TPTP Problem File: DAT190^1.p
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%------------------------------------------------------------------------------
% File : DAT190^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Data Structures
% Problem : Lazy list mirror 44
% 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__44.p [Bla16]
% Status : Theorem
% Rating : 0.67 v8.1.0, 0.75 v7.5.0, 1.00 v7.3.0, 0.67 v7.2.0, 0.75 v7.1.0
% Syntax : Number of formulae : 300 ( 138 unt; 42 typ; 0 def)
% Number of atoms : 675 ( 294 equ; 0 cnn)
% Maximal formula atoms : 8 ( 2 avg)
% Number of connectives : 4198 ( 157 ~; 54 |; 84 &;3641 @)
% ( 0 <=>; 262 =>; 0 <=; 0 <~>)
% Maximal formula depth : 28 ( 8 avg)
% Number of types : 3 ( 2 usr)
% Number of type conns : 248 ( 248 >; 0 *; 0 +; 0 <<)
% Number of symbols : 43 ( 40 usr; 6 con; 0-6 aty)
% Number of variables : 1117 ( 30 ^; 987 !; 60 ?;1117 :)
% ( 40 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 14:40:54.235
%------------------------------------------------------------------------------
%----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_Set_Oset,type,
set: $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 (36)
thf(sy_c_Coinductive__List_Ofinite__lprefix,type,
coindu328551480prefix:
!>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List_Ogen__lset,type,
coinductive_gen_lset:
!>[A: $tType] : ( ( set @ A ) > ( coinductive_llist @ A ) > ( set @ A ) ) ).
thf(sy_c_Coinductive__List_Olappend,type,
coinductive_lappend:
!>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
thf(sy_c_Coinductive__List_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_Olset,type,
coinductive_lset:
!>[A: $tType] : ( ( coinductive_llist @ A ) > ( set @ A ) ) ).
thf(sy_c_Coinductive__List_Ollist_Oltl,type,
coinductive_ltl:
!>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
thf(sy_c_Coinductive__List_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_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_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_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_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_acc,type,
acc: coinductive_llist @ a ).
thf(sy_v_acca____,type,
acca: coinductive_llist @ a ).
thf(sy_v_xs,type,
xs: coinductive_llist @ a ).
thf(sy_v_xsa____,type,
xsa: coinductive_llist @ a ).
%----Relevant facts (254)
thf(fact_0__092_060open_062lfinite_A_Ilmirror__aux_Aacc_Axs_J_092_060close_062,axiom,
coinductive_lfinite @ a @ ( lMirro999291890or_aux @ a @ acc @ xs ) ).
% \<open>lfinite (lmirror_aux acc xs)\<close>
thf(fact_1_LCons_Ohyps_I2_J,axiom,
~ ( coinductive_lnull @ a @ ( lMirro999291890or_aux @ a @ acca @ xsa ) ) ).
% LCons.hyps(2)
thf(fact_2_LCons_Ohyps_I1_J,axiom,
coinductive_lfinite @ a @ ( lMirro999291890or_aux @ a @ acca @ xsa ) ).
% LCons.hyps(1)
thf(fact_3_LCons_Ohyps_I3_J,axiom,
! [Acc: coinductive_llist @ a,Xs: coinductive_llist @ a] :
( ( ( coinductive_ltl @ a @ ( lMirro999291890or_aux @ a @ acca @ xsa ) )
= ( lMirro999291890or_aux @ a @ Acc @ Xs ) )
=> ( ( coinductive_lfinite @ a @ Xs )
& ( coinductive_lfinite @ a @ Acc ) ) ) ).
% LCons.hyps(3)
thf(fact_4_lfinite__ltl,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ ( coinductive_ltl @ A @ Xs ) )
= ( coinductive_lfinite @ A @ Xs ) ) ).
% lfinite_ltl
thf(fact_5_lnull__imp__lfinite,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ Xs )
=> ( coinductive_lfinite @ A @ Xs ) ) ).
% lnull_imp_lfinite
thf(fact_6_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_7_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_8_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_9_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_10_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_11_lfinite__code_I1_J,axiom,
! [A: $tType] : ( coinductive_lfinite @ A @ ( coinductive_LNil @ A ) ) ).
% lfinite_code(1)
thf(fact_12_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_13_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_14_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_15_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_16_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_17_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_18_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_19_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_20_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_21_ldropn__LNil,axiom,
! [A: $tType,N: nat] :
( ( coinductive_ldropn @ A @ N @ ( coinductive_LNil @ A ) )
= ( coinductive_LNil @ A ) ) ).
% ldropn_LNil
thf(fact_22_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_23_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_24_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_25_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_26_lstrict__prefix__code_I1_J,axiom,
! [A: $tType] :
~ ( coindu1478340336prefix @ A @ ( coinductive_LNil @ A ) @ ( coinductive_LNil @ A ) ) ).
% lstrict_prefix_code(1)
thf(fact_27_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_28_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_29_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_30_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_31_ltl__simps_I1_J,axiom,
! [A: $tType] :
( ( coinductive_ltl @ A @ ( coinductive_LNil @ A ) )
= ( coinductive_LNil @ A ) ) ).
% ltl_simps(1)
thf(fact_32_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_33_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_34_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_35_llist_Odisc_I1_J,axiom,
! [A: $tType] : ( coinductive_lnull @ A @ ( coinductive_LNil @ A ) ) ).
% llist.disc(1)
thf(fact_36_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_37_llist_OdiscI_I1_J,axiom,
! [A: $tType,Llist: coinductive_llist @ A] :
( ( Llist
= ( coinductive_LNil @ A ) )
=> ( coinductive_lnull @ A @ Llist ) ) ).
% llist.discI(1)
thf(fact_38_llist_Ocollapse_I1_J,axiom,
! [A: $tType,Llist: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ Llist )
=> ( Llist
= ( coinductive_LNil @ A ) ) ) ).
% llist.collapse(1)
thf(fact_39_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_40_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_41_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_42_lnull__ltlI,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ Xs )
=> ( coinductive_lnull @ A @ ( coinductive_ltl @ A @ Xs ) ) ) ).
% lnull_ltlI
thf(fact_43_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_44_mem__Collect__eq,axiom,
! [A: $tType,A2: A,P: A > $o] :
( ( member @ A @ A2 @ ( collect @ A @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_45_Collect__mem__eq,axiom,
! [A: $tType,A3: set @ A] :
( ( collect @ A
@ ^ [X2: A] : ( member @ A @ X2 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_46_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X3: A] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_47_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X3: A] :
( ( F @ X3 )
= ( G @ X3 ) )
=> ( F = G ) ) ).
% ext
thf(fact_48_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_49_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_50_lfinite_Ocases,axiom,
! [A: $tType,A2: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ A2 )
=> ( ( A2
!= ( coinductive_LNil @ A ) )
=> ~ ! [Xs2: coinductive_llist @ A] :
( ? [X3: A] :
( A2
= ( coinductive_LCons @ A @ X3 @ Xs2 ) )
=> ~ ( coinductive_lfinite @ A @ Xs2 ) ) ) ) ).
% lfinite.cases
thf(fact_51_lfinite_Osimps,axiom,
! [A: $tType] :
( ( coinductive_lfinite @ A )
= ( ^ [A4: coinductive_llist @ A] :
( ( A4
= ( coinductive_LNil @ A ) )
| ? [Xs3: coinductive_llist @ A,X2: A] :
( ( A4
= ( coinductive_LCons @ A @ X2 @ Xs3 ) )
& ( coinductive_lfinite @ A @ Xs3 ) ) ) ) ) ).
% lfinite.simps
thf(fact_52_llimit__induct,axiom,
! [A: $tType,P: ( coinductive_llist @ A ) > $o,Xs: coinductive_llist @ A] :
( ( P @ ( coinductive_LNil @ A ) )
=> ( ! [X3: A,Xs2: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ Xs2 )
=> ( ( P @ Xs2 )
=> ( P @ ( coinductive_LCons @ A @ X3 @ Xs2 ) ) ) )
=> ( ( ! [Ys2: coinductive_llist @ A] :
( ( coindu1478340336prefix @ A @ Ys2 @ Xs )
=> ( P @ Ys2 ) )
=> ( P @ Xs ) )
=> ( P @ Xs ) ) ) ) ).
% llimit_induct
thf(fact_53_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_54_neq__LNil__conv,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( Xs
!= ( coinductive_LNil @ A ) )
= ( ? [X2: A,Xs4: coinductive_llist @ A] :
( Xs
= ( coinductive_LCons @ A @ X2 @ Xs4 ) ) ) ) ).
% neq_LNil_conv
thf(fact_55_not__lnull__conv,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( ~ ( coinductive_lnull @ A @ Xs ) )
= ( ? [X2: A,Xs4: coinductive_llist @ A] :
( Xs
= ( coinductive_LCons @ A @ X2 @ Xs4 ) ) ) ) ).
% not_lnull_conv
thf(fact_56_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_57_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,X3: A] :
( ( coinductive_lfinite @ A @ Xs2 )
=> ( ( P @ Xs2 )
=> ( P @ ( coinductive_LCons @ A @ X3 @ Xs2 ) ) ) )
=> ( P @ X ) ) ) ) ).
% lfinite.inducts
thf(fact_58_lnull__def,axiom,
! [A: $tType] :
( ( coinductive_lnull @ A )
= ( ^ [Llist2: coinductive_llist @ A] :
( Llist2
= ( coinductive_LNil @ A ) ) ) ) ).
% lnull_def
thf(fact_59_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_60_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_61_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_62_lfinite__LNil,axiom,
! [A: $tType] : ( coinductive_lfinite @ A @ ( coinductive_LNil @ A ) ) ).
% lfinite_LNil
thf(fact_63_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_64_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_65_lmirror__def,axiom,
! [A: $tType] :
( ( lMirro427583474mirror @ A )
= ( lMirro999291890or_aux @ A @ ( coinductive_LNil @ A ) ) ) ).
% lmirror_def
thf(fact_66_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_67_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_68_llast__singleton,axiom,
! [A: $tType,X: A] :
( ( coinductive_llast @ A @ ( coinductive_LCons @ A @ X @ ( coinductive_LNil @ A ) ) )
= X ) ).
% llast_singleton
thf(fact_69_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_70_lmember__code_I1_J,axiom,
! [A: $tType,X: A] :
~ ( coinductive_lmember @ A @ X @ ( coinductive_LNil @ A ) ) ).
% lmember_code(1)
thf(fact_71_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_72_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_73_gen__lset__code_I1_J,axiom,
! [A: $tType,A3: set @ A] :
( ( coinductive_gen_lset @ A @ A3 @ ( coinductive_LNil @ A ) )
= A3 ) ).
% gen_lset_code(1)
thf(fact_74_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_75_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_76_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_77_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_78_lappend__LNil2,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( coinductive_lappend @ A @ Xs @ ( coinductive_LNil @ A ) )
= Xs ) ).
% lappend_LNil2
thf(fact_79_lappend__code_I1_J,axiom,
! [A: $tType,Ys: coinductive_llist @ A] :
( ( coinductive_lappend @ A @ ( coinductive_LNil @ A ) @ Ys )
= Ys ) ).
% lappend_code(1)
thf(fact_80_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_81_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_82_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_83_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_84_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_85_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_86_lhd__LCons,axiom,
! [A: $tType,X21: A,X22: coinductive_llist @ A] :
( ( coinductive_lhd @ A @ ( coinductive_LCons @ A @ X21 @ X22 ) )
= X21 ) ).
% lhd_LCons
thf(fact_87_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_88_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_89_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_90_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_91_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_92_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_93_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_94_lappend__LNil__LNil,axiom,
! [A: $tType] :
( ( coinductive_lappend @ A @ ( coinductive_LNil @ A ) @ ( coinductive_LNil @ A ) )
= ( coinductive_LNil @ A ) ) ).
% lappend_LNil_LNil
thf(fact_95_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_96_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_97_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_98_llist_Ocoinduct__strong,axiom,
! [A: $tType,R: ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o,Llist: coinductive_llist @ A,Llist3: coinductive_llist @ A] :
( ( R @ Llist @ Llist3 )
=> ( ! [Llist4: coinductive_llist @ A,Llist5: coinductive_llist @ A] :
( ( R @ Llist4 @ Llist5 )
=> ( ( ( coinductive_lnull @ A @ Llist4 )
= ( coinductive_lnull @ A @ Llist5 ) )
& ( ~ ( coinductive_lnull @ A @ Llist4 )
=> ( ~ ( coinductive_lnull @ A @ Llist5 )
=> ( ( ( coinductive_lhd @ A @ Llist4 )
= ( coinductive_lhd @ A @ Llist5 ) )
& ( ( R @ ( coinductive_ltl @ A @ Llist4 ) @ ( coinductive_ltl @ A @ Llist5 ) )
| ( ( coinductive_ltl @ A @ Llist4 )
= ( coinductive_ltl @ A @ Llist5 ) ) ) ) ) ) ) )
=> ( Llist = Llist3 ) ) ) ).
% llist.coinduct_strong
thf(fact_99_llist_Ocoinduct,axiom,
! [A: $tType,R: ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o,Llist: coinductive_llist @ A,Llist3: coinductive_llist @ A] :
( ( R @ Llist @ Llist3 )
=> ( ! [Llist4: coinductive_llist @ A,Llist5: coinductive_llist @ A] :
( ( R @ Llist4 @ Llist5 )
=> ( ( ( coinductive_lnull @ A @ Llist4 )
= ( coinductive_lnull @ A @ Llist5 ) )
& ( ~ ( coinductive_lnull @ A @ Llist4 )
=> ( ~ ( coinductive_lnull @ A @ Llist5 )
=> ( ( ( coinductive_lhd @ A @ Llist4 )
= ( coinductive_lhd @ A @ Llist5 ) )
& ( R @ ( coinductive_ltl @ A @ Llist4 ) @ ( coinductive_ltl @ A @ Llist5 ) ) ) ) ) ) )
=> ( Llist = Llist3 ) ) ) ).
% llist.coinduct
thf(fact_100_llist_Oexpand,axiom,
! [A: $tType,Llist: coinductive_llist @ A,Llist3: coinductive_llist @ A] :
( ( ( coinductive_lnull @ A @ Llist )
= ( coinductive_lnull @ A @ Llist3 ) )
=> ( ( ~ ( coinductive_lnull @ A @ Llist )
=> ( ~ ( coinductive_lnull @ A @ Llist3 )
=> ( ( ( coinductive_lhd @ A @ Llist )
= ( coinductive_lhd @ A @ Llist3 ) )
& ( ( coinductive_ltl @ A @ Llist )
= ( coinductive_ltl @ A @ Llist3 ) ) ) ) )
=> ( Llist = Llist3 ) ) ) ).
% llist.expand
thf(fact_101_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_102_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_103_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_104_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_105_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_106_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_107_lfinite__rev__induct,axiom,
! [A: $tType,Xs: coinductive_llist @ A,P: ( coinductive_llist @ A ) > $o] :
( ( coinductive_lfinite @ A @ Xs )
=> ( ( P @ ( coinductive_LNil @ A ) )
=> ( ! [X3: A,Xs2: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ Xs2 )
=> ( ( P @ Xs2 )
=> ( P @ ( coinductive_lappend @ A @ Xs2 @ ( coinductive_LCons @ A @ X3 @ ( coinductive_LNil @ A ) ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% lfinite_rev_induct
thf(fact_108_lzip_Ocode,axiom,
! [B: $tType,A: $tType] :
( ( coinductive_lzip @ A @ B )
= ( ^ [Xs3: coinductive_llist @ A,Ys3: coinductive_llist @ B] :
( if @ ( coinductive_llist @ ( product_prod @ A @ B ) )
@ ( ( coinductive_lnull @ A @ Xs3 )
| ( coinductive_lnull @ B @ Ys3 ) )
@ ( coinductive_LNil @ ( product_prod @ A @ B ) )
@ ( coinductive_LCons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ ( coinductive_lhd @ A @ Xs3 ) @ ( coinductive_lhd @ B @ Ys3 ) ) @ ( coinductive_lzip @ A @ B @ ( coinductive_ltl @ A @ Xs3 ) @ ( coinductive_ltl @ B @ Ys3 ) ) ) ) ) ) ).
% lzip.code
thf(fact_109_llexord__conv,axiom,
! [A: $tType] :
( ( coinductive_llexord @ A )
= ( ^ [R2: A > A > $o,Xs3: coinductive_llist @ A,Ys3: coinductive_llist @ A] :
( ( Xs3 = Ys3 )
| ? [Zs2: coinductive_llist @ A,Xs4: coinductive_llist @ A,Y2: A,Ys4: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ Zs2 )
& ( Xs3
= ( coinductive_lappend @ A @ Zs2 @ Xs4 ) )
& ( Ys3
= ( coinductive_lappend @ A @ Zs2 @ ( coinductive_LCons @ A @ Y2 @ Ys4 ) ) )
& ( ( Xs4
= ( coinductive_LNil @ A ) )
| ( R2 @ ( coinductive_lhd @ A @ Xs4 ) @ Y2 ) ) ) ) ) ) ).
% llexord_conv
thf(fact_110_llist_Osplit__sel__asm,axiom,
! [B: $tType,A: $tType,P: B > $o,F1: B,F2: A > ( coinductive_llist @ A ) > B,Llist: coinductive_llist @ A] :
( ( P @ ( coindu1381640503_llist @ B @ A @ F1 @ F2 @ Llist ) )
= ( ~ ( ( ( Llist
= ( coinductive_LNil @ A ) )
& ~ ( P @ F1 ) )
| ( ( Llist
= ( coinductive_LCons @ A @ ( coinductive_lhd @ A @ Llist ) @ ( coinductive_ltl @ A @ Llist ) ) )
& ~ ( P @ ( F2 @ ( coinductive_lhd @ A @ Llist ) @ ( coinductive_ltl @ A @ Llist ) ) ) ) ) ) ) ).
% llist.split_sel_asm
thf(fact_111_llist_Osplit__sel,axiom,
! [B: $tType,A: $tType,P: B > $o,F1: B,F2: A > ( coinductive_llist @ A ) > B,Llist: coinductive_llist @ A] :
( ( P @ ( coindu1381640503_llist @ B @ A @ F1 @ F2 @ Llist ) )
= ( ( ( Llist
= ( coinductive_LNil @ A ) )
=> ( P @ F1 ) )
& ( ( Llist
= ( coinductive_LCons @ A @ ( coinductive_lhd @ A @ Llist ) @ ( coinductive_ltl @ A @ Llist ) ) )
=> ( P @ ( F2 @ ( coinductive_lhd @ A @ Llist ) @ ( coinductive_ltl @ A @ Llist ) ) ) ) ) ) ).
% llist.split_sel
thf(fact_112_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_113_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_114_llexord__refl,axiom,
! [A: $tType,R3: A > A > $o,Xs: coinductive_llist @ A] : ( coinductive_llexord @ A @ R3 @ Xs @ Xs ) ).
% llexord_refl
thf(fact_115_ltakeWhile__LNil,axiom,
! [A: $tType,P: A > $o] :
( ( coindu501562517eWhile @ A @ P @ ( coinductive_LNil @ A ) )
= ( coinductive_LNil @ A ) ) ).
% ltakeWhile_LNil
thf(fact_116_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_117_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_118_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_119_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_120_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_121_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_122_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_123_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_124_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_125_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_126_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_127_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,B3: A] :
( ( R3 @ A5 @ B3 )
=> ~ ( R3 @ B3 @ A5 ) )
=> ( Xs = Ys ) ) ) ) ).
% llexord_antisym
thf(fact_128_llexord__linear,axiom,
! [A: $tType,R3: A > A > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ! [X3: A,Y3: A] :
( ( R3 @ X3 @ Y3 )
| ( X3 = Y3 )
| ( R3 @ Y3 @ X3 ) )
=> ( ( coinductive_llexord @ A @ R3 @ Xs @ Ys )
| ( coinductive_llexord @ A @ R3 @ Ys @ Xs ) ) ) ).
% llexord_linear
thf(fact_129_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,B3: A,C2: A] :
( ( R3 @ A5 @ B3 )
=> ( ( R3 @ B3 @ C2 )
=> ( R3 @ A5 @ C2 ) ) )
=> ( coinductive_llexord @ A @ R3 @ Xs @ Zs ) ) ) ) ).
% llexord_trans
thf(fact_130_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_131_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,Ys4: coinductive_llist @ A] :
( ( Ys
= ( coinductive_LCons @ A @ Y2 @ Ys4 ) )
& ( ( ( X = Y2 )
& ( coinductive_llexord @ A @ R3 @ Xs @ Ys4 ) )
| ( R3 @ X @ Y2 ) ) ) ) ) ).
% llexord_LCons_left
thf(fact_132_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_133_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_134_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_135_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_136_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_137_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_138_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_139_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_140_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_141_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_142_llist_Osimps_I5_J,axiom,
! [B: $tType,A: $tType,F1: B,F2: A > ( coinductive_llist @ A ) > B,X21: A,X22: coinductive_llist @ A] :
( ( coindu1381640503_llist @ B @ A @ F1 @ F2 @ ( coinductive_LCons @ A @ X21 @ X22 ) )
= ( F2 @ X21 @ X22 ) ) ).
% llist.simps(5)
thf(fact_143_llist_Osimps_I4_J,axiom,
! [A: $tType,B: $tType,F1: B,F2: A > ( coinductive_llist @ A ) > B] :
( ( coindu1381640503_llist @ B @ A @ F1 @ F2 @ ( coinductive_LNil @ A ) )
= F1 ) ).
% llist.simps(4)
thf(fact_144_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_145_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_146_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_147_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_148_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_149_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_150_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 ) )
= ( ? [X2: A,Xs4: coinductive_llist @ A,Y2: B,Ys4: coinductive_llist @ B] :
( ( Xs
= ( coinductive_LCons @ A @ X2 @ Xs4 ) )
& ( Ys
= ( coinductive_LCons @ B @ Y2 @ Ys4 ) )
& ( Z
= ( product_Pair @ A @ B @ X2 @ Y2 ) )
& ( Zs
= ( coinductive_lzip @ A @ B @ Xs4 @ Ys4 ) ) ) ) ) ).
% lzip_eq_LCons_conv
thf(fact_151_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_152_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,X3: A] :
( ( A1
= ( coinductive_LCons @ A @ X3 @ Xs2 ) )
=> ( ( A22
= ( coinductive_LCons @ A @ X3 @ Ys5 ) )
=> ~ ( coinductive_llexord @ A @ R3 @ Xs2 @ Ys5 ) ) )
=> ( ! [X3: A] :
( ? [Xs2: coinductive_llist @ A] :
( A1
= ( coinductive_LCons @ A @ X3 @ Xs2 ) )
=> ! [Y3: A] :
( ? [Ys5: coinductive_llist @ A] :
( A22
= ( coinductive_LCons @ A @ Y3 @ Ys5 ) )
=> ~ ( R3 @ X3 @ Y3 ) ) )
=> ~ ( ( A1
= ( coinductive_LNil @ A ) )
=> ! [Ys5: coinductive_llist @ A] : A22 != Ys5 ) ) ) ) ).
% llexord.cases
thf(fact_153_llexord_Osimps,axiom,
! [A: $tType] :
( ( coinductive_llexord @ A )
= ( ^ [R2: A > A > $o,A12: coinductive_llist @ A,A23: coinductive_llist @ A] :
( ? [Xs3: coinductive_llist @ A,Ys3: coinductive_llist @ A,X2: A] :
( ( A12
= ( coinductive_LCons @ A @ X2 @ Xs3 ) )
& ( A23
= ( coinductive_LCons @ A @ X2 @ Ys3 ) )
& ( coinductive_llexord @ A @ R2 @ Xs3 @ Ys3 ) )
| ? [X2: A,Y2: A,Xs3: coinductive_llist @ A,Ys3: coinductive_llist @ A] :
( ( A12
= ( coinductive_LCons @ A @ X2 @ Xs3 ) )
& ( A23
= ( coinductive_LCons @ A @ Y2 @ Ys3 ) )
& ( R2 @ X2 @ Y2 ) )
| ? [Ys3: coinductive_llist @ A] :
( ( A12
= ( coinductive_LNil @ A ) )
& ( A23 = Ys3 ) ) ) ) ) ).
% llexord.simps
thf(fact_154_llexord_Ocoinduct,axiom,
! [A: $tType,X4: ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o,X: coinductive_llist @ A,Xa: coinductive_llist @ A,R3: A > A > $o] :
( ( X4 @ X @ Xa )
=> ( ! [X3: coinductive_llist @ A,Xa2: coinductive_llist @ A] :
( ( X4 @ X3 @ Xa2 )
=> ( ? [Xs5: coinductive_llist @ A,Ys2: coinductive_llist @ A,Xb: A] :
( ( X3
= ( coinductive_LCons @ A @ Xb @ Xs5 ) )
& ( Xa2
= ( 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] :
( ( X3
= ( coinductive_LCons @ A @ Xb @ Xs5 ) )
& ( Xa2
= ( coinductive_LCons @ A @ Y4 @ Ys2 ) )
& ( R3 @ Xb @ Y4 ) )
| ? [Ys2: coinductive_llist @ A] :
( ( X3
= ( coinductive_LNil @ A ) )
& ( Xa2 = Ys2 ) ) ) )
=> ( coinductive_llexord @ A @ R3 @ X @ Xa ) ) ) ).
% llexord.coinduct
thf(fact_155_unfold__llist_Ocode,axiom,
! [B: $tType,A: $tType] :
( ( coindu1441602521_llist @ A @ B )
= ( ^ [P4: A > $o,G212: A > B,G222: A > A,A4: A] : ( if @ ( coinductive_llist @ B ) @ ( P4 @ A4 ) @ ( coinductive_LNil @ B ) @ ( coinductive_LCons @ B @ ( G212 @ A4 ) @ ( coindu1441602521_llist @ A @ B @ P4 @ G212 @ G222 @ ( G222 @ A4 ) ) ) ) ) ) ).
% unfold_llist.code
thf(fact_156_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_157_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_158_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_159_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_160_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_161_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_162_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_163_llist_Ocase__eq__if,axiom,
! [A: $tType,B: $tType] :
( ( coindu1381640503_llist @ B @ A )
= ( ^ [F12: B,F22: A > ( coinductive_llist @ A ) > B,Llist2: coinductive_llist @ A] : ( if @ B @ ( coinductive_lnull @ A @ Llist2 ) @ F12 @ ( F22 @ ( coinductive_lhd @ A @ Llist2 ) @ ( coinductive_ltl @ A @ Llist2 ) ) ) ) ) ).
% llist.case_eq_if
thf(fact_164_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_165_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_166_old_Oprod_Oinject,axiom,
! [A: $tType,B: $tType,A2: A,B2: B,A6: A,B4: B] :
( ( ( product_Pair @ A @ B @ A2 @ B2 )
= ( product_Pair @ A @ B @ A6 @ B4 ) )
= ( ( A2 = A6 )
& ( B2 = B4 ) ) ) ).
% old.prod.inject
thf(fact_167_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_168_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_169_lconcat__LNil,axiom,
! [A: $tType] :
( ( coinductive_lconcat @ A @ ( coinductive_LNil @ ( coinductive_llist @ A ) ) )
= ( coinductive_LNil @ A ) ) ).
% lconcat_LNil
thf(fact_170_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_171_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_172_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_173_llast__LNil,axiom,
! [A: $tType] :
( ( coinductive_llast @ A @ ( coinductive_LNil @ A ) )
= ( undefined @ A ) ) ).
% llast_LNil
thf(fact_174_llast__linfinite,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ~ ( coinductive_lfinite @ A @ Xs )
=> ( ( coinductive_llast @ A @ Xs )
= ( undefined @ A ) ) ) ).
% llast_linfinite
thf(fact_175_old_Oprod_Oinducts,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
( ! [A5: A,B3: B] : ( P @ ( product_Pair @ A @ B @ A5 @ B3 ) )
=> ( P @ Prod ) ) ).
% old.prod.inducts
thf(fact_176_old_Oprod_Oexhaust,axiom,
! [A: $tType,B: $tType,Y: product_prod @ A @ B] :
~ ! [A5: A,B3: B] :
( Y
!= ( product_Pair @ A @ B @ A5 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_177_prod__induct7,axiom,
! [G2: $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 @ G2 ) ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G2 ) ) ) ) )] :
( ! [A5: A,B3: B,C2: C,D2: D,E2: E,F4: F3,G3: G2] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G2 ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G2 ) ) ) ) @ B3 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G2 ) ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G2 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F3 @ G2 ) @ E2 @ ( product_Pair @ F3 @ G2 @ F4 @ G3 ) ) ) ) ) ) )
=> ( P @ X ) ) ).
% prod_induct7
thf(fact_178_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,B3: 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 ) ) ) @ B3 @ ( 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_179_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,B3: 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 ) ) @ B3 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C2 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) )
=> ( P @ X ) ) ).
% prod_induct5
thf(fact_180_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,B3: B,C2: C,D2: D] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B3 @ ( product_Pair @ C @ D @ C2 @ D2 ) ) ) )
=> ( P @ X ) ) ).
% prod_induct4
thf(fact_181_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,B3: B,C2: C] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B3 @ C2 ) ) )
=> ( P @ X ) ) ).
% prod_induct3
thf(fact_182_prod__cases7,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F3: $tType,G2: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G2 ) ) ) ) )] :
~ ! [A5: A,B3: B,C2: C,D2: D,E2: E,F4: F3,G3: G2] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G2 ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G2 ) ) ) ) @ B3 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G2 ) ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G2 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F3 @ G2 ) @ E2 @ ( product_Pair @ F3 @ G2 @ F4 @ G3 ) ) ) ) ) ) ) ).
% prod_cases7
thf(fact_183_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,B3: 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 ) ) ) @ B3 @ ( 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_184_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,B3: 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 ) ) @ B3 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C2 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) ) ).
% prod_cases5
thf(fact_185_prod__cases4,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
~ ! [A5: A,B3: B,C2: C,D2: D] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B3 @ ( product_Pair @ C @ D @ C2 @ D2 ) ) ) ) ).
% prod_cases4
thf(fact_186_prod__cases3,axiom,
! [A: $tType,B: $tType,C: $tType,Y: product_prod @ A @ ( product_prod @ B @ C )] :
~ ! [A5: A,B3: B,C2: C] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B3 @ C2 ) ) ) ).
% prod_cases3
thf(fact_187_Pair__inject,axiom,
! [A: $tType,B: $tType,A2: A,B2: B,A6: A,B4: B] :
( ( ( product_Pair @ A @ B @ A2 @ B2 )
= ( product_Pair @ A @ B @ A6 @ B4 ) )
=> ~ ( ( A2 = A6 )
=> ( B2 != B4 ) ) ) ).
% Pair_inject
thf(fact_188_prod__cases,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P3: product_prod @ A @ B] :
( ! [A5: A,B3: B] : ( P @ ( product_Pair @ A @ B @ A5 @ B3 ) )
=> ( P @ P3 ) ) ).
% prod_cases
thf(fact_189_surj__pair,axiom,
! [A: $tType,B: $tType,P3: product_prod @ A @ B] :
? [X3: A,Y3: B] :
( P3
= ( product_Pair @ A @ B @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_190_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_191_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_192_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_193_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_194_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_195_ord_OLNil,axiom,
! [A: $tType,Less_eq: A > A > $o] : ( coinductive_lsorted @ A @ Less_eq @ ( coinductive_LNil @ A ) ) ).
% ord.LNil
thf(fact_196_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_197_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_198_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_199_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_200_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 ) )
=> ( ! [X3: A] :
( A2
!= ( coinductive_LCons @ A @ X3 @ ( coinductive_LNil @ A ) ) )
=> ~ ! [X3: A,Y3: A,Xs2: coinductive_llist @ A] :
( ( A2
= ( coinductive_LCons @ A @ X3 @ ( coinductive_LCons @ A @ Y3 @ Xs2 ) ) )
=> ( ( Less_eq @ X3 @ Y3 )
=> ~ ( coinductive_lsorted @ A @ Less_eq @ ( coinductive_LCons @ A @ Y3 @ Xs2 ) ) ) ) ) ) ) ).
% ord.lsorted.cases
thf(fact_201_ord_Olsorted_Osimps,axiom,
! [A: $tType] :
( ( coinductive_lsorted @ A )
= ( ^ [Less_eq2: A > A > $o,A4: coinductive_llist @ A] :
( ( A4
= ( coinductive_LNil @ A ) )
| ? [X2: A] :
( A4
= ( coinductive_LCons @ A @ X2 @ ( coinductive_LNil @ A ) ) )
| ? [X2: A,Y2: A,Xs3: coinductive_llist @ A] :
( ( A4
= ( coinductive_LCons @ A @ X2 @ ( coinductive_LCons @ A @ Y2 @ Xs3 ) ) )
& ( Less_eq2 @ X2 @ Y2 )
& ( coinductive_lsorted @ A @ Less_eq2 @ ( coinductive_LCons @ A @ Y2 @ Xs3 ) ) ) ) ) ) ).
% ord.lsorted.simps
thf(fact_202_ord_Olsorted_Ocoinduct,axiom,
! [A: $tType,X4: ( coinductive_llist @ A ) > $o,X: coinductive_llist @ A,Less_eq: A > A > $o] :
( ( X4 @ X )
=> ( ! [X3: coinductive_llist @ A] :
( ( X4 @ X3 )
=> ( ( X3
= ( coinductive_LNil @ A ) )
| ? [Xa3: A] :
( X3
= ( coinductive_LCons @ A @ Xa3 @ ( coinductive_LNil @ A ) ) )
| ? [Xa3: A,Y4: A,Xs5: coinductive_llist @ A] :
( ( X3
= ( coinductive_LCons @ A @ Xa3 @ ( coinductive_LCons @ A @ Y4 @ Xs5 ) ) )
& ( Less_eq @ Xa3 @ 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_203_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_204_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_205_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_206_ssubst__Pair__rhs,axiom,
! [B: $tType,A: $tType,R3: A,S: B,R: set @ ( product_prod @ A @ B ),S2: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R3 @ S ) @ R )
=> ( ( S2 = S )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R3 @ S2 ) @ R ) ) ) ).
% ssubst_Pair_rhs
thf(fact_207_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_208_lprefix__refl,axiom,
! [A: $tType,Xs: coinductive_llist @ A] : ( coinductive_lprefix @ A @ Xs @ Xs ) ).
% lprefix_refl
thf(fact_209_llist_Oleq__refl,axiom,
! [A: $tType,X: coinductive_llist @ A] : ( coinductive_lprefix @ A @ X @ X ) ).
% llist.leq_refl
thf(fact_210_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_211_lprefix__code_I1_J,axiom,
! [A: $tType,Ys: coinductive_llist @ A] : ( coinductive_lprefix @ A @ ( coinductive_LNil @ A ) @ Ys ) ).
% lprefix_code(1)
thf(fact_212_lprefix__LNil,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( coinductive_lprefix @ A @ Xs @ ( coinductive_LNil @ A ) )
= ( coinductive_lnull @ A @ Xs ) ) ).
% lprefix_LNil
thf(fact_213_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_214_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_215_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_216_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_217_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_218_lprefix__conv__lappend,axiom,
! [A: $tType] :
( ( coinductive_lprefix @ A )
= ( ^ [Xs3: coinductive_llist @ A,Ys3: coinductive_llist @ A] :
? [Zs2: coinductive_llist @ A] :
( Ys3
= ( coinductive_lappend @ A @ Xs3 @ Zs2 ) ) ) ) ).
% lprefix_conv_lappend
thf(fact_219_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_220_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_221_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_222_LNil__lprefix,axiom,
! [A: $tType,Xs: coinductive_llist @ A] : ( coinductive_lprefix @ A @ ( coinductive_LNil @ A ) @ Xs ) ).
% LNil_lprefix
thf(fact_223_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 )
= ( ? [Ys4: coinductive_llist @ A] :
( ( Ys
= ( coinductive_LCons @ A @ X @ Ys4 ) )
& ( coinductive_lprefix @ A @ Xs @ Ys4 ) ) ) ) ).
% LCons_lprefix_conv
thf(fact_224_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_225_lprefixI,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,X4: set @ ( product_prod @ ( coinductive_llist @ A ) @ ( coinductive_llist @ A ) )] :
( ( member @ ( product_prod @ ( coinductive_llist @ A ) @ ( coinductive_llist @ A ) ) @ ( product_Pair @ ( coinductive_llist @ A ) @ ( coinductive_llist @ A ) @ Xs @ Ys ) @ X4 )
=> ( ! [Xs2: coinductive_llist @ A,Ys5: coinductive_llist @ A] :
( ( member @ ( product_prod @ ( coinductive_llist @ A ) @ ( coinductive_llist @ A ) ) @ ( product_Pair @ ( coinductive_llist @ A ) @ ( coinductive_llist @ A ) @ Xs2 @ Ys5 ) @ X4 )
=> ( ( coinductive_lnull @ A @ Xs2 )
| ? [X5: A,Xs6: coinductive_llist @ A,Ys6: coinductive_llist @ A] :
( ( Xs2
= ( coinductive_LCons @ A @ X5 @ Xs6 ) )
& ( Ys5
= ( coinductive_LCons @ A @ X5 @ Ys6 ) )
& ( ( member @ ( product_prod @ ( coinductive_llist @ A ) @ ( coinductive_llist @ A ) ) @ ( product_Pair @ ( coinductive_llist @ A ) @ ( coinductive_llist @ A ) @ Xs6 @ Ys6 ) @ X4 )
| ( coinductive_lprefix @ A @ Xs6 @ Ys6 ) ) ) ) )
=> ( coinductive_lprefix @ A @ Xs @ Ys ) ) ) ).
% lprefixI
thf(fact_226_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_227_lprefix_Ocases,axiom,
! [A: $tType,A1: coinductive_llist @ A,A22: coinductive_llist @ A] :
( ( coinductive_lprefix @ A @ A1 @ A22 )
=> ( ( ( A1
= ( coinductive_LNil @ A ) )
=> ! [Xs2: coinductive_llist @ A] : A22 != Xs2 )
=> ~ ! [Xs2: coinductive_llist @ A,Ys5: coinductive_llist @ A,X3: A] :
( ( A1
= ( coinductive_LCons @ A @ X3 @ Xs2 ) )
=> ( ( A22
= ( coinductive_LCons @ A @ X3 @ Ys5 ) )
=> ~ ( coinductive_lprefix @ A @ Xs2 @ Ys5 ) ) ) ) ) ).
% lprefix.cases
thf(fact_228_lprefix_Osimps,axiom,
! [A: $tType] :
( ( coinductive_lprefix @ A )
= ( ^ [A12: coinductive_llist @ A,A23: coinductive_llist @ A] :
( ? [Xs3: coinductive_llist @ A] :
( ( A12
= ( coinductive_LNil @ A ) )
& ( A23 = Xs3 ) )
| ? [Xs3: coinductive_llist @ A,Ys3: coinductive_llist @ A,X2: A] :
( ( A12
= ( coinductive_LCons @ A @ X2 @ Xs3 ) )
& ( A23
= ( coinductive_LCons @ A @ X2 @ Ys3 ) )
& ( coinductive_lprefix @ A @ Xs3 @ Ys3 ) ) ) ) ) ).
% lprefix.simps
thf(fact_229_lprefix_Ocoinduct,axiom,
! [A: $tType,X4: ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o,X: coinductive_llist @ A,Xa: coinductive_llist @ A] :
( ( X4 @ X @ Xa )
=> ( ! [X3: coinductive_llist @ A,Xa2: coinductive_llist @ A] :
( ( X4 @ X3 @ Xa2 )
=> ( ? [Xs5: coinductive_llist @ A] :
( ( X3
= ( coinductive_LNil @ A ) )
& ( Xa2 = Xs5 ) )
| ? [Xs5: coinductive_llist @ A,Ys2: coinductive_llist @ A,Xb: A] :
( ( X3
= ( coinductive_LCons @ A @ Xb @ Xs5 ) )
& ( Xa2
= ( coinductive_LCons @ A @ Xb @ Ys2 ) )
& ( ( X4 @ Xs5 @ Ys2 )
| ( coinductive_lprefix @ A @ Xs5 @ Ys2 ) ) ) ) )
=> ( coinductive_lprefix @ A @ X @ Xa ) ) ) ).
% lprefix.coinduct
thf(fact_230_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 ) )
| ? [Xs4: coinductive_llist @ A] :
( ( Xs
= ( coinductive_LCons @ A @ Y @ Xs4 ) )
& ( coinductive_lprefix @ A @ Xs4 @ Ys ) ) ) ) ).
% lprefix_LCons_conv
thf(fact_231_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_232_lprefix__imp__llexord,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,R3: A > A > $o] :
( ( coinductive_lprefix @ A @ Xs @ Ys )
=> ( coinductive_llexord @ A @ R3 @ Xs @ Ys ) ) ).
% lprefix_imp_llexord
thf(fact_233_lprefix__ltakeWhile,axiom,
! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] : ( coinductive_lprefix @ A @ ( coindu501562517eWhile @ A @ P @ Xs ) @ Xs ) ).
% lprefix_ltakeWhile
thf(fact_234_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_235_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_236_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_237_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_238_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_239_Coinductive__List_Ofinite__lprefix__def,axiom,
! [A: $tType] :
( ( coindu328551480prefix @ A )
= ( coinductive_lprefix @ A ) ) ).
% Coinductive_List.finite_lprefix_def
thf(fact_240_lstrict__prefix__def,axiom,
! [A: $tType] :
( ( coindu1478340336prefix @ A )
= ( ^ [Xs3: coinductive_llist @ A,Ys3: coinductive_llist @ A] :
( ( coinductive_lprefix @ A @ Xs3 @ Ys3 )
& ( Xs3 != Ys3 ) ) ) ) ).
% lstrict_prefix_def
thf(fact_241_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_242_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_243_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_244_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_245_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_246_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_247_Coinductive__List_Olprefix__nitpick__simps,axiom,
! [A: $tType] :
( ( coinductive_lprefix @ A )
= ( ^ [Xs3: coinductive_llist @ A,Ys3: coinductive_llist @ A] :
( ( ( coinductive_lfinite @ A @ Xs3 )
=> ( coindu328551480prefix @ A @ Xs3 @ Ys3 ) )
& ( ~ ( coinductive_lfinite @ A @ Xs3 )
=> ( Xs3 = Ys3 ) ) ) ) ) ).
% Coinductive_List.lprefix_nitpick_simps
thf(fact_248_lprefix__coinduct,axiom,
! [A: $tType,P: ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( P @ Xs @ Ys )
=> ( ! [Xs2: coinductive_llist @ A,Ys5: coinductive_llist @ A] :
( ( P @ Xs2 @ Ys5 )
=> ( ( ( coinductive_lnull @ A @ Ys5 )
=> ( coinductive_lnull @ A @ Xs2 ) )
& ( ~ ( coinductive_lnull @ A @ Xs2 )
=> ( ~ ( coinductive_lnull @ A @ Ys5 )
=> ( ( ( coinductive_lhd @ A @ Xs2 )
= ( coinductive_lhd @ A @ Ys5 ) )
& ( ( P @ ( coinductive_ltl @ A @ Xs2 ) @ ( coinductive_ltl @ A @ Ys5 ) )
| ( coinductive_lprefix @ A @ ( coinductive_ltl @ A @ Xs2 ) @ ( coinductive_ltl @ A @ Ys5 ) ) ) ) ) ) ) )
=> ( coinductive_lprefix @ A @ Xs @ Ys ) ) ) ).
% lprefix_coinduct
thf(fact_249_llexord__lappend__left,axiom,
! [A: $tType,Xs: coinductive_llist @ A,R3: A > A > $o,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ Xs )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs ) )
=> ~ ( R3 @ X3 @ X3 ) )
=> ( ( coinductive_llexord @ A @ R3 @ ( coinductive_lappend @ A @ Xs @ Ys ) @ ( coinductive_lappend @ A @ Xs @ Zs ) )
= ( coinductive_llexord @ A @ R3 @ Ys @ Zs ) ) ) ) ).
% llexord_lappend_left
thf(fact_250_llexord__lappend__leftD,axiom,
! [A: $tType,R3: A > A > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A] :
( ( coinductive_llexord @ A @ R3 @ ( coinductive_lappend @ A @ Xs @ Ys ) @ ( coinductive_lappend @ A @ Xs @ Zs ) )
=> ( ( coinductive_lfinite @ A @ Xs )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs ) )
=> ~ ( R3 @ X3 @ X3 ) )
=> ( coinductive_llexord @ A @ R3 @ Ys @ Zs ) ) ) ) ).
% llexord_lappend_leftD
thf(fact_251_in__lset__ltlD,axiom,
! [A: $tType,X: A,Xs: coinductive_llist @ A] :
( ( member @ A @ X @ ( coinductive_lset @ A @ ( coinductive_ltl @ A @ Xs ) ) )
=> ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) ) ) ).
% in_lset_ltlD
thf(fact_252_lset__lmember,axiom,
! [A: $tType,X: A,Xs: coinductive_llist @ A] :
( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
= ( coinductive_lmember @ A @ X @ Xs ) ) ).
% lset_lmember
thf(fact_253_in__lset__ldropnD,axiom,
! [A: $tType,X: A,N: nat,Xs: coinductive_llist @ A] :
( ( member @ A @ X @ ( coinductive_lset @ A @ ( coinductive_ldropn @ A @ N @ Xs ) ) )
=> ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) ) ) ).
% in_lset_ldropnD
%----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 (1)
thf(conj_0,conjecture,
( ( coinductive_lfinite @ a @ xsa )
& ( coinductive_lfinite @ a @ acca ) ) ).
%------------------------------------------------------------------------------