TPTP Problem File: DAT146^1.p
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%------------------------------------------------------------------------------
% File : DAT146^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Data Structures
% Problem : Coinductive stream 133
% Version : [Bla16] axioms : Especial.
% English :
% Refs : [Loc10] Lochbihler (2010), Coinductive
% : [RB15] Reynolds & Blanchette (2015), A Decision Procedure for
% : [Bla16] Blanchette (2016), Email to Geoff Sutcliffe
% Source : [Bla16]
% Names : coinductive_stream__133.p [Bla16]
% Status : Theorem
% Rating : 1.00 v7.5.0, 0.33 v7.2.0, 0.50 v7.1.0
% Syntax : Number of formulae : 299 ( 162 unt; 48 typ; 0 def)
% Number of atoms : 562 ( 316 equ; 4 cnn)
% Maximal formula atoms : 8 ( 2 avg)
% Number of connectives : 4112 ( 149 ~; 43 |; 72 &;3636 @)
% ( 0 <=>; 212 =>; 0 <=; 0 <~>)
% Maximal formula depth : 28 ( 8 avg)
% Number of types : 3 ( 2 usr)
% Number of type conns : 329 ( 329 >; 0 *; 0 +; 0 <<)
% Number of symbols : 50 ( 46 usr; 4 con; 0-6 aty)
% Number of variables : 1135 ( 41 ^; 991 !; 46 ?;1135 :)
% ( 57 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 15:13:15.956
%------------------------------------------------------------------------------
%----Could-be-implicit typings (6)
thf(ty_t_Coinductive__List_Ollist,type,
coinductive_llist: $tType > $tType ).
thf(ty_t_Product__Type_Oprod,type,
product_prod: $tType > $tType > $tType ).
thf(ty_t_Stream_Ostream,type,
stream: $tType > $tType ).
thf(ty_t_Nat_Onat,type,
nat: $tType ).
thf(ty_t_itself,type,
itself: $tType > $tType ).
thf(ty_tf_a,type,
a: $tType ).
%----Explicit typings (42)
thf(sy_c_Coinductive__List_Ofinite__lprefix,type,
coindu328551480prefix:
!>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List_Oiterates,type,
coinductive_iterates:
!>[A: $tType] : ( ( A > A ) > A > ( coinductive_llist @ A ) ) ).
thf(sy_c_Coinductive__List_Olappend,type,
coinductive_lappend:
!>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
thf(sy_c_Coinductive__List_Olconcat,type,
coinductive_lconcat:
!>[A: $tType] : ( ( coinductive_llist @ ( coinductive_llist @ A ) ) > ( coinductive_llist @ A ) ) ).
thf(sy_c_Coinductive__List_Oldropn,type,
coinductive_ldropn:
!>[A: $tType] : ( nat > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
thf(sy_c_Coinductive__List_Olfinite,type,
coinductive_lfinite:
!>[A: $tType] : ( ( coinductive_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List_Ollast,type,
coinductive_llast:
!>[A: $tType] : ( ( coinductive_llist @ A ) > A ) ).
thf(sy_c_Coinductive__List_Ollexord,type,
coinductive_llexord:
!>[A: $tType] : ( ( A > A > $o ) > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List_Ollist_OLCons,type,
coinductive_LCons:
!>[A: $tType] : ( A > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
thf(sy_c_Coinductive__List_Ollist_OLNil,type,
coinductive_LNil:
!>[A: $tType] : ( coinductive_llist @ A ) ).
thf(sy_c_Coinductive__List_Ollist_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_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_Olzip,type,
coinductive_lzip:
!>[A: $tType,B: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ B ) > ( coinductive_llist @ ( product_prod @ A @ B ) ) ) ).
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__Stream__Mirabelle__dydkjoctes_Ollist__of__stream,type,
coindu1724414836stream:
!>[A: $tType] : ( ( stream @ A ) > ( coinductive_llist @ A ) ) ).
thf(sy_c_Coinductive__Stream__Mirabelle__dydkjoctes_Ostream__of__llist,type,
coindu2010755910_llist:
!>[A: $tType] : ( ( coinductive_llist @ A ) > ( stream @ A ) ) ).
thf(sy_c_Coinductive__Stream__Mirabelle__dydkjoctes_Ounfold__stream,type,
coindu139217191stream:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( A > A ) > A > ( stream @ B ) ) ).
thf(sy_c_Fun_Ocomp,type,
comp:
!>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( A > B ) > A > C ) ).
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_Linear__Temporal__Logic__on__Streams_Onxt,type,
linear1494993505on_nxt:
!>[A: $tType,B: $tType] : ( ( ( stream @ A ) > B ) > ( stream @ A ) > B ) ).
thf(sy_c_Product__Type_OPair,type,
product_Pair:
!>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
thf(sy_c_Product__Type_Ointernal__case__prod,type,
produc2004651681e_prod:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
product_rec_prod:
!>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
thf(sy_c_Pure_Otype,type,
type:
!>[A: $tType] : ( itself @ A ) ).
thf(sy_c_Stream_Osdrop__while,type,
sdrop_while:
!>[A: $tType] : ( ( A > $o ) > ( stream @ A ) > ( stream @ A ) ) ).
thf(sy_c_Stream_Osfilter,type,
sfilter:
!>[A: $tType] : ( ( A > $o ) > ( stream @ A ) > ( stream @ A ) ) ).
thf(sy_c_Stream_Osinterleave,type,
sinterleave:
!>[A: $tType] : ( ( stream @ A ) > ( stream @ A ) > ( stream @ A ) ) ).
thf(sy_c_Stream_Osmap2,type,
smap2:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( stream @ A ) > ( stream @ B ) > ( stream @ C ) ) ).
thf(sy_c_Stream_Osmember,type,
smember:
!>[A: $tType] : ( A > ( stream @ A ) > $o ) ).
thf(sy_c_Stream_Ostream_OSCons,type,
sCons:
!>[A: $tType] : ( A > ( stream @ A ) > ( stream @ A ) ) ).
thf(sy_c_Stream_Ostream_Ocase__stream,type,
case_stream:
!>[A: $tType,B: $tType] : ( ( A > ( stream @ A ) > B ) > ( stream @ A ) > B ) ).
thf(sy_c_Stream_Ostream_Oshd,type,
shd:
!>[A: $tType] : ( ( stream @ A ) > A ) ).
thf(sy_c_Stream_Ostream_Ostl,type,
stl:
!>[A: $tType] : ( ( stream @ A ) > ( stream @ A ) ) ).
thf(sy_c_Stream_Oszip,type,
szip:
!>[A: $tType,B: $tType] : ( ( stream @ A ) > ( stream @ B ) > ( stream @ ( product_prod @ A @ B ) ) ) ).
thf(sy_v_xs,type,
xs: coinductive_llist @ a ).
%----Relevant facts (246)
thf(fact_0_stream__of__llist__llist__of__stream,axiom,
! [A: $tType,Xs: stream @ A] :
( ( coindu2010755910_llist @ A @ ( coindu1724414836stream @ A @ Xs ) )
= Xs ) ).
% stream_of_llist_llist_of_stream
thf(fact_1_lnull__llist__of__stream,axiom,
! [A: $tType,Xs: stream @ A] :
~ ( coinductive_lnull @ A @ ( coindu1724414836stream @ A @ Xs ) ) ).
% lnull_llist_of_stream
thf(fact_2_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_3_shd__stream__of__llist,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( shd @ A @ ( coindu2010755910_llist @ A @ Xs ) )
= ( coinductive_lhd @ A @ Xs ) ) ).
% shd_stream_of_llist
thf(fact_4_stl__stream__of__llist,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( stl @ A @ ( coindu2010755910_llist @ A @ Xs ) )
= ( coindu2010755910_llist @ A @ ( coinductive_ltl @ A @ Xs ) ) ) ).
% stl_stream_of_llist
thf(fact_5_lhd__llist__of__stream,axiom,
! [A: $tType,Xs: stream @ A] :
( ( coinductive_lhd @ A @ ( coindu1724414836stream @ A @ Xs ) )
= ( shd @ A @ Xs ) ) ).
% lhd_llist_of_stream
thf(fact_6_lfinite__code_I1_J,axiom,
! [A: $tType] : ( coinductive_lfinite @ A @ ( coinductive_LNil @ A ) ) ).
% lfinite_code(1)
thf(fact_7_ltl__llist__of__stream,axiom,
! [A: $tType,Xs: stream @ A] :
( ( coinductive_ltl @ A @ ( coindu1724414836stream @ A @ Xs ) )
= ( coindu1724414836stream @ A @ ( stl @ A @ Xs ) ) ) ).
% ltl_llist_of_stream
thf(fact_8_lfinite__ldropn,axiom,
! [A: $tType,N: nat,Xs: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ ( coinductive_ldropn @ A @ N @ Xs ) )
= ( coinductive_lfinite @ A @ Xs ) ) ).
% lfinite_ldropn
thf(fact_9_lfinite__lzip,axiom,
! [A: $tType,B: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B] :
( ( coinductive_lfinite @ ( product_prod @ A @ B ) @ ( coinductive_lzip @ A @ B @ Xs @ Ys ) )
= ( ( coinductive_lfinite @ A @ Xs )
| ( coinductive_lfinite @ B @ Ys ) ) ) ).
% lfinite_lzip
thf(fact_10_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_11_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_12_lfinite__iterates,axiom,
! [A: $tType,F: A > A,X: A] :
~ ( coinductive_lfinite @ A @ ( coinductive_iterates @ A @ F @ X ) ) ).
% lfinite_iterates
thf(fact_13_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_14_lfinite__ltl,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ ( coinductive_ltl @ A @ Xs ) )
= ( coinductive_lfinite @ A @ Xs ) ) ).
% lfinite_ltl
thf(fact_15_lzip_Odisc__iff_I2_J,axiom,
! [A: $tType,B: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B] :
( ( ~ ( coinductive_lnull @ ( product_prod @ A @ B ) @ ( coinductive_lzip @ A @ B @ Xs @ Ys ) ) )
= ( ~ ( coinductive_lnull @ A @ Xs )
& ~ ( coinductive_lnull @ B @ Ys ) ) ) ).
% lzip.disc_iff(2)
thf(fact_16_lnull__lzip,axiom,
! [A: $tType,B: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B] :
( ( coinductive_lnull @ ( product_prod @ A @ B ) @ ( coinductive_lzip @ A @ B @ Xs @ Ys ) )
= ( ( coinductive_lnull @ A @ Xs )
| ( coinductive_lnull @ B @ Ys ) ) ) ).
% lnull_lzip
thf(fact_17_lzip__simps_I2_J,axiom,
! [D: $tType,C: $tType,Xs: coinductive_llist @ C] :
( ( coinductive_lzip @ C @ D @ Xs @ ( coinductive_LNil @ D ) )
= ( coinductive_LNil @ ( product_prod @ C @ D ) ) ) ).
% lzip_simps(2)
thf(fact_18_lzip__simps_I1_J,axiom,
! [B: $tType,A: $tType,Ys: coinductive_llist @ B] :
( ( coinductive_lzip @ A @ B @ ( coinductive_LNil @ A ) @ Ys )
= ( coinductive_LNil @ ( product_prod @ A @ B ) ) ) ).
% lzip_simps(1)
thf(fact_19_ldropn__LNil,axiom,
! [A: $tType,N: nat] :
( ( coinductive_ldropn @ A @ N @ ( coinductive_LNil @ A ) )
= ( coinductive_LNil @ A ) ) ).
% ldropn_LNil
thf(fact_20_ldropn__lzip,axiom,
! [A: $tType,B: $tType,N: nat,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B] :
( ( coinductive_ldropn @ ( product_prod @ A @ B ) @ N @ ( coinductive_lzip @ A @ B @ Xs @ Ys ) )
= ( coinductive_lzip @ A @ B @ ( coinductive_ldropn @ A @ N @ Xs ) @ ( coinductive_ldropn @ B @ N @ Ys ) ) ) ).
% ldropn_lzip
thf(fact_21_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_22_lstrict__prefix__code_I1_J,axiom,
! [A: $tType] :
~ ( coindu1478340336prefix @ A @ ( coinductive_LNil @ A ) @ ( coinductive_LNil @ A ) ) ).
% lstrict_prefix_code(1)
thf(fact_23_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_24_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_25_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_26_lzip_Octr_I1_J,axiom,
! [B: $tType,A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B] :
( ( ( coinductive_lnull @ A @ Xs )
| ( coinductive_lnull @ B @ Ys ) )
=> ( ( coinductive_lzip @ A @ B @ Xs @ Ys )
= ( coinductive_LNil @ ( product_prod @ A @ B ) ) ) ) ).
% lzip.ctr(1)
thf(fact_27_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_28_ltl__simps_I1_J,axiom,
! [A: $tType] :
( ( coinductive_ltl @ A @ ( coinductive_LNil @ A ) )
= ( coinductive_LNil @ A ) ) ).
% ltl_simps(1)
thf(fact_29_lzip_Odisc_I2_J,axiom,
! [A: $tType,B: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B] :
( ~ ( coinductive_lnull @ A @ Xs )
=> ( ~ ( coinductive_lnull @ B @ Ys )
=> ~ ( coinductive_lnull @ ( product_prod @ A @ B ) @ ( coinductive_lzip @ A @ B @ Xs @ Ys ) ) ) ) ).
% lzip.disc(2)
thf(fact_30_lzip_Odisc_I1_J,axiom,
! [A: $tType,B: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B] :
( ( ( coinductive_lnull @ A @ Xs )
| ( coinductive_lnull @ B @ Ys ) )
=> ( coinductive_lnull @ ( product_prod @ A @ B ) @ ( coinductive_lzip @ A @ B @ Xs @ Ys ) ) ) ).
% lzip.disc(1)
thf(fact_31_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_32_llist_Odisc_I1_J,axiom,
! [A: $tType] : ( coinductive_lnull @ A @ ( coinductive_LNil @ A ) ) ).
% llist.disc(1)
thf(fact_33_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_34_llist_OdiscI_I1_J,axiom,
! [A: $tType,Llist: coinductive_llist @ A] :
( ( Llist
= ( coinductive_LNil @ A ) )
=> ( coinductive_lnull @ A @ Llist ) ) ).
% llist.discI(1)
thf(fact_35_llist_Ocollapse_I1_J,axiom,
! [A: $tType,Llist: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ Llist )
=> ( Llist
= ( coinductive_LNil @ A ) ) ) ).
% llist.collapse(1)
thf(fact_36_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_37_ltl__lzip,axiom,
! [A: $tType,B: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B] :
( ~ ( coinductive_lnull @ A @ Xs )
=> ( ~ ( coinductive_lnull @ B @ Ys )
=> ( ( coinductive_ltl @ ( product_prod @ A @ B ) @ ( coinductive_lzip @ A @ B @ Xs @ Ys ) )
= ( coinductive_lzip @ A @ B @ ( coinductive_ltl @ A @ Xs ) @ ( coinductive_ltl @ B @ Ys ) ) ) ) ) ).
% ltl_lzip
thf(fact_38_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_39_lhd__LCons,axiom,
! [A: $tType,X21: A,X22: coinductive_llist @ A] :
( ( coinductive_lhd @ A @ ( coinductive_LCons @ A @ X21 @ X22 ) )
= X21 ) ).
% lhd_LCons
thf(fact_40_lnull__ltlI,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ Xs )
=> ( coinductive_lnull @ A @ ( coinductive_ltl @ A @ Xs ) ) ) ).
% lnull_ltlI
thf(fact_41_ltl__ldropn,axiom,
! [A: $tType,N: nat,Xs: coinductive_llist @ A] :
( ( coinductive_ltl @ A @ ( coinductive_ldropn @ A @ N @ Xs ) )
= ( coinductive_ldropn @ A @ N @ ( coinductive_ltl @ A @ Xs ) ) ) ).
% ltl_ldropn
thf(fact_42_ldropn__lnull,axiom,
! [A: $tType,Xs: coinductive_llist @ A,N: nat] :
( ( coinductive_lnull @ A @ Xs )
=> ( ( coinductive_ldropn @ A @ N @ Xs )
= ( coinductive_LNil @ A ) ) ) ).
% ldropn_lnull
thf(fact_43_lhd__iterates,axiom,
! [A: $tType,F: A > A,X: A] :
( ( coinductive_lhd @ A @ ( coinductive_iterates @ A @ F @ X ) )
= X ) ).
% lhd_iterates
thf(fact_44_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X2: A] :
( ( F @ X2 )
= ( G @ X2 ) )
=> ( F = G ) ) ).
% ext
thf(fact_45_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_46_ltl__iterates,axiom,
! [A: $tType,F: A > A,X: A] :
( ( coinductive_ltl @ A @ ( coinductive_iterates @ A @ F @ X ) )
= ( coinductive_iterates @ A @ F @ ( F @ X ) ) ) ).
% ltl_iterates
thf(fact_47_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_48_iterates_Ocode,axiom,
! [A: $tType] :
( ( coinductive_iterates @ A )
= ( ^ [F2: A > A,X3: A] : ( coinductive_LCons @ A @ X3 @ ( coinductive_iterates @ A @ F2 @ ( F2 @ X3 ) ) ) ) ) ).
% iterates.code
thf(fact_49_lfinite_Ocases,axiom,
! [A: $tType,A2: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ A2 )
=> ( ( A2
!= ( coinductive_LNil @ A ) )
=> ~ ! [Xs2: coinductive_llist @ A] :
( ? [X2: A] :
( A2
= ( coinductive_LCons @ A @ X2 @ Xs2 ) )
=> ~ ( coinductive_lfinite @ A @ Xs2 ) ) ) ) ).
% lfinite.cases
thf(fact_50_lfinite_Osimps,axiom,
! [A: $tType] :
( ( coinductive_lfinite @ A )
= ( ^ [A3: coinductive_llist @ A] :
( ( A3
= ( coinductive_LNil @ A ) )
| ? [Xs3: coinductive_llist @ A,X3: A] :
( ( A3
= ( coinductive_LCons @ A @ X3 @ Xs3 ) )
& ( coinductive_lfinite @ A @ Xs3 ) ) ) ) ) ).
% lfinite.simps
thf(fact_51_llimit__induct,axiom,
! [A: $tType,P: ( coinductive_llist @ A ) > $o,Xs: coinductive_llist @ A] :
( ( P @ ( coinductive_LNil @ A ) )
=> ( ! [X2: A,Xs2: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ Xs2 )
=> ( ( P @ Xs2 )
=> ( P @ ( coinductive_LCons @ A @ X2 @ Xs2 ) ) ) )
=> ( ( ! [Ys2: coinductive_llist @ A] :
( ( coindu1478340336prefix @ A @ Ys2 @ Xs )
=> ( P @ Ys2 ) )
=> ( P @ Xs ) )
=> ( P @ Xs ) ) ) ) ).
% llimit_induct
thf(fact_52_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_53_neq__LNil__conv,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( Xs
!= ( coinductive_LNil @ A ) )
= ( ? [X3: A,Xs4: coinductive_llist @ A] :
( Xs
= ( coinductive_LCons @ A @ X3 @ Xs4 ) ) ) ) ).
% neq_LNil_conv
thf(fact_54_lfinite__induct,axiom,
! [A: $tType,Xs: coinductive_llist @ A,P: ( coinductive_llist @ A ) > $o] :
( ( coinductive_lfinite @ A @ Xs )
=> ( ! [Xs2: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ Xs2 )
=> ( P @ Xs2 ) )
=> ( ! [Xs2: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ Xs2 )
=> ( ~ ( coinductive_lnull @ A @ Xs2 )
=> ( ( P @ ( coinductive_ltl @ A @ Xs2 ) )
=> ( P @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% lfinite_induct
thf(fact_55_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_56_not__lnull__conv,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( ~ ( coinductive_lnull @ A @ Xs ) )
= ( ? [X3: A,Xs4: coinductive_llist @ A] :
( Xs
= ( coinductive_LCons @ A @ X3 @ Xs4 ) ) ) ) ).
% not_lnull_conv
thf(fact_57_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_58_lfinite_Oinducts,axiom,
! [A: $tType,X: coinductive_llist @ A,P: ( coinductive_llist @ A ) > $o] :
( ( coinductive_lfinite @ A @ X )
=> ( ( P @ ( coinductive_LNil @ A ) )
=> ( ! [Xs2: coinductive_llist @ A,X2: A] :
( ( coinductive_lfinite @ A @ Xs2 )
=> ( ( P @ Xs2 )
=> ( P @ ( coinductive_LCons @ A @ X2 @ Xs2 ) ) ) )
=> ( P @ X ) ) ) ) ).
% lfinite.inducts
thf(fact_59_lnull__def,axiom,
! [A: $tType] :
( ( coinductive_lnull @ A )
= ( ^ [Llist5: coinductive_llist @ A] :
( Llist5
= ( coinductive_LNil @ A ) ) ) ) ).
% lnull_def
thf(fact_60_iterates_Odisc__iff,axiom,
! [A: $tType,F: A > A,X: A] :
~ ( coinductive_lnull @ A @ ( coinductive_iterates @ A @ F @ X ) ) ).
% iterates.disc_iff
thf(fact_61_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_62_llist__less__induct,axiom,
! [A: $tType,P: ( coinductive_llist @ A ) > $o,Xs: coinductive_llist @ A] :
( ! [Xs2: coinductive_llist @ A] :
( ! [Ys2: coinductive_llist @ A] :
( ( coindu1478340336prefix @ A @ Ys2 @ Xs2 )
=> ( P @ Ys2 ) )
=> ( P @ Xs2 ) )
=> ( P @ Xs ) ) ).
% llist_less_induct
thf(fact_63_lzip__eq__LNil__conv,axiom,
! [A: $tType,B: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B] :
( ( ( coinductive_lzip @ A @ B @ Xs @ Ys )
= ( coinductive_LNil @ ( product_prod @ A @ B ) ) )
= ( ( Xs
= ( coinductive_LNil @ A ) )
| ( Ys
= ( coinductive_LNil @ B ) ) ) ) ).
% lzip_eq_LNil_conv
thf(fact_64_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_65_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_66_lnull__imp__lfinite,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ Xs )
=> ( coinductive_lfinite @ A @ Xs ) ) ).
% lnull_imp_lfinite
thf(fact_67_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_68_lfinite__LNil,axiom,
! [A: $tType] : ( coinductive_lfinite @ A @ ( coinductive_LNil @ A ) ) ).
% lfinite_LNil
thf(fact_69_stream_Oexpand,axiom,
! [A: $tType,Stream: stream @ A,Stream2: stream @ A] :
( ( ( ( shd @ A @ Stream )
= ( shd @ A @ Stream2 ) )
& ( ( stl @ A @ Stream )
= ( stl @ A @ Stream2 ) ) )
=> ( Stream = Stream2 ) ) ).
% stream.expand
thf(fact_70_stream_Ocoinduct,axiom,
! [A: $tType,R: ( stream @ A ) > ( stream @ A ) > $o,Stream: stream @ A,Stream2: stream @ A] :
( ( R @ Stream @ Stream2 )
=> ( ! [Stream3: stream @ A,Stream4: stream @ A] :
( ( R @ Stream3 @ Stream4 )
=> ( ( ( shd @ A @ Stream3 )
= ( shd @ A @ Stream4 ) )
& ( R @ ( stl @ A @ Stream3 ) @ ( stl @ A @ Stream4 ) ) ) )
=> ( Stream = Stream2 ) ) ) ).
% stream.coinduct
thf(fact_71_stream_Ocoinduct__strong,axiom,
! [A: $tType,R: ( stream @ A ) > ( stream @ A ) > $o,Stream: stream @ A,Stream2: stream @ A] :
( ( R @ Stream @ Stream2 )
=> ( ! [Stream3: stream @ A,Stream4: stream @ A] :
( ( R @ Stream3 @ Stream4 )
=> ( ( ( shd @ A @ Stream3 )
= ( shd @ A @ Stream4 ) )
& ( ( R @ ( stl @ A @ Stream3 ) @ ( stl @ A @ Stream4 ) )
| ( ( stl @ A @ Stream3 )
= ( stl @ A @ Stream4 ) ) ) ) )
=> ( Stream = Stream2 ) ) ) ).
% stream.coinduct_strong
thf(fact_72_stream__of__llist__def,axiom,
! [A: $tType] :
( ( coindu2010755910_llist @ A )
= ( coindu139217191stream @ ( coinductive_llist @ A ) @ A @ ( coinductive_lhd @ A ) @ ( coinductive_ltl @ A ) ) ) ).
% stream_of_llist_def
thf(fact_73_llist_Osplit__sel,axiom,
! [B: $tType,A: $tType,P: B > $o,F1: B,F22: A > ( coinductive_llist @ A ) > B,Llist: coinductive_llist @ A] :
( ( P @ ( coindu1381640503_llist @ B @ A @ F1 @ F22 @ Llist ) )
= ( ( ( Llist
= ( coinductive_LNil @ A ) )
=> ( P @ F1 ) )
& ( ( 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_74_llist_Osplit__sel__asm,axiom,
! [B: $tType,A: $tType,P: B > $o,F1: B,F22: A > ( coinductive_llist @ A ) > B,Llist: coinductive_llist @ A] :
( ( P @ ( coindu1381640503_llist @ B @ A @ F1 @ F22 @ Llist ) )
= ( ~ ( ( ( Llist
= ( coinductive_LNil @ A ) )
& ~ ( P @ F1 ) )
| ( ( 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_75_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_76_unfold__stream__id,axiom,
! [A: $tType,Xs: stream @ A] :
( ( coindu139217191stream @ ( stream @ A ) @ A @ ( shd @ A ) @ ( stl @ A ) @ Xs )
= Xs ) ).
% unfold_stream_id
thf(fact_77_ltakeWhile_Ocode,axiom,
! [A: $tType] :
( ( coindu501562517eWhile @ A )
= ( ^ [P2: A > $o,Xs3: coinductive_llist @ A] :
( if @ ( coinductive_llist @ A )
@ ( ( coinductive_lnull @ A @ Xs3 )
| ~ ( P2 @ ( coinductive_lhd @ A @ Xs3 ) ) )
@ ( coinductive_LNil @ A )
@ ( coinductive_LCons @ A @ ( coinductive_lhd @ A @ Xs3 ) @ ( coindu501562517eWhile @ A @ P2 @ ( coinductive_ltl @ A @ Xs3 ) ) ) ) ) ) ).
% ltakeWhile.code
thf(fact_78_Coinductive__List_Ofinite__lprefix__nitpick__simps_I3_J,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Y: A,Ys: coinductive_llist @ A] :
( ( coindu328551480prefix @ A @ Xs @ ( coinductive_LCons @ A @ Y @ Ys ) )
= ( ( Xs
= ( coinductive_LNil @ A ) )
| ? [Xs4: coinductive_llist @ A] :
( ( Xs
= ( coinductive_LCons @ A @ Y @ Xs4 ) )
& ( coindu328551480prefix @ A @ Xs4 @ Ys ) ) ) ) ).
% Coinductive_List.finite_lprefix_nitpick_simps(3)
thf(fact_79_lzip_Octr_I2_J,axiom,
! [A: $tType,B: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B] :
( ~ ( coinductive_lnull @ A @ Xs )
=> ( ~ ( coinductive_lnull @ B @ Ys )
=> ( ( coinductive_lzip @ A @ B @ Xs @ Ys )
= ( coinductive_LCons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ ( coinductive_lhd @ A @ Xs ) @ ( coinductive_lhd @ B @ Ys ) ) @ ( coinductive_lzip @ A @ B @ ( coinductive_ltl @ A @ Xs ) @ ( coinductive_ltl @ B @ Ys ) ) ) ) ) ) ).
% lzip.ctr(2)
thf(fact_80_ltakeWhile__LNil,axiom,
! [A: $tType,P: A > $o] :
( ( coindu501562517eWhile @ A @ P @ ( coinductive_LNil @ A ) )
= ( coinductive_LNil @ A ) ) ).
% ltakeWhile_LNil
thf(fact_81_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_82_unfold__llist_Odisc__iff_I1_J,axiom,
! [B: $tType,A: $tType,P3: A > $o,G21: A > B,G22: A > A,A2: A] :
( ( coinductive_lnull @ B @ ( coindu1441602521_llist @ A @ B @ P3 @ G21 @ G22 @ A2 ) )
= ( P3 @ A2 ) ) ).
% unfold_llist.disc_iff(1)
thf(fact_83_unfold__llist_Odisc__iff_I2_J,axiom,
! [B: $tType,A: $tType,P3: A > $o,G21: A > B,G22: A > A,A2: A] :
( ( ~ ( coinductive_lnull @ B @ ( coindu1441602521_llist @ A @ B @ P3 @ G21 @ G22 @ A2 ) ) )
= ( ~ ( P3 @ A2 ) ) ) ).
% unfold_llist.disc_iff(2)
thf(fact_84_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_85_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_86_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_87_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_88_lzip__simps_I3_J,axiom,
! [C: $tType,B: $tType,X: C,Xs: coinductive_llist @ C,Y: B,Ys: coinductive_llist @ B] :
( ( coinductive_lzip @ C @ B @ ( coinductive_LCons @ C @ X @ Xs ) @ ( coinductive_LCons @ B @ Y @ Ys ) )
= ( coinductive_LCons @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ X @ Y ) @ ( coinductive_lzip @ C @ B @ Xs @ Ys ) ) ) ).
% lzip_simps(3)
thf(fact_89_unfold__llist_Octr_I2_J,axiom,
! [B: $tType,A: $tType,P3: A > $o,A2: A,G21: A > B,G22: A > A] :
( ~ ( P3 @ A2 )
=> ( ( coindu1441602521_llist @ A @ B @ P3 @ G21 @ G22 @ A2 )
= ( coinductive_LCons @ B @ ( G21 @ A2 ) @ ( coindu1441602521_llist @ A @ B @ P3 @ G21 @ G22 @ ( G22 @ A2 ) ) ) ) ) ).
% unfold_llist.ctr(2)
thf(fact_90_unfold__llist_Odisc_I1_J,axiom,
! [B: $tType,A: $tType,P3: A > $o,A2: A,G21: A > B,G22: A > A] :
( ( P3 @ A2 )
=> ( coinductive_lnull @ B @ ( coindu1441602521_llist @ A @ B @ P3 @ G21 @ G22 @ A2 ) ) ) ).
% unfold_llist.disc(1)
thf(fact_91_unfold__llist_Odisc_I2_J,axiom,
! [B: $tType,A: $tType,P3: A > $o,A2: A,G21: A > B,G22: A > A] :
( ~ ( P3 @ A2 )
=> ~ ( coinductive_lnull @ B @ ( coindu1441602521_llist @ A @ B @ P3 @ G21 @ G22 @ A2 ) ) ) ).
% unfold_llist.disc(2)
thf(fact_92_unfold__llist_Octr_I1_J,axiom,
! [A: $tType,B: $tType,P3: A > $o,A2: A,G21: A > B,G22: A > A] :
( ( P3 @ A2 )
=> ( ( coindu1441602521_llist @ A @ B @ P3 @ G21 @ G22 @ A2 )
= ( coinductive_LNil @ B ) ) ) ).
% unfold_llist.ctr(1)
thf(fact_93_unfold__llist_Osimps_I4_J,axiom,
! [B: $tType,A: $tType,P3: A > $o,A2: A,G21: A > B,G22: A > A] :
( ~ ( P3 @ A2 )
=> ( ( coinductive_ltl @ B @ ( coindu1441602521_llist @ A @ B @ P3 @ G21 @ G22 @ A2 ) )
= ( coindu1441602521_llist @ A @ B @ P3 @ G21 @ G22 @ ( G22 @ A2 ) ) ) ) ).
% unfold_llist.simps(4)
thf(fact_94_unfold__llist_Osimps_I3_J,axiom,
! [B: $tType,A: $tType,P3: A > $o,A2: A,G21: A > B,G22: A > A] :
( ~ ( P3 @ A2 )
=> ( ( coinductive_lhd @ B @ ( coindu1441602521_llist @ A @ B @ P3 @ G21 @ G22 @ A2 ) )
= ( G21 @ A2 ) ) ) ).
% unfold_llist.simps(3)
thf(fact_95_llist_Osimps_I5_J,axiom,
! [B: $tType,A: $tType,F1: B,F22: A > ( coinductive_llist @ A ) > B,X21: A,X22: coinductive_llist @ A] :
( ( coindu1381640503_llist @ B @ A @ F1 @ F22 @ ( coinductive_LCons @ A @ X21 @ X22 ) )
= ( F22 @ X21 @ X22 ) ) ).
% llist.simps(5)
thf(fact_96_llist_Osimps_I4_J,axiom,
! [A: $tType,B: $tType,F1: B,F22: A > ( coinductive_llist @ A ) > B] :
( ( coindu1381640503_llist @ B @ A @ F1 @ F22 @ ( coinductive_LNil @ A ) )
= F1 ) ).
% llist.simps(4)
thf(fact_97_unfold__stream_Osimps_I2_J,axiom,
! [B: $tType,A: $tType,G1: A > B,G2: A > A,A2: A] :
( ( stl @ B @ ( coindu139217191stream @ A @ B @ G1 @ G2 @ A2 ) )
= ( coindu139217191stream @ A @ B @ G1 @ G2 @ ( G2 @ A2 ) ) ) ).
% unfold_stream.simps(2)
thf(fact_98_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_99_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_100_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_101_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_102_unfold__stream_Osimps_I1_J,axiom,
! [B: $tType,A: $tType,G1: A > B,G2: A > A,A2: A] :
( ( shd @ B @ ( coindu139217191stream @ A @ B @ G1 @ G2 @ A2 ) )
= ( G1 @ A2 ) ) ).
% unfold_stream.simps(1)
thf(fact_103_lzip__eq__LCons__conv,axiom,
! [B: $tType,A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B,Z: product_prod @ A @ B,Zs: coinductive_llist @ ( product_prod @ A @ B )] :
( ( ( coinductive_lzip @ A @ B @ Xs @ Ys )
= ( coinductive_LCons @ ( product_prod @ A @ B ) @ Z @ Zs ) )
= ( ? [X3: A,Xs4: coinductive_llist @ A,Y2: B,Ys3: coinductive_llist @ B] :
( ( Xs
= ( coinductive_LCons @ A @ X3 @ Xs4 ) )
& ( Ys
= ( coinductive_LCons @ B @ Y2 @ Ys3 ) )
& ( Z
= ( product_Pair @ A @ B @ X3 @ Y2 ) )
& ( Zs
= ( coinductive_lzip @ A @ B @ Xs4 @ Ys3 ) ) ) ) ) ).
% lzip_eq_LCons_conv
thf(fact_104_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_105_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_106_unfold__llist_Ocode,axiom,
! [B: $tType,A: $tType] :
( ( coindu1441602521_llist @ A @ B )
= ( ^ [P4: A > $o,G212: A > B,G222: A > A,A3: A] : ( if @ ( coinductive_llist @ B ) @ ( P4 @ A3 ) @ ( coinductive_LNil @ B ) @ ( coinductive_LCons @ B @ ( G212 @ A3 ) @ ( coindu1441602521_llist @ A @ B @ P4 @ G212 @ G222 @ ( G222 @ A3 ) ) ) ) ) ) ).
% unfold_llist.code
thf(fact_107_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_108_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_109_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_110_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_111_lhd__lzip,axiom,
! [A: $tType,B: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B] :
( ~ ( coinductive_lnull @ A @ Xs )
=> ( ~ ( coinductive_lnull @ B @ Ys )
=> ( ( coinductive_lhd @ ( product_prod @ A @ B ) @ ( coinductive_lzip @ A @ B @ Xs @ Ys ) )
= ( product_Pair @ A @ B @ ( coinductive_lhd @ A @ Xs ) @ ( coinductive_lhd @ B @ Ys ) ) ) ) ) ).
% lhd_lzip
thf(fact_112_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_113_llist_Ocase__eq__if,axiom,
! [A: $tType,B: $tType] :
( ( coindu1381640503_llist @ B @ A )
= ( ^ [F12: B,F23: A > ( coinductive_llist @ A ) > B,Llist5: coinductive_llist @ A] : ( if @ B @ ( coinductive_lnull @ A @ Llist5 ) @ F12 @ ( F23 @ ( coinductive_lhd @ A @ Llist5 ) @ ( coinductive_ltl @ A @ Llist5 ) ) ) ) ) ).
% llist.case_eq_if
thf(fact_114_lzip_Ocode,axiom,
! [B: $tType,A: $tType] :
( ( coinductive_lzip @ A @ B )
= ( ^ [Xs3: coinductive_llist @ A,Ys4: coinductive_llist @ B] :
( if @ ( coinductive_llist @ ( product_prod @ A @ B ) )
@ ( ( coinductive_lnull @ A @ Xs3 )
| ( coinductive_lnull @ B @ Ys4 ) )
@ ( coinductive_LNil @ ( product_prod @ A @ B ) )
@ ( coinductive_LCons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ ( coinductive_lhd @ A @ Xs3 ) @ ( coinductive_lhd @ B @ Ys4 ) ) @ ( coinductive_lzip @ A @ B @ ( coinductive_ltl @ A @ Xs3 ) @ ( coinductive_ltl @ B @ Ys4 ) ) ) ) ) ) ).
% lzip.code
thf(fact_115_old_Oprod_Oinject,axiom,
! [A: $tType,B: $tType,A2: A,B2: B,A4: A,B3: B] :
( ( ( product_Pair @ A @ B @ A2 @ B2 )
= ( product_Pair @ A @ B @ A4 @ B3 ) )
= ( ( A2 = A4 )
& ( B2 = B3 ) ) ) ).
% old.prod.inject
thf(fact_116_prod_Oinject,axiom,
! [A: $tType,B: $tType,X1: A,X23: B,Y1: A,Y23: B] :
( ( ( product_Pair @ A @ B @ X1 @ X23 )
= ( product_Pair @ A @ B @ Y1 @ Y23 ) )
= ( ( X1 = Y1 )
& ( X23 = Y23 ) ) ) ).
% prod.inject
thf(fact_117_stream_Ocase__eq__if,axiom,
! [B: $tType,A: $tType] :
( ( case_stream @ A @ B )
= ( ^ [F2: A > ( stream @ A ) > B,Stream5: stream @ A] : ( F2 @ ( shd @ A @ Stream5 ) @ ( stl @ A @ Stream5 ) ) ) ) ).
% stream.case_eq_if
thf(fact_118_llast__singleton,axiom,
! [A: $tType,X: A] :
( ( coinductive_llast @ A @ ( coinductive_LCons @ A @ X @ ( coinductive_LNil @ A ) ) )
= X ) ).
% llast_singleton
thf(fact_119_sdrop__while_Osimps,axiom,
! [A: $tType] :
( ( sdrop_while @ A )
= ( ^ [P2: A > $o,S: stream @ A] : ( if @ ( stream @ A ) @ ( P2 @ ( shd @ A @ S ) ) @ ( sdrop_while @ A @ P2 @ ( stl @ A @ S ) ) @ S ) ) ) ).
% sdrop_while.simps
thf(fact_120_ord_Olsorted__coinduct_H,axiom,
! [A: $tType,X4: ( coinductive_llist @ A ) > $o,Xs: coinductive_llist @ A,Less_eq: A > A > $o] :
( ( X4 @ Xs )
=> ( ! [Xs2: coinductive_llist @ A] :
( ( X4 @ Xs2 )
=> ( ~ ( coinductive_lnull @ A @ Xs2 )
=> ( ~ ( coinductive_lnull @ A @ ( coinductive_ltl @ A @ Xs2 ) )
=> ( ( Less_eq @ ( coinductive_lhd @ A @ Xs2 ) @ ( coinductive_lhd @ A @ ( coinductive_ltl @ A @ Xs2 ) ) )
& ( ( X4 @ ( coinductive_ltl @ A @ Xs2 ) )
| ( coinductive_lsorted @ A @ Less_eq @ ( coinductive_ltl @ A @ Xs2 ) ) ) ) ) ) )
=> ( coinductive_lsorted @ A @ Less_eq @ Xs ) ) ) ).
% ord.lsorted_coinduct'
thf(fact_121_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_122_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_123_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_124_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_125_ord_OLNil,axiom,
! [A: $tType,Less_eq: A > A > $o] : ( coinductive_lsorted @ A @ Less_eq @ ( coinductive_LNil @ A ) ) ).
% ord.LNil
thf(fact_126_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_127_ord_Olsorted__ldropn,axiom,
! [A: $tType,Less_eq: A > A > $o,Xs: coinductive_llist @ A,N: nat] :
( ( coinductive_lsorted @ A @ Less_eq @ Xs )
=> ( coinductive_lsorted @ A @ Less_eq @ ( coinductive_ldropn @ A @ N @ Xs ) ) ) ).
% ord.lsorted_ldropn
thf(fact_128_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_129_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 ) )
=> ( ! [X2: A] :
( A2
!= ( coinductive_LCons @ A @ X2 @ ( coinductive_LNil @ A ) ) )
=> ~ ! [X2: A,Y3: A,Xs2: coinductive_llist @ A] :
( ( A2
= ( coinductive_LCons @ A @ X2 @ ( coinductive_LCons @ A @ Y3 @ Xs2 ) ) )
=> ( ( Less_eq @ X2 @ Y3 )
=> ~ ( coinductive_lsorted @ A @ Less_eq @ ( coinductive_LCons @ A @ Y3 @ Xs2 ) ) ) ) ) ) ) ).
% ord.lsorted.cases
thf(fact_130_ord_Olsorted_Osimps,axiom,
! [A: $tType] :
( ( coinductive_lsorted @ A )
= ( ^ [Less_eq2: A > A > $o,A3: coinductive_llist @ A] :
( ( A3
= ( coinductive_LNil @ A ) )
| ? [X3: A] :
( A3
= ( coinductive_LCons @ A @ X3 @ ( coinductive_LNil @ A ) ) )
| ? [X3: A,Y2: A,Xs3: coinductive_llist @ A] :
( ( A3
= ( coinductive_LCons @ A @ X3 @ ( coinductive_LCons @ A @ Y2 @ Xs3 ) ) )
& ( Less_eq2 @ X3 @ Y2 )
& ( coinductive_lsorted @ A @ Less_eq2 @ ( coinductive_LCons @ A @ Y2 @ Xs3 ) ) ) ) ) ) ).
% ord.lsorted.simps
thf(fact_131_ord_Olsorted_Ocoinduct,axiom,
! [A: $tType,X4: ( coinductive_llist @ A ) > $o,X: coinductive_llist @ A,Less_eq: A > A > $o] :
( ( X4 @ X )
=> ( ! [X2: coinductive_llist @ A] :
( ( X4 @ X2 )
=> ( ( X2
= ( coinductive_LNil @ A ) )
| ? [Xa: A] :
( X2
= ( coinductive_LCons @ A @ Xa @ ( coinductive_LNil @ A ) ) )
| ? [Xa: A,Y4: A,Xs5: coinductive_llist @ A] :
( ( X2
= ( coinductive_LCons @ A @ Xa @ ( coinductive_LCons @ A @ Y4 @ Xs5 ) ) )
& ( Less_eq @ Xa @ Y4 )
& ( ( X4 @ ( coinductive_LCons @ A @ Y4 @ Xs5 ) )
| ( coinductive_lsorted @ A @ Less_eq @ ( coinductive_LCons @ A @ Y4 @ Xs5 ) ) ) ) ) )
=> ( coinductive_lsorted @ A @ Less_eq @ X ) ) ) ).
% ord.lsorted.coinduct
thf(fact_132_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_133_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_134_surj__pair,axiom,
! [A: $tType,B: $tType,P3: product_prod @ A @ B] :
? [X2: A,Y3: B] :
( P3
= ( product_Pair @ A @ B @ X2 @ Y3 ) ) ).
% surj_pair
thf(fact_135_prod__cases,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P3: product_prod @ A @ B] :
( ! [A5: A,B4: B] : ( P @ ( product_Pair @ A @ B @ A5 @ B4 ) )
=> ( P @ P3 ) ) ).
% prod_cases
thf(fact_136_Pair__inject,axiom,
! [A: $tType,B: $tType,A2: A,B2: B,A4: A,B3: B] :
( ( ( product_Pair @ A @ B @ A2 @ B2 )
= ( product_Pair @ A @ B @ A4 @ B3 ) )
=> ~ ( ( A2 = A4 )
=> ( B2 != B3 ) ) ) ).
% Pair_inject
thf(fact_137_prod__cases3,axiom,
! [A: $tType,B: $tType,C: $tType,Y: product_prod @ A @ ( product_prod @ B @ C )] :
~ ! [A5: A,B4: B,C2: C] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B4 @ C2 ) ) ) ).
% prod_cases3
thf(fact_138_prod__cases4,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
~ ! [A5: A,B4: B,C2: C,D2: D] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B4 @ ( product_Pair @ C @ D @ C2 @ D2 ) ) ) ) ).
% prod_cases4
thf(fact_139_prod__cases5,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
~ ! [A5: A,B4: B,C2: C,D2: D,E2: E] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C2 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) ) ).
% prod_cases5
thf(fact_140_prod__cases6,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F3: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) )] :
~ ! [A5: A,B4: B,C2: C,D2: D,E2: E,F4: F3] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ F3 ) @ D2 @ ( product_Pair @ E @ F3 @ E2 @ F4 ) ) ) ) ) ) ).
% prod_cases6
thf(fact_141_prod__cases7,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F3: $tType,G3: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) )] :
~ ! [A5: A,B4: B,C2: C,D2: D,E2: E,F4: F3,G4: G3] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F3 @ G3 ) @ E2 @ ( product_Pair @ F3 @ G3 @ F4 @ G4 ) ) ) ) ) ) ) ).
% prod_cases7
thf(fact_142_prod__induct3,axiom,
! [C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ C )] :
( ! [A5: A,B4: B,C2: C] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B4 @ C2 ) ) )
=> ( P @ X ) ) ).
% prod_induct3
thf(fact_143_prod__induct4,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
( ! [A5: A,B4: B,C2: C,D2: D] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B4 @ ( product_Pair @ C @ D @ C2 @ D2 ) ) ) )
=> ( P @ X ) ) ).
% prod_induct4
thf(fact_144_prod__induct5,axiom,
! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
( ! [A5: A,B4: B,C2: C,D2: D,E2: E] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C2 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) )
=> ( P @ X ) ) ).
% prod_induct5
thf(fact_145_prod__induct6,axiom,
! [F3: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) )] :
( ! [A5: A,B4: B,C2: C,D2: D,E2: E,F4: F3] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ F3 ) @ D2 @ ( product_Pair @ E @ F3 @ E2 @ F4 ) ) ) ) ) )
=> ( P @ X ) ) ).
% prod_induct6
thf(fact_146_prod__induct7,axiom,
! [G3: $tType,F3: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) )] :
( ! [A5: A,B4: B,C2: C,D2: D,E2: E,F4: F3,G4: G3] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F3 @ G3 ) @ E2 @ ( product_Pair @ F3 @ G3 @ F4 @ G4 ) ) ) ) ) ) )
=> ( P @ X ) ) ).
% prod_induct7
thf(fact_147_old_Oprod_Oexhaust,axiom,
! [A: $tType,B: $tType,Y: product_prod @ A @ B] :
~ ! [A5: A,B4: B] :
( Y
!= ( product_Pair @ A @ B @ A5 @ B4 ) ) ).
% old.prod.exhaust
thf(fact_148_old_Oprod_Oinducts,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
( ! [A5: A,B4: B] : ( P @ ( product_Pair @ A @ B @ A5 @ B4 ) )
=> ( P @ Prod ) ) ).
% old.prod.inducts
thf(fact_149_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_150_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_151_old_Oprod_Orec,axiom,
! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A2: A,B2: B] :
( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A2 @ B2 ) )
= ( F1 @ A2 @ B2 ) ) ).
% old.prod.rec
thf(fact_152_internal__case__prod__conv,axiom,
! [B: $tType,A: $tType,C: $tType,C3: B > C > A,A2: B,B2: C] :
( ( produc2004651681e_prod @ B @ C @ A @ C3 @ ( product_Pair @ B @ C @ A2 @ B2 ) )
= ( C3 @ A2 @ B2 ) ) ).
% internal_case_prod_conv
thf(fact_153_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_154_sdrop__while_Oraw__induct,axiom,
! [A: $tType,Pa: ( product_prod @ ( A > $o ) @ ( stream @ A ) ) > ( stream @ A ) > $o,P: A > $o,S2: stream @ A,Y: stream @ A] :
( ! [Sdrop_while: ( A > $o ) > ( stream @ A ) > ( stream @ A )] :
( ! [S3: A > $o,B5: stream @ A] :
( ( ( Sdrop_while @ S3 @ B5 )
!= ( undefined @ ( stream @ A ) ) )
=> ( Pa @ ( product_Pair @ ( A > $o ) @ ( stream @ A ) @ S3 @ B5 ) @ ( Sdrop_while @ S3 @ B5 ) ) )
=> ! [P5: A > $o,S4: stream @ A,Pa2: stream @ A] :
( ( ( ( P5 @ ( shd @ A @ S4 ) )
=> ( ( Sdrop_while @ P5 @ ( stl @ A @ S4 ) )
= Pa2 ) )
& ( ~ ( P5 @ ( shd @ A @ S4 ) )
=> ( S4 = Pa2 ) ) )
=> ( ( Pa2
!= ( undefined @ ( stream @ A ) ) )
=> ( Pa @ ( product_Pair @ ( A > $o ) @ ( stream @ A ) @ P5 @ S4 ) @ Pa2 ) ) ) )
=> ( ( ( sdrop_while @ A @ P @ S2 )
= Y )
=> ( ( Y
!= ( undefined @ ( stream @ A ) ) )
=> ( Pa @ ( product_Pair @ ( A > $o ) @ ( stream @ A ) @ P @ S2 ) @ Y ) ) ) ) ).
% sdrop_while.raw_induct
thf(fact_155_lappend__code_I2_J,axiom,
! [A: $tType,Xa2: A,X: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_lappend @ A @ ( coinductive_LCons @ A @ Xa2 @ X ) @ Ys )
= ( coinductive_LCons @ A @ Xa2 @ ( coinductive_lappend @ A @ X @ Ys ) ) ) ).
% lappend_code(2)
thf(fact_156_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_157_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_158_lappend__code_I1_J,axiom,
! [A: $tType,Ys: coinductive_llist @ A] :
( ( coinductive_lappend @ A @ ( coinductive_LNil @ A ) @ Ys )
= Ys ) ).
% lappend_code(1)
thf(fact_159_lappend__LNil2,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( coinductive_lappend @ A @ Xs @ ( coinductive_LNil @ A ) )
= Xs ) ).
% lappend_LNil2
thf(fact_160_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_161_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_162_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_163_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_164_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_165_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_166_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_167_lappend__LNil__LNil,axiom,
! [A: $tType] :
( ( coinductive_lappend @ A @ ( coinductive_LNil @ A ) @ ( coinductive_LNil @ A ) )
= ( coinductive_LNil @ A ) ) ).
% lappend_LNil_LNil
thf(fact_168_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_169_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_170_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_171_lappend__iterates,axiom,
! [A: $tType,F: A > A,X: A,Xs: coinductive_llist @ A] :
( ( coinductive_lappend @ A @ ( coinductive_iterates @ A @ F @ X ) @ Xs )
= ( coinductive_iterates @ A @ F @ X ) ) ).
% lappend_iterates
thf(fact_172_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_173_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_174_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_175_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_176_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_177_llast__LNil,axiom,
! [A: $tType] :
( ( coinductive_llast @ A @ ( coinductive_LNil @ A ) )
= ( undefined @ A ) ) ).
% llast_LNil
thf(fact_178_llast__linfinite,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ~ ( coinductive_lfinite @ A @ Xs )
=> ( ( coinductive_llast @ A @ Xs )
= ( undefined @ A ) ) ) ).
% llast_linfinite
thf(fact_179_lfinite__rev__induct,axiom,
! [A: $tType,Xs: coinductive_llist @ A,P: ( coinductive_llist @ A ) > $o] :
( ( coinductive_lfinite @ A @ Xs )
=> ( ( P @ ( coinductive_LNil @ A ) )
=> ( ! [X2: A,Xs2: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ Xs2 )
=> ( ( P @ Xs2 )
=> ( P @ ( coinductive_lappend @ A @ Xs2 @ ( coinductive_LCons @ A @ X2 @ ( coinductive_LNil @ A ) ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% lfinite_rev_induct
thf(fact_180_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_181_llexord__conv,axiom,
! [A: $tType] :
( ( coinductive_llexord @ A )
= ( ^ [R2: A > A > $o,Xs3: coinductive_llist @ A,Ys4: coinductive_llist @ A] :
( ( Xs3 = Ys4 )
| ? [Zs2: coinductive_llist @ A,Xs4: coinductive_llist @ A,Y2: A,Ys3: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ Zs2 )
& ( Xs3
= ( coinductive_lappend @ A @ Zs2 @ Xs4 ) )
& ( Ys4
= ( coinductive_lappend @ A @ Zs2 @ ( coinductive_LCons @ A @ Y2 @ Ys3 ) ) )
& ( ( Xs4
= ( coinductive_LNil @ A ) )
| ( R2 @ ( coinductive_lhd @ A @ Xs4 ) @ Y2 ) ) ) ) ) ) ).
% llexord_conv
thf(fact_182_holds_Ocases,axiom,
! [A: $tType,X: product_prod @ ( A > $o ) @ ( stream @ A )] :
~ ! [P5: A > $o,Xs2: stream @ A] :
( X
!= ( product_Pair @ ( A > $o ) @ ( stream @ A ) @ P5 @ Xs2 ) ) ).
% holds.cases
thf(fact_183_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_184_llexord__refl,axiom,
! [A: $tType,R3: A > A > $o,Xs: coinductive_llist @ A] : ( coinductive_llexord @ A @ R3 @ Xs @ Xs ) ).
% llexord_refl
thf(fact_185_lconcat__LNil,axiom,
! [A: $tType] :
( ( coinductive_lconcat @ A @ ( coinductive_LNil @ ( coinductive_llist @ A ) ) )
= ( coinductive_LNil @ A ) ) ).
% lconcat_LNil
thf(fact_186_llexord__LCons__LCons,axiom,
! [A: $tType,R3: A > A > $o,X: A,Xs: coinductive_llist @ A,Y: A,Ys: coinductive_llist @ A] :
( ( coinductive_llexord @ A @ R3 @ ( coinductive_LCons @ A @ X @ Xs ) @ ( coinductive_LCons @ A @ Y @ Ys ) )
= ( ( ( X = Y )
& ( coinductive_llexord @ A @ R3 @ Xs @ Ys ) )
| ( R3 @ X @ Y ) ) ) ).
% llexord_LCons_LCons
thf(fact_187_llexord__LNil__right,axiom,
! [A: $tType,Ys: coinductive_llist @ A,R3: A > A > $o,Xs: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ Ys )
=> ( ( coinductive_llexord @ A @ R3 @ Xs @ Ys )
= ( coinductive_lnull @ A @ Xs ) ) ) ).
% llexord_LNil_right
thf(fact_188_llexord__code_I1_J,axiom,
! [A: $tType,R3: A > A > $o,Ys: coinductive_llist @ A] : ( coinductive_llexord @ A @ R3 @ ( coinductive_LNil @ A ) @ Ys ) ).
% llexord_code(1)
thf(fact_189_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_190_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_191_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_192_llexord__append__right,axiom,
! [A: $tType,R3: A > A > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] : ( coinductive_llexord @ A @ R3 @ Xs @ ( coinductive_lappend @ A @ Xs @ Ys ) ) ).
% llexord_append_right
thf(fact_193_llexord__lappend__leftI,axiom,
! [A: $tType,R3: A > A > $o,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A,Xs: coinductive_llist @ A] :
( ( coinductive_llexord @ A @ R3 @ Ys @ Zs )
=> ( coinductive_llexord @ A @ R3 @ ( coinductive_lappend @ A @ Xs @ Ys ) @ ( coinductive_lappend @ A @ Xs @ Zs ) ) ) ).
% llexord_lappend_leftI
thf(fact_194_llexord__antisym,axiom,
! [A: $tType,R3: A > A > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_llexord @ A @ R3 @ Xs @ Ys )
=> ( ( coinductive_llexord @ A @ R3 @ Ys @ Xs )
=> ( ! [A5: A,B4: A] :
( ( R3 @ A5 @ B4 )
=> ~ ( R3 @ B4 @ A5 ) )
=> ( Xs = Ys ) ) ) ) ).
% llexord_antisym
thf(fact_195_llexord__linear,axiom,
! [A: $tType,R3: A > A > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ! [X2: A,Y3: A] :
( ( R3 @ X2 @ Y3 )
| ( X2 = Y3 )
| ( R3 @ Y3 @ X2 ) )
=> ( ( coinductive_llexord @ A @ R3 @ Xs @ Ys )
| ( coinductive_llexord @ A @ R3 @ Ys @ Xs ) ) ) ).
% llexord_linear
thf(fact_196_llexord__trans,axiom,
! [A: $tType,R3: A > A > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A] :
( ( coinductive_llexord @ A @ R3 @ Xs @ Ys )
=> ( ( coinductive_llexord @ A @ R3 @ Ys @ Zs )
=> ( ! [A5: A,B4: A,C2: A] :
( ( R3 @ A5 @ B4 )
=> ( ( R3 @ B4 @ C2 )
=> ( R3 @ A5 @ C2 ) ) )
=> ( coinductive_llexord @ A @ R3 @ Xs @ Zs ) ) ) ) ).
% llexord_trans
thf(fact_197_llexord__LNil,axiom,
! [A: $tType,R3: A > A > $o,Ys: coinductive_llist @ A] : ( coinductive_llexord @ A @ R3 @ ( coinductive_LNil @ A ) @ Ys ) ).
% llexord_LNil
thf(fact_198_lnull__llexord,axiom,
! [A: $tType,Xs: coinductive_llist @ A,R3: A > A > $o,Ys: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ Xs )
=> ( coinductive_llexord @ A @ R3 @ Xs @ Ys ) ) ).
% lnull_llexord
thf(fact_199_llexord__LCons__less,axiom,
! [A: $tType,R3: A > A > $o,X: A,Y: A,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( R3 @ X @ Y )
=> ( coinductive_llexord @ A @ R3 @ ( coinductive_LCons @ A @ X @ Xs ) @ ( coinductive_LCons @ A @ Y @ Ys ) ) ) ).
% llexord_LCons_less
thf(fact_200_llexord__LCons__eq,axiom,
! [A: $tType,R3: A > A > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,X: A] :
( ( coinductive_llexord @ A @ R3 @ Xs @ Ys )
=> ( coinductive_llexord @ A @ R3 @ ( coinductive_LCons @ A @ X @ Xs ) @ ( coinductive_LCons @ A @ X @ Ys ) ) ) ).
% llexord_LCons_eq
thf(fact_201_llexord__LCons__left,axiom,
! [A: $tType,R3: A > A > $o,X: A,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_llexord @ A @ R3 @ ( coinductive_LCons @ A @ X @ Xs ) @ Ys )
= ( ? [Y2: A,Ys3: coinductive_llist @ A] :
( ( Ys
= ( coinductive_LCons @ A @ Y2 @ Ys3 ) )
& ( ( ( X = Y2 )
& ( coinductive_llexord @ A @ R3 @ Xs @ Ys3 ) )
| ( R3 @ X @ Y2 ) ) ) ) ) ).
% llexord_LCons_left
thf(fact_202_llexord__code_I3_J,axiom,
! [A: $tType,R3: A > A > $o,X: A,Xs: coinductive_llist @ A,Y: A,Ys: coinductive_llist @ A] :
( ( coinductive_llexord @ A @ R3 @ ( coinductive_LCons @ A @ X @ Xs ) @ ( coinductive_LCons @ A @ Y @ Ys ) )
= ( ( R3 @ X @ Y )
| ( ( X = Y )
& ( coinductive_llexord @ A @ R3 @ Xs @ Ys ) ) ) ) ).
% llexord_code(3)
thf(fact_203_llexord__code_I2_J,axiom,
! [A: $tType,R3: A > A > $o,X: A,Xs: coinductive_llist @ A] :
~ ( coinductive_llexord @ A @ R3 @ ( coinductive_LCons @ A @ X @ Xs ) @ ( coinductive_LNil @ A ) ) ).
% llexord_code(2)
thf(fact_204_llexord_Ocases,axiom,
! [A: $tType,R3: A > A > $o,A1: coinductive_llist @ A,A22: coinductive_llist @ A] :
( ( coinductive_llexord @ A @ R3 @ A1 @ A22 )
=> ( ! [Xs2: coinductive_llist @ A,Ys5: coinductive_llist @ A,X2: A] :
( ( A1
= ( coinductive_LCons @ A @ X2 @ Xs2 ) )
=> ( ( A22
= ( coinductive_LCons @ A @ X2 @ Ys5 ) )
=> ~ ( coinductive_llexord @ A @ R3 @ Xs2 @ Ys5 ) ) )
=> ( ! [X2: A] :
( ? [Xs2: coinductive_llist @ A] :
( A1
= ( coinductive_LCons @ A @ X2 @ Xs2 ) )
=> ! [Y3: A] :
( ? [Ys5: coinductive_llist @ A] :
( A22
= ( coinductive_LCons @ A @ Y3 @ Ys5 ) )
=> ~ ( R3 @ X2 @ Y3 ) ) )
=> ~ ( ( A1
= ( coinductive_LNil @ A ) )
=> ! [Ys5: coinductive_llist @ A] : A22 != Ys5 ) ) ) ) ).
% llexord.cases
thf(fact_205_llexord_Osimps,axiom,
! [A: $tType] :
( ( coinductive_llexord @ A )
= ( ^ [R2: A > A > $o,A12: coinductive_llist @ A,A23: coinductive_llist @ A] :
( ? [Xs3: coinductive_llist @ A,Ys4: coinductive_llist @ A,X3: A] :
( ( A12
= ( coinductive_LCons @ A @ X3 @ Xs3 ) )
& ( A23
= ( coinductive_LCons @ A @ X3 @ Ys4 ) )
& ( coinductive_llexord @ A @ R2 @ Xs3 @ Ys4 ) )
| ? [X3: A,Y2: A,Xs3: coinductive_llist @ A,Ys4: coinductive_llist @ A] :
( ( A12
= ( coinductive_LCons @ A @ X3 @ Xs3 ) )
& ( A23
= ( coinductive_LCons @ A @ Y2 @ Ys4 ) )
& ( R2 @ X3 @ Y2 ) )
| ? [Ys4: coinductive_llist @ A] :
( ( A12
= ( coinductive_LNil @ A ) )
& ( A23 = Ys4 ) ) ) ) ) ).
% llexord.simps
thf(fact_206_llexord_Ocoinduct,axiom,
! [A: $tType,X4: ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o,X: coinductive_llist @ A,Xa2: coinductive_llist @ A,R3: A > A > $o] :
( ( X4 @ X @ Xa2 )
=> ( ! [X2: coinductive_llist @ A,Xa3: coinductive_llist @ A] :
( ( X4 @ X2 @ Xa3 )
=> ( ? [Xs5: coinductive_llist @ A,Ys2: coinductive_llist @ A,Xb: A] :
( ( X2
= ( coinductive_LCons @ A @ Xb @ Xs5 ) )
& ( Xa3
= ( coinductive_LCons @ A @ Xb @ Ys2 ) )
& ( ( X4 @ Xs5 @ Ys2 )
| ( coinductive_llexord @ A @ R3 @ Xs5 @ Ys2 ) ) )
| ? [Xb: A,Y4: A,Xs5: coinductive_llist @ A,Ys2: coinductive_llist @ A] :
( ( X2
= ( coinductive_LCons @ A @ Xb @ Xs5 ) )
& ( Xa3
= ( coinductive_LCons @ A @ Y4 @ Ys2 ) )
& ( R3 @ Xb @ Y4 ) )
| ? [Ys2: coinductive_llist @ A] :
( ( X2
= ( coinductive_LNil @ A ) )
& ( Xa3 = Ys2 ) ) ) )
=> ( coinductive_llexord @ A @ R3 @ X @ Xa2 ) ) ) ).
% llexord.coinduct
thf(fact_207_llexord__coinduct,axiom,
! [A: $tType,X4: ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,R3: A > A > $o] :
( ( X4 @ Xs @ Ys )
=> ( ! [Xs2: coinductive_llist @ A,Ys5: coinductive_llist @ A] :
( ( X4 @ Xs2 @ Ys5 )
=> ( ~ ( coinductive_lnull @ A @ Xs2 )
=> ( ~ ( coinductive_lnull @ A @ Ys5 )
& ( ~ ( coinductive_lnull @ A @ Ys5 )
=> ( ( R3 @ ( coinductive_lhd @ A @ Xs2 ) @ ( coinductive_lhd @ A @ Ys5 ) )
| ( ( ( coinductive_lhd @ A @ Xs2 )
= ( coinductive_lhd @ A @ Ys5 ) )
& ( ( X4 @ ( coinductive_ltl @ A @ Xs2 ) @ ( coinductive_ltl @ A @ Ys5 ) )
| ( coinductive_llexord @ A @ R3 @ ( coinductive_ltl @ A @ Xs2 ) @ ( coinductive_ltl @ A @ Ys5 ) ) ) ) ) ) ) ) )
=> ( coinductive_llexord @ A @ R3 @ Xs @ Ys ) ) ) ).
% llexord_coinduct
thf(fact_208_sfilter__not__P,axiom,
! [A: $tType,P: A > $o,S2: stream @ A] :
( ~ ( P @ ( shd @ A @ S2 ) )
=> ( ( sfilter @ A @ P @ S2 )
= ( sfilter @ A @ P @ ( stl @ A @ S2 ) ) ) ) ).
% sfilter_not_P
thf(fact_209_nxt_Osimps,axiom,
! [B: $tType,A: $tType] :
( ( linear1494993505on_nxt @ A @ B )
= ( ^ [Phi: ( stream @ A ) > B,Xs3: stream @ A] : ( Phi @ ( stl @ A @ Xs3 ) ) ) ) ).
% nxt.simps
thf(fact_210_nxt_Oelims,axiom,
! [B: $tType,A: $tType,X: ( stream @ A ) > B,Xa2: stream @ A,Y: B] :
( ( ( linear1494993505on_nxt @ A @ B @ X @ Xa2 )
= Y )
=> ( Y
= ( X @ ( stl @ A @ Xa2 ) ) ) ) ).
% nxt.elims
thf(fact_211_sfilter__P,axiom,
! [A: $tType,P: A > $o,S2: stream @ A] :
( ( P @ ( shd @ A @ S2 ) )
=> ( ( sfilter @ A @ P @ S2 )
= ( sCons @ A @ ( shd @ A @ S2 ) @ ( sfilter @ A @ P @ ( stl @ A @ S2 ) ) ) ) ) ).
% sfilter_P
thf(fact_212_sfilter_Osimps_I1_J,axiom,
! [A: $tType,P: A > $o,S2: stream @ A] :
( ( shd @ A @ ( sfilter @ A @ P @ S2 ) )
= ( shd @ A @ ( sdrop_while @ A @ ( comp @ $o @ $o @ A @ (~) @ P ) @ S2 ) ) ) ).
% sfilter.simps(1)
thf(fact_213_stream_Oinject,axiom,
! [A: $tType,X1: A,X23: stream @ A,Y1: A,Y23: stream @ A] :
( ( ( sCons @ A @ X1 @ X23 )
= ( sCons @ A @ Y1 @ Y23 ) )
= ( ( X1 = Y1 )
& ( X23 = Y23 ) ) ) ).
% stream.inject
thf(fact_214_unfold__stream__eq__SCons,axiom,
! [A: $tType,B: $tType,SHD: B > A,STL: B > B,B2: B,X: A,Xs: stream @ A] :
( ( ( coindu139217191stream @ B @ A @ SHD @ STL @ B2 )
= ( sCons @ A @ X @ Xs ) )
= ( ( X
= ( SHD @ B2 ) )
& ( Xs
= ( coindu139217191stream @ B @ A @ SHD @ STL @ ( STL @ B2 ) ) ) ) ) ).
% unfold_stream_eq_SCons
thf(fact_215_stream_Ocollapse,axiom,
! [A: $tType,Stream: stream @ A] :
( ( sCons @ A @ ( shd @ A @ Stream ) @ ( stl @ A @ Stream ) )
= Stream ) ).
% stream.collapse
thf(fact_216_sfilter_Ocode,axiom,
! [A: $tType] :
( ( sfilter @ A )
= ( ^ [P2: A > $o,S: stream @ A] : ( sCons @ A @ ( shd @ A @ ( sdrop_while @ A @ ( comp @ $o @ $o @ A @ (~) @ P2 ) @ S ) ) @ ( sfilter @ A @ P2 @ ( stl @ A @ ( sdrop_while @ A @ ( comp @ $o @ $o @ A @ (~) @ P2 ) @ S ) ) ) ) ) ) ).
% sfilter.code
thf(fact_217_sfilter__Stream,axiom,
! [A: $tType,P: A > $o,X: A,S2: stream @ A] :
( ( ( P @ X )
=> ( ( sfilter @ A @ P @ ( sCons @ A @ X @ S2 ) )
= ( sCons @ A @ X @ ( sfilter @ A @ P @ S2 ) ) ) )
& ( ~ ( P @ X )
=> ( ( sfilter @ A @ P @ ( sCons @ A @ X @ S2 ) )
= ( sfilter @ A @ P @ S2 ) ) ) ) ).
% sfilter_Stream
thf(fact_218_unfold__stream_Ocode,axiom,
! [B: $tType,A: $tType] :
( ( coindu139217191stream @ A @ B )
= ( ^ [G12: A > B,G23: A > A,A3: A] : ( sCons @ B @ ( G12 @ A3 ) @ ( coindu139217191stream @ A @ B @ G12 @ G23 @ ( G23 @ A3 ) ) ) ) ) ).
% unfold_stream.code
thf(fact_219_stream_Ocase,axiom,
! [B: $tType,A: $tType,F: A > ( stream @ A ) > B,X1: A,X23: stream @ A] :
( ( case_stream @ A @ B @ F @ ( sCons @ A @ X1 @ X23 ) )
= ( F @ X1 @ X23 ) ) ).
% stream.case
thf(fact_220_sdrop__while__SCons,axiom,
! [A: $tType,P: A > $o,A2: A,S2: stream @ A] :
( ( ( P @ A2 )
=> ( ( sdrop_while @ A @ P @ ( sCons @ A @ A2 @ S2 ) )
= ( sdrop_while @ A @ P @ S2 ) ) )
& ( ~ ( P @ A2 )
=> ( ( sdrop_while @ A @ P @ ( sCons @ A @ A2 @ S2 ) )
= ( sCons @ A @ A2 @ S2 ) ) ) ) ).
% sdrop_while_SCons
thf(fact_221_stream_Oexhaust,axiom,
! [A: $tType,Y: stream @ A] :
~ ! [X12: A,X24: stream @ A] :
( Y
!= ( sCons @ A @ X12 @ X24 ) ) ).
% stream.exhaust
thf(fact_222_unfold__stream__ltl__unroll,axiom,
! [A: $tType,B: $tType,SHD: B > A,STL: B > B,B2: B] :
( ( coindu139217191stream @ B @ A @ SHD @ STL @ ( STL @ B2 ) )
= ( coindu139217191stream @ B @ A @ ( comp @ B @ A @ B @ SHD @ STL ) @ STL @ B2 ) ) ).
% unfold_stream_ltl_unroll
thf(fact_223_stream_Osel_I1_J,axiom,
! [A: $tType,X1: A,X23: stream @ A] :
( ( shd @ A @ ( sCons @ A @ X1 @ X23 ) )
= X1 ) ).
% stream.sel(1)
thf(fact_224_stream_Osel_I2_J,axiom,
! [A: $tType,X1: A,X23: stream @ A] :
( ( stl @ A @ ( sCons @ A @ X1 @ X23 ) )
= X23 ) ).
% stream.sel(2)
thf(fact_225_unfold__llist__ltl__unroll,axiom,
! [A: $tType,B: $tType,IS_LNIL: B > $o,LHD: B > A,LTL: B > B,B2: B] :
( ( coindu1441602521_llist @ B @ A @ IS_LNIL @ LHD @ LTL @ ( LTL @ B2 ) )
= ( coindu1441602521_llist @ B @ A @ ( comp @ B @ $o @ B @ IS_LNIL @ LTL ) @ ( comp @ B @ A @ B @ LHD @ LTL ) @ LTL @ B2 ) ) ).
% unfold_llist_ltl_unroll
thf(fact_226_stream_Oexhaust__sel,axiom,
! [A: $tType,Stream: stream @ A] :
( Stream
= ( sCons @ A @ ( shd @ A @ Stream ) @ ( stl @ A @ Stream ) ) ) ).
% stream.exhaust_sel
thf(fact_227_eq__SConsD,axiom,
! [A: $tType,Xs: stream @ A,Y: A,Ys: stream @ A] :
( ( Xs
= ( sCons @ A @ Y @ Ys ) )
=> ( ( ( shd @ A @ Xs )
= Y )
& ( ( stl @ A @ Xs )
= Ys ) ) ) ).
% eq_SConsD
thf(fact_228_stream_Osplit__sel,axiom,
! [B: $tType,A: $tType,P: B > $o,F: A > ( stream @ A ) > B,Stream: stream @ A] :
( ( P @ ( case_stream @ A @ B @ F @ Stream ) )
= ( ( Stream
= ( sCons @ A @ ( shd @ A @ Stream ) @ ( stl @ A @ Stream ) ) )
=> ( P @ ( F @ ( shd @ A @ Stream ) @ ( stl @ A @ Stream ) ) ) ) ) ).
% stream.split_sel
thf(fact_229_stream_Osplit__sel__asm,axiom,
! [B: $tType,A: $tType,P: B > $o,F: A > ( stream @ A ) > B,Stream: stream @ A] :
( ( P @ ( case_stream @ A @ B @ F @ Stream ) )
= ( ~ ( ( Stream
= ( sCons @ A @ ( shd @ A @ Stream ) @ ( stl @ A @ Stream ) ) )
& ~ ( P @ ( F @ ( shd @ A @ Stream ) @ ( stl @ A @ Stream ) ) ) ) ) ) ).
% stream.split_sel_asm
thf(fact_230_sfilter_Osimps_I2_J,axiom,
! [A: $tType,P: A > $o,S2: stream @ A] :
( ( stl @ A @ ( sfilter @ A @ P @ S2 ) )
= ( sfilter @ A @ P @ ( stl @ A @ ( sdrop_while @ A @ ( comp @ $o @ $o @ A @ (~) @ P ) @ S2 ) ) ) ) ).
% sfilter.simps(2)
thf(fact_231_smember__code,axiom,
! [A: $tType,X: A,Y: A,S2: stream @ A] :
( ( smember @ A @ X @ ( sCons @ A @ Y @ S2 ) )
= ( ( X != Y )
=> ( smember @ A @ X @ S2 ) ) ) ).
% smember_code
thf(fact_232_sinterleave_Ocode,axiom,
! [A: $tType] :
( ( sinterleave @ A )
= ( ^ [S1: stream @ A,S22: stream @ A] : ( sCons @ A @ ( shd @ A @ S1 ) @ ( sinterleave @ A @ S22 @ ( stl @ A @ S1 ) ) ) ) ) ).
% sinterleave.code
thf(fact_233_sinterleave__code,axiom,
! [A: $tType,X: A,S12: stream @ A,S23: stream @ A] :
( ( sinterleave @ A @ ( sCons @ A @ X @ S12 ) @ S23 )
= ( sCons @ A @ X @ ( sinterleave @ A @ S23 @ S12 ) ) ) ).
% sinterleave_code
thf(fact_234_sinterleave_Osimps_I1_J,axiom,
! [A: $tType,S12: stream @ A,S23: stream @ A] :
( ( shd @ A @ ( sinterleave @ A @ S12 @ S23 ) )
= ( shd @ A @ S12 ) ) ).
% sinterleave.simps(1)
thf(fact_235_sinterleave_Osimps_I2_J,axiom,
! [A: $tType,S12: stream @ A,S23: stream @ A] :
( ( stl @ A @ ( sinterleave @ A @ S12 @ S23 ) )
= ( sinterleave @ A @ S23 @ ( stl @ A @ S12 ) ) ) ).
% sinterleave.simps(2)
thf(fact_236_smap2_Ocode,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( smap2 @ A @ B @ C )
= ( ^ [F2: A > B > C,S1: stream @ A,S22: stream @ B] : ( sCons @ C @ ( F2 @ ( shd @ A @ S1 ) @ ( shd @ B @ S22 ) ) @ ( smap2 @ A @ B @ C @ F2 @ ( stl @ A @ S1 ) @ ( stl @ B @ S22 ) ) ) ) ) ).
% smap2.code
thf(fact_237_szip_Ocode,axiom,
! [B: $tType,A: $tType] :
( ( szip @ A @ B )
= ( ^ [S1: stream @ A,S22: stream @ B] : ( sCons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ ( shd @ A @ S1 ) @ ( shd @ B @ S22 ) ) @ ( szip @ A @ B @ ( stl @ A @ S1 ) @ ( stl @ B @ S22 ) ) ) ) ) ).
% szip.code
thf(fact_238_szip__unfold,axiom,
! [A: $tType,B: $tType,A2: A,S12: stream @ A,B2: B,S23: stream @ B] :
( ( szip @ A @ B @ ( sCons @ A @ A2 @ S12 ) @ ( sCons @ B @ B2 @ S23 ) )
= ( sCons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ ( szip @ A @ B @ S12 @ S23 ) ) ) ).
% szip_unfold
thf(fact_239_szip_Osimps_I1_J,axiom,
! [A: $tType,B: $tType,S12: stream @ A,S23: stream @ B] :
( ( shd @ ( product_prod @ A @ B ) @ ( szip @ A @ B @ S12 @ S23 ) )
= ( product_Pair @ A @ B @ ( shd @ A @ S12 ) @ ( shd @ B @ S23 ) ) ) ).
% szip.simps(1)
thf(fact_240_szip_Osimps_I2_J,axiom,
! [A: $tType,B: $tType,S12: stream @ A,S23: stream @ B] :
( ( stl @ ( product_prod @ A @ B ) @ ( szip @ A @ B @ S12 @ S23 ) )
= ( szip @ A @ B @ ( stl @ A @ S12 ) @ ( stl @ B @ S23 ) ) ) ).
% szip.simps(2)
thf(fact_241_smap2_Osimps_I1_J,axiom,
! [A: $tType,C: $tType,B: $tType,F: A > B > C,S12: stream @ A,S23: stream @ B] :
( ( shd @ C @ ( smap2 @ A @ B @ C @ F @ S12 @ S23 ) )
= ( F @ ( shd @ A @ S12 ) @ ( shd @ B @ S23 ) ) ) ).
% smap2.simps(1)
thf(fact_242_smap2_Osimps_I2_J,axiom,
! [A: $tType,C: $tType,B: $tType,F: A > B > C,S12: stream @ A,S23: stream @ B] :
( ( stl @ C @ ( smap2 @ A @ B @ C @ F @ S12 @ S23 ) )
= ( smap2 @ A @ B @ C @ F @ ( stl @ A @ S12 ) @ ( stl @ B @ S23 ) ) ) ).
% smap2.simps(2)
thf(fact_243_smap2__unfold,axiom,
! [B: $tType,A: $tType,C: $tType,F: B > C > A,A2: B,S12: stream @ B,B2: C,S23: stream @ C] :
( ( smap2 @ B @ C @ A @ F @ ( sCons @ B @ A2 @ S12 ) @ ( sCons @ C @ B2 @ S23 ) )
= ( sCons @ A @ ( F @ A2 @ B2 ) @ ( smap2 @ B @ C @ A @ F @ S12 @ S23 ) ) ) ).
% smap2_unfold
thf(fact_244_lmember__code_I1_J,axiom,
! [A: $tType,X: A] :
~ ( coinductive_lmember @ A @ X @ ( coinductive_LNil @ A ) ) ).
% lmember_code(1)
thf(fact_245_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)
%----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 (2)
thf(conj_0,hypothesis,
~ ( coinductive_lfinite @ a @ xs ) ).
thf(conj_1,conjecture,
( ( coindu1724414836stream @ a @ ( coindu2010755910_llist @ a @ xs ) )
= xs ) ).
%------------------------------------------------------------------------------