TPTP Problem File: DAT193^1.p
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%------------------------------------------------------------------------------
% File : DAT193^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Data Structures
% Problem : Lazy list mirror 79
% Version : [Bla16] axioms : Especial.
% English :
% Refs : [Loc10] Lochbihler (2010), Coinductive
% : [RB15] Reynolds & Blanchette (2015), A Decision Procedure for
% : [Bla16] Blanchette (2016), Email to Geoff Sutcliffe
% Source : [Bla16]
% Names : lmirror__79.p [Bla16]
% Status : Theorem
% Rating : 1.00 v7.2.0, 0.75 v7.1.0
% Syntax : Number of formulae : 313 ( 130 unt; 48 typ; 0 def)
% Number of atoms : 707 ( 328 equ; 0 cnn)
% Maximal formula atoms : 8 ( 2 avg)
% Number of connectives : 3764 ( 135 ~; 48 |; 77 &;3228 @)
% ( 0 <=>; 276 =>; 0 <=; 0 <~>)
% Maximal formula depth : 22 ( 7 avg)
% Number of types : 3 ( 2 usr)
% Number of type conns : 229 ( 229 >; 0 *; 0 +; 0 <<)
% Number of symbols : 49 ( 46 usr; 4 con; 0-6 aty)
% Number of variables : 987 ( 35 ^; 858 !; 51 ?; 987 :)
% ( 43 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 14:41:16.997
%------------------------------------------------------------------------------
%----Could-be-implicit typings (4)
thf(ty_t_Coinductive__List_Ollist,type,
coinductive_llist: $tType > $tType ).
thf(ty_t_Extended__Nat_Oenat,type,
extended_enat: $tType ).
thf(ty_t_itself,type,
itself: $tType > $tType ).
thf(ty_tf_a,type,
a: $tType ).
%----Explicit typings (44)
thf(sy_cl_Groups_Ozero,type,
zero:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Num_Oneg__numeral,type,
neg_numeral:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ogroup__add,type,
group_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Omonoid__add,type,
monoid_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Osemigroup__add,type,
semigroup_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Ocomm__semiring__1,type,
comm_semiring_1:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Fields_Olinordered__field,type,
linordered_field:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Oab__semigroup__add,type,
ab_semigroup_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ocancel__semigroup__add,type,
cancel_semigroup_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ocancel__comm__monoid__add,type,
cancel1352612707id_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Olinordered__ab__group__add,type,
linord219039673up_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Oordered__ab__semigroup__add,type,
ordere779506340up_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_c_Coinductive__List_Ofinite__lprefix,type,
coindu328551480prefix:
!>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List_Olappend,type,
coinductive_lappend:
!>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
thf(sy_c_Coinductive__List_Olconcat,type,
coinductive_lconcat:
!>[A: $tType] : ( ( coinductive_llist @ ( coinductive_llist @ A ) ) > ( coinductive_llist @ A ) ) ).
thf(sy_c_Coinductive__List_Olfinite,type,
coinductive_lfinite:
!>[A: $tType] : ( ( coinductive_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List_Ollast,type,
coinductive_llast:
!>[A: $tType] : ( ( coinductive_llist @ A ) > A ) ).
thf(sy_c_Coinductive__List_Ollcp,type,
coinductive_llcp:
!>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > extended_enat ) ).
thf(sy_c_Coinductive__List_Ollength,type,
coinductive_llength:
!>[A: $tType] : ( ( coinductive_llist @ A ) > extended_enat ) ).
thf(sy_c_Coinductive__List_Ollexord,type,
coinductive_llexord:
!>[A: $tType] : ( ( A > A > $o ) > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List_Ollist_OLCons,type,
coinductive_LCons:
!>[A: $tType] : ( A > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
thf(sy_c_Coinductive__List_Ollist_OLNil,type,
coinductive_LNil:
!>[A: $tType] : ( coinductive_llist @ A ) ).
thf(sy_c_Coinductive__List_Ollist_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_Oltl,type,
coinductive_ltl:
!>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
thf(sy_c_Coinductive__List_Olmember,type,
coinductive_lmember:
!>[A: $tType] : ( A > ( coinductive_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List_Olprefix,type,
coinductive_lprefix:
!>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List_Olstrict__prefix,type,
coindu1478340336prefix:
!>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List_OltakeWhile,type,
coindu501562517eWhile:
!>[A: $tType] : ( ( A > $o ) > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
thf(sy_c_Coinductive__List_Omonoid__add__class_Ollistsum,type,
coindu780009021istsum:
!>[A: $tType] : ( ( coinductive_llist @ A ) > A ) ).
thf(sy_c_Coinductive__List_Oord_Olsorted,type,
coinductive_lsorted:
!>[A: $tType] : ( ( A > A > $o ) > ( coinductive_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List_Ounfold__llist,type,
coindu1441602521_llist:
!>[A: $tType,B: $tType] : ( ( A > $o ) > ( A > B ) > ( A > A ) > A > ( coinductive_llist @ B ) ) ).
thf(sy_c_Coinductive__Nat_Oco_Oenat_Oepred,type,
coinductive_epred: extended_enat > extended_enat ).
thf(sy_c_Coinductive__Nat_Oenat__unfold,type,
coindu1491768222unfold:
!>[A: $tType] : ( ( A > $o ) > ( A > A ) > A > extended_enat ) ).
thf(sy_c_Groups_Oplus__class_Oplus,type,
plus_plus:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Groups_Ozero__class_Ozero,type,
zero_zero:
!>[A: $tType] : A ).
thf(sy_c_HOL_Oundefined,type,
undefined:
!>[A: $tType] : A ).
thf(sy_c_If,type,
if:
!>[A: $tType] : ( $o > A > A > A ) ).
thf(sy_c_LMirror__Mirabelle__wyovfcktfy_Olmirror,type,
lMirro427583474mirror:
!>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
thf(sy_c_LMirror__Mirabelle__wyovfcktfy_Olmirror__aux,type,
lMirro999291890or_aux:
!>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
thf(sy_c_Pure_Otype,type,
type:
!>[A: $tType] : ( itself @ A ) ).
thf(sy_v_acca,type,
acca: coinductive_llist @ a ).
thf(sy_v_xsa,type,
xsa: coinductive_llist @ a ).
%----Relevant facts (253)
thf(fact_0_lmirror__aux__simps_I2_J,axiom,
! [A: $tType,Acc: coinductive_llist @ A,Xa: A,X: coinductive_llist @ A] :
( ( lMirro999291890or_aux @ A @ Acc @ ( coinductive_LCons @ A @ Xa @ X ) )
= ( coinductive_LCons @ A @ Xa @ ( lMirro999291890or_aux @ A @ ( coinductive_LCons @ A @ Xa @ Acc ) @ X ) ) ) ).
% lmirror_aux_simps(2)
thf(fact_1_lmirror__aux_Odisc__iff_I2_J,axiom,
! [A: $tType,Acc: coinductive_llist @ A,Xs: coinductive_llist @ A] :
( ( ~ ( coinductive_lnull @ A @ ( lMirro999291890or_aux @ A @ Acc @ Xs ) ) )
= ( ~ ( coinductive_lnull @ A @ Xs )
| ~ ( coinductive_lnull @ A @ Acc ) ) ) ).
% lmirror_aux.disc_iff(2)
thf(fact_2_lnull__lmirror__aux,axiom,
! [A: $tType,Acc: coinductive_llist @ A,Xs: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ ( lMirro999291890or_aux @ A @ Acc @ Xs ) )
= ( ( coinductive_lnull @ A @ Xs )
& ( coinductive_lnull @ A @ Acc ) ) ) ).
% lnull_lmirror_aux
thf(fact_3_lmirror__aux_Odisc_I2_J,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Acc: coinductive_llist @ A] :
( ( ~ ( coinductive_lnull @ A @ Xs )
| ~ ( coinductive_lnull @ A @ Acc ) )
=> ~ ( coinductive_lnull @ A @ ( lMirro999291890or_aux @ A @ Acc @ Xs ) ) ) ).
% lmirror_aux.disc(2)
thf(fact_4_lmirror__aux_Odisc_I1_J,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Acc: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ Xs )
=> ( ( coinductive_lnull @ A @ Acc )
=> ( coinductive_lnull @ A @ ( lMirro999291890or_aux @ A @ Acc @ Xs ) ) ) ) ).
% lmirror_aux.disc(1)
thf(fact_5_lhd__lmirror__aux,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Acc: coinductive_llist @ A] :
( ( ( coinductive_lnull @ A @ Xs )
=> ( ( coinductive_lhd @ A @ ( lMirro999291890or_aux @ A @ Acc @ Xs ) )
= ( coinductive_lhd @ A @ Acc ) ) )
& ( ~ ( coinductive_lnull @ A @ Xs )
=> ( ( coinductive_lhd @ A @ ( lMirro999291890or_aux @ A @ Acc @ Xs ) )
= ( coinductive_lhd @ A @ Xs ) ) ) ) ).
% lhd_lmirror_aux
thf(fact_6_ltl__lmirror__aux,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Acc: coinductive_llist @ A] :
( ( ( coinductive_lnull @ A @ Xs )
=> ( ( coinductive_ltl @ A @ ( lMirro999291890or_aux @ A @ Acc @ Xs ) )
= ( coinductive_ltl @ A @ Acc ) ) )
& ( ~ ( coinductive_lnull @ A @ Xs )
=> ( ( coinductive_ltl @ A @ ( lMirro999291890or_aux @ A @ Acc @ Xs ) )
= ( lMirro999291890or_aux @ A @ ( coinductive_LCons @ A @ ( coinductive_lhd @ A @ Xs ) @ Acc ) @ ( coinductive_ltl @ A @ Xs ) ) ) ) ) ).
% ltl_lmirror_aux
thf(fact_7_lmirror__aux_Oexhaust,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Acc: coinductive_llist @ A] :
( ( ( coinductive_lnull @ A @ Xs )
=> ~ ( coinductive_lnull @ A @ Acc ) )
=> ( ~ ( coinductive_lnull @ A @ Xs )
| ~ ( coinductive_lnull @ A @ Acc ) ) ) ).
% lmirror_aux.exhaust
thf(fact_8_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_9_llist_Oexpand,axiom,
! [A: $tType,Llist: coinductive_llist @ A,Llist2: coinductive_llist @ A] :
( ( ( coinductive_lnull @ A @ Llist )
= ( coinductive_lnull @ A @ Llist2 ) )
=> ( ( ~ ( coinductive_lnull @ A @ Llist )
=> ( ~ ( coinductive_lnull @ A @ Llist2 )
=> ( ( ( coinductive_lhd @ A @ Llist )
= ( coinductive_lhd @ A @ Llist2 ) )
& ( ( coinductive_ltl @ A @ Llist )
= ( coinductive_ltl @ A @ Llist2 ) ) ) ) )
=> ( Llist = Llist2 ) ) ) ).
% llist.expand
thf(fact_10_llist_Ocoinduct,axiom,
! [A: $tType,R: ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o,Llist: coinductive_llist @ A,Llist2: coinductive_llist @ A] :
( ( R @ Llist @ Llist2 )
=> ( ! [Llist3: coinductive_llist @ A,Llist4: coinductive_llist @ A] :
( ( R @ Llist3 @ Llist4 )
=> ( ( ( coinductive_lnull @ A @ Llist3 )
= ( coinductive_lnull @ A @ Llist4 ) )
& ( ~ ( coinductive_lnull @ A @ Llist3 )
=> ( ~ ( coinductive_lnull @ A @ Llist4 )
=> ( ( ( coinductive_lhd @ A @ Llist3 )
= ( coinductive_lhd @ A @ Llist4 ) )
& ( R @ ( coinductive_ltl @ A @ Llist3 ) @ ( coinductive_ltl @ A @ Llist4 ) ) ) ) ) ) )
=> ( Llist = Llist2 ) ) ) ).
% llist.coinduct
thf(fact_11_llist_Ocoinduct__strong,axiom,
! [A: $tType,R: ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o,Llist: coinductive_llist @ A,Llist2: coinductive_llist @ A] :
( ( R @ Llist @ Llist2 )
=> ( ! [Llist3: coinductive_llist @ A,Llist4: coinductive_llist @ A] :
( ( R @ Llist3 @ Llist4 )
=> ( ( ( coinductive_lnull @ A @ Llist3 )
= ( coinductive_lnull @ A @ Llist4 ) )
& ( ~ ( coinductive_lnull @ A @ Llist3 )
=> ( ~ ( coinductive_lnull @ A @ Llist4 )
=> ( ( ( coinductive_lhd @ A @ Llist3 )
= ( coinductive_lhd @ A @ Llist4 ) )
& ( ( R @ ( coinductive_ltl @ A @ Llist3 ) @ ( coinductive_ltl @ A @ Llist4 ) )
| ( ( coinductive_ltl @ A @ Llist3 )
= ( coinductive_ltl @ A @ Llist4 ) ) ) ) ) ) ) )
=> ( Llist = Llist2 ) ) ) ).
% llist.coinduct_strong
thf(fact_12_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_13_add__left__cancel,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A @ ( type @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( ( plus_plus @ A @ A2 @ B2 )
= ( plus_plus @ A @ A2 @ C ) )
= ( B2 = C ) ) ) ).
% add_left_cancel
thf(fact_14_add__right__cancel,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A @ ( type @ A ) )
=> ! [B2: A,A2: A,C: A] :
( ( ( plus_plus @ A @ B2 @ A2 )
= ( plus_plus @ A @ C @ A2 ) )
= ( B2 = C ) ) ) ).
% add_right_cancel
thf(fact_15_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_16_lhd__LCons,axiom,
! [A: $tType,X21: A,X22: coinductive_llist @ A] :
( ( coinductive_lhd @ A @ ( coinductive_LCons @ A @ X21 @ X22 ) )
= X21 ) ).
% lhd_LCons
thf(fact_17_lnull__ltlI,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ Xs )
=> ( coinductive_lnull @ A @ ( coinductive_ltl @ A @ Xs ) ) ) ).
% lnull_ltlI
thf(fact_18_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_19_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_20_add__right__imp__eq,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A @ ( type @ A ) )
=> ! [B2: A,A2: A,C: A] :
( ( ( plus_plus @ A @ B2 @ A2 )
= ( plus_plus @ A @ C @ A2 ) )
=> ( B2 = C ) ) ) ).
% add_right_imp_eq
thf(fact_21_add__left__imp__eq,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A @ ( type @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( ( plus_plus @ A @ A2 @ B2 )
= ( plus_plus @ A @ A2 @ C ) )
=> ( B2 = C ) ) ) ).
% add_left_imp_eq
thf(fact_22_add_Oleft__commute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A @ ( type @ A ) )
=> ! [B2: A,A2: A,C: A] :
( ( plus_plus @ A @ B2 @ ( plus_plus @ A @ A2 @ C ) )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C ) ) ) ) ).
% add.left_commute
thf(fact_23_add_Ocommute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A @ ( type @ A ) )
=> ( ( plus_plus @ A )
= ( ^ [A3: A,B3: A] : ( plus_plus @ A @ B3 @ A3 ) ) ) ) ).
% add.commute
thf(fact_24_add_Oright__cancel,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type @ A ) )
=> ! [B2: A,A2: A,C: A] :
( ( ( plus_plus @ A @ B2 @ A2 )
= ( plus_plus @ A @ C @ A2 ) )
= ( B2 = C ) ) ) ).
% add.right_cancel
thf(fact_25_add_Oleft__cancel,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( ( plus_plus @ A @ A2 @ B2 )
= ( plus_plus @ A @ A2 @ C ) )
= ( B2 = C ) ) ) ).
% add.left_cancel
thf(fact_26_add_Oassoc,axiom,
! [A: $tType] :
( ( semigroup_add @ A @ ( type @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C ) ) ) ) ).
% add.assoc
thf(fact_27_add__mono__thms__linordered__semiring_I4_J,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A @ ( type @ A ) )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus @ A @ I @ K )
= ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_28_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A @ ( type @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C ) ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_29_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_30_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_31_not__lnull__conv,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( ~ ( coinductive_lnull @ A @ Xs ) )
= ( ? [X2: A,Xs2: coinductive_llist @ A] :
( Xs
= ( coinductive_LCons @ A @ X2 @ Xs2 ) ) ) ) ).
% not_lnull_conv
thf(fact_32_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_33_llength__def,axiom,
! [A: $tType] :
( ( coinductive_llength @ A )
= ( coindu1491768222unfold @ ( coinductive_llist @ A ) @ ( coinductive_lnull @ A ) @ ( coinductive_ltl @ A ) ) ) ).
% llength_def
thf(fact_34_unfold__llist__id,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( coindu1441602521_llist @ ( coinductive_llist @ A ) @ A @ ( coinductive_lnull @ A ) @ ( coinductive_lhd @ A ) @ ( coinductive_ltl @ A ) @ Xs )
= Xs ) ).
% unfold_llist_id
thf(fact_35_llist_Ocase__eq__if,axiom,
! [A: $tType,B: $tType] :
( ( coindu1381640503_llist @ B @ A )
= ( ^ [F1: B,F2: A > ( coinductive_llist @ A ) > B,Llist5: coinductive_llist @ A] : ( if @ B @ ( coinductive_lnull @ A @ Llist5 ) @ F1 @ ( F2 @ ( coinductive_lhd @ A @ Llist5 ) @ ( coinductive_ltl @ A @ Llist5 ) ) ) ) ) ).
% llist.case_eq_if
thf(fact_36_ltakeWhile_Octr_I2_J,axiom,
! [A: $tType,Xs: coinductive_llist @ A,P: A > $o] :
( ~ ( coinductive_lnull @ A @ Xs )
=> ( ( P @ ( coinductive_lhd @ A @ Xs ) )
=> ( ( coindu501562517eWhile @ A @ P @ Xs )
= ( coinductive_LCons @ A @ ( coinductive_lhd @ A @ Xs ) @ ( coindu501562517eWhile @ A @ P @ ( coinductive_ltl @ A @ Xs ) ) ) ) ) ) ).
% ltakeWhile.ctr(2)
thf(fact_37_ord_Olsorted__coinduct_H,axiom,
! [A: $tType,X3: ( coinductive_llist @ A ) > $o,Xs: coinductive_llist @ A,Less_eq: A > A > $o] :
( ( X3 @ Xs )
=> ( ! [Xs3: coinductive_llist @ A] :
( ( X3 @ Xs3 )
=> ( ~ ( coinductive_lnull @ A @ Xs3 )
=> ( ~ ( coinductive_lnull @ A @ ( coinductive_ltl @ A @ Xs3 ) )
=> ( ( Less_eq @ ( coinductive_lhd @ A @ Xs3 ) @ ( coinductive_lhd @ A @ ( coinductive_ltl @ A @ Xs3 ) ) )
& ( ( X3 @ ( coinductive_ltl @ A @ Xs3 ) )
| ( coinductive_lsorted @ A @ Less_eq @ ( coinductive_ltl @ A @ Xs3 ) ) ) ) ) ) )
=> ( coinductive_lsorted @ A @ Less_eq @ Xs ) ) ) ).
% ord.lsorted_coinduct'
thf(fact_38_ord_Olsorted__lhdD,axiom,
! [A: $tType,Less_eq: A > A > $o,Xs: coinductive_llist @ A] :
( ( coinductive_lsorted @ A @ Less_eq @ Xs )
=> ( ~ ( coinductive_lnull @ A @ Xs )
=> ( ~ ( coinductive_lnull @ A @ ( coinductive_ltl @ A @ Xs ) )
=> ( Less_eq @ ( coinductive_lhd @ A @ Xs ) @ ( coinductive_lhd @ A @ ( coinductive_ltl @ A @ Xs ) ) ) ) ) ) ).
% ord.lsorted_lhdD
thf(fact_39_ord_Olsorted__LCons_H,axiom,
! [A: $tType,Less_eq: A > A > $o,X: A,Xs: coinductive_llist @ A] :
( ( coinductive_lsorted @ A @ Less_eq @ ( coinductive_LCons @ A @ X @ Xs ) )
= ( ~ ( coinductive_lnull @ A @ Xs )
=> ( ( Less_eq @ X @ ( coinductive_lhd @ A @ Xs ) )
& ( coinductive_lsorted @ A @ Less_eq @ Xs ) ) ) ) ).
% ord.lsorted_LCons'
thf(fact_40_llexord__coinduct,axiom,
! [A: $tType,X3: ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,R2: A > A > $o] :
( ( X3 @ Xs @ Ys )
=> ( ! [Xs3: coinductive_llist @ A,Ys2: coinductive_llist @ A] :
( ( X3 @ Xs3 @ Ys2 )
=> ( ~ ( coinductive_lnull @ A @ Xs3 )
=> ( ~ ( coinductive_lnull @ A @ Ys2 )
& ( ~ ( coinductive_lnull @ A @ Ys2 )
=> ( ( R2 @ ( coinductive_lhd @ A @ Xs3 ) @ ( coinductive_lhd @ A @ Ys2 ) )
| ( ( ( coinductive_lhd @ A @ Xs3 )
= ( coinductive_lhd @ A @ Ys2 ) )
& ( ( X3 @ ( coinductive_ltl @ A @ Xs3 ) @ ( coinductive_ltl @ A @ Ys2 ) )
| ( coinductive_llexord @ A @ R2 @ ( coinductive_ltl @ A @ Xs3 ) @ ( coinductive_ltl @ A @ Ys2 ) ) ) ) ) ) ) ) )
=> ( coinductive_llexord @ A @ R2 @ Xs @ Ys ) ) ) ).
% llexord_coinduct
thf(fact_41_lmember__code_I2_J,axiom,
! [A: $tType,X: A,Y: A,Ys: coinductive_llist @ A] :
( ( coinductive_lmember @ A @ X @ ( coinductive_LCons @ A @ Y @ Ys ) )
= ( ( X = Y )
| ( coinductive_lmember @ A @ X @ Ys ) ) ) ).
% lmember_code(2)
thf(fact_42_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_43_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X4: A] :
( ( F @ X4 )
= ( G @ X4 ) )
=> ( F = G ) ) ).
% ext
thf(fact_44_llexord__refl,axiom,
! [A: $tType,R2: A > A > $o,Xs: coinductive_llist @ A] : ( coinductive_llexord @ A @ R2 @ Xs @ Xs ) ).
% llexord_refl
thf(fact_45_ltakeWhile__LNil,axiom,
! [A: $tType,P: A > $o] :
( ( coindu501562517eWhile @ A @ P @ ( coinductive_LNil @ A ) )
= ( coinductive_LNil @ A ) ) ).
% ltakeWhile_LNil
thf(fact_46_llexord__LCons__LCons,axiom,
! [A: $tType,R2: A > A > $o,X: A,Xs: coinductive_llist @ A,Y: A,Ys: coinductive_llist @ A] :
( ( coinductive_llexord @ A @ R2 @ ( coinductive_LCons @ A @ X @ Xs ) @ ( coinductive_LCons @ A @ Y @ Ys ) )
= ( ( ( X = Y )
& ( coinductive_llexord @ A @ R2 @ Xs @ Ys ) )
| ( R2 @ X @ Y ) ) ) ).
% llexord_LCons_LCons
thf(fact_47_llexord__LNil__right,axiom,
! [A: $tType,Ys: coinductive_llist @ A,R2: A > A > $o,Xs: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ Ys )
=> ( ( coinductive_llexord @ A @ R2 @ Xs @ Ys )
= ( coinductive_lnull @ A @ Xs ) ) ) ).
% llexord_LNil_right
thf(fact_48_llexord__code_I1_J,axiom,
! [A: $tType,R2: A > A > $o,Ys: coinductive_llist @ A] : ( coinductive_llexord @ A @ R2 @ ( coinductive_LNil @ A ) @ Ys ) ).
% llexord_code(1)
thf(fact_49_unfold__llist__eq__LCons,axiom,
! [A: $tType,B: $tType,IS_LNIL: B > $o,LHD: B > A,LTL: B > B,B2: B,X: A,Xs: coinductive_llist @ A] :
( ( ( coindu1441602521_llist @ B @ A @ IS_LNIL @ LHD @ LTL @ B2 )
= ( coinductive_LCons @ A @ X @ Xs ) )
= ( ~ ( IS_LNIL @ B2 )
& ( X
= ( LHD @ B2 ) )
& ( Xs
= ( coindu1441602521_llist @ B @ A @ IS_LNIL @ LHD @ LTL @ ( LTL @ B2 ) ) ) ) ) ).
% unfold_llist_eq_LCons
thf(fact_50_unfold__llist_Odisc__iff_I2_J,axiom,
! [B: $tType,A: $tType,P2: A > $o,G21: A > B,G22: A > A,A2: A] :
( ( ~ ( coinductive_lnull @ B @ ( coindu1441602521_llist @ A @ B @ P2 @ G21 @ G22 @ A2 ) ) )
= ( ~ ( P2 @ A2 ) ) ) ).
% unfold_llist.disc_iff(2)
thf(fact_51_unfold__llist_Odisc__iff_I1_J,axiom,
! [B: $tType,A: $tType,P2: A > $o,G21: A > B,G22: A > A,A2: A] :
( ( coinductive_lnull @ B @ ( coindu1441602521_llist @ A @ B @ P2 @ G21 @ G22 @ A2 ) )
= ( P2 @ A2 ) ) ).
% unfold_llist.disc_iff(1)
thf(fact_52_lmirror__aux__simps_I1_J,axiom,
! [A: $tType,Acc: coinductive_llist @ A] :
( ( lMirro999291890or_aux @ A @ Acc @ ( coinductive_LNil @ A ) )
= Acc ) ).
% lmirror_aux_simps(1)
thf(fact_53_ltakeWhile__LCons,axiom,
! [A: $tType,P: A > $o,X: A,Xs: coinductive_llist @ A] :
( ( ( P @ X )
=> ( ( coindu501562517eWhile @ A @ P @ ( coinductive_LCons @ A @ X @ Xs ) )
= ( coinductive_LCons @ A @ X @ ( coindu501562517eWhile @ A @ P @ Xs ) ) ) )
& ( ~ ( P @ X )
=> ( ( coindu501562517eWhile @ A @ P @ ( coinductive_LCons @ A @ X @ Xs ) )
= ( coinductive_LNil @ A ) ) ) ) ).
% ltakeWhile_LCons
thf(fact_54_ltakeWhile_Odisc__iff_I2_J,axiom,
! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
( ( ~ ( coinductive_lnull @ A @ ( coindu501562517eWhile @ A @ P @ Xs ) ) )
= ( ~ ( coinductive_lnull @ A @ Xs )
& ( P @ ( coinductive_lhd @ A @ Xs ) ) ) ) ).
% ltakeWhile.disc_iff(2)
thf(fact_55_ltakeWhile_Odisc__iff_I1_J,axiom,
! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ ( coindu501562517eWhile @ A @ P @ Xs ) )
= ( ( coinductive_lnull @ A @ Xs )
| ~ ( P @ ( coinductive_lhd @ A @ Xs ) ) ) ) ).
% ltakeWhile.disc_iff(1)
thf(fact_56_lnull__ltakeWhile,axiom,
! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ ( coindu501562517eWhile @ A @ P @ Xs ) )
= ( ~ ( coinductive_lnull @ A @ Xs )
=> ~ ( P @ ( coinductive_lhd @ A @ Xs ) ) ) ) ).
% lnull_ltakeWhile
thf(fact_57_llist_Osimps_I4_J,axiom,
! [A: $tType,B: $tType,F12: B,F22: A > ( coinductive_llist @ A ) > B] :
( ( coindu1381640503_llist @ B @ A @ F12 @ F22 @ ( coinductive_LNil @ A ) )
= F12 ) ).
% llist.simps(4)
thf(fact_58_lmember__code_I1_J,axiom,
! [A: $tType,X: A] :
~ ( coinductive_lmember @ A @ X @ ( coinductive_LNil @ A ) ) ).
% lmember_code(1)
thf(fact_59_ord_Olsorted__code_I1_J,axiom,
! [A: $tType,Less_eq: A > A > $o] : ( coinductive_lsorted @ A @ Less_eq @ ( coinductive_LNil @ A ) ) ).
% ord.lsorted_code(1)
thf(fact_60_unfold__llist_Octr_I1_J,axiom,
! [A: $tType,B: $tType,P2: A > $o,A2: A,G21: A > B,G22: A > A] :
( ( P2 @ A2 )
=> ( ( coindu1441602521_llist @ A @ B @ P2 @ G21 @ G22 @ A2 )
= ( coinductive_LNil @ B ) ) ) ).
% unfold_llist.ctr(1)
thf(fact_61_llexord__trans,axiom,
! [A: $tType,R2: A > A > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A] :
( ( coinductive_llexord @ A @ R2 @ Xs @ Ys )
=> ( ( coinductive_llexord @ A @ R2 @ Ys @ Zs )
=> ( ! [A4: A,B4: A,C2: A] :
( ( R2 @ A4 @ B4 )
=> ( ( R2 @ B4 @ C2 )
=> ( R2 @ A4 @ C2 ) ) )
=> ( coinductive_llexord @ A @ R2 @ Xs @ Zs ) ) ) ) ).
% llexord_trans
thf(fact_62_llexord__linear,axiom,
! [A: $tType,R2: A > A > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ! [X4: A,Y2: A] :
( ( R2 @ X4 @ Y2 )
| ( X4 = Y2 )
| ( R2 @ Y2 @ X4 ) )
=> ( ( coinductive_llexord @ A @ R2 @ Xs @ Ys )
| ( coinductive_llexord @ A @ R2 @ Ys @ Xs ) ) ) ).
% llexord_linear
thf(fact_63_llexord__antisym,axiom,
! [A: $tType,R2: A > A > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_llexord @ A @ R2 @ Xs @ Ys )
=> ( ( coinductive_llexord @ A @ R2 @ Ys @ Xs )
=> ( ! [A4: A,B4: A] :
( ( R2 @ A4 @ B4 )
=> ~ ( R2 @ B4 @ A4 ) )
=> ( Xs = Ys ) ) ) ) ).
% llexord_antisym
thf(fact_64_ord_OLNil,axiom,
! [A: $tType,Less_eq: A > A > $o] : ( coinductive_lsorted @ A @ Less_eq @ ( coinductive_LNil @ A ) ) ).
% ord.LNil
thf(fact_65_llexord__LNil,axiom,
! [A: $tType,R2: A > A > $o,Ys: coinductive_llist @ A] : ( coinductive_llexord @ A @ R2 @ ( coinductive_LNil @ A ) @ Ys ) ).
% llexord_LNil
thf(fact_66_ltakeWhile__eq__LNil__iff,axiom,
! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
( ( ( coindu501562517eWhile @ A @ P @ Xs )
= ( coinductive_LNil @ A ) )
= ( ( Xs
!= ( coinductive_LNil @ A ) )
=> ~ ( P @ ( coinductive_lhd @ A @ Xs ) ) ) ) ).
% ltakeWhile_eq_LNil_iff
thf(fact_67_llexord_Ocoinduct,axiom,
! [A: $tType,X3: ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o,X: coinductive_llist @ A,Xa: coinductive_llist @ A,R2: A > A > $o] :
( ( X3 @ X @ Xa )
=> ( ! [X4: coinductive_llist @ A,Xa2: coinductive_llist @ A] :
( ( X3 @ X4 @ Xa2 )
=> ( ? [Xs4: coinductive_llist @ A,Ys3: coinductive_llist @ A,Xb: A] :
( ( X4
= ( coinductive_LCons @ A @ Xb @ Xs4 ) )
& ( Xa2
= ( coinductive_LCons @ A @ Xb @ Ys3 ) )
& ( ( X3 @ Xs4 @ Ys3 )
| ( coinductive_llexord @ A @ R2 @ Xs4 @ Ys3 ) ) )
| ? [Xb: A,Y3: A,Xs4: coinductive_llist @ A,Ys3: coinductive_llist @ A] :
( ( X4
= ( coinductive_LCons @ A @ Xb @ Xs4 ) )
& ( Xa2
= ( coinductive_LCons @ A @ Y3 @ Ys3 ) )
& ( R2 @ Xb @ Y3 ) )
| ? [Ys3: coinductive_llist @ A] :
( ( X4
= ( coinductive_LNil @ A ) )
& ( Xa2 = Ys3 ) ) ) )
=> ( coinductive_llexord @ A @ R2 @ X @ Xa ) ) ) ).
% llexord.coinduct
thf(fact_68_llexord_Osimps,axiom,
! [A: $tType] :
( ( coinductive_llexord @ A )
= ( ^ [R3: A > A > $o,A1: coinductive_llist @ A,A22: coinductive_llist @ A] :
( ? [Xs5: coinductive_llist @ A,Ys4: coinductive_llist @ A,X2: A] :
( ( A1
= ( coinductive_LCons @ A @ X2 @ Xs5 ) )
& ( A22
= ( coinductive_LCons @ A @ X2 @ Ys4 ) )
& ( coinductive_llexord @ A @ R3 @ Xs5 @ Ys4 ) )
| ? [X2: A,Y4: A,Xs5: coinductive_llist @ A,Ys4: coinductive_llist @ A] :
( ( A1
= ( coinductive_LCons @ A @ X2 @ Xs5 ) )
& ( A22
= ( coinductive_LCons @ A @ Y4 @ Ys4 ) )
& ( R3 @ X2 @ Y4 ) )
| ? [Ys4: coinductive_llist @ A] :
( ( A1
= ( coinductive_LNil @ A ) )
& ( A22 = Ys4 ) ) ) ) ) ).
% llexord.simps
thf(fact_69_llexord_Ocases,axiom,
! [A: $tType,R2: A > A > $o,A12: coinductive_llist @ A,A23: coinductive_llist @ A] :
( ( coinductive_llexord @ A @ R2 @ A12 @ A23 )
=> ( ! [Xs3: coinductive_llist @ A,Ys2: coinductive_llist @ A,X4: A] :
( ( A12
= ( coinductive_LCons @ A @ X4 @ Xs3 ) )
=> ( ( A23
= ( coinductive_LCons @ A @ X4 @ Ys2 ) )
=> ~ ( coinductive_llexord @ A @ R2 @ Xs3 @ Ys2 ) ) )
=> ( ! [X4: A] :
( ? [Xs3: coinductive_llist @ A] :
( A12
= ( coinductive_LCons @ A @ X4 @ Xs3 ) )
=> ! [Y2: A] :
( ? [Ys2: coinductive_llist @ A] :
( A23
= ( coinductive_LCons @ A @ Y2 @ Ys2 ) )
=> ~ ( R2 @ X4 @ Y2 ) ) )
=> ~ ( ( A12
= ( coinductive_LNil @ A ) )
=> ! [Ys2: coinductive_llist @ A] : A23 != Ys2 ) ) ) ) ).
% llexord.cases
thf(fact_70_llexord__code_I2_J,axiom,
! [A: $tType,R2: A > A > $o,X: A,Xs: coinductive_llist @ A] :
~ ( coinductive_llexord @ A @ R2 @ ( coinductive_LCons @ A @ X @ Xs ) @ ( coinductive_LNil @ A ) ) ).
% llexord_code(2)
thf(fact_71_ord_OSingleton,axiom,
! [A: $tType,Less_eq: A > A > $o,X: A] : ( coinductive_lsorted @ A @ Less_eq @ ( coinductive_LCons @ A @ X @ ( coinductive_LNil @ A ) ) ) ).
% ord.Singleton
thf(fact_72_ord_Olsorted_Ocoinduct,axiom,
! [A: $tType,X3: ( coinductive_llist @ A ) > $o,X: coinductive_llist @ A,Less_eq: A > A > $o] :
( ( X3 @ X )
=> ( ! [X4: coinductive_llist @ A] :
( ( X3 @ X4 )
=> ( ( X4
= ( coinductive_LNil @ A ) )
| ? [Xa3: A] :
( X4
= ( coinductive_LCons @ A @ Xa3 @ ( coinductive_LNil @ A ) ) )
| ? [Xa3: A,Y3: A,Xs4: coinductive_llist @ A] :
( ( X4
= ( coinductive_LCons @ A @ Xa3 @ ( coinductive_LCons @ A @ Y3 @ Xs4 ) ) )
& ( Less_eq @ Xa3 @ Y3 )
& ( ( X3 @ ( coinductive_LCons @ A @ Y3 @ Xs4 ) )
| ( coinductive_lsorted @ A @ Less_eq @ ( coinductive_LCons @ A @ Y3 @ Xs4 ) ) ) ) ) )
=> ( coinductive_lsorted @ A @ Less_eq @ X ) ) ) ).
% ord.lsorted.coinduct
thf(fact_73_ord_Olsorted_Osimps,axiom,
! [A: $tType] :
( ( coinductive_lsorted @ A )
= ( ^ [Less_eq2: A > A > $o,A3: coinductive_llist @ A] :
( ( A3
= ( coinductive_LNil @ A ) )
| ? [X2: A] :
( A3
= ( coinductive_LCons @ A @ X2 @ ( coinductive_LNil @ A ) ) )
| ? [X2: A,Y4: A,Xs5: coinductive_llist @ A] :
( ( A3
= ( coinductive_LCons @ A @ X2 @ ( coinductive_LCons @ A @ Y4 @ Xs5 ) ) )
& ( Less_eq2 @ X2 @ Y4 )
& ( coinductive_lsorted @ A @ Less_eq2 @ ( coinductive_LCons @ A @ Y4 @ Xs5 ) ) ) ) ) ) ).
% ord.lsorted.simps
thf(fact_74_ord_Olsorted_Ocases,axiom,
! [A: $tType,Less_eq: A > A > $o,A2: coinductive_llist @ A] :
( ( coinductive_lsorted @ A @ Less_eq @ A2 )
=> ( ( A2
!= ( coinductive_LNil @ A ) )
=> ( ! [X4: A] :
( A2
!= ( coinductive_LCons @ A @ X4 @ ( coinductive_LNil @ A ) ) )
=> ~ ! [X4: A,Y2: A,Xs3: coinductive_llist @ A] :
( ( A2
= ( coinductive_LCons @ A @ X4 @ ( coinductive_LCons @ A @ Y2 @ Xs3 ) ) )
=> ( ( Less_eq @ X4 @ Y2 )
=> ~ ( coinductive_lsorted @ A @ Less_eq @ ( coinductive_LCons @ A @ Y2 @ Xs3 ) ) ) ) ) ) ) ).
% ord.lsorted.cases
thf(fact_75_ord_Olsorted__code_I2_J,axiom,
! [A: $tType,Less_eq: A > A > $o,X: A] : ( coinductive_lsorted @ A @ Less_eq @ ( coinductive_LCons @ A @ X @ ( coinductive_LNil @ A ) ) ) ).
% ord.lsorted_code(2)
thf(fact_76_unfold__llist_Ocode,axiom,
! [B: $tType,A: $tType] :
( ( coindu1441602521_llist @ A @ B )
= ( ^ [P3: A > $o,G212: A > B,G222: A > A,A3: A] : ( if @ ( coinductive_llist @ B ) @ ( P3 @ A3 ) @ ( coinductive_LNil @ B ) @ ( coinductive_LCons @ B @ ( G212 @ A3 ) @ ( coindu1441602521_llist @ A @ B @ P3 @ G212 @ G222 @ ( G222 @ A3 ) ) ) ) ) ) ).
% unfold_llist.code
thf(fact_77_ltl__unfold__llist,axiom,
! [A: $tType,B: $tType,IS_LNIL: B > $o,A2: B,LHD: B > A,LTL: B > B] :
( ( ( IS_LNIL @ A2 )
=> ( ( coinductive_ltl @ A @ ( coindu1441602521_llist @ B @ A @ IS_LNIL @ LHD @ LTL @ A2 ) )
= ( coinductive_LNil @ A ) ) )
& ( ~ ( IS_LNIL @ A2 )
=> ( ( coinductive_ltl @ A @ ( coindu1441602521_llist @ B @ A @ IS_LNIL @ LHD @ LTL @ A2 ) )
= ( coindu1441602521_llist @ B @ A @ IS_LNIL @ LHD @ LTL @ ( LTL @ A2 ) ) ) ) ) ).
% ltl_unfold_llist
thf(fact_78_ltakeWhile_Octr_I1_J,axiom,
! [A: $tType,Xs: coinductive_llist @ A,P: A > $o] :
( ( ( coinductive_lnull @ A @ Xs )
| ~ ( P @ ( coinductive_lhd @ A @ Xs ) ) )
=> ( ( coindu501562517eWhile @ A @ P @ Xs )
= ( coinductive_LNil @ A ) ) ) ).
% ltakeWhile.ctr(1)
thf(fact_79_ltl__ltakeWhile,axiom,
! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
( ( ( P @ ( coinductive_lhd @ A @ Xs ) )
=> ( ( coinductive_ltl @ A @ ( coindu501562517eWhile @ A @ P @ Xs ) )
= ( coindu501562517eWhile @ A @ P @ ( coinductive_ltl @ A @ Xs ) ) ) )
& ( ~ ( P @ ( coinductive_lhd @ A @ Xs ) )
=> ( ( coinductive_ltl @ A @ ( coindu501562517eWhile @ A @ P @ Xs ) )
= ( coinductive_LNil @ A ) ) ) ) ).
% ltl_ltakeWhile
thf(fact_80_llength__ltakeWhile__all,axiom,
! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
( ( ( coinductive_llength @ A @ ( coindu501562517eWhile @ A @ P @ Xs ) )
= ( coinductive_llength @ A @ Xs ) )
= ( ( coindu501562517eWhile @ A @ P @ Xs )
= Xs ) ) ).
% llength_ltakeWhile_all
thf(fact_81_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_82_llist_Oexhaust,axiom,
! [A: $tType,Y: coinductive_llist @ A] :
( ( Y
!= ( coinductive_LNil @ A ) )
=> ~ ! [X212: A,X222: coinductive_llist @ A] :
( Y
!= ( coinductive_LCons @ A @ X212 @ X222 ) ) ) ).
% llist.exhaust
thf(fact_83_neq__LNil__conv,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( Xs
!= ( coinductive_LNil @ A ) )
= ( ? [X2: A,Xs2: coinductive_llist @ A] :
( Xs
= ( coinductive_LCons @ A @ X2 @ Xs2 ) ) ) ) ).
% neq_LNil_conv
thf(fact_84_llist_Odisc_I1_J,axiom,
! [A: $tType] : ( coinductive_lnull @ A @ ( coinductive_LNil @ A ) ) ).
% llist.disc(1)
thf(fact_85_llist_OdiscI_I1_J,axiom,
! [A: $tType,Llist: coinductive_llist @ A] :
( ( Llist
= ( coinductive_LNil @ A ) )
=> ( coinductive_lnull @ A @ Llist ) ) ).
% llist.discI(1)
thf(fact_86_llist_Ocollapse_I1_J,axiom,
! [A: $tType,Llist: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ Llist )
=> ( Llist
= ( coinductive_LNil @ A ) ) ) ).
% llist.collapse(1)
thf(fact_87_lnull__def,axiom,
! [A: $tType] :
( ( coinductive_lnull @ A )
= ( ^ [Llist5: coinductive_llist @ A] :
( Llist5
= ( coinductive_LNil @ A ) ) ) ) ).
% lnull_def
thf(fact_88_ltl__simps_I1_J,axiom,
! [A: $tType] :
( ( coinductive_ltl @ A @ ( coinductive_LNil @ A ) )
= ( coinductive_LNil @ A ) ) ).
% ltl_simps(1)
thf(fact_89_llexord__code_I3_J,axiom,
! [A: $tType,R2: A > A > $o,X: A,Xs: coinductive_llist @ A,Y: A,Ys: coinductive_llist @ A] :
( ( coinductive_llexord @ A @ R2 @ ( coinductive_LCons @ A @ X @ Xs ) @ ( coinductive_LCons @ A @ Y @ Ys ) )
= ( ( R2 @ X @ Y )
| ( ( X = Y )
& ( coinductive_llexord @ A @ R2 @ Xs @ Ys ) ) ) ) ).
% llexord_code(3)
thf(fact_90_llexord__LCons__left,axiom,
! [A: $tType,R2: A > A > $o,X: A,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_llexord @ A @ R2 @ ( coinductive_LCons @ A @ X @ Xs ) @ Ys )
= ( ? [Y4: A,Ys5: coinductive_llist @ A] :
( ( Ys
= ( coinductive_LCons @ A @ Y4 @ Ys5 ) )
& ( ( ( X = Y4 )
& ( coinductive_llexord @ A @ R2 @ Xs @ Ys5 ) )
| ( R2 @ X @ Y4 ) ) ) ) ) ).
% llexord_LCons_left
thf(fact_91_llexord__LCons__eq,axiom,
! [A: $tType,R2: A > A > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,X: A] :
( ( coinductive_llexord @ A @ R2 @ Xs @ Ys )
=> ( coinductive_llexord @ A @ R2 @ ( coinductive_LCons @ A @ X @ Xs ) @ ( coinductive_LCons @ A @ X @ Ys ) ) ) ).
% llexord_LCons_eq
thf(fact_92_llexord__LCons__less,axiom,
! [A: $tType,R2: A > A > $o,X: A,Y: A,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( R2 @ X @ Y )
=> ( coinductive_llexord @ A @ R2 @ ( coinductive_LCons @ A @ X @ Xs ) @ ( coinductive_LCons @ A @ Y @ Ys ) ) ) ).
% llexord_LCons_less
thf(fact_93_lnull__llexord,axiom,
! [A: $tType,Xs: coinductive_llist @ A,R2: A > A > $o,Ys: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ Xs )
=> ( coinductive_llexord @ A @ R2 @ Xs @ Ys ) ) ).
% lnull_llexord
thf(fact_94_ord_OLCons__LCons,axiom,
! [A: $tType,Less_eq: A > A > $o,X: A,Y: A,Xs: coinductive_llist @ A] :
( ( Less_eq @ X @ Y )
=> ( ( coinductive_lsorted @ A @ Less_eq @ ( coinductive_LCons @ A @ Y @ Xs ) )
=> ( coinductive_lsorted @ A @ Less_eq @ ( coinductive_LCons @ A @ X @ ( coinductive_LCons @ A @ Y @ Xs ) ) ) ) ) ).
% ord.LCons_LCons
thf(fact_95_ord_Olsorted__LCons__LCons,axiom,
! [A: $tType,Less_eq: A > A > $o,X: A,Y: A,Xs: coinductive_llist @ A] :
( ( coinductive_lsorted @ A @ Less_eq @ ( coinductive_LCons @ A @ X @ ( coinductive_LCons @ A @ Y @ Xs ) ) )
= ( ( Less_eq @ X @ Y )
& ( coinductive_lsorted @ A @ Less_eq @ ( coinductive_LCons @ A @ Y @ Xs ) ) ) ) ).
% ord.lsorted_LCons_LCons
thf(fact_96_ord_Olsorted__ltlI,axiom,
! [A: $tType,Less_eq: A > A > $o,Xs: coinductive_llist @ A] :
( ( coinductive_lsorted @ A @ Less_eq @ Xs )
=> ( coinductive_lsorted @ A @ Less_eq @ ( coinductive_ltl @ A @ Xs ) ) ) ).
% ord.lsorted_ltlI
thf(fact_97_unfold__llist_Octr_I2_J,axiom,
! [B: $tType,A: $tType,P2: A > $o,A2: A,G21: A > B,G22: A > A] :
( ~ ( P2 @ A2 )
=> ( ( coindu1441602521_llist @ A @ B @ P2 @ G21 @ G22 @ A2 )
= ( coinductive_LCons @ B @ ( G21 @ A2 ) @ ( coindu1441602521_llist @ A @ B @ P2 @ G21 @ G22 @ ( G22 @ A2 ) ) ) ) ) ).
% unfold_llist.ctr(2)
thf(fact_98_unfold__llist_Odisc_I2_J,axiom,
! [B: $tType,A: $tType,P2: A > $o,A2: A,G21: A > B,G22: A > A] :
( ~ ( P2 @ A2 )
=> ~ ( coinductive_lnull @ B @ ( coindu1441602521_llist @ A @ B @ P2 @ G21 @ G22 @ A2 ) ) ) ).
% unfold_llist.disc(2)
thf(fact_99_unfold__llist_Odisc_I1_J,axiom,
! [B: $tType,A: $tType,P2: A > $o,A2: A,G21: A > B,G22: A > A] :
( ( P2 @ A2 )
=> ( coinductive_lnull @ B @ ( coindu1441602521_llist @ A @ B @ P2 @ G21 @ G22 @ A2 ) ) ) ).
% unfold_llist.disc(1)
thf(fact_100_llist_Osimps_I5_J,axiom,
! [B: $tType,A: $tType,F12: B,F22: A > ( coinductive_llist @ A ) > B,X21: A,X22: coinductive_llist @ A] :
( ( coindu1381640503_llist @ B @ A @ F12 @ F22 @ ( coinductive_LCons @ A @ X21 @ X22 ) )
= ( F22 @ X21 @ X22 ) ) ).
% llist.simps(5)
thf(fact_101_unfold__llist_Osimps_I4_J,axiom,
! [B: $tType,A: $tType,P2: A > $o,A2: A,G21: A > B,G22: A > A] :
( ~ ( P2 @ A2 )
=> ( ( coinductive_ltl @ B @ ( coindu1441602521_llist @ A @ B @ P2 @ G21 @ G22 @ A2 ) )
= ( coindu1441602521_llist @ A @ B @ P2 @ G21 @ G22 @ ( G22 @ A2 ) ) ) ) ).
% unfold_llist.simps(4)
thf(fact_102_unfold__llist_Osimps_I3_J,axiom,
! [B: $tType,A: $tType,P2: A > $o,A2: A,G21: A > B,G22: A > A] :
( ~ ( P2 @ A2 )
=> ( ( coinductive_lhd @ B @ ( coindu1441602521_llist @ A @ B @ P2 @ G21 @ G22 @ A2 ) )
= ( G21 @ A2 ) ) ) ).
% unfold_llist.simps(3)
thf(fact_103_llist_Osplit__sel,axiom,
! [B: $tType,A: $tType,P: B > $o,F12: B,F22: A > ( coinductive_llist @ A ) > B,Llist: coinductive_llist @ A] :
( ( P @ ( coindu1381640503_llist @ B @ A @ F12 @ F22 @ Llist ) )
= ( ( ( Llist
= ( coinductive_LNil @ A ) )
=> ( P @ F12 ) )
& ( ( Llist
= ( coinductive_LCons @ A @ ( coinductive_lhd @ A @ Llist ) @ ( coinductive_ltl @ A @ Llist ) ) )
=> ( P @ ( F22 @ ( coinductive_lhd @ A @ Llist ) @ ( coinductive_ltl @ A @ Llist ) ) ) ) ) ) ).
% llist.split_sel
thf(fact_104_llist_Osplit__sel__asm,axiom,
! [B: $tType,A: $tType,P: B > $o,F12: B,F22: A > ( coinductive_llist @ A ) > B,Llist: coinductive_llist @ A] :
( ( P @ ( coindu1381640503_llist @ B @ A @ F12 @ F22 @ Llist ) )
= ( ~ ( ( ( Llist
= ( coinductive_LNil @ A ) )
& ~ ( P @ F12 ) )
| ( ( Llist
= ( coinductive_LCons @ A @ ( coinductive_lhd @ A @ Llist ) @ ( coinductive_ltl @ A @ Llist ) ) )
& ~ ( P @ ( F22 @ ( coinductive_lhd @ A @ Llist ) @ ( coinductive_ltl @ A @ Llist ) ) ) ) ) ) ) ).
% llist.split_sel_asm
thf(fact_105_ltakeWhile_Ocode,axiom,
! [A: $tType] :
( ( coindu501562517eWhile @ A )
= ( ^ [P4: A > $o,Xs5: coinductive_llist @ A] :
( if @ ( coinductive_llist @ A )
@ ( ( coinductive_lnull @ A @ Xs5 )
| ~ ( P4 @ ( coinductive_lhd @ A @ Xs5 ) ) )
@ ( coinductive_LNil @ A )
@ ( coinductive_LCons @ A @ ( coinductive_lhd @ A @ Xs5 ) @ ( coindu501562517eWhile @ A @ P4 @ ( coinductive_ltl @ A @ Xs5 ) ) ) ) ) ) ).
% ltakeWhile.code
thf(fact_106_ltakeWhile_Odisc_I2_J,axiom,
! [A: $tType,Xs: coinductive_llist @ A,P: A > $o] :
( ~ ( coinductive_lnull @ A @ Xs )
=> ( ( P @ ( coinductive_lhd @ A @ Xs ) )
=> ~ ( coinductive_lnull @ A @ ( coindu501562517eWhile @ A @ P @ Xs ) ) ) ) ).
% ltakeWhile.disc(2)
thf(fact_107_ltakeWhile_Odisc_I1_J,axiom,
! [A: $tType,Xs: coinductive_llist @ A,P: A > $o] :
( ( ( coinductive_lnull @ A @ Xs )
| ~ ( P @ ( coinductive_lhd @ A @ Xs ) ) )
=> ( coinductive_lnull @ A @ ( coindu501562517eWhile @ A @ P @ Xs ) ) ) ).
% ltakeWhile.disc(1)
thf(fact_108_lhd__ltakeWhile,axiom,
! [A: $tType,Xs: coinductive_llist @ A,P: A > $o] :
( ~ ( coinductive_lnull @ A @ Xs )
=> ( ( P @ ( coinductive_lhd @ A @ Xs ) )
=> ( ( coinductive_lhd @ A @ ( coindu501562517eWhile @ A @ P @ Xs ) )
= ( coinductive_lhd @ A @ Xs ) ) ) ) ).
% lhd_ltakeWhile
thf(fact_109_lmirror__aux_Octr_I1_J,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Acc: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ Xs )
=> ( ( coinductive_lnull @ A @ Acc )
=> ( ( lMirro999291890or_aux @ A @ Acc @ Xs )
= ( coinductive_LNil @ A ) ) ) ) ).
% lmirror_aux.ctr(1)
thf(fact_110_ltakeWhile_Osimps_I4_J,axiom,
! [A: $tType,Xs: coinductive_llist @ A,P: A > $o] :
( ~ ( coinductive_lnull @ A @ Xs )
=> ( ( P @ ( coinductive_lhd @ A @ Xs ) )
=> ( ( coinductive_ltl @ A @ ( coindu501562517eWhile @ A @ P @ Xs ) )
= ( coindu501562517eWhile @ A @ P @ ( coinductive_ltl @ A @ Xs ) ) ) ) ) ).
% ltakeWhile.simps(4)
thf(fact_111_eq__LConsD,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Y: A,Ys: coinductive_llist @ A] :
( ( Xs
= ( coinductive_LCons @ A @ Y @ Ys ) )
=> ( ( Xs
!= ( coinductive_LNil @ A ) )
& ( ( coinductive_lhd @ A @ Xs )
= Y )
& ( ( coinductive_ltl @ A @ Xs )
= Ys ) ) ) ).
% eq_LConsD
thf(fact_112_lmirror__def,axiom,
! [A: $tType] :
( ( lMirro427583474mirror @ A )
= ( lMirro999291890or_aux @ A @ ( coinductive_LNil @ A ) ) ) ).
% lmirror_def
thf(fact_113_Coinductive__List_Ofinite__lprefix__nitpick__simps_I3_J,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Y: A,Ys: coinductive_llist @ A] :
( ( coindu328551480prefix @ A @ Xs @ ( coinductive_LCons @ A @ Y @ Ys ) )
= ( ( Xs
= ( coinductive_LNil @ A ) )
| ? [Xs2: coinductive_llist @ A] :
( ( Xs
= ( coinductive_LCons @ A @ Y @ Xs2 ) )
& ( coindu328551480prefix @ A @ Xs2 @ Ys ) ) ) ) ).
% Coinductive_List.finite_lprefix_nitpick_simps(3)
thf(fact_114_lstrict__prefix__code_I3_J,axiom,
! [B: $tType,X: B,Xs: coinductive_llist @ B] :
~ ( coindu1478340336prefix @ B @ ( coinductive_LCons @ B @ X @ Xs ) @ ( coinductive_LNil @ B ) ) ).
% lstrict_prefix_code(3)
thf(fact_115_lstrict__prefix__code_I2_J,axiom,
! [B: $tType,Y: B,Ys: coinductive_llist @ B] : ( coindu1478340336prefix @ B @ ( coinductive_LNil @ B ) @ ( coinductive_LCons @ B @ Y @ Ys ) ) ).
% lstrict_prefix_code(2)
thf(fact_116_llast__singleton,axiom,
! [A: $tType,X: A] :
( ( coinductive_llast @ A @ ( coinductive_LCons @ A @ X @ ( coinductive_LNil @ A ) ) )
= X ) ).
% llast_singleton
thf(fact_117_is__num__normalize_I1_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A @ ( type @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C ) ) ) ) ).
% is_num_normalize(1)
thf(fact_118_linordered__field__class_Osign__simps_I28_J,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type @ A ) )
=> ! [B2: A,A2: A,C: A] :
( ( plus_plus @ A @ B2 @ ( plus_plus @ A @ A2 @ C ) )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C ) ) ) ) ).
% linordered_field_class.sign_simps(28)
thf(fact_119_linordered__field__class_Osign__simps_I27_J,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type @ A ) )
=> ( ( plus_plus @ A )
= ( ^ [A3: A,B3: A] : ( plus_plus @ A @ B3 @ A3 ) ) ) ) ).
% linordered_field_class.sign_simps(27)
thf(fact_120_llast__LCons2,axiom,
! [A: $tType,X: A,Y: A,Xs: coinductive_llist @ A] :
( ( coinductive_llast @ A @ ( coinductive_LCons @ A @ X @ ( coinductive_LCons @ A @ Y @ Xs ) ) )
= ( coinductive_llast @ A @ ( coinductive_LCons @ A @ Y @ Xs ) ) ) ).
% llast_LCons2
thf(fact_121_lstrict__prefix__code_I4_J,axiom,
! [B: $tType,X: B,Xs: coinductive_llist @ B,Y: B,Ys: coinductive_llist @ B] :
( ( coindu1478340336prefix @ B @ ( coinductive_LCons @ B @ X @ Xs ) @ ( coinductive_LCons @ B @ Y @ Ys ) )
= ( ( X = Y )
& ( coindu1478340336prefix @ B @ Xs @ Ys ) ) ) ).
% lstrict_prefix_code(4)
thf(fact_122_lstrict__prefix__code_I1_J,axiom,
! [A: $tType] :
~ ( coindu1478340336prefix @ A @ ( coinductive_LNil @ A ) @ ( coinductive_LNil @ A ) ) ).
% lstrict_prefix_code(1)
thf(fact_123_llist__less__induct,axiom,
! [A: $tType,P: ( coinductive_llist @ A ) > $o,Xs: coinductive_llist @ A] :
( ! [Xs3: coinductive_llist @ A] :
( ! [Ys3: coinductive_llist @ A] :
( ( coindu1478340336prefix @ A @ Ys3 @ Xs3 )
=> ( P @ Ys3 ) )
=> ( P @ Xs3 ) )
=> ( P @ Xs ) ) ).
% llist_less_induct
thf(fact_124_Coinductive__List_Ofinite__lprefix__nitpick__simps_I1_J,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( coindu328551480prefix @ A @ Xs @ ( coinductive_LNil @ A ) )
= ( Xs
= ( coinductive_LNil @ A ) ) ) ).
% Coinductive_List.finite_lprefix_nitpick_simps(1)
thf(fact_125_Coinductive__List_Ofinite__lprefix__nitpick__simps_I2_J,axiom,
! [A: $tType,Xs: coinductive_llist @ A] : ( coindu328551480prefix @ A @ ( coinductive_LNil @ A ) @ Xs ) ).
% Coinductive_List.finite_lprefix_nitpick_simps(2)
thf(fact_126_llast__LCons,axiom,
! [A: $tType,Xs: coinductive_llist @ A,X: A] :
( ( ( coinductive_lnull @ A @ Xs )
=> ( ( coinductive_llast @ A @ ( coinductive_LCons @ A @ X @ Xs ) )
= X ) )
& ( ~ ( coinductive_lnull @ A @ Xs )
=> ( ( coinductive_llast @ A @ ( coinductive_LCons @ A @ X @ Xs ) )
= ( coinductive_llast @ A @ Xs ) ) ) ) ).
% llast_LCons
thf(fact_127_linordered__field__class_Osign__simps_I26_J,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C ) ) ) ) ).
% linordered_field_class.sign_simps(26)
thf(fact_128_semiring__normalization__rules_I20_J,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A @ ( type @ A ) )
=> ! [A2: A,B2: A,C: A,D: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( plus_plus @ A @ C @ D ) )
= ( plus_plus @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ D ) ) ) ) ).
% semiring_normalization_rules(20)
thf(fact_129_semiring__normalization__rules_I21_J,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A @ ( type @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C ) ) ) ) ).
% semiring_normalization_rules(21)
thf(fact_130_semiring__normalization__rules_I22_J,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A @ ( type @ A ) )
=> ! [A2: A,C: A,D: A] :
( ( plus_plus @ A @ A2 @ ( plus_plus @ A @ C @ D ) )
= ( plus_plus @ A @ C @ ( plus_plus @ A @ A2 @ D ) ) ) ) ).
% semiring_normalization_rules(22)
thf(fact_131_semiring__normalization__rules_I23_J,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A @ ( type @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C )
= ( plus_plus @ A @ ( plus_plus @ A @ A2 @ C ) @ B2 ) ) ) ).
% semiring_normalization_rules(23)
thf(fact_132_semiring__normalization__rules_I25_J,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A @ ( type @ A ) )
=> ! [A2: A,C: A,D: A] :
( ( plus_plus @ A @ A2 @ ( plus_plus @ A @ C @ D ) )
= ( plus_plus @ A @ ( plus_plus @ A @ A2 @ C ) @ D ) ) ) ).
% semiring_normalization_rules(25)
thf(fact_133_semiring__normalization__rules_I24_J,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A @ ( type @ A ) )
=> ( ( plus_plus @ A )
= ( ^ [A3: A,C3: A] : ( plus_plus @ A @ C3 @ A3 ) ) ) ) ).
% semiring_normalization_rules(24)
thf(fact_134_llimit__induct,axiom,
! [A: $tType,P: ( coinductive_llist @ A ) > $o,Xs: coinductive_llist @ A] :
( ( P @ ( coinductive_LNil @ A ) )
=> ( ! [X4: A,Xs3: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ Xs3 )
=> ( ( P @ Xs3 )
=> ( P @ ( coinductive_LCons @ A @ X4 @ Xs3 ) ) ) )
=> ( ( ! [Ys3: coinductive_llist @ A] :
( ( coindu1478340336prefix @ A @ Ys3 @ Xs )
=> ( P @ Ys3 ) )
=> ( P @ Xs ) )
=> ( P @ Xs ) ) ) ) ).
% llimit_induct
thf(fact_135_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_136_lprefix__coinduct,axiom,
! [A: $tType,P: ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( P @ Xs @ Ys )
=> ( ! [Xs3: coinductive_llist @ A,Ys2: coinductive_llist @ A] :
( ( P @ Xs3 @ Ys2 )
=> ( ( ( coinductive_lnull @ A @ Ys2 )
=> ( coinductive_lnull @ A @ Xs3 ) )
& ( ~ ( coinductive_lnull @ A @ Xs3 )
=> ( ~ ( coinductive_lnull @ A @ Ys2 )
=> ( ( ( coinductive_lhd @ A @ Xs3 )
= ( coinductive_lhd @ A @ Ys2 ) )
& ( ( P @ ( coinductive_ltl @ A @ Xs3 ) @ ( coinductive_ltl @ A @ Ys2 ) )
| ( coinductive_lprefix @ A @ ( coinductive_ltl @ A @ Xs3 ) @ ( coinductive_ltl @ A @ Ys2 ) ) ) ) ) ) ) )
=> ( coinductive_lprefix @ A @ Xs @ Ys ) ) ) ).
% lprefix_coinduct
thf(fact_137_lmirror__aux__LCons,axiom,
! [A: $tType,Acc: coinductive_llist @ A,X: A,Xs: coinductive_llist @ A] :
( ( lMirro999291890or_aux @ A @ Acc @ ( coinductive_LCons @ A @ X @ Xs ) )
= ( coinductive_LCons @ A @ X @ ( coinductive_lappend @ A @ ( lMirro999291890or_aux @ A @ ( coinductive_LNil @ A ) @ Xs ) @ ( coinductive_LCons @ A @ X @ Acc ) ) ) ) ).
% lmirror_aux_LCons
thf(fact_138_llist_Oleq__refl,axiom,
! [A: $tType,X: coinductive_llist @ A] : ( coinductive_lprefix @ A @ X @ X ) ).
% llist.leq_refl
thf(fact_139_lprefix__refl,axiom,
! [A: $tType,Xs: coinductive_llist @ A] : ( coinductive_lprefix @ A @ Xs @ Xs ) ).
% lprefix_refl
thf(fact_140_LCons__lprefix__LCons,axiom,
! [A: $tType,X: A,Xs: coinductive_llist @ A,Y: A,Ys: coinductive_llist @ A] :
( ( coinductive_lprefix @ A @ ( coinductive_LCons @ A @ X @ Xs ) @ ( coinductive_LCons @ A @ Y @ Ys ) )
= ( ( X = Y )
& ( coinductive_lprefix @ A @ Xs @ Ys ) ) ) ).
% LCons_lprefix_LCons
thf(fact_141_lprefix__code_I1_J,axiom,
! [A: $tType,Ys: coinductive_llist @ A] : ( coinductive_lprefix @ A @ ( coinductive_LNil @ A ) @ Ys ) ).
% lprefix_code(1)
thf(fact_142_lappend__code_I2_J,axiom,
! [A: $tType,Xa: A,X: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_lappend @ A @ ( coinductive_LCons @ A @ Xa @ X ) @ Ys )
= ( coinductive_LCons @ A @ Xa @ ( coinductive_lappend @ A @ X @ Ys ) ) ) ).
% lappend_code(2)
thf(fact_143_lfinite__code_I2_J,axiom,
! [B: $tType,X: B,Xs: coinductive_llist @ B] :
( ( coinductive_lfinite @ B @ ( coinductive_LCons @ B @ X @ Xs ) )
= ( coinductive_lfinite @ B @ Xs ) ) ).
% lfinite_code(2)
thf(fact_144_lfinite__LCons,axiom,
! [A: $tType,X: A,Xs: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ ( coinductive_LCons @ A @ X @ Xs ) )
= ( coinductive_lfinite @ A @ Xs ) ) ).
% lfinite_LCons
thf(fact_145_lappend_Odisc__iff_I2_J,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( ~ ( coinductive_lnull @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) ) )
= ( ~ ( coinductive_lnull @ A @ Xs )
| ~ ( coinductive_lnull @ A @ Ys ) ) ) ).
% lappend.disc_iff(2)
thf(fact_146_lnull__lappend,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
= ( ( coinductive_lnull @ A @ Xs )
& ( coinductive_lnull @ A @ Ys ) ) ) ).
% lnull_lappend
thf(fact_147_lappend__code_I1_J,axiom,
! [A: $tType,Ys: coinductive_llist @ A] :
( ( coinductive_lappend @ A @ ( coinductive_LNil @ A ) @ Ys )
= Ys ) ).
% lappend_code(1)
thf(fact_148_lappend__LNil2,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( coinductive_lappend @ A @ Xs @ ( coinductive_LNil @ A ) )
= Xs ) ).
% lappend_LNil2
thf(fact_149_lfinite__code_I1_J,axiom,
! [A: $tType] : ( coinductive_lfinite @ A @ ( coinductive_LNil @ A ) ) ).
% lfinite_code(1)
thf(fact_150_lfinite__ltl,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ ( coinductive_ltl @ A @ Xs ) )
= ( coinductive_lfinite @ A @ Xs ) ) ).
% lfinite_ltl
thf(fact_151_lfinite__lappend,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
= ( ( coinductive_lfinite @ A @ Xs )
& ( coinductive_lfinite @ A @ Ys ) ) ) ).
% lfinite_lappend
thf(fact_152_lfinite__lmirror__aux,axiom,
! [A: $tType,Acc: coinductive_llist @ A,Xs: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ ( lMirro999291890or_aux @ A @ Acc @ Xs ) )
= ( ( coinductive_lfinite @ A @ Xs )
& ( coinductive_lfinite @ A @ Acc ) ) ) ).
% lfinite_lmirror_aux
thf(fact_153_lprefix__LNil,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( coinductive_lprefix @ A @ Xs @ ( coinductive_LNil @ A ) )
= ( coinductive_lnull @ A @ Xs ) ) ).
% lprefix_LNil
thf(fact_154_lprefix__lappend__same,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A] :
( ( coinductive_lprefix @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) @ ( coinductive_lappend @ A @ Xs @ Zs ) )
= ( ( coinductive_lfinite @ A @ Xs )
=> ( coinductive_lprefix @ A @ Ys @ Zs ) ) ) ).
% lprefix_lappend_same
thf(fact_155_ltl__lappend,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( ( coinductive_lnull @ A @ Xs )
=> ( ( coinductive_ltl @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
= ( coinductive_ltl @ A @ Ys ) ) )
& ( ~ ( coinductive_lnull @ A @ Xs )
=> ( ( coinductive_ltl @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
= ( coinductive_lappend @ A @ ( coinductive_ltl @ A @ Xs ) @ Ys ) ) ) ) ).
% ltl_lappend
thf(fact_156_lhd__lappend,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( ( coinductive_lnull @ A @ Xs )
=> ( ( coinductive_lhd @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
= ( coinductive_lhd @ A @ Ys ) ) )
& ( ~ ( coinductive_lnull @ A @ Xs )
=> ( ( coinductive_lhd @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
= ( coinductive_lhd @ A @ Xs ) ) ) ) ).
% lhd_lappend
thf(fact_157_llength__lappend,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_llength @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
= ( plus_plus @ extended_enat @ ( coinductive_llength @ A @ Xs ) @ ( coinductive_llength @ A @ Ys ) ) ) ).
% llength_lappend
thf(fact_158_llast__lappend__LCons,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Y: A,Ys: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ Xs )
=> ( ( coinductive_llast @ A @ ( coinductive_lappend @ A @ Xs @ ( coinductive_LCons @ A @ Y @ Ys ) ) )
= ( coinductive_llast @ A @ ( coinductive_LCons @ A @ Y @ Ys ) ) ) ) ).
% llast_lappend_LCons
thf(fact_159_lstrict__prefix__def,axiom,
! [A: $tType] :
( ( coindu1478340336prefix @ A )
= ( ^ [Xs5: coinductive_llist @ A,Ys4: coinductive_llist @ A] :
( ( coinductive_lprefix @ A @ Xs5 @ Ys4 )
& ( Xs5 != Ys4 ) ) ) ) ).
% lstrict_prefix_def
thf(fact_160_lappend__eq__lappend__conv,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Us: coinductive_llist @ A,Ys: coinductive_llist @ A,Vs: coinductive_llist @ A] :
( ( ( coinductive_llength @ A @ Xs )
= ( coinductive_llength @ A @ Us ) )
=> ( ( ( coinductive_lappend @ A @ Xs @ Ys )
= ( coinductive_lappend @ A @ Us @ Vs ) )
= ( ( Xs = Us )
& ( ( coinductive_lfinite @ A @ Xs )
=> ( Ys = Vs ) ) ) ) ) ).
% lappend_eq_lappend_conv
thf(fact_161_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_162_lprefix__lappend__sameI,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A] :
( ( coinductive_lprefix @ A @ Xs @ Ys )
=> ( coinductive_lprefix @ A @ ( coinductive_lappend @ A @ Zs @ Xs ) @ ( coinductive_lappend @ A @ Zs @ Ys ) ) ) ).
% lprefix_lappend_sameI
thf(fact_163_lprefix__conv__lappend,axiom,
! [A: $tType] :
( ( coinductive_lprefix @ A )
= ( ^ [Xs5: coinductive_llist @ A,Ys4: coinductive_llist @ A] :
? [Zs2: coinductive_llist @ A] :
( Ys4
= ( coinductive_lappend @ A @ Xs5 @ Zs2 ) ) ) ) ).
% lprefix_conv_lappend
thf(fact_164_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_165_llist_Oleq__antisym,axiom,
! [A: $tType,X: coinductive_llist @ A,Y: coinductive_llist @ A] :
( ( coinductive_lprefix @ A @ X @ Y )
=> ( ( coinductive_lprefix @ A @ Y @ X )
=> ( X = Y ) ) ) ).
% llist.leq_antisym
thf(fact_166_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_167_lprefix__lappendD,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A] :
( ( coinductive_lprefix @ A @ Xs @ ( coinductive_lappend @ A @ Ys @ Zs ) )
=> ( ( coinductive_lprefix @ A @ Xs @ Ys )
| ( coinductive_lprefix @ A @ Ys @ Xs ) ) ) ).
% lprefix_lappendD
thf(fact_168_lappend__lprefixE,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A] :
( ( coinductive_lprefix @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) @ Zs )
=> ~ ! [Zs3: coinductive_llist @ A] :
( Zs
!= ( coinductive_lappend @ A @ Xs @ Zs3 ) ) ) ).
% lappend_lprefixE
thf(fact_169_lprefix__lappend,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] : ( coinductive_lprefix @ A @ Xs @ ( coinductive_lappend @ A @ Xs @ Ys ) ) ).
% lprefix_lappend
thf(fact_170_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_171_llist_Oleq__trans,axiom,
! [A: $tType,X: coinductive_llist @ A,Y: coinductive_llist @ A,Z: coinductive_llist @ A] :
( ( coinductive_lprefix @ A @ X @ Y )
=> ( ( coinductive_lprefix @ A @ Y @ Z )
=> ( coinductive_lprefix @ A @ X @ Z ) ) ) ).
% llist.leq_trans
thf(fact_172_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_173_lappend__assoc,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A] :
( ( coinductive_lappend @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) @ Zs )
= ( coinductive_lappend @ A @ Xs @ ( coinductive_lappend @ A @ Ys @ Zs ) ) ) ).
% lappend_assoc
thf(fact_174_lappend__inf,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ~ ( coinductive_lfinite @ A @ Xs )
=> ( ( coinductive_lappend @ A @ Xs @ Ys )
= Xs ) ) ).
% lappend_inf
thf(fact_175_lprefix__imp__llexord,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,R2: A > A > $o] :
( ( coinductive_lprefix @ A @ Xs @ Ys )
=> ( coinductive_llexord @ A @ R2 @ Xs @ Ys ) ) ).
% lprefix_imp_llexord
thf(fact_176_lmirror__aux__acc,axiom,
! [A: $tType,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A,Xs: coinductive_llist @ A] :
( ( lMirro999291890or_aux @ A @ ( coinductive_lappend @ A @ Ys @ Zs ) @ Xs )
= ( coinductive_lappend @ A @ ( lMirro999291890or_aux @ A @ Ys @ Xs ) @ Zs ) ) ).
% lmirror_aux_acc
thf(fact_177_lmirror__aux__inf,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Acc: coinductive_llist @ A] :
( ~ ( coinductive_lfinite @ A @ Xs )
=> ( ( lMirro999291890or_aux @ A @ Acc @ Xs )
= Xs ) ) ).
% lmirror_aux_inf
thf(fact_178_lprefix__ltakeWhile,axiom,
! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] : ( coinductive_lprefix @ A @ ( coindu501562517eWhile @ A @ P @ Xs ) @ Xs ) ).
% lprefix_ltakeWhile
thf(fact_179_ord_Olsorted__lprefixD,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Less_eq: A > A > $o] :
( ( coinductive_lprefix @ A @ Xs @ Ys )
=> ( ( coinductive_lsorted @ A @ Less_eq @ Ys )
=> ( coinductive_lsorted @ A @ Less_eq @ Xs ) ) ) ).
% ord.lsorted_lprefixD
thf(fact_180_llexord__append__right,axiom,
! [A: $tType,R2: A > A > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] : ( coinductive_llexord @ A @ R2 @ Xs @ ( coinductive_lappend @ A @ Xs @ Ys ) ) ).
% llexord_append_right
thf(fact_181_llexord__lappend__leftI,axiom,
! [A: $tType,R2: A > A > $o,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A,Xs: coinductive_llist @ A] :
( ( coinductive_llexord @ A @ R2 @ Ys @ Zs )
=> ( coinductive_llexord @ A @ R2 @ ( coinductive_lappend @ A @ Xs @ Ys ) @ ( coinductive_lappend @ A @ Xs @ Zs ) ) ) ).
% llexord_lappend_leftI
thf(fact_182_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_183_lprefix__llength__eq__imp__eq,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_lprefix @ A @ Xs @ Ys )
=> ( ( ( coinductive_llength @ A @ Xs )
= ( coinductive_llength @ A @ Ys ) )
=> ( Xs = Ys ) ) ) ).
% lprefix_llength_eq_imp_eq
thf(fact_184_LNil__lprefix,axiom,
! [A: $tType,Xs: coinductive_llist @ A] : ( coinductive_lprefix @ A @ ( coinductive_LNil @ A ) @ Xs ) ).
% LNil_lprefix
thf(fact_185_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_186_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_187_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_188_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_189_Le__LCons,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,X: A] :
( ( coinductive_lprefix @ A @ Xs @ Ys )
=> ( coinductive_lprefix @ A @ ( coinductive_LCons @ A @ X @ Xs ) @ ( coinductive_LCons @ A @ X @ Ys ) ) ) ).
% Le_LCons
thf(fact_190_LCons__lprefix__conv,axiom,
! [A: $tType,X: A,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_lprefix @ A @ ( coinductive_LCons @ A @ X @ Xs ) @ Ys )
= ( ? [Ys5: coinductive_llist @ A] :
( ( Ys
= ( coinductive_LCons @ A @ X @ Ys5 ) )
& ( coinductive_lprefix @ A @ Xs @ Ys5 ) ) ) ) ).
% LCons_lprefix_conv
thf(fact_191_lappend__LNil__LNil,axiom,
! [A: $tType] :
( ( coinductive_lappend @ A @ ( coinductive_LNil @ A ) @ ( coinductive_LNil @ A ) )
= ( coinductive_LNil @ A ) ) ).
% lappend_LNil_LNil
thf(fact_192_LNil__eq__lappend__iff,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( ( coinductive_LNil @ A )
= ( coinductive_lappend @ A @ Xs @ Ys ) )
= ( ( Xs
= ( coinductive_LNil @ A ) )
& ( Ys
= ( coinductive_LNil @ A ) ) ) ) ).
% LNil_eq_lappend_iff
thf(fact_193_lappend__eq__LNil__iff,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( ( coinductive_lappend @ A @ Xs @ Ys )
= ( coinductive_LNil @ A ) )
= ( ( Xs
= ( coinductive_LNil @ A ) )
& ( Ys
= ( coinductive_LNil @ A ) ) ) ) ).
% lappend_eq_LNil_iff
thf(fact_194_lfinite__LNil,axiom,
! [A: $tType] : ( coinductive_lfinite @ A @ ( coinductive_LNil @ A ) ) ).
% lfinite_LNil
thf(fact_195_lappend_Odisc_I2_J,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( ~ ( coinductive_lnull @ A @ Xs )
| ~ ( coinductive_lnull @ A @ Ys ) )
=> ~ ( coinductive_lnull @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) ) ) ).
% lappend.disc(2)
thf(fact_196_lappend_Odisc_I1_J,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ Xs )
=> ( ( coinductive_lnull @ A @ Ys )
=> ( coinductive_lnull @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) ) ) ) ).
% lappend.disc(1)
thf(fact_197_lappend__lnull1,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ Xs )
=> ( ( coinductive_lappend @ A @ Xs @ Ys )
= Ys ) ) ).
% lappend_lnull1
thf(fact_198_lappend__lnull2,axiom,
! [A: $tType,Ys: coinductive_llist @ A,Xs: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ Ys )
=> ( ( coinductive_lappend @ A @ Xs @ Ys )
= Xs ) ) ).
% lappend_lnull2
thf(fact_199_lnull__imp__lfinite,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ Xs )
=> ( coinductive_lfinite @ A @ Xs ) ) ).
% lnull_imp_lfinite
thf(fact_200_lfinite__LConsI,axiom,
! [A: $tType,Xs: coinductive_llist @ A,X: A] :
( ( coinductive_lfinite @ A @ Xs )
=> ( coinductive_lfinite @ A @ ( coinductive_LCons @ A @ X @ Xs ) ) ) ).
% lfinite_LConsI
thf(fact_201_lfinite__rev__induct,axiom,
! [A: $tType,Xs: coinductive_llist @ A,P: ( coinductive_llist @ A ) > $o] :
( ( coinductive_lfinite @ A @ Xs )
=> ( ( P @ ( coinductive_LNil @ A ) )
=> ( ! [X4: A,Xs3: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ Xs3 )
=> ( ( P @ Xs3 )
=> ( P @ ( coinductive_lappend @ A @ Xs3 @ ( coinductive_LCons @ A @ X4 @ ( coinductive_LNil @ A ) ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% lfinite_rev_induct
thf(fact_202_Coinductive__List_Olprefix__nitpick__simps,axiom,
! [A: $tType] :
( ( coinductive_lprefix @ A )
= ( ^ [Xs5: coinductive_llist @ A,Ys4: coinductive_llist @ A] :
( ( ( coinductive_lfinite @ A @ Xs5 )
=> ( coindu328551480prefix @ A @ Xs5 @ Ys4 ) )
& ( ~ ( coinductive_lfinite @ A @ Xs5 )
=> ( Xs5 = Ys4 ) ) ) ) ) ).
% Coinductive_List.lprefix_nitpick_simps
thf(fact_203_lstrict__prefix__lfinite1,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coindu1478340336prefix @ A @ Xs @ Ys )
=> ( coinductive_lfinite @ A @ Xs ) ) ).
% lstrict_prefix_lfinite1
thf(fact_204_lstrict__prefix__lappend__conv,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coindu1478340336prefix @ A @ Xs @ ( coinductive_lappend @ A @ Xs @ Ys ) )
= ( ( coinductive_lfinite @ A @ Xs )
& ~ ( coinductive_lnull @ A @ Ys ) ) ) ).
% lstrict_prefix_lappend_conv
thf(fact_205_Coinductive__List_Ofinite__lprefix__def,axiom,
! [A: $tType] :
( ( coindu328551480prefix @ A )
= ( coinductive_lprefix @ A ) ) ).
% Coinductive_List.finite_lprefix_def
thf(fact_206_llexord__conv,axiom,
! [A: $tType] :
( ( coinductive_llexord @ A )
= ( ^ [R3: A > A > $o,Xs5: coinductive_llist @ A,Ys4: coinductive_llist @ A] :
( ( Xs5 = Ys4 )
| ? [Zs2: coinductive_llist @ A,Xs2: coinductive_llist @ A,Y4: A,Ys5: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ Zs2 )
& ( Xs5
= ( coinductive_lappend @ A @ Zs2 @ Xs2 ) )
& ( Ys4
= ( coinductive_lappend @ A @ Zs2 @ ( coinductive_LCons @ A @ Y4 @ Ys5 ) ) )
& ( ( Xs2
= ( coinductive_LNil @ A ) )
| ( R3 @ ( coinductive_lhd @ A @ Xs2 ) @ Y4 ) ) ) ) ) ) ).
% llexord_conv
thf(fact_207_lfinite_Oinducts,axiom,
! [A: $tType,X: coinductive_llist @ A,P: ( coinductive_llist @ A ) > $o] :
( ( coinductive_lfinite @ A @ X )
=> ( ( P @ ( coinductive_LNil @ A ) )
=> ( ! [Xs3: coinductive_llist @ A,X4: A] :
( ( coinductive_lfinite @ A @ Xs3 )
=> ( ( P @ Xs3 )
=> ( P @ ( coinductive_LCons @ A @ X4 @ Xs3 ) ) ) )
=> ( P @ X ) ) ) ) ).
% lfinite.inducts
thf(fact_208_lfinite_Osimps,axiom,
! [A: $tType] :
( ( coinductive_lfinite @ A )
= ( ^ [A3: coinductive_llist @ A] :
( ( A3
= ( coinductive_LNil @ A ) )
| ? [Xs5: coinductive_llist @ A,X2: A] :
( ( A3
= ( coinductive_LCons @ A @ X2 @ Xs5 ) )
& ( coinductive_lfinite @ A @ Xs5 ) ) ) ) ) ).
% lfinite.simps
thf(fact_209_lfinite_Ocases,axiom,
! [A: $tType,A2: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ A2 )
=> ( ( A2
!= ( coinductive_LNil @ A ) )
=> ~ ! [Xs3: coinductive_llist @ A] :
( ? [X4: A] :
( A2
= ( coinductive_LCons @ A @ X4 @ Xs3 ) )
=> ~ ( coinductive_lfinite @ A @ Xs3 ) ) ) ) ).
% lfinite.cases
thf(fact_210_lappend__snocL1__conv__LCons2,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Y: A,Ys: coinductive_llist @ A] :
( ( coinductive_lappend @ A @ ( coinductive_lappend @ A @ Xs @ ( coinductive_LCons @ A @ Y @ ( coinductive_LNil @ A ) ) ) @ Ys )
= ( coinductive_lappend @ A @ Xs @ ( coinductive_LCons @ A @ Y @ Ys ) ) ) ).
% lappend_snocL1_conv_LCons2
thf(fact_211_lappend_Octr_I1_J,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ Xs )
=> ( ( coinductive_lnull @ A @ Ys )
=> ( ( coinductive_lappend @ A @ Xs @ Ys )
= ( coinductive_LNil @ A ) ) ) ) ).
% lappend.ctr(1)
thf(fact_212_lfinite__induct,axiom,
! [A: $tType,Xs: coinductive_llist @ A,P: ( coinductive_llist @ A ) > $o] :
( ( coinductive_lfinite @ A @ Xs )
=> ( ! [Xs3: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ Xs3 )
=> ( P @ Xs3 ) )
=> ( ! [Xs3: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ Xs3 )
=> ( ~ ( coinductive_lnull @ A @ Xs3 )
=> ( ( P @ ( coinductive_ltl @ A @ Xs3 ) )
=> ( P @ Xs3 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% lfinite_induct
thf(fact_213_lappend__ltl,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ~ ( coinductive_lnull @ A @ Xs )
=> ( ( coinductive_lappend @ A @ ( coinductive_ltl @ A @ Xs ) @ Ys )
= ( coinductive_ltl @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) ) ) ) ).
% lappend_ltl
thf(fact_214_lprefix__LCons__conv,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Y: A,Ys: coinductive_llist @ A] :
( ( coinductive_lprefix @ A @ Xs @ ( coinductive_LCons @ A @ Y @ Ys ) )
= ( ( Xs
= ( coinductive_LNil @ A ) )
| ? [Xs2: coinductive_llist @ A] :
( ( Xs
= ( coinductive_LCons @ A @ Y @ Xs2 ) )
& ( coinductive_lprefix @ A @ Xs2 @ Ys ) ) ) ) ).
% lprefix_LCons_conv
thf(fact_215_lprefix_Ocoinduct,axiom,
! [A: $tType,X3: ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o,X: coinductive_llist @ A,Xa: coinductive_llist @ A] :
( ( X3 @ X @ Xa )
=> ( ! [X4: coinductive_llist @ A,Xa2: coinductive_llist @ A] :
( ( X3 @ X4 @ Xa2 )
=> ( ? [Xs4: coinductive_llist @ A] :
( ( X4
= ( coinductive_LNil @ A ) )
& ( Xa2 = Xs4 ) )
| ? [Xs4: coinductive_llist @ A,Ys3: coinductive_llist @ A,Xb: A] :
( ( X4
= ( coinductive_LCons @ A @ Xb @ Xs4 ) )
& ( Xa2
= ( coinductive_LCons @ A @ Xb @ Ys3 ) )
& ( ( X3 @ Xs4 @ Ys3 )
| ( coinductive_lprefix @ A @ Xs4 @ Ys3 ) ) ) ) )
=> ( coinductive_lprefix @ A @ X @ Xa ) ) ) ).
% lprefix.coinduct
thf(fact_216_lprefix_Osimps,axiom,
! [A: $tType] :
( ( coinductive_lprefix @ A )
= ( ^ [A1: coinductive_llist @ A,A22: coinductive_llist @ A] :
( ? [Xs5: coinductive_llist @ A] :
( ( A1
= ( coinductive_LNil @ A ) )
& ( A22 = Xs5 ) )
| ? [Xs5: coinductive_llist @ A,Ys4: coinductive_llist @ A,X2: A] :
( ( A1
= ( coinductive_LCons @ A @ X2 @ Xs5 ) )
& ( A22
= ( coinductive_LCons @ A @ X2 @ Ys4 ) )
& ( coinductive_lprefix @ A @ Xs5 @ Ys4 ) ) ) ) ) ).
% lprefix.simps
thf(fact_217_lprefix_Ocases,axiom,
! [A: $tType,A12: coinductive_llist @ A,A23: coinductive_llist @ A] :
( ( coinductive_lprefix @ A @ A12 @ A23 )
=> ( ( ( A12
= ( coinductive_LNil @ A ) )
=> ! [Xs3: coinductive_llist @ A] : A23 != Xs3 )
=> ~ ! [Xs3: coinductive_llist @ A,Ys2: coinductive_llist @ A,X4: A] :
( ( A12
= ( coinductive_LCons @ A @ X4 @ Xs3 ) )
=> ( ( A23
= ( coinductive_LCons @ A @ X4 @ Ys2 ) )
=> ~ ( coinductive_lprefix @ A @ Xs3 @ Ys2 ) ) ) ) ) ).
% lprefix.cases
thf(fact_218_lprefix__code_I2_J,axiom,
! [A: $tType,X: A,Xs: coinductive_llist @ A] :
~ ( coinductive_lprefix @ A @ ( coinductive_LCons @ A @ X @ Xs ) @ ( coinductive_LNil @ A ) ) ).
% lprefix_code(2)
thf(fact_219_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_220_llistsum__LCons,axiom,
! [A: $tType] :
( ( monoid_add @ A @ ( type @ A ) )
=> ! [Xs: coinductive_llist @ A,X: A] :
( ( coinductive_lfinite @ A @ Xs )
=> ( ( coindu780009021istsum @ A @ ( coinductive_LCons @ A @ X @ Xs ) )
= ( plus_plus @ A @ X @ ( coindu780009021istsum @ A @ Xs ) ) ) ) ) ).
% llistsum_LCons
thf(fact_221_llcp__lappend__same,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A] :
( ( coinductive_llcp @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) @ ( coinductive_lappend @ A @ Xs @ Zs ) )
= ( plus_plus @ extended_enat @ ( coinductive_llength @ A @ Xs ) @ ( coinductive_llcp @ A @ Ys @ Zs ) ) ) ).
% llcp_lappend_same
thf(fact_222_llistsum__lappend,axiom,
! [A: $tType] :
( ( monoid_add @ A @ ( type @ A ) )
=> ! [Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ Xs )
=> ( ( coinductive_lfinite @ A @ Ys )
=> ( ( coindu780009021istsum @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
= ( plus_plus @ A @ ( coindu780009021istsum @ A @ Xs ) @ ( coindu780009021istsum @ A @ Ys ) ) ) ) ) ) ).
% llistsum_lappend
thf(fact_223_llast__lappend,axiom,
! [A: $tType,Ys: coinductive_llist @ A,Xs: coinductive_llist @ A] :
( ( ( coinductive_lnull @ A @ Ys )
=> ( ( coinductive_llast @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
= ( coinductive_llast @ A @ Xs ) ) )
& ( ~ ( coinductive_lnull @ A @ Ys )
=> ( ( ( coinductive_lfinite @ A @ Xs )
=> ( ( coinductive_llast @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
= ( coinductive_llast @ A @ Ys ) ) )
& ( ~ ( coinductive_lfinite @ A @ Xs )
=> ( ( coinductive_llast @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
= ( undefined @ A ) ) ) ) ) ) ).
% llast_lappend
thf(fact_224_llcp__same__conv__length,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( coinductive_llcp @ A @ Xs @ Xs )
= ( coinductive_llength @ A @ Xs ) ) ).
% llcp_same_conv_length
thf(fact_225_llcp__lprefix1,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_lprefix @ A @ Xs @ Ys )
=> ( ( coinductive_llcp @ A @ Xs @ Ys )
= ( coinductive_llength @ A @ Xs ) ) ) ).
% llcp_lprefix1
thf(fact_226_llcp__lprefix2,axiom,
! [A: $tType,Ys: coinductive_llist @ A,Xs: coinductive_llist @ A] :
( ( coinductive_lprefix @ A @ Ys @ Xs )
=> ( ( coinductive_llcp @ A @ Xs @ Ys )
= ( coinductive_llength @ A @ Ys ) ) ) ).
% llcp_lprefix2
thf(fact_227_llcp__commute,axiom,
! [A: $tType] :
( ( coinductive_llcp @ A )
= ( ^ [Xs5: coinductive_llist @ A,Ys4: coinductive_llist @ A] : ( coinductive_llcp @ A @ Ys4 @ Xs5 ) ) ) ).
% llcp_commute
thf(fact_228_llast__LNil,axiom,
! [A: $tType] :
( ( coinductive_llast @ A @ ( coinductive_LNil @ A ) )
= ( undefined @ A ) ) ).
% llast_LNil
thf(fact_229_llast__linfinite,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ~ ( coinductive_lfinite @ A @ Xs )
=> ( ( coinductive_llast @ A @ Xs )
= ( undefined @ A ) ) ) ).
% llast_linfinite
thf(fact_230_epred__llcp,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_epred @ ( coinductive_llcp @ A @ Xs @ Ys ) )
= ( coinductive_llcp @ A @ ( coinductive_ltl @ A @ Xs ) @ ( coinductive_ltl @ A @ Ys ) ) ) ) ) ) ).
% epred_llcp
thf(fact_231_ltl__lconcat,axiom,
! [A: $tType,Xss: coinductive_llist @ ( coinductive_llist @ A )] :
( ~ ( coinductive_lnull @ ( coinductive_llist @ A ) @ Xss )
=> ( ~ ( coinductive_lnull @ A @ ( coinductive_lhd @ ( coinductive_llist @ A ) @ Xss ) )
=> ( ( coinductive_ltl @ A @ ( coinductive_lconcat @ A @ Xss ) )
= ( coinductive_lappend @ A @ ( coinductive_ltl @ A @ ( coinductive_lhd @ ( coinductive_llist @ A ) @ Xss ) ) @ ( coinductive_lconcat @ A @ ( coinductive_ltl @ ( coinductive_llist @ A ) @ Xss ) ) ) ) ) ) ).
% ltl_lconcat
thf(fact_232_lconcat__LNil,axiom,
! [A: $tType] :
( ( coinductive_lconcat @ A @ ( coinductive_LNil @ ( coinductive_llist @ A ) ) )
= ( coinductive_LNil @ A ) ) ).
% lconcat_LNil
thf(fact_233_lconcat__LCons,axiom,
! [B: $tType,Xs: coinductive_llist @ B,Xss: coinductive_llist @ ( coinductive_llist @ B )] :
( ( coinductive_lconcat @ B @ ( coinductive_LCons @ ( coinductive_llist @ B ) @ Xs @ Xss ) )
= ( coinductive_lappend @ B @ Xs @ ( coinductive_lconcat @ B @ Xss ) ) ) ).
% lconcat_LCons
thf(fact_234_lconcat__lappend,axiom,
! [A: $tType,Xss: coinductive_llist @ ( coinductive_llist @ A ),Yss: coinductive_llist @ ( coinductive_llist @ A )] :
( ( coinductive_lfinite @ ( coinductive_llist @ A ) @ Xss )
=> ( ( coinductive_lconcat @ A @ ( coinductive_lappend @ ( coinductive_llist @ A ) @ Xss @ Yss ) )
= ( coinductive_lappend @ A @ ( coinductive_lconcat @ A @ Xss ) @ ( coinductive_lconcat @ A @ Yss ) ) ) ) ).
% lconcat_lappend
thf(fact_235_lhd__lconcat,axiom,
! [A: $tType,Xss: coinductive_llist @ ( coinductive_llist @ A )] :
( ~ ( coinductive_lnull @ ( coinductive_llist @ A ) @ Xss )
=> ( ~ ( coinductive_lnull @ A @ ( coinductive_lhd @ ( coinductive_llist @ A ) @ Xss ) )
=> ( ( coinductive_lhd @ A @ ( coinductive_lconcat @ A @ Xss ) )
= ( coinductive_lhd @ A @ ( coinductive_lhd @ ( coinductive_llist @ A ) @ Xss ) ) ) ) ) ).
% lhd_lconcat
thf(fact_236_epred__llength,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( coinductive_epred @ ( coinductive_llength @ A @ Xs ) )
= ( coinductive_llength @ A @ ( coinductive_ltl @ A @ Xs ) ) ) ).
% epred_llength
thf(fact_237_lprefix__lconcatI,axiom,
! [A: $tType,Xss: coinductive_llist @ ( coinductive_llist @ A ),Yss: coinductive_llist @ ( coinductive_llist @ A )] :
( ( coinductive_lprefix @ ( coinductive_llist @ A ) @ Xss @ Yss )
=> ( coinductive_lprefix @ A @ ( coinductive_lconcat @ A @ Xss ) @ ( coinductive_lconcat @ A @ Yss ) ) ) ).
% lprefix_lconcatI
thf(fact_238_llcp__eq__0__iff,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( ( coinductive_llcp @ A @ Xs @ Ys )
= ( zero_zero @ extended_enat ) )
= ( ( coinductive_lnull @ A @ Xs )
| ( coinductive_lnull @ A @ Ys )
| ( ( coinductive_lhd @ A @ Xs )
!= ( coinductive_lhd @ A @ Ys ) ) ) ) ).
% llcp_eq_0_iff
thf(fact_239_llistsum__inf,axiom,
! [A: $tType] :
( ( monoid_add @ A @ ( type @ A ) )
=> ! [Xs: coinductive_llist @ A] :
( ~ ( coinductive_lfinite @ A @ Xs )
=> ( ( coindu780009021istsum @ A @ Xs )
= ( zero_zero @ A ) ) ) ) ).
% llistsum_inf
thf(fact_240_add__cancel__right__right,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A @ ( type @ A ) )
=> ! [A2: A,B2: A] :
( ( A2
= ( plus_plus @ A @ A2 @ B2 ) )
= ( B2
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_right_right
thf(fact_241_add__cancel__right__left,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A @ ( type @ A ) )
=> ! [A2: A,B2: A] :
( ( A2
= ( plus_plus @ A @ B2 @ A2 ) )
= ( B2
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_right_left
thf(fact_242_add__cancel__left__right,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A @ ( type @ A ) )
=> ! [A2: A,B2: A] :
( ( ( plus_plus @ A @ A2 @ B2 )
= A2 )
= ( B2
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_left_right
thf(fact_243_add__cancel__left__left,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A @ ( type @ A ) )
=> ! [B2: A,A2: A] :
( ( ( plus_plus @ A @ B2 @ A2 )
= A2 )
= ( B2
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_left_left
thf(fact_244_double__zero__sym,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A @ ( type @ A ) )
=> ! [A2: A] :
( ( ( zero_zero @ A )
= ( plus_plus @ A @ A2 @ A2 ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% double_zero_sym
thf(fact_245_double__zero,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A @ ( type @ A ) )
=> ! [A2: A] :
( ( ( plus_plus @ A @ A2 @ A2 )
= ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% double_zero
thf(fact_246_add_Oright__neutral,axiom,
! [A: $tType] :
( ( monoid_add @ A @ ( type @ A ) )
=> ! [A2: A] :
( ( plus_plus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% add.right_neutral
thf(fact_247_add_Oleft__neutral,axiom,
! [A: $tType] :
( ( monoid_add @ A @ ( type @ A ) )
=> ! [A2: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
= A2 ) ) ).
% add.left_neutral
thf(fact_248_llength__eq__0,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( ( coinductive_llength @ A @ Xs )
= ( zero_zero @ extended_enat ) )
= ( coinductive_lnull @ A @ Xs ) ) ).
% llength_eq_0
thf(fact_249_llength__lnull,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ Xs )
=> ( ( coinductive_llength @ A @ Xs )
= ( zero_zero @ extended_enat ) ) ) ).
% llength_lnull
thf(fact_250_llength__LNil,axiom,
! [A: $tType] :
( ( coinductive_llength @ A @ ( coinductive_LNil @ A ) )
= ( zero_zero @ extended_enat ) ) ).
% llength_LNil
thf(fact_251_llistsum__LNil,axiom,
! [A: $tType] :
( ( monoid_add @ A @ ( type @ A ) )
=> ( ( coindu780009021istsum @ A @ ( coinductive_LNil @ A ) )
= ( zero_zero @ A ) ) ) ).
% llistsum_LNil
thf(fact_252_llcp__LNil2,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( coinductive_llcp @ A @ Xs @ ( coinductive_LNil @ A ) )
= ( zero_zero @ extended_enat ) ) ).
% llcp_LNil2
%----Type constructors (6)
thf(tcon_Extended__Nat_Oenat___Groups_Oordered__ab__semigroup__add,axiom,
ordere779506340up_add @ extended_enat @ ( type @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Groups_Oab__semigroup__add,axiom,
ab_semigroup_add @ extended_enat @ ( type @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring__1,axiom,
comm_semiring_1 @ extended_enat @ ( type @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Groups_Osemigroup__add,axiom,
semigroup_add @ extended_enat @ ( type @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Groups_Omonoid__add,axiom,
monoid_add @ extended_enat @ ( type @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Groups_Ozero,axiom,
zero @ extended_enat @ ( type @ extended_enat ) ).
%----Helper facts (3)
thf(help_If_3_1_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_T,axiom,
! [A: $tType,X: A,Y: A] :
( ( if @ A @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X: A,Y: A] :
( ( if @ A @ $true @ X @ Y )
= X ) ).
%----Conjectures (3)
thf(conj_0,hypothesis,
~ ( coinductive_lnull @ a @ xsa ) ).
thf(conj_1,hypothesis,
( ( coinductive_llength @ a @ ( lMirro999291890or_aux @ a @ ( coinductive_LCons @ a @ ( coinductive_lhd @ a @ xsa ) @ acca ) @ ( coinductive_ltl @ a @ xsa ) ) )
!= ( plus_plus @ extended_enat @ ( plus_plus @ extended_enat @ ( coinductive_llength @ a @ ( coinductive_ltl @ a @ xsa ) ) @ ( coinductive_llength @ a @ xsa ) ) @ ( coinductive_llength @ a @ acca ) ) ) ).
thf(conj_2,conjecture,
( ( plus_plus @ extended_enat @ ( plus_plus @ extended_enat @ ( coinductive_llength @ a @ ( coinductive_ltl @ a @ xsa ) ) @ ( coinductive_llength @ a @ xsa ) ) @ ( coinductive_llength @ a @ acca ) )
= ( plus_plus @ extended_enat @ ( plus_plus @ extended_enat @ ( coinductive_llength @ a @ ( coinductive_ltl @ a @ xsa ) ) @ ( coinductive_llength @ a @ ( coinductive_ltl @ a @ xsa ) ) ) @ ( coinductive_llength @ a @ ( coinductive_LCons @ a @ ( coinductive_lhd @ a @ xsa ) @ acca ) ) ) ) ).
%------------------------------------------------------------------------------