TPTP Problem File: DAT156^1.p
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%------------------------------------------------------------------------------
% File : DAT156^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Data Structures
% Problem : Hamming stream 122
% 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 : hamming_stream__122.p [Bla16]
% Status : Theorem
% Rating : 1.00 v7.1.0
% Syntax : Number of formulae : 295 ( 108 unt; 35 typ; 0 def)
% Number of atoms : 874 ( 261 equ; 0 cnn)
% Maximal formula atoms : 13 ( 3 avg)
% Number of connectives : 5021 ( 191 ~; 40 |; 81 &;4348 @)
% ( 0 <=>; 361 =>; 0 <=; 0 <~>)
% Maximal formula depth : 23 ( 9 avg)
% Number of types : 2 ( 1 usr)
% Number of type conns : 345 ( 345 >; 0 *; 0 +; 0 <<)
% Number of symbols : 37 ( 34 usr; 3 con; 0-7 aty)
% Number of variables : 1187 ( 121 ^; 972 !; 63 ?;1187 :)
% ( 31 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 14:41:31.616
%------------------------------------------------------------------------------
%----Could-be-implicit typings (5)
thf(ty_t_Coinductive__List_Ollist,type,
coinductive_llist: $tType > $tType ).
thf(ty_t_List_Olist,type,
list: $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 (30)
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_c_Coinductive__List_OlSup,type,
coinductive_lSup:
!>[A: $tType] : ( ( set @ ( 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_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_Ocase__llist,type,
coindu1381640503_llist:
!>[B: $tType,A: $tType] : ( B > ( A > ( coinductive_llist @ A ) > B ) > ( coinductive_llist @ A ) > B ) ).
thf(sy_c_Coinductive__List_Ollist_Olhd,type,
coinductive_lhd:
!>[A: $tType] : ( ( coinductive_llist @ A ) > 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_Ollist_Oltl,type,
coinductive_ltl:
!>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
thf(sy_c_Coinductive__List_Olprefix,type,
coinductive_lprefix:
!>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o ) ).
thf(sy_c_Complete__Partial__Order2_Omcont,type,
comple1403672095_mcont:
!>[A: $tType,B: $tType] : ( ( ( set @ A ) > A ) > ( A > A > $o ) > ( ( set @ B ) > B ) > ( B > B > $o ) > ( A > B ) > $o ) ).
thf(sy_c_Complete__Partial__Order_Omonotone,type,
comple1396247847notone:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( B > B > $o ) > ( A > B ) > $o ) ).
thf(sy_c_HOL_Oundefined,type,
undefined:
!>[A: $tType] : A ).
thf(sy_c_Hamming__Stream__Mirabelle__rwekfkwckg_Oord_Olmerge,type,
hammin1328233080lmerge:
!>[A: $tType] : ( ( A > A > $o ) > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
thf(sy_c_If,type,
if:
!>[A: $tType] : ( $o > A > A > A ) ).
thf(sy_c_Lattices_Oinf__class_Oinf,type,
inf_inf:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_List_Olist_OCons,type,
cons:
!>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Olist_ONil,type,
nil:
!>[A: $tType] : ( list @ A ) ).
thf(sy_c_List_Oord_Olexordp,type,
lexordp:
!>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Oord_Olexordp__eq,type,
lexordp_eq:
!>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > ( list @ 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_Set__Interval_Oord_OgreaterThan,type,
set_greaterThan:
!>[A: $tType] : ( ( A > A > $o ) > A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord_OgreaterThanLessThan,type,
set_gr1161524159ssThan:
!>[A: $tType] : ( ( A > A > $o ) > A > A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord_OlessThan,type,
set_lessThan:
!>[A: $tType] : ( ( A > A > $o ) > A > ( set @ A ) ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_less,type,
less: a > a > $o ).
thf(sy_v_xs,type,
xs: coinductive_llist @ a ).
%----Relevant facts (253)
thf(fact_0_local_Oord__eq__less__trans,axiom,
! [A2: a,B2: a,C: a] :
( ( A2 = B2 )
=> ( ( less @ B2 @ C )
=> ( less @ A2 @ C ) ) ) ).
% local.ord_eq_less_trans
thf(fact_1_local_Oord__less__eq__trans,axiom,
! [A2: a,B2: a,C: a] :
( ( less @ A2 @ B2 )
=> ( ( B2 = C )
=> ( less @ A2 @ C ) ) ) ).
% local.ord_less_eq_trans
thf(fact_2_lprefix__lmergeI,axiom,
! [Xs: coinductive_llist @ a,Xs2: coinductive_llist @ a,Ys: coinductive_llist @ a,Ys2: coinductive_llist @ a] :
( ( coinductive_lprefix @ a @ Xs @ Xs2 )
=> ( ( coinductive_lprefix @ a @ Ys @ Ys2 )
=> ( coinductive_lprefix @ a @ ( hammin1328233080lmerge @ a @ less @ Xs @ Ys ) @ ( hammin1328233080lmerge @ a @ less @ Xs2 @ Ys2 ) ) ) ) ).
% lprefix_lmergeI
thf(fact_3_mcont2mcont__lmerge1,axiom,
! [C2: $tType,Lub: ( set @ C2 ) > C2,Ord: C2 > C2 > $o,T: C2 > ( coinductive_llist @ a ),Ys: coinductive_llist @ a] :
( ( comple1403672095_mcont @ C2 @ ( coinductive_llist @ a ) @ Lub @ Ord @ ( coinductive_lSup @ a ) @ ( coinductive_lprefix @ a ) @ T )
=> ( comple1403672095_mcont @ C2 @ ( coinductive_llist @ a ) @ Lub @ Ord @ ( coinductive_lSup @ a ) @ ( coinductive_lprefix @ a )
@ ^ [X: C2] : ( hammin1328233080lmerge @ a @ less @ ( T @ X ) @ Ys ) ) ) ).
% mcont2mcont_lmerge1
thf(fact_4_mcont__lmerge1,axiom,
! [Ys: coinductive_llist @ a] :
( comple1403672095_mcont @ ( coinductive_llist @ a ) @ ( coinductive_llist @ a ) @ ( coinductive_lSup @ a ) @ ( coinductive_lprefix @ a ) @ ( coinductive_lSup @ a ) @ ( coinductive_lprefix @ a )
@ ^ [Xs3: coinductive_llist @ a] : ( hammin1328233080lmerge @ a @ less @ Xs3 @ Ys ) ) ).
% mcont_lmerge1
thf(fact_5_local_OgreaterThan__def,axiom,
! [L: a] :
( ( set_greaterThan @ a @ less @ L )
= ( collect @ a @ ( less @ L ) ) ) ).
% local.greaterThan_def
thf(fact_6_local_OlessThan__def,axiom,
! [U: a] :
( ( set_lessThan @ a @ less @ U )
= ( collect @ a
@ ^ [X: a] : ( less @ X @ U ) ) ) ).
% local.lessThan_def
thf(fact_7_ord_Olprefix__lmergeI,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Xs2: coinductive_llist @ A,Ys: coinductive_llist @ A,Ys2: coinductive_llist @ A,Less: A > A > $o] :
( ( coinductive_lprefix @ A @ Xs @ Xs2 )
=> ( ( coinductive_lprefix @ A @ Ys @ Ys2 )
=> ( coinductive_lprefix @ A @ ( hammin1328233080lmerge @ A @ Less @ Xs @ Ys ) @ ( hammin1328233080lmerge @ A @ Less @ Xs2 @ Ys2 ) ) ) ) ).
% ord.lprefix_lmergeI
thf(fact_8_ord_Omcont2mcont__lmerge1,axiom,
! [A: $tType,Less: A > A > $o,Ys: coinductive_llist @ A] :
( comple1403672095_mcont @ ( coinductive_llist @ A ) @ ( coinductive_llist @ A ) @ ( coinductive_lSup @ A ) @ ( coinductive_lprefix @ A ) @ ( coinductive_lSup @ A ) @ ( coinductive_lprefix @ A )
@ ^ [Xs3: coinductive_llist @ A] : ( hammin1328233080lmerge @ A @ Less @ Xs3 @ Ys ) ) ).
% ord.mcont2mcont_lmerge1
thf(fact_9_llist_Omcont2mcont,axiom,
! [B: $tType,A: $tType,C2: $tType,Lub2: ( set @ B ) > B,Ord2: B > B > $o,F: B > ( coinductive_llist @ A ),Lub: ( set @ C2 ) > C2,Ord: C2 > C2 > $o,T: C2 > B] :
( ( comple1403672095_mcont @ B @ ( coinductive_llist @ A ) @ Lub2 @ Ord2 @ ( coinductive_lSup @ A ) @ ( coinductive_lprefix @ A ) @ F )
=> ( ( comple1403672095_mcont @ C2 @ B @ Lub @ Ord @ Lub2 @ Ord2 @ T )
=> ( comple1403672095_mcont @ C2 @ ( coinductive_llist @ A ) @ Lub @ Ord @ ( coinductive_lSup @ A ) @ ( coinductive_lprefix @ A )
@ ^ [X: C2] : ( F @ ( T @ X ) ) ) ) ) ).
% llist.mcont2mcont
thf(fact_10_llist_Omcont2mcont_H,axiom,
! [C2: $tType,A: $tType,B: $tType,Lub2: ( set @ C2 ) > C2,Ord2: C2 > C2 > $o,F: B > C2 > ( coinductive_llist @ A ),Lub: ( set @ B ) > B,Ord: B > B > $o,T: B > C2] :
( ! [X2: B] : ( comple1403672095_mcont @ C2 @ ( coinductive_llist @ A ) @ Lub2 @ Ord2 @ ( coinductive_lSup @ A ) @ ( coinductive_lprefix @ A ) @ ( F @ X2 ) )
=> ( ! [Y: C2] :
( comple1403672095_mcont @ B @ ( coinductive_llist @ A ) @ Lub @ Ord @ ( coinductive_lSup @ A ) @ ( coinductive_lprefix @ A )
@ ^ [X: B] : ( F @ X @ Y ) )
=> ( ( comple1403672095_mcont @ B @ C2 @ Lub @ Ord @ Lub2 @ Ord2 @ T )
=> ( comple1403672095_mcont @ B @ ( coinductive_llist @ A ) @ Lub @ Ord @ ( coinductive_lSup @ A ) @ ( coinductive_lprefix @ A )
@ ^ [X: B] : ( F @ X @ ( T @ X ) ) ) ) ) ) ).
% llist.mcont2mcont'
thf(fact_11_llist_Omcont__const,axiom,
! [A: $tType,B: $tType,Luba: ( set @ B ) > B,Orda: B > B > $o,C: coinductive_llist @ A] :
( comple1403672095_mcont @ B @ ( coinductive_llist @ A ) @ Luba @ Orda @ ( coinductive_lSup @ A ) @ ( coinductive_lprefix @ A )
@ ^ [X: B] : C ) ).
% llist.mcont_const
thf(fact_12_monotone__lmerge1,axiom,
! [Ys: coinductive_llist @ a] :
( comple1396247847notone @ ( coinductive_llist @ a ) @ ( coinductive_llist @ a ) @ ( coinductive_lprefix @ a ) @ ( coinductive_lprefix @ a )
@ ^ [Xs3: coinductive_llist @ a] : ( hammin1328233080lmerge @ a @ less @ Xs3 @ Ys ) ) ).
% monotone_lmerge1
thf(fact_13_monotone__lmerge2,axiom,
! [Xs: coinductive_llist @ a] : ( comple1396247847notone @ ( coinductive_llist @ a ) @ ( coinductive_llist @ a ) @ ( coinductive_lprefix @ a ) @ ( coinductive_lprefix @ a ) @ ( hammin1328233080lmerge @ a @ less @ Xs ) ) ).
% monotone_lmerge2
thf(fact_14_mono2mono__lmerge1,axiom,
! [C2: $tType,Orda: C2 > C2 > $o,T: C2 > ( coinductive_llist @ a ),Ys: coinductive_llist @ a] :
( ( comple1396247847notone @ C2 @ ( coinductive_llist @ a ) @ Orda @ ( coinductive_lprefix @ a ) @ T )
=> ( comple1396247847notone @ C2 @ ( coinductive_llist @ a ) @ Orda @ ( coinductive_lprefix @ a )
@ ^ [X: C2] : ( hammin1328233080lmerge @ a @ less @ ( T @ X ) @ Ys ) ) ) ).
% mono2mono_lmerge1
thf(fact_15_mono2mono__lmerge2,axiom,
! [C2: $tType,Orda: C2 > C2 > $o,T: C2 > ( coinductive_llist @ a ),Xs: coinductive_llist @ a] :
( ( comple1396247847notone @ C2 @ ( coinductive_llist @ a ) @ Orda @ ( coinductive_lprefix @ a ) @ T )
=> ( comple1396247847notone @ C2 @ ( coinductive_llist @ a ) @ Orda @ ( coinductive_lprefix @ a )
@ ^ [X: C2] : ( hammin1328233080lmerge @ a @ less @ Xs @ ( T @ X ) ) ) ) ).
% mono2mono_lmerge2
thf(fact_16_lprefix__refl,axiom,
! [A: $tType,Xs: coinductive_llist @ A] : ( coinductive_lprefix @ A @ Xs @ Xs ) ).
% lprefix_refl
thf(fact_17_llist_Oleq__refl,axiom,
! [A: $tType,X3: coinductive_llist @ A] : ( coinductive_lprefix @ A @ X3 @ X3 ) ).
% llist.leq_refl
thf(fact_18_lmerge__eq__LNil__iff,axiom,
! [Xs: coinductive_llist @ a,Ys: coinductive_llist @ a] :
( ( ( hammin1328233080lmerge @ a @ less @ Xs @ Ys )
= ( coinductive_LNil @ a ) )
= ( ( Xs
= ( coinductive_LNil @ a ) )
| ( Ys
= ( coinductive_LNil @ a ) ) ) ) ).
% lmerge_eq_LNil_iff
thf(fact_19_lmerge__simps,axiom,
! [X3: a,Y2: a,Xs: coinductive_llist @ a,Ys: coinductive_llist @ a] :
( ( ( less @ X3 @ Y2 )
=> ( ( hammin1328233080lmerge @ a @ less @ ( coinductive_LCons @ a @ X3 @ Xs ) @ ( coinductive_LCons @ a @ Y2 @ Ys ) )
= ( coinductive_LCons @ a @ X3 @ ( hammin1328233080lmerge @ a @ less @ Xs @ ( coinductive_LCons @ a @ Y2 @ Ys ) ) ) ) )
& ( ~ ( less @ X3 @ Y2 )
=> ( ( ( less @ Y2 @ X3 )
=> ( ( hammin1328233080lmerge @ a @ less @ ( coinductive_LCons @ a @ X3 @ Xs ) @ ( coinductive_LCons @ a @ Y2 @ Ys ) )
= ( coinductive_LCons @ a @ Y2 @ ( hammin1328233080lmerge @ a @ less @ ( coinductive_LCons @ a @ X3 @ Xs ) @ Ys ) ) ) )
& ( ~ ( less @ Y2 @ X3 )
=> ( ( hammin1328233080lmerge @ a @ less @ ( coinductive_LCons @ a @ X3 @ Xs ) @ ( coinductive_LCons @ a @ Y2 @ Ys ) )
= ( coinductive_LCons @ a @ Y2 @ ( hammin1328233080lmerge @ a @ less @ Xs @ Ys ) ) ) ) ) ) ) ).
% lmerge_simps
thf(fact_20_lmerge_Odisc_I2_J,axiom,
! [Xs: coinductive_llist @ a,Ys: coinductive_llist @ a] :
( ~ ( coinductive_lnull @ a @ Xs )
=> ( ~ ( coinductive_lnull @ a @ Ys )
=> ~ ( coinductive_lnull @ a @ ( hammin1328233080lmerge @ a @ less @ Xs @ Ys ) ) ) ) ).
% lmerge.disc(2)
thf(fact_21_lmerge_Oexhaust,axiom,
! [Xs: coinductive_llist @ a,Ys: coinductive_llist @ a] :
( ~ ( ( coinductive_lnull @ a @ Xs )
| ( coinductive_lnull @ a @ Ys ) )
=> ~ ( ~ ( coinductive_lnull @ a @ Xs )
=> ( coinductive_lnull @ a @ Ys ) ) ) ).
% lmerge.exhaust
thf(fact_22_lmerge_Odisc_I1_J,axiom,
! [Xs: coinductive_llist @ a,Ys: coinductive_llist @ a] :
( ( ( coinductive_lnull @ a @ Xs )
| ( coinductive_lnull @ a @ Ys ) )
=> ( coinductive_lnull @ a @ ( hammin1328233080lmerge @ a @ less @ Xs @ Ys ) ) ) ).
% lmerge.disc(1)
thf(fact_23_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_24_lmerge_Octr_I1_J,axiom,
! [Xs: coinductive_llist @ a,Ys: coinductive_llist @ a] :
( ( ( coinductive_lnull @ a @ Xs )
| ( coinductive_lnull @ a @ Ys ) )
=> ( ( hammin1328233080lmerge @ a @ less @ Xs @ Ys )
= ( coinductive_LNil @ a ) ) ) ).
% lmerge.ctr(1)
thf(fact_25_LCons__lprefix__LCons,axiom,
! [A: $tType,X3: A,Xs: coinductive_llist @ A,Y2: A,Ys: coinductive_llist @ A] :
( ( coinductive_lprefix @ A @ ( coinductive_LCons @ A @ X3 @ Xs ) @ ( coinductive_LCons @ A @ Y2 @ Ys ) )
= ( ( X3 = Y2 )
& ( coinductive_lprefix @ A @ Xs @ Ys ) ) ) ).
% LCons_lprefix_LCons
thf(fact_26_lprefix__code_I1_J,axiom,
! [A: $tType,Ys: coinductive_llist @ A] : ( coinductive_lprefix @ A @ ( coinductive_LNil @ A ) @ Ys ) ).
% lprefix_code(1)
thf(fact_27_lnull__lSup,axiom,
! [A: $tType,A3: set @ ( coinductive_llist @ A )] :
( ( coinductive_lnull @ A @ ( coinductive_lSup @ A @ A3 ) )
= ( ! [X: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ X @ A3 )
=> ( coinductive_lnull @ A @ X ) ) ) ) ).
% lnull_lSup
thf(fact_28_lprefix__LNil,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( coinductive_lprefix @ A @ Xs @ ( coinductive_LNil @ A ) )
= ( coinductive_lnull @ A @ Xs ) ) ).
% lprefix_LNil
thf(fact_29_local_OlessThan__iff,axiom,
! [I: a,K: a] :
( ( member @ a @ I @ ( set_lessThan @ a @ less @ K ) )
= ( less @ I @ K ) ) ).
% local.lessThan_iff
thf(fact_30_local_OgreaterThan__iff,axiom,
! [I: a,K: a] :
( ( member @ a @ I @ ( set_greaterThan @ a @ less @ K ) )
= ( less @ K @ I ) ) ).
% local.greaterThan_iff
thf(fact_31_lnull__lmerge,axiom,
! [Xs: coinductive_llist @ a,Ys: coinductive_llist @ a] :
( ( coinductive_lnull @ a @ ( hammin1328233080lmerge @ a @ less @ Xs @ Ys ) )
= ( ( coinductive_lnull @ a @ Xs )
| ( coinductive_lnull @ a @ Ys ) ) ) ).
% lnull_lmerge
thf(fact_32_lmerge_Osimps_I2_J,axiom,
! [Xs: coinductive_llist @ a,Ys: coinductive_llist @ a] :
( ( ~ ( coinductive_lnull @ a @ ( hammin1328233080lmerge @ a @ less @ Xs @ Ys ) ) )
= ( ~ ( coinductive_lnull @ a @ Xs )
& ~ ( coinductive_lnull @ a @ Ys ) ) ) ).
% lmerge.simps(2)
thf(fact_33_lmerge__LNil_I1_J,axiom,
! [Ys: coinductive_llist @ a] :
( ( hammin1328233080lmerge @ a @ less @ ( coinductive_LNil @ a ) @ Ys )
= ( coinductive_LNil @ a ) ) ).
% lmerge_LNil(1)
thf(fact_34_lmerge__LNil_I2_J,axiom,
! [Xs: coinductive_llist @ a] :
( ( hammin1328233080lmerge @ a @ less @ Xs @ ( coinductive_LNil @ a ) )
= ( coinductive_LNil @ a ) ) ).
% lmerge_LNil(2)
thf(fact_35_lSup_Octr_I1_J,axiom,
! [A: $tType,A3: set @ ( coinductive_llist @ A )] :
( ! [X2: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ X2 @ A3 )
=> ( coinductive_lnull @ A @ X2 ) )
=> ( ( coinductive_lSup @ A @ A3 )
= ( coinductive_LNil @ A ) ) ) ).
% lSup.ctr(1)
thf(fact_36_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_37_llist_Odisc_I1_J,axiom,
! [A: $tType] : ( coinductive_lnull @ A @ ( coinductive_LNil @ A ) ) ).
% llist.disc(1)
thf(fact_38_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_39_llist_OdiscI_I1_J,axiom,
! [A: $tType,Llist: coinductive_llist @ A] :
( ( Llist
= ( coinductive_LNil @ A ) )
=> ( coinductive_lnull @ A @ Llist ) ) ).
% llist.discI(1)
thf(fact_40_lprefix__code_I2_J,axiom,
! [A: $tType,X3: A,Xs: coinductive_llist @ A] :
~ ( coinductive_lprefix @ A @ ( coinductive_LCons @ A @ X3 @ Xs ) @ ( coinductive_LNil @ A ) ) ).
% lprefix_code(2)
thf(fact_41_llist_Ocollapse_I1_J,axiom,
! [A: $tType,Llist: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ Llist )
=> ( Llist
= ( coinductive_LNil @ A ) ) ) ).
% llist.collapse(1)
thf(fact_42_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_43_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_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,A3: set @ A] :
( ( collect @ A
@ ^ [X: A] : ( member @ A @ X @ A3 ) )
= A3 ) ).
% 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_llist_Oexhaust,axiom,
! [A: $tType,Y2: coinductive_llist @ A] :
( ( Y2
!= ( coinductive_LNil @ A ) )
=> ~ ! [X212: A,X222: coinductive_llist @ A] :
( Y2
!= ( coinductive_LCons @ A @ X212 @ X222 ) ) ) ).
% llist.exhaust
thf(fact_49_lprefix_Ocases,axiom,
! [A: $tType,A1: coinductive_llist @ A,A22: coinductive_llist @ A] :
( ( coinductive_lprefix @ A @ A1 @ A22 )
=> ( ( ( A1
= ( coinductive_LNil @ A ) )
=> ! [Xs4: coinductive_llist @ A] : A22 != Xs4 )
=> ~ ! [Xs4: coinductive_llist @ A,Ys3: coinductive_llist @ A,X2: A] :
( ( A1
= ( coinductive_LCons @ A @ X2 @ Xs4 ) )
=> ( ( A22
= ( coinductive_LCons @ A @ X2 @ Ys3 ) )
=> ~ ( coinductive_lprefix @ A @ Xs4 @ Ys3 ) ) ) ) ) ).
% lprefix.cases
thf(fact_50_lprefix_Osimps,axiom,
! [A: $tType] :
( ( coinductive_lprefix @ A )
= ( ^ [A12: coinductive_llist @ A,A23: coinductive_llist @ A] :
( ? [Xs3: coinductive_llist @ A] :
( ( A12
= ( coinductive_LNil @ A ) )
& ( A23 = Xs3 ) )
| ? [Xs3: coinductive_llist @ A,Ys4: coinductive_llist @ A,X: A] :
( ( A12
= ( coinductive_LCons @ A @ X @ Xs3 ) )
& ( A23
= ( coinductive_LCons @ A @ X @ Ys4 ) )
& ( coinductive_lprefix @ A @ Xs3 @ Ys4 ) ) ) ) ) ).
% lprefix.simps
thf(fact_51_neq__LNil__conv,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( Xs
!= ( coinductive_LNil @ A ) )
= ( ? [X: A,Xs5: coinductive_llist @ A] :
( Xs
= ( coinductive_LCons @ A @ X @ Xs5 ) ) ) ) ).
% neq_LNil_conv
thf(fact_52_monotone__LCons,axiom,
! [A: $tType,X3: A] : ( comple1396247847notone @ ( coinductive_llist @ A ) @ ( coinductive_llist @ A ) @ ( coinductive_lprefix @ A ) @ ( coinductive_lprefix @ A ) @ ( coinductive_LCons @ A @ X3 ) ) ).
% monotone_LCons
thf(fact_53_not__lnull__conv,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( ~ ( coinductive_lnull @ A @ Xs ) )
= ( ? [X: A,Xs5: coinductive_llist @ A] :
( Xs
= ( coinductive_LCons @ A @ X @ Xs5 ) ) ) ) ).
% not_lnull_conv
thf(fact_54_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_55_lnull__def,axiom,
! [A: $tType] :
( ( coinductive_lnull @ A )
= ( ^ [Llist2: coinductive_llist @ A] :
( Llist2
= ( coinductive_LNil @ A ) ) ) ) ).
% lnull_def
thf(fact_56_llist_Omono2mono,axiom,
! [B: $tType,A: $tType,C2: $tType,Ordb: B > B > $o,F: B > ( coinductive_llist @ A ),Orda: C2 > C2 > $o,T: C2 > B] :
( ( comple1396247847notone @ B @ ( coinductive_llist @ A ) @ Ordb @ ( coinductive_lprefix @ A ) @ F )
=> ( ( comple1396247847notone @ C2 @ B @ Orda @ Ordb @ T )
=> ( comple1396247847notone @ C2 @ ( coinductive_llist @ A ) @ Orda @ ( coinductive_lprefix @ A )
@ ^ [X: C2] : ( F @ ( T @ X ) ) ) ) ) ).
% llist.mono2mono
thf(fact_57_lprefix_Ocoinduct,axiom,
! [A: $tType,X4: ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o,X3: coinductive_llist @ A,Xa: coinductive_llist @ A] :
( ( X4 @ X3 @ Xa )
=> ( ! [X2: coinductive_llist @ A,Xa2: coinductive_llist @ A] :
( ( X4 @ X2 @ Xa2 )
=> ( ? [Xs6: coinductive_llist @ A] :
( ( X2
= ( coinductive_LNil @ A ) )
& ( Xa2 = Xs6 ) )
| ? [Xs6: coinductive_llist @ A,Ys5: coinductive_llist @ A,Xb: A] :
( ( X2
= ( coinductive_LCons @ A @ Xb @ Xs6 ) )
& ( Xa2
= ( coinductive_LCons @ A @ Xb @ Ys5 ) )
& ( ( X4 @ Xs6 @ Ys5 )
| ( coinductive_lprefix @ A @ Xs6 @ Ys5 ) ) ) ) )
=> ( coinductive_lprefix @ A @ X3 @ Xa ) ) ) ).
% lprefix.coinduct
thf(fact_58_lprefix__LCons__conv,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Y2: A,Ys: coinductive_llist @ A] :
( ( coinductive_lprefix @ A @ Xs @ ( coinductive_LCons @ A @ Y2 @ Ys ) )
= ( ( Xs
= ( coinductive_LNil @ A ) )
| ? [Xs5: coinductive_llist @ A] :
( ( Xs
= ( coinductive_LCons @ A @ Y2 @ Xs5 ) )
& ( coinductive_lprefix @ A @ Xs5 @ Ys ) ) ) ) ).
% lprefix_LCons_conv
thf(fact_59_ord_Olmerge_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 ) ) ) ).
% ord.lmerge.exhaust
thf(fact_60_ord__class_Olmerge_Oexhaust,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ~ ( ( coinductive_lnull @ A @ Xs )
| ( coinductive_lnull @ A @ Ys ) )
=> ~ ( ~ ( coinductive_lnull @ A @ Xs )
=> ( coinductive_lnull @ A @ Ys ) ) ) ) ).
% ord_class.lmerge.exhaust
thf(fact_61_mono2mono__LCons,axiom,
! [A: $tType,C2: $tType,Orda: C2 > C2 > $o,T: C2 > ( coinductive_llist @ A ),X5: A] :
( ( comple1396247847notone @ C2 @ ( coinductive_llist @ A ) @ Orda @ ( coinductive_lprefix @ A ) @ T )
=> ( comple1396247847notone @ C2 @ ( coinductive_llist @ A ) @ Orda @ ( coinductive_lprefix @ A )
@ ^ [X: C2] : ( coinductive_LCons @ A @ X5 @ ( T @ X ) ) ) ) ).
% mono2mono_LCons
thf(fact_62_llist_Omonotone__if__bot,axiom,
! [B: $tType,A: $tType,Bound: coinductive_llist @ A,G: ( coinductive_llist @ A ) > B,Bot: B,F: ( coinductive_llist @ A ) > B,Ord: B > B > $o] :
( ! [X2: coinductive_llist @ A] :
( ( ( coinductive_lprefix @ A @ X2 @ Bound )
=> ( ( G @ X2 )
= Bot ) )
& ( ~ ( coinductive_lprefix @ A @ X2 @ Bound )
=> ( ( G @ X2 )
= ( F @ X2 ) ) ) )
=> ( ! [X2: coinductive_llist @ A,Y: coinductive_llist @ A] :
( ( coinductive_lprefix @ A @ X2 @ Y )
=> ( ~ ( coinductive_lprefix @ A @ X2 @ Bound )
=> ( Ord @ ( F @ X2 ) @ ( F @ Y ) ) ) )
=> ( ! [X2: coinductive_llist @ A] :
( ~ ( coinductive_lprefix @ A @ X2 @ Bound )
=> ( Ord @ Bot @ ( F @ X2 ) ) )
=> ( ( Ord @ Bot @ Bot )
=> ( comple1396247847notone @ ( coinductive_llist @ A ) @ B @ ( coinductive_lprefix @ A ) @ Ord @ G ) ) ) ) ) ).
% llist.monotone_if_bot
thf(fact_63_ord_Olmerge_Octr_I1_J,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Less: A > A > $o] :
( ( ( coinductive_lnull @ A @ Xs )
| ( coinductive_lnull @ A @ Ys ) )
=> ( ( hammin1328233080lmerge @ A @ Less @ Xs @ Ys )
= ( coinductive_LNil @ A ) ) ) ).
% ord.lmerge.ctr(1)
thf(fact_64_lprefix__not__lnullD,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_lprefix @ A @ Xs @ Ys )
=> ( ~ ( coinductive_lnull @ A @ Xs )
=> ~ ( coinductive_lnull @ A @ Ys ) ) ) ).
% lprefix_not_lnullD
thf(fact_65_lprefix__lnullD,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_lprefix @ A @ Xs @ Ys )
=> ( ( coinductive_lnull @ A @ Ys )
=> ( coinductive_lnull @ A @ Xs ) ) ) ).
% lprefix_lnullD
thf(fact_66_lprefix__lnull,axiom,
! [A: $tType,Ys: coinductive_llist @ A,Xs: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ Ys )
=> ( ( coinductive_lprefix @ A @ Xs @ Ys )
= ( coinductive_lnull @ A @ Xs ) ) ) ).
% lprefix_lnull
thf(fact_67_lnull__lprefix,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ Xs )
=> ( coinductive_lprefix @ A @ Xs @ Ys ) ) ).
% lnull_lprefix
thf(fact_68_LCons__lprefix__conv,axiom,
! [A: $tType,X3: A,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_lprefix @ A @ ( coinductive_LCons @ A @ X3 @ Xs ) @ Ys )
= ( ? [Ys6: coinductive_llist @ A] :
( ( Ys
= ( coinductive_LCons @ A @ X3 @ Ys6 ) )
& ( coinductive_lprefix @ A @ Xs @ Ys6 ) ) ) ) ).
% LCons_lprefix_conv
thf(fact_69_Le__LCons,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,X3: A] :
( ( coinductive_lprefix @ A @ Xs @ Ys )
=> ( coinductive_lprefix @ A @ ( coinductive_LCons @ A @ X3 @ Xs ) @ ( coinductive_LCons @ A @ X3 @ Ys ) ) ) ).
% Le_LCons
thf(fact_70_lSup_Odisc__iff_I2_J,axiom,
! [A: $tType,A3: set @ ( coinductive_llist @ A )] :
( ( coinductive_lnull @ A @ ( coinductive_lSup @ A @ A3 ) )
= ( ! [X: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ X @ A3 )
=> ( coinductive_lnull @ A @ X ) ) ) ) ).
% lSup.disc_iff(2)
thf(fact_71_lSup_Odisc_I1_J,axiom,
! [A: $tType,A3: set @ ( coinductive_llist @ A )] :
( ! [X2: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ X2 @ A3 )
=> ( coinductive_lnull @ A @ X2 ) )
=> ( coinductive_lnull @ A @ ( coinductive_lSup @ A @ A3 ) ) ) ).
% lSup.disc(1)
thf(fact_72_lSup_Odisc_I2_J,axiom,
! [A: $tType,A3: set @ ( coinductive_llist @ A )] :
( ~ ! [X6: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ X6 @ A3 )
=> ( coinductive_lnull @ A @ X6 ) )
=> ~ ( coinductive_lnull @ A @ ( coinductive_lSup @ A @ A3 ) ) ) ).
% lSup.disc(2)
thf(fact_73_LNil__lprefix,axiom,
! [A: $tType,Xs: coinductive_llist @ A] : ( coinductive_lprefix @ A @ ( coinductive_LNil @ A ) @ Xs ) ).
% LNil_lprefix
thf(fact_74_llist_Oconst__mono,axiom,
! [A: $tType,B: $tType,Ord: B > B > $o,C: coinductive_llist @ A] :
( comple1396247847notone @ B @ ( coinductive_llist @ A ) @ Ord @ ( coinductive_lprefix @ A )
@ ^ [F2: B] : C ) ).
% llist.const_mono
thf(fact_75_ord_Olnull__lmerge,axiom,
! [A: $tType,Less: A > A > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ ( hammin1328233080lmerge @ A @ Less @ Xs @ Ys ) )
= ( ( coinductive_lnull @ A @ Xs )
| ( coinductive_lnull @ A @ Ys ) ) ) ).
% ord.lnull_lmerge
thf(fact_76_ord_Olmerge_Odisc__iff_I2_J,axiom,
! [A: $tType,Less: A > A > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( ~ ( coinductive_lnull @ A @ ( hammin1328233080lmerge @ A @ Less @ Xs @ Ys ) ) )
= ( ~ ( coinductive_lnull @ A @ Xs )
& ~ ( coinductive_lnull @ A @ Ys ) ) ) ).
% ord.lmerge.disc_iff(2)
thf(fact_77_ord_Olmerge_Odisc_I1_J,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Less: A > A > $o] :
( ( ( coinductive_lnull @ A @ Xs )
| ( coinductive_lnull @ A @ Ys ) )
=> ( coinductive_lnull @ A @ ( hammin1328233080lmerge @ A @ Less @ Xs @ Ys ) ) ) ).
% ord.lmerge.disc(1)
thf(fact_78_ord_Olmerge_Odisc_I2_J,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Less: A > A > $o] :
( ~ ( coinductive_lnull @ A @ Xs )
=> ( ~ ( coinductive_lnull @ A @ Ys )
=> ~ ( coinductive_lnull @ A @ ( hammin1328233080lmerge @ A @ Less @ Xs @ Ys ) ) ) ) ).
% ord.lmerge.disc(2)
thf(fact_79_ord_Olmerge__simps,axiom,
! [A: $tType,Less: A > A > $o,X3: A,Y2: A,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( ( Less @ X3 @ Y2 )
=> ( ( hammin1328233080lmerge @ A @ Less @ ( coinductive_LCons @ A @ X3 @ Xs ) @ ( coinductive_LCons @ A @ Y2 @ Ys ) )
= ( coinductive_LCons @ A @ X3 @ ( hammin1328233080lmerge @ A @ Less @ Xs @ ( coinductive_LCons @ A @ Y2 @ Ys ) ) ) ) )
& ( ~ ( Less @ X3 @ Y2 )
=> ( ( ( Less @ Y2 @ X3 )
=> ( ( hammin1328233080lmerge @ A @ Less @ ( coinductive_LCons @ A @ X3 @ Xs ) @ ( coinductive_LCons @ A @ Y2 @ Ys ) )
= ( coinductive_LCons @ A @ Y2 @ ( hammin1328233080lmerge @ A @ Less @ ( coinductive_LCons @ A @ X3 @ Xs ) @ Ys ) ) ) )
& ( ~ ( Less @ Y2 @ X3 )
=> ( ( hammin1328233080lmerge @ A @ Less @ ( coinductive_LCons @ A @ X3 @ Xs ) @ ( coinductive_LCons @ A @ Y2 @ Ys ) )
= ( coinductive_LCons @ A @ Y2 @ ( hammin1328233080lmerge @ A @ Less @ Xs @ Ys ) ) ) ) ) ) ) ).
% ord.lmerge_simps
thf(fact_80_ord_Olmerge__eq__LNil__iff,axiom,
! [A: $tType,Less: A > A > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( ( hammin1328233080lmerge @ A @ Less @ Xs @ Ys )
= ( coinductive_LNil @ A ) )
= ( ( Xs
= ( coinductive_LNil @ A ) )
| ( Ys
= ( coinductive_LNil @ A ) ) ) ) ).
% ord.lmerge_eq_LNil_iff
thf(fact_81_ord_Olmerge__LNil_I1_J,axiom,
! [A: $tType,Less: A > A > $o,Ys: coinductive_llist @ A] :
( ( hammin1328233080lmerge @ A @ Less @ ( coinductive_LNil @ A ) @ Ys )
= ( coinductive_LNil @ A ) ) ).
% ord.lmerge_LNil(1)
thf(fact_82_ord_Olmerge__LNil_I2_J,axiom,
! [A: $tType,Less: A > A > $o,Xs: coinductive_llist @ A] :
( ( hammin1328233080lmerge @ A @ Less @ Xs @ ( coinductive_LNil @ A ) )
= ( coinductive_LNil @ A ) ) ).
% ord.lmerge_LNil(2)
thf(fact_83_mcont__LCons,axiom,
! [A: $tType,X3: A] : ( comple1403672095_mcont @ ( coinductive_llist @ A ) @ ( coinductive_llist @ A ) @ ( coinductive_lSup @ A ) @ ( coinductive_lprefix @ A ) @ ( coinductive_lSup @ A ) @ ( coinductive_lprefix @ A ) @ ( coinductive_LCons @ A @ X3 ) ) ).
% mcont_LCons
thf(fact_84_ord_Omono2mono__lmerge2,axiom,
! [A: $tType,Less: A > A > $o,Xs: coinductive_llist @ A] : ( comple1396247847notone @ ( coinductive_llist @ A ) @ ( coinductive_llist @ A ) @ ( coinductive_lprefix @ A ) @ ( coinductive_lprefix @ A ) @ ( hammin1328233080lmerge @ A @ Less @ Xs ) ) ).
% ord.mono2mono_lmerge2
thf(fact_85_ord_Omono2mono__lmerge1,axiom,
! [A: $tType,Less: A > A > $o,Ys: coinductive_llist @ A] :
( comple1396247847notone @ ( coinductive_llist @ A ) @ ( coinductive_llist @ A ) @ ( coinductive_lprefix @ A ) @ ( coinductive_lprefix @ A )
@ ^ [Xs3: coinductive_llist @ A] : ( hammin1328233080lmerge @ A @ Less @ Xs3 @ Ys ) ) ).
% ord.mono2mono_lmerge1
thf(fact_86_mcont2mcont__LCons,axiom,
! [A: $tType,C2: $tType,Lub: ( set @ C2 ) > C2,Ord: C2 > C2 > $o,T: C2 > ( coinductive_llist @ A ),X5: A] :
( ( comple1403672095_mcont @ C2 @ ( coinductive_llist @ A ) @ Lub @ Ord @ ( coinductive_lSup @ A ) @ ( coinductive_lprefix @ A ) @ T )
=> ( comple1403672095_mcont @ C2 @ ( coinductive_llist @ A ) @ Lub @ Ord @ ( coinductive_lSup @ A ) @ ( coinductive_lprefix @ A )
@ ^ [X: C2] : ( coinductive_LCons @ A @ X5 @ ( T @ X ) ) ) ) ).
% mcont2mcont_LCons
thf(fact_87_lprefix__down__linear,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Zs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_lprefix @ A @ Xs @ Zs )
=> ( ( coinductive_lprefix @ A @ Ys @ Zs )
=> ( ( coinductive_lprefix @ A @ Xs @ Ys )
| ( coinductive_lprefix @ A @ Ys @ Xs ) ) ) ) ).
% lprefix_down_linear
thf(fact_88_llist_Oleq__antisym,axiom,
! [A: $tType,X3: coinductive_llist @ A,Y2: coinductive_llist @ A] :
( ( coinductive_lprefix @ A @ X3 @ Y2 )
=> ( ( coinductive_lprefix @ A @ Y2 @ X3 )
=> ( X3 = Y2 ) ) ) ).
% llist.leq_antisym
thf(fact_89_lprefix__antisym,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_lprefix @ A @ Xs @ Ys )
=> ( ( coinductive_lprefix @ A @ Ys @ Xs )
=> ( Xs = Ys ) ) ) ).
% lprefix_antisym
thf(fact_90_llist_Oleq__trans,axiom,
! [A: $tType,X3: coinductive_llist @ A,Y2: coinductive_llist @ A,Z: coinductive_llist @ A] :
( ( coinductive_lprefix @ A @ X3 @ Y2 )
=> ( ( coinductive_lprefix @ A @ Y2 @ Z )
=> ( coinductive_lprefix @ A @ X3 @ Z ) ) ) ).
% llist.leq_trans
thf(fact_91_lprefix__trans,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A] :
( ( coinductive_lprefix @ A @ Xs @ Ys )
=> ( ( coinductive_lprefix @ A @ Ys @ Zs )
=> ( coinductive_lprefix @ A @ Xs @ Zs ) ) ) ).
% lprefix_trans
thf(fact_92_lhd__lmerge,axiom,
! [Xs: coinductive_llist @ a,Ys: coinductive_llist @ a] :
( ~ ( coinductive_lnull @ a @ Xs )
=> ( ~ ( coinductive_lnull @ a @ Ys )
=> ( ( ( less @ ( coinductive_lhd @ a @ Xs ) @ ( coinductive_lhd @ a @ Ys ) )
=> ( ( coinductive_lhd @ a @ ( hammin1328233080lmerge @ a @ less @ Xs @ Ys ) )
= ( coinductive_lhd @ a @ Xs ) ) )
& ( ~ ( less @ ( coinductive_lhd @ a @ Xs ) @ ( coinductive_lhd @ a @ Ys ) )
=> ( ( coinductive_lhd @ a @ ( hammin1328233080lmerge @ a @ less @ Xs @ Ys ) )
= ( coinductive_lhd @ a @ Ys ) ) ) ) ) ) ).
% lhd_lmerge
thf(fact_93_mcont__mono,axiom,
! [B: $tType,A: $tType,Luba: ( set @ A ) > A,Orda: A > A > $o,Lubb: ( set @ B ) > B,Ordb: B > B > $o,F: A > B] :
( ( comple1403672095_mcont @ A @ B @ Luba @ Orda @ Lubb @ Ordb @ F )
=> ( comple1396247847notone @ A @ B @ Orda @ Ordb @ F ) ) ).
% mcont_mono
thf(fact_94_in__lset__lmergeD,axiom,
! [X3: a,Xs: coinductive_llist @ a,Ys: coinductive_llist @ a] :
( ( member @ a @ X3 @ ( coinductive_lset @ a @ ( hammin1328233080lmerge @ a @ less @ Xs @ Ys ) ) )
=> ( ( member @ a @ X3 @ ( coinductive_lset @ a @ Xs ) )
| ( member @ a @ X3 @ ( coinductive_lset @ a @ Ys ) ) ) ) ).
% in_lset_lmergeD
thf(fact_95_local_Olexordp__eq__refl,axiom,
! [Xs: list @ a] : ( lexordp_eq @ a @ less @ Xs @ Xs ) ).
% local.lexordp_eq_refl
thf(fact_96_lfinite__lmergeD,axiom,
! [Xs: coinductive_llist @ a,Ys: coinductive_llist @ a] :
( ( coinductive_lfinite @ a @ ( hammin1328233080lmerge @ a @ less @ Xs @ Ys ) )
=> ( ( coinductive_lfinite @ a @ Xs )
| ( coinductive_lfinite @ a @ Ys ) ) ) ).
% lfinite_lmergeD
thf(fact_97_local_OgreaterThanLessThan__iff,axiom,
! [I: a,L: a,U: a] :
( ( member @ a @ I @ ( set_gr1161524159ssThan @ a @ less @ L @ U ) )
= ( ( less @ L @ I )
& ( less @ I @ U ) ) ) ).
% local.greaterThanLessThan_iff
thf(fact_98_ord_OlessThan__def,axiom,
! [A: $tType] :
( ( set_lessThan @ A )
= ( ^ [Less2: A > A > $o,U2: A] :
( collect @ A
@ ^ [X: A] : ( Less2 @ X @ U2 ) ) ) ) ).
% ord.lessThan_def
thf(fact_99_ord_OgreaterThan__def,axiom,
! [A: $tType] :
( ( set_greaterThan @ A )
= ( ^ [Less2: A > A > $o,L2: A] : ( collect @ A @ ( Less2 @ L2 ) ) ) ) ).
% ord.greaterThan_def
thf(fact_100_local_OgreaterThanLessThan__eq,axiom,
! [A2: a,B2: a] :
( ( set_gr1161524159ssThan @ a @ less @ A2 @ B2 )
= ( inf_inf @ ( set @ a ) @ ( set_greaterThan @ a @ less @ A2 ) @ ( set_lessThan @ a @ less @ B2 ) ) ) ).
% local.greaterThanLessThan_eq
thf(fact_101_local_OgreaterThanLessThan__def,axiom,
! [L: a,U: a] :
( ( set_gr1161524159ssThan @ a @ less @ L @ U )
= ( inf_inf @ ( set @ a ) @ ( set_greaterThan @ a @ less @ L ) @ ( set_lessThan @ a @ less @ U ) ) ) ).
% local.greaterThanLessThan_def
thf(fact_102_lfinite__LCons,axiom,
! [A: $tType,X3: A,Xs: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ ( coinductive_LCons @ A @ X3 @ Xs ) )
= ( coinductive_lfinite @ A @ Xs ) ) ).
% lfinite_LCons
thf(fact_103_lfinite__code_I2_J,axiom,
! [B: $tType,X3: B,Xs: coinductive_llist @ B] :
( ( coinductive_lfinite @ B @ ( coinductive_LCons @ B @ X3 @ Xs ) )
= ( coinductive_lfinite @ B @ Xs ) ) ).
% lfinite_code(2)
thf(fact_104_lfinite__code_I1_J,axiom,
! [A: $tType] : ( coinductive_lfinite @ A @ ( coinductive_LNil @ A ) ) ).
% lfinite_code(1)
thf(fact_105_llist_Oset__sel_I1_J,axiom,
! [A: $tType,A2: coinductive_llist @ A] :
( ~ ( coinductive_lnull @ A @ A2 )
=> ( member @ A @ ( coinductive_lhd @ A @ A2 ) @ ( coinductive_lset @ A @ A2 ) ) ) ).
% llist.set_sel(1)
thf(fact_106_ord_OgreaterThanLessThan__iff,axiom,
! [A: $tType,I: A,Less: A > A > $o,L: A,U: A] :
( ( member @ A @ I @ ( set_gr1161524159ssThan @ A @ Less @ L @ U ) )
= ( ( Less @ L @ I )
& ( Less @ I @ U ) ) ) ).
% ord.greaterThanLessThan_iff
thf(fact_107_lprefix__lfiniteD,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_lprefix @ A @ Xs @ Ys )
=> ( ( coinductive_lfinite @ A @ Ys )
=> ( coinductive_lfinite @ A @ Xs ) ) ) ).
% lprefix_lfiniteD
thf(fact_108_not__lfinite__lprefix__conv__eq,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ~ ( coinductive_lfinite @ A @ Xs )
=> ( ( coinductive_lprefix @ A @ Xs @ Ys )
= ( Xs = Ys ) ) ) ).
% not_lfinite_lprefix_conv_eq
thf(fact_109_lfinite__LConsI,axiom,
! [A: $tType,Xs: coinductive_llist @ A,X3: A] :
( ( coinductive_lfinite @ A @ Xs )
=> ( coinductive_lfinite @ A @ ( coinductive_LCons @ A @ X3 @ Xs ) ) ) ).
% lfinite_LConsI
thf(fact_110_lnull__imp__lfinite,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ Xs )
=> ( coinductive_lfinite @ A @ Xs ) ) ).
% lnull_imp_lfinite
thf(fact_111_lfinite__LNil,axiom,
! [A: $tType] : ( coinductive_lfinite @ A @ ( coinductive_LNil @ A ) ) ).
% lfinite_LNil
thf(fact_112_lfinite__lSupD,axiom,
! [A: $tType,A3: set @ ( coinductive_llist @ A )] :
( ( coinductive_lfinite @ A @ ( coinductive_lSup @ A @ A3 ) )
=> ! [X6: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ X6 @ A3 )
=> ( coinductive_lfinite @ A @ X6 ) ) ) ).
% lfinite_lSupD
thf(fact_113_lhd__LCons,axiom,
! [A: $tType,X21: A,X22: coinductive_llist @ A] :
( ( coinductive_lhd @ A @ ( coinductive_LCons @ A @ X21 @ X22 ) )
= X21 ) ).
% lhd_LCons
thf(fact_114_ltakeWhile_Oexhaust,axiom,
! [A: $tType,Xs: coinductive_llist @ A,P: A > $o] :
( ~ ( ( coinductive_lnull @ A @ Xs )
| ~ ( P @ ( coinductive_lhd @ A @ Xs ) ) )
=> ~ ( ~ ( coinductive_lnull @ A @ Xs )
=> ~ ( P @ ( coinductive_lhd @ A @ Xs ) ) ) ) ).
% ltakeWhile.exhaust
thf(fact_115_llist_Oset__induct,axiom,
! [A: $tType,X3: A,A2: coinductive_llist @ A,P: A > ( coinductive_llist @ A ) > $o] :
( ( member @ A @ X3 @ ( 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 @ X3 @ A2 ) ) ) ) ).
% llist.set_induct
thf(fact_116_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_117_lset__induct_H,axiom,
! [A: $tType,X3: A,Xs: coinductive_llist @ A,P: ( coinductive_llist @ A ) > $o] :
( ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs ) )
=> ( ! [Xs4: coinductive_llist @ A] : ( P @ ( coinductive_LCons @ A @ X3 @ Xs4 ) )
=> ( ! [X7: A,Xs4: coinductive_llist @ A] :
( ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs4 ) )
=> ( ( P @ Xs4 )
=> ( P @ ( coinductive_LCons @ A @ X7 @ Xs4 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% lset_induct'
thf(fact_118_lset__induct,axiom,
! [A: $tType,X3: A,Xs: coinductive_llist @ A,P: ( coinductive_llist @ A ) > $o] :
( ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs ) )
=> ( ! [Xs4: coinductive_llist @ A] : ( P @ ( coinductive_LCons @ A @ X3 @ Xs4 ) )
=> ( ! [X7: A,Xs4: coinductive_llist @ A] :
( ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs4 ) )
=> ( ( X3 != X7 )
=> ( ( P @ Xs4 )
=> ( P @ ( coinductive_LCons @ A @ X7 @ Xs4 ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% lset_induct
thf(fact_119_lset__cases,axiom,
! [A: $tType,X3: A,Xs: coinductive_llist @ A] :
( ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs ) )
=> ( ! [Xs7: coinductive_llist @ A] :
( Xs
!= ( coinductive_LCons @ A @ X3 @ Xs7 ) )
=> ~ ! [X7: A,Xs7: coinductive_llist @ A] :
( ( Xs
= ( coinductive_LCons @ A @ X7 @ Xs7 ) )
=> ~ ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs7 ) ) ) ) ) ).
% lset_cases
thf(fact_120_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_121_llist_Oset__intros_I2_J,axiom,
! [A: $tType,X3: A,A22: coinductive_llist @ A,A1: A] :
( ( member @ A @ X3 @ ( coinductive_lset @ A @ A22 ) )
=> ( member @ A @ X3 @ ( coinductive_lset @ A @ ( coinductive_LCons @ A @ A1 @ A22 ) ) ) ) ).
% llist.set_intros(2)
thf(fact_122_lset__intros_I1_J,axiom,
! [A: $tType,X3: A,Xs: coinductive_llist @ A] : ( member @ A @ X3 @ ( coinductive_lset @ A @ ( coinductive_LCons @ A @ X3 @ Xs ) ) ) ).
% lset_intros(1)
thf(fact_123_lset__intros_I2_J,axiom,
! [A: $tType,X3: A,Xs: coinductive_llist @ A,X8: A] :
( ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs ) )
=> ( member @ A @ X3 @ ( coinductive_lset @ A @ ( coinductive_LCons @ A @ X8 @ Xs ) ) ) ) ).
% lset_intros(2)
thf(fact_124_ord_Olfinite__lmergeD,axiom,
! [A: $tType,Less: A > A > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ ( hammin1328233080lmerge @ A @ Less @ Xs @ Ys ) )
=> ( ( coinductive_lfinite @ A @ Xs )
| ( coinductive_lfinite @ A @ Ys ) ) ) ).
% ord.lfinite_lmergeD
thf(fact_125_ord_Oin__lset__lmergeD,axiom,
! [A: $tType,X3: A,Less: A > A > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( member @ A @ X3 @ ( coinductive_lset @ A @ ( hammin1328233080lmerge @ A @ Less @ Xs @ Ys ) ) )
=> ( ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs ) )
| ( member @ A @ X3 @ ( coinductive_lset @ A @ Ys ) ) ) ) ).
% ord.in_lset_lmergeD
thf(fact_126_ord_OgreaterThanLessThan__def,axiom,
! [A: $tType] :
( ( set_gr1161524159ssThan @ A )
= ( ^ [Less2: A > A > $o,L2: A,U2: A] : ( inf_inf @ ( set @ A ) @ ( set_greaterThan @ A @ Less2 @ L2 ) @ ( set_lessThan @ A @ Less2 @ U2 ) ) ) ) ).
% ord.greaterThanLessThan_def
thf(fact_127_ord_OgreaterThanLessThan__eq,axiom,
! [A: $tType] :
( ( set_gr1161524159ssThan @ A )
= ( ^ [Less2: A > A > $o,A4: A,B3: A] : ( inf_inf @ ( set @ A ) @ ( set_greaterThan @ A @ Less2 @ A4 ) @ ( set_lessThan @ A @ Less2 @ B3 ) ) ) ) ).
% ord.greaterThanLessThan_eq
thf(fact_128_lfinite_Oinducts,axiom,
! [A: $tType,X3: coinductive_llist @ A,P: ( coinductive_llist @ A ) > $o] :
( ( coinductive_lfinite @ A @ X3 )
=> ( ( P @ ( coinductive_LNil @ A ) )
=> ( ! [Xs4: coinductive_llist @ A,X2: A] :
( ( coinductive_lfinite @ A @ Xs4 )
=> ( ( P @ Xs4 )
=> ( P @ ( coinductive_LCons @ A @ X2 @ Xs4 ) ) ) )
=> ( P @ X3 ) ) ) ) ).
% lfinite.inducts
thf(fact_129_lfinite_Osimps,axiom,
! [A: $tType] :
( ( coinductive_lfinite @ A )
= ( ^ [A4: coinductive_llist @ A] :
( ( A4
= ( coinductive_LNil @ A ) )
| ? [Xs3: coinductive_llist @ A,X: A] :
( ( A4
= ( coinductive_LCons @ A @ X @ Xs3 ) )
& ( coinductive_lfinite @ A @ Xs3 ) ) ) ) ) ).
% lfinite.simps
thf(fact_130_lfinite_Ocases,axiom,
! [A: $tType,A2: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ A2 )
=> ( ( A2
!= ( coinductive_LNil @ A ) )
=> ~ ! [Xs4: coinductive_llist @ A] :
( ? [X2: A] :
( A2
= ( coinductive_LCons @ A @ X2 @ Xs4 ) )
=> ~ ( coinductive_lfinite @ A @ Xs4 ) ) ) ) ).
% lfinite.cases
thf(fact_131_lprefix__lhdD,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_lprefix @ A @ Xs @ Ys )
=> ( ~ ( coinductive_lnull @ A @ Xs )
=> ( ( coinductive_lhd @ A @ Xs )
= ( coinductive_lhd @ A @ Ys ) ) ) ) ).
% lprefix_lhdD
thf(fact_132_ord_Olhd__lmerge,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Less: A > A > $o] :
( ~ ( coinductive_lnull @ A @ Xs )
=> ( ~ ( coinductive_lnull @ A @ Ys )
=> ( ( ( Less @ ( coinductive_lhd @ A @ Xs ) @ ( coinductive_lhd @ A @ Ys ) )
=> ( ( coinductive_lhd @ A @ ( hammin1328233080lmerge @ A @ Less @ Xs @ Ys ) )
= ( coinductive_lhd @ A @ Xs ) ) )
& ( ~ ( Less @ ( coinductive_lhd @ A @ Xs ) @ ( coinductive_lhd @ A @ Ys ) )
=> ( ( coinductive_lhd @ A @ ( hammin1328233080lmerge @ A @ Less @ Xs @ Ys ) )
= ( coinductive_lhd @ A @ Ys ) ) ) ) ) ) ).
% ord.lhd_lmerge
thf(fact_133_lSup__finite__prefixes,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( coinductive_lSup @ A
@ ( collect @ ( coinductive_llist @ A )
@ ^ [Ys4: coinductive_llist @ A] :
( ( coinductive_lprefix @ A @ Ys4 @ Xs )
& ( coinductive_lfinite @ A @ Ys4 ) ) ) )
= Xs ) ).
% lSup_finite_prefixes
thf(fact_134_lSup__strict__prefixes,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ~ ( coinductive_lfinite @ A @ Xs )
=> ( ( coinductive_lSup @ A
@ ( collect @ ( coinductive_llist @ A )
@ ^ [Ys4: coinductive_llist @ A] :
( ( coinductive_lprefix @ A @ Ys4 @ Xs )
& ( Ys4 != Xs ) ) ) )
= Xs ) ) ).
% lSup_strict_prefixes
thf(fact_135_lSup__finite__gen__prefixes,axiom,
! [A: $tType,Zs: coinductive_llist @ A,Xs: coinductive_llist @ A] :
( ( coinductive_lprefix @ A @ Zs @ Xs )
=> ( ( coinductive_lfinite @ A @ Zs )
=> ( ( coinductive_lSup @ A
@ ( collect @ ( coinductive_llist @ A )
@ ^ [Ys4: coinductive_llist @ A] :
( ( coinductive_lprefix @ A @ Ys4 @ Xs )
& ( coinductive_lprefix @ A @ Zs @ Ys4 )
& ( coinductive_lfinite @ A @ Ys4 ) ) ) )
= Xs ) ) ) ).
% lSup_finite_gen_prefixes
thf(fact_136_mcont__monoD,axiom,
! [B: $tType,A: $tType,Luba: ( set @ A ) > A,Orda: A > A > $o,Lubb: ( set @ B ) > B,Ordb: B > B > $o,F: A > B,X3: A,Y2: A] :
( ( comple1403672095_mcont @ A @ B @ Luba @ Orda @ Lubb @ Ordb @ F )
=> ( ( Orda @ X3 @ Y2 )
=> ( Ordb @ ( F @ X3 ) @ ( F @ Y2 ) ) ) ) ).
% mcont_monoD
thf(fact_137_ord_OgreaterThan__iff,axiom,
! [A: $tType,I: A,Less: A > A > $o,K: A] :
( ( member @ A @ I @ ( set_greaterThan @ A @ Less @ K ) )
= ( Less @ K @ I ) ) ).
% ord.greaterThan_iff
thf(fact_138_ord_OlessThan__iff,axiom,
! [A: $tType,I: A,Less: A > A > $o,K: A] :
( ( member @ A @ I @ ( set_lessThan @ A @ Less @ K ) )
= ( Less @ I @ K ) ) ).
% ord.lessThan_iff
thf(fact_139_monotone__id_H,axiom,
! [A: $tType,Ord: A > A > $o] :
( comple1396247847notone @ A @ A @ Ord @ Ord
@ ^ [X: A] : X ) ).
% monotone_id'
thf(fact_140_mcont__if,axiom,
! [B: $tType,A: $tType,Luba: ( set @ A ) > A,Orda: A > A > $o,Lubb: ( set @ B ) > B,Ordb: B > B > $o,F: A > B,G: A > B,C: $o] :
( ( comple1403672095_mcont @ A @ B @ Luba @ Orda @ Lubb @ Ordb @ F )
=> ( ( comple1403672095_mcont @ A @ B @ Luba @ Orda @ Lubb @ Ordb @ G )
=> ( comple1403672095_mcont @ A @ B @ Luba @ Orda @ Lubb @ Ordb
@ ^ [X: A] : ( if @ B @ C @ ( F @ X ) @ ( G @ X ) ) ) ) ) ).
% mcont_if
thf(fact_141_mcont__id_H,axiom,
! [A: $tType,Lub: ( set @ A ) > A,Ord: A > A > $o] :
( comple1403672095_mcont @ A @ A @ Lub @ Ord @ Lub @ Ord
@ ^ [X: A] : X ) ).
% mcont_id'
thf(fact_142_lmerge_Ocode,axiom,
! [Xs: coinductive_llist @ a,Ys: coinductive_llist @ a] :
( ( hammin1328233080lmerge @ a @ less @ Xs @ Ys )
= ( coindu1381640503_llist @ ( coinductive_llist @ a ) @ a @ ( coinductive_LNil @ a )
@ ^ [X: a,Xs5: coinductive_llist @ a] :
( coindu1381640503_llist @ ( coinductive_llist @ a ) @ a @ ( coinductive_LNil @ a )
@ ^ [Y3: a,Ys6: coinductive_llist @ a] : ( if @ ( coinductive_llist @ a ) @ ( less @ ( coinductive_lhd @ a @ Xs ) @ ( coinductive_lhd @ a @ Ys ) ) @ ( coinductive_LCons @ a @ X @ ( hammin1328233080lmerge @ a @ less @ Xs5 @ Ys ) ) @ ( coinductive_LCons @ a @ Y3 @ ( if @ ( coinductive_llist @ a ) @ ( less @ ( coinductive_lhd @ a @ Ys ) @ ( coinductive_lhd @ a @ Xs ) ) @ ( hammin1328233080lmerge @ a @ less @ Xs @ Ys6 ) @ ( hammin1328233080lmerge @ a @ less @ Xs5 @ Ys6 ) ) ) )
@ Ys )
@ Xs ) ) ).
% lmerge.code
thf(fact_143_ltl__lmerge,axiom,
! [Xs: coinductive_llist @ a,Ys: coinductive_llist @ a] :
( ~ ( coinductive_lnull @ a @ Xs )
=> ( ~ ( coinductive_lnull @ a @ Ys )
=> ( ( ( less @ ( coinductive_lhd @ a @ Xs ) @ ( coinductive_lhd @ a @ Ys ) )
=> ( ( coinductive_ltl @ a @ ( hammin1328233080lmerge @ a @ less @ Xs @ Ys ) )
= ( hammin1328233080lmerge @ a @ less @ ( coinductive_ltl @ a @ Xs ) @ Ys ) ) )
& ( ~ ( less @ ( coinductive_lhd @ a @ Xs ) @ ( coinductive_lhd @ a @ Ys ) )
=> ( ( ( less @ ( coinductive_lhd @ a @ Ys ) @ ( coinductive_lhd @ a @ Xs ) )
=> ( ( coinductive_ltl @ a @ ( hammin1328233080lmerge @ a @ less @ Xs @ Ys ) )
= ( hammin1328233080lmerge @ a @ less @ Xs @ ( coinductive_ltl @ a @ Ys ) ) ) )
& ( ~ ( less @ ( coinductive_lhd @ a @ Ys ) @ ( coinductive_lhd @ a @ Xs ) )
=> ( ( coinductive_ltl @ a @ ( hammin1328233080lmerge @ a @ less @ Xs @ Ys ) )
= ( hammin1328233080lmerge @ a @ less @ ( coinductive_ltl @ a @ Xs ) @ ( coinductive_ltl @ a @ Ys ) ) ) ) ) ) ) ) ) ).
% ltl_lmerge
thf(fact_144_local_Olexordp__eq_ONil,axiom,
! [Ys: list @ a] : ( lexordp_eq @ a @ less @ ( nil @ a ) @ Ys ) ).
% local.lexordp_eq.Nil
thf(fact_145_Int__iff,axiom,
! [A: $tType,C: A,A3: set @ A,B4: set @ A] :
( ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A3 @ B4 ) )
= ( ( member @ A @ C @ A3 )
& ( member @ A @ C @ B4 ) ) ) ).
% Int_iff
thf(fact_146_IntI,axiom,
! [A: $tType,C: A,A3: set @ A,B4: set @ A] :
( ( member @ A @ C @ A3 )
=> ( ( member @ A @ C @ B4 )
=> ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A3 @ B4 ) ) ) ) ).
% IntI
thf(fact_147_lfinite__ltl,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ ( coinductive_ltl @ A @ Xs ) )
= ( coinductive_lfinite @ A @ Xs ) ) ).
% lfinite_ltl
thf(fact_148_lhd__LCons__ltl,axiom,
! [A: $tType,Llist: coinductive_llist @ A] :
( ~ ( coinductive_lnull @ A @ Llist )
=> ( ( coinductive_LCons @ A @ ( coinductive_lhd @ A @ Llist ) @ ( coinductive_ltl @ A @ Llist ) )
= Llist ) ) ).
% lhd_LCons_ltl
thf(fact_149_local_Olexordp__eq__simps_I1_J,axiom,
! [Ys: list @ a] : ( lexordp_eq @ a @ less @ ( nil @ a ) @ Ys ) ).
% local.lexordp_eq_simps(1)
thf(fact_150_local_Olexordp__eq__simps_I2_J,axiom,
! [Xs: list @ a] :
( ( lexordp_eq @ a @ less @ Xs @ ( nil @ a ) )
= ( Xs
= ( nil @ a ) ) ) ).
% local.lexordp_eq_simps(2)
thf(fact_151_in__lset__ltlD,axiom,
! [A: $tType,X3: A,Xs: coinductive_llist @ A] :
( ( member @ A @ X3 @ ( coinductive_lset @ A @ ( coinductive_ltl @ A @ Xs ) ) )
=> ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs ) ) ) ).
% in_lset_ltlD
thf(fact_152_llist_Ocase__eq__if,axiom,
! [A: $tType,B: $tType] :
( ( coindu1381640503_llist @ B @ A )
= ( ^ [F1: B,F22: A > ( coinductive_llist @ A ) > B,Llist2: coinductive_llist @ A] : ( if @ B @ ( coinductive_lnull @ A @ Llist2 ) @ F1 @ ( F22 @ ( coinductive_lhd @ A @ Llist2 ) @ ( coinductive_ltl @ A @ Llist2 ) ) ) ) ) ).
% llist.case_eq_if
thf(fact_153_ltl__def,axiom,
! [A: $tType] :
( ( coinductive_ltl @ A )
= ( coindu1381640503_llist @ ( coinductive_llist @ A ) @ A @ ( coinductive_LNil @ A )
@ ^ [X213: A,X223: coinductive_llist @ A] : X223 ) ) ).
% ltl_def
thf(fact_154_llist_Ocase__distrib,axiom,
! [B: $tType,C2: $tType,A: $tType,H: B > C2,F12: B,F23: A > ( coinductive_llist @ A ) > B,Llist: coinductive_llist @ A] :
( ( H @ ( coindu1381640503_llist @ B @ A @ F12 @ F23 @ Llist ) )
= ( coindu1381640503_llist @ C2 @ A @ ( H @ F12 )
@ ^ [X1: A,X23: coinductive_llist @ A] : ( H @ ( F23 @ X1 @ X23 ) )
@ Llist ) ) ).
% llist.case_distrib
thf(fact_155_lprefix__ltlI,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_lprefix @ A @ Xs @ Ys )
=> ( coinductive_lprefix @ A @ ( coinductive_ltl @ A @ Xs ) @ ( coinductive_ltl @ A @ Ys ) ) ) ).
% lprefix_ltlI
thf(fact_156_ltl__simps_I2_J,axiom,
! [A: $tType,X21: A,X22: coinductive_llist @ A] :
( ( coinductive_ltl @ A @ ( coinductive_LCons @ A @ X21 @ X22 ) )
= X22 ) ).
% ltl_simps(2)
thf(fact_157_lnull__ltlI,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ Xs )
=> ( coinductive_lnull @ A @ ( coinductive_ltl @ A @ Xs ) ) ) ).
% lnull_ltlI
thf(fact_158_ltl__simps_I1_J,axiom,
! [A: $tType] :
( ( coinductive_ltl @ A @ ( coinductive_LNil @ A ) )
= ( coinductive_LNil @ A ) ) ).
% ltl_simps(1)
thf(fact_159_llist_Osimps_I5_J,axiom,
! [B: $tType,A: $tType,F12: B,F23: A > ( coinductive_llist @ A ) > B,X21: A,X22: coinductive_llist @ A] :
( ( coindu1381640503_llist @ B @ A @ F12 @ F23 @ ( coinductive_LCons @ A @ X21 @ X22 ) )
= ( F23 @ X21 @ X22 ) ) ).
% llist.simps(5)
thf(fact_160_llist_Osimps_I4_J,axiom,
! [A: $tType,B: $tType,F12: B,F23: A > ( coinductive_llist @ A ) > B] :
( ( coindu1381640503_llist @ B @ A @ F12 @ F23 @ ( coinductive_LNil @ A ) )
= F12 ) ).
% llist.simps(4)
thf(fact_161_llist_Osplit__sel,axiom,
! [B: $tType,A: $tType,P: B > $o,F12: B,F23: A > ( coinductive_llist @ A ) > B,Llist: coinductive_llist @ A] :
( ( P @ ( coindu1381640503_llist @ B @ A @ F12 @ F23 @ Llist ) )
= ( ( ( Llist
= ( coinductive_LNil @ A ) )
=> ( P @ F12 ) )
& ( ( Llist
= ( coinductive_LCons @ A @ ( coinductive_lhd @ A @ Llist ) @ ( coinductive_ltl @ A @ Llist ) ) )
=> ( P @ ( F23 @ ( coinductive_lhd @ A @ Llist ) @ ( coinductive_ltl @ A @ Llist ) ) ) ) ) ) ).
% llist.split_sel
thf(fact_162_llist_Osplit__sel__asm,axiom,
! [B: $tType,A: $tType,P: B > $o,F12: B,F23: A > ( coinductive_llist @ A ) > B,Llist: coinductive_llist @ A] :
( ( P @ ( coindu1381640503_llist @ B @ A @ F12 @ F23 @ Llist ) )
= ( ~ ( ( ( Llist
= ( coinductive_LNil @ A ) )
& ~ ( P @ F12 ) )
| ( ( Llist
= ( coinductive_LCons @ A @ ( coinductive_lhd @ A @ Llist ) @ ( coinductive_ltl @ A @ Llist ) ) )
& ~ ( P @ ( F23 @ ( coinductive_lhd @ A @ Llist ) @ ( coinductive_ltl @ A @ Llist ) ) ) ) ) ) ) ).
% llist.split_sel_asm
thf(fact_163_llist__case__mono,axiom,
! [C2: $tType,B: $tType,A: $tType,Orda: A > A > $o,Ordb: B > B > $o,Lnil: A > B,Lcons: A > C2 > ( coinductive_llist @ C2 ) > B,X3: coinductive_llist @ C2] :
( ( comple1396247847notone @ A @ B @ Orda @ Ordb @ Lnil )
=> ( ! [X2: C2,Xs4: coinductive_llist @ C2] :
( comple1396247847notone @ A @ B @ Orda @ Ordb
@ ^ [F2: A] : ( Lcons @ F2 @ X2 @ Xs4 ) )
=> ( comple1396247847notone @ A @ B @ Orda @ Ordb
@ ^ [F2: A] : ( coindu1381640503_llist @ B @ C2 @ ( Lnil @ F2 ) @ ( Lcons @ F2 ) @ X3 ) ) ) ) ).
% llist_case_mono
thf(fact_164_mcont__llist__case,axiom,
! [C2: $tType,B: $tType,A: $tType,Luba: ( set @ A ) > A,Orda: A > A > $o,Lubb: ( set @ B ) > B,Ordb: B > B > $o,F: A > B,G: C2 > ( coinductive_llist @ C2 ) > A > B,Xs: coinductive_llist @ C2] :
( ( comple1403672095_mcont @ A @ B @ Luba @ Orda @ Lubb @ Ordb @ F )
=> ( ! [X2: C2,Xs4: coinductive_llist @ C2] : ( comple1403672095_mcont @ A @ B @ Luba @ Orda @ Lubb @ Ordb @ ( G @ X2 @ Xs4 ) )
=> ( comple1403672095_mcont @ A @ B @ Luba @ Orda @ Lubb @ Ordb
@ ^ [Y3: A] :
( coindu1381640503_llist @ B @ C2 @ ( F @ Y3 )
@ ^ [X: C2,Xs5: coinductive_llist @ C2] : ( G @ X @ Xs5 @ Y3 )
@ Xs ) ) ) ) ).
% mcont_llist_case
thf(fact_165_llist_Ocoinduct__strong,axiom,
! [A: $tType,R: ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o,Llist: coinductive_llist @ A,Llist3: coinductive_llist @ A] :
( ( R @ Llist @ Llist3 )
=> ( ! [Llist4: coinductive_llist @ A,Llist5: coinductive_llist @ A] :
( ( R @ Llist4 @ Llist5 )
=> ( ( ( coinductive_lnull @ A @ Llist4 )
= ( coinductive_lnull @ A @ Llist5 ) )
& ( ~ ( coinductive_lnull @ A @ Llist4 )
=> ( ~ ( coinductive_lnull @ A @ Llist5 )
=> ( ( ( coinductive_lhd @ A @ Llist4 )
= ( coinductive_lhd @ A @ Llist5 ) )
& ( ( R @ ( coinductive_ltl @ A @ Llist4 ) @ ( coinductive_ltl @ A @ Llist5 ) )
| ( ( coinductive_ltl @ A @ Llist4 )
= ( coinductive_ltl @ A @ Llist5 ) ) ) ) ) ) ) )
=> ( Llist = Llist3 ) ) ) ).
% llist.coinduct_strong
thf(fact_166_llist_Ocoinduct,axiom,
! [A: $tType,R: ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o,Llist: coinductive_llist @ A,Llist3: coinductive_llist @ A] :
( ( R @ Llist @ Llist3 )
=> ( ! [Llist4: coinductive_llist @ A,Llist5: coinductive_llist @ A] :
( ( R @ Llist4 @ Llist5 )
=> ( ( ( coinductive_lnull @ A @ Llist4 )
= ( coinductive_lnull @ A @ Llist5 ) )
& ( ~ ( coinductive_lnull @ A @ Llist4 )
=> ( ~ ( coinductive_lnull @ A @ Llist5 )
=> ( ( ( coinductive_lhd @ A @ Llist4 )
= ( coinductive_lhd @ A @ Llist5 ) )
& ( R @ ( coinductive_ltl @ A @ Llist4 ) @ ( coinductive_ltl @ A @ Llist5 ) ) ) ) ) ) )
=> ( Llist = Llist3 ) ) ) ).
% llist.coinduct
thf(fact_167_llist_Oexpand,axiom,
! [A: $tType,Llist: coinductive_llist @ A,Llist3: coinductive_llist @ A] :
( ( ( coinductive_lnull @ A @ Llist )
= ( coinductive_lnull @ A @ Llist3 ) )
=> ( ( ~ ( coinductive_lnull @ A @ Llist )
=> ( ~ ( coinductive_lnull @ A @ Llist3 )
=> ( ( ( coinductive_lhd @ A @ Llist )
= ( coinductive_lhd @ A @ Llist3 ) )
& ( ( coinductive_ltl @ A @ Llist )
= ( coinductive_ltl @ A @ Llist3 ) ) ) ) )
=> ( Llist = Llist3 ) ) ) ).
% llist.expand
thf(fact_168_llist_Oset__sel_I2_J,axiom,
! [A: $tType,A2: coinductive_llist @ A,X3: A] :
( ~ ( coinductive_lnull @ A @ A2 )
=> ( ( member @ A @ X3 @ ( coinductive_lset @ A @ ( coinductive_ltl @ A @ A2 ) ) )
=> ( member @ A @ X3 @ ( coinductive_lset @ A @ A2 ) ) ) ) ).
% llist.set_sel(2)
thf(fact_169_lfinite__induct,axiom,
! [A: $tType,Xs: coinductive_llist @ A,P: ( coinductive_llist @ A ) > $o] :
( ( coinductive_lfinite @ A @ Xs )
=> ( ! [Xs4: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ Xs4 )
=> ( P @ Xs4 ) )
=> ( ! [Xs4: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ Xs4 )
=> ( ~ ( coinductive_lnull @ A @ Xs4 )
=> ( ( P @ ( coinductive_ltl @ A @ Xs4 ) )
=> ( P @ Xs4 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% lfinite_induct
thf(fact_170_monotone__ltl,axiom,
! [A: $tType] : ( comple1396247847notone @ ( coinductive_llist @ A ) @ ( coinductive_llist @ A ) @ ( coinductive_lprefix @ A ) @ ( coinductive_lprefix @ A ) @ ( coinductive_ltl @ A ) ) ).
% monotone_ltl
thf(fact_171_mono2mono__ltl,axiom,
! [A: $tType,C2: $tType,Orda: C2 > C2 > $o,T: C2 > ( coinductive_llist @ A )] :
( ( comple1396247847notone @ C2 @ ( coinductive_llist @ A ) @ Orda @ ( coinductive_lprefix @ A ) @ T )
=> ( comple1396247847notone @ C2 @ ( coinductive_llist @ A ) @ Orda @ ( coinductive_lprefix @ A )
@ ^ [X: C2] : ( coinductive_ltl @ A @ ( T @ X ) ) ) ) ).
% mono2mono_ltl
thf(fact_172_lprefix__expand,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( ~ ( coinductive_lnull @ A @ Xs )
=> ( ~ ( coinductive_lnull @ A @ Ys )
& ( ( coinductive_lhd @ A @ Xs )
= ( coinductive_lhd @ A @ Ys ) )
& ( coinductive_lprefix @ A @ ( coinductive_ltl @ A @ Xs ) @ ( coinductive_ltl @ A @ Ys ) ) ) )
=> ( coinductive_lprefix @ A @ Xs @ Ys ) ) ).
% lprefix_expand
thf(fact_173_lprefix__coinduct,axiom,
! [A: $tType,P: ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( P @ Xs @ Ys )
=> ( ! [Xs4: coinductive_llist @ A,Ys3: coinductive_llist @ A] :
( ( P @ Xs4 @ Ys3 )
=> ( ( ( coinductive_lnull @ A @ Ys3 )
=> ( coinductive_lnull @ A @ Xs4 ) )
& ( ~ ( coinductive_lnull @ A @ Xs4 )
=> ( ~ ( coinductive_lnull @ A @ Ys3 )
=> ( ( ( coinductive_lhd @ A @ Xs4 )
= ( coinductive_lhd @ A @ Ys3 ) )
& ( ( P @ ( coinductive_ltl @ A @ Xs4 ) @ ( coinductive_ltl @ A @ Ys3 ) )
| ( coinductive_lprefix @ A @ ( coinductive_ltl @ A @ Xs4 ) @ ( coinductive_ltl @ A @ Ys3 ) ) ) ) ) ) ) )
=> ( coinductive_lprefix @ A @ Xs @ Ys ) ) ) ).
% lprefix_coinduct
thf(fact_174_eq__LConsD,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Y2: A,Ys: coinductive_llist @ A] :
( ( Xs
= ( coinductive_LCons @ A @ Y2 @ Ys ) )
=> ( ( Xs
!= ( coinductive_LNil @ A ) )
& ( ( coinductive_lhd @ A @ Xs )
= Y2 )
& ( ( coinductive_ltl @ A @ Xs )
= Ys ) ) ) ).
% eq_LConsD
thf(fact_175_llist_Oexhaust__sel,axiom,
! [A: $tType,Llist: coinductive_llist @ A] :
( ( Llist
!= ( coinductive_LNil @ A ) )
=> ( Llist
= ( coinductive_LCons @ A @ ( coinductive_lhd @ A @ Llist ) @ ( coinductive_ltl @ A @ Llist ) ) ) ) ).
% llist.exhaust_sel
thf(fact_176_llist__set__induct,axiom,
! [A: $tType,X3: A,Xs: coinductive_llist @ A,P: A > ( coinductive_llist @ A ) > $o] :
( ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs ) )
=> ( ! [Xs4: coinductive_llist @ A] :
( ~ ( coinductive_lnull @ A @ Xs4 )
=> ( P @ ( coinductive_lhd @ A @ Xs4 ) @ Xs4 ) )
=> ( ! [Xs4: coinductive_llist @ A,Y: A] :
( ~ ( coinductive_lnull @ A @ Xs4 )
=> ( ( member @ A @ Y @ ( coinductive_lset @ A @ ( coinductive_ltl @ A @ Xs4 ) ) )
=> ( ( P @ Y @ ( coinductive_ltl @ A @ Xs4 ) )
=> ( P @ Y @ Xs4 ) ) ) )
=> ( P @ X3 @ Xs ) ) ) ) ).
% llist_set_induct
thf(fact_177_ord_Oltl__lmerge,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Less: A > A > $o] :
( ~ ( coinductive_lnull @ A @ Xs )
=> ( ~ ( coinductive_lnull @ A @ Ys )
=> ( ( ( Less @ ( coinductive_lhd @ A @ Xs ) @ ( coinductive_lhd @ A @ Ys ) )
=> ( ( coinductive_ltl @ A @ ( hammin1328233080lmerge @ A @ Less @ Xs @ Ys ) )
= ( hammin1328233080lmerge @ A @ Less @ ( coinductive_ltl @ A @ Xs ) @ Ys ) ) )
& ( ~ ( Less @ ( coinductive_lhd @ A @ Xs ) @ ( coinductive_lhd @ A @ Ys ) )
=> ( ( ( Less @ ( coinductive_lhd @ A @ Ys ) @ ( coinductive_lhd @ A @ Xs ) )
=> ( ( coinductive_ltl @ A @ ( hammin1328233080lmerge @ A @ Less @ Xs @ Ys ) )
= ( hammin1328233080lmerge @ A @ Less @ Xs @ ( coinductive_ltl @ A @ Ys ) ) ) )
& ( ~ ( Less @ ( coinductive_lhd @ A @ Ys ) @ ( coinductive_lhd @ A @ Xs ) )
=> ( ( coinductive_ltl @ A @ ( hammin1328233080lmerge @ A @ Less @ Xs @ Ys ) )
= ( hammin1328233080lmerge @ A @ Less @ ( coinductive_ltl @ A @ Xs ) @ ( coinductive_ltl @ A @ Ys ) ) ) ) ) ) ) ) ) ).
% ord.ltl_lmerge
thf(fact_178_mcont__ltl,axiom,
! [A: $tType] : ( comple1403672095_mcont @ ( coinductive_llist @ A ) @ ( coinductive_llist @ A ) @ ( coinductive_lSup @ A ) @ ( coinductive_lprefix @ A ) @ ( coinductive_lSup @ A ) @ ( coinductive_lprefix @ A ) @ ( coinductive_ltl @ A ) ) ).
% mcont_ltl
thf(fact_179_mcont2mcont__ltl,axiom,
! [A: $tType,C2: $tType,Lub: ( set @ C2 ) > C2,Ord: C2 > C2 > $o,T: C2 > ( coinductive_llist @ A )] :
( ( comple1403672095_mcont @ C2 @ ( coinductive_llist @ A ) @ Lub @ Ord @ ( coinductive_lSup @ A ) @ ( coinductive_lprefix @ A ) @ T )
=> ( comple1403672095_mcont @ C2 @ ( coinductive_llist @ A ) @ Lub @ Ord @ ( coinductive_lSup @ A ) @ ( coinductive_lprefix @ A )
@ ^ [X: C2] : ( coinductive_ltl @ A @ ( T @ X ) ) ) ) ).
% mcont2mcont_ltl
thf(fact_180_IntE,axiom,
! [A: $tType,C: A,A3: set @ A,B4: set @ A] :
( ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A3 @ B4 ) )
=> ~ ( ( member @ A @ C @ A3 )
=> ~ ( member @ A @ C @ B4 ) ) ) ).
% IntE
thf(fact_181_IntD1,axiom,
! [A: $tType,C: A,A3: set @ A,B4: set @ A] :
( ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A3 @ B4 ) )
=> ( member @ A @ C @ A3 ) ) ).
% IntD1
thf(fact_182_IntD2,axiom,
! [A: $tType,C: A,A3: set @ A,B4: set @ A] :
( ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A3 @ B4 ) )
=> ( member @ A @ C @ B4 ) ) ).
% IntD2
thf(fact_183_Int__assoc,axiom,
! [A: $tType,A3: set @ A,B4: set @ A,C3: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ B4 ) @ C3 )
= ( inf_inf @ ( set @ A ) @ A3 @ ( inf_inf @ ( set @ A ) @ B4 @ C3 ) ) ) ).
% Int_assoc
thf(fact_184_Int__absorb,axiom,
! [A: $tType,A3: set @ A] :
( ( inf_inf @ ( set @ A ) @ A3 @ A3 )
= A3 ) ).
% Int_absorb
thf(fact_185_Int__commute,axiom,
! [A: $tType] :
( ( inf_inf @ ( set @ A ) )
= ( ^ [A5: set @ A,B5: set @ A] : ( inf_inf @ ( set @ A ) @ B5 @ A5 ) ) ) ).
% Int_commute
thf(fact_186_Int__left__absorb,axiom,
! [A: $tType,A3: set @ A,B4: set @ A] :
( ( inf_inf @ ( set @ A ) @ A3 @ ( inf_inf @ ( set @ A ) @ A3 @ B4 ) )
= ( inf_inf @ ( set @ A ) @ A3 @ B4 ) ) ).
% Int_left_absorb
thf(fact_187_Int__left__commute,axiom,
! [A: $tType,A3: set @ A,B4: set @ A,C3: set @ A] :
( ( inf_inf @ ( set @ A ) @ A3 @ ( inf_inf @ ( set @ A ) @ B4 @ C3 ) )
= ( inf_inf @ ( set @ A ) @ B4 @ ( inf_inf @ ( set @ A ) @ A3 @ C3 ) ) ) ).
% Int_left_commute
thf(fact_188_ord_Olmerge_Ocode,axiom,
! [A: $tType] :
( ( hammin1328233080lmerge @ A )
= ( ^ [Less2: A > A > $o,Xs3: coinductive_llist @ A,Ys4: coinductive_llist @ A] :
( coindu1381640503_llist @ ( coinductive_llist @ A ) @ A @ ( coinductive_LNil @ A )
@ ^ [X: A,Xs5: coinductive_llist @ A] :
( coindu1381640503_llist @ ( coinductive_llist @ A ) @ A @ ( coinductive_LNil @ A )
@ ^ [Y3: A,Ys6: coinductive_llist @ A] : ( if @ ( coinductive_llist @ A ) @ ( Less2 @ ( coinductive_lhd @ A @ Xs3 ) @ ( coinductive_lhd @ A @ Ys4 ) ) @ ( coinductive_LCons @ A @ X @ ( hammin1328233080lmerge @ A @ Less2 @ Xs5 @ Ys4 ) ) @ ( coinductive_LCons @ A @ Y3 @ ( if @ ( coinductive_llist @ A ) @ ( Less2 @ ( coinductive_lhd @ A @ Ys4 ) @ ( coinductive_lhd @ A @ Xs3 ) ) @ ( hammin1328233080lmerge @ A @ Less2 @ Xs3 @ Ys6 ) @ ( hammin1328233080lmerge @ A @ Less2 @ Xs5 @ Ys6 ) ) ) )
@ Ys4 )
@ Xs3 ) ) ) ).
% ord.lmerge.code
thf(fact_189_monotone__lprefix__case,axiom,
! [B: $tType,A: $tType,F: A > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > ( coinductive_llist @ B )] :
( ! [X2: A] :
( comple1396247847notone @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ ( coinductive_lprefix @ A ) @ ( coinductive_lprefix @ B )
@ ^ [Xs3: coinductive_llist @ A] : ( F @ X2 @ Xs3 @ ( coinductive_LCons @ A @ X2 @ Xs3 ) ) )
=> ( comple1396247847notone @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ ( coinductive_lprefix @ A ) @ ( coinductive_lprefix @ B )
@ ^ [Xs3: coinductive_llist @ A] :
( coindu1381640503_llist @ ( coinductive_llist @ B ) @ A @ ( coinductive_LNil @ B )
@ ^ [X: A,Xs5: coinductive_llist @ A] : ( F @ X @ Xs5 @ Xs3 )
@ Xs3 ) ) ) ).
% monotone_lprefix_case
thf(fact_190_Int__def,axiom,
! [A: $tType] :
( ( inf_inf @ ( set @ A ) )
= ( ^ [A5: set @ A,B5: set @ A] :
( collect @ A
@ ^ [X: A] :
( ( member @ A @ X @ A5 )
& ( member @ A @ X @ B5 ) ) ) ) ) ).
% Int_def
thf(fact_191_Int__Collect,axiom,
! [A: $tType,X3: A,A3: set @ A,P: A > $o] :
( ( member @ A @ X3 @ ( inf_inf @ ( set @ A ) @ A3 @ ( collect @ A @ P ) ) )
= ( ( member @ A @ X3 @ A3 )
& ( P @ X3 ) ) ) ).
% Int_Collect
thf(fact_192_Collect__conj__eq,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( collect @ A
@ ^ [X: A] :
( ( P @ X )
& ( Q @ X ) ) )
= ( inf_inf @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_193_mcont__lprefix__case,axiom,
! [B: $tType,A: $tType,F: A > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > ( coinductive_llist @ B )] :
( ! [X2: A] :
( comple1403672095_mcont @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ ( coinductive_lSup @ A ) @ ( coinductive_lprefix @ A ) @ ( coinductive_lSup @ B ) @ ( coinductive_lprefix @ B )
@ ^ [Xs3: coinductive_llist @ A] : ( F @ X2 @ Xs3 @ ( coinductive_LCons @ A @ X2 @ Xs3 ) ) )
=> ( comple1403672095_mcont @ ( coinductive_llist @ A ) @ ( coinductive_llist @ B ) @ ( coinductive_lSup @ A ) @ ( coinductive_lprefix @ A ) @ ( coinductive_lSup @ B ) @ ( coinductive_lprefix @ B )
@ ^ [Xs3: coinductive_llist @ A] :
( coindu1381640503_llist @ ( coinductive_llist @ B ) @ A @ ( coinductive_LNil @ B )
@ ^ [X: A,Xs5: coinductive_llist @ A] : ( F @ X @ Xs5 @ Xs3 )
@ Xs3 ) ) ) ).
% mcont_lprefix_case
thf(fact_194_lmerge_Osimps_I4_J,axiom,
! [Xs: coinductive_llist @ a,Ys: coinductive_llist @ a] :
( ~ ( coinductive_lnull @ a @ Xs )
=> ( ~ ( coinductive_lnull @ a @ Ys )
=> ( ( coinductive_ltl @ a @ ( hammin1328233080lmerge @ a @ less @ Xs @ Ys ) )
= ( coindu1381640503_llist @ ( coinductive_llist @ a ) @ a @ ( undefined @ ( coinductive_llist @ a ) )
@ ^ [X: a,Xs5: coinductive_llist @ a] :
( coindu1381640503_llist @ ( coinductive_llist @ a ) @ a @ ( undefined @ ( coinductive_llist @ a ) )
@ ^ [Y3: a,Ys6: coinductive_llist @ a] : ( if @ ( coinductive_llist @ a ) @ ( less @ ( coinductive_lhd @ a @ Xs ) @ ( coinductive_lhd @ a @ Ys ) ) @ ( hammin1328233080lmerge @ a @ less @ Xs5 @ Ys ) @ ( if @ ( coinductive_llist @ a ) @ ( less @ ( coinductive_lhd @ a @ Ys ) @ ( coinductive_lhd @ a @ Xs ) ) @ ( hammin1328233080lmerge @ a @ less @ Xs @ Ys6 ) @ ( hammin1328233080lmerge @ a @ less @ Xs5 @ Ys6 ) ) )
@ Ys )
@ Xs ) ) ) ) ).
% lmerge.simps(4)
thf(fact_195_ord_Olexordp__eq__simps_I1_J,axiom,
! [A: $tType,Less: A > A > $o,Ys: list @ A] : ( lexordp_eq @ A @ Less @ ( nil @ A ) @ Ys ) ).
% ord.lexordp_eq_simps(1)
thf(fact_196_ord_Olexordp__eq__simps_I2_J,axiom,
! [A: $tType,Less: A > A > $o,Xs: list @ A] :
( ( lexordp_eq @ A @ Less @ Xs @ ( nil @ A ) )
= ( Xs
= ( nil @ A ) ) ) ).
% ord.lexordp_eq_simps(2)
thf(fact_197_llist_Odisc__eq__case_I1_J,axiom,
! [A: $tType] :
( ( coinductive_lnull @ A )
= ( coindu1381640503_llist @ $o @ A @ $true
@ ^ [Uu: A,Uv: coinductive_llist @ A] : $false ) ) ).
% llist.disc_eq_case(1)
thf(fact_198_llist_Odisc__eq__case_I2_J,axiom,
! [A: $tType,Llist: coinductive_llist @ A] :
( ( ~ ( coinductive_lnull @ A @ Llist ) )
= ( coindu1381640503_llist @ $o @ A @ $false
@ ^ [Uu: A,Uv: coinductive_llist @ A] : $true
@ Llist ) ) ).
% llist.disc_eq_case(2)
thf(fact_199_lhd__def,axiom,
! [A: $tType] :
( ( coinductive_lhd @ A )
= ( coindu1381640503_llist @ A @ A @ ( undefined @ A )
@ ^ [X213: A,X223: coinductive_llist @ A] : X213 ) ) ).
% lhd_def
thf(fact_200_ord_Olmerge_Octr_I2_J,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Less: A > A > $o] :
( ~ ( coinductive_lnull @ A @ Xs )
=> ( ~ ( coinductive_lnull @ A @ Ys )
=> ( ( hammin1328233080lmerge @ A @ Less @ Xs @ Ys )
= ( coinductive_LCons @ A
@ ( coindu1381640503_llist @ A @ A @ ( undefined @ A )
@ ^ [X: A,Xs5: coinductive_llist @ A] :
( coindu1381640503_llist @ A @ A @ ( undefined @ A )
@ ^ [Y3: A,Ys6: coinductive_llist @ A] : ( if @ A @ ( Less @ ( coinductive_lhd @ A @ Xs ) @ ( coinductive_lhd @ A @ Ys ) ) @ X @ Y3 )
@ Ys )
@ Xs )
@ ( coindu1381640503_llist @ ( coinductive_llist @ A ) @ A @ ( undefined @ ( coinductive_llist @ A ) )
@ ^ [X: A,Xs5: coinductive_llist @ A] :
( coindu1381640503_llist @ ( coinductive_llist @ A ) @ A @ ( undefined @ ( coinductive_llist @ A ) )
@ ^ [Y3: A,Ys6: coinductive_llist @ A] : ( if @ ( coinductive_llist @ A ) @ ( Less @ ( coinductive_lhd @ A @ Xs ) @ ( coinductive_lhd @ A @ Ys ) ) @ ( hammin1328233080lmerge @ A @ Less @ Xs5 @ Ys ) @ ( if @ ( coinductive_llist @ A ) @ ( Less @ ( coinductive_lhd @ A @ Ys ) @ ( coinductive_lhd @ A @ Xs ) ) @ ( hammin1328233080lmerge @ A @ Less @ Xs @ Ys6 ) @ ( hammin1328233080lmerge @ A @ Less @ Xs5 @ Ys6 ) ) )
@ Ys )
@ Xs ) ) ) ) ) ).
% ord.lmerge.ctr(2)
thf(fact_201_ord_Olexordp__eq__refl,axiom,
! [A: $tType,Less: A > A > $o,Xs: list @ A] : ( lexordp_eq @ A @ Less @ Xs @ Xs ) ).
% ord.lexordp_eq_refl
thf(fact_202_ord_Olmerge_Osimps_I3_J,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Less: A > A > $o] :
( ~ ( coinductive_lnull @ A @ Xs )
=> ( ~ ( coinductive_lnull @ A @ Ys )
=> ( ( coinductive_lhd @ A @ ( hammin1328233080lmerge @ A @ Less @ Xs @ Ys ) )
= ( coindu1381640503_llist @ A @ A @ ( undefined @ A )
@ ^ [X: A,Xs5: coinductive_llist @ A] :
( coindu1381640503_llist @ A @ A @ ( undefined @ A )
@ ^ [Y3: A,Ys6: coinductive_llist @ A] : ( if @ A @ ( Less @ ( coinductive_lhd @ A @ Xs ) @ ( coinductive_lhd @ A @ Ys ) ) @ X @ Y3 )
@ Ys )
@ Xs ) ) ) ) ).
% ord.lmerge.simps(3)
thf(fact_203_ord_Olmerge_Osimps_I4_J,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Less: A > A > $o] :
( ~ ( coinductive_lnull @ A @ Xs )
=> ( ~ ( coinductive_lnull @ A @ Ys )
=> ( ( coinductive_ltl @ A @ ( hammin1328233080lmerge @ A @ Less @ Xs @ Ys ) )
= ( coindu1381640503_llist @ ( coinductive_llist @ A ) @ A @ ( undefined @ ( coinductive_llist @ A ) )
@ ^ [X: A,Xs5: coinductive_llist @ A] :
( coindu1381640503_llist @ ( coinductive_llist @ A ) @ A @ ( undefined @ ( coinductive_llist @ A ) )
@ ^ [Y3: A,Ys6: coinductive_llist @ A] : ( if @ ( coinductive_llist @ A ) @ ( Less @ ( coinductive_lhd @ A @ Xs ) @ ( coinductive_lhd @ A @ Ys ) ) @ ( hammin1328233080lmerge @ A @ Less @ Xs5 @ Ys ) @ ( if @ ( coinductive_llist @ A ) @ ( Less @ ( coinductive_lhd @ A @ Ys ) @ ( coinductive_lhd @ A @ Xs ) ) @ ( hammin1328233080lmerge @ A @ Less @ Xs @ Ys6 ) @ ( hammin1328233080lmerge @ A @ Less @ Xs5 @ Ys6 ) ) )
@ Ys )
@ Xs ) ) ) ) ).
% ord.lmerge.simps(4)
thf(fact_204_ord_Olexordp__eq_ONil,axiom,
! [A: $tType,Less: A > A > $o,Ys: list @ A] : ( lexordp_eq @ A @ Less @ ( nil @ A ) @ Ys ) ).
% ord.lexordp_eq.Nil
thf(fact_205_lmerge_Octr_I2_J,axiom,
! [Xs: coinductive_llist @ a,Ys: coinductive_llist @ a] :
( ~ ( coinductive_lnull @ a @ Xs )
=> ( ~ ( coinductive_lnull @ a @ Ys )
=> ( ( hammin1328233080lmerge @ a @ less @ Xs @ Ys )
= ( coinductive_LCons @ a
@ ( coindu1381640503_llist @ a @ a @ ( undefined @ a )
@ ^ [X: a,Xs5: coinductive_llist @ a] :
( coindu1381640503_llist @ a @ a @ ( undefined @ a )
@ ^ [Y3: a,Ys6: coinductive_llist @ a] : ( if @ a @ ( less @ ( coinductive_lhd @ a @ Xs ) @ ( coinductive_lhd @ a @ Ys ) ) @ X @ Y3 )
@ Ys )
@ Xs )
@ ( coindu1381640503_llist @ ( coinductive_llist @ a ) @ a @ ( undefined @ ( coinductive_llist @ a ) )
@ ^ [X: a,Xs5: coinductive_llist @ a] :
( coindu1381640503_llist @ ( coinductive_llist @ a ) @ a @ ( undefined @ ( coinductive_llist @ a ) )
@ ^ [Y3: a,Ys6: coinductive_llist @ a] : ( if @ ( coinductive_llist @ a ) @ ( less @ ( coinductive_lhd @ a @ Xs ) @ ( coinductive_lhd @ a @ Ys ) ) @ ( hammin1328233080lmerge @ a @ less @ Xs5 @ Ys ) @ ( if @ ( coinductive_llist @ a ) @ ( less @ ( coinductive_lhd @ a @ Ys ) @ ( coinductive_lhd @ a @ Xs ) ) @ ( hammin1328233080lmerge @ a @ less @ Xs @ Ys6 ) @ ( hammin1328233080lmerge @ a @ less @ Xs5 @ Ys6 ) ) )
@ Ys )
@ Xs ) ) ) ) ) ).
% lmerge.ctr(2)
thf(fact_206_lmerge_Osimps_I3_J,axiom,
! [Xs: coinductive_llist @ a,Ys: coinductive_llist @ a] :
( ~ ( coinductive_lnull @ a @ Xs )
=> ( ~ ( coinductive_lnull @ a @ Ys )
=> ( ( coinductive_lhd @ a @ ( hammin1328233080lmerge @ a @ less @ Xs @ Ys ) )
= ( coindu1381640503_llist @ a @ a @ ( undefined @ a )
@ ^ [X: a,Xs5: coinductive_llist @ a] :
( coindu1381640503_llist @ a @ a @ ( undefined @ a )
@ ^ [Y3: a,Ys6: coinductive_llist @ a] : ( if @ a @ ( less @ ( coinductive_lhd @ a @ Xs ) @ ( coinductive_lhd @ a @ Ys ) ) @ X @ Y3 )
@ Ys )
@ Xs ) ) ) ) ).
% lmerge.simps(3)
thf(fact_207_local_Olexordp__eq_Osimps,axiom,
! [A1: list @ a,A22: list @ a] :
( ( lexordp_eq @ a @ less @ A1 @ A22 )
= ( ? [Ys4: list @ a] :
( ( A1
= ( nil @ a ) )
& ( A22 = Ys4 ) )
| ? [X: a,Y3: a,Xs3: list @ a,Ys4: list @ a] :
( ( A1
= ( cons @ a @ X @ Xs3 ) )
& ( A22
= ( cons @ a @ Y3 @ Ys4 ) )
& ( less @ X @ Y3 ) )
| ? [X: a,Y3: a,Xs3: list @ a,Ys4: list @ a] :
( ( A1
= ( cons @ a @ X @ Xs3 ) )
& ( A22
= ( cons @ a @ Y3 @ Ys4 ) )
& ~ ( less @ X @ Y3 )
& ~ ( less @ Y3 @ X )
& ( lexordp_eq @ a @ less @ Xs3 @ Ys4 ) ) ) ) ).
% local.lexordp_eq.simps
thf(fact_208_local_Olexordp__eq_Oinducts,axiom,
! [X12: list @ a,X24: list @ a,P: ( list @ a ) > ( list @ a ) > $o] :
( ( lexordp_eq @ a @ less @ X12 @ X24 )
=> ( ! [X13: list @ a] : ( P @ ( nil @ a ) @ X13 )
=> ( ! [X2: a,Y: a,Xs4: list @ a,Ys3: list @ a] :
( ( less @ X2 @ Y )
=> ( P @ ( cons @ a @ X2 @ Xs4 ) @ ( cons @ a @ Y @ Ys3 ) ) )
=> ( ! [X2: a,Y: a,Xs4: list @ a,Ys3: list @ a] :
( ~ ( less @ X2 @ Y )
=> ( ~ ( less @ Y @ X2 )
=> ( ( lexordp_eq @ a @ less @ Xs4 @ Ys3 )
=> ( ( P @ Xs4 @ Ys3 )
=> ( P @ ( cons @ a @ X2 @ Xs4 ) @ ( cons @ a @ Y @ Ys3 ) ) ) ) ) )
=> ( P @ X12 @ X24 ) ) ) ) ) ).
% local.lexordp_eq.inducts
thf(fact_209_local_Olexordp__eq_OCons,axiom,
! [X3: a,Y2: a,Xs: list @ a,Ys: list @ a] :
( ( less @ X3 @ Y2 )
=> ( lexordp_eq @ a @ less @ ( cons @ a @ X3 @ Xs ) @ ( cons @ a @ Y2 @ Ys ) ) ) ).
% local.lexordp_eq.Cons
thf(fact_210_local_Olexordp__eq_OCons__eq,axiom,
! [X3: a,Y2: a,Xs: list @ a,Ys: list @ a] :
( ~ ( less @ X3 @ Y2 )
=> ( ~ ( less @ Y2 @ X3 )
=> ( ( lexordp_eq @ a @ less @ Xs @ Ys )
=> ( lexordp_eq @ a @ less @ ( cons @ a @ X3 @ Xs ) @ ( cons @ a @ Y2 @ Ys ) ) ) ) ) ).
% local.lexordp_eq.Cons_eq
thf(fact_211_list_Oinject,axiom,
! [A: $tType,X21: A,X22: list @ A,Y21: A,Y22: list @ A] :
( ( ( cons @ A @ X21 @ X22 )
= ( cons @ A @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_212_local_Olexordp__eq_Ocases,axiom,
! [A1: list @ a,A22: list @ a] :
( ( lexordp_eq @ a @ less @ A1 @ A22 )
=> ( ( ( A1
= ( nil @ a ) )
=> ! [Ys3: list @ a] : A22 != Ys3 )
=> ( ! [X2: a] :
( ? [Xs4: list @ a] :
( A1
= ( cons @ a @ X2 @ Xs4 ) )
=> ! [Y: a] :
( ? [Ys3: list @ a] :
( A22
= ( cons @ a @ Y @ Ys3 ) )
=> ~ ( less @ X2 @ Y ) ) )
=> ~ ! [X2: a,Y: a,Xs4: list @ a] :
( ( A1
= ( cons @ a @ X2 @ Xs4 ) )
=> ! [Ys3: list @ a] :
( ( A22
= ( cons @ a @ Y @ Ys3 ) )
=> ( ~ ( less @ X2 @ Y )
=> ( ~ ( less @ Y @ X2 )
=> ~ ( lexordp_eq @ a @ less @ Xs4 @ Ys3 ) ) ) ) ) ) ) ) ).
% local.lexordp_eq.cases
thf(fact_213_ord_Olexordp__eq__simps_I4_J,axiom,
! [A: $tType,Less: A > A > $o,X3: A,Xs: list @ A,Y2: A,Ys: list @ A] :
( ( lexordp_eq @ A @ Less @ ( cons @ A @ X3 @ Xs ) @ ( cons @ A @ Y2 @ Ys ) )
= ( ( Less @ X3 @ Y2 )
| ( ~ ( Less @ Y2 @ X3 )
& ( lexordp_eq @ A @ Less @ Xs @ Ys ) ) ) ) ).
% ord.lexordp_eq_simps(4)
thf(fact_214_ord_Olexordp__eq__simps_I3_J,axiom,
! [A: $tType,Less: A > A > $o,X3: A,Xs: list @ A] :
~ ( lexordp_eq @ A @ Less @ ( cons @ A @ X3 @ Xs ) @ ( nil @ A ) ) ).
% ord.lexordp_eq_simps(3)
thf(fact_215_local_Olexordp__eq__simps_I4_J,axiom,
! [X3: a,Xs: list @ a,Y2: a,Ys: list @ a] :
( ( lexordp_eq @ a @ less @ ( cons @ a @ X3 @ Xs ) @ ( cons @ a @ Y2 @ Ys ) )
= ( ( less @ X3 @ Y2 )
| ( ~ ( less @ Y2 @ X3 )
& ( lexordp_eq @ a @ less @ Xs @ Ys ) ) ) ) ).
% local.lexordp_eq_simps(4)
thf(fact_216_local_Olexordp__eq__simps_I3_J,axiom,
! [X3: a,Xs: list @ a] :
~ ( lexordp_eq @ a @ less @ ( cons @ a @ X3 @ Xs ) @ ( nil @ a ) ) ).
% local.lexordp_eq_simps(3)
thf(fact_217_ord_Olexordp__eq_OCons__eq,axiom,
! [A: $tType,Less: A > A > $o,X3: A,Y2: A,Xs: list @ A,Ys: list @ A] :
( ~ ( Less @ X3 @ Y2 )
=> ( ~ ( Less @ Y2 @ X3 )
=> ( ( lexordp_eq @ A @ Less @ Xs @ Ys )
=> ( lexordp_eq @ A @ Less @ ( cons @ A @ X3 @ Xs ) @ ( cons @ A @ Y2 @ Ys ) ) ) ) ) ).
% ord.lexordp_eq.Cons_eq
thf(fact_218_ord_Olexordp__eq_OCons,axiom,
! [A: $tType,Less: A > A > $o,X3: A,Y2: A,Xs: list @ A,Ys: list @ A] :
( ( Less @ X3 @ Y2 )
=> ( lexordp_eq @ A @ Less @ ( cons @ A @ X3 @ Xs ) @ ( cons @ A @ Y2 @ Ys ) ) ) ).
% ord.lexordp_eq.Cons
thf(fact_219_not__Cons__self2,axiom,
! [A: $tType,X3: A,Xs: list @ A] :
( ( cons @ A @ X3 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_220_map__tailrec__rev_Oinduct,axiom,
! [A: $tType,B: $tType,P: ( A > B ) > ( list @ A ) > ( list @ B ) > $o,A0: A > B,A1: list @ A,A22: list @ B] :
( ! [F3: A > B,X13: list @ B] : ( P @ F3 @ ( nil @ A ) @ X13 )
=> ( ! [F3: A > B,A6: A,As: list @ A,Bs: list @ B] :
( ( P @ F3 @ As @ ( cons @ B @ ( F3 @ A6 ) @ Bs ) )
=> ( P @ F3 @ ( cons @ A @ A6 @ As ) @ Bs ) )
=> ( P @ A0 @ A1 @ A22 ) ) ) ).
% map_tailrec_rev.induct
thf(fact_221_list__nonempty__induct,axiom,
! [A: $tType,Xs: list @ A,P: ( list @ A ) > $o] :
( ( Xs
!= ( nil @ A ) )
=> ( ! [X2: A] : ( P @ ( cons @ A @ X2 @ ( nil @ A ) ) )
=> ( ! [X2: A,Xs4: list @ A] :
( ( Xs4
!= ( nil @ A ) )
=> ( ( P @ Xs4 )
=> ( P @ ( cons @ A @ X2 @ Xs4 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_222_remdups__adj_Oinduct,axiom,
! [A: $tType,P: ( list @ A ) > $o,A0: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [X2: A] : ( P @ ( cons @ A @ X2 @ ( nil @ A ) ) )
=> ( ! [X2: A,Y: A,Xs4: list @ A] :
( ( ( X2 = Y )
=> ( P @ ( cons @ A @ X2 @ Xs4 ) ) )
=> ( ( ( X2 != Y )
=> ( P @ ( cons @ A @ Y @ Xs4 ) ) )
=> ( P @ ( cons @ A @ X2 @ ( cons @ A @ Y @ Xs4 ) ) ) ) )
=> ( P @ A0 ) ) ) ) ).
% remdups_adj.induct
thf(fact_223_remdups__adj_Ocases,axiom,
! [A: $tType,X3: list @ A] :
( ( X3
!= ( nil @ A ) )
=> ( ! [X2: A] :
( X3
!= ( cons @ A @ X2 @ ( nil @ A ) ) )
=> ~ ! [X2: A,Y: A,Xs4: list @ A] :
( X3
!= ( cons @ A @ X2 @ ( cons @ A @ Y @ Xs4 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_224_transpose_Ocases,axiom,
! [A: $tType,X3: list @ ( list @ A )] :
( ( X3
!= ( nil @ ( list @ A ) ) )
=> ( ! [Xss: list @ ( list @ A )] :
( X3
!= ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss ) )
=> ~ ! [X2: A,Xs4: list @ A,Xss: list @ ( list @ A )] :
( X3
!= ( cons @ ( list @ A ) @ ( cons @ A @ X2 @ Xs4 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_225_splice_Oinduct,axiom,
! [A: $tType,P: ( list @ A ) > ( list @ A ) > $o,A0: list @ A,A1: list @ A] :
( ! [X13: list @ A] : ( P @ ( nil @ A ) @ X13 )
=> ( ! [V: A,Va: list @ A] : ( P @ ( cons @ A @ V @ Va ) @ ( nil @ A ) )
=> ( ! [X2: A,Xs4: list @ A,Y: A,Ys3: list @ A] :
( ( P @ Xs4 @ Ys3 )
=> ( P @ ( cons @ A @ X2 @ Xs4 ) @ ( cons @ A @ Y @ Ys3 ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% splice.induct
thf(fact_226_list__induct2_H,axiom,
! [A: $tType,B: $tType,P: ( list @ A ) > ( list @ B ) > $o,Xs: list @ A,Ys: list @ B] :
( ( P @ ( nil @ A ) @ ( nil @ B ) )
=> ( ! [X2: A,Xs4: list @ A] : ( P @ ( cons @ A @ X2 @ Xs4 ) @ ( nil @ B ) )
=> ( ! [Y: B,Ys3: list @ B] : ( P @ ( nil @ A ) @ ( cons @ B @ Y @ Ys3 ) )
=> ( ! [X2: A,Xs4: list @ A,Y: B,Ys3: list @ B] :
( ( P @ Xs4 @ Ys3 )
=> ( P @ ( cons @ A @ X2 @ Xs4 ) @ ( cons @ B @ Y @ Ys3 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_227_neq__Nil__conv,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
= ( ? [Y3: A,Ys4: list @ A] :
( Xs
= ( cons @ A @ Y3 @ Ys4 ) ) ) ) ).
% neq_Nil_conv
thf(fact_228_list_Oinducts,axiom,
! [A: $tType,P: ( list @ A ) > $o,List: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [X13: A,X25: list @ A] :
( ( P @ X25 )
=> ( P @ ( cons @ A @ X13 @ X25 ) ) )
=> ( P @ List ) ) ) ).
% list.inducts
thf(fact_229_list_Oexhaust,axiom,
! [A: $tType,Y2: list @ A] :
( ( Y2
!= ( nil @ A ) )
=> ~ ! [X212: A,X222: list @ A] :
( Y2
!= ( cons @ A @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_230_list_OdiscI,axiom,
! [A: $tType,List: list @ A,X21: A,X22: list @ A] :
( ( List
= ( cons @ A @ X21 @ X22 ) )
=> ( List
!= ( nil @ A ) ) ) ).
% list.discI
thf(fact_231_list_Odistinct_I1_J,axiom,
! [A: $tType,X21: A,X22: list @ A] :
( ( nil @ A )
!= ( cons @ A @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_232_ord_Olexordp__eq_Ocases,axiom,
! [A: $tType,Less: A > A > $o,A1: list @ A,A22: list @ A] :
( ( lexordp_eq @ A @ Less @ A1 @ A22 )
=> ( ( ( A1
= ( nil @ A ) )
=> ! [Ys3: list @ A] : A22 != Ys3 )
=> ( ! [X2: A] :
( ? [Xs4: list @ A] :
( A1
= ( cons @ A @ X2 @ Xs4 ) )
=> ! [Y: A] :
( ? [Ys3: list @ A] :
( A22
= ( cons @ A @ Y @ Ys3 ) )
=> ~ ( Less @ X2 @ Y ) ) )
=> ~ ! [X2: A,Y: A,Xs4: list @ A] :
( ( A1
= ( cons @ A @ X2 @ Xs4 ) )
=> ! [Ys3: list @ A] :
( ( A22
= ( cons @ A @ Y @ Ys3 ) )
=> ( ~ ( Less @ X2 @ Y )
=> ( ~ ( Less @ Y @ X2 )
=> ~ ( lexordp_eq @ A @ Less @ Xs4 @ Ys3 ) ) ) ) ) ) ) ) ).
% ord.lexordp_eq.cases
thf(fact_233_ord_Olexordp__eq_Osimps,axiom,
! [A: $tType] :
( ( lexordp_eq @ A )
= ( ^ [Less2: A > A > $o,A12: list @ A,A23: list @ A] :
( ? [Ys4: list @ A] :
( ( A12
= ( nil @ A ) )
& ( A23 = Ys4 ) )
| ? [X: A,Y3: A,Xs3: list @ A,Ys4: list @ A] :
( ( A12
= ( cons @ A @ X @ Xs3 ) )
& ( A23
= ( cons @ A @ Y3 @ Ys4 ) )
& ( Less2 @ X @ Y3 ) )
| ? [X: A,Y3: A,Xs3: list @ A,Ys4: list @ A] :
( ( A12
= ( cons @ A @ X @ Xs3 ) )
& ( A23
= ( cons @ A @ Y3 @ Ys4 ) )
& ~ ( Less2 @ X @ Y3 )
& ~ ( Less2 @ Y3 @ X )
& ( lexordp_eq @ A @ Less2 @ Xs3 @ Ys4 ) ) ) ) ) ).
% ord.lexordp_eq.simps
thf(fact_234_ord_Olexordp__eq_Oinducts,axiom,
! [A: $tType,Less: A > A > $o,X12: list @ A,X24: list @ A,P: ( list @ A ) > ( list @ A ) > $o] :
( ( lexordp_eq @ A @ Less @ X12 @ X24 )
=> ( ! [X13: list @ A] : ( P @ ( nil @ A ) @ X13 )
=> ( ! [X2: A,Y: A,Xs4: list @ A,Ys3: list @ A] :
( ( Less @ X2 @ Y )
=> ( P @ ( cons @ A @ X2 @ Xs4 ) @ ( cons @ A @ Y @ Ys3 ) ) )
=> ( ! [X2: A,Y: A,Xs4: list @ A,Ys3: list @ A] :
( ~ ( Less @ X2 @ Y )
=> ( ~ ( Less @ Y @ X2 )
=> ( ( lexordp_eq @ A @ Less @ Xs4 @ Ys3 )
=> ( ( P @ Xs4 @ Ys3 )
=> ( P @ ( cons @ A @ X2 @ Xs4 ) @ ( cons @ A @ Y @ Ys3 ) ) ) ) ) )
=> ( P @ X12 @ X24 ) ) ) ) ) ).
% ord.lexordp_eq.inducts
thf(fact_235_local_Olexordp_ONil,axiom,
! [Y2: a,Ys: list @ a] : ( lexordp @ a @ less @ ( nil @ a ) @ ( cons @ a @ Y2 @ Ys ) ) ).
% local.lexordp.Nil
thf(fact_236_local_Olexordp_Ocases,axiom,
! [A1: list @ a,A22: list @ a] :
( ( lexordp @ a @ less @ A1 @ A22 )
=> ( ( ( A1
= ( nil @ a ) )
=> ! [Y: a,Ys3: list @ a] :
( A22
!= ( cons @ a @ Y @ Ys3 ) ) )
=> ( ! [X2: a] :
( ? [Xs4: list @ a] :
( A1
= ( cons @ a @ X2 @ Xs4 ) )
=> ! [Y: a] :
( ? [Ys3: list @ a] :
( A22
= ( cons @ a @ Y @ Ys3 ) )
=> ~ ( less @ X2 @ Y ) ) )
=> ~ ! [X2: a,Y: a,Xs4: list @ a] :
( ( A1
= ( cons @ a @ X2 @ Xs4 ) )
=> ! [Ys3: list @ a] :
( ( A22
= ( cons @ a @ Y @ Ys3 ) )
=> ( ~ ( less @ X2 @ Y )
=> ( ~ ( less @ Y @ X2 )
=> ~ ( lexordp @ a @ less @ Xs4 @ Ys3 ) ) ) ) ) ) ) ) ).
% local.lexordp.cases
thf(fact_237_local_Olexordp__irreflexive,axiom,
! [Xs: list @ a] :
( ! [X2: a] :
~ ( less @ X2 @ X2 )
=> ~ ( lexordp @ a @ less @ Xs @ Xs ) ) ).
% local.lexordp_irreflexive
thf(fact_238_local_Olexordp_OCons,axiom,
! [X3: a,Y2: a,Xs: list @ a,Ys: list @ a] :
( ( less @ X3 @ Y2 )
=> ( lexordp @ a @ less @ ( cons @ a @ X3 @ Xs ) @ ( cons @ a @ Y2 @ Ys ) ) ) ).
% local.lexordp.Cons
thf(fact_239_local_Olexordp_OCons__eq,axiom,
! [X3: a,Y2: a,Xs: list @ a,Ys: list @ a] :
( ~ ( less @ X3 @ Y2 )
=> ( ~ ( less @ Y2 @ X3 )
=> ( ( lexordp @ a @ less @ Xs @ Ys )
=> ( lexordp @ a @ less @ ( cons @ a @ X3 @ Xs ) @ ( cons @ a @ Y2 @ Ys ) ) ) ) ) ).
% local.lexordp.Cons_eq
thf(fact_240_local_Olexordp__into__lexordp__eq,axiom,
! [Xs: list @ a,Ys: list @ a] :
( ( lexordp @ a @ less @ Xs @ Ys )
=> ( lexordp_eq @ a @ less @ Xs @ Ys ) ) ).
% local.lexordp_into_lexordp_eq
thf(fact_241_local_Olexordp_Osimps,axiom,
! [A1: list @ a,A22: list @ a] :
( ( lexordp @ a @ less @ A1 @ A22 )
= ( ? [Y3: a,Ys4: list @ a] :
( ( A1
= ( nil @ a ) )
& ( A22
= ( cons @ a @ Y3 @ Ys4 ) ) )
| ? [X: a,Y3: a,Xs3: list @ a,Ys4: list @ a] :
( ( A1
= ( cons @ a @ X @ Xs3 ) )
& ( A22
= ( cons @ a @ Y3 @ Ys4 ) )
& ( less @ X @ Y3 ) )
| ? [X: a,Y3: a,Xs3: list @ a,Ys4: list @ a] :
( ( A1
= ( cons @ a @ X @ Xs3 ) )
& ( A22
= ( cons @ a @ Y3 @ Ys4 ) )
& ~ ( less @ X @ Y3 )
& ~ ( less @ Y3 @ X )
& ( lexordp @ a @ less @ Xs3 @ Ys4 ) ) ) ) ).
% local.lexordp.simps
thf(fact_242_local_Olexordp_Oinducts,axiom,
! [X12: list @ a,X24: list @ a,P: ( list @ a ) > ( list @ a ) > $o] :
( ( lexordp @ a @ less @ X12 @ X24 )
=> ( ! [Y: a,Ys3: list @ a] : ( P @ ( nil @ a ) @ ( cons @ a @ Y @ Ys3 ) )
=> ( ! [X2: a,Y: a,Xs4: list @ a,Ys3: list @ a] :
( ( less @ X2 @ Y )
=> ( P @ ( cons @ a @ X2 @ Xs4 ) @ ( cons @ a @ Y @ Ys3 ) ) )
=> ( ! [X2: a,Y: a,Xs4: list @ a,Ys3: list @ a] :
( ~ ( less @ X2 @ Y )
=> ( ~ ( less @ Y @ X2 )
=> ( ( lexordp @ a @ less @ Xs4 @ Ys3 )
=> ( ( P @ Xs4 @ Ys3 )
=> ( P @ ( cons @ a @ X2 @ Xs4 ) @ ( cons @ a @ Y @ Ys3 ) ) ) ) ) )
=> ( P @ X12 @ X24 ) ) ) ) ) ).
% local.lexordp.inducts
thf(fact_243_ord_Olexordp__simps_I3_J,axiom,
! [A: $tType,Less: A > A > $o,X3: A,Xs: list @ A,Y2: A,Ys: list @ A] :
( ( lexordp @ A @ Less @ ( cons @ A @ X3 @ Xs ) @ ( cons @ A @ Y2 @ Ys ) )
= ( ( Less @ X3 @ Y2 )
| ( ~ ( Less @ Y2 @ X3 )
& ( lexordp @ A @ Less @ Xs @ Ys ) ) ) ) ).
% ord.lexordp_simps(3)
thf(fact_244_ord_Olexordp__simps_I2_J,axiom,
! [A: $tType,Less: A > A > $o,Xs: list @ A] :
~ ( lexordp @ A @ Less @ Xs @ ( nil @ A ) ) ).
% ord.lexordp_simps(2)
thf(fact_245_ord_Olexordp__simps_I1_J,axiom,
! [A: $tType,Less: A > A > $o,Ys: list @ A] :
( ( lexordp @ A @ Less @ ( nil @ A ) @ Ys )
= ( Ys
!= ( nil @ A ) ) ) ).
% ord.lexordp_simps(1)
thf(fact_246_local_Olexordp__simps_I3_J,axiom,
! [X3: a,Xs: list @ a,Y2: a,Ys: list @ a] :
( ( lexordp @ a @ less @ ( cons @ a @ X3 @ Xs ) @ ( cons @ a @ Y2 @ Ys ) )
= ( ( less @ X3 @ Y2 )
| ( ~ ( less @ Y2 @ X3 )
& ( lexordp @ a @ less @ Xs @ Ys ) ) ) ) ).
% local.lexordp_simps(3)
thf(fact_247_local_Olexordp__simps_I2_J,axiom,
! [Xs: list @ a] :
~ ( lexordp @ a @ less @ Xs @ ( nil @ a ) ) ).
% local.lexordp_simps(2)
thf(fact_248_local_Olexordp__simps_I1_J,axiom,
! [Ys: list @ a] :
( ( lexordp @ a @ less @ ( nil @ a ) @ Ys )
= ( Ys
!= ( nil @ a ) ) ) ).
% local.lexordp_simps(1)
thf(fact_249_ord_Olexordp_OCons,axiom,
! [A: $tType,Less: A > A > $o,X3: A,Y2: A,Xs: list @ A,Ys: list @ A] :
( ( Less @ X3 @ Y2 )
=> ( lexordp @ A @ Less @ ( cons @ A @ X3 @ Xs ) @ ( cons @ A @ Y2 @ Ys ) ) ) ).
% ord.lexordp.Cons
thf(fact_250_ord_Olexordp_OCons__eq,axiom,
! [A: $tType,Less: A > A > $o,X3: A,Y2: A,Xs: list @ A,Ys: list @ A] :
( ~ ( Less @ X3 @ Y2 )
=> ( ~ ( Less @ Y2 @ X3 )
=> ( ( lexordp @ A @ Less @ Xs @ Ys )
=> ( lexordp @ A @ Less @ ( cons @ A @ X3 @ Xs ) @ ( cons @ A @ Y2 @ Ys ) ) ) ) ) ).
% ord.lexordp.Cons_eq
thf(fact_251_ord_Olexordp_Oinducts,axiom,
! [A: $tType,Less: A > A > $o,X12: list @ A,X24: list @ A,P: ( list @ A ) > ( list @ A ) > $o] :
( ( lexordp @ A @ Less @ X12 @ X24 )
=> ( ! [Y: A,Ys3: list @ A] : ( P @ ( nil @ A ) @ ( cons @ A @ Y @ Ys3 ) )
=> ( ! [X2: A,Y: A,Xs4: list @ A,Ys3: list @ A] :
( ( Less @ X2 @ Y )
=> ( P @ ( cons @ A @ X2 @ Xs4 ) @ ( cons @ A @ Y @ Ys3 ) ) )
=> ( ! [X2: A,Y: A,Xs4: list @ A,Ys3: list @ A] :
( ~ ( Less @ X2 @ Y )
=> ( ~ ( Less @ Y @ X2 )
=> ( ( lexordp @ A @ Less @ Xs4 @ Ys3 )
=> ( ( P @ Xs4 @ Ys3 )
=> ( P @ ( cons @ A @ X2 @ Xs4 ) @ ( cons @ A @ Y @ Ys3 ) ) ) ) ) )
=> ( P @ X12 @ X24 ) ) ) ) ) ).
% ord.lexordp.inducts
thf(fact_252_ord_Olexordp_Osimps,axiom,
! [A: $tType] :
( ( lexordp @ A )
= ( ^ [Less2: A > A > $o,A12: list @ A,A23: list @ A] :
( ? [Y3: A,Ys4: list @ A] :
( ( A12
= ( nil @ A ) )
& ( A23
= ( cons @ A @ Y3 @ Ys4 ) ) )
| ? [X: A,Y3: A,Xs3: list @ A,Ys4: list @ A] :
( ( A12
= ( cons @ A @ X @ Xs3 ) )
& ( A23
= ( cons @ A @ Y3 @ Ys4 ) )
& ( Less2 @ X @ Y3 ) )
| ? [X: A,Y3: A,Xs3: list @ A,Ys4: list @ A] :
( ( A12
= ( cons @ A @ X @ Xs3 ) )
& ( A23
= ( cons @ A @ Y3 @ Ys4 ) )
& ~ ( Less2 @ X @ Y3 )
& ~ ( Less2 @ Y3 @ X )
& ( lexordp @ A @ Less2 @ Xs3 @ Ys4 ) ) ) ) ) ).
% ord.lexordp.simps
%----Type constructors (3)
thf(tcon_fun___Orderings_Oord,axiom,
! [A7: $tType,A8: $tType] :
( ( ord @ A8 @ ( type2 @ A8 ) )
=> ( ord @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_1,axiom,
! [A7: $tType] : ( ord @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_HOL_Obool___Orderings_Oord_2,axiom,
ord @ $o @ ( type2 @ $o ) ).
%----Helper facts (3)
thf(help_If_3_1_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_T,axiom,
! [A: $tType,X3: A,Y2: A] :
( ( if @ A @ $false @ X3 @ Y2 )
= Y2 ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X3: A,Y2: A] :
( ( if @ A @ $true @ X3 @ Y2 )
= X3 ) ).
%----Conjectures (1)
thf(conj_0,conjecture,
comple1403672095_mcont @ ( coinductive_llist @ a ) @ ( coinductive_llist @ a ) @ ( coinductive_lSup @ a ) @ ( coinductive_lprefix @ a ) @ ( coinductive_lSup @ a ) @ ( coinductive_lprefix @ a ) @ ( hammin1328233080lmerge @ a @ less @ xs ) ).
%------------------------------------------------------------------------------