TPTP Problem File: DAT195^1.p
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%------------------------------------------------------------------------------
% File : DAT195^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Data Structures
% Problem : Lazy list mirror 104
% 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 : lmirror__104.p [Bla16]
% Status : Theorem
% Rating : 1.00 v7.1.0
% Syntax : Number of formulae : 306 ( 110 unt; 35 typ; 0 def)
% Number of atoms : 740 ( 229 equ; 0 cnn)
% Maximal formula atoms : 13 ( 2 avg)
% Number of connectives : 3696 ( 90 ~; 28 |; 59 &;3165 @)
% ( 0 <=>; 354 =>; 0 <=; 0 <~>)
% Maximal formula depth : 22 ( 8 avg)
% Number of types : 2 ( 1 usr)
% Number of type conns : 162 ( 162 >; 0 *; 0 +; 0 <<)
% Number of symbols : 35 ( 34 usr; 2 con; 0-4 aty)
% Number of variables : 1024 ( 47 ^; 905 !; 43 ?;1024 :)
% ( 29 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 14:41:34.406
%------------------------------------------------------------------------------
%----Could-be-implicit typings (4)
thf(ty_t_Coinductive__List_Ollist,type,
coinductive_llist: $tType > $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_t_itself,type,
itself: $tType > $tType ).
thf(ty_tf_a,type,
a: $tType ).
%----Explicit typings (31)
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Oorder,type,
order:
!>[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_Orderings_Opreorder,type,
preorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Lattices_Osemilattice__sup,type,
semilattice_sup:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_c_Coinductive__List_Ofinite__lprefix,type,
coindu328551480prefix:
!>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List_Ogen__lset,type,
coinductive_gen_lset:
!>[A: $tType] : ( ( set @ A ) > ( coinductive_llist @ A ) > ( set @ A ) ) ).
thf(sy_c_Coinductive__List_Olappend,type,
coinductive_lappend:
!>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
thf(sy_c_Coinductive__List_Oldistinct,type,
coindu351974385stinct:
!>[A: $tType] : ( ( coinductive_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List_Olfilter,type,
coinductive_lfilter:
!>[A: $tType] : ( ( A > $o ) > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
thf(sy_c_Coinductive__List_Olfinite,type,
coinductive_lfinite:
!>[A: $tType] : ( ( coinductive_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List_Ollast,type,
coinductive_llast:
!>[A: $tType] : ( ( coinductive_llist @ A ) > A ) ).
thf(sy_c_Coinductive__List_Ollexord,type,
coinductive_llexord:
!>[A: $tType] : ( ( A > A > $o ) > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List_Ollist_OLCons,type,
coinductive_LCons:
!>[A: $tType] : ( A > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
thf(sy_c_Coinductive__List_Ollist_OLNil,type,
coinductive_LNil:
!>[A: $tType] : ( coinductive_llist @ A ) ).
thf(sy_c_Coinductive__List_Ollist_Olnull,type,
coinductive_lnull:
!>[A: $tType] : ( ( coinductive_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List_Ollist_Olset,type,
coinductive_lset:
!>[A: $tType] : ( ( coinductive_llist @ A ) > ( set @ A ) ) ).
thf(sy_c_Coinductive__List_Olmember,type,
coinductive_lmember:
!>[A: $tType] : ( A > ( coinductive_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List_Olstrict__prefix,type,
coindu1478340336prefix:
!>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o ) ).
thf(sy_c_HOL_Oundefined,type,
undefined:
!>[A: $tType] : A ).
thf(sy_c_LMirror__Mirabelle__wyovfcktfy_Olmirror,type,
lMirro427583474mirror:
!>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
thf(sy_c_LMirror__Mirabelle__wyovfcktfy_Olmirror__aux,type,
lMirro999291890or_aux:
!>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > ( coinductive_llist @ 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_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_acc,type,
acc: coinductive_llist @ a ).
thf(sy_v_xs,type,
xs: coinductive_llist @ a ).
%----Relevant facts (254)
thf(fact_0_True,axiom,
coinductive_lfinite @ a @ xs ).
% True
thf(fact_1_lmirror__aux__acc,axiom,
! [A: $tType,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A,Xs: coinductive_llist @ A] :
( ( lMirro999291890or_aux @ A @ ( coinductive_lappend @ A @ Ys @ Zs ) @ Xs )
= ( coinductive_lappend @ A @ ( lMirro999291890or_aux @ A @ Ys @ Xs ) @ Zs ) ) ).
% lmirror_aux_acc
thf(fact_2_lfinite__lmirror__aux,axiom,
! [A: $tType,Acc: coinductive_llist @ A,Xs: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ ( lMirro999291890or_aux @ A @ Acc @ Xs ) )
= ( ( coinductive_lfinite @ A @ Xs )
& ( coinductive_lfinite @ A @ Acc ) ) ) ).
% lfinite_lmirror_aux
thf(fact_3_lmirror__aux__inf,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Acc: coinductive_llist @ A] :
( ~ ( coinductive_lfinite @ A @ Xs )
=> ( ( lMirro999291890or_aux @ A @ Acc @ Xs )
= Xs ) ) ).
% lmirror_aux_inf
thf(fact_4_in__lset__lappend__iff,axiom,
! [A: $tType,X: A,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( member @ A @ X @ ( coinductive_lset @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) ) )
= ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
| ( ( coinductive_lfinite @ A @ Xs )
& ( member @ A @ X @ ( coinductive_lset @ A @ Ys ) ) ) ) ) ).
% in_lset_lappend_iff
thf(fact_5_lappend__assoc,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A] :
( ( coinductive_lappend @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) @ Zs )
= ( coinductive_lappend @ A @ Xs @ ( coinductive_lappend @ A @ Ys @ Zs ) ) ) ).
% lappend_assoc
thf(fact_6_lfinite__lappend,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
= ( ( coinductive_lfinite @ A @ Xs )
& ( coinductive_lfinite @ A @ Ys ) ) ) ).
% lfinite_lappend
thf(fact_7_lappend__inf,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ~ ( coinductive_lfinite @ A @ Xs )
=> ( ( coinductive_lappend @ A @ Xs @ Ys )
= Xs ) ) ).
% lappend_inf
thf(fact_8_lmirror__aux__simps_I1_J,axiom,
! [A: $tType,Acc: coinductive_llist @ A] :
( ( lMirro999291890or_aux @ A @ Acc @ ( coinductive_LNil @ A ) )
= Acc ) ).
% lmirror_aux_simps(1)
thf(fact_9_lmirror__aux__simps_I2_J,axiom,
! [A: $tType,Acc: coinductive_llist @ A,Xa: A,X: coinductive_llist @ A] :
( ( lMirro999291890or_aux @ A @ Acc @ ( coinductive_LCons @ A @ Xa @ X ) )
= ( coinductive_LCons @ A @ Xa @ ( lMirro999291890or_aux @ A @ ( coinductive_LCons @ A @ Xa @ Acc ) @ X ) ) ) ).
% lmirror_aux_simps(2)
thf(fact_10_lmirror__aux_Odisc__iff_I2_J,axiom,
! [A: $tType,Acc: coinductive_llist @ A,Xs: coinductive_llist @ A] :
( ( ~ ( coinductive_lnull @ A @ ( lMirro999291890or_aux @ A @ Acc @ Xs ) ) )
= ( ~ ( coinductive_lnull @ A @ Xs )
| ~ ( coinductive_lnull @ A @ Acc ) ) ) ).
% lmirror_aux.disc_iff(2)
thf(fact_11_lnull__lmirror__aux,axiom,
! [A: $tType,Acc: coinductive_llist @ A,Xs: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ ( lMirro999291890or_aux @ A @ Acc @ Xs ) )
= ( ( coinductive_lnull @ A @ Xs )
& ( coinductive_lnull @ A @ Acc ) ) ) ).
% lnull_lmirror_aux
thf(fact_12_lset__lappend1,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] : ( ord_less_eq @ ( set @ A ) @ ( coinductive_lset @ A @ Xs ) @ ( coinductive_lset @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) ) ) ).
% lset_lappend1
thf(fact_13_lset__lmember,axiom,
! [A: $tType,X: A,Xs: coinductive_llist @ A] :
( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
= ( coinductive_lmember @ A @ X @ Xs ) ) ).
% lset_lmember
thf(fact_14_llist_Oinject,axiom,
! [A: $tType,X21: A,X22: coinductive_llist @ A,Y21: A,Y22: coinductive_llist @ A] :
( ( ( coinductive_LCons @ A @ X21 @ X22 )
= ( coinductive_LCons @ A @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% llist.inject
thf(fact_15_lappend__code_I2_J,axiom,
! [A: $tType,Xa: A,X: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_lappend @ A @ ( coinductive_LCons @ A @ Xa @ X ) @ Ys )
= ( coinductive_LCons @ A @ Xa @ ( coinductive_lappend @ A @ X @ Ys ) ) ) ).
% lappend_code(2)
thf(fact_16_lfinite__code_I2_J,axiom,
! [B: $tType,X: B,Xs: coinductive_llist @ B] :
( ( coinductive_lfinite @ B @ ( coinductive_LCons @ B @ X @ Xs ) )
= ( coinductive_lfinite @ B @ Xs ) ) ).
% lfinite_code(2)
thf(fact_17_lfinite__LCons,axiom,
! [A: $tType,X: A,Xs: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ ( coinductive_LCons @ A @ X @ Xs ) )
= ( coinductive_lfinite @ A @ Xs ) ) ).
% lfinite_LCons
thf(fact_18_lnull__lappend,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
= ( ( coinductive_lnull @ A @ Xs )
& ( coinductive_lnull @ A @ Ys ) ) ) ).
% lnull_lappend
thf(fact_19_lappend_Odisc__iff_I2_J,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( ~ ( coinductive_lnull @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) ) )
= ( ~ ( coinductive_lnull @ A @ Xs )
| ~ ( coinductive_lnull @ A @ Ys ) ) ) ).
% lappend.disc_iff(2)
thf(fact_20_lappend__LNil2,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( coinductive_lappend @ A @ Xs @ ( coinductive_LNil @ A ) )
= Xs ) ).
% lappend_LNil2
thf(fact_21_lappend__code_I1_J,axiom,
! [A: $tType,Ys: coinductive_llist @ A] :
( ( coinductive_lappend @ A @ ( coinductive_LNil @ A ) @ Ys )
= Ys ) ).
% lappend_code(1)
thf(fact_22_lfinite__code_I1_J,axiom,
! [A: $tType] : ( coinductive_lfinite @ A @ ( coinductive_LNil @ A ) ) ).
% lfinite_code(1)
thf(fact_23_llist_Odisc_I2_J,axiom,
! [A: $tType,X21: A,X22: coinductive_llist @ A] :
~ ( coinductive_lnull @ A @ ( coinductive_LCons @ A @ X21 @ X22 ) ) ).
% llist.disc(2)
thf(fact_24_llist_Odisc_I1_J,axiom,
! [A: $tType] : ( coinductive_lnull @ A @ ( coinductive_LNil @ A ) ) ).
% llist.disc(1)
thf(fact_25_lappend_Octr_I1_J,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ Xs )
=> ( ( coinductive_lnull @ A @ Ys )
=> ( ( coinductive_lappend @ A @ Xs @ Ys )
= ( coinductive_LNil @ A ) ) ) ) ).
% lappend.ctr(1)
thf(fact_26_llist_OdiscI_I2_J,axiom,
! [A: $tType,Llist: coinductive_llist @ A,X21: A,X22: coinductive_llist @ A] :
( ( Llist
= ( coinductive_LCons @ A @ X21 @ X22 ) )
=> ~ ( coinductive_lnull @ A @ Llist ) ) ).
% llist.discI(2)
thf(fact_27_llist_OdiscI_I1_J,axiom,
! [A: $tType,Llist: coinductive_llist @ A] :
( ( Llist
= ( coinductive_LNil @ A ) )
=> ( coinductive_lnull @ A @ Llist ) ) ).
% llist.discI(1)
thf(fact_28_lmember__code_I2_J,axiom,
! [A: $tType,X: A,Y: A,Ys: coinductive_llist @ A] :
( ( coinductive_lmember @ A @ X @ ( coinductive_LCons @ A @ Y @ Ys ) )
= ( ( X = Y )
| ( coinductive_lmember @ A @ X @ Ys ) ) ) ).
% lmember_code(2)
thf(fact_29_lmember__code_I1_J,axiom,
! [A: $tType,X: A] :
~ ( coinductive_lmember @ A @ X @ ( coinductive_LNil @ A ) ) ).
% lmember_code(1)
thf(fact_30_llist_Ocollapse_I1_J,axiom,
! [A: $tType,Llist: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ Llist )
=> ( Llist
= ( coinductive_LNil @ A ) ) ) ).
% llist.collapse(1)
thf(fact_31_llist_Odistinct_I1_J,axiom,
! [A: $tType,X21: A,X22: coinductive_llist @ A] :
( ( coinductive_LNil @ A )
!= ( coinductive_LCons @ A @ X21 @ X22 ) ) ).
% llist.distinct(1)
thf(fact_32_lzip_Oexhaust,axiom,
! [A: $tType,B: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B] :
( ~ ( ( coinductive_lnull @ A @ Xs )
| ( coinductive_lnull @ B @ Ys ) )
=> ~ ( ~ ( coinductive_lnull @ A @ Xs )
=> ( coinductive_lnull @ B @ Ys ) ) ) ).
% lzip.exhaust
thf(fact_33_lfinite_Ocases,axiom,
! [A: $tType,A2: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ A2 )
=> ( ( A2
!= ( coinductive_LNil @ A ) )
=> ~ ! [Xs2: coinductive_llist @ A] :
( ? [X2: A] :
( A2
= ( coinductive_LCons @ A @ X2 @ Xs2 ) )
=> ~ ( coinductive_lfinite @ A @ Xs2 ) ) ) ) ).
% lfinite.cases
thf(fact_34_lfinite_Osimps,axiom,
! [A: $tType] :
( ( coinductive_lfinite @ A )
= ( ^ [A3: coinductive_llist @ A] :
( ( A3
= ( coinductive_LNil @ A ) )
| ? [Xs3: coinductive_llist @ A,X3: A] :
( ( A3
= ( coinductive_LCons @ A @ X3 @ Xs3 ) )
& ( coinductive_lfinite @ A @ Xs3 ) ) ) ) ) ).
% lfinite.simps
thf(fact_35_llist_Oexhaust,axiom,
! [A: $tType,Y: coinductive_llist @ A] :
( ( Y
!= ( coinductive_LNil @ A ) )
=> ~ ! [X212: A,X222: coinductive_llist @ A] :
( Y
!= ( coinductive_LCons @ A @ X212 @ X222 ) ) ) ).
% llist.exhaust
thf(fact_36_neq__LNil__conv,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( Xs
!= ( coinductive_LNil @ A ) )
= ( ? [X3: A,Xs4: coinductive_llist @ A] :
( Xs
= ( coinductive_LCons @ A @ X3 @ Xs4 ) ) ) ) ).
% neq_LNil_conv
thf(fact_37_not__lnull__conv,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( ~ ( coinductive_lnull @ A @ Xs ) )
= ( ? [X3: A,Xs4: coinductive_llist @ A] :
( Xs
= ( coinductive_LCons @ A @ X3 @ Xs4 ) ) ) ) ).
% not_lnull_conv
thf(fact_38_lappend_Oexhaust,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( ( coinductive_lnull @ A @ Xs )
=> ~ ( coinductive_lnull @ A @ Ys ) )
=> ( ~ ( coinductive_lnull @ A @ Xs )
| ~ ( coinductive_lnull @ A @ Ys ) ) ) ).
% lappend.exhaust
thf(fact_39_lfinite_Oinducts,axiom,
! [A: $tType,X: coinductive_llist @ A,P: ( coinductive_llist @ A ) > $o] :
( ( coinductive_lfinite @ A @ X )
=> ( ( P @ ( coinductive_LNil @ A ) )
=> ( ! [Xs2: coinductive_llist @ A,X2: A] :
( ( coinductive_lfinite @ A @ Xs2 )
=> ( ( P @ Xs2 )
=> ( P @ ( coinductive_LCons @ A @ X2 @ Xs2 ) ) ) )
=> ( P @ X ) ) ) ) ).
% lfinite.inducts
thf(fact_40_lnull__def,axiom,
! [A: $tType] :
( ( coinductive_lnull @ A )
= ( ^ [Llist2: coinductive_llist @ A] :
( Llist2
= ( coinductive_LNil @ A ) ) ) ) ).
% lnull_def
thf(fact_41_wlog__linorder__le,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > A > $o,B2: A,A2: A] :
( ! [A4: A,B3: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ( ( P @ B2 @ A2 )
=> ( P @ A2 @ B2 ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% wlog_linorder_le
thf(fact_42_lnull__imp__lfinite,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ Xs )
=> ( coinductive_lfinite @ A @ Xs ) ) ).
% lnull_imp_lfinite
thf(fact_43_lfinite__rev__induct,axiom,
! [A: $tType,Xs: coinductive_llist @ A,P: ( coinductive_llist @ A ) > $o] :
( ( coinductive_lfinite @ A @ Xs )
=> ( ( P @ ( coinductive_LNil @ A ) )
=> ( ! [X2: A,Xs2: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ Xs2 )
=> ( ( P @ Xs2 )
=> ( P @ ( coinductive_lappend @ A @ Xs2 @ ( coinductive_LCons @ A @ X2 @ ( coinductive_LNil @ A ) ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% lfinite_rev_induct
thf(fact_44_mem__Collect__eq,axiom,
! [A: $tType,A2: A,P: A > $o] :
( ( member @ A @ A2 @ ( collect @ A @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_45_Collect__mem__eq,axiom,
! [A: $tType,A5: set @ A] :
( ( collect @ A
@ ^ [X3: A] : ( member @ A @ X3 @ A5 ) )
= A5 ) ).
% Collect_mem_eq
thf(fact_46_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_47_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X2: A] :
( ( F @ X2 )
= ( G @ X2 ) )
=> ( F = G ) ) ).
% ext
thf(fact_48_lfinite__LNil,axiom,
! [A: $tType] : ( coinductive_lfinite @ A @ ( coinductive_LNil @ A ) ) ).
% lfinite_LNil
thf(fact_49_lfinite__LConsI,axiom,
! [A: $tType,Xs: coinductive_llist @ A,X: A] :
( ( coinductive_lfinite @ A @ Xs )
=> ( coinductive_lfinite @ A @ ( coinductive_LCons @ A @ X @ Xs ) ) ) ).
% lfinite_LConsI
thf(fact_50_lappend__snocL1__conv__LCons2,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Y: A,Ys: coinductive_llist @ A] :
( ( coinductive_lappend @ A @ ( coinductive_lappend @ A @ Xs @ ( coinductive_LCons @ A @ Y @ ( coinductive_LNil @ A ) ) ) @ Ys )
= ( coinductive_lappend @ A @ Xs @ ( coinductive_LCons @ A @ Y @ Ys ) ) ) ).
% lappend_snocL1_conv_LCons2
thf(fact_51_lmirror__aux_Oexhaust,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Acc: coinductive_llist @ A] :
( ( ( coinductive_lnull @ A @ Xs )
=> ~ ( coinductive_lnull @ A @ Acc ) )
=> ( ~ ( coinductive_lnull @ A @ Xs )
| ~ ( coinductive_lnull @ A @ Acc ) ) ) ).
% lmirror_aux.exhaust
thf(fact_52_lmirror__aux_Octr_I1_J,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Acc: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ Xs )
=> ( ( coinductive_lnull @ A @ Acc )
=> ( ( lMirro999291890or_aux @ A @ Acc @ Xs )
= ( coinductive_LNil @ A ) ) ) ) ).
% lmirror_aux.ctr(1)
thf(fact_53_llist_Oset__induct,axiom,
! [A: $tType,X: A,A2: coinductive_llist @ A,P: A > ( coinductive_llist @ A ) > $o] :
( ( member @ A @ X @ ( coinductive_lset @ A @ A2 ) )
=> ( ! [Z1: A,Z2: coinductive_llist @ A] : ( P @ Z1 @ ( coinductive_LCons @ A @ Z1 @ Z2 ) )
=> ( ! [Z1: A,Z2: coinductive_llist @ A,Xa2: A] :
( ( member @ A @ Xa2 @ ( coinductive_lset @ A @ Z2 ) )
=> ( ( P @ Xa2 @ Z2 )
=> ( P @ Xa2 @ ( coinductive_LCons @ A @ Z1 @ Z2 ) ) ) )
=> ( P @ X @ A2 ) ) ) ) ).
% llist.set_induct
thf(fact_54_llist_Oset__cases,axiom,
! [A: $tType,E: A,A2: coinductive_llist @ A] :
( ( member @ A @ E @ ( coinductive_lset @ A @ A2 ) )
=> ( ! [Z2: coinductive_llist @ A] :
( A2
!= ( coinductive_LCons @ A @ E @ Z2 ) )
=> ~ ! [Z1: A,Z2: coinductive_llist @ A] :
( ( A2
= ( coinductive_LCons @ A @ Z1 @ Z2 ) )
=> ~ ( member @ A @ E @ ( coinductive_lset @ A @ Z2 ) ) ) ) ) ).
% llist.set_cases
thf(fact_55_lset__induct_H,axiom,
! [A: $tType,X: A,Xs: coinductive_llist @ A,P: ( coinductive_llist @ A ) > $o] :
( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
=> ( ! [Xs2: coinductive_llist @ A] : ( P @ ( coinductive_LCons @ A @ X @ Xs2 ) )
=> ( ! [X4: A,Xs2: coinductive_llist @ A] :
( ( member @ A @ X @ ( coinductive_lset @ A @ Xs2 ) )
=> ( ( P @ Xs2 )
=> ( P @ ( coinductive_LCons @ A @ X4 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% lset_induct'
thf(fact_56_lset__induct,axiom,
! [A: $tType,X: A,Xs: coinductive_llist @ A,P: ( coinductive_llist @ A ) > $o] :
( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
=> ( ! [Xs2: coinductive_llist @ A] : ( P @ ( coinductive_LCons @ A @ X @ Xs2 ) )
=> ( ! [X4: A,Xs2: coinductive_llist @ A] :
( ( member @ A @ X @ ( coinductive_lset @ A @ Xs2 ) )
=> ( ( X != X4 )
=> ( ( P @ Xs2 )
=> ( P @ ( coinductive_LCons @ A @ X4 @ Xs2 ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% lset_induct
thf(fact_57_lset__cases,axiom,
! [A: $tType,X: A,Xs: coinductive_llist @ A] :
( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
=> ( ! [Xs5: coinductive_llist @ A] :
( Xs
!= ( coinductive_LCons @ A @ X @ Xs5 ) )
=> ~ ! [X4: A,Xs5: coinductive_llist @ A] :
( ( Xs
= ( coinductive_LCons @ A @ X4 @ Xs5 ) )
=> ~ ( member @ A @ X @ ( coinductive_lset @ A @ Xs5 ) ) ) ) ) ).
% lset_cases
thf(fact_58_llist_Oset__intros_I1_J,axiom,
! [A: $tType,A1: A,A22: coinductive_llist @ A] : ( member @ A @ A1 @ ( coinductive_lset @ A @ ( coinductive_LCons @ A @ A1 @ A22 ) ) ) ).
% llist.set_intros(1)
thf(fact_59_llist_Oset__intros_I2_J,axiom,
! [A: $tType,X: A,A22: coinductive_llist @ A,A1: A] :
( ( member @ A @ X @ ( coinductive_lset @ A @ A22 ) )
=> ( member @ A @ X @ ( coinductive_lset @ A @ ( coinductive_LCons @ A @ A1 @ A22 ) ) ) ) ).
% llist.set_intros(2)
thf(fact_60_lset__intros_I1_J,axiom,
! [A: $tType,X: A,Xs: coinductive_llist @ A] : ( member @ A @ X @ ( coinductive_lset @ A @ ( coinductive_LCons @ A @ X @ Xs ) ) ) ).
% lset_intros(1)
thf(fact_61_lset__intros_I2_J,axiom,
! [A: $tType,X: A,Xs: coinductive_llist @ A,X5: A] :
( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
=> ( member @ A @ X @ ( coinductive_lset @ A @ ( coinductive_LCons @ A @ X5 @ Xs ) ) ) ) ).
% lset_intros(2)
thf(fact_62_lappend__lnull2,axiom,
! [A: $tType,Ys: coinductive_llist @ A,Xs: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ Ys )
=> ( ( coinductive_lappend @ A @ Xs @ Ys )
= Xs ) ) ).
% lappend_lnull2
thf(fact_63_lappend__lnull1,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ Xs )
=> ( ( coinductive_lappend @ A @ Xs @ Ys )
= Ys ) ) ).
% lappend_lnull1
thf(fact_64_lappend_Odisc_I1_J,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ Xs )
=> ( ( coinductive_lnull @ A @ Ys )
=> ( coinductive_lnull @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) ) ) ) ).
% lappend.disc(1)
thf(fact_65_lappend_Odisc_I2_J,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( ~ ( coinductive_lnull @ A @ Xs )
| ~ ( coinductive_lnull @ A @ Ys ) )
=> ~ ( coinductive_lnull @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) ) ) ).
% lappend.disc(2)
thf(fact_66_lappend__eq__LNil__iff,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( ( coinductive_lappend @ A @ Xs @ Ys )
= ( coinductive_LNil @ A ) )
= ( ( Xs
= ( coinductive_LNil @ A ) )
& ( Ys
= ( coinductive_LNil @ A ) ) ) ) ).
% lappend_eq_LNil_iff
thf(fact_67_LNil__eq__lappend__iff,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( ( coinductive_LNil @ A )
= ( coinductive_lappend @ A @ Xs @ Ys ) )
= ( ( Xs
= ( coinductive_LNil @ A ) )
& ( Ys
= ( coinductive_LNil @ A ) ) ) ) ).
% LNil_eq_lappend_iff
thf(fact_68_lappend__LNil__LNil,axiom,
! [A: $tType] :
( ( coinductive_lappend @ A @ ( coinductive_LNil @ A ) @ ( coinductive_LNil @ A ) )
= ( coinductive_LNil @ A ) ) ).
% lappend_LNil_LNil
thf(fact_69_lmirror__aux__LCons,axiom,
! [A: $tType,Acc: coinductive_llist @ A,X: A,Xs: coinductive_llist @ A] :
( ( lMirro999291890or_aux @ A @ Acc @ ( coinductive_LCons @ A @ X @ Xs ) )
= ( coinductive_LCons @ A @ X @ ( coinductive_lappend @ A @ ( lMirro999291890or_aux @ A @ ( coinductive_LNil @ A ) @ Xs ) @ ( coinductive_LCons @ A @ X @ Acc ) ) ) ) ).
% lmirror_aux_LCons
thf(fact_70_split__llist__first,axiom,
! [A: $tType,X: A,Xs: coinductive_llist @ A] :
( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
=> ? [Ys2: coinductive_llist @ A,Zs2: coinductive_llist @ A] :
( ( Xs
= ( coinductive_lappend @ A @ Ys2 @ ( coinductive_LCons @ A @ X @ Zs2 ) ) )
& ( coinductive_lfinite @ A @ Ys2 )
& ~ ( member @ A @ X @ ( coinductive_lset @ A @ Ys2 ) ) ) ) ).
% split_llist_first
thf(fact_71_split__llist,axiom,
! [A: $tType,X: A,Xs: coinductive_llist @ A] :
( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
=> ? [Ys2: coinductive_llist @ A,Zs2: coinductive_llist @ A] :
( ( Xs
= ( coinductive_lappend @ A @ Ys2 @ ( coinductive_LCons @ A @ X @ Zs2 ) ) )
& ( coinductive_lfinite @ A @ Ys2 ) ) ) ).
% split_llist
thf(fact_72_lmirror__aux_Odisc_I1_J,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Acc: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ Xs )
=> ( ( coinductive_lnull @ A @ Acc )
=> ( coinductive_lnull @ A @ ( lMirro999291890or_aux @ A @ Acc @ Xs ) ) ) ) ).
% lmirror_aux.disc(1)
thf(fact_73_lmirror__aux_Odisc_I2_J,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Acc: coinductive_llist @ A] :
( ( ~ ( coinductive_lnull @ A @ Xs )
| ~ ( coinductive_lnull @ A @ Acc ) )
=> ~ ( coinductive_lnull @ A @ ( lMirro999291890or_aux @ A @ Acc @ Xs ) ) ) ).
% lmirror_aux.disc(2)
thf(fact_74_lmirror__LCons,axiom,
! [A: $tType,X: A,Xs: coinductive_llist @ A] :
( ( lMirro427583474mirror @ A @ ( coinductive_LCons @ A @ X @ Xs ) )
= ( coinductive_LCons @ A @ X @ ( coinductive_lappend @ A @ ( lMirro427583474mirror @ A @ Xs ) @ ( coinductive_LCons @ A @ X @ ( coinductive_LNil @ A ) ) ) ) ) ).
% lmirror_LCons
thf(fact_75_subsetI,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ! [X2: A] :
( ( member @ A @ X2 @ A5 )
=> ( member @ A @ X2 @ B4 ) )
=> ( ord_less_eq @ ( set @ A ) @ A5 @ B4 ) ) ).
% subsetI
thf(fact_76_subset__antisym,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ A5 )
=> ( A5 = B4 ) ) ) ).
% subset_antisym
thf(fact_77_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% order_refl
thf(fact_78_llast__lappend__LCons,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Y: A,Ys: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ Xs )
=> ( ( coinductive_llast @ A @ ( coinductive_lappend @ A @ Xs @ ( coinductive_LCons @ A @ Y @ Ys ) ) )
= ( coinductive_llast @ A @ ( coinductive_LCons @ A @ Y @ Ys ) ) ) ) ).
% llast_lappend_LCons
thf(fact_79_lstrict__prefix__lappend__conv,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coindu1478340336prefix @ A @ Xs @ ( coinductive_lappend @ A @ Xs @ Ys ) )
= ( ( coinductive_lfinite @ A @ Xs )
& ~ ( coinductive_lnull @ A @ Ys ) ) ) ).
% lstrict_prefix_lappend_conv
thf(fact_80_llimit__induct,axiom,
! [A: $tType,P: ( coinductive_llist @ A ) > $o,Xs: coinductive_llist @ A] :
( ( P @ ( coinductive_LNil @ A ) )
=> ( ! [X2: A,Xs2: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ Xs2 )
=> ( ( P @ Xs2 )
=> ( P @ ( coinductive_LCons @ A @ X2 @ Xs2 ) ) ) )
=> ( ( ! [Ys3: coinductive_llist @ A] :
( ( coindu1478340336prefix @ A @ Ys3 @ Xs )
=> ( P @ Ys3 ) )
=> ( P @ Xs ) )
=> ( P @ Xs ) ) ) ) ).
% llimit_induct
thf(fact_81_Coinductive__List_Ofinite__lprefix__nitpick__simps_I3_J,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Y: A,Ys: coinductive_llist @ A] :
( ( coindu328551480prefix @ A @ Xs @ ( coinductive_LCons @ A @ Y @ Ys ) )
= ( ( Xs
= ( coinductive_LNil @ A ) )
| ? [Xs4: coinductive_llist @ A] :
( ( Xs
= ( coinductive_LCons @ A @ Y @ Xs4 ) )
& ( coindu328551480prefix @ A @ Xs4 @ Ys ) ) ) ) ).
% Coinductive_List.finite_lprefix_nitpick_simps(3)
thf(fact_82_lstrict__prefix__code_I3_J,axiom,
! [B: $tType,X: B,Xs: coinductive_llist @ B] :
~ ( coindu1478340336prefix @ B @ ( coinductive_LCons @ B @ X @ Xs ) @ ( coinductive_LNil @ B ) ) ).
% lstrict_prefix_code(3)
thf(fact_83_lstrict__prefix__code_I2_J,axiom,
! [B: $tType,Y: B,Ys: coinductive_llist @ B] : ( coindu1478340336prefix @ B @ ( coinductive_LNil @ B ) @ ( coinductive_LCons @ B @ Y @ Ys ) ) ).
% lstrict_prefix_code(2)
thf(fact_84_lmirror__def,axiom,
! [A: $tType] :
( ( lMirro427583474mirror @ A )
= ( lMirro999291890or_aux @ A @ ( coinductive_LNil @ A ) ) ) ).
% lmirror_def
thf(fact_85_llast__LCons2,axiom,
! [A: $tType,X: A,Y: A,Xs: coinductive_llist @ A] :
( ( coinductive_llast @ A @ ( coinductive_LCons @ A @ X @ ( coinductive_LCons @ A @ Y @ Xs ) ) )
= ( coinductive_llast @ A @ ( coinductive_LCons @ A @ Y @ Xs ) ) ) ).
% llast_LCons2
thf(fact_86_lstrict__prefix__code_I4_J,axiom,
! [B: $tType,X: B,Xs: coinductive_llist @ B,Y: B,Ys: coinductive_llist @ B] :
( ( coindu1478340336prefix @ B @ ( coinductive_LCons @ B @ X @ Xs ) @ ( coinductive_LCons @ B @ Y @ Ys ) )
= ( ( X = Y )
& ( coindu1478340336prefix @ B @ Xs @ Ys ) ) ) ).
% lstrict_prefix_code(4)
thf(fact_87_lstrict__prefix__code_I1_J,axiom,
! [A: $tType] :
~ ( coindu1478340336prefix @ A @ ( coinductive_LNil @ A ) @ ( coinductive_LNil @ A ) ) ).
% lstrict_prefix_code(1)
thf(fact_88_lnull__lmirror,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ ( lMirro427583474mirror @ A @ Xs ) )
= ( coinductive_lnull @ A @ Xs ) ) ).
% lnull_lmirror
thf(fact_89_lmirror__LNil,axiom,
! [A: $tType] :
( ( lMirro427583474mirror @ A @ ( coinductive_LNil @ A ) )
= ( coinductive_LNil @ A ) ) ).
% lmirror_LNil
thf(fact_90_llast__singleton,axiom,
! [A: $tType,X: A] :
( ( coinductive_llast @ A @ ( coinductive_LCons @ A @ X @ ( coinductive_LNil @ A ) ) )
= X ) ).
% llast_singleton
thf(fact_91_llist__less__induct,axiom,
! [A: $tType,P: ( coinductive_llist @ A ) > $o,Xs: coinductive_llist @ A] :
( ! [Xs2: coinductive_llist @ A] :
( ! [Ys3: coinductive_llist @ A] :
( ( coindu1478340336prefix @ A @ Ys3 @ Xs2 )
=> ( P @ Ys3 ) )
=> ( P @ Xs2 ) )
=> ( P @ Xs ) ) ).
% llist_less_induct
thf(fact_92_lstrict__prefix__lfinite1,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coindu1478340336prefix @ A @ Xs @ Ys )
=> ( coinductive_lfinite @ A @ Xs ) ) ).
% lstrict_prefix_lfinite1
thf(fact_93_Coinductive__List_Ofinite__lprefix__nitpick__simps_I1_J,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( coindu328551480prefix @ A @ Xs @ ( coinductive_LNil @ A ) )
= ( Xs
= ( coinductive_LNil @ A ) ) ) ).
% Coinductive_List.finite_lprefix_nitpick_simps(1)
thf(fact_94_Coinductive__List_Ofinite__lprefix__nitpick__simps_I2_J,axiom,
! [A: $tType,Xs: coinductive_llist @ A] : ( coindu328551480prefix @ A @ ( coinductive_LNil @ A ) @ Xs ) ).
% Coinductive_List.finite_lprefix_nitpick_simps(2)
thf(fact_95_llast__LCons,axiom,
! [A: $tType,Xs: coinductive_llist @ A,X: A] :
( ( ( coinductive_lnull @ A @ Xs )
=> ( ( coinductive_llast @ A @ ( coinductive_LCons @ A @ X @ Xs ) )
= X ) )
& ( ~ ( coinductive_lnull @ A @ Xs )
=> ( ( coinductive_llast @ A @ ( coinductive_LCons @ A @ X @ Xs ) )
= ( coinductive_llast @ A @ Xs ) ) ) ) ).
% llast_LCons
thf(fact_96_dual__order_Oantisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ) ).
% dual_order.antisym
thf(fact_97_dual__order_Otrans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A,C: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ C @ B2 )
=> ( ord_less_eq @ A @ C @ A2 ) ) ) ) ).
% dual_order.trans
thf(fact_98_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > A > $o,A2: A,B2: A] :
( ! [A4: A,B3: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: A,B3: A] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% linorder_wlog
thf(fact_99_dual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% dual_order.refl
thf(fact_100_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ Z )
=> ( ord_less_eq @ A @ X @ Z ) ) ) ) ).
% order_trans
thf(fact_101_order__class_Oorder_Oantisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ) ).
% order_class.order.antisym
thf(fact_102_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% ord_le_eq_trans
thf(fact_103_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( A2 = B2 )
=> ( ( ord_less_eq @ A @ B2 @ C )
=> ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% ord_eq_le_trans
thf(fact_104_antisym__conv,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( ord_less_eq @ A @ X @ Y )
= ( X = Y ) ) ) ) ).
% antisym_conv
thf(fact_105_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z: A] :
( ( ( ord_less_eq @ A @ X @ Y )
=> ~ ( ord_less_eq @ A @ Y @ Z ) )
=> ( ( ( ord_less_eq @ A @ Y @ X )
=> ~ ( ord_less_eq @ A @ X @ Z ) )
=> ( ( ( ord_less_eq @ A @ X @ Z )
=> ~ ( ord_less_eq @ A @ Z @ Y ) )
=> ( ( ( ord_less_eq @ A @ Z @ Y )
=> ~ ( ord_less_eq @ A @ Y @ X ) )
=> ( ( ( ord_less_eq @ A @ Y @ Z )
=> ~ ( ord_less_eq @ A @ Z @ X ) )
=> ~ ( ( ord_less_eq @ A @ Z @ X )
=> ~ ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_106_order_Otrans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ C )
=> ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% order.trans
thf(fact_107_le__cases,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ~ ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ A @ Y @ X ) ) ) ).
% le_cases
thf(fact_108_eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( X = Y )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ).
% eq_refl
thf(fact_109_linear,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
| ( ord_less_eq @ A @ Y @ X ) ) ) ).
% linear
thf(fact_110_antisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ X )
=> ( X = Y ) ) ) ) ).
% antisym
thf(fact_111_eq__iff,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ^ [Y2: A,Z3: A] : Y2 = Z3 )
= ( ^ [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
& ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ) ) ).
% eq_iff
thf(fact_112_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A2: A,B2: A,F: A > B,C: B] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X2: A,Y4: A] :
( ( ord_less_eq @ A @ X2 @ Y4 )
=> ( ord_less_eq @ B @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ B @ ( F @ A2 ) @ C ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_113_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A2: A,F: B > A,B2: B,C: B] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C )
=> ( ! [X2: B,Y4: B] :
( ( ord_less_eq @ B @ X2 @ Y4 )
=> ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_114_order__subst2,axiom,
! [A: $tType,C2: $tType] :
( ( ( order @ C2 @ ( type2 @ C2 ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,B2: A,F: A > C2,C: C2] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ C2 @ ( F @ B2 ) @ C )
=> ( ! [X2: A,Y4: A] :
( ( ord_less_eq @ A @ X2 @ Y4 )
=> ( ord_less_eq @ C2 @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% order_subst2
thf(fact_115_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B @ ( type2 @ B ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,F: B > A,B2: B,C: B] :
( ( ord_less_eq @ A @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C )
=> ( ! [X2: B,Y4: B] :
( ( ord_less_eq @ B @ X2 @ Y4 )
=> ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% order_subst1
thf(fact_116_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ( ( ord_less_eq @ ( A > B ) )
= ( ^ [F2: A > B,G2: A > B] :
! [X3: A] : ( ord_less_eq @ B @ ( F2 @ X3 ) @ ( G2 @ X3 ) ) ) ) ) ).
% le_fun_def
thf(fact_117_le__funI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [F: A > B,G: A > B] :
( ! [X2: A] : ( ord_less_eq @ B @ ( F @ X2 ) @ ( G @ X2 ) )
=> ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% le_funI
thf(fact_118_le__funE,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [F: A > B,G: A > B,X: A] :
( ( ord_less_eq @ ( A > B ) @ F @ G )
=> ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% le_funE
thf(fact_119_le__funD,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [F: A > B,G: A > B,X: A] :
( ( ord_less_eq @ ( A > B ) @ F @ G )
=> ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% le_funD
thf(fact_120_Collect__mono__iff,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
= ( ! [X3: A] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_121_contra__subsetD,axiom,
! [A: $tType,A5: set @ A,B4: set @ A,C: A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ( ~ ( member @ A @ C @ B4 )
=> ~ ( member @ A @ C @ A5 ) ) ) ).
% contra_subsetD
thf(fact_122_set__eq__subset,axiom,
! [A: $tType] :
( ( ^ [Y2: set @ A,Z3: set @ A] : Y2 = Z3 )
= ( ^ [A6: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B5 )
& ( ord_less_eq @ ( set @ A ) @ B5 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_123_subset__trans,axiom,
! [A: $tType,A5: set @ A,B4: set @ A,C3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ C3 )
=> ( ord_less_eq @ ( set @ A ) @ A5 @ C3 ) ) ) ).
% subset_trans
thf(fact_124_Collect__mono,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X2: A] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% Collect_mono
thf(fact_125_subset__refl,axiom,
! [A: $tType,A5: set @ A] : ( ord_less_eq @ ( set @ A ) @ A5 @ A5 ) ).
% subset_refl
thf(fact_126_rev__subsetD,axiom,
! [A: $tType,C: A,A5: set @ A,B4: set @ A] :
( ( member @ A @ C @ A5 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ( member @ A @ C @ B4 ) ) ) ).
% rev_subsetD
thf(fact_127_subset__iff,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A6: set @ A,B5: set @ A] :
! [T: A] :
( ( member @ A @ T @ A6 )
=> ( member @ A @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_128_set__rev__mp,axiom,
! [A: $tType,X: A,A5: set @ A,B4: set @ A] :
( ( member @ A @ X @ A5 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ( member @ A @ X @ B4 ) ) ) ).
% set_rev_mp
thf(fact_129_equalityD2,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( A5 = B4 )
=> ( ord_less_eq @ ( set @ A ) @ B4 @ A5 ) ) ).
% equalityD2
thf(fact_130_equalityD1,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( A5 = B4 )
=> ( ord_less_eq @ ( set @ A ) @ A5 @ B4 ) ) ).
% equalityD1
thf(fact_131_subset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A6: set @ A,B5: set @ A] :
! [X3: A] :
( ( member @ A @ X3 @ A6 )
=> ( member @ A @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_132_equalityE,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( A5 = B4 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ~ ( ord_less_eq @ ( set @ A ) @ B4 @ A5 ) ) ) ).
% equalityE
thf(fact_133_subsetCE,axiom,
! [A: $tType,A5: set @ A,B4: set @ A,C: A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ( ( member @ A @ C @ A5 )
=> ( member @ A @ C @ B4 ) ) ) ).
% subsetCE
thf(fact_134_subsetD,axiom,
! [A: $tType,A5: set @ A,B4: set @ A,C: A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ( ( member @ A @ C @ A5 )
=> ( member @ A @ C @ B4 ) ) ) ).
% subsetD
thf(fact_135_in__mono,axiom,
! [A: $tType,A5: set @ A,B4: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ( ( member @ A @ X @ A5 )
=> ( member @ A @ X @ B4 ) ) ) ).
% in_mono
thf(fact_136_set__mp,axiom,
! [A: $tType,A5: set @ A,B4: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ( ( member @ A @ X @ A5 )
=> ( member @ A @ X @ B4 ) ) ) ).
% set_mp
thf(fact_137_llast__lappend,axiom,
! [A: $tType,Ys: coinductive_llist @ A,Xs: coinductive_llist @ A] :
( ( ( coinductive_lnull @ A @ Ys )
=> ( ( coinductive_llast @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
= ( coinductive_llast @ A @ Xs ) ) )
& ( ~ ( coinductive_lnull @ A @ Ys )
=> ( ( ( coinductive_lfinite @ A @ Xs )
=> ( ( coinductive_llast @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
= ( coinductive_llast @ A @ Ys ) ) )
& ( ~ ( coinductive_lfinite @ A @ Xs )
=> ( ( coinductive_llast @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
= ( undefined @ A ) ) ) ) ) ) ).
% llast_lappend
thf(fact_138_lfilter__eq__LConsD,axiom,
! [A: $tType,P: A > $o,Ys: coinductive_llist @ A,X: A,Xs: coinductive_llist @ A] :
( ( ( coinductive_lfilter @ A @ P @ Ys )
= ( coinductive_LCons @ A @ X @ Xs ) )
=> ? [Us: coinductive_llist @ A,Vs: coinductive_llist @ A] :
( ( Ys
= ( coinductive_lappend @ A @ Us @ ( coinductive_LCons @ A @ X @ Vs ) ) )
& ( coinductive_lfinite @ A @ Us )
& ! [X6: A] :
( ( member @ A @ X6 @ ( coinductive_lset @ A @ Us ) )
=> ~ ( P @ X6 ) )
& ( P @ X )
& ( Xs
= ( coinductive_lfilter @ A @ P @ Vs ) ) ) ) ).
% lfilter_eq_LConsD
thf(fact_139_lset__lappend__lfinite,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ Xs )
=> ( ( coinductive_lset @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
= ( sup_sup @ ( set @ A ) @ ( coinductive_lset @ A @ Xs ) @ ( coinductive_lset @ A @ Ys ) ) ) ) ).
% lset_lappend_lfinite
thf(fact_140_ldistinct_Ocoinduct,axiom,
! [A: $tType,X7: ( coinductive_llist @ A ) > $o,X: coinductive_llist @ A] :
( ( X7 @ X )
=> ( ! [X2: coinductive_llist @ A] :
( ( X7 @ X2 )
=> ( ( X2
= ( coinductive_LNil @ A ) )
| ? [Xa3: A,Xs6: coinductive_llist @ A] :
( ( X2
= ( coinductive_LCons @ A @ Xa3 @ Xs6 ) )
& ~ ( member @ A @ Xa3 @ ( coinductive_lset @ A @ Xs6 ) )
& ( ( X7 @ Xs6 )
| ( coindu351974385stinct @ A @ Xs6 ) ) ) ) )
=> ( coindu351974385stinct @ A @ X ) ) ) ).
% ldistinct.coinduct
thf(fact_141_ldistinct_Osimps,axiom,
! [A: $tType] :
( ( coindu351974385stinct @ A )
= ( ^ [A3: coinductive_llist @ A] :
( ( A3
= ( coinductive_LNil @ A ) )
| ? [X3: A,Xs3: coinductive_llist @ A] :
( ( A3
= ( coinductive_LCons @ A @ X3 @ Xs3 ) )
& ~ ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs3 ) )
& ( coindu351974385stinct @ A @ Xs3 ) ) ) ) ) ).
% ldistinct.simps
thf(fact_142_ldistinct_Ocases,axiom,
! [A: $tType,A2: coinductive_llist @ A] :
( ( coindu351974385stinct @ A @ A2 )
=> ( ( A2
!= ( coinductive_LNil @ A ) )
=> ~ ! [X2: A,Xs2: coinductive_llist @ A] :
( ( A2
= ( coinductive_LCons @ A @ X2 @ Xs2 ) )
=> ( ~ ( member @ A @ X2 @ ( coinductive_lset @ A @ Xs2 ) )
=> ~ ( coindu351974385stinct @ A @ Xs2 ) ) ) ) ) ).
% ldistinct.cases
thf(fact_143_Un__iff,axiom,
! [A: $tType,C: A,A5: set @ A,B4: set @ A] :
( ( member @ A @ C @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) )
= ( ( member @ A @ C @ A5 )
| ( member @ A @ C @ B4 ) ) ) ).
% Un_iff
thf(fact_144_UnCI,axiom,
! [A: $tType,C: A,B4: set @ A,A5: set @ A] :
( ( ~ ( member @ A @ C @ B4 )
=> ( member @ A @ C @ A5 ) )
=> ( member @ A @ C @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) ) ) ).
% UnCI
thf(fact_145_lfilter__idem,axiom,
! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
( ( coinductive_lfilter @ A @ P @ ( coinductive_lfilter @ A @ P @ Xs ) )
= ( coinductive_lfilter @ A @ P @ Xs ) ) ).
% lfilter_idem
thf(fact_146_Un__subset__iff,axiom,
! [A: $tType,A5: set @ A,B4: set @ A,C3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) @ C3 )
= ( ( ord_less_eq @ ( set @ A ) @ A5 @ C3 )
& ( ord_less_eq @ ( set @ A ) @ B4 @ C3 ) ) ) ).
% Un_subset_iff
thf(fact_147_lfilter__LCons,axiom,
! [A: $tType,P: A > $o,X: A,Xs: coinductive_llist @ A] :
( ( ( P @ X )
=> ( ( coinductive_lfilter @ A @ P @ ( coinductive_LCons @ A @ X @ Xs ) )
= ( coinductive_LCons @ A @ X @ ( coinductive_lfilter @ A @ P @ Xs ) ) ) )
& ( ~ ( P @ X )
=> ( ( coinductive_lfilter @ A @ P @ ( coinductive_LCons @ A @ X @ Xs ) )
= ( coinductive_lfilter @ A @ P @ Xs ) ) ) ) ).
% lfilter_LCons
thf(fact_148_lfilter__LNil,axiom,
! [A: $tType,P: A > $o] :
( ( coinductive_lfilter @ A @ P @ ( coinductive_LNil @ A ) )
= ( coinductive_LNil @ A ) ) ).
% lfilter_LNil
thf(fact_149_ldistinct__LNil__code,axiom,
! [A: $tType] : ( coindu351974385stinct @ A @ ( coinductive_LNil @ A ) ) ).
% ldistinct_LNil_code
thf(fact_150_lnull__lfilter,axiom,
! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ ( coinductive_lfilter @ A @ P @ Xs ) )
= ( ! [X3: A] :
( ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs ) )
=> ~ ( P @ X3 ) ) ) ) ).
% lnull_lfilter
thf(fact_151_diverge__lfilter__LNil,axiom,
! [A: $tType,Xs: coinductive_llist @ A,P: A > $o] :
( ! [X2: A] :
( ( member @ A @ X2 @ ( coinductive_lset @ A @ Xs ) )
=> ~ ( P @ X2 ) )
=> ( ( coinductive_lfilter @ A @ P @ Xs )
= ( coinductive_LNil @ A ) ) ) ).
% diverge_lfilter_LNil
thf(fact_152_ldistinct__LCons,axiom,
! [A: $tType,X: A,Xs: coinductive_llist @ A] :
( ( coindu351974385stinct @ A @ ( coinductive_LCons @ A @ X @ Xs ) )
= ( ~ ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
& ( coindu351974385stinct @ A @ Xs ) ) ) ).
% ldistinct_LCons
thf(fact_153_lfilter__lappend__lfinite,axiom,
! [A: $tType,Xs: coinductive_llist @ A,P: A > $o,Ys: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ Xs )
=> ( ( coinductive_lfilter @ A @ P @ ( coinductive_lappend @ A @ Xs @ Ys ) )
= ( coinductive_lappend @ A @ ( coinductive_lfilter @ A @ P @ Xs ) @ ( coinductive_lfilter @ A @ P @ Ys ) ) ) ) ).
% lfilter_lappend_lfinite
thf(fact_154_Un__left__commute,axiom,
! [A: $tType,A5: set @ A,B4: set @ A,C3: set @ A] :
( ( sup_sup @ ( set @ A ) @ A5 @ ( sup_sup @ ( set @ A ) @ B4 @ C3 ) )
= ( sup_sup @ ( set @ A ) @ B4 @ ( sup_sup @ ( set @ A ) @ A5 @ C3 ) ) ) ).
% Un_left_commute
thf(fact_155_Un__left__absorb,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( sup_sup @ ( set @ A ) @ A5 @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) )
= ( sup_sup @ ( set @ A ) @ A5 @ B4 ) ) ).
% Un_left_absorb
thf(fact_156_Un__commute,axiom,
! [A: $tType] :
( ( sup_sup @ ( set @ A ) )
= ( ^ [A6: set @ A,B5: set @ A] : ( sup_sup @ ( set @ A ) @ B5 @ A6 ) ) ) ).
% Un_commute
thf(fact_157_Un__absorb,axiom,
! [A: $tType,A5: set @ A] :
( ( sup_sup @ ( set @ A ) @ A5 @ A5 )
= A5 ) ).
% Un_absorb
thf(fact_158_Un__assoc,axiom,
! [A: $tType,A5: set @ A,B4: set @ A,C3: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) @ C3 )
= ( sup_sup @ ( set @ A ) @ A5 @ ( sup_sup @ ( set @ A ) @ B4 @ C3 ) ) ) ).
% Un_assoc
thf(fact_159_ball__Un,axiom,
! [A: $tType,A5: set @ A,B4: set @ A,P: A > $o] :
( ( ! [X3: A] :
( ( member @ A @ X3 @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) )
=> ( P @ X3 ) ) )
= ( ! [X3: A] :
( ( member @ A @ X3 @ A5 )
=> ( P @ X3 ) )
& ! [X3: A] :
( ( member @ A @ X3 @ B4 )
=> ( P @ X3 ) ) ) ) ).
% ball_Un
thf(fact_160_bex__Un,axiom,
! [A: $tType,A5: set @ A,B4: set @ A,P: A > $o] :
( ( ? [X3: A] :
( ( member @ A @ X3 @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) )
& ( P @ X3 ) ) )
= ( ? [X3: A] :
( ( member @ A @ X3 @ A5 )
& ( P @ X3 ) )
| ? [X3: A] :
( ( member @ A @ X3 @ B4 )
& ( P @ X3 ) ) ) ) ).
% bex_Un
thf(fact_161_UnI2,axiom,
! [A: $tType,C: A,B4: set @ A,A5: set @ A] :
( ( member @ A @ C @ B4 )
=> ( member @ A @ C @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) ) ) ).
% UnI2
thf(fact_162_UnI1,axiom,
! [A: $tType,C: A,A5: set @ A,B4: set @ A] :
( ( member @ A @ C @ A5 )
=> ( member @ A @ C @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) ) ) ).
% UnI1
thf(fact_163_UnE,axiom,
! [A: $tType,C: A,A5: set @ A,B4: set @ A] :
( ( member @ A @ C @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) )
=> ( ~ ( member @ A @ C @ A5 )
=> ( member @ A @ C @ B4 ) ) ) ).
% UnE
thf(fact_164_ldistinct__lfilterI,axiom,
! [A: $tType,Xs: coinductive_llist @ A,P: A > $o] :
( ( coindu351974385stinct @ A @ Xs )
=> ( coindu351974385stinct @ A @ ( coinductive_lfilter @ A @ P @ Xs ) ) ) ).
% ldistinct_lfilterI
thf(fact_165_lfilter__LCons__seek,axiom,
! [A: $tType,P2: A > $o,X: A,L: coinductive_llist @ A] :
( ~ ( P2 @ X )
=> ( ( coinductive_lfilter @ A @ P2 @ ( coinductive_LCons @ A @ X @ L ) )
= ( coinductive_lfilter @ A @ P2 @ L ) ) ) ).
% lfilter_LCons_seek
thf(fact_166_lfilter__LCons__found,axiom,
! [A: $tType,P: A > $o,X: A,Xs: coinductive_llist @ A] :
( ( P @ X )
=> ( ( coinductive_lfilter @ A @ P @ ( coinductive_LCons @ A @ X @ Xs ) )
= ( coinductive_LCons @ A @ X @ ( coinductive_lfilter @ A @ P @ Xs ) ) ) ) ).
% lfilter_LCons_found
thf(fact_167_lfilter__id__conv,axiom,
! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
( ( ( coinductive_lfilter @ A @ P @ Xs )
= Xs )
= ( ! [X3: A] :
( ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs ) )
=> ( P @ X3 ) ) ) ) ).
% lfilter_id_conv
thf(fact_168_lfilter__cong,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,P: A > $o,Q: A > $o] :
( ( Xs = Ys )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ ( coinductive_lset @ A @ Ys ) )
=> ( ( P @ X2 )
= ( Q @ X2 ) ) )
=> ( ( coinductive_lfilter @ A @ P @ Xs )
= ( coinductive_lfilter @ A @ Q @ Ys ) ) ) ) ).
% lfilter_cong
thf(fact_169_Un__mono,axiom,
! [A: $tType,A5: set @ A,C3: set @ A,B4: set @ A,D: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ C3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ D )
=> ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) @ ( sup_sup @ ( set @ A ) @ C3 @ D ) ) ) ) ).
% Un_mono
thf(fact_170_Un__least,axiom,
! [A: $tType,A5: set @ A,C3: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ C3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ C3 )
=> ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) @ C3 ) ) ) ).
% Un_least
thf(fact_171_Un__upper1,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] : ( ord_less_eq @ ( set @ A ) @ A5 @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) ) ).
% Un_upper1
thf(fact_172_Un__upper2,axiom,
! [A: $tType,B4: set @ A,A5: set @ A] : ( ord_less_eq @ ( set @ A ) @ B4 @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) ) ).
% Un_upper2
thf(fact_173_Un__absorb1,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ( ( sup_sup @ ( set @ A ) @ A5 @ B4 )
= B4 ) ) ).
% Un_absorb1
thf(fact_174_Un__absorb2,axiom,
! [A: $tType,B4: set @ A,A5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B4 @ A5 )
=> ( ( sup_sup @ ( set @ A ) @ A5 @ B4 )
= A5 ) ) ).
% Un_absorb2
thf(fact_175_subset__Un__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A6: set @ A,B5: set @ A] :
( ( sup_sup @ ( set @ A ) @ A6 @ B5 )
= B5 ) ) ) ).
% subset_Un_eq
thf(fact_176_lfinite__lfilterI,axiom,
! [A: $tType,Xs: coinductive_llist @ A,P: A > $o] :
( ( coinductive_lfinite @ A @ Xs )
=> ( coinductive_lfinite @ A @ ( coinductive_lfilter @ A @ P @ Xs ) ) ) ).
% lfinite_lfilterI
thf(fact_177_ldistinct_OLNil,axiom,
! [A: $tType] : ( coindu351974385stinct @ A @ ( coinductive_LNil @ A ) ) ).
% ldistinct.LNil
thf(fact_178_lfilter__empty__conv,axiom,
! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
( ( ( coinductive_lfilter @ A @ P @ Xs )
= ( coinductive_LNil @ A ) )
= ( ! [X3: A] :
( ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs ) )
=> ~ ( P @ X3 ) ) ) ) ).
% lfilter_empty_conv
thf(fact_179_lfilter__eq__lappend__lfiniteD,axiom,
! [A: $tType,P: A > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A] :
( ( ( coinductive_lfilter @ A @ P @ Xs )
= ( coinductive_lappend @ A @ Ys @ Zs ) )
=> ( ( coinductive_lfinite @ A @ Ys )
=> ? [Us: coinductive_llist @ A,Vs: coinductive_llist @ A] :
( ( Xs
= ( coinductive_lappend @ A @ Us @ Vs ) )
& ( coinductive_lfinite @ A @ Us )
& ( Ys
= ( coinductive_lfilter @ A @ P @ Us ) )
& ( Zs
= ( coinductive_lfilter @ A @ P @ Vs ) ) ) ) ) ).
% lfilter_eq_lappend_lfiniteD
thf(fact_180_llast__LNil,axiom,
! [A: $tType] :
( ( coinductive_llast @ A @ ( coinductive_LNil @ A ) )
= ( undefined @ A ) ) ).
% llast_LNil
thf(fact_181_llast__linfinite,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ~ ( coinductive_lfinite @ A @ Xs )
=> ( ( coinductive_llast @ A @ Xs )
= ( undefined @ A ) ) ) ).
% llast_linfinite
thf(fact_182_ldistinct_OLCons,axiom,
! [A: $tType,X: A,Xs: coinductive_llist @ A] :
( ~ ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
=> ( ( coindu351974385stinct @ A @ Xs )
=> ( coindu351974385stinct @ A @ ( coinductive_LCons @ A @ X @ Xs ) ) ) ) ).
% ldistinct.LCons
thf(fact_183_lset__lappend,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] : ( ord_less_eq @ ( set @ A ) @ ( coinductive_lset @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) ) @ ( sup_sup @ ( set @ A ) @ ( coinductive_lset @ A @ Xs ) @ ( coinductive_lset @ A @ Ys ) ) ) ).
% lset_lappend
thf(fact_184_lset__lappend__conv,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( ( coinductive_lfinite @ A @ Xs )
=> ( ( coinductive_lset @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
= ( sup_sup @ ( set @ A ) @ ( coinductive_lset @ A @ Xs ) @ ( coinductive_lset @ A @ Ys ) ) ) )
& ( ~ ( coinductive_lfinite @ A @ Xs )
=> ( ( coinductive_lset @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
= ( coinductive_lset @ A @ Xs ) ) ) ) ).
% lset_lappend_conv
thf(fact_185_sup_Obounded__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [B2: A,C: A,A2: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ B2 @ C ) @ A2 )
= ( ( ord_less_eq @ A @ B2 @ A2 )
& ( ord_less_eq @ A @ C @ A2 ) ) ) ) ).
% sup.bounded_iff
thf(fact_186_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_187_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_188_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_189_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_190_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_191_sup__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [X: A] :
( ( sup_sup @ A @ X @ X )
= X ) ) ).
% sup_idem
thf(fact_192_sup_Oidem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( sup_sup @ A @ A2 @ A2 )
= A2 ) ) ).
% sup.idem
thf(fact_193_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_194_sup_Oleft__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A,C: A] :
( ( sup_sup @ A @ B2 @ ( sup_sup @ A @ A2 @ C ) )
= ( sup_sup @ A @ A2 @ ( sup_sup @ A @ B2 @ C ) ) ) ) ).
% sup.left_commute
thf(fact_195_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_196_sup_Ocommute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ( ( sup_sup @ A )
= ( ^ [A3: A,B6: A] : ( sup_sup @ A @ B6 @ A3 ) ) ) ) ).
% sup.commute
thf(fact_197_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_198_sup_Oassoc,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ A2 @ B2 ) @ C )
= ( sup_sup @ A @ A2 @ ( sup_sup @ A @ B2 @ C ) ) ) ) ).
% sup.assoc
thf(fact_199_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_200_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_201_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_202_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_203_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_204_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_205_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_206_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_207_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_208_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_209_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_210_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_211_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_212_sup_Omono,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [C: A,A2: A,D2: A,B2: A] :
( ( ord_less_eq @ A @ C @ A2 )
=> ( ( ord_less_eq @ A @ D2 @ B2 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ C @ D2 ) @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ) ).
% sup.mono
thf(fact_213_sup__mono,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [A2: A,C: A,B2: A,D2: A] :
( ( ord_less_eq @ A @ A2 @ C )
=> ( ( ord_less_eq @ A @ B2 @ D2 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ A2 @ B2 ) @ ( sup_sup @ A @ C @ D2 ) ) ) ) ) ).
% sup_mono
thf(fact_214_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_215_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_216_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_217_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_218_sup__unique,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [F: A > A > A,X: A,Y: A] :
( ! [X2: A,Y4: A] : ( ord_less_eq @ A @ X2 @ ( F @ X2 @ Y4 ) )
=> ( ! [X2: A,Y4: A] : ( ord_less_eq @ A @ Y4 @ ( F @ X2 @ Y4 ) )
=> ( ! [X2: A,Y4: A,Z4: A] :
( ( ord_less_eq @ A @ Y4 @ X2 )
=> ( ( ord_less_eq @ A @ Z4 @ X2 )
=> ( ord_less_eq @ A @ ( F @ Y4 @ Z4 ) @ X2 ) ) )
=> ( ( sup_sup @ A @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ) ).
% sup_unique
thf(fact_219_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_220_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_221_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_222_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_223_sup_OboundedE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [B2: A,C: A,A2: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ B2 @ C ) @ A2 )
=> ~ ( ( ord_less_eq @ A @ B2 @ A2 )
=> ~ ( ord_less_eq @ A @ C @ A2 ) ) ) ) ).
% sup.boundedE
thf(fact_224_sup_OboundedI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A,C: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ C @ A2 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ B2 @ C ) @ A2 ) ) ) ) ).
% sup.boundedI
thf(fact_225_sup_Oorder__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ( ( ord_less_eq @ A )
= ( ^ [B6: A,A3: A] :
( A3
= ( sup_sup @ A @ A3 @ B6 ) ) ) ) ) ).
% sup.order_iff
thf(fact_226_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_227_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_228_sup_Oabsorb__iff1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ( ( ord_less_eq @ A )
= ( ^ [B6: A,A3: A] :
( ( sup_sup @ A @ A3 @ B6 )
= A3 ) ) ) ) ).
% sup.absorb_iff1
thf(fact_229_sup_Oabsorb__iff2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ( ( ord_less_eq @ A )
= ( ^ [A3: A,B6: A] :
( ( sup_sup @ A @ A3 @ B6 )
= B6 ) ) ) ) ).
% sup.absorb_iff2
thf(fact_230_sup_OcoboundedI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [C: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ C @ A2 )
=> ( ord_less_eq @ A @ C @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% sup.coboundedI1
thf(fact_231_sup_OcoboundedI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [C: A,B2: A,A2: A] :
( ( ord_less_eq @ A @ C @ B2 )
=> ( ord_less_eq @ A @ C @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% sup.coboundedI2
thf(fact_232_gen__lset__def,axiom,
! [A: $tType] :
( ( coinductive_gen_lset @ A )
= ( ^ [A6: set @ A,Xs3: coinductive_llist @ A] : ( sup_sup @ ( set @ A ) @ A6 @ ( coinductive_lset @ A @ Xs3 ) ) ) ) ).
% gen_lset_def
thf(fact_233_llexord__lappend__left,axiom,
! [A: $tType,Xs: coinductive_llist @ A,R: A > A > $o,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ Xs )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ ( coinductive_lset @ A @ Xs ) )
=> ~ ( R @ X2 @ X2 ) )
=> ( ( coinductive_llexord @ A @ R @ ( coinductive_lappend @ A @ Xs @ Ys ) @ ( coinductive_lappend @ A @ Xs @ Zs ) )
= ( coinductive_llexord @ A @ R @ Ys @ Zs ) ) ) ) ).
% llexord_lappend_left
thf(fact_234_llexord__refl,axiom,
! [A: $tType,R: A > A > $o,Xs: coinductive_llist @ A] : ( coinductive_llexord @ A @ R @ Xs @ Xs ) ).
% llexord_refl
thf(fact_235_llexord__LCons__LCons,axiom,
! [A: $tType,R: A > A > $o,X: A,Xs: coinductive_llist @ A,Y: A,Ys: coinductive_llist @ A] :
( ( coinductive_llexord @ A @ R @ ( coinductive_LCons @ A @ X @ Xs ) @ ( coinductive_LCons @ A @ Y @ Ys ) )
= ( ( ( X = Y )
& ( coinductive_llexord @ A @ R @ Xs @ Ys ) )
| ( R @ X @ Y ) ) ) ).
% llexord_LCons_LCons
thf(fact_236_llexord__LNil__right,axiom,
! [A: $tType,Ys: coinductive_llist @ A,R: A > A > $o,Xs: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ Ys )
=> ( ( coinductive_llexord @ A @ R @ Xs @ Ys )
= ( coinductive_lnull @ A @ Xs ) ) ) ).
% llexord_LNil_right
thf(fact_237_llexord__code_I1_J,axiom,
! [A: $tType,R: A > A > $o,Ys: coinductive_llist @ A] : ( coinductive_llexord @ A @ R @ ( coinductive_LNil @ A ) @ Ys ) ).
% llexord_code(1)
thf(fact_238_llexord__antisym,axiom,
! [A: $tType,R: A > A > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_llexord @ A @ R @ Xs @ Ys )
=> ( ( coinductive_llexord @ A @ R @ Ys @ Xs )
=> ( ! [A4: A,B3: A] :
( ( R @ A4 @ B3 )
=> ~ ( R @ B3 @ A4 ) )
=> ( Xs = Ys ) ) ) ) ).
% llexord_antisym
thf(fact_239_llexord__linear,axiom,
! [A: $tType,R: A > A > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ! [X2: A,Y4: A] :
( ( R @ X2 @ Y4 )
| ( X2 = Y4 )
| ( R @ Y4 @ X2 ) )
=> ( ( coinductive_llexord @ A @ R @ Xs @ Ys )
| ( coinductive_llexord @ A @ R @ Ys @ Xs ) ) ) ).
% llexord_linear
thf(fact_240_llexord__trans,axiom,
! [A: $tType,R: A > A > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A] :
( ( coinductive_llexord @ A @ R @ Xs @ Ys )
=> ( ( coinductive_llexord @ A @ R @ Ys @ Zs )
=> ( ! [A4: A,B3: A,C4: A] :
( ( R @ A4 @ B3 )
=> ( ( R @ B3 @ C4 )
=> ( R @ A4 @ C4 ) ) )
=> ( coinductive_llexord @ A @ R @ Xs @ Zs ) ) ) ) ).
% llexord_trans
thf(fact_241_llexord__append__right,axiom,
! [A: $tType,R: A > A > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] : ( coinductive_llexord @ A @ R @ Xs @ ( coinductive_lappend @ A @ Xs @ Ys ) ) ).
% llexord_append_right
thf(fact_242_llexord__lappend__leftI,axiom,
! [A: $tType,R: A > A > $o,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A,Xs: coinductive_llist @ A] :
( ( coinductive_llexord @ A @ R @ Ys @ Zs )
=> ( coinductive_llexord @ A @ R @ ( coinductive_lappend @ A @ Xs @ Ys ) @ ( coinductive_lappend @ A @ Xs @ Zs ) ) ) ).
% llexord_lappend_leftI
thf(fact_243_llexord__LNil,axiom,
! [A: $tType,R: A > A > $o,Ys: coinductive_llist @ A] : ( coinductive_llexord @ A @ R @ ( coinductive_LNil @ A ) @ Ys ) ).
% llexord_LNil
thf(fact_244_lnull__llexord,axiom,
! [A: $tType,Xs: coinductive_llist @ A,R: A > A > $o,Ys: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ Xs )
=> ( coinductive_llexord @ A @ R @ Xs @ Ys ) ) ).
% lnull_llexord
thf(fact_245_llexord__LCons__less,axiom,
! [A: $tType,R: A > A > $o,X: A,Y: A,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( R @ X @ Y )
=> ( coinductive_llexord @ A @ R @ ( coinductive_LCons @ A @ X @ Xs ) @ ( coinductive_LCons @ A @ Y @ Ys ) ) ) ).
% llexord_LCons_less
thf(fact_246_llexord__LCons__eq,axiom,
! [A: $tType,R: A > A > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,X: A] :
( ( coinductive_llexord @ A @ R @ Xs @ Ys )
=> ( coinductive_llexord @ A @ R @ ( coinductive_LCons @ A @ X @ Xs ) @ ( coinductive_LCons @ A @ X @ Ys ) ) ) ).
% llexord_LCons_eq
thf(fact_247_llexord__LCons__left,axiom,
! [A: $tType,R: A > A > $o,X: A,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_llexord @ A @ R @ ( coinductive_LCons @ A @ X @ Xs ) @ Ys )
= ( ? [Y3: A,Ys4: coinductive_llist @ A] :
( ( Ys
= ( coinductive_LCons @ A @ Y3 @ Ys4 ) )
& ( ( ( X = Y3 )
& ( coinductive_llexord @ A @ R @ Xs @ Ys4 ) )
| ( R @ X @ Y3 ) ) ) ) ) ).
% llexord_LCons_left
thf(fact_248_llexord__code_I3_J,axiom,
! [A: $tType,R: A > A > $o,X: A,Xs: coinductive_llist @ A,Y: A,Ys: coinductive_llist @ A] :
( ( coinductive_llexord @ A @ R @ ( coinductive_LCons @ A @ X @ Xs ) @ ( coinductive_LCons @ A @ Y @ Ys ) )
= ( ( R @ X @ Y )
| ( ( X = Y )
& ( coinductive_llexord @ A @ R @ Xs @ Ys ) ) ) ) ).
% llexord_code(3)
thf(fact_249_llexord__code_I2_J,axiom,
! [A: $tType,R: A > A > $o,X: A,Xs: coinductive_llist @ A] :
~ ( coinductive_llexord @ A @ R @ ( coinductive_LCons @ A @ X @ Xs ) @ ( coinductive_LNil @ A ) ) ).
% llexord_code(2)
thf(fact_250_llexord_Ocases,axiom,
! [A: $tType,R: A > A > $o,A1: coinductive_llist @ A,A22: coinductive_llist @ A] :
( ( coinductive_llexord @ A @ R @ A1 @ A22 )
=> ( ! [Xs2: coinductive_llist @ A,Ys2: coinductive_llist @ A,X2: A] :
( ( A1
= ( coinductive_LCons @ A @ X2 @ Xs2 ) )
=> ( ( A22
= ( coinductive_LCons @ A @ X2 @ Ys2 ) )
=> ~ ( coinductive_llexord @ A @ R @ Xs2 @ Ys2 ) ) )
=> ( ! [X2: A] :
( ? [Xs2: coinductive_llist @ A] :
( A1
= ( coinductive_LCons @ A @ X2 @ Xs2 ) )
=> ! [Y4: A] :
( ? [Ys2: coinductive_llist @ A] :
( A22
= ( coinductive_LCons @ A @ Y4 @ Ys2 ) )
=> ~ ( R @ X2 @ Y4 ) ) )
=> ~ ( ( A1
= ( coinductive_LNil @ A ) )
=> ! [Ys2: coinductive_llist @ A] : A22 != Ys2 ) ) ) ) ).
% llexord.cases
thf(fact_251_llexord_Osimps,axiom,
! [A: $tType] :
( ( coinductive_llexord @ A )
= ( ^ [R2: A > A > $o,A12: coinductive_llist @ A,A23: coinductive_llist @ A] :
( ? [Xs3: coinductive_llist @ A,Ys5: coinductive_llist @ A,X3: A] :
( ( A12
= ( coinductive_LCons @ A @ X3 @ Xs3 ) )
& ( A23
= ( coinductive_LCons @ A @ X3 @ Ys5 ) )
& ( coinductive_llexord @ A @ R2 @ Xs3 @ Ys5 ) )
| ? [X3: A,Y3: A,Xs3: coinductive_llist @ A,Ys5: coinductive_llist @ A] :
( ( A12
= ( coinductive_LCons @ A @ X3 @ Xs3 ) )
& ( A23
= ( coinductive_LCons @ A @ Y3 @ Ys5 ) )
& ( R2 @ X3 @ Y3 ) )
| ? [Ys5: coinductive_llist @ A] :
( ( A12
= ( coinductive_LNil @ A ) )
& ( A23 = Ys5 ) ) ) ) ) ).
% llexord.simps
thf(fact_252_llexord_Ocoinduct,axiom,
! [A: $tType,X7: ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o,X: coinductive_llist @ A,Xa: coinductive_llist @ A,R: A > A > $o] :
( ( X7 @ X @ Xa )
=> ( ! [X2: coinductive_llist @ A,Xa2: coinductive_llist @ A] :
( ( X7 @ X2 @ Xa2 )
=> ( ? [Xs6: coinductive_llist @ A,Ys3: coinductive_llist @ A,Xb: A] :
( ( X2
= ( coinductive_LCons @ A @ Xb @ Xs6 ) )
& ( Xa2
= ( coinductive_LCons @ A @ Xb @ Ys3 ) )
& ( ( X7 @ Xs6 @ Ys3 )
| ( coinductive_llexord @ A @ R @ Xs6 @ Ys3 ) ) )
| ? [Xb: A,Y5: A,Xs6: coinductive_llist @ A,Ys3: coinductive_llist @ A] :
( ( X2
= ( coinductive_LCons @ A @ Xb @ Xs6 ) )
& ( Xa2
= ( coinductive_LCons @ A @ Y5 @ Ys3 ) )
& ( R @ Xb @ Y5 ) )
| ? [Ys3: coinductive_llist @ A] :
( ( X2
= ( coinductive_LNil @ A ) )
& ( Xa2 = Ys3 ) ) ) )
=> ( coinductive_llexord @ A @ R @ X @ Xa ) ) ) ).
% llexord.coinduct
thf(fact_253_gen__lset__code_I1_J,axiom,
! [A: $tType,A5: set @ A] :
( ( coinductive_gen_lset @ A @ A5 @ ( coinductive_LNil @ A ) )
= A5 ) ).
% gen_lset_code(1)
%----Type constructors (16)
thf(tcon_fun___Lattices_Osemilattice__sup,axiom,
! [A7: $tType,A8: $tType] :
( ( semilattice_sup @ A8 @ ( type2 @ A8 ) )
=> ( semilattice_sup @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A7: $tType,A8: $tType] :
( ( preorder @ A8 @ ( type2 @ A8 ) )
=> ( preorder @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
thf(tcon_fun___Lattices_Olattice,axiom,
! [A7: $tType,A8: $tType] :
( ( lattice @ A8 @ ( type2 @ A8 ) )
=> ( lattice @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A7: $tType,A8: $tType] :
( ( order @ A8 @ ( type2 @ A8 ) )
=> ( order @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A7: $tType,A8: $tType] :
( ( ord @ A8 @ ( type2 @ A8 ) )
=> ( ord @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
thf(tcon_Set_Oset___Lattices_Osemilattice__sup_1,axiom,
! [A7: $tType] : ( semilattice_sup @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_2,axiom,
! [A7: $tType] : ( preorder @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_Set_Oset___Lattices_Olattice_3,axiom,
! [A7: $tType] : ( lattice @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_4,axiom,
! [A7: $tType] : ( order @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_5,axiom,
! [A7: $tType] : ( ord @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_6,axiom,
semilattice_sup @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Opreorder_7,axiom,
preorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
linorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Lattices_Olattice_8,axiom,
lattice @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oorder_9,axiom,
order @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oord_10,axiom,
ord @ $o @ ( type2 @ $o ) ).
%----Conjectures (1)
thf(conj_0,conjecture,
( ( coinductive_lset @ a @ ( lMirro999291890or_aux @ a @ acc @ xs ) )
= ( coinductive_lset @ a @ ( coinductive_lappend @ a @ xs @ acc ) ) ) ).
%------------------------------------------------------------------------------