TPTP Problem File: DAT154^1.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : DAT154^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Data Structures
% Problem : Hamming stream 35
% 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__35.p [Bla16]
% Status : Theorem
% Rating : 0.00 v7.2.0, 0.25 v7.1.0
% Syntax : Number of formulae : 322 ( 139 unt; 59 typ; 0 def)
% Number of atoms : 753 ( 384 equ; 0 cnn)
% Maximal formula atoms : 13 ( 2 avg)
% Number of connectives : 4035 ( 152 ~; 33 |; 76 &;3501 @)
% ( 0 <=>; 273 =>; 0 <=; 0 <~>)
% Maximal formula depth : 23 ( 7 avg)
% Number of types : 2 ( 1 usr)
% Number of type conns : 307 ( 307 >; 0 *; 0 +; 0 <<)
% Number of symbols : 59 ( 58 usr; 2 con; 0-5 aty)
% Number of variables : 1107 ( 48 ^; 938 !; 65 ?;1107 :)
% ( 56 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 14:40:50.764
%------------------------------------------------------------------------------
%----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 (54)
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_Lattices_Olattice,type,
lattice:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Lattices_Osemilattice__inf,type,
semilattice_inf:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_c_BNF__Greatest__Fixpoint_OShift,type,
bNF_Greatest_Shift:
!>[A: $tType] : ( ( set @ ( list @ A ) ) > A > ( set @ ( list @ A ) ) ) ).
thf(sy_c_BNF__Greatest__Fixpoint_OSucc,type,
bNF_Greatest_Succ:
!>[A: $tType] : ( ( set @ ( list @ A ) ) > ( list @ A ) > ( set @ A ) ) ).
thf(sy_c_BNF__Greatest__Fixpoint_Oshift,type,
bNF_Greatest_shift:
!>[A: $tType,B: $tType] : ( ( ( list @ A ) > B ) > A > ( list @ A ) > B ) ).
thf(sy_c_Coinductive__List_Olist__of__aux,type,
coindu1384447384of_aux:
!>[A: $tType] : ( ( list @ A ) > ( coinductive_llist @ A ) > ( list @ 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_Oord_Olsorted,type,
coinductive_lsorted:
!>[A: $tType] : ( ( A > A > $o ) > ( coinductive_llist @ A ) > $o ) ).
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_Hamming__Stream__Mirabelle__rwekfkwckg_Oord__class_Olmerge,type,
hammin1138290136lmerge:
!>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
thf(sy_c_Lattices_Oinf__class_Oinf,type,
inf_inf:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_List_Oappend,type,
append:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Obind,type,
bind:
!>[A: $tType,B: $tType] : ( ( list @ A ) > ( A > ( list @ B ) ) > ( list @ B ) ) ).
thf(sy_c_List_Obutlast,type,
butlast:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Oinsert,type,
insert:
!>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Olast,type,
last:
!>[A: $tType] : ( ( list @ 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_Olist_Ohd,type,
hd:
!>[A: $tType] : ( ( list @ A ) > A ) ).
thf(sy_c_List_Olist_Omap,type,
map:
!>[A: $tType,Aa: $tType] : ( ( A > Aa ) > ( list @ A ) > ( list @ Aa ) ) ).
thf(sy_c_List_Olist__ex1,type,
list_ex1:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Omap__tailrec,type,
map_tailrec:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( list @ A ) > ( list @ B ) ) ).
thf(sy_c_List_Omap__tailrec__rev,type,
map_tailrec_rev:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( list @ A ) > ( list @ B ) > ( list @ B ) ) ).
thf(sy_c_List_Omaps,type,
maps:
!>[A: $tType,B: $tType] : ( ( A > ( list @ B ) ) > ( list @ A ) > ( list @ B ) ) ).
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_List_Oproduct__lists,type,
product_lists:
!>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ ( list @ A ) ) ) ).
thf(sy_c_List_Orev,type,
rev:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Orotate1,type,
rotate1:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Osublists,type,
sublists:
!>[A: $tType] : ( ( list @ A ) > ( list @ ( list @ A ) ) ) ).
thf(sy_c_Orderings_Oord_Omax,type,
max:
!>[A: $tType] : ( ( A > A > $o ) > A > A > A ) ).
thf(sy_c_Orderings_Oord_Omin,type,
min:
!>[A: $tType] : ( ( A > A > $o ) > A > A > A ) ).
thf(sy_c_Pure_Otype,type,
type2:
!>[A: $tType] : ( itself @ A ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord_OatLeast,type,
set_atLeast:
!>[A: $tType] : ( ( A > A > $o ) > A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord_OatLeastAtMost,type,
set_atLeastAtMost:
!>[A: $tType] : ( ( A > A > $o ) > A > A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord_OatLeastLessThan,type,
set_atLeastLessThan:
!>[A: $tType] : ( ( A > A > $o ) > ( A > A > $o ) > A > A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord_OatMost,type,
set_atMost:
!>[A: $tType] : ( ( A > A > $o ) > A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord_OgreaterThan,type,
set_greaterThan:
!>[A: $tType] : ( ( A > A > $o ) > A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord_OgreaterThanAtMost,type,
set_gr323396891AtMost:
!>[A: $tType] : ( ( A > A > $o ) > ( A > A > $o ) > A > 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_Sublist_Oprefixes,type,
prefixes:
!>[A: $tType] : ( ( list @ A ) > ( list @ ( list @ A ) ) ) ).
thf(sy_c_Sublist_Oprefixes__rel,type,
prefixes_rel:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > $o ) ).
thf(sy_c_Sublist_Ostrict__suffix,type,
strict_suffix:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > $o ) ).
thf(sy_c_Wellfounded_Oaccp,type,
accp:
!>[A: $tType] : ( ( A > A > $o ) > A > $o ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_less,type,
less: a > a > $o ).
thf(sy_v_less__eq,type,
less_eq: a > a > $o ).
thf(sy_v_xs,type,
xs: coinductive_llist @ a ).
thf(sy_v_ys,type,
ys: coinductive_llist @ a ).
%----Relevant facts (253)
thf(fact_0_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_1_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_2_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_3_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_4_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_5_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_6_lmerge_Osimps_I1_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(1)
thf(fact_7_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_8_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_9_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_10_ord_Olmerge_Odisc__iff_I1_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(1)
thf(fact_11_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_12_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_13_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_14_local_Olexordp__eq__refl,axiom,
! [Xs: list @ a] : ( lexordp_eq @ a @ less @ Xs @ Xs ) ).
% local.lexordp_eq_refl
thf(fact_15_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_16_local_Olexordp__irreflexive,axiom,
! [Xs: list @ a] :
( ! [X: a] :
~ ( less @ X @ X )
=> ~ ( lexordp @ a @ less @ Xs @ Xs ) ) ).
% local.lexordp_irreflexive
thf(fact_17_local_OgreaterThan__iff,axiom,
! [I: a,K: a] :
( ( member @ a @ I @ ( set_greaterThan @ a @ less @ K ) )
= ( less @ K @ I ) ) ).
% local.greaterThan_iff
thf(fact_18_local_OlessThan__iff,axiom,
! [I: a,K: a] :
( ( member @ a @ I @ ( set_lessThan @ a @ less @ K ) )
= ( less @ I @ K ) ) ).
% local.lessThan_iff
thf(fact_19_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_20_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_21_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_22_ord__class_Olmerge_Odisc_I2_J,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 @ ( hammin1138290136lmerge @ A @ Xs @ Ys ) ) ) ) ) ).
% ord_class.lmerge.disc(2)
thf(fact_23_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_24_lnull__def,axiom,
! [A: $tType] :
( ( coinductive_lnull @ A )
= ( ^ [Llist: coinductive_llist @ A] :
( Llist
= ( coinductive_LNil @ A ) ) ) ) ).
% lnull_def
thf(fact_25_llist_Ocollapse_I1_J,axiom,
! [A: $tType,Llist2: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ Llist2 )
=> ( Llist2
= ( coinductive_LNil @ A ) ) ) ).
% llist.collapse(1)
thf(fact_26_llist_OdiscI_I1_J,axiom,
! [A: $tType,Llist2: coinductive_llist @ A] :
( ( Llist2
= ( coinductive_LNil @ A ) )
=> ( coinductive_lnull @ A @ Llist2 ) ) ).
% llist.discI(1)
thf(fact_27_llist_Odisc_I1_J,axiom,
! [A: $tType] : ( coinductive_lnull @ A @ ( coinductive_LNil @ A ) ) ).
% llist.disc(1)
thf(fact_28_ord__class_Olmerge_Octr_I1_J,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( ( coinductive_lnull @ A @ Xs )
| ( coinductive_lnull @ A @ Ys ) )
=> ( ( hammin1138290136lmerge @ A @ Xs @ Ys )
= ( coinductive_LNil @ A ) ) ) ) ).
% ord_class.lmerge.ctr(1)
thf(fact_29_ord__class_Olmerge_Odisc__iff_I1_J,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ ( hammin1138290136lmerge @ A @ Xs @ Ys ) )
= ( ( coinductive_lnull @ A @ Xs )
| ( coinductive_lnull @ A @ Ys ) ) ) ) ).
% ord_class.lmerge.disc_iff(1)
thf(fact_30_ord__class_Olmerge_Odisc__iff_I2_J,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( ~ ( coinductive_lnull @ A @ ( hammin1138290136lmerge @ A @ Xs @ Ys ) ) )
= ( ~ ( coinductive_lnull @ A @ Xs )
& ~ ( coinductive_lnull @ A @ Ys ) ) ) ) ).
% ord_class.lmerge.disc_iff(2)
thf(fact_31_ord__class_Olmerge_Odisc_I1_J,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 @ ( hammin1138290136lmerge @ A @ Xs @ Ys ) ) ) ) ).
% ord_class.lmerge.disc(1)
thf(fact_32_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_33_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_34_local_Olexordp__eq_ONil,axiom,
! [Ys: list @ a] : ( lexordp_eq @ a @ less @ ( nil @ a ) @ Ys ) ).
% local.lexordp_eq.Nil
thf(fact_35_local_Olexordp__eq_OCons__eq,axiom,
! [X2: a,Y: a,Xs: list @ a,Ys: list @ a] :
( ~ ( less @ X2 @ Y )
=> ( ~ ( less @ Y @ X2 )
=> ( ( lexordp_eq @ a @ less @ Xs @ Ys )
=> ( lexordp_eq @ a @ less @ ( cons @ a @ X2 @ Xs ) @ ( cons @ a @ Y @ Ys ) ) ) ) ) ).
% local.lexordp_eq.Cons_eq
thf(fact_36_local_Olexordp__eq_OCons,axiom,
! [X2: a,Y: a,Xs: list @ a,Ys: list @ a] :
( ( less @ X2 @ Y )
=> ( lexordp_eq @ a @ less @ ( cons @ a @ X2 @ Xs ) @ ( cons @ a @ Y @ Ys ) ) ) ).
% local.lexordp_eq.Cons
thf(fact_37_ord_Olexordp__into__lexordp__eq,axiom,
! [A: $tType,Less: A > A > $o,Xs: list @ A,Ys: list @ A] :
( ( lexordp @ A @ Less @ Xs @ Ys )
=> ( lexordp_eq @ A @ Less @ Xs @ Ys ) ) ).
% ord.lexordp_into_lexordp_eq
thf(fact_38_local_Olexordp_OCons__eq,axiom,
! [X2: a,Y: a,Xs: list @ a,Ys: list @ a] :
( ~ ( less @ X2 @ Y )
=> ( ~ ( less @ Y @ X2 )
=> ( ( lexordp @ a @ less @ Xs @ Ys )
=> ( lexordp @ a @ less @ ( cons @ a @ X2 @ Xs ) @ ( cons @ a @ Y @ Ys ) ) ) ) ) ).
% local.lexordp.Cons_eq
thf(fact_39_local_Olexordp_OCons,axiom,
! [X2: a,Y: a,Xs: list @ a,Ys: list @ a] :
( ( less @ X2 @ Y )
=> ( lexordp @ a @ less @ ( cons @ a @ X2 @ Xs ) @ ( cons @ a @ Y @ Ys ) ) ) ).
% local.lexordp.Cons
thf(fact_40_local_Olexordp__append__leftI,axiom,
! [Us: list @ a,Vs: list @ a,Xs: list @ a] :
( ( lexordp @ a @ less @ Us @ Vs )
=> ( lexordp @ a @ less @ ( append @ a @ Xs @ Us ) @ ( append @ a @ Xs @ Vs ) ) ) ).
% local.lexordp_append_leftI
thf(fact_41_local_Olexordp__append__leftD,axiom,
! [Xs: list @ a,Us: list @ a,Vs: list @ a] :
( ( lexordp @ a @ less @ ( append @ a @ Xs @ Us ) @ ( append @ a @ Xs @ Vs ) )
=> ( ! [A3: a] :
~ ( less @ A3 @ A3 )
=> ( lexordp @ a @ less @ Us @ Vs ) ) ) ).
% local.lexordp_append_leftD
thf(fact_42_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_43_mem__Collect__eq,axiom,
! [A: $tType,A2: A,P: A > $o] :
( ( member @ A @ A2 @ ( collect @ A @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_44_Collect__mem__eq,axiom,
! [A: $tType,A4: set @ A] :
( ( collect @ A
@ ^ [X3: A] : ( member @ A @ X3 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_45_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X: A] :
( ( P @ X )
= ( Q @ X ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_46_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X: A] :
( ( F @ X )
= ( G @ X ) )
=> ( F = G ) ) ).
% ext
thf(fact_47_same__append__eq,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( ( append @ A @ Xs @ Ys )
= ( append @ A @ Xs @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_48_append__same__eq,axiom,
! [A: $tType,Ys: list @ A,Xs: list @ A,Zs: list @ A] :
( ( ( append @ A @ Ys @ Xs )
= ( append @ A @ Zs @ Xs ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_49_append__assoc,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( append @ A @ ( append @ A @ Xs @ Ys ) @ Zs )
= ( append @ A @ Xs @ ( append @ A @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_50_local_Olexordp_ONil,axiom,
! [Y: a,Ys: list @ a] : ( lexordp @ a @ less @ ( nil @ a ) @ ( cons @ a @ Y @ Ys ) ) ).
% local.lexordp.Nil
thf(fact_51_local_Olexordp_Ocases,axiom,
! [A1: list @ a,A22: list @ a] :
( ( lexordp @ a @ less @ A1 @ A22 )
=> ( ( ( A1
= ( nil @ a ) )
=> ! [Y2: a,Ys2: list @ a] :
( A22
!= ( cons @ a @ Y2 @ Ys2 ) ) )
=> ( ! [X: a] :
( ? [Xs2: list @ a] :
( A1
= ( cons @ a @ X @ Xs2 ) )
=> ! [Y2: a] :
( ? [Ys2: list @ a] :
( A22
= ( cons @ a @ Y2 @ Ys2 ) )
=> ~ ( less @ X @ Y2 ) ) )
=> ~ ! [X: a,Y2: a,Xs2: list @ a] :
( ( A1
= ( cons @ a @ X @ Xs2 ) )
=> ! [Ys2: list @ a] :
( ( A22
= ( cons @ a @ Y2 @ Ys2 ) )
=> ( ~ ( less @ X @ Y2 )
=> ( ~ ( less @ Y2 @ X )
=> ~ ( lexordp @ a @ less @ Xs2 @ Ys2 ) ) ) ) ) ) ) ) ).
% local.lexordp.cases
thf(fact_52_local_Olexordp_Oinducts,axiom,
! [X1: list @ a,X23: list @ a,P: ( list @ a ) > ( list @ a ) > $o] :
( ( lexordp @ a @ less @ X1 @ X23 )
=> ( ! [Y2: a,Ys2: list @ a] : ( P @ ( nil @ a ) @ ( cons @ a @ Y2 @ Ys2 ) )
=> ( ! [X: a,Y2: a,Xs2: list @ a,Ys2: list @ a] :
( ( less @ X @ Y2 )
=> ( P @ ( cons @ a @ X @ Xs2 ) @ ( cons @ a @ Y2 @ Ys2 ) ) )
=> ( ! [X: a,Y2: a,Xs2: list @ a,Ys2: list @ a] :
( ~ ( less @ X @ Y2 )
=> ( ~ ( less @ Y2 @ X )
=> ( ( lexordp @ a @ less @ Xs2 @ Ys2 )
=> ( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons @ a @ X @ Xs2 ) @ ( cons @ a @ Y2 @ Ys2 ) ) ) ) ) )
=> ( P @ X1 @ X23 ) ) ) ) ) ).
% local.lexordp.inducts
thf(fact_53_local_Olexordp_Osimps,axiom,
! [A1: list @ a,A22: list @ a] :
( ( lexordp @ a @ less @ A1 @ A22 )
= ( ? [Y3: a,Ys3: list @ a] :
( ( A1
= ( nil @ a ) )
& ( A22
= ( cons @ a @ Y3 @ Ys3 ) ) )
| ? [X3: a,Y3: a,Xs3: list @ a,Ys3: list @ a] :
( ( A1
= ( cons @ a @ X3 @ Xs3 ) )
& ( A22
= ( cons @ a @ Y3 @ Ys3 ) )
& ( less @ X3 @ Y3 ) )
| ? [X3: a,Y3: a,Xs3: list @ a,Ys3: list @ a] :
( ( A1
= ( cons @ a @ X3 @ Xs3 ) )
& ( A22
= ( cons @ a @ Y3 @ Ys3 ) )
& ~ ( less @ X3 @ Y3 )
& ~ ( less @ Y3 @ X3 )
& ( lexordp @ a @ less @ Xs3 @ Ys3 ) ) ) ) ).
% local.lexordp.simps
thf(fact_54_local_Olexordp__eq_Ocases,axiom,
! [A1: list @ a,A22: list @ a] :
( ( lexordp_eq @ a @ less @ A1 @ A22 )
=> ( ( ( A1
= ( nil @ a ) )
=> ! [Ys2: list @ a] : A22 != Ys2 )
=> ( ! [X: a] :
( ? [Xs2: list @ a] :
( A1
= ( cons @ a @ X @ Xs2 ) )
=> ! [Y2: a] :
( ? [Ys2: list @ a] :
( A22
= ( cons @ a @ Y2 @ Ys2 ) )
=> ~ ( less @ X @ Y2 ) ) )
=> ~ ! [X: a,Y2: a,Xs2: list @ a] :
( ( A1
= ( cons @ a @ X @ Xs2 ) )
=> ! [Ys2: list @ a] :
( ( A22
= ( cons @ a @ Y2 @ Ys2 ) )
=> ( ~ ( less @ X @ Y2 )
=> ( ~ ( less @ Y2 @ X )
=> ~ ( lexordp_eq @ a @ less @ Xs2 @ Ys2 ) ) ) ) ) ) ) ) ).
% local.lexordp_eq.cases
thf(fact_55_local_Olexordp__eq_Oinducts,axiom,
! [X1: list @ a,X23: list @ a,P: ( list @ a ) > ( list @ a ) > $o] :
( ( lexordp_eq @ a @ less @ X1 @ X23 )
=> ( ! [X12: list @ a] : ( P @ ( nil @ a ) @ X12 )
=> ( ! [X: a,Y2: a,Xs2: list @ a,Ys2: list @ a] :
( ( less @ X @ Y2 )
=> ( P @ ( cons @ a @ X @ Xs2 ) @ ( cons @ a @ Y2 @ Ys2 ) ) )
=> ( ! [X: a,Y2: a,Xs2: list @ a,Ys2: list @ a] :
( ~ ( less @ X @ Y2 )
=> ( ~ ( less @ Y2 @ X )
=> ( ( lexordp_eq @ a @ less @ Xs2 @ Ys2 )
=> ( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons @ a @ X @ Xs2 ) @ ( cons @ a @ Y2 @ Ys2 ) ) ) ) ) )
=> ( P @ X1 @ X23 ) ) ) ) ) ).
% local.lexordp_eq.inducts
thf(fact_56_local_Olexordp__eq_Osimps,axiom,
! [A1: list @ a,A22: list @ a] :
( ( lexordp_eq @ a @ less @ A1 @ A22 )
= ( ? [Ys3: list @ a] :
( ( A1
= ( nil @ a ) )
& ( A22 = Ys3 ) )
| ? [X3: a,Y3: a,Xs3: list @ a,Ys3: list @ a] :
( ( A1
= ( cons @ a @ X3 @ Xs3 ) )
& ( A22
= ( cons @ a @ Y3 @ Ys3 ) )
& ( less @ X3 @ Y3 ) )
| ? [X3: a,Y3: a,Xs3: list @ a,Ys3: list @ a] :
( ( A1
= ( cons @ a @ X3 @ Xs3 ) )
& ( A22
= ( cons @ a @ Y3 @ Ys3 ) )
& ~ ( less @ X3 @ Y3 )
& ~ ( less @ Y3 @ X3 )
& ( lexordp_eq @ a @ less @ Xs3 @ Ys3 ) ) ) ) ).
% local.lexordp_eq.simps
thf(fact_57_local_Olexordp__append__left__rightI,axiom,
! [X2: a,Y: a,Us: list @ a,Xs: list @ a,Ys: list @ a] :
( ( less @ X2 @ Y )
=> ( lexordp @ a @ less @ ( append @ a @ Us @ ( cons @ a @ X2 @ Xs ) ) @ ( append @ a @ Us @ ( cons @ a @ Y @ Ys ) ) ) ) ).
% local.lexordp_append_left_rightI
thf(fact_58_local_Olexordp__append__rightI,axiom,
! [Ys: list @ a,Xs: list @ a] :
( ( Ys
!= ( nil @ a ) )
=> ( lexordp @ a @ less @ Xs @ ( append @ a @ Xs @ Ys ) ) ) ).
% local.lexordp_append_rightI
thf(fact_59_append__is__Nil__conv,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( append @ A @ Xs @ Ys )
= ( nil @ A ) )
= ( ( Xs
= ( nil @ A ) )
& ( Ys
= ( nil @ A ) ) ) ) ).
% append_is_Nil_conv
thf(fact_60_Nil__is__append__conv,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( nil @ A )
= ( append @ A @ Xs @ Ys ) )
= ( ( Xs
= ( nil @ A ) )
& ( Ys
= ( nil @ A ) ) ) ) ).
% Nil_is_append_conv
thf(fact_61_self__append__conv2,axiom,
! [A: $tType,Ys: list @ A,Xs: list @ A] :
( ( Ys
= ( append @ A @ Xs @ Ys ) )
= ( Xs
= ( nil @ A ) ) ) ).
% self_append_conv2
thf(fact_62_append__self__conv2,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( append @ A @ Xs @ Ys )
= Ys )
= ( Xs
= ( nil @ A ) ) ) ).
% append_self_conv2
thf(fact_63_self__append__conv,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( Xs
= ( append @ A @ Xs @ Ys ) )
= ( Ys
= ( nil @ A ) ) ) ).
% self_append_conv
thf(fact_64_append__self__conv,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( append @ A @ Xs @ Ys )
= Xs )
= ( Ys
= ( nil @ A ) ) ) ).
% append_self_conv
thf(fact_65_append__Nil2,axiom,
! [A: $tType,Xs: list @ A] :
( ( append @ A @ Xs @ ( nil @ A ) )
= Xs ) ).
% append_Nil2
thf(fact_66_ord_Olexordp__simps_I3_J,axiom,
! [A: $tType,Less: A > A > $o,X2: A,Xs: list @ A,Y: A,Ys: list @ A] :
( ( lexordp @ A @ Less @ ( cons @ A @ X2 @ Xs ) @ ( cons @ A @ Y @ Ys ) )
= ( ( Less @ X2 @ Y )
| ( ~ ( Less @ Y @ X2 )
& ( lexordp @ A @ Less @ Xs @ Ys ) ) ) ) ).
% ord.lexordp_simps(3)
thf(fact_67_ord_Olexordp__eq__simps_I4_J,axiom,
! [A: $tType,Less: A > A > $o,X2: A,Xs: list @ A,Y: A,Ys: list @ A] :
( ( lexordp_eq @ A @ Less @ ( cons @ A @ X2 @ Xs ) @ ( cons @ A @ Y @ Ys ) )
= ( ( Less @ X2 @ Y )
| ( ~ ( Less @ Y @ X2 )
& ( lexordp_eq @ A @ Less @ Xs @ Ys ) ) ) ) ).
% ord.lexordp_eq_simps(4)
thf(fact_68_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_69_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_70_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_71_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_72_append1__eq__conv,axiom,
! [A: $tType,Xs: list @ A,X2: A,Ys: list @ A,Y: A] :
( ( ( append @ A @ Xs @ ( cons @ A @ X2 @ ( nil @ A ) ) )
= ( append @ A @ Ys @ ( cons @ A @ Y @ ( nil @ A ) ) ) )
= ( ( Xs = Ys )
& ( X2 = Y ) ) ) ).
% append1_eq_conv
thf(fact_73_ord_Olexordp__eq__simps_I3_J,axiom,
! [A: $tType,Less: A > A > $o,X2: A,Xs: list @ A] :
~ ( lexordp_eq @ A @ Less @ ( cons @ A @ X2 @ Xs ) @ ( nil @ A ) ) ).
% ord.lexordp_eq_simps(3)
thf(fact_74_local_Olexordp__simps_I3_J,axiom,
! [X2: a,Xs: list @ a,Y: a,Ys: list @ a] :
( ( lexordp @ a @ less @ ( cons @ a @ X2 @ Xs ) @ ( cons @ a @ Y @ Ys ) )
= ( ( less @ X2 @ Y )
| ( ~ ( less @ Y @ X2 )
& ( lexordp @ a @ less @ Xs @ Ys ) ) ) ) ).
% local.lexordp_simps(3)
thf(fact_75_local_Olexordp__eq__simps_I4_J,axiom,
! [X2: a,Xs: list @ a,Y: a,Ys: list @ a] :
( ( lexordp_eq @ a @ less @ ( cons @ a @ X2 @ Xs ) @ ( cons @ a @ Y @ Ys ) )
= ( ( less @ X2 @ Y )
| ( ~ ( less @ Y @ X2 )
& ( lexordp_eq @ a @ less @ Xs @ Ys ) ) ) ) ).
% local.lexordp_eq_simps(4)
thf(fact_76_local_Olexordp__simps_I2_J,axiom,
! [Xs: list @ a] :
~ ( lexordp @ a @ less @ Xs @ ( nil @ a ) ) ).
% local.lexordp_simps(2)
thf(fact_77_local_Olexordp__simps_I1_J,axiom,
! [Ys: list @ a] :
( ( lexordp @ a @ less @ ( nil @ a ) @ Ys )
= ( Ys
!= ( nil @ a ) ) ) ).
% local.lexordp_simps(1)
thf(fact_78_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_79_local_Olexordp__eq__simps_I1_J,axiom,
! [Ys: list @ a] : ( lexordp_eq @ a @ less @ ( nil @ a ) @ Ys ) ).
% local.lexordp_eq_simps(1)
thf(fact_80_local_Olexordp__eq__simps_I3_J,axiom,
! [X2: a,Xs: list @ a] :
~ ( lexordp_eq @ a @ less @ ( cons @ a @ X2 @ Xs ) @ ( nil @ a ) ) ).
% local.lexordp_eq_simps(3)
thf(fact_81_ord_Olexordp_ONil,axiom,
! [A: $tType,Less: A > A > $o,Y: A,Ys: list @ A] : ( lexordp @ A @ Less @ ( nil @ A ) @ ( cons @ A @ Y @ Ys ) ) ).
% ord.lexordp.Nil
thf(fact_82_ord_Olexordp_Ocases,axiom,
! [A: $tType,Less: A > A > $o,A1: list @ A,A22: list @ A] :
( ( lexordp @ A @ Less @ A1 @ A22 )
=> ( ( ( A1
= ( nil @ A ) )
=> ! [Y2: A,Ys2: list @ A] :
( A22
!= ( cons @ A @ Y2 @ Ys2 ) ) )
=> ( ! [X: A] :
( ? [Xs2: list @ A] :
( A1
= ( cons @ A @ X @ Xs2 ) )
=> ! [Y2: A] :
( ? [Ys2: list @ A] :
( A22
= ( cons @ A @ Y2 @ Ys2 ) )
=> ~ ( Less @ X @ Y2 ) ) )
=> ~ ! [X: A,Y2: A,Xs2: list @ A] :
( ( A1
= ( cons @ A @ X @ Xs2 ) )
=> ! [Ys2: list @ A] :
( ( A22
= ( cons @ A @ Y2 @ Ys2 ) )
=> ( ~ ( Less @ X @ Y2 )
=> ( ~ ( Less @ Y2 @ X )
=> ~ ( lexordp @ A @ Less @ Xs2 @ Ys2 ) ) ) ) ) ) ) ) ).
% ord.lexordp.cases
thf(fact_83_ord_Olexordp_Osimps,axiom,
! [A: $tType] :
( ( lexordp @ A )
= ( ^ [Less2: A > A > $o,A12: list @ A,A23: list @ A] :
( ? [Y3: A,Ys3: list @ A] :
( ( A12
= ( nil @ A ) )
& ( A23
= ( cons @ A @ Y3 @ Ys3 ) ) )
| ? [X3: A,Y3: A,Xs3: list @ A,Ys3: list @ A] :
( ( A12
= ( cons @ A @ X3 @ Xs3 ) )
& ( A23
= ( cons @ A @ Y3 @ Ys3 ) )
& ( Less2 @ X3 @ Y3 ) )
| ? [X3: A,Y3: A,Xs3: list @ A,Ys3: list @ A] :
( ( A12
= ( cons @ A @ X3 @ Xs3 ) )
& ( A23
= ( cons @ A @ Y3 @ Ys3 ) )
& ~ ( Less2 @ X3 @ Y3 )
& ~ ( Less2 @ Y3 @ X3 )
& ( lexordp @ A @ Less2 @ Xs3 @ Ys3 ) ) ) ) ) ).
% ord.lexordp.simps
thf(fact_84_ord_Olexordp_Oinducts,axiom,
! [A: $tType,Less: A > A > $o,X1: list @ A,X23: list @ A,P: ( list @ A ) > ( list @ A ) > $o] :
( ( lexordp @ A @ Less @ X1 @ X23 )
=> ( ! [Y2: A,Ys2: list @ A] : ( P @ ( nil @ A ) @ ( cons @ A @ Y2 @ Ys2 ) )
=> ( ! [X: A,Y2: A,Xs2: list @ A,Ys2: list @ A] :
( ( Less @ X @ Y2 )
=> ( P @ ( cons @ A @ X @ Xs2 ) @ ( cons @ A @ Y2 @ Ys2 ) ) )
=> ( ! [X: A,Y2: A,Xs2: list @ A,Ys2: list @ A] :
( ~ ( Less @ X @ Y2 )
=> ( ~ ( Less @ Y2 @ X )
=> ( ( lexordp @ A @ Less @ Xs2 @ Ys2 )
=> ( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons @ A @ X @ Xs2 ) @ ( cons @ A @ Y2 @ Ys2 ) ) ) ) ) )
=> ( P @ X1 @ X23 ) ) ) ) ) ).
% ord.lexordp.inducts
thf(fact_85_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 ) )
=> ! [Ys2: list @ A] : A22 != Ys2 )
=> ( ! [X: A] :
( ? [Xs2: list @ A] :
( A1
= ( cons @ A @ X @ Xs2 ) )
=> ! [Y2: A] :
( ? [Ys2: list @ A] :
( A22
= ( cons @ A @ Y2 @ Ys2 ) )
=> ~ ( Less @ X @ Y2 ) ) )
=> ~ ! [X: A,Y2: A,Xs2: list @ A] :
( ( A1
= ( cons @ A @ X @ Xs2 ) )
=> ! [Ys2: list @ A] :
( ( A22
= ( cons @ A @ Y2 @ Ys2 ) )
=> ( ~ ( Less @ X @ Y2 )
=> ( ~ ( Less @ Y2 @ X )
=> ~ ( lexordp_eq @ A @ Less @ Xs2 @ Ys2 ) ) ) ) ) ) ) ) ).
% ord.lexordp_eq.cases
thf(fact_86_ord_Olexordp__eq_Osimps,axiom,
! [A: $tType] :
( ( lexordp_eq @ A )
= ( ^ [Less2: A > A > $o,A12: list @ A,A23: list @ A] :
( ? [Ys3: list @ A] :
( ( A12
= ( nil @ A ) )
& ( A23 = Ys3 ) )
| ? [X3: A,Y3: A,Xs3: list @ A,Ys3: list @ A] :
( ( A12
= ( cons @ A @ X3 @ Xs3 ) )
& ( A23
= ( cons @ A @ Y3 @ Ys3 ) )
& ( Less2 @ X3 @ Y3 ) )
| ? [X3: A,Y3: A,Xs3: list @ A,Ys3: list @ A] :
( ( A12
= ( cons @ A @ X3 @ Xs3 ) )
& ( A23
= ( cons @ A @ Y3 @ Ys3 ) )
& ~ ( Less2 @ X3 @ Y3 )
& ~ ( Less2 @ Y3 @ X3 )
& ( lexordp_eq @ A @ Less2 @ Xs3 @ Ys3 ) ) ) ) ) ).
% ord.lexordp_eq.simps
thf(fact_87_ord_Olexordp__eq_Oinducts,axiom,
! [A: $tType,Less: A > A > $o,X1: list @ A,X23: list @ A,P: ( list @ A ) > ( list @ A ) > $o] :
( ( lexordp_eq @ A @ Less @ X1 @ X23 )
=> ( ! [X12: list @ A] : ( P @ ( nil @ A ) @ X12 )
=> ( ! [X: A,Y2: A,Xs2: list @ A,Ys2: list @ A] :
( ( Less @ X @ Y2 )
=> ( P @ ( cons @ A @ X @ Xs2 ) @ ( cons @ A @ Y2 @ Ys2 ) ) )
=> ( ! [X: A,Y2: A,Xs2: list @ A,Ys2: list @ A] :
( ~ ( Less @ X @ Y2 )
=> ( ~ ( Less @ Y2 @ X )
=> ( ( lexordp_eq @ A @ Less @ Xs2 @ Ys2 )
=> ( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons @ A @ X @ Xs2 ) @ ( cons @ A @ Y2 @ Ys2 ) ) ) ) ) )
=> ( P @ X1 @ X23 ) ) ) ) ) ).
% ord.lexordp_eq.inducts
thf(fact_88_ord_Olexordp__append__rightI,axiom,
! [A: $tType,Ys: list @ A,Less: A > A > $o,Xs: list @ A] :
( ( Ys
!= ( nil @ A ) )
=> ( lexordp @ A @ Less @ Xs @ ( append @ A @ Xs @ Ys ) ) ) ).
% ord.lexordp_append_rightI
thf(fact_89_ord_Olexordp__append__left__rightI,axiom,
! [A: $tType,Less: A > A > $o,X2: A,Y: A,Us: list @ A,Xs: list @ A,Ys: list @ A] :
( ( Less @ X2 @ Y )
=> ( lexordp @ A @ Less @ ( append @ A @ Us @ ( cons @ A @ X2 @ Xs ) ) @ ( append @ A @ Us @ ( cons @ A @ Y @ Ys ) ) ) ) ).
% ord.lexordp_append_left_rightI
thf(fact_90_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] :
( ! [F2: A > B,X12: list @ B] : ( P @ F2 @ ( nil @ A ) @ X12 )
=> ( ! [F2: A > B,A3: A,As: list @ A,Bs: list @ B] :
( ( P @ F2 @ As @ ( cons @ B @ ( F2 @ A3 ) @ Bs ) )
=> ( P @ F2 @ ( cons @ A @ A3 @ As ) @ Bs ) )
=> ( P @ A0 @ A1 @ A22 ) ) ) ).
% map_tailrec_rev.induct
thf(fact_91_append__eq__append__conv2,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A,Ts: list @ A] :
( ( ( append @ A @ Xs @ Ys )
= ( append @ A @ Zs @ Ts ) )
= ( ? [Us2: list @ A] :
( ( ( Xs
= ( append @ A @ Zs @ Us2 ) )
& ( ( append @ A @ Us2 @ Ys )
= Ts ) )
| ( ( ( append @ A @ Xs @ Us2 )
= Zs )
& ( Ys
= ( append @ A @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_92_list__nonempty__induct,axiom,
! [A: $tType,Xs: list @ A,P: ( list @ A ) > $o] :
( ( Xs
!= ( nil @ A ) )
=> ( ! [X: A] : ( P @ ( cons @ A @ X @ ( nil @ A ) ) )
=> ( ! [X: A,Xs2: list @ A] :
( ( Xs2
!= ( nil @ A ) )
=> ( ( P @ Xs2 )
=> ( P @ ( cons @ A @ X @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_93_rev__nonempty__induct,axiom,
! [A: $tType,Xs: list @ A,P: ( list @ A ) > $o] :
( ( Xs
!= ( nil @ A ) )
=> ( ! [X: A] : ( P @ ( cons @ A @ X @ ( nil @ A ) ) )
=> ( ! [X: A,Xs2: list @ A] :
( ( Xs2
!= ( nil @ A ) )
=> ( ( P @ Xs2 )
=> ( P @ ( append @ A @ Xs2 @ ( cons @ A @ X @ ( nil @ A ) ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_94_append__eq__Cons__conv,axiom,
! [A: $tType,Ys: list @ A,Zs: list @ A,X2: A,Xs: list @ A] :
( ( ( append @ A @ Ys @ Zs )
= ( cons @ A @ X2 @ Xs ) )
= ( ( ( Ys
= ( nil @ A ) )
& ( Zs
= ( cons @ A @ X2 @ Xs ) ) )
| ? [Ys4: list @ A] :
( ( Ys
= ( cons @ A @ X2 @ Ys4 ) )
& ( ( append @ A @ Ys4 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_95_Cons__eq__append__conv,axiom,
! [A: $tType,X2: A,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( ( cons @ A @ X2 @ Xs )
= ( append @ A @ Ys @ Zs ) )
= ( ( ( Ys
= ( nil @ A ) )
& ( ( cons @ A @ X2 @ Xs )
= Zs ) )
| ? [Ys4: list @ A] :
( ( ( cons @ A @ X2 @ Ys4 )
= Ys )
& ( Xs
= ( append @ A @ Ys4 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_96_remdups__adj_Oinduct,axiom,
! [A: $tType,P: ( list @ A ) > $o,A0: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [X: A] : ( P @ ( cons @ A @ X @ ( nil @ A ) ) )
=> ( ! [X: A,Y2: A,Xs2: list @ A] :
( ( ( X = Y2 )
=> ( P @ ( cons @ A @ X @ Xs2 ) ) )
=> ( ( ( X != Y2 )
=> ( P @ ( cons @ A @ Y2 @ Xs2 ) ) )
=> ( P @ ( cons @ A @ X @ ( cons @ A @ Y2 @ Xs2 ) ) ) ) )
=> ( P @ A0 ) ) ) ) ).
% remdups_adj.induct
thf(fact_97_append__Cons,axiom,
! [A: $tType,X2: A,Xs: list @ A,Ys: list @ A] :
( ( append @ A @ ( cons @ A @ X2 @ Xs ) @ Ys )
= ( cons @ A @ X2 @ ( append @ A @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_98_remdups__adj_Ocases,axiom,
! [A: $tType,X2: list @ A] :
( ( X2
!= ( nil @ A ) )
=> ( ! [X: A] :
( X2
!= ( cons @ A @ X @ ( nil @ A ) ) )
=> ~ ! [X: A,Y2: A,Xs2: list @ A] :
( X2
!= ( cons @ A @ X @ ( cons @ A @ Y2 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_99_append__eq__appendI,axiom,
! [A: $tType,Xs: list @ A,Xs1: list @ A,Zs: list @ A,Ys: list @ A,Us: list @ A] :
( ( ( append @ A @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append @ A @ Xs1 @ Us ) )
=> ( ( append @ A @ Xs @ Ys )
= ( append @ A @ Zs @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_100_append__Nil,axiom,
! [A: $tType,Ys: list @ A] :
( ( append @ A @ ( nil @ A ) @ Ys )
= Ys ) ).
% append_Nil
thf(fact_101_transpose_Ocases,axiom,
! [A: $tType,X2: list @ ( list @ A )] :
( ( X2
!= ( nil @ ( list @ A ) ) )
=> ( ! [Xss: list @ ( list @ A )] :
( X2
!= ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss ) )
=> ~ ! [X: A,Xs2: list @ A,Xss: list @ ( list @ A )] :
( X2
!= ( cons @ ( list @ A ) @ ( cons @ A @ X @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_102_Cons__eq__appendI,axiom,
! [A: $tType,X2: A,Xs1: list @ A,Ys: list @ A,Xs: list @ A,Zs: list @ A] :
( ( ( cons @ A @ X2 @ Xs1 )
= Ys )
=> ( ( Xs
= ( append @ A @ Xs1 @ Zs ) )
=> ( ( cons @ A @ X2 @ Xs )
= ( append @ A @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_103_not__Cons__self2,axiom,
! [A: $tType,X2: A,Xs: list @ A] :
( ( cons @ A @ X2 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_104_eq__Nil__appendI,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( Xs = Ys )
=> ( Xs
= ( append @ A @ ( nil @ A ) @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_105_splice_Oinduct,axiom,
! [A: $tType,P: ( list @ A ) > ( list @ A ) > $o,A0: list @ A,A1: list @ A] :
( ! [X12: list @ A] : ( P @ ( nil @ A ) @ X12 )
=> ( ! [V: A,Va: list @ A] : ( P @ ( cons @ A @ V @ Va ) @ ( nil @ A ) )
=> ( ! [X: A,Xs2: list @ A,Y2: A,Ys2: list @ A] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons @ A @ X @ Xs2 ) @ ( cons @ A @ Y2 @ Ys2 ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% splice.induct
thf(fact_106_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 ) )
=> ( ! [X: A,Xs2: list @ A] : ( P @ ( cons @ A @ X @ Xs2 ) @ ( nil @ B ) )
=> ( ! [Y2: B,Ys2: list @ B] : ( P @ ( nil @ A ) @ ( cons @ B @ Y2 @ Ys2 ) )
=> ( ! [X: A,Xs2: list @ A,Y2: B,Ys2: list @ B] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons @ A @ X @ Xs2 ) @ ( cons @ B @ Y2 @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_107_neq__Nil__conv,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
= ( ? [Y3: A,Ys3: list @ A] :
( Xs
= ( cons @ A @ Y3 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_108_list_Oinducts,axiom,
! [A: $tType,P: ( list @ A ) > $o,List: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [X12: A,X24: list @ A] :
( ( P @ X24 )
=> ( P @ ( cons @ A @ X12 @ X24 ) ) )
=> ( P @ List ) ) ) ).
% list.inducts
thf(fact_109_list_Oexhaust,axiom,
! [A: $tType,Y: list @ A] :
( ( Y
!= ( nil @ A ) )
=> ~ ! [X212: A,X222: list @ A] :
( Y
!= ( cons @ A @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_110_rev__exhaust,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ~ ! [Ys2: list @ A,Y2: A] :
( Xs
!= ( append @ A @ Ys2 @ ( cons @ A @ Y2 @ ( nil @ A ) ) ) ) ) ).
% rev_exhaust
thf(fact_111_rev__induct,axiom,
! [A: $tType,P: ( list @ A ) > $o,Xs: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [X: A,Xs2: list @ A] :
( ( P @ Xs2 )
=> ( P @ ( append @ A @ Xs2 @ ( cons @ A @ X @ ( nil @ A ) ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_112_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_113_list_Odistinct_I1_J,axiom,
! [A: $tType,X21: A,X22: list @ A] :
( ( nil @ A )
!= ( cons @ A @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_114_ord_Olexordp__append__leftI,axiom,
! [A: $tType,Less: A > A > $o,Us: list @ A,Vs: list @ A,Xs: list @ A] :
( ( lexordp @ A @ Less @ Us @ Vs )
=> ( lexordp @ A @ Less @ ( append @ A @ Xs @ Us ) @ ( append @ A @ Xs @ Vs ) ) ) ).
% ord.lexordp_append_leftI
thf(fact_115_ord_Olexordp__append__leftD,axiom,
! [A: $tType,Less: A > A > $o,Xs: list @ A,Us: list @ A,Vs: list @ A] :
( ( lexordp @ A @ Less @ ( append @ A @ Xs @ Us ) @ ( append @ A @ Xs @ Vs ) )
=> ( ! [A3: A] :
~ ( Less @ A3 @ A3 )
=> ( lexordp @ A @ Less @ Us @ Vs ) ) ) ).
% ord.lexordp_append_leftD
thf(fact_116_ord_Olexordp_OCons__eq,axiom,
! [A: $tType,Less: A > A > $o,X2: A,Y: A,Xs: list @ A,Ys: list @ A] :
( ~ ( Less @ X2 @ Y )
=> ( ~ ( Less @ Y @ X2 )
=> ( ( lexordp @ A @ Less @ Xs @ Ys )
=> ( lexordp @ A @ Less @ ( cons @ A @ X2 @ Xs ) @ ( cons @ A @ Y @ Ys ) ) ) ) ) ).
% ord.lexordp.Cons_eq
thf(fact_117_ord_Olexordp_OCons,axiom,
! [A: $tType,Less: A > A > $o,X2: A,Y: A,Xs: list @ A,Ys: list @ A] :
( ( Less @ X2 @ Y )
=> ( lexordp @ A @ Less @ ( cons @ A @ X2 @ Xs ) @ ( cons @ A @ Y @ Ys ) ) ) ).
% ord.lexordp.Cons
thf(fact_118_ord_Olexordp__eq_OCons__eq,axiom,
! [A: $tType,Less: A > A > $o,X2: A,Y: A,Xs: list @ A,Ys: list @ A] :
( ~ ( Less @ X2 @ Y )
=> ( ~ ( Less @ Y @ X2 )
=> ( ( lexordp_eq @ A @ Less @ Xs @ Ys )
=> ( lexordp_eq @ A @ Less @ ( cons @ A @ X2 @ Xs ) @ ( cons @ A @ Y @ Ys ) ) ) ) ) ).
% ord.lexordp_eq.Cons_eq
thf(fact_119_ord_Olexordp__eq_OCons,axiom,
! [A: $tType,Less: A > A > $o,X2: A,Y: A,Xs: list @ A,Ys: list @ A] :
( ( Less @ X2 @ Y )
=> ( lexordp_eq @ A @ Less @ ( cons @ A @ X2 @ Xs ) @ ( cons @ A @ Y @ Ys ) ) ) ).
% ord.lexordp_eq.Cons
thf(fact_120_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_121_ord_Olexordp__irreflexive,axiom,
! [A: $tType,Less: A > A > $o,Xs: list @ A] :
( ! [X: A] :
~ ( Less @ X @ X )
=> ~ ( lexordp @ A @ Less @ Xs @ Xs ) ) ).
% ord.lexordp_irreflexive
thf(fact_122_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_123_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_124_ord_OgreaterThanLessThan__eq,axiom,
! [A: $tType] :
( ( set_gr1161524159ssThan @ A )
= ( ^ [Less2: A > A > $o,A5: A,B3: A] : ( inf_inf @ ( set @ A ) @ ( set_greaterThan @ A @ Less2 @ A5 ) @ ( set_lessThan @ A @ Less2 @ B3 ) ) ) ) ).
% ord.greaterThanLessThan_eq
thf(fact_125_Int__iff,axiom,
! [A: $tType,C: A,A4: set @ A,B4: set @ A] :
( ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A4 @ B4 ) )
= ( ( member @ A @ C @ A4 )
& ( member @ A @ C @ B4 ) ) ) ).
% Int_iff
thf(fact_126_IntI,axiom,
! [A: $tType,C: A,A4: set @ A,B4: set @ A] :
( ( member @ A @ C @ A4 )
=> ( ( member @ A @ C @ B4 )
=> ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A4 @ B4 ) ) ) ) ).
% IntI
thf(fact_127_inf__right__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A @ ( type2 @ A ) )
=> ! [X2: A,Y: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ X2 @ Y ) @ Y )
= ( inf_inf @ A @ X2 @ Y ) ) ) ).
% inf_right_idem
thf(fact_128_inf_Oright__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ A2 @ B2 ) @ B2 )
= ( inf_inf @ A @ A2 @ B2 ) ) ) ).
% inf.right_idem
thf(fact_129_inf__left__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A @ ( type2 @ A ) )
=> ! [X2: A,Y: A] :
( ( inf_inf @ A @ X2 @ ( inf_inf @ A @ X2 @ Y ) )
= ( inf_inf @ A @ X2 @ Y ) ) ) ).
% inf_left_idem
thf(fact_130_inf_Oleft__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( inf_inf @ A @ A2 @ ( inf_inf @ A @ A2 @ B2 ) )
= ( inf_inf @ A @ A2 @ B2 ) ) ) ).
% inf.left_idem
thf(fact_131_inf__apply,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_inf @ B @ ( type2 @ B ) )
=> ( ( inf_inf @ ( A > B ) )
= ( ^ [F3: A > B,G2: A > B,X3: A] : ( inf_inf @ B @ ( F3 @ X3 ) @ ( G2 @ X3 ) ) ) ) ) ).
% inf_apply
thf(fact_132_inf_Oidem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( inf_inf @ A @ A2 @ A2 )
= A2 ) ) ).
% inf.idem
thf(fact_133_inf__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A @ ( type2 @ A ) )
=> ! [X2: A] :
( ( inf_inf @ A @ X2 @ X2 )
= X2 ) ) ).
% inf_idem
thf(fact_134_inf__sup__aci_I4_J,axiom,
! [A: $tType] :
( ( lattice @ A @ ( type2 @ A ) )
=> ! [X2: A,Y: A] :
( ( inf_inf @ A @ X2 @ ( inf_inf @ A @ X2 @ Y ) )
= ( inf_inf @ A @ X2 @ Y ) ) ) ).
% inf_sup_aci(4)
thf(fact_135_inf__sup__aci_I3_J,axiom,
! [A: $tType] :
( ( lattice @ A @ ( type2 @ A ) )
=> ! [X2: A,Y: A,Z: A] :
( ( inf_inf @ A @ X2 @ ( inf_inf @ A @ Y @ Z ) )
= ( inf_inf @ A @ Y @ ( inf_inf @ A @ X2 @ Z ) ) ) ) ).
% inf_sup_aci(3)
thf(fact_136_inf__sup__aci_I2_J,axiom,
! [A: $tType] :
( ( lattice @ A @ ( type2 @ A ) )
=> ! [X2: A,Y: A,Z: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ X2 @ Y ) @ Z )
= ( inf_inf @ A @ X2 @ ( inf_inf @ A @ Y @ Z ) ) ) ) ).
% inf_sup_aci(2)
thf(fact_137_inf__sup__aci_I1_J,axiom,
! [A: $tType] :
( ( lattice @ A @ ( type2 @ A ) )
=> ( ( inf_inf @ A )
= ( ^ [X3: A,Y3: A] : ( inf_inf @ A @ Y3 @ X3 ) ) ) ) ).
% inf_sup_aci(1)
thf(fact_138_inf__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_inf @ B @ ( type2 @ B ) )
=> ( ( inf_inf @ ( A > B ) )
= ( ^ [F3: A > B,G2: A > B,X3: A] : ( inf_inf @ B @ ( F3 @ X3 ) @ ( G2 @ X3 ) ) ) ) ) ).
% inf_fun_def
thf(fact_139_inf_Oassoc,axiom,
! [A: $tType] :
( ( semilattice_inf @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ A2 @ B2 ) @ C )
= ( inf_inf @ A @ A2 @ ( inf_inf @ A @ B2 @ C ) ) ) ) ).
% inf.assoc
thf(fact_140_inf__assoc,axiom,
! [A: $tType] :
( ( semilattice_inf @ A @ ( type2 @ A ) )
=> ! [X2: A,Y: A,Z: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ X2 @ Y ) @ Z )
= ( inf_inf @ A @ X2 @ ( inf_inf @ A @ Y @ Z ) ) ) ) ).
% inf_assoc
thf(fact_141_inf_Ocommute,axiom,
! [A: $tType] :
( ( semilattice_inf @ A @ ( type2 @ A ) )
=> ( ( inf_inf @ A )
= ( ^ [A5: A,B3: A] : ( inf_inf @ A @ B3 @ A5 ) ) ) ) ).
% inf.commute
thf(fact_142_inf__commute,axiom,
! [A: $tType] :
( ( semilattice_inf @ A @ ( type2 @ A ) )
=> ( ( inf_inf @ A )
= ( ^ [X3: A,Y3: A] : ( inf_inf @ A @ Y3 @ X3 ) ) ) ) ).
% inf_commute
thf(fact_143_inf_Oleft__commute,axiom,
! [A: $tType] :
( ( semilattice_inf @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A,C: A] :
( ( inf_inf @ A @ B2 @ ( inf_inf @ A @ A2 @ C ) )
= ( inf_inf @ A @ A2 @ ( inf_inf @ A @ B2 @ C ) ) ) ) ).
% inf.left_commute
thf(fact_144_inf__left__commute,axiom,
! [A: $tType] :
( ( semilattice_inf @ A @ ( type2 @ A ) )
=> ! [X2: A,Y: A,Z: A] :
( ( inf_inf @ A @ X2 @ ( inf_inf @ A @ Y @ Z ) )
= ( inf_inf @ A @ Y @ ( inf_inf @ A @ X2 @ Z ) ) ) ) ).
% inf_left_commute
thf(fact_145_IntE,axiom,
! [A: $tType,C: A,A4: set @ A,B4: set @ A] :
( ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A4 @ B4 ) )
=> ~ ( ( member @ A @ C @ A4 )
=> ~ ( member @ A @ C @ B4 ) ) ) ).
% IntE
thf(fact_146_IntD1,axiom,
! [A: $tType,C: A,A4: set @ A,B4: set @ A] :
( ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A4 @ B4 ) )
=> ( member @ A @ C @ A4 ) ) ).
% IntD1
thf(fact_147_IntD2,axiom,
! [A: $tType,C: A,A4: set @ A,B4: set @ A] :
( ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A4 @ B4 ) )
=> ( member @ A @ C @ B4 ) ) ).
% IntD2
thf(fact_148_Int__assoc,axiom,
! [A: $tType,A4: set @ A,B4: set @ A,C2: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A4 @ B4 ) @ C2 )
= ( inf_inf @ ( set @ A ) @ A4 @ ( inf_inf @ ( set @ A ) @ B4 @ C2 ) ) ) ).
% Int_assoc
thf(fact_149_Int__absorb,axiom,
! [A: $tType,A4: set @ A] :
( ( inf_inf @ ( set @ A ) @ A4 @ A4 )
= A4 ) ).
% Int_absorb
thf(fact_150_Int__commute,axiom,
! [A: $tType] :
( ( inf_inf @ ( set @ A ) )
= ( ^ [A6: set @ A,B5: set @ A] : ( inf_inf @ ( set @ A ) @ B5 @ A6 ) ) ) ).
% Int_commute
thf(fact_151_Int__left__absorb,axiom,
! [A: $tType,A4: set @ A,B4: set @ A] :
( ( inf_inf @ ( set @ A ) @ A4 @ ( inf_inf @ ( set @ A ) @ A4 @ B4 ) )
= ( inf_inf @ ( set @ A ) @ A4 @ B4 ) ) ).
% Int_left_absorb
thf(fact_152_Int__left__commute,axiom,
! [A: $tType,A4: set @ A,B4: set @ A,C2: set @ A] :
( ( inf_inf @ ( set @ A ) @ A4 @ ( inf_inf @ ( set @ A ) @ B4 @ C2 ) )
= ( inf_inf @ ( set @ A ) @ B4 @ ( inf_inf @ ( set @ A ) @ A4 @ C2 ) ) ) ).
% Int_left_commute
thf(fact_153_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_154_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_155_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_156_sublists_Osimps_I1_J,axiom,
! [A: $tType] :
( ( sublists @ A @ ( nil @ A ) )
= ( cons @ ( list @ A ) @ ( nil @ A ) @ ( nil @ ( list @ A ) ) ) ) ).
% sublists.simps(1)
thf(fact_157_prefixes_Ocases,axiom,
! [A: $tType,X2: list @ A] :
( ( X2
!= ( nil @ A ) )
=> ~ ! [X: A,Xs2: list @ A] :
( X2
!= ( cons @ A @ X @ Xs2 ) ) ) ).
% prefixes.cases
thf(fact_158_prefixes_Oinduct,axiom,
! [A: $tType,P: ( list @ A ) > $o,A0: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [X: A,Xs2: list @ A] :
( ( P @ Xs2 )
=> ( P @ ( cons @ A @ X @ Xs2 ) ) )
=> ( P @ A0 ) ) ) ).
% prefixes.induct
thf(fact_159_longest__common__prefix_Oinduct,axiom,
! [A: $tType,P: ( list @ A ) > ( list @ A ) > $o,A0: list @ A,A1: list @ A] :
( ! [X: A,Xs2: list @ A,Y2: A,Ys2: list @ A] :
( ( ( X = Y2 )
=> ( P @ Xs2 @ Ys2 ) )
=> ( P @ ( cons @ A @ X @ Xs2 ) @ ( cons @ A @ Y2 @ Ys2 ) ) )
=> ( ! [X12: list @ A] : ( P @ ( nil @ A ) @ X12 )
=> ( ! [Uu: list @ A] : ( P @ Uu @ ( nil @ A ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% longest_common_prefix.induct
thf(fact_160_prefixes__snoc,axiom,
! [A: $tType,Xs: list @ A,X2: A] :
( ( prefixes @ A @ ( append @ A @ Xs @ ( cons @ A @ X2 @ ( nil @ A ) ) ) )
= ( append @ ( list @ A ) @ ( prefixes @ A @ Xs ) @ ( cons @ ( list @ A ) @ ( append @ A @ Xs @ ( cons @ A @ X2 @ ( nil @ A ) ) ) @ ( nil @ ( list @ A ) ) ) ) ) ).
% prefixes_snoc
thf(fact_161_prefixes__eq__Snoc,axiom,
! [A: $tType,Ys: list @ A,Xs: list @ ( list @ A ),X2: list @ A] :
( ( ( prefixes @ A @ Ys )
= ( append @ ( list @ A ) @ Xs @ ( cons @ ( list @ A ) @ X2 @ ( nil @ ( list @ A ) ) ) ) )
= ( ( ( ( Ys
= ( nil @ A ) )
& ( Xs
= ( nil @ ( list @ A ) ) ) )
| ? [Z2: A,Zs2: list @ A] :
( ( Ys
= ( append @ A @ Zs2 @ ( cons @ A @ Z2 @ ( nil @ A ) ) ) )
& ( Xs
= ( prefixes @ A @ Zs2 ) ) ) )
& ( X2 = Ys ) ) ) ).
% prefixes_eq_Snoc
thf(fact_162_bind__simps_I2_J,axiom,
! [A: $tType,B: $tType,X2: B,Xs: list @ B,F: B > ( list @ A )] :
( ( bind @ B @ A @ ( cons @ B @ X2 @ Xs ) @ F )
= ( append @ A @ ( F @ X2 ) @ ( bind @ B @ A @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_163_bind__simps_I1_J,axiom,
! [B: $tType,A: $tType,F: B > ( list @ A )] :
( ( bind @ B @ A @ ( nil @ B ) @ F )
= ( nil @ A ) ) ).
% bind_simps(1)
thf(fact_164_prefixes_Osimps_I1_J,axiom,
! [A: $tType] :
( ( prefixes @ A @ ( nil @ A ) )
= ( cons @ ( list @ A ) @ ( nil @ A ) @ ( nil @ ( list @ A ) ) ) ) ).
% prefixes.simps(1)
thf(fact_165_product__lists_Osimps_I1_J,axiom,
! [A: $tType] :
( ( product_lists @ A @ ( nil @ ( list @ A ) ) )
= ( cons @ ( list @ A ) @ ( nil @ A ) @ ( nil @ ( list @ A ) ) ) ) ).
% product_lists.simps(1)
thf(fact_166_prefixes_Oelims,axiom,
! [A: $tType,X2: list @ A,Y: list @ ( list @ A )] :
( ( ( prefixes @ A @ X2 )
= Y )
=> ( ( ( X2
= ( nil @ A ) )
=> ( Y
!= ( cons @ ( list @ A ) @ ( nil @ A ) @ ( nil @ ( list @ A ) ) ) ) )
=> ~ ! [X: A,Xs2: list @ A] :
( ( X2
= ( cons @ A @ X @ Xs2 ) )
=> ( Y
!= ( cons @ ( list @ A ) @ ( nil @ A ) @ ( map @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X ) @ ( prefixes @ A @ Xs2 ) ) ) ) ) ) ) ).
% prefixes.elims
thf(fact_167_maps__simps_I1_J,axiom,
! [A: $tType,B: $tType,F: B > ( list @ A ),X2: B,Xs: list @ B] :
( ( maps @ B @ A @ F @ ( cons @ B @ X2 @ Xs ) )
= ( append @ A @ ( F @ X2 ) @ ( maps @ B @ A @ F @ Xs ) ) ) ).
% maps_simps(1)
thf(fact_168_Nil__is__map__conv,axiom,
! [A: $tType,B: $tType,F: B > A,Xs: list @ B] :
( ( ( nil @ A )
= ( map @ B @ A @ F @ Xs ) )
= ( Xs
= ( nil @ B ) ) ) ).
% Nil_is_map_conv
thf(fact_169_map__is__Nil__conv,axiom,
! [A: $tType,B: $tType,F: B > A,Xs: list @ B] :
( ( ( map @ B @ A @ F @ Xs )
= ( nil @ A ) )
= ( Xs
= ( nil @ B ) ) ) ).
% map_is_Nil_conv
thf(fact_170_list_Omap__disc__iff,axiom,
! [B: $tType,A: $tType,F: A > B,A2: list @ A] :
( ( ( map @ A @ B @ F @ A2 )
= ( nil @ B ) )
= ( A2
= ( nil @ A ) ) ) ).
% list.map_disc_iff
thf(fact_171_map__append,axiom,
! [A: $tType,B: $tType,F: B > A,Xs: list @ B,Ys: list @ B] :
( ( map @ B @ A @ F @ ( append @ B @ Xs @ Ys ) )
= ( append @ A @ ( map @ B @ A @ F @ Xs ) @ ( map @ B @ A @ F @ Ys ) ) ) ).
% map_append
thf(fact_172_list_Osimps_I9_J,axiom,
! [B: $tType,A: $tType,F: A > B,X21: A,X22: list @ A] :
( ( map @ A @ B @ F @ ( cons @ A @ X21 @ X22 ) )
= ( cons @ B @ ( F @ X21 ) @ ( map @ A @ B @ F @ X22 ) ) ) ).
% list.simps(9)
thf(fact_173_Cons__eq__map__D,axiom,
! [A: $tType,B: $tType,X2: A,Xs: list @ A,F: B > A,Ys: list @ B] :
( ( ( cons @ A @ X2 @ Xs )
= ( map @ B @ A @ F @ Ys ) )
=> ? [Z3: B,Zs3: list @ B] :
( ( Ys
= ( cons @ B @ Z3 @ Zs3 ) )
& ( X2
= ( F @ Z3 ) )
& ( Xs
= ( map @ B @ A @ F @ Zs3 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_174_map__eq__Cons__D,axiom,
! [B: $tType,A: $tType,F: B > A,Xs: list @ B,Y: A,Ys: list @ A] :
( ( ( map @ B @ A @ F @ Xs )
= ( cons @ A @ Y @ Ys ) )
=> ? [Z3: B,Zs3: list @ B] :
( ( Xs
= ( cons @ B @ Z3 @ Zs3 ) )
& ( ( F @ Z3 )
= Y )
& ( ( map @ B @ A @ F @ Zs3 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_175_Cons__eq__map__conv,axiom,
! [A: $tType,B: $tType,X2: A,Xs: list @ A,F: B > A,Ys: list @ B] :
( ( ( cons @ A @ X2 @ Xs )
= ( map @ B @ A @ F @ Ys ) )
= ( ? [Z2: B,Zs2: list @ B] :
( ( Ys
= ( cons @ B @ Z2 @ Zs2 ) )
& ( X2
= ( F @ Z2 ) )
& ( Xs
= ( map @ B @ A @ F @ Zs2 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_176_map__eq__Cons__conv,axiom,
! [A: $tType,B: $tType,F: B > A,Xs: list @ B,Y: A,Ys: list @ A] :
( ( ( map @ B @ A @ F @ Xs )
= ( cons @ A @ Y @ Ys ) )
= ( ? [Z2: B,Zs2: list @ B] :
( ( Xs
= ( cons @ B @ Z2 @ Zs2 ) )
& ( ( F @ Z2 )
= Y )
& ( ( map @ B @ A @ F @ Zs2 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_177_list_Osimps_I8_J,axiom,
! [A: $tType,B: $tType,F: A > B] :
( ( map @ A @ B @ F @ ( nil @ A ) )
= ( nil @ B ) ) ).
% list.simps(8)
thf(fact_178_maps__simps_I2_J,axiom,
! [B: $tType,A: $tType,F: B > ( list @ A )] :
( ( maps @ B @ A @ F @ ( nil @ B ) )
= ( nil @ A ) ) ).
% maps_simps(2)
thf(fact_179_prefixes_Osimps_I2_J,axiom,
! [A: $tType,X2: A,Xs: list @ A] :
( ( prefixes @ A @ ( cons @ A @ X2 @ Xs ) )
= ( cons @ ( list @ A ) @ ( nil @ A ) @ ( map @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X2 ) @ ( prefixes @ A @ Xs ) ) ) ) ).
% prefixes.simps(2)
thf(fact_180_prefixes_Opelims,axiom,
! [A: $tType,X2: list @ A,Y: list @ ( list @ A )] :
( ( ( prefixes @ A @ X2 )
= Y )
=> ( ( accp @ ( list @ A ) @ ( prefixes_rel @ A ) @ X2 )
=> ( ( ( X2
= ( nil @ A ) )
=> ( ( Y
= ( cons @ ( list @ A ) @ ( nil @ A ) @ ( nil @ ( list @ A ) ) ) )
=> ~ ( accp @ ( list @ A ) @ ( prefixes_rel @ A ) @ ( nil @ A ) ) ) )
=> ~ ! [X: A,Xs2: list @ A] :
( ( X2
= ( cons @ A @ X @ Xs2 ) )
=> ( ( Y
= ( cons @ ( list @ A ) @ ( nil @ A ) @ ( map @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X ) @ ( prefixes @ A @ Xs2 ) ) ) )
=> ~ ( accp @ ( list @ A ) @ ( prefixes_rel @ A ) @ ( cons @ A @ X @ Xs2 ) ) ) ) ) ) ) ).
% prefixes.pelims
thf(fact_181_insert__Nil,axiom,
! [A: $tType,X2: A] :
( ( insert @ A @ X2 @ ( nil @ A ) )
= ( cons @ A @ X2 @ ( nil @ A ) ) ) ).
% insert_Nil
thf(fact_182_rotate1_Osimps_I2_J,axiom,
! [A: $tType,X2: A,Xs: list @ A] :
( ( rotate1 @ A @ ( cons @ A @ X2 @ Xs ) )
= ( append @ A @ Xs @ ( cons @ A @ X2 @ ( nil @ A ) ) ) ) ).
% rotate1.simps(2)
thf(fact_183_rotate1__is__Nil__conv,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( rotate1 @ A @ Xs )
= ( nil @ A ) )
= ( Xs
= ( nil @ A ) ) ) ).
% rotate1_is_Nil_conv
thf(fact_184_rotate1_Osimps_I1_J,axiom,
! [A: $tType] :
( ( rotate1 @ A @ ( nil @ A ) )
= ( nil @ A ) ) ).
% rotate1.simps(1)
thf(fact_185_map__eq__map__tailrec,axiom,
! [B: $tType,A: $tType] :
( ( map @ A @ B )
= ( map_tailrec @ A @ B ) ) ).
% map_eq_map_tailrec
thf(fact_186_last__snoc,axiom,
! [A: $tType,Xs: list @ A,X2: A] :
( ( last @ A @ ( append @ A @ Xs @ ( cons @ A @ X2 @ ( nil @ A ) ) ) )
= X2 ) ).
% last_snoc
thf(fact_187_butlast__snoc,axiom,
! [A: $tType,Xs: list @ A,X2: A] :
( ( butlast @ A @ ( append @ A @ Xs @ ( cons @ A @ X2 @ ( nil @ A ) ) ) )
= Xs ) ).
% butlast_snoc
thf(fact_188_last__appendR,axiom,
! [A: $tType,Ys: list @ A,Xs: list @ A] :
( ( Ys
!= ( nil @ A ) )
=> ( ( last @ A @ ( append @ A @ Xs @ Ys ) )
= ( last @ A @ Ys ) ) ) ).
% last_appendR
thf(fact_189_last__appendL,axiom,
! [A: $tType,Ys: list @ A,Xs: list @ A] :
( ( Ys
= ( nil @ A ) )
=> ( ( last @ A @ ( append @ A @ Xs @ Ys ) )
= ( last @ A @ Xs ) ) ) ).
% last_appendL
thf(fact_190_append__butlast__last__id,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( append @ A @ ( butlast @ A @ Xs ) @ ( cons @ A @ ( last @ A @ Xs ) @ ( nil @ A ) ) )
= Xs ) ) ).
% append_butlast_last_id
thf(fact_191_map__butlast,axiom,
! [A: $tType,B: $tType,F: B > A,Xs: list @ B] :
( ( map @ B @ A @ F @ ( butlast @ B @ Xs ) )
= ( butlast @ A @ ( map @ B @ A @ F @ Xs ) ) ) ).
% map_butlast
thf(fact_192_snoc__eq__iff__butlast,axiom,
! [A: $tType,Xs: list @ A,X2: A,Ys: list @ A] :
( ( ( append @ A @ Xs @ ( cons @ A @ X2 @ ( nil @ A ) ) )
= Ys )
= ( ( Ys
!= ( nil @ A ) )
& ( ( butlast @ A @ Ys )
= Xs )
& ( ( last @ A @ Ys )
= X2 ) ) ) ).
% snoc_eq_iff_butlast
thf(fact_193_butlast_Osimps_I1_J,axiom,
! [A: $tType] :
( ( butlast @ A @ ( nil @ A ) )
= ( nil @ A ) ) ).
% butlast.simps(1)
thf(fact_194_last__ConsR,axiom,
! [A: $tType,Xs: list @ A,X2: A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( last @ A @ ( cons @ A @ X2 @ Xs ) )
= ( last @ A @ Xs ) ) ) ).
% last_ConsR
thf(fact_195_last__ConsL,axiom,
! [A: $tType,Xs: list @ A,X2: A] :
( ( Xs
= ( nil @ A ) )
=> ( ( last @ A @ ( cons @ A @ X2 @ Xs ) )
= X2 ) ) ).
% last_ConsL
thf(fact_196_last_Osimps,axiom,
! [A: $tType,Xs: list @ A,X2: A] :
( ( ( Xs
= ( nil @ A ) )
=> ( ( last @ A @ ( cons @ A @ X2 @ Xs ) )
= X2 ) )
& ( ( Xs
!= ( nil @ A ) )
=> ( ( last @ A @ ( cons @ A @ X2 @ Xs ) )
= ( last @ A @ Xs ) ) ) ) ).
% last.simps
thf(fact_197_butlast_Osimps_I2_J,axiom,
! [A: $tType,Xs: list @ A,X2: A] :
( ( ( Xs
= ( nil @ A ) )
=> ( ( butlast @ A @ ( cons @ A @ X2 @ Xs ) )
= ( nil @ A ) ) )
& ( ( Xs
!= ( nil @ A ) )
=> ( ( butlast @ A @ ( cons @ A @ X2 @ Xs ) )
= ( cons @ A @ X2 @ ( butlast @ A @ Xs ) ) ) ) ) ).
% butlast.simps(2)
thf(fact_198_last__map,axiom,
! [B: $tType,A: $tType,Xs: list @ A,F: A > B] :
( ( Xs
!= ( nil @ A ) )
=> ( ( last @ B @ ( map @ A @ B @ F @ Xs ) )
= ( F @ ( last @ A @ Xs ) ) ) ) ).
% last_map
thf(fact_199_longest__common__suffix,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
? [Ss: list @ A,Xs4: list @ A,Ys5: list @ A] :
( ( Xs
= ( append @ A @ Xs4 @ Ss ) )
& ( Ys
= ( append @ A @ Ys5 @ Ss ) )
& ( ( Xs4
= ( nil @ A ) )
| ( Ys5
= ( nil @ A ) )
| ( ( last @ A @ Xs4 )
!= ( last @ A @ Ys5 ) ) ) ) ).
% longest_common_suffix
thf(fact_200_butlast__append,axiom,
! [A: $tType,Ys: list @ A,Xs: list @ A] :
( ( ( Ys
= ( nil @ A ) )
=> ( ( butlast @ A @ ( append @ A @ Xs @ Ys ) )
= ( butlast @ A @ Xs ) ) )
& ( ( Ys
!= ( nil @ A ) )
=> ( ( butlast @ A @ ( append @ A @ Xs @ Ys ) )
= ( append @ A @ Xs @ ( butlast @ A @ Ys ) ) ) ) ) ).
% butlast_append
thf(fact_201_last__append,axiom,
! [A: $tType,Ys: list @ A,Xs: list @ A] :
( ( ( Ys
= ( nil @ A ) )
=> ( ( last @ A @ ( append @ A @ Xs @ Ys ) )
= ( last @ A @ Xs ) ) )
& ( ( Ys
!= ( nil @ A ) )
=> ( ( last @ A @ ( append @ A @ Xs @ Ys ) )
= ( last @ A @ Ys ) ) ) ) ).
% last_append
thf(fact_202_list__ex1__simps_I1_J,axiom,
! [A: $tType,P: A > $o] :
~ ( list_ex1 @ A @ P @ ( nil @ A ) ) ).
% list_ex1_simps(1)
thf(fact_203_strict__suffix__def,axiom,
! [A: $tType] :
( ( strict_suffix @ A )
= ( ^ [Xs3: list @ A,Ys3: list @ A] :
? [Us2: list @ A] :
( ( Ys3
= ( append @ A @ Us2 @ Xs3 ) )
& ( Us2
!= ( nil @ A ) ) ) ) ) ).
% strict_suffix_def
thf(fact_204_strict__suffix__trans,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( strict_suffix @ A @ Xs @ Ys )
=> ( ( strict_suffix @ A @ Ys @ Zs )
=> ( strict_suffix @ A @ Xs @ Zs ) ) ) ).
% strict_suffix_trans
thf(fact_205_map__tailrec__rev_Oelims,axiom,
! [A: $tType,B: $tType,X2: A > B,Xa: list @ A,Xb: list @ B,Y: list @ B] :
( ( ( map_tailrec_rev @ A @ B @ X2 @ Xa @ Xb )
= Y )
=> ( ( ( Xa
= ( nil @ A ) )
=> ( Y != Xb ) )
=> ~ ! [A3: A,As: list @ A] :
( ( Xa
= ( cons @ A @ A3 @ As ) )
=> ( Y
!= ( map_tailrec_rev @ A @ B @ X2 @ As @ ( cons @ B @ ( X2 @ A3 ) @ Xb ) ) ) ) ) ) ).
% map_tailrec_rev.elims
thf(fact_206_SuccD,axiom,
! [A: $tType,K: A,Kl: set @ ( list @ A ),Kl2: list @ A] :
( ( member @ A @ K @ ( bNF_Greatest_Succ @ A @ Kl @ Kl2 ) )
=> ( member @ ( list @ A ) @ ( append @ A @ Kl2 @ ( cons @ A @ K @ ( nil @ A ) ) ) @ Kl ) ) ).
% SuccD
thf(fact_207_map__tailrec__rev_Osimps_I2_J,axiom,
! [A: $tType,B: $tType,F: A > B,A2: A,As2: list @ A,Bs2: list @ B] :
( ( map_tailrec_rev @ A @ B @ F @ ( cons @ A @ A2 @ As2 ) @ Bs2 )
= ( map_tailrec_rev @ A @ B @ F @ As2 @ ( cons @ B @ ( F @ A2 ) @ Bs2 ) ) ) ).
% map_tailrec_rev.simps(2)
thf(fact_208_map__tailrec__rev_Osimps_I1_J,axiom,
! [A: $tType,B: $tType,F: A > B,Bs2: list @ B] :
( ( map_tailrec_rev @ A @ B @ F @ ( nil @ A ) @ Bs2 )
= Bs2 ) ).
% map_tailrec_rev.simps(1)
thf(fact_209_SuccI,axiom,
! [A: $tType,Kl2: list @ A,K: A,Kl: set @ ( list @ A )] :
( ( member @ ( list @ A ) @ ( append @ A @ Kl2 @ ( cons @ A @ K @ ( nil @ A ) ) ) @ Kl )
=> ( member @ A @ K @ ( bNF_Greatest_Succ @ A @ Kl @ Kl2 ) ) ) ).
% SuccI
thf(fact_210_shift__def,axiom,
! [B: $tType,A: $tType] :
( ( bNF_Greatest_shift @ A @ B )
= ( ^ [Lab: ( list @ A ) > B,K2: A,Kl3: list @ A] : ( Lab @ ( cons @ A @ K2 @ Kl3 ) ) ) ) ).
% shift_def
thf(fact_211_empty__Shift,axiom,
! [A: $tType,Kl: set @ ( list @ A ),K: A] :
( ( member @ ( list @ A ) @ ( nil @ A ) @ Kl )
=> ( ( member @ A @ K @ ( bNF_Greatest_Succ @ A @ Kl @ ( nil @ A ) ) )
=> ( member @ ( list @ A ) @ ( nil @ A ) @ ( bNF_Greatest_Shift @ A @ Kl @ K ) ) ) ) ).
% empty_Shift
thf(fact_212_ShiftD,axiom,
! [A: $tType,Kl2: list @ A,Kl: set @ ( list @ A ),K: A] :
( ( member @ ( list @ A ) @ Kl2 @ ( bNF_Greatest_Shift @ A @ Kl @ K ) )
=> ( member @ ( list @ A ) @ ( cons @ A @ K @ Kl2 ) @ Kl ) ) ).
% ShiftD
thf(fact_213_Succ__Shift,axiom,
! [A: $tType,Kl: set @ ( list @ A ),K: A,Kl2: list @ A] :
( ( bNF_Greatest_Succ @ A @ ( bNF_Greatest_Shift @ A @ Kl @ K ) @ Kl2 )
= ( bNF_Greatest_Succ @ A @ Kl @ ( cons @ A @ K @ Kl2 ) ) ) ).
% Succ_Shift
thf(fact_214_map__tailrec__def,axiom,
! [B: $tType,A: $tType] :
( ( map_tailrec @ A @ B )
= ( ^ [F3: A > B,As3: list @ A] : ( rev @ B @ ( map_tailrec_rev @ A @ B @ F3 @ As3 @ ( nil @ B ) ) ) ) ) ).
% map_tailrec_def
thf(fact_215_ord_OgreaterThanAtMost__def,axiom,
! [A: $tType] :
( ( set_gr323396891AtMost @ A )
= ( ^ [Less_eq: A > A > $o,Less2: A > A > $o,L2: A,U2: A] : ( inf_inf @ ( set @ A ) @ ( set_greaterThan @ A @ Less2 @ L2 ) @ ( set_atMost @ A @ Less_eq @ U2 ) ) ) ) ).
% ord.greaterThanAtMost_def
thf(fact_216_rev__rev__ident,axiom,
! [A: $tType,Xs: list @ A] :
( ( rev @ A @ ( rev @ A @ Xs ) )
= Xs ) ).
% rev_rev_ident
thf(fact_217_rev__is__rev__conv,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( rev @ A @ Xs )
= ( rev @ A @ Ys ) )
= ( Xs = Ys ) ) ).
% rev_is_rev_conv
thf(fact_218_Nil__is__rev__conv,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( nil @ A )
= ( rev @ A @ Xs ) )
= ( Xs
= ( nil @ A ) ) ) ).
% Nil_is_rev_conv
thf(fact_219_rev__is__Nil__conv,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( rev @ A @ Xs )
= ( nil @ A ) )
= ( Xs
= ( nil @ A ) ) ) ).
% rev_is_Nil_conv
thf(fact_220_rev__append,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( rev @ A @ ( append @ A @ Xs @ Ys ) )
= ( append @ A @ ( rev @ A @ Ys ) @ ( rev @ A @ Xs ) ) ) ).
% rev_append
thf(fact_221_rev__singleton__conv,axiom,
! [A: $tType,Xs: list @ A,X2: A] :
( ( ( rev @ A @ Xs )
= ( cons @ A @ X2 @ ( nil @ A ) ) )
= ( Xs
= ( cons @ A @ X2 @ ( nil @ A ) ) ) ) ).
% rev_singleton_conv
thf(fact_222_singleton__rev__conv,axiom,
! [A: $tType,X2: A,Xs: list @ A] :
( ( ( cons @ A @ X2 @ ( nil @ A ) )
= ( rev @ A @ Xs ) )
= ( Xs
= ( cons @ A @ X2 @ ( nil @ A ) ) ) ) ).
% singleton_rev_conv
thf(fact_223_rev__eq__Cons__iff,axiom,
! [A: $tType,Xs: list @ A,Y: A,Ys: list @ A] :
( ( ( rev @ A @ Xs )
= ( cons @ A @ Y @ Ys ) )
= ( Xs
= ( append @ A @ ( rev @ A @ Ys ) @ ( cons @ A @ Y @ ( nil @ A ) ) ) ) ) ).
% rev_eq_Cons_iff
thf(fact_224_rev__map,axiom,
! [A: $tType,B: $tType,F: B > A,Xs: list @ B] :
( ( rev @ A @ ( map @ B @ A @ F @ Xs ) )
= ( map @ B @ A @ F @ ( rev @ B @ Xs ) ) ) ).
% rev_map
thf(fact_225_ord_OatMost__iff,axiom,
! [A: $tType,I: A,Less_eq2: A > A > $o,K: A] :
( ( member @ A @ I @ ( set_atMost @ A @ Less_eq2 @ K ) )
= ( Less_eq2 @ I @ K ) ) ).
% ord.atMost_iff
thf(fact_226_ord_OgreaterThanAtMost__iff,axiom,
! [A: $tType,I: A,Less_eq2: A > A > $o,Less: A > A > $o,L: A,U: A] :
( ( member @ A @ I @ ( set_gr323396891AtMost @ A @ Less_eq2 @ Less @ L @ U ) )
= ( ( Less @ L @ I )
& ( Less_eq2 @ I @ U ) ) ) ).
% ord.greaterThanAtMost_iff
thf(fact_227_rev_Osimps_I1_J,axiom,
! [A: $tType] :
( ( rev @ A @ ( nil @ A ) )
= ( nil @ A ) ) ).
% rev.simps(1)
thf(fact_228_rev__swap,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( rev @ A @ Xs )
= Ys )
= ( Xs
= ( rev @ A @ Ys ) ) ) ).
% rev_swap
thf(fact_229_rev_Osimps_I2_J,axiom,
! [A: $tType,X2: A,Xs: list @ A] :
( ( rev @ A @ ( cons @ A @ X2 @ Xs ) )
= ( append @ A @ ( rev @ A @ Xs ) @ ( cons @ A @ X2 @ ( nil @ A ) ) ) ) ).
% rev.simps(2)
thf(fact_230_map__tailrec__rev,axiom,
! [A: $tType,B: $tType] :
( ( map_tailrec_rev @ B @ A )
= ( ^ [F3: B > A,As3: list @ B] : ( append @ A @ ( rev @ A @ ( map @ B @ A @ F3 @ As3 ) ) ) ) ) ).
% map_tailrec_rev
thf(fact_231_list__of__aux__code_I1_J,axiom,
! [A: $tType,Xs: list @ A] :
( ( coindu1384447384of_aux @ A @ Xs @ ( coinductive_LNil @ A ) )
= ( rev @ A @ Xs ) ) ).
% list_of_aux_code(1)
thf(fact_232_local_OgreaterThanAtMost__def,axiom,
! [L: a,U: a] :
( ( set_gr323396891AtMost @ a @ less_eq @ less @ L @ U )
= ( inf_inf @ ( set @ a ) @ ( set_greaterThan @ a @ less @ L ) @ ( set_atMost @ a @ less_eq @ U ) ) ) ).
% local.greaterThanAtMost_def
thf(fact_233_local_Oord__eq__le__trans,axiom,
! [A2: a,B2: a,C: a] :
( ( A2 = B2 )
=> ( ( less_eq @ B2 @ C )
=> ( less_eq @ A2 @ C ) ) ) ).
% local.ord_eq_le_trans
thf(fact_234_local_Oord__le__eq__trans,axiom,
! [A2: a,B2: a,C: a] :
( ( less_eq @ A2 @ B2 )
=> ( ( B2 = C )
=> ( less_eq @ A2 @ C ) ) ) ).
% local.ord_le_eq_trans
thf(fact_235_local_Omax__def,axiom,
! [A2: a,B2: a] :
( ( ( less_eq @ A2 @ B2 )
=> ( ( max @ a @ less_eq @ A2 @ B2 )
= B2 ) )
& ( ~ ( less_eq @ A2 @ B2 )
=> ( ( max @ a @ less_eq @ A2 @ B2 )
= A2 ) ) ) ).
% local.max_def
thf(fact_236_local_Omin__def,axiom,
! [A2: a,B2: a] :
( ( ( less_eq @ A2 @ B2 )
=> ( ( min @ a @ less_eq @ A2 @ B2 )
= A2 ) )
& ( ~ ( less_eq @ A2 @ B2 )
=> ( ( min @ a @ less_eq @ A2 @ B2 )
= B2 ) ) ) ).
% local.min_def
thf(fact_237_local_OatMost__iff,axiom,
! [I: a,K: a] :
( ( member @ a @ I @ ( set_atMost @ a @ less_eq @ K ) )
= ( less_eq @ I @ K ) ) ).
% local.atMost_iff
thf(fact_238_local_OgreaterThanAtMost__iff,axiom,
! [I: a,L: a,U: a] :
( ( member @ a @ I @ ( set_gr323396891AtMost @ a @ less_eq @ less @ L @ U ) )
= ( ( less @ L @ I )
& ( less_eq @ I @ U ) ) ) ).
% local.greaterThanAtMost_iff
thf(fact_239_local_OatLeastLessThan__def,axiom,
! [L: a,U: a] :
( ( set_atLeastLessThan @ a @ less_eq @ less @ L @ U )
= ( inf_inf @ ( set @ a ) @ ( set_atLeast @ a @ less_eq @ L ) @ ( set_lessThan @ a @ less @ U ) ) ) ).
% local.atLeastLessThan_def
thf(fact_240_local_OatLeastAtMost__def,axiom,
! [L: a,U: a] :
( ( set_atLeastAtMost @ a @ less_eq @ L @ U )
= ( inf_inf @ ( set @ a ) @ ( set_atLeast @ a @ less_eq @ L ) @ ( set_atMost @ a @ less_eq @ U ) ) ) ).
% local.atLeastAtMost_def
thf(fact_241_local_OatLeast__iff,axiom,
! [I: a,K: a] :
( ( member @ a @ I @ ( set_atLeast @ a @ less_eq @ K ) )
= ( less_eq @ K @ I ) ) ).
% local.atLeast_iff
thf(fact_242_local_OatLeastAtMost__iff,axiom,
! [I: a,L: a,U: a] :
( ( member @ a @ I @ ( set_atLeastAtMost @ a @ less_eq @ L @ U ) )
= ( ( less_eq @ L @ I )
& ( less_eq @ I @ U ) ) ) ).
% local.atLeastAtMost_iff
thf(fact_243_local_OatLeastLessThan__iff,axiom,
! [I: a,L: a,U: a] :
( ( member @ a @ I @ ( set_atLeastLessThan @ a @ less_eq @ less @ L @ U ) )
= ( ( less_eq @ L @ I )
& ( less @ I @ U ) ) ) ).
% local.atLeastLessThan_iff
thf(fact_244_ord_OatLeast__iff,axiom,
! [A: $tType,I: A,Less_eq2: A > A > $o,K: A] :
( ( member @ A @ I @ ( set_atLeast @ A @ Less_eq2 @ K ) )
= ( Less_eq2 @ K @ I ) ) ).
% ord.atLeast_iff
thf(fact_245_ord_OatLeastAtMost__iff,axiom,
! [A: $tType,I: A,Less_eq2: A > A > $o,L: A,U: A] :
( ( member @ A @ I @ ( set_atLeastAtMost @ A @ Less_eq2 @ L @ U ) )
= ( ( Less_eq2 @ L @ I )
& ( Less_eq2 @ I @ U ) ) ) ).
% ord.atLeastAtMost_iff
thf(fact_246_ord_OatLeastLessThan__iff,axiom,
! [A: $tType,I: A,Less_eq2: A > A > $o,Less: A > A > $o,L: A,U: A] :
( ( member @ A @ I @ ( set_atLeastLessThan @ A @ Less_eq2 @ Less @ L @ U ) )
= ( ( Less_eq2 @ L @ I )
& ( Less @ I @ U ) ) ) ).
% ord.atLeastLessThan_iff
thf(fact_247_ord_OatLeastLessThan__def,axiom,
! [A: $tType] :
( ( set_atLeastLessThan @ A )
= ( ^ [Less_eq: A > A > $o,Less2: A > A > $o,L2: A,U2: A] : ( inf_inf @ ( set @ A ) @ ( set_atLeast @ A @ Less_eq @ L2 ) @ ( set_lessThan @ A @ Less2 @ U2 ) ) ) ) ).
% ord.atLeastLessThan_def
thf(fact_248_ord_OatLeastAtMost__def,axiom,
! [A: $tType] :
( ( set_atLeastAtMost @ A )
= ( ^ [Less_eq: A > A > $o,L2: A,U2: A] : ( inf_inf @ ( set @ A ) @ ( set_atLeast @ A @ Less_eq @ L2 ) @ ( set_atMost @ A @ Less_eq @ U2 ) ) ) ) ).
% ord.atLeastAtMost_def
thf(fact_249_local_Olsorted_OLNil,axiom,
coinductive_lsorted @ a @ less_eq @ ( coinductive_LNil @ a ) ).
% local.lsorted.LNil
thf(fact_250_hd__rev,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( hd @ A @ ( rev @ A @ Xs ) )
= ( last @ A @ Xs ) ) ) ).
% hd_rev
thf(fact_251_hd__append2,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( hd @ A @ ( append @ A @ Xs @ Ys ) )
= ( hd @ A @ Xs ) ) ) ).
% hd_append2
thf(fact_252_local_Olsorted__code_I1_J,axiom,
coinductive_lsorted @ a @ less_eq @ ( coinductive_LNil @ a ) ).
% local.lsorted_code(1)
%----Type constructors (9)
thf(tcon_fun___Lattices_Osemilattice__inf,axiom,
! [A7: $tType,A8: $tType] :
( ( semilattice_inf @ A8 @ ( type2 @ A8 ) )
=> ( semilattice_inf @ ( 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_Oord,axiom,
! [A7: $tType,A8: $tType] :
( ( ord @ A8 @ ( type2 @ A8 ) )
=> ( ord @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
thf(tcon_Set_Oset___Lattices_Osemilattice__inf_1,axiom,
! [A7: $tType] : ( semilattice_inf @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_Set_Oset___Lattices_Olattice_2,axiom,
! [A7: $tType] : ( lattice @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_3,axiom,
! [A7: $tType] : ( ord @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_HOL_Obool___Lattices_Osemilattice__inf_4,axiom,
semilattice_inf @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Lattices_Olattice_5,axiom,
lattice @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oord_6,axiom,
ord @ $o @ ( type2 @ $o ) ).
%----Conjectures (1)
thf(conj_0,conjecture,
( ( coinductive_lnull @ a @ ( hammin1328233080lmerge @ a @ less @ xs @ ys ) )
= ( ( coinductive_lnull @ a @ xs )
| ( coinductive_lnull @ a @ ys ) ) ) ).
%------------------------------------------------------------------------------