( collect @ A
@ ^ [X3: A] : ( member @ A @ X3 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X2: A] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_48_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X2: A] :
( ( F @ X2 )
= ( G @ X2 ) )
=> ( F = G ) ) ).
% ext
thf(fact_49_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_50_llist_Orel__cases,axiom,
! [A: $tType,B: $tType,R: A > B > $o,A2: coindu1593790203_llist @ A,B2: coindu1593790203_llist @ B] :
( ( coindu987187967t_all2 @ A @ B @ R @ A2 @ B2 )
=> ( ( ( A2
= ( coindu1598213697e_LNil @ A ) )
=> ( B2
!= ( coindu1598213697e_LNil @ B ) ) )
=> ~ ! [X12: A,X24: coindu1593790203_llist @ A] :
( ( A2
= ( coindu1121789889_LCons @ A @ X12 @ X24 ) )
=> ! [Y1: B,Y23: coindu1593790203_llist @ B] :
( ( B2
= ( coindu1121789889_LCons @ B @ Y1 @ Y23 ) )
=> ( ( R @ X12 @ Y1 )
=> ~ ( coindu987187967t_all2 @ A @ B @ R @ X24 @ Y23 ) ) ) ) ) ) ).
% llist.rel_cases
thf(fact_51_llist_Orel__distinct_I1_J,axiom,
! [A: $tType,B: $tType,R: A > B > $o,Y21: B,Y22: coindu1593790203_llist @ B] :
~ ( coindu987187967t_all2 @ A @ B @ R @ ( coindu1598213697e_LNil @ A ) @ ( coindu1121789889_LCons @ B @ Y21 @ Y22 ) ) ).
% llist.rel_distinct(1)
thf(fact_52_llist_Orel__inject_I2_J,axiom,
! [A: $tType,B: $tType,R: A > B > $o,X21: A,X22: coindu1593790203_llist @ A,Y21: B,Y22: coindu1593790203_llist @ B] :
( ( coindu987187967t_all2 @ A @ B @ R @ ( coindu1121789889_LCons @ A @ X21 @ X22 ) @ ( coindu1121789889_LCons @ B @ Y21 @ Y22 ) )
= ( ( R @ X21 @ Y21 )
& ( coindu987187967t_all2 @ A @ B @ R @ X22 @ Y22 ) ) ) ).
% llist.rel_inject(2)
thf(fact_53_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_54_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_55_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_56_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_57_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_58_lappend__LNil2,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( ( coindu268472904append @ A @ Xs @ ( coindu1598213697e_LNil @ A ) )
= Xs ) ).
% lappend_LNil2
thf(fact_59_lappend__code_I1_J,axiom,
! [A: $tType,Ys: coindu1593790203_llist @ A] :
( ( coindu268472904append @ A @ ( coindu1598213697e_LNil @ A ) @ Ys )
= Ys ) ).
% lappend_code(1)
thf(fact_60_lfinite__code_I1_J,axiom,
! [A: $tType] : ( coindu1213758845finite @ A @ ( coindu1598213697e_LNil @ A ) ) ).
% lfinite_code(1)
thf(fact_61_lfinite__ltl,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( ( coindu1213758845finite @ A @ ( coindu1047225960le_ltl @ A @ Xs ) )
= ( coindu1213758845finite @ A @ Xs ) ) ).
% lfinite_ltl
thf(fact_62_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_63_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_64_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_65_llist_Ocorec__disc__iff_I1_J,axiom,
! [A: $tType,C: $tType,P2: C > $o,G21: C > A,Q222: C > $o,G2212: C > ( coindu1593790203_llist @ A ),G2223: C > C,A2: C] :
( ( coindu335574135_lnull @ A @ ( coindu1244876290_llist @ C @ A @ P2 @ G21 @ Q222 @ G2212 @ G2223 @ A2 ) )
= ( P2 @ A2 ) ) ).
% llist.corec_disc_iff(1)
thf(fact_66_llist_Ocorec__disc__iff_I2_J,axiom,
! [A: $tType,C: $tType,P2: C > $o,G21: C > A,Q222: C > $o,G2212: C > ( coindu1593790203_llist @ A ),G2223: C > C,A2: C] :
( ( ~ ( coindu335574135_lnull @ A @ ( coindu1244876290_llist @ C @ A @ P2 @ G21 @ Q222 @ G2212 @ G2223 @ A2 ) ) )
= ( ~ ( P2 @ A2 ) ) ) ).
% llist.corec_disc_iff(2)
thf(fact_67_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_68_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_69_llist_Orel__refl,axiom,
! [B: $tType,Ra: B > B > $o,X: coindu1593790203_llist @ B] :
( ! [X2: B] : ( Ra @ X2 @ X2 )
=> ( coindu987187967t_all2 @ B @ B @ Ra @ X @ X ) ) ).
% llist.rel_refl
thf(fact_70_llist_Orel__eq,axiom,
! [A: $tType] :
( ( coindu987187967t_all2 @ A @ A
@ ^ [Y4: A,Z2: A] : Y4 = Z2 )
= ( ^ [Y4: coindu1593790203_llist @ A,Z2: coindu1593790203_llist @ A] : Y4 = Z2 ) ) ).
% llist.rel_eq
thf(fact_71_lnull__imp__lfinite,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ A @ Xs )
=> ( coindu1213758845finite @ A @ Xs ) ) ).
% lnull_imp_lfinite
thf(fact_72_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_73_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_74_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_75_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_76_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_77_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_78_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_79_lnull__ltlI,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ A @ Xs )
=> ( coindu335574135_lnull @ A @ ( coindu1047225960le_ltl @ A @ Xs ) ) ) ).
% lnull_ltlI
thf(fact_80_llist_Ocorec__disc_I1_J,axiom,
! [A: $tType,C: $tType,P2: C > $o,A2: C,G21: C > A,Q222: C > $o,G2212: C > ( coindu1593790203_llist @ A ),G2223: C > C] :
( ( P2 @ A2 )
=> ( coindu335574135_lnull @ A @ ( coindu1244876290_llist @ C @ A @ P2 @ G21 @ Q222 @ G2212 @ G2223 @ A2 ) ) ) ).
% llist.corec_disc(1)
thf(fact_81_llist_Ocorec__disc_I2_J,axiom,
! [A: $tType,C: $tType,P2: C > $o,A2: C,G21: C > A,Q222: C > $o,G2212: C > ( coindu1593790203_llist @ A ),G2223: C > C] :
( ~ ( P2 @ A2 )
=> ~ ( coindu335574135_lnull @ A @ ( coindu1244876290_llist @ C @ A @ P2 @ G21 @ Q222 @ G2212 @ G2223 @ A2 ) ) ) ).
% llist.corec_disc(2)
thf(fact_82_llist_Ocorec__sel_I2_J,axiom,
! [A: $tType,C: $tType,P2: C > $o,A2: C,Q222: C > $o,G21: C > A,G2212: C > ( coindu1593790203_llist @ A ),G2223: C > C] :
( ~ ( P2 @ A2 )
=> ( ( ( Q222 @ A2 )
=> ( ( coindu1047225960le_ltl @ A @ ( coindu1244876290_llist @ C @ A @ P2 @ G21 @ Q222 @ G2212 @ G2223 @ A2 ) )
= ( G2212 @ A2 ) ) )
& ( ~ ( Q222 @ A2 )
=> ( ( coindu1047225960le_ltl @ A @ ( coindu1244876290_llist @ C @ A @ P2 @ G21 @ Q222 @ G2212 @ G2223 @ A2 ) )
= ( coindu1244876290_llist @ C @ A @ P2 @ G21 @ Q222 @ G2212 @ G2223 @ ( G2223 @ A2 ) ) ) ) ) ) ).
% llist.corec_sel(2)
thf(fact_83_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_84_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_85_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_86_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_87_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_88_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_89_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_90_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_91_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_92_not__lnull__conv,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( ( ~ ( coindu335574135_lnull @ A @ Xs ) )
= ( ? [X3: A,Xs5: coindu1593790203_llist @ A] :
( Xs
= ( coindu1121789889_LCons @ A @ X3 @ Xs5 ) ) ) ) ).
% not_lnull_conv
thf(fact_93_lnull__def,axiom,
! [A: $tType] :
( ( coindu335574135_lnull @ A )
= ( ^ [Llist2: coindu1593790203_llist @ A] :
( Llist2
= ( coindu1598213697e_LNil @ A ) ) ) ) ).
% lnull_def
thf(fact_94_llist_Ocollapse_I1_J,axiom,
! [A: $tType,Llist: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ A @ Llist )
=> ( Llist
= ( coindu1598213697e_LNil @ A ) ) ) ).
% llist.collapse(1)
thf(fact_95_llist_OdiscI_I1_J,axiom,
! [A: $tType,Llist: coindu1593790203_llist @ A] :
( ( Llist
= ( coindu1598213697e_LNil @ A ) )
=> ( coindu335574135_lnull @ A @ Llist ) ) ).
% llist.discI(1)
thf(fact_96_llist_Odisc_I1_J,axiom,
! [A: $tType] : ( coindu335574135_lnull @ A @ ( coindu1598213697e_LNil @ A ) ) ).
% llist.disc(1)
thf(fact_97_llist_Orel__intros_I2_J,axiom,
! [A: $tType,B: $tType,R: A > B > $o,X21: A,Y21: B,X22: coindu1593790203_llist @ A,Y22: coindu1593790203_llist @ B] :
( ( R @ X21 @ Y21 )
=> ( ( coindu987187967t_all2 @ A @ B @ R @ X22 @ Y22 )
=> ( coindu987187967t_all2 @ A @ B @ R @ ( coindu1121789889_LCons @ A @ X21 @ X22 ) @ ( coindu1121789889_LCons @ B @ Y21 @ Y22 ) ) ) ) ).
% llist.rel_intros(2)
thf(fact_98_llist_Octr__transfer_I1_J,axiom,
! [A: $tType,B: $tType,R: A > B > $o] : ( coindu987187967t_all2 @ A @ B @ R @ ( coindu1598213697e_LNil @ A ) @ ( coindu1598213697e_LNil @ B ) ) ).
% llist.ctr_transfer(1)
thf(fact_99_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_100_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_101_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_102_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_103_ltl__simps_I1_J,axiom,
! [A: $tType] :
( ( coindu1047225960le_ltl @ A @ ( coindu1598213697e_LNil @ A ) )
= ( coindu1598213697e_LNil @ A ) ) ).
% ltl_simps(1)
thf(fact_104_lprefix__lappend,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] : ( coindu1696667936prefix @ A @ Xs @ ( coindu268472904append @ A @ Xs @ Ys ) ) ).
% lprefix_lappend
thf(fact_105_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_106_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_107_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_108_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_109_lappend__LNil__LNil,axiom,
! [A: $tType] :
( ( coindu268472904append @ A @ ( coindu1598213697e_LNil @ A ) @ ( coindu1598213697e_LNil @ A ) )
= ( coindu1598213697e_LNil @ A ) ) ).
% lappend_LNil_LNil
thf(fact_110_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_111_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_112_lfinite__LNil,axiom,
! [A: $tType] : ( coindu1213758845finite @ A @ ( coindu1598213697e_LNil @ A ) ) ).
% lfinite_LNil
thf(fact_113_llist_Ocorec_I2_J,axiom,
! [A: $tType,C: $tType,P2: C > $o,A2: C,G21: C > A,Q222: C > $o,G2212: C > ( coindu1593790203_llist @ A ),G2223: C > C] :
( ~ ( P2 @ A2 )
=> ( ( coindu1244876290_llist @ C @ A @ P2 @ G21 @ Q222 @ G2212 @ G2223 @ A2 )
= ( coindu1121789889_LCons @ A @ ( G21 @ A2 ) @ ( if @ ( coindu1593790203_llist @ A ) @ ( Q222 @ A2 ) @ ( G2212 @ A2 ) @ ( coindu1244876290_llist @ C @ A @ P2 @ G21 @ Q222 @ G2212 @ G2223 @ ( G2223 @ A2 ) ) ) ) ) ) ).
% llist.corec(2)
thf(fact_114_llist_Ocorec_I1_J,axiom,
! [C: $tType,A: $tType,P2: C > $o,A2: C,G21: C > A,Q222: C > $o,G2212: C > ( coindu1593790203_llist @ A ),G2223: C > C] :
( ( P2 @ A2 )
=> ( ( coindu1244876290_llist @ C @ A @ P2 @ G21 @ Q222 @ G2212 @ G2223 @ A2 )
= ( coindu1598213697e_LNil @ A ) ) ) ).
% llist.corec(1)
thf(fact_115_llist_Orel__distinct_I2_J,axiom,
! [A: $tType,B: $tType,R: A > B > $o,Y21: A,Y22: coindu1593790203_llist @ A] :
~ ( coindu987187967t_all2 @ A @ B @ R @ ( coindu1121789889_LCons @ A @ Y21 @ Y22 ) @ ( coindu1598213697e_LNil @ B ) ) ).
% llist.rel_distinct(2)
thf(fact_116_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_117_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_118_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_119_llist_Orel__coinduct,axiom,
! [A: $tType,B: $tType,P: ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ B ) > $o,X: coindu1593790203_llist @ A,Y: coindu1593790203_llist @ B,R: A > B > $o] :
( ( P @ X @ Y )
=> ( ! [Llist3: coindu1593790203_llist @ A,Llist4: coindu1593790203_llist @ B] :
( ( P @ Llist3 @ Llist4 )
=> ( ( ( coindu335574135_lnull @ A @ Llist3 )
= ( coindu335574135_lnull @ B @ Llist4 ) )
& ( ~ ( coindu335574135_lnull @ A @ Llist3 )
=> ( ~ ( coindu335574135_lnull @ B @ Llist4 )
=> ( ( R @ ( coindu1046438764le_lhd @ A @ Llist3 ) @ ( coindu1046438764le_lhd @ B @ Llist4 ) )
& ( P @ ( coindu1047225960le_ltl @ A @ Llist3 ) @ ( coindu1047225960le_ltl @ B @ Llist4 ) ) ) ) ) ) )
=> ( coindu987187967t_all2 @ A @ B @ R @ X @ Y ) ) ) ).
% llist.rel_coinduct
thf(fact_120_llist_Orel__sel,axiom,
! [B: $tType,A: $tType] :
( ( coindu987187967t_all2 @ A @ B )
= ( ^ [R2: A > B > $o,A3: coindu1593790203_llist @ A,B3: coindu1593790203_llist @ B] :
( ( ( coindu335574135_lnull @ A @ A3 )
= ( coindu335574135_lnull @ B @ B3 ) )
& ( ~ ( coindu335574135_lnull @ A @ A3 )
=> ( ~ ( coindu335574135_lnull @ B @ B3 )
=> ( ( R2 @ ( coindu1046438764le_lhd @ A @ A3 ) @ ( coindu1046438764le_lhd @ B @ B3 ) )
& ( coindu987187967t_all2 @ A @ B @ R2 @ ( coindu1047225960le_ltl @ A @ A3 ) @ ( coindu1047225960le_ltl @ B @ B3 ) ) ) ) ) ) ) ) ).
% llist.rel_sel
thf(fact_121_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_122_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,Ys3: coindu1593790203_llist @ A] :
( ( P @ Xs2 @ Ys3 )
=> ( ( ( coindu335574135_lnull @ A @ Ys3 )
=> ( coindu335574135_lnull @ A @ Xs2 ) )
& ( ~ ( coindu335574135_lnull @ A @ Xs2 )
=> ( ~ ( coindu335574135_lnull @ A @ Ys3 )
=> ( ( ( coindu1046438764le_lhd @ A @ Xs2 )
= ( coindu1046438764le_lhd @ A @ Ys3 ) )
& ( ( P @ ( coindu1047225960le_ltl @ A @ Xs2 ) @ ( coindu1047225960le_ltl @ A @ Ys3 ) )
| ( coindu1696667936prefix @ A @ ( coindu1047225960le_ltl @ A @ Xs2 ) @ ( coindu1047225960le_ltl @ A @ Ys3 ) ) ) ) ) ) ) )
=> ( coindu1696667936prefix @ A @ Xs @ Ys ) ) ) ).
% lprefix_coinduct
thf(fact_123_split__llist,axiom,
! [A: $tType,X: A,Xs: coindu1593790203_llist @ A] :
( ( member @ A @ X @ ( coindu1112613586e_lset @ A @ Xs ) )
=> ? [Ys3: coindu1593790203_llist @ A,Zs2: coindu1593790203_llist @ A] :
( ( Xs
= ( coindu268472904append @ A @ Ys3 @ ( coindu1121789889_LCons @ A @ X @ Zs2 ) ) )
& ( coindu1213758845finite @ A @ Ys3 ) ) ) ).
% split_llist
thf(fact_124_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_125_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_126_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_127_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,Y2: A] :
( ~ ( coindu335574135_lnull @ A @ Xs2 )
=> ( ( member @ A @ Y2 @ ( coindu1112613586e_lset @ A @ ( coindu1047225960le_ltl @ A @ Xs2 ) ) )
=> ( ( P @ Y2 @ ( coindu1047225960le_ltl @ A @ Xs2 ) )
=> ( P @ Y2 @ Xs2 ) ) ) )
=> ( P @ X @ Xs ) ) ) ) ).
% llist_set_induct
thf(fact_128_llist_Oset__induct,axiom,
! [A: $tType,X: A,A2: coindu1593790203_llist @ A,P: A > ( coindu1593790203_llist @ A ) > $o] :
( ( member @ A @ X @ ( coindu1112613586e_lset @ A @ A2 ) )
=> ( ! [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 @ A2 ) ) ) ) ).
% llist.set_induct
thf(fact_129_llist_Oset__cases,axiom,
! [A: $tType,E: A,A2: coindu1593790203_llist @ A] :
( ( member @ A @ E @ ( coindu1112613586e_lset @ A @ A2 ) )
=> ( ! [Z22: coindu1593790203_llist @ A] :
( A2
!= ( coindu1121789889_LCons @ A @ E @ Z22 ) )
=> ~ ! [Z1: A,Z22: coindu1593790203_llist @ A] :
( ( A2
= ( coindu1121789889_LCons @ A @ Z1 @ Z22 ) )
=> ~ ( member @ A @ E @ ( coindu1112613586e_lset @ A @ Z22 ) ) ) ) ) ).
% llist.set_cases
thf(fact_130_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 ) )
=> ( ! [X6: A,Xs2: coindu1593790203_llist @ A] :
( ( member @ A @ X @ ( coindu1112613586e_lset @ A @ Xs2 ) )
=> ( ( P @ Xs2 )
=> ( P @ ( coindu1121789889_LCons @ A @ X6 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% lset_induct'
thf(fact_131_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 ) )
=> ( ! [X6: A,Xs2: coindu1593790203_llist @ A] :
( ( member @ A @ X @ ( coindu1112613586e_lset @ A @ Xs2 ) )
=> ( ( X != X6 )
=> ( ( P @ Xs2 )
=> ( P @ ( coindu1121789889_LCons @ A @ X6 @ Xs2 ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% lset_induct
thf(fact_132_lset__cases,axiom,
! [A: $tType,X: A,Xs: coindu1593790203_llist @ A] :
( ( member @ A @ X @ ( coindu1112613586e_lset @ A @ Xs ) )
=> ( ! [Xs6: coindu1593790203_llist @ A] :
( Xs
!= ( coindu1121789889_LCons @ A @ X @ Xs6 ) )
=> ~ ! [X6: A,Xs6: coindu1593790203_llist @ A] :
( ( Xs
= ( coindu1121789889_LCons @ A @ X6 @ Xs6 ) )
=> ~ ( member @ A @ X @ ( coindu1112613586e_lset @ A @ Xs6 ) ) ) ) ) ).
% lset_cases
thf(fact_133_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_134_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_135_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_136_lset__intros_I2_J,axiom,
! [A: $tType,X: A,Xs: coindu1593790203_llist @ A,X5: A] :
( ( member @ A @ X @ ( coindu1112613586e_lset @ A @ Xs ) )
=> ( member @ A @ X @ ( coindu1112613586e_lset @ A @ ( coindu1121789889_LCons @ A @ X5 @ Xs ) ) ) ) ).
% lset_intros(2)
thf(fact_137_llist_Orel__cong,axiom,
! [A: $tType,B: $tType,X: coindu1593790203_llist @ A,Ya: coindu1593790203_llist @ A,Y: coindu1593790203_llist @ B,Xa: coindu1593790203_llist @ B,R: A > B > $o,Ra: A > B > $o] :
( ( X = Ya )
=> ( ( Y = Xa )
=> ( ! [Z3: A,Yb: B] :
( ( member @ A @ Z3 @ ( coindu1112613586e_lset @ A @ Ya ) )
=> ( ( member @ B @ Yb @ ( coindu1112613586e_lset @ B @ Xa ) )
=> ( ( R @ Z3 @ Yb )
= ( Ra @ Z3 @ Yb ) ) ) )
=> ( ( coindu987187967t_all2 @ A @ B @ R @ X @ Y )
= ( coindu987187967t_all2 @ A @ B @ Ra @ Ya @ Xa ) ) ) ) ) ).
% llist.rel_cong
thf(fact_138_llist_Orel__mono__strong,axiom,
! [A: $tType,B: $tType,R: A > B > $o,X: coindu1593790203_llist @ A,Y: coindu1593790203_llist @ B,Ra: A > B > $o] :
( ( coindu987187967t_all2 @ A @ B @ R @ X @ Y )
=> ( ! [Z3: A,Yb: B] :
( ( member @ A @ Z3 @ ( coindu1112613586e_lset @ A @ X ) )
=> ( ( member @ B @ Yb @ ( coindu1112613586e_lset @ B @ Y ) )
=> ( ( R @ Z3 @ Yb )
=> ( Ra @ Z3 @ Yb ) ) ) )
=> ( coindu987187967t_all2 @ A @ B @ Ra @ X @ Y ) ) ) ).
% llist.rel_mono_strong
thf(fact_139_llist_Orel__refl__strong,axiom,
! [A: $tType,X: coindu1593790203_llist @ A,Ra: A > A > $o] :
( ! [Z3: A] :
( ( member @ A @ Z3 @ ( coindu1112613586e_lset @ A @ X ) )
=> ( Ra @ Z3 @ Z3 ) )
=> ( coindu987187967t_all2 @ A @ A @ Ra @ X @ X ) ) ).
% llist.rel_refl_strong
thf(fact_140_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_141_lhd__LCons,axiom,
! [A: $tType,X21: A,X22: coindu1593790203_llist @ A] :
( ( coindu1046438764le_lhd @ A @ ( coindu1121789889_LCons @ A @ X21 @ X22 ) )
= X21 ) ).
% lhd_LCons
thf(fact_142_llist_Opred__cong,axiom,
! [A: $tType,X: coindu1593790203_llist @ A,Ya: coindu1593790203_llist @ A,P: A > $o,Pa: A > $o] :
( ( X = Ya )
=> ( ! [Z3: A] :
( ( member @ A @ Z3 @ ( coindu1112613586e_lset @ A @ Ya ) )
=> ( ( P @ Z3 )
= ( Pa @ Z3 ) ) )
=> ( ( coindu44415597_llist @ A @ P @ X )
= ( coindu44415597_llist @ A @ Pa @ Ya ) ) ) ) ).
% llist.pred_cong
thf(fact_143_llist_Opred__mono__strong,axiom,
! [A: $tType,P: A > $o,X: coindu1593790203_llist @ A,Pa: A > $o] :
( ( coindu44415597_llist @ A @ P @ X )
=> ( ! [Z3: A] :
( ( member @ A @ Z3 @ ( coindu1112613586e_lset @ A @ X ) )
=> ( ( P @ Z3 )
=> ( Pa @ Z3 ) ) )
=> ( coindu44415597_llist @ A @ Pa @ X ) ) ) ).
% llist.pred_mono_strong
thf(fact_144_Coinductive__List__Mirabelle__kmikjhschf_Olset__into__lsetp,axiom,
! [A: $tType,X: A,Xs: coindu1593790203_llist @ A] :
( ( member @ A @ X @ ( coindu1112613586e_lset @ A @ Xs ) )
=> ( coindu1055736764_lsetp @ A @ Xs @ X ) ) ).
% Coinductive_List_Mirabelle_kmikjhschf.lset_into_lsetp
thf(fact_145_lsetp__into__lset,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,X: A] :
( ( coindu1055736764_lsetp @ A @ Xs @ X )
=> ( member @ A @ X @ ( coindu1112613586e_lset @ A @ Xs ) ) ) ).
% lsetp_into_lset
thf(fact_146_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_147_llist_Ocorec__sel_I1_J,axiom,
! [A: $tType,C: $tType,P2: C > $o,A2: C,G21: C > A,Q222: C > $o,G2212: C > ( coindu1593790203_llist @ A ),G2223: C > C] :
( ~ ( P2 @ A2 )
=> ( ( coindu1046438764le_lhd @ A @ ( coindu1244876290_llist @ C @ A @ P2 @ G21 @ Q222 @ G2212 @ G2223 @ A2 ) )
= ( G21 @ A2 ) ) ) ).
% llist.corec_sel(1)
thf(fact_148_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_149_llist_Oset__sel_I2_J,axiom,
! [A: $tType,A2: coindu1593790203_llist @ A,X: A] :
( ~ ( coindu335574135_lnull @ A @ A2 )
=> ( ( member @ A @ X @ ( coindu1112613586e_lset @ A @ ( coindu1047225960le_ltl @ A @ A2 ) ) )
=> ( member @ A @ X @ ( coindu1112613586e_lset @ A @ A2 ) ) ) ) ).
% llist.set_sel(2)
thf(fact_150_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_151_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_152_llist_Oexpand,axiom,
! [A: $tType,Llist: coindu1593790203_llist @ A,Llist5: coindu1593790203_llist @ A] :
( ( ( coindu335574135_lnull @ A @ Llist )
= ( coindu335574135_lnull @ A @ Llist5 ) )
=> ( ( ~ ( coindu335574135_lnull @ A @ Llist )
=> ( ~ ( coindu335574135_lnull @ A @ Llist5 )
=> ( ( ( coindu1046438764le_lhd @ A @ Llist )
= ( coindu1046438764le_lhd @ A @ Llist5 ) )
& ( ( coindu1047225960le_ltl @ A @ Llist )
= ( coindu1047225960le_ltl @ A @ Llist5 ) ) ) ) )
=> ( Llist = Llist5 ) ) ) ).
% llist.expand
thf(fact_153_llist_Ocoinduct,axiom,
! [A: $tType,R: ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) > $o,Llist: coindu1593790203_llist @ A,Llist5: coindu1593790203_llist @ A] :
( ( R @ Llist @ Llist5 )
=> ( ! [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 = Llist5 ) ) ) ).
% llist.coinduct
thf(fact_154_llist_Ocoinduct__strong,axiom,
! [A: $tType,R: ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) > $o,Llist: coindu1593790203_llist @ A,Llist5: coindu1593790203_llist @ A] :
( ( R @ Llist @ Llist5 )
=> ( ! [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 = Llist5 ) ) ) ).
% llist.coinduct_strong
thf(fact_155_split__llist__first,axiom,
! [A: $tType,X: A,Xs: coindu1593790203_llist @ A] :
( ( member @ A @ X @ ( coindu1112613586e_lset @ A @ Xs ) )
=> ? [Ys3: coindu1593790203_llist @ A,Zs2: coindu1593790203_llist @ A] :
( ( Xs
= ( coindu268472904append @ A @ Ys3 @ ( coindu1121789889_LCons @ A @ X @ Zs2 ) ) )
& ( coindu1213758845finite @ A @ Ys3 )
& ~ ( member @ A @ X @ ( coindu1112613586e_lset @ A @ Ys3 ) ) ) ) ).
% split_llist_first
thf(fact_156_llist_Osplit__sel,axiom,
! [B: $tType,A: $tType,P: B > $o,F1: B,F22: A > ( coindu1593790203_llist @ A ) > B,Llist: coindu1593790203_llist @ A] :
( ( P @ ( coindu882539134_llist @ B @ A @ F1 @ F22 @ Llist ) )
= ( ( ( Llist
= ( coindu1598213697e_LNil @ A ) )
=> ( P @ F1 ) )
& ( ( Llist
= ( coindu1121789889_LCons @ A @ ( coindu1046438764le_lhd @ A @ Llist ) @ ( coindu1047225960le_ltl @ A @ Llist ) ) )
=> ( P @ ( F22 @ ( coindu1046438764le_lhd @ A @ Llist ) @ ( coindu1047225960le_ltl @ A @ Llist ) ) ) ) ) ) ).
% llist.split_sel
thf(fact_157_llist_Osplit__sel__asm,axiom,
! [B: $tType,A: $tType,P: B > $o,F1: B,F22: A > ( coindu1593790203_llist @ A ) > B,Llist: coindu1593790203_llist @ A] :
( ( P @ ( coindu882539134_llist @ B @ A @ F1 @ F22 @ Llist ) )
= ( ~ ( ( ( Llist
= ( coindu1598213697e_LNil @ A ) )
& ~ ( P @ F1 ) )
| ( ( Llist
= ( coindu1121789889_LCons @ A @ ( coindu1046438764le_lhd @ A @ Llist ) @ ( coindu1047225960le_ltl @ A @ Llist ) ) )
& ~ ( P @ ( F22 @ ( coindu1046438764le_lhd @ A @ Llist ) @ ( coindu1047225960le_ltl @ A @ Llist ) ) ) ) ) ) ) ).
% llist.split_sel_asm
thf(fact_158_llist_Ocase__eq__if,axiom,
! [A: $tType,B: $tType] :
( ( coindu882539134_llist @ B @ A )
= ( ^ [F12: B,F23: A > ( coindu1593790203_llist @ A ) > B,Llist2: coindu1593790203_llist @ A] : ( if @ B @ ( coindu335574135_lnull @ A @ Llist2 ) @ F12 @ ( F23 @ ( coindu1046438764le_lhd @ A @ Llist2 ) @ ( coindu1047225960le_ltl @ A @ Llist2 ) ) ) ) ) ).
% llist.case_eq_if
thf(fact_159_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_160_llist_Osimps_I5_J,axiom,
! [B: $tType,A: $tType,F1: B,F22: A > ( coindu1593790203_llist @ A ) > B,X21: A,X22: coindu1593790203_llist @ A] :
( ( coindu882539134_llist @ B @ A @ F1 @ F22 @ ( coindu1121789889_LCons @ A @ X21 @ X22 ) )
= ( F22 @ X21 @ X22 ) ) ).
% llist.simps(5)
thf(fact_161_llist_Osimps_I4_J,axiom,
! [A: $tType,B: $tType,F1: B,F22: A > ( coindu1593790203_llist @ A ) > B] :
( ( coindu882539134_llist @ B @ A @ F1 @ F22 @ ( coindu1598213697e_LNil @ A ) )
= F1 ) ).
% llist.simps(4)
thf(fact_162_gen__lset__def,axiom,
! [A: $tType] :
( ( coindu1928975208n_lset @ A )
= ( ^ [A5: set @ A,Xs3: coindu1593790203_llist @ A] : ( sup_sup @ ( set @ A ) @ A5 @ ( coindu1112613586e_lset @ A @ Xs3 ) ) ) ) ).
% gen_lset_def
thf(fact_163_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_164_Un__iff,axiom,
! [A: $tType,C2: A,A4: set @ A,B4: set @ A] :
( ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A4 @ B4 ) )
= ( ( member @ A @ C2 @ A4 )
| ( member @ A @ C2 @ B4 ) ) ) ).
% Un_iff
thf(fact_165_UnCI,axiom,
! [A: $tType,C2: A,B4: set @ A,A4: set @ A] :
( ( ~ ( member @ A @ C2 @ B4 )
=> ( member @ A @ C2 @ A4 ) )
=> ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A4 @ B4 ) ) ) ).
% UnCI
thf(fact_166_sup_Oright__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ A2 @ B2 ) @ B2 )
= ( sup_sup @ A @ A2 @ B2 ) ) ) ).
% sup.right_idem
thf(fact_167_sup__apply,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_sup @ B @ ( type2 @ B ) )
=> ( ( sup_sup @ ( A > B ) )
= ( ^ [F2: A > B,G2: A > B,X3: A] : ( sup_sup @ B @ ( F2 @ X3 ) @ ( G2 @ X3 ) ) ) ) ) ).
% sup_apply
thf(fact_168_sup_Oidem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( sup_sup @ A @ A2 @ A2 )
= A2 ) ) ).
% sup.idem
thf(fact_169_sup__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [X: A] :
( ( sup_sup @ A @ X @ X )
= X ) ) ).
% sup_idem
thf(fact_170_sup_Oleft__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( sup_sup @ A @ A2 @ ( sup_sup @ A @ A2 @ B2 ) )
= ( sup_sup @ A @ A2 @ B2 ) ) ) ).
% sup.left_idem
thf(fact_171_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_172_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_173_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_174_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_175_inf__sup__aci_I5_J,axiom,
! [A: $tType] :
( ( lattice @ A @ ( type2 @ A ) )
=> ( ( sup_sup @ A )
= ( ^ [X3: A,Y3: A] : ( sup_sup @ A @ Y3 @ X3 ) ) ) ) ).
% inf_sup_aci(5)
thf(fact_176_sup__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_sup @ B @ ( type2 @ B ) )
=> ( ( sup_sup @ ( A > B ) )
= ( ^ [F2: A > B,G2: A > B,X3: A] : ( sup_sup @ B @ ( F2 @ X3 ) @ ( G2 @ X3 ) ) ) ) ) ).
% sup_fun_def
thf(fact_177_sup_Oassoc,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C2: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ A2 @ B2 ) @ C2 )
= ( sup_sup @ A @ A2 @ ( sup_sup @ A @ B2 @ C2 ) ) ) ) ).
% sup.assoc
thf(fact_178_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_179_sup_Ocommute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ( ( sup_sup @ A )
= ( ^ [A3: A,B3: A] : ( sup_sup @ A @ B3 @ A3 ) ) ) ) ).
% sup.commute
thf(fact_180_sup__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ( ( sup_sup @ A )
= ( ^ [X3: A,Y3: A] : ( sup_sup @ A @ Y3 @ X3 ) ) ) ) ).
% sup_commute
thf(fact_181_sup_Oleft__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A,C2: A] :
( ( sup_sup @ A @ B2 @ ( sup_sup @ A @ A2 @ C2 ) )
= ( sup_sup @ A @ A2 @ ( sup_sup @ A @ B2 @ C2 ) ) ) ) ).
% sup.left_commute
thf(fact_182_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_183_UnE,axiom,
! [A: $tType,C2: A,A4: set @ A,B4: set @ A] :
( ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A4 @ B4 ) )
=> ( ~ ( member @ A @ C2 @ A4 )
=> ( member @ A @ C2 @ B4 ) ) ) ).
% UnE
thf(fact_184_UnI1,axiom,
! [A: $tType,C2: A,A4: set @ A,B4: set @ A] :
( ( member @ A @ C2 @ A4 )
=> ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A4 @ B4 ) ) ) ).
% UnI1
thf(fact_185_UnI2,axiom,
! [A: $tType,C2: A,B4: set @ A,A4: set @ A] :
( ( member @ A @ C2 @ B4 )
=> ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A4 @ B4 ) ) ) ).
% UnI2
thf(fact_186_bex__Un,axiom,
! [A: $tType,A4: set @ A,B4: set @ A,P: A > $o] :
( ( ? [X3: A] :
( ( member @ A @ X3 @ ( sup_sup @ ( set @ A ) @ A4 @ B4 ) )
& ( P @ X3 ) ) )
= ( ? [X3: A] :
( ( member @ A @ X3 @ A4 )
& ( P @ X3 ) )
| ? [X3: A] :
( ( member @ A @ X3 @ B4 )
& ( P @ X3 ) ) ) ) ).
% bex_Un
thf(fact_187_ball__Un,axiom,
! [A: $tType,A4: set @ A,B4: set @ A,P: A > $o] :
( ( ! [X3: A] :
( ( member @ A @ X3 @ ( sup_sup @ ( set @ A ) @ A4 @ B4 ) )
=> ( P @ X3 ) ) )
= ( ! [X3: A] :
( ( member @ A @ X3 @ A4 )
=> ( P @ X3 ) )
& ! [X3: A] :
( ( member @ A @ X3 @ B4 )
=> ( P @ X3 ) ) ) ) ).
% ball_Un
thf(fact_188_Un__assoc,axiom,
! [A: $tType,A4: set @ A,B4: set @ A,C3: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A4 @ B4 ) @ C3 )
= ( sup_sup @ ( set @ A ) @ A4 @ ( sup_sup @ ( set @ A ) @ B4 @ C3 ) ) ) ).
% Un_assoc
thf(fact_189_Un__absorb,axiom,
! [A: $tType,A4: set @ A] :
( ( sup_sup @ ( set @ A ) @ A4 @ A4 )
= A4 ) ).
% Un_absorb
thf(fact_190_Un__commute,axiom,
! [A: $tType] :
( ( sup_sup @ ( set @ A ) )
= ( ^ [A5: set @ A,B5: set @ A] : ( sup_sup @ ( set @ A ) @ B5 @ A5 ) ) ) ).
% Un_commute
thf(fact_191_Un__left__absorb,axiom,
! [A: $tType,A4: set @ A,B4: set @ A] :
( ( sup_sup @ ( set @ A ) @ A4 @ ( sup_sup @ ( set @ A ) @ A4 @ B4 ) )
= ( sup_sup @ ( set @ A ) @ A4 @ B4 ) ) ).
% Un_left_absorb
thf(fact_192_Un__left__commute,axiom,
! [A: $tType,A4: set @ A,B4: set @ A,C3: set @ A] :
( ( sup_sup @ ( set @ A ) @ A4 @ ( sup_sup @ ( set @ A ) @ B4 @ C3 ) )
= ( sup_sup @ ( set @ A ) @ B4 @ ( sup_sup @ ( set @ A ) @ A4 @ C3 ) ) ) ).
% Un_left_commute
thf(fact_193_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_194_lprefixI,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A,X4: 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 ) @ X4 )
=> ( ! [Xs2: coindu1593790203_llist @ A,Ys3: coindu1593790203_llist @ A] :
( ( member @ ( product_prod @ ( coindu1593790203_llist @ A ) @ ( coindu1593790203_llist @ A ) ) @ ( product_Pair @ ( coindu1593790203_llist @ A ) @ ( coindu1593790203_llist @ A ) @ Xs2 @ Ys3 ) @ X4 )
=> ( ( coindu335574135_lnull @ A @ Xs2 )
| ? [X8: A,Xs7: coindu1593790203_llist @ A,Ys6: coindu1593790203_llist @ A] :
( ( Xs2
= ( coindu1121789889_LCons @ A @ X8 @ Xs7 ) )
& ( Ys3
= ( coindu1121789889_LCons @ A @ X8 @ Ys6 ) )
& ( ( member @ ( product_prod @ ( coindu1593790203_llist @ A ) @ ( coindu1593790203_llist @ A ) ) @ ( product_Pair @ ( coindu1593790203_llist @ A ) @ ( coindu1593790203_llist @ A ) @ Xs7 @ Ys6 ) @ X4 )
| ( coindu1696667936prefix @ A @ Xs7 @ Ys6 ) ) ) ) )
=> ( coindu1696667936prefix @ A @ Xs @ Ys ) ) ) ).
% lprefixI
thf(fact_195_lset__eq__empty,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( ( ( coindu1112613586e_lset @ A @ Xs )
= ( bot_bot @ ( set @ A ) ) )
= ( coindu335574135_lnull @ A @ Xs ) ) ).
% lset_eq_empty
thf(fact_196_empty__iff,axiom,
! [A: $tType,C2: A] :
~ ( member @ A @ C2 @ ( bot_bot @ ( set @ A ) ) ) ).
% empty_iff
thf(fact_197_all__not__in__conv,axiom,
! [A: $tType,A4: set @ A] :
( ( ! [X3: A] :
~ ( member @ A @ X3 @ A4 ) )
= ( A4
= ( bot_bot @ ( set @ A ) ) ) ) ).
% all_not_in_conv
thf(fact_198_Collect__empty__eq,axiom,
! [A: $tType,P: A > $o] :
( ( ( collect @ A @ P )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X3: A] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_199_empty__Collect__eq,axiom,
! [A: $tType,P: A > $o] :
( ( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ P ) )
= ( ! [X3: A] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_200_subsetI,axiom,
! [A: $tType,A4: set @ A,B4: set @ A] :
( ! [X2: A] :
( ( member @ A @ X2 @ A4 )
=> ( member @ A @ X2 @ B4 ) )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ B4 ) ) ).
% subsetI
thf(fact_201_subset__antisym,axiom,
! [A: $tType,A4: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ A4 )
=> ( A4 = B4 ) ) ) ).
% subset_antisym
thf(fact_202_sup_Obounded__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [B2: A,C2: A,A2: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ B2 @ C2 ) @ A2 )
= ( ( ord_less_eq @ A @ B2 @ A2 )
& ( ord_less_eq @ A @ C2 @ A2 ) ) ) ) ).
% sup.bounded_iff
thf(fact_203_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_204_subset__empty,axiom,
! [A: $tType,A4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ ( bot_bot @ ( set @ A ) ) )
= ( A4
= ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_empty
thf(fact_205_empty__subsetI,axiom,
! [A: $tType,A4: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A4 ) ).
% empty_subsetI
thf(fact_206_sup__bot_Oright__neutral,axiom,
! [A: $tType] :
( ( bounde1808546759up_bot @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( sup_sup @ A @ A2 @ ( bot_bot @ A ) )
= A2 ) ) ).
% sup_bot.right_neutral
thf(fact_207_sup__bot_Oleft__neutral,axiom,
! [A: $tType] :
( ( bounde1808546759up_bot @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( sup_sup @ A @ ( bot_bot @ A ) @ A2 )
= A2 ) ) ).
% sup_bot.left_neutral
thf(fact_208_sup__eq__bot__iff,axiom,
! [A: $tType] :
( ( bounded_lattice_bot @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ( sup_sup @ A @ X @ Y )
= ( bot_bot @ A ) )
= ( ( X
= ( bot_bot @ A ) )
& ( Y
= ( bot_bot @ A ) ) ) ) ) ).
% sup_eq_bot_iff
thf(fact_209_bot__eq__sup__iff,axiom,
! [A: $tType] :
( ( bounded_lattice_bot @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ( bot_bot @ A )
= ( sup_sup @ A @ X @ Y ) )
= ( ( X
= ( bot_bot @ A ) )
& ( Y
= ( bot_bot @ A ) ) ) ) ) ).
% bot_eq_sup_iff
thf(fact_210_Un__empty,axiom,
! [A: $tType,A4: set @ A,B4: set @ A] :
( ( ( sup_sup @ ( set @ A ) @ A4 @ B4 )
= ( bot_bot @ ( set @ A ) ) )
= ( ( A4
= ( bot_bot @ ( set @ A ) ) )
& ( B4
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Un_empty
thf(fact_211_Un__subset__iff,axiom,
! [A: $tType,A4: set @ A,B4: set @ A,C3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A4 @ B4 ) @ C3 )
= ( ( ord_less_eq @ ( set @ A ) @ A4 @ C3 )
& ( ord_less_eq @ ( set @ A ) @ B4 @ C3 ) ) ) ).
% Un_subset_iff
thf(fact_212_sup__bot__right,axiom,
! [A: $tType] :
( ( bounded_lattice_bot @ A @ ( type2 @ A ) )
=> ! [X: A] :
( ( sup_sup @ A @ X @ ( bot_bot @ A ) )
= X ) ) ).
% sup_bot_right
thf(fact_213_sup__bot__left,axiom,
! [A: $tType] :
( ( bounded_lattice_bot @ A @ ( type2 @ A ) )
=> ! [X: A] :
( ( sup_sup @ A @ ( bot_bot @ A ) @ X )
= X ) ) ).
% sup_bot_left
thf(fact_214_sup_OcoboundedI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [C2: A,B2: A,A2: A] :
( ( ord_less_eq @ A @ C2 @ B2 )
=> ( ord_less_eq @ A @ C2 @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% sup.coboundedI2
thf(fact_215_sup_OcoboundedI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ C2 @ A2 )
=> ( ord_less_eq @ A @ C2 @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% sup.coboundedI1
thf(fact_216_sup_Oabsorb__iff2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ( ( ord_less_eq @ A )
= ( ^ [A3: A,B3: A] :
( ( sup_sup @ A @ A3 @ B3 )
= B3 ) ) ) ) ).
% sup.absorb_iff2
thf(fact_217_sup_Oabsorb__iff1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ( ( ord_less_eq @ A )
= ( ^ [B3: A,A3: A] :
( ( sup_sup @ A @ A3 @ B3 )
= A3 ) ) ) ) ).
% sup.absorb_iff1
thf(fact_218_sup_Ocobounded2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A] : ( ord_less_eq @ A @ B2 @ ( sup_sup @ A @ A2 @ B2 ) ) ) ).
% sup.cobounded2
thf(fact_219_sup_Ocobounded1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] : ( ord_less_eq @ A @ A2 @ ( sup_sup @ A @ A2 @ B2 ) ) ) ).
% sup.cobounded1
thf(fact_220_sup_Oorder__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ( ( ord_less_eq @ A )
= ( ^ [B3: A,A3: A] :
( A3
= ( sup_sup @ A @ A3 @ B3 ) ) ) ) ) ).
% sup.order_iff
thf(fact_221_sup_OboundedI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A,C2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ C2 @ A2 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ B2 @ C2 ) @ A2 ) ) ) ) ).
% sup.boundedI
thf(fact_222_sup_OboundedE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [B2: A,C2: A,A2: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ B2 @ C2 ) @ A2 )
=> ~ ( ( ord_less_eq @ A @ B2 @ A2 )
=> ~ ( ord_less_eq @ A @ C2 @ A2 ) ) ) ) ).
% sup.boundedE
thf(fact_223_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_224_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_225_sup_Oabsorb2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( sup_sup @ A @ A2 @ B2 )
= B2 ) ) ) ).
% sup.absorb2
thf(fact_226_sup_Oabsorb1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( sup_sup @ A @ A2 @ B2 )
= A2 ) ) ) ).
% sup.absorb1
thf(fact_227_sup__unique,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [F: A > A > A,X: A,Y: A] :
( ! [X2: A,Y2: A] : ( ord_less_eq @ A @ X2 @ ( F @ X2 @ Y2 ) )
=> ( ! [X2: A,Y2: A] : ( ord_less_eq @ A @ Y2 @ ( F @ X2 @ Y2 ) )
=> ( ! [X2: A,Y2: A,Z3: A] :
( ( ord_less_eq @ A @ Y2 @ X2 )
=> ( ( ord_less_eq @ A @ Z3 @ X2 )
=> ( ord_less_eq @ A @ ( F @ Y2 @ Z3 ) @ X2 ) ) )
=> ( ( sup_sup @ A @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ) ).
% sup_unique
thf(fact_228_sup_OorderI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( A2
= ( sup_sup @ A @ A2 @ B2 ) )
=> ( ord_less_eq @ A @ B2 @ A2 ) ) ) ).
% sup.orderI
thf(fact_229_sup_OorderE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( A2
= ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% sup.orderE
thf(fact_230_le__iff__sup,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ( ( ord_less_eq @ A )
= ( ^ [X3: A,Y3: A] :
( ( sup_sup @ A @ X3 @ Y3 )
= Y3 ) ) ) ) ).
% le_iff_sup
thf(fact_231_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_232_sup__mono,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [A2: A,C2: A,B2: A,D: A] :
( ( ord_less_eq @ A @ A2 @ C2 )
=> ( ( ord_less_eq @ A @ B2 @ D )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ A2 @ B2 ) @ ( sup_sup @ A @ C2 @ D ) ) ) ) ) ).
% sup_mono
thf(fact_233_sup_Omono,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [C2: A,A2: A,D: A,B2: A] :
( ( ord_less_eq @ A @ C2 @ A2 )
=> ( ( ord_less_eq @ A @ D @ B2 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ C2 @ D ) @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ) ).
% sup.mono
thf(fact_234_le__supI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [X: A,B2: A,A2: A] :
( ( ord_less_eq @ A @ X @ B2 )
=> ( ord_less_eq @ A @ X @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% le_supI2
thf(fact_235_le__supI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [X: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ X @ A2 )
=> ( ord_less_eq @ A @ X @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% le_supI1
thf(fact_236_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_237_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_238_le__supI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [A2: A,X: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ X )
=> ( ( ord_less_eq @ A @ B2 @ X )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ A2 @ B2 ) @ X ) ) ) ) ).
% le_supI
thf(fact_239_le__supE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,X: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ A2 @ B2 ) @ X )
=> ~ ( ( ord_less_eq @ A @ A2 @ X )
=> ~ ( ord_less_eq @ A @ B2 @ X ) ) ) ) ).
% le_supE
thf(fact_240_Un__empty__right,axiom,
! [A: $tType,A4: set @ A] :
( ( sup_sup @ ( set @ A ) @ A4 @ ( bot_bot @ ( set @ A ) ) )
= A4 ) ).
% Un_empty_right
thf(fact_241_Un__empty__left,axiom,
! [A: $tType,B4: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ B4 )
= B4 ) ).
% Un_empty_left
thf(fact_242_subset__Un__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A5: set @ A,B5: set @ A] :
( ( sup_sup @ ( set @ A ) @ A5 @ B5 )
= B5 ) ) ) ).
% subset_Un_eq
thf(fact_243_Un__absorb2,axiom,
! [A: $tType,B4: set @ A,A4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B4 @ A4 )
=> ( ( sup_sup @ ( set @ A ) @ A4 @ B4 )
= A4 ) ) ).
% Un_absorb2
thf(fact_244_Un__absorb1,axiom,
! [A: $tType,A4: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
=> ( ( sup_sup @ ( set @ A ) @ A4 @ B4 )
= B4 ) ) ).
% Un_absorb1
thf(fact_245_Un__upper2,axiom,
! [A: $tType,B4: set @ A,A4: set @ A] : ( ord_less_eq @ ( set @ A ) @ B4 @ ( sup_sup @ ( set @ A ) @ A4 @ B4 ) ) ).
% Un_upper2
thf(fact_246_Un__upper1,axiom,
! [A: $tType,A4: set @ A,B4: set @ A] : ( ord_less_eq @ ( set @ A ) @ A4 @ ( sup_sup @ ( set @ A ) @ A4 @ B4 ) ) ).
% Un_upper1
thf(fact_247_Un__least,axiom,
! [A: $tType,A4: set @ A,C3: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ C3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ C3 )
=> ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A4 @ B4 ) @ C3 ) ) ) ).
% Un_least
thf(fact_248_Un__mono,axiom,
! [A: $tType,A4: set @ A,C3: set @ A,B4: set @ A,D2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ C3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ D2 )
=> ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A4 @ B4 ) @ ( sup_sup @ ( set @ A ) @ C3 @ D2 ) ) ) ) ).
% Un_mono
thf(fact_249_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_250_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_251_emptyE,axiom,
! [A: $tType,A2: A] :
~ ( member @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ).
% emptyE
thf(fact_252_set__mp,axiom,
! [A: $tType,A4: set @ A,B4: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
=> ( ( member @ A @ X @ A4 )
=> ( member @ A @ X @ B4 ) ) ) ).
% set_mp
thf(fact_253_in__mono,axiom,
! [A: $tType,A4: set @ A,B4: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
=> ( ( member @ A @ X @ A4 )
=> ( member @ A @ X @ B4 ) ) ) ).
% in_mono
%----Type constructors (15)
thf(tcon_HOL_Obool___Lattices_Obounded__lattice,axiom,
bounded_lattice @ $o @ ( type2 @ $o ) ).
thf(tcon_Set_Oset___Lattices_Obounded__lattice_1,axiom,
! [A6: $tType] : ( bounded_lattice @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
thf(tcon_fun___Lattices_Obounded__lattice_2,axiom,
! [A6: $tType,A7: $tType] :
( ( bounded_lattice @ A7 @ ( type2 @ A7 ) )
=> ( bounded_lattice @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
thf(tcon_fun___Lattices_Obounded__semilattice__sup__bot,axiom,
! [A6: $tType,A7: $tType] :
( ( bounded_lattice @ A7 @ ( type2 @ A7 ) )
=> ( bounde1808546759up_bot @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
thf(tcon_fun___Lattices_Obounded__lattice__bot,axiom,
! [A6: $tType,A7: $tType] :
( ( bounded_lattice @ A7 @ ( type2 @ A7 ) )
=> ( bounded_lattice_bot @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
thf(tcon_fun___Lattices_Osemilattice__sup,axiom,
! [A6: $tType,A7: $tType] :
( ( semilattice_sup @ A7 @ ( type2 @ A7 ) )
=> ( semilattice_sup @ ( 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_Obounded__semilattice__sup__bot_3,axiom,
! [A6: $tType] : ( bounde1808546759up_bot @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
thf(tcon_Set_Oset___Lattices_Obounded__lattice__bot_4,axiom,
! [A6: $tType] : ( bounded_lattice_bot @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
thf(tcon_Set_Oset___Lattices_Osemilattice__sup_5,axiom,
! [A6: $tType] : ( semilattice_sup @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
thf(tcon_Set_Oset___Lattices_Olattice_6,axiom,
! [A6: $tType] : ( lattice @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
thf(tcon_HOL_Obool___Lattices_Obounded__semilattice__sup__bot_7,axiom,
bounde1808546759up_bot @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Lattices_Obounded__lattice__bot_8,axiom,
bounded_lattice_bot @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_9,axiom,
semilattice_sup @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Lattices_Olattice_10,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,
comple1396247847notone @ ( coindu1593790203_llist @ a ) @ ( coindu1593790203_llist @ a ) @ ( coindu1696667936prefix @ a ) @ ( coindu1696667936prefix @ a ) @ ( coindu1121789889_LCons @ a @ x ) ).
%------------------------------------------------------------------------------