TPTP Problem File: COM178^1.p
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%------------------------------------------------------------------------------
% File : COM178^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Computing Theory
% Problem : Koenig's lemma 206
% 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 : koenigslemma__206.p [Bla16]
% Status : Theorem
% Rating : 0.00 v7.2.0, 0.25 v7.1.0
% Syntax : Number of formulae : 305 ( 140 unt; 44 typ; 0 def)
% Number of atoms : 673 ( 290 equ; 0 cnn)
% Maximal formula atoms : 8 ( 2 avg)
% Number of connectives : 4179 ( 161 ~; 53 |; 82 &;3599 @)
% ( 0 <=>; 284 =>; 0 <=; 0 <~>)
% Maximal formula depth : 28 ( 8 avg)
% Number of types : 2 ( 1 usr)
% Number of type conns : 287 ( 287 >; 0 *; 0 +; 0 <<)
% Number of symbols : 46 ( 43 usr; 4 con; 0-6 aty)
% Number of variables : 1150 ( 37 ^;1001 !; 68 ?;1150 :)
% ( 44 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 14:47:17.094
%------------------------------------------------------------------------------
%----Could-be-implicit typings (5)
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_itself,type,
itself: $tType > $tType ).
thf(ty_tf_node,type,
node: $tType ).
%----Explicit typings (39)
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Obot,type,
bot:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_c_Coinductive__List_Ofinite__lprefix,type,
coindu328551480prefix:
!>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List_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_Oldistinct,type,
coindu351974385stinct:
!>[A: $tType] : ( ( coinductive_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List_OldropWhile,type,
coindu218763757pWhile:
!>[A: $tType] : ( ( A > $o ) > ( 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_If,type,
if:
!>[A: $tType] : ( $o > A > A > A ) ).
thf(sy_c_Koenigslemma__Mirabelle__aepjeeakgn_Oconnected,type,
koenig793108494nected:
!>[Node: $tType] : ( ( Node > Node > $o ) > $o ) ).
thf(sy_c_Orderings_Obot__class_Obot,type,
bot_bot:
!>[A: $tType] : 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,
type2:
!>[A: $tType] : ( itself @ A ) ).
thf(sy_c_Relation_Oinv__imagep,type,
inv_imagep:
!>[B: $tType,A: $tType] : ( ( B > B > $o ) > ( A > B ) > A > A > $o ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_Set_Ois__empty,type,
is_empty:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_f____,type,
f: ( product_prod @ node @ ( set @ node ) ) > ( coinductive_llist @ node ) ).
thf(sy_v_graph,type,
graph: node > node > $o ).
thf(sy_v_n,type,
n: node ).
thf(sy_v_ns____,type,
ns: set @ node ).
%----Relevant facts (254)
thf(fact_0__092_060open_062lhd_A_If_A_In_M_Ans_J_J_A_092_060in_062_Alset_A_If_A_In_M_Ans_J_J_092_060close_062,axiom,
member @ node @ ( coinductive_lhd @ node @ ( f @ ( product_Pair @ node @ ( set @ node ) @ n @ ns ) ) ) @ ( coinductive_lset @ node @ ( f @ ( product_Pair @ node @ ( set @ node ) @ n @ ns ) ) ) ).
% \<open>lhd (f (n, ns)) \<in> lset (f (n, ns))\<close>
thf(fact_1_ns__def,axiom,
( ns
= ( bot_bot @ ( set @ node ) ) ) ).
% ns_def
thf(fact_2_f__simps_I2_J,axiom,
! [Na: node,Nsa: set @ node] :
( ( coinductive_lhd @ node @ ( f @ ( product_Pair @ node @ ( set @ node ) @ Na @ Nsa ) ) )
= Na ) ).
% f_simps(2)
thf(fact_3_f__simps_I1_J,axiom,
! [Na: node,Nsa: set @ node] :
~ ( coinductive_lnull @ node @ ( f @ ( product_Pair @ node @ ( set @ node ) @ Na @ Nsa ) ) ) ).
% f_simps(1)
thf(fact_4_prod_Oinject,axiom,
! [A: $tType,B: $tType,X1: A,X2: B,Y1: A,Y2: B] :
( ( ( product_Pair @ A @ B @ X1 @ X2 )
= ( product_Pair @ A @ B @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_5_old_Oprod_Oinject,axiom,
! [A: $tType,B: $tType,A2: A,B2: B,A3: A,B3: B] :
( ( ( product_Pair @ A @ B @ A2 @ B2 )
= ( product_Pair @ A @ B @ A3 @ B3 ) )
= ( ( A2 = A3 )
& ( B2 = B3 ) ) ) ).
% old.prod.inject
thf(fact_6_connected,axiom,
koenig793108494nected @ node @ graph ).
% connected
thf(fact_7_surj__pair,axiom,
! [A: $tType,B: $tType,P: product_prod @ A @ B] :
? [X: A,Y: B] :
( P
= ( product_Pair @ A @ B @ X @ Y ) ) ).
% surj_pair
thf(fact_8_prod__cases,axiom,
! [B: $tType,A: $tType,P2: ( product_prod @ A @ B ) > $o,P: product_prod @ A @ B] :
( ! [A4: A,B4: B] : ( P2 @ ( product_Pair @ A @ B @ A4 @ B4 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_9_Pair__inject,axiom,
! [A: $tType,B: $tType,A2: A,B2: B,A3: A,B3: B] :
( ( ( product_Pair @ A @ B @ A2 @ B2 )
= ( product_Pair @ A @ B @ A3 @ B3 ) )
=> ~ ( ( A2 = A3 )
=> ( B2 != B3 ) ) ) ).
% Pair_inject
thf(fact_10_prod__cases3,axiom,
! [A: $tType,B: $tType,C: $tType,Y3: product_prod @ A @ ( product_prod @ B @ C )] :
~ ! [A4: A,B4: B,C2: C] :
( Y3
!= ( product_Pair @ A @ ( product_prod @ B @ C ) @ A4 @ ( product_Pair @ B @ C @ B4 @ C2 ) ) ) ).
% prod_cases3
thf(fact_11_prod__cases4,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
~ ! [A4: A,B4: B,C2: C,D2: D] :
( Y3
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B4 @ ( product_Pair @ C @ D @ C2 @ D2 ) ) ) ) ).
% prod_cases4
thf(fact_12_prod__cases5,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
~ ! [A4: A,B4: B,C2: C,D2: D,E2: E] :
( Y3
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A4 @ ( 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_13_prod__cases6,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) )] :
~ ! [A4: A,B4: B,C2: C,D2: D,E2: E,F2: F] :
( Y3
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ F ) @ D2 @ ( product_Pair @ E @ F @ E2 @ F2 ) ) ) ) ) ) ).
% prod_cases6
thf(fact_14_old_Oprod_Oinducts,axiom,
! [B: $tType,A: $tType,P2: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
( ! [A4: A,B4: B] : ( P2 @ ( product_Pair @ A @ B @ A4 @ B4 ) )
=> ( P2 @ Prod ) ) ).
% old.prod.inducts
thf(fact_15_old_Oprod_Oexhaust,axiom,
! [A: $tType,B: $tType,Y3: product_prod @ A @ B] :
~ ! [A4: A,B4: B] :
( Y3
!= ( product_Pair @ A @ B @ A4 @ B4 ) ) ).
% old.prod.exhaust
thf(fact_16_prod__induct7,axiom,
! [G: $tType,F: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) )] :
( ! [A4: A,B4: B,C2: C,D2: D,E2: E,F2: F,G2: G] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F @ G ) @ E2 @ ( product_Pair @ F @ G @ F2 @ G2 ) ) ) ) ) ) )
=> ( P2 @ X3 ) ) ).
% prod_induct7
thf(fact_17_prod__induct6,axiom,
! [F: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) )] :
( ! [A4: A,B4: B,C2: C,D2: D,E2: E,F2: F] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ F ) @ D2 @ ( product_Pair @ E @ F @ E2 @ F2 ) ) ) ) ) )
=> ( P2 @ X3 ) ) ).
% prod_induct6
thf(fact_18_prod__induct5,axiom,
! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
( ! [A4: A,B4: B,C2: C,D2: D,E2: E] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A4 @ ( 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 ) ) ) ) )
=> ( P2 @ X3 ) ) ).
% prod_induct5
thf(fact_19_prod__induct4,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
( ! [A4: A,B4: B,C2: C,D2: D] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B4 @ ( product_Pair @ C @ D @ C2 @ D2 ) ) ) )
=> ( P2 @ X3 ) ) ).
% prod_induct4
thf(fact_20_prod__induct3,axiom,
! [C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ C )] :
( ! [A4: A,B4: B,C2: C] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A4 @ ( product_Pair @ B @ C @ B4 @ C2 ) ) )
=> ( P2 @ X3 ) ) ).
% prod_induct3
thf(fact_21_prod__cases7,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F: $tType,G: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) )] :
~ ! [A4: A,B4: B,C2: C,D2: D,E2: E,F2: F,G2: G] :
( Y3
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F @ G ) @ E2 @ ( product_Pair @ F @ G @ F2 @ G2 ) ) ) ) ) ) ) ).
% prod_cases7
thf(fact_22_lset__eq__empty,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( ( coinductive_lset @ A @ Xs )
= ( bot_bot @ ( set @ A ) ) )
= ( coinductive_lnull @ A @ Xs ) ) ).
% lset_eq_empty
thf(fact_23_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_24_llist_Oset__sel_I1_J,axiom,
! [A: $tType,A2: coinductive_llist @ A] :
( ~ ( coinductive_lnull @ A @ A2 )
=> ( member @ A @ ( coinductive_lhd @ A @ A2 ) @ ( coinductive_lset @ A @ A2 ) ) ) ).
% llist.set_sel(1)
thf(fact_25_lset__lnull,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ Xs )
=> ( ( coinductive_lset @ A @ Xs )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% lset_lnull
thf(fact_26_empty__iff,axiom,
! [A: $tType,C3: A] :
~ ( member @ A @ C3 @ ( bot_bot @ ( set @ A ) ) ) ).
% empty_iff
thf(fact_27_all__not__in__conv,axiom,
! [A: $tType,A5: set @ A] :
( ( ! [X4: A] :
~ ( member @ A @ X4 @ A5 ) )
= ( A5
= ( bot_bot @ ( set @ A ) ) ) ) ).
% all_not_in_conv
thf(fact_28_Collect__empty__eq,axiom,
! [A: $tType,P2: A > $o] :
( ( ( collect @ A @ P2 )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X4: A] :
~ ( P2 @ X4 ) ) ) ).
% Collect_empty_eq
thf(fact_29_empty__Collect__eq,axiom,
! [A: $tType,P2: A > $o] :
( ( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ P2 ) )
= ( ! [X4: A] :
~ ( P2 @ X4 ) ) ) ).
% empty_Collect_eq
thf(fact_30_bot__apply,axiom,
! [C: $tType,D: $tType] :
( ( bot @ C @ ( type2 @ C ) )
=> ( ( bot_bot @ ( D > C ) )
= ( ^ [X4: D] : ( bot_bot @ C ) ) ) ) ).
% bot_apply
thf(fact_31_ltakeWhile_Oexhaust,axiom,
! [A: $tType,Xs: coinductive_llist @ A,P2: A > $o] :
( ~ ( ( coinductive_lnull @ A @ Xs )
| ~ ( P2 @ ( coinductive_lhd @ A @ Xs ) ) )
=> ~ ( ~ ( coinductive_lnull @ A @ Xs )
=> ~ ( P2 @ ( coinductive_lhd @ A @ Xs ) ) ) ) ).
% ltakeWhile.exhaust
thf(fact_32_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_33_bot__set__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% bot_set_def
thf(fact_34_bot__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( bot @ B @ ( type2 @ B ) )
=> ( ( bot_bot @ ( A > B ) )
= ( ^ [X4: A] : ( bot_bot @ B ) ) ) ) ).
% bot_fun_def
thf(fact_35_ex__in__conv,axiom,
! [A: $tType,A5: set @ A] :
( ( ? [X4: A] : ( member @ A @ X4 @ A5 ) )
= ( A5
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% ex_in_conv
thf(fact_36_equals0I,axiom,
! [A: $tType,A5: set @ A] :
( ! [Y: A] :
~ ( member @ A @ Y @ A5 )
=> ( A5
= ( bot_bot @ ( set @ A ) ) ) ) ).
% equals0I
thf(fact_37_equals0D,axiom,
! [A: $tType,A5: set @ A,A2: A] :
( ( A5
= ( bot_bot @ ( set @ A ) ) )
=> ~ ( member @ A @ A2 @ A5 ) ) ).
% equals0D
thf(fact_38_emptyE,axiom,
! [A: $tType,A2: A] :
~ ( member @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ).
% emptyE
thf(fact_39_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_40_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_41_lset__code,axiom,
! [A: $tType] :
( ( coinductive_lset @ A )
= ( coinductive_gen_lset @ A @ ( bot_bot @ ( set @ A ) ) ) ) ).
% lset_code
thf(fact_42_Set_Ois__empty__def,axiom,
! [A: $tType] :
( ( is_empty @ A )
= ( ^ [A6: set @ A] :
( A6
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Set.is_empty_def
thf(fact_43_mem__Collect__eq,axiom,
! [A: $tType,A2: A,P2: A > $o] :
( ( member @ A @ A2 @ ( collect @ A @ P2 ) )
= ( P2 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_44_Collect__mem__eq,axiom,
! [A: $tType,A5: set @ A] :
( ( collect @ A
@ ^ [X4: A] : ( member @ A @ X4 @ A5 ) )
= A5 ) ).
% Collect_mem_eq
thf(fact_45_Collect__cong,axiom,
! [A: $tType,P2: A > $o,Q: A > $o] :
( ! [X: A] :
( ( P2 @ X )
= ( Q @ X ) )
=> ( ( collect @ A @ P2 )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_46_ext,axiom,
! [B: $tType,A: $tType,F3: A > B,G3: A > B] :
( ! [X: A] :
( ( F3 @ X )
= ( G3 @ X ) )
=> ( F3 = G3 ) ) ).
% ext
thf(fact_47_lset__lmember,axiom,
! [A: $tType,X3: A,Xs: coinductive_llist @ A] :
( ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs ) )
= ( coinductive_lmember @ A @ X3 @ Xs ) ) ).
% lset_lmember
thf(fact_48_Collect__empty__eq__bot,axiom,
! [A: $tType,P2: A > $o] :
( ( ( collect @ A @ P2 )
= ( bot_bot @ ( set @ A ) ) )
= ( P2
= ( bot_bot @ ( A > $o ) ) ) ) ).
% Collect_empty_eq_bot
thf(fact_49_bot__empty__eq,axiom,
! [A: $tType] :
( ( bot_bot @ ( A > $o ) )
= ( ^ [X4: A] : ( member @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% bot_empty_eq
thf(fact_50_ssubst__Pair__rhs,axiom,
! [B: $tType,A: $tType,R: A,S: B,R2: set @ ( product_prod @ A @ B ),S2: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R @ S ) @ R2 )
=> ( ( S2 = S )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R @ S2 ) @ R2 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_51_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_52_llist__set__induct,axiom,
! [A: $tType,X3: A,Xs: coinductive_llist @ A,P2: A > ( coinductive_llist @ A ) > $o] :
( ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs ) )
=> ( ! [Xs2: coinductive_llist @ A] :
( ~ ( coinductive_lnull @ A @ Xs2 )
=> ( P2 @ ( coinductive_lhd @ A @ Xs2 ) @ Xs2 ) )
=> ( ! [Xs2: coinductive_llist @ A,Y: A] :
( ~ ( coinductive_lnull @ A @ Xs2 )
=> ( ( member @ A @ Y @ ( coinductive_lset @ A @ ( coinductive_ltl @ A @ Xs2 ) ) )
=> ( ( P2 @ Y @ ( coinductive_ltl @ A @ Xs2 ) )
=> ( P2 @ Y @ Xs2 ) ) ) )
=> ( P2 @ X3 @ Xs ) ) ) ) ).
% llist_set_induct
thf(fact_53_lnull__ldropWhile,axiom,
! [A: $tType,P2: A > $o,Xs: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ ( coindu218763757pWhile @ A @ P2 @ Xs ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ ( coinductive_lset @ A @ Xs ) )
=> ( P2 @ X4 ) ) ) ) ).
% lnull_ldropWhile
thf(fact_54_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_55_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_56_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_57_lnull__ltlI,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ Xs )
=> ( coinductive_lnull @ A @ ( coinductive_ltl @ A @ Xs ) ) ) ).
% lnull_ltlI
thf(fact_58_in__lset__ltlD,axiom,
! [A: $tType,X3: A,Xs: coinductive_llist @ A] :
( ( member @ A @ X3 @ ( coinductive_lset @ A @ ( coinductive_ltl @ A @ Xs ) ) )
=> ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs ) ) ) ).
% in_lset_ltlD
thf(fact_59_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_60_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_61_in__lset__ldropWhileD,axiom,
! [A: $tType,X3: A,P2: A > $o,Xs: coinductive_llist @ A] :
( ( member @ A @ X3 @ ( coinductive_lset @ A @ ( coindu218763757pWhile @ A @ P2 @ Xs ) ) )
=> ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs ) ) ) ).
% in_lset_ldropWhileD
thf(fact_62_ldropWhile__cong,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,P2: A > $o,Q: A > $o] :
( ( Xs = Ys )
=> ( ! [X: A] :
( ( member @ A @ X @ ( coinductive_lset @ A @ Ys ) )
=> ( ( P2 @ X )
= ( Q @ X ) ) )
=> ( ( coindu218763757pWhile @ A @ P2 @ Xs )
= ( coindu218763757pWhile @ A @ Q @ Ys ) ) ) ) ).
% ldropWhile_cong
thf(fact_63_llist_Oset__sel_I2_J,axiom,
! [A: $tType,A2: coinductive_llist @ A,X3: A] :
( ~ ( coinductive_lnull @ A @ A2 )
=> ( ( member @ A @ X3 @ ( coinductive_lset @ A @ ( coinductive_ltl @ A @ A2 ) ) )
=> ( member @ A @ X3 @ ( coinductive_lset @ A @ A2 ) ) ) ) ).
% llist.set_sel(2)
thf(fact_64_lset__lzipD2,axiom,
! [A: $tType,B: $tType,X3: A,Y3: B,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ ( coinductive_lset @ ( product_prod @ A @ B ) @ ( coinductive_lzip @ A @ B @ Xs @ Ys ) ) )
=> ( member @ B @ Y3 @ ( coinductive_lset @ B @ Ys ) ) ) ).
% lset_lzipD2
thf(fact_65_lset__lzipD1,axiom,
! [B: $tType,A: $tType,X3: A,Y3: B,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ ( coinductive_lset @ ( product_prod @ A @ B ) @ ( coinductive_lzip @ A @ B @ Xs @ Ys ) ) )
=> ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs ) ) ) ).
% lset_lzipD1
thf(fact_66_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_67_llist_Ocoinduct,axiom,
! [A: $tType,R2: ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o,Llist: coinductive_llist @ A,Llist2: coinductive_llist @ A] :
( ( R2 @ Llist @ Llist2 )
=> ( ! [Llist3: coinductive_llist @ A,Llist4: coinductive_llist @ A] :
( ( R2 @ 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 ) )
& ( R2 @ ( coinductive_ltl @ A @ Llist3 ) @ ( coinductive_ltl @ A @ Llist4 ) ) ) ) ) ) )
=> ( Llist = Llist2 ) ) ) ).
% llist.coinduct
thf(fact_68_llist_Ocoinduct__strong,axiom,
! [A: $tType,R2: ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o,Llist: coinductive_llist @ A,Llist2: coinductive_llist @ A] :
( ( R2 @ Llist @ Llist2 )
=> ( ! [Llist3: coinductive_llist @ A,Llist4: coinductive_llist @ A] :
( ( R2 @ 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 ) )
& ( ( R2 @ ( coinductive_ltl @ A @ Llist3 ) @ ( coinductive_ltl @ A @ Llist4 ) )
| ( ( coinductive_ltl @ A @ Llist3 )
= ( coinductive_ltl @ A @ Llist4 ) ) ) ) ) ) ) )
=> ( Llist = Llist2 ) ) ) ).
% llist.coinduct_strong
thf(fact_69_lhd__ldropWhile__in__lset,axiom,
! [A: $tType,Xs: coinductive_llist @ A,P2: A > $o] :
( ? [X5: A] :
( ( member @ A @ X5 @ ( coinductive_lset @ A @ Xs ) )
& ~ ( P2 @ X5 ) )
=> ( member @ A @ ( coinductive_lhd @ A @ ( coindu218763757pWhile @ A @ P2 @ Xs ) ) @ ( coinductive_lset @ A @ Xs ) ) ) ).
% lhd_ldropWhile_in_lset
thf(fact_70_lhd__ldropWhile,axiom,
! [A: $tType,Xs: coinductive_llist @ A,P2: A > $o] :
( ? [X5: A] :
( ( member @ A @ X5 @ ( coinductive_lset @ A @ Xs ) )
& ~ ( P2 @ X5 ) )
=> ~ ( P2 @ ( coinductive_lhd @ A @ ( coindu218763757pWhile @ A @ P2 @ Xs ) ) ) ) ).
% lhd_ldropWhile
thf(fact_71_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_72_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_73_ldistinct__lhdD,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( coindu351974385stinct @ A @ Xs )
=> ( ~ ( coinductive_lnull @ A @ Xs )
=> ~ ( member @ A @ ( coinductive_lhd @ A @ Xs ) @ ( coinductive_lset @ A @ ( coinductive_ltl @ A @ Xs ) ) ) ) ) ).
% ldistinct_lhdD
thf(fact_74_ldistinct__coinduct,axiom,
! [A: $tType,X6: ( coinductive_llist @ A ) > $o,Xs: coinductive_llist @ A] :
( ( X6 @ Xs )
=> ( ! [Xs2: coinductive_llist @ A] :
( ( X6 @ Xs2 )
=> ( ~ ( coinductive_lnull @ A @ Xs2 )
=> ( ~ ( member @ A @ ( coinductive_lhd @ A @ Xs2 ) @ ( coinductive_lset @ A @ ( coinductive_ltl @ A @ Xs2 ) ) )
& ( ( X6 @ ( coinductive_ltl @ A @ Xs2 ) )
| ( coindu351974385stinct @ A @ ( coinductive_ltl @ A @ Xs2 ) ) ) ) ) )
=> ( coindu351974385stinct @ A @ Xs ) ) ) ).
% ldistinct_coinduct
thf(fact_75_llist_Ocase__eq__if,axiom,
! [A: $tType,B: $tType] :
( ( coindu1381640503_llist @ B @ A )
= ( ^ [F12: B,F22: A > ( coinductive_llist @ A ) > B,Llist5: coinductive_llist @ A] : ( if @ B @ ( coinductive_lnull @ A @ Llist5 ) @ F12 @ ( F22 @ ( coinductive_lhd @ A @ Llist5 ) @ ( coinductive_ltl @ A @ Llist5 ) ) ) ) ) ).
% llist.case_eq_if
thf(fact_76_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_77_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_78_ord_Olsorted__coinduct_H,axiom,
! [A: $tType,X6: ( coinductive_llist @ A ) > $o,Xs: coinductive_llist @ A,Less_eq: A > A > $o] :
( ( X6 @ Xs )
=> ( ! [Xs2: coinductive_llist @ A] :
( ( X6 @ 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 ) ) )
& ( ( X6 @ ( coinductive_ltl @ A @ Xs2 ) )
| ( coinductive_lsorted @ A @ Less_eq @ ( coinductive_ltl @ A @ Xs2 ) ) ) ) ) ) )
=> ( coinductive_lsorted @ A @ Less_eq @ Xs ) ) ) ).
% ord.lsorted_coinduct'
thf(fact_79_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_80_ldropWhile__LCons,axiom,
! [A: $tType,P2: A > $o,X3: A,Xs: coinductive_llist @ A] :
( ( ( P2 @ X3 )
=> ( ( coindu218763757pWhile @ A @ P2 @ ( coinductive_LCons @ A @ X3 @ Xs ) )
= ( coindu218763757pWhile @ A @ P2 @ Xs ) ) )
& ( ~ ( P2 @ X3 )
=> ( ( coindu218763757pWhile @ A @ P2 @ ( coinductive_LCons @ A @ X3 @ Xs ) )
= ( coinductive_LCons @ A @ X3 @ Xs ) ) ) ) ).
% ldropWhile_LCons
thf(fact_81_unfold__llist__eq__LCons,axiom,
! [A: $tType,B: $tType,IS_LNIL: B > $o,LHD: B > A,LTL: B > B,B2: B,X3: A,Xs: coinductive_llist @ A] :
( ( ( coindu1441602521_llist @ B @ A @ IS_LNIL @ LHD @ LTL @ B2 )
= ( coinductive_LCons @ A @ X3 @ Xs ) )
= ( ~ ( IS_LNIL @ B2 )
& ( X3
= ( 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,P: A > $o,G21: A > B,G22: A > A,A2: A] :
( ( coinductive_lnull @ B @ ( coindu1441602521_llist @ A @ B @ P @ G21 @ G22 @ A2 ) )
= ( P @ A2 ) ) ).
% unfold_llist.disc_iff(1)
thf(fact_83_unfold__llist_Odisc__iff_I2_J,axiom,
! [B: $tType,A: $tType,P: A > $o,G21: A > B,G22: A > A,A2: A] :
( ( ~ ( coinductive_lnull @ B @ ( coindu1441602521_llist @ A @ B @ P @ G21 @ G22 @ A2 ) ) )
= ( ~ ( P @ A2 ) ) ) ).
% unfold_llist.disc_iff(2)
thf(fact_84_ldistinct__LCons,axiom,
! [A: $tType,X3: A,Xs: coinductive_llist @ A] :
( ( coindu351974385stinct @ A @ ( coinductive_LCons @ A @ X3 @ Xs ) )
= ( ~ ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs ) )
& ( coindu351974385stinct @ A @ Xs ) ) ) ).
% ldistinct_LCons
thf(fact_85_lzip__simps_I3_J,axiom,
! [C: $tType,B: $tType,X3: C,Xs: coinductive_llist @ C,Y3: B,Ys: coinductive_llist @ B] :
( ( coinductive_lzip @ C @ B @ ( coinductive_LCons @ C @ X3 @ Xs ) @ ( coinductive_LCons @ B @ Y3 @ Ys ) )
= ( coinductive_LCons @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ X3 @ Y3 ) @ ( coinductive_lzip @ C @ B @ Xs @ Ys ) ) ) ).
% lzip_simps(3)
thf(fact_86_ldistinct__lzipI2,axiom,
! [B: $tType,A: $tType,Ys: coinductive_llist @ A,Xs: coinductive_llist @ B] :
( ( coindu351974385stinct @ A @ Ys )
=> ( coindu351974385stinct @ ( product_prod @ B @ A ) @ ( coinductive_lzip @ B @ A @ Xs @ Ys ) ) ) ).
% ldistinct_lzipI2
thf(fact_87_ldistinct__lzipI1,axiom,
! [A: $tType,B: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B] :
( ( coindu351974385stinct @ A @ Xs )
=> ( coindu351974385stinct @ ( product_prod @ A @ B ) @ ( coinductive_lzip @ A @ B @ Xs @ Ys ) ) ) ).
% ldistinct_lzipI1
thf(fact_88_llist_Osimps_I5_J,axiom,
! [B: $tType,A: $tType,F1: B,F23: A > ( coinductive_llist @ A ) > B,X21: A,X22: coinductive_llist @ A] :
( ( coindu1381640503_llist @ B @ A @ F1 @ F23 @ ( coinductive_LCons @ A @ X21 @ X22 ) )
= ( F23 @ X21 @ X22 ) ) ).
% llist.simps(5)
thf(fact_89_unfold__llist_Octr_I2_J,axiom,
! [B: $tType,A: $tType,P: A > $o,A2: A,G21: A > B,G22: A > A] :
( ~ ( P @ A2 )
=> ( ( coindu1441602521_llist @ A @ B @ P @ G21 @ G22 @ A2 )
= ( coinductive_LCons @ B @ ( G21 @ A2 ) @ ( coindu1441602521_llist @ A @ B @ P @ G21 @ G22 @ ( G22 @ A2 ) ) ) ) ) ).
% unfold_llist.ctr(2)
thf(fact_90_ldistinct_OLCons,axiom,
! [A: $tType,X3: A,Xs: coinductive_llist @ A] :
( ~ ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs ) )
=> ( ( coindu351974385stinct @ A @ Xs )
=> ( coindu351974385stinct @ A @ ( coinductive_LCons @ A @ X3 @ Xs ) ) ) ) ).
% ldistinct.LCons
thf(fact_91_ord_OLCons__LCons,axiom,
! [A: $tType,Less_eq: A > A > $o,X3: A,Y3: A,Xs: coinductive_llist @ A] :
( ( Less_eq @ X3 @ Y3 )
=> ( ( coinductive_lsorted @ A @ Less_eq @ ( coinductive_LCons @ A @ Y3 @ Xs ) )
=> ( coinductive_lsorted @ A @ Less_eq @ ( coinductive_LCons @ A @ X3 @ ( coinductive_LCons @ A @ Y3 @ Xs ) ) ) ) ) ).
% ord.LCons_LCons
thf(fact_92_ord_Olsorted__LCons__LCons,axiom,
! [A: $tType,Less_eq: A > A > $o,X3: A,Y3: A,Xs: coinductive_llist @ A] :
( ( coinductive_lsorted @ A @ Less_eq @ ( coinductive_LCons @ A @ X3 @ ( coinductive_LCons @ A @ Y3 @ Xs ) ) )
= ( ( Less_eq @ X3 @ Y3 )
& ( coinductive_lsorted @ A @ Less_eq @ ( coinductive_LCons @ A @ Y3 @ Xs ) ) ) ) ).
% ord.lsorted_LCons_LCons
thf(fact_93_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 ) )
= ( ? [X4: A,Xs3: coinductive_llist @ A,Y4: B,Ys2: coinductive_llist @ B] :
( ( Xs
= ( coinductive_LCons @ A @ X4 @ Xs3 ) )
& ( Ys
= ( coinductive_LCons @ B @ Y4 @ Ys2 ) )
& ( Z
= ( product_Pair @ A @ B @ X4 @ Y4 ) )
& ( Zs
= ( coinductive_lzip @ A @ B @ Xs3 @ Ys2 ) ) ) ) ) ).
% lzip_eq_LCons_conv
thf(fact_94_ord_Olsorted__LCons_H,axiom,
! [A: $tType,Less_eq: A > A > $o,X3: A,Xs: coinductive_llist @ A] :
( ( coinductive_lsorted @ A @ Less_eq @ ( coinductive_LCons @ A @ X3 @ Xs ) )
= ( ~ ( coinductive_lnull @ A @ Xs )
=> ( ( Less_eq @ X3 @ ( coinductive_lhd @ A @ Xs ) )
& ( coinductive_lsorted @ A @ Less_eq @ Xs ) ) ) ) ).
% ord.lsorted_LCons'
thf(fact_95_not__lnull__conv,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( ~ ( coinductive_lnull @ A @ Xs ) )
= ( ? [X4: A,Xs3: coinductive_llist @ A] :
( Xs
= ( coinductive_LCons @ A @ X4 @ Xs3 ) ) ) ) ).
% not_lnull_conv
thf(fact_96_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_97_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_98_lset__intros_I2_J,axiom,
! [A: $tType,X3: A,Xs: coinductive_llist @ A,X7: A] :
( ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs ) )
=> ( member @ A @ X3 @ ( coinductive_lset @ A @ ( coinductive_LCons @ A @ X7 @ Xs ) ) ) ) ).
% lset_intros(2)
thf(fact_99_lset__intros_I1_J,axiom,
! [A: $tType,X3: A,Xs: coinductive_llist @ A] : ( member @ A @ X3 @ ( coinductive_lset @ A @ ( coinductive_LCons @ A @ X3 @ Xs ) ) ) ).
% lset_intros(1)
thf(fact_100_llist_Oset__intros_I2_J,axiom,
! [A: $tType,X3: A,A22: coinductive_llist @ A,A1: A] :
( ( member @ A @ X3 @ ( coinductive_lset @ A @ A22 ) )
=> ( member @ A @ X3 @ ( coinductive_lset @ A @ ( coinductive_LCons @ A @ A1 @ A22 ) ) ) ) ).
% llist.set_intros(2)
thf(fact_101_llist_Oset__intros_I1_J,axiom,
! [A: $tType,A1: A,A22: coinductive_llist @ A] : ( member @ A @ A1 @ ( coinductive_lset @ A @ ( coinductive_LCons @ A @ A1 @ A22 ) ) ) ).
% llist.set_intros(1)
thf(fact_102_lset__cases,axiom,
! [A: $tType,X3: A,Xs: coinductive_llist @ A] :
( ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs ) )
=> ( ! [Xs4: coinductive_llist @ A] :
( Xs
!= ( coinductive_LCons @ A @ X3 @ Xs4 ) )
=> ~ ! [X8: A,Xs4: coinductive_llist @ A] :
( ( Xs
= ( coinductive_LCons @ A @ X8 @ Xs4 ) )
=> ~ ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs4 ) ) ) ) ) ).
% lset_cases
thf(fact_103_lset__induct,axiom,
! [A: $tType,X3: A,Xs: coinductive_llist @ A,P2: ( coinductive_llist @ A ) > $o] :
( ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs ) )
=> ( ! [Xs2: coinductive_llist @ A] : ( P2 @ ( coinductive_LCons @ A @ X3 @ Xs2 ) )
=> ( ! [X8: A,Xs2: coinductive_llist @ A] :
( ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs2 ) )
=> ( ( X3 != X8 )
=> ( ( P2 @ Xs2 )
=> ( P2 @ ( coinductive_LCons @ A @ X8 @ Xs2 ) ) ) ) )
=> ( P2 @ Xs ) ) ) ) ).
% lset_induct
thf(fact_104_lset__induct_H,axiom,
! [A: $tType,X3: A,Xs: coinductive_llist @ A,P2: ( coinductive_llist @ A ) > $o] :
( ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs ) )
=> ( ! [Xs2: coinductive_llist @ A] : ( P2 @ ( coinductive_LCons @ A @ X3 @ Xs2 ) )
=> ( ! [X8: A,Xs2: coinductive_llist @ A] :
( ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs2 ) )
=> ( ( P2 @ Xs2 )
=> ( P2 @ ( coinductive_LCons @ A @ X8 @ Xs2 ) ) ) )
=> ( P2 @ Xs ) ) ) ) ).
% lset_induct'
thf(fact_105_llist_Oset__cases,axiom,
! [A: $tType,E3: A,A2: coinductive_llist @ A] :
( ( member @ A @ E3 @ ( coinductive_lset @ A @ A2 ) )
=> ( ! [Z2: coinductive_llist @ A] :
( A2
!= ( coinductive_LCons @ A @ E3 @ Z2 ) )
=> ~ ! [Z1: A,Z2: coinductive_llist @ A] :
( ( A2
= ( coinductive_LCons @ A @ Z1 @ Z2 ) )
=> ~ ( member @ A @ E3 @ ( coinductive_lset @ A @ Z2 ) ) ) ) ) ).
% llist.set_cases
thf(fact_106_llist_Oset__induct,axiom,
! [A: $tType,X3: A,A2: coinductive_llist @ A,P2: A > ( coinductive_llist @ A ) > $o] :
( ( member @ A @ X3 @ ( coinductive_lset @ A @ A2 ) )
=> ( ! [Z1: A,Z2: coinductive_llist @ A] : ( P2 @ Z1 @ ( coinductive_LCons @ A @ Z1 @ Z2 ) )
=> ( ! [Z1: A,Z2: coinductive_llist @ A,Xa: A] :
( ( member @ A @ Xa @ ( coinductive_lset @ A @ Z2 ) )
=> ( ( P2 @ Xa @ Z2 )
=> ( P2 @ Xa @ ( coinductive_LCons @ A @ Z1 @ Z2 ) ) ) )
=> ( P2 @ X3 @ A2 ) ) ) ) ).
% llist.set_induct
thf(fact_107_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_108_lhd__LCons,axiom,
! [A: $tType,X21: A,X22: coinductive_llist @ A] :
( ( coinductive_lhd @ A @ ( coinductive_LCons @ A @ X21 @ X22 ) )
= X21 ) ).
% lhd_LCons
thf(fact_109_ldistinct__ltlI,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( coindu351974385stinct @ A @ Xs )
=> ( coindu351974385stinct @ A @ ( coinductive_ltl @ A @ Xs ) ) ) ).
% ldistinct_ltlI
thf(fact_110_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_111_unfold__llist_Odisc_I1_J,axiom,
! [B: $tType,A: $tType,P: A > $o,A2: A,G21: A > B,G22: A > A] :
( ( P @ A2 )
=> ( coinductive_lnull @ B @ ( coindu1441602521_llist @ A @ B @ P @ G21 @ G22 @ A2 ) ) ) ).
% unfold_llist.disc(1)
thf(fact_112_unfold__llist_Odisc_I2_J,axiom,
! [B: $tType,A: $tType,P: A > $o,A2: A,G21: A > B,G22: A > A] :
( ~ ( P @ A2 )
=> ~ ( coinductive_lnull @ B @ ( coindu1441602521_llist @ A @ B @ P @ G21 @ G22 @ A2 ) ) ) ).
% unfold_llist.disc(2)
thf(fact_113_unfold__llist_Osimps_I4_J,axiom,
! [B: $tType,A: $tType,P: A > $o,A2: A,G21: A > B,G22: A > A] :
( ~ ( P @ A2 )
=> ( ( coinductive_ltl @ B @ ( coindu1441602521_llist @ A @ B @ P @ G21 @ G22 @ A2 ) )
= ( coindu1441602521_llist @ A @ B @ P @ G21 @ G22 @ ( G22 @ A2 ) ) ) ) ).
% unfold_llist.simps(4)
thf(fact_114_unfold__llist_Osimps_I3_J,axiom,
! [B: $tType,A: $tType,P: A > $o,A2: A,G21: A > B,G22: A > A] :
( ~ ( P @ A2 )
=> ( ( coinductive_lhd @ B @ ( coindu1441602521_llist @ A @ B @ P @ G21 @ G22 @ A2 ) )
= ( G21 @ A2 ) ) ) ).
% unfold_llist.simps(3)
thf(fact_115_lmember__code_I2_J,axiom,
! [A: $tType,X3: A,Y3: A,Ys: coinductive_llist @ A] :
( ( coinductive_lmember @ A @ X3 @ ( coinductive_LCons @ A @ Y3 @ Ys ) )
= ( ( X3 = Y3 )
| ( coinductive_lmember @ A @ X3 @ Ys ) ) ) ).
% lmember_code(2)
thf(fact_116_lzip_Ocode,axiom,
! [B: $tType,A: $tType] :
( ( coinductive_lzip @ A @ B )
= ( ^ [Xs5: coinductive_llist @ A,Ys3: coinductive_llist @ B] :
( if @ ( coinductive_llist @ ( product_prod @ A @ B ) )
@ ( ( coinductive_lnull @ A @ Xs5 )
| ( coinductive_lnull @ B @ Ys3 ) )
@ ( coinductive_LNil @ ( product_prod @ A @ B ) )
@ ( coinductive_LCons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ ( coinductive_lhd @ A @ Xs5 ) @ ( coinductive_lhd @ B @ Ys3 ) ) @ ( coinductive_lzip @ A @ B @ ( coinductive_ltl @ A @ Xs5 ) @ ( coinductive_ltl @ B @ Ys3 ) ) ) ) ) ) ).
% lzip.code
thf(fact_117_llist_Osplit__sel,axiom,
! [B: $tType,A: $tType,P2: B > $o,F1: B,F23: A > ( coinductive_llist @ A ) > B,Llist: coinductive_llist @ A] :
( ( P2 @ ( coindu1381640503_llist @ B @ A @ F1 @ F23 @ Llist ) )
= ( ( ( Llist
= ( coinductive_LNil @ A ) )
=> ( P2 @ F1 ) )
& ( ( Llist
= ( coinductive_LCons @ A @ ( coinductive_lhd @ A @ Llist ) @ ( coinductive_ltl @ A @ Llist ) ) )
=> ( P2 @ ( F23 @ ( coinductive_lhd @ A @ Llist ) @ ( coinductive_ltl @ A @ Llist ) ) ) ) ) ) ).
% llist.split_sel
thf(fact_118_llist_Osplit__sel__asm,axiom,
! [B: $tType,A: $tType,P2: B > $o,F1: B,F23: A > ( coinductive_llist @ A ) > B,Llist: coinductive_llist @ A] :
( ( P2 @ ( coindu1381640503_llist @ B @ A @ F1 @ F23 @ Llist ) )
= ( ~ ( ( ( Llist
= ( coinductive_LNil @ A ) )
& ~ ( P2 @ F1 ) )
| ( ( Llist
= ( coinductive_LCons @ A @ ( coinductive_lhd @ A @ Llist ) @ ( coinductive_ltl @ A @ Llist ) ) )
& ~ ( P2 @ ( F23 @ ( coinductive_lhd @ A @ Llist ) @ ( coinductive_ltl @ A @ Llist ) ) ) ) ) ) ) ).
% llist.split_sel_asm
thf(fact_119_ltakeWhile_Octr_I2_J,axiom,
! [A: $tType,Xs: coinductive_llist @ A,P2: A > $o] :
( ~ ( coinductive_lnull @ A @ Xs )
=> ( ( P2 @ ( coinductive_lhd @ A @ Xs ) )
=> ( ( coindu501562517eWhile @ A @ P2 @ Xs )
= ( coinductive_LCons @ A @ ( coinductive_lhd @ A @ Xs ) @ ( coindu501562517eWhile @ A @ P2 @ ( coinductive_ltl @ A @ Xs ) ) ) ) ) ) ).
% ltakeWhile.ctr(2)
thf(fact_120_llexord__coinduct,axiom,
! [A: $tType,X6: ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,R: A > A > $o] :
( ( X6 @ Xs @ Ys )
=> ( ! [Xs2: coinductive_llist @ A,Ys4: coinductive_llist @ A] :
( ( X6 @ Xs2 @ Ys4 )
=> ( ~ ( coinductive_lnull @ A @ Xs2 )
=> ( ~ ( coinductive_lnull @ A @ Ys4 )
& ( ~ ( coinductive_lnull @ A @ Ys4 )
=> ( ( R @ ( coinductive_lhd @ A @ Xs2 ) @ ( coinductive_lhd @ A @ Ys4 ) )
| ( ( ( coinductive_lhd @ A @ Xs2 )
= ( coinductive_lhd @ A @ Ys4 ) )
& ( ( X6 @ ( coinductive_ltl @ A @ Xs2 ) @ ( coinductive_ltl @ A @ Ys4 ) )
| ( coinductive_llexord @ A @ R @ ( coinductive_ltl @ A @ Xs2 ) @ ( coinductive_ltl @ A @ Ys4 ) ) ) ) ) ) ) ) )
=> ( coinductive_llexord @ A @ R @ Xs @ Ys ) ) ) ).
% llexord_coinduct
thf(fact_121_ldistinct_Ocases,axiom,
! [A: $tType,A2: coinductive_llist @ A] :
( ( coindu351974385stinct @ A @ A2 )
=> ( ( A2
!= ( coinductive_LNil @ A ) )
=> ~ ! [X: A,Xs2: coinductive_llist @ A] :
( ( A2
= ( coinductive_LCons @ A @ X @ Xs2 ) )
=> ( ~ ( member @ A @ X @ ( coinductive_lset @ A @ Xs2 ) )
=> ~ ( coindu351974385stinct @ A @ Xs2 ) ) ) ) ) ).
% ldistinct.cases
thf(fact_122_llexord__refl,axiom,
! [A: $tType,R: A > A > $o,Xs: coinductive_llist @ A] : ( coinductive_llexord @ A @ R @ Xs @ Xs ) ).
% llexord_refl
thf(fact_123_ltakeWhile__LNil,axiom,
! [A: $tType,P2: A > $o] :
( ( coindu501562517eWhile @ A @ P2 @ ( coinductive_LNil @ A ) )
= ( coinductive_LNil @ A ) ) ).
% ltakeWhile_LNil
thf(fact_124_llexord__LCons__LCons,axiom,
! [A: $tType,R: A > A > $o,X3: A,Xs: coinductive_llist @ A,Y3: A,Ys: coinductive_llist @ A] :
( ( coinductive_llexord @ A @ R @ ( coinductive_LCons @ A @ X3 @ Xs ) @ ( coinductive_LCons @ A @ Y3 @ Ys ) )
= ( ( ( X3 = Y3 )
& ( coinductive_llexord @ A @ R @ Xs @ Ys ) )
| ( R @ X3 @ Y3 ) ) ) ).
% llexord_LCons_LCons
thf(fact_125_ldistinct__LNil__code,axiom,
! [A: $tType] : ( coindu351974385stinct @ A @ ( coinductive_LNil @ A ) ) ).
% ldistinct_LNil_code
thf(fact_126_llexord__code_I1_J,axiom,
! [A: $tType,R: A > A > $o,Ys: coinductive_llist @ A] : ( coinductive_llexord @ A @ R @ ( coinductive_LNil @ A ) @ Ys ) ).
% llexord_code(1)
thf(fact_127_llexord__LNil__right,axiom,
! [A: $tType,Ys: coinductive_llist @ A,R: A > A > $o,Xs: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ Ys )
=> ( ( coinductive_llexord @ A @ R @ Xs @ Ys )
= ( coinductive_lnull @ A @ Xs ) ) ) ).
% llexord_LNil_right
thf(fact_128_ldropWhile__LNil,axiom,
! [A: $tType,P2: A > $o] :
( ( coindu218763757pWhile @ A @ P2 @ ( coinductive_LNil @ A ) )
= ( coinductive_LNil @ A ) ) ).
% ldropWhile_LNil
thf(fact_129_ltakeWhile__LCons,axiom,
! [A: $tType,P2: A > $o,X3: A,Xs: coinductive_llist @ A] :
( ( ( P2 @ X3 )
=> ( ( coindu501562517eWhile @ A @ P2 @ ( coinductive_LCons @ A @ X3 @ Xs ) )
= ( coinductive_LCons @ A @ X3 @ ( coindu501562517eWhile @ A @ P2 @ Xs ) ) ) )
& ( ~ ( P2 @ X3 )
=> ( ( coindu501562517eWhile @ A @ P2 @ ( coinductive_LCons @ A @ X3 @ Xs ) )
= ( coinductive_LNil @ A ) ) ) ) ).
% ltakeWhile_LCons
thf(fact_130_ltakeWhile_Odisc__iff_I2_J,axiom,
! [A: $tType,P2: A > $o,Xs: coinductive_llist @ A] :
( ( ~ ( coinductive_lnull @ A @ ( coindu501562517eWhile @ A @ P2 @ Xs ) ) )
= ( ~ ( coinductive_lnull @ A @ Xs )
& ( P2 @ ( coinductive_lhd @ A @ Xs ) ) ) ) ).
% ltakeWhile.disc_iff(2)
thf(fact_131_ltakeWhile_Odisc__iff_I1_J,axiom,
! [A: $tType,P2: A > $o,Xs: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ ( coindu501562517eWhile @ A @ P2 @ Xs ) )
= ( ( coinductive_lnull @ A @ Xs )
| ~ ( P2 @ ( coinductive_lhd @ A @ Xs ) ) ) ) ).
% ltakeWhile.disc_iff(1)
thf(fact_132_lnull__ltakeWhile,axiom,
! [A: $tType,P2: A > $o,Xs: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ ( coindu501562517eWhile @ A @ P2 @ Xs ) )
= ( ~ ( coinductive_lnull @ A @ Xs )
=> ~ ( P2 @ ( coinductive_lhd @ A @ Xs ) ) ) ) ).
% lnull_ltakeWhile
thf(fact_133_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_134_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_135_llexord__code_I2_J,axiom,
! [A: $tType,R: A > A > $o,X3: A,Xs: coinductive_llist @ A] :
~ ( coinductive_llexord @ A @ R @ ( coinductive_LCons @ A @ X3 @ Xs ) @ ( coinductive_LNil @ A ) ) ).
% llexord_code(2)
thf(fact_136_llexord_Ocases,axiom,
! [A: $tType,R: A > A > $o,A1: coinductive_llist @ A,A22: coinductive_llist @ A] :
( ( coinductive_llexord @ A @ R @ A1 @ A22 )
=> ( ! [Xs2: coinductive_llist @ A,Ys4: coinductive_llist @ A,X: A] :
( ( A1
= ( coinductive_LCons @ A @ X @ Xs2 ) )
=> ( ( A22
= ( coinductive_LCons @ A @ X @ Ys4 ) )
=> ~ ( coinductive_llexord @ A @ R @ Xs2 @ Ys4 ) ) )
=> ( ! [X: A] :
( ? [Xs2: coinductive_llist @ A] :
( A1
= ( coinductive_LCons @ A @ X @ Xs2 ) )
=> ! [Y: A] :
( ? [Ys4: coinductive_llist @ A] :
( A22
= ( coinductive_LCons @ A @ Y @ Ys4 ) )
=> ~ ( R @ X @ Y ) ) )
=> ~ ( ( A1
= ( coinductive_LNil @ A ) )
=> ! [Ys4: coinductive_llist @ A] : A22 != Ys4 ) ) ) ) ).
% llexord.cases
thf(fact_137_llexord_Osimps,axiom,
! [A: $tType] :
( ( coinductive_llexord @ A )
= ( ^ [R3: A > A > $o,A12: coinductive_llist @ A,A23: coinductive_llist @ A] :
( ? [Xs5: coinductive_llist @ A,Ys3: coinductive_llist @ A,X4: A] :
( ( A12
= ( coinductive_LCons @ A @ X4 @ Xs5 ) )
& ( A23
= ( coinductive_LCons @ A @ X4 @ Ys3 ) )
& ( coinductive_llexord @ A @ R3 @ Xs5 @ Ys3 ) )
| ? [X4: A,Y4: A,Xs5: coinductive_llist @ A,Ys3: coinductive_llist @ A] :
( ( A12
= ( coinductive_LCons @ A @ X4 @ Xs5 ) )
& ( A23
= ( coinductive_LCons @ A @ Y4 @ Ys3 ) )
& ( R3 @ X4 @ Y4 ) )
| ? [Ys3: coinductive_llist @ A] :
( ( A12
= ( coinductive_LNil @ A ) )
& ( A23 = Ys3 ) ) ) ) ) ).
% llexord.simps
thf(fact_138_llexord_Ocoinduct,axiom,
! [A: $tType,X6: ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o,X3: coinductive_llist @ A,Xa2: coinductive_llist @ A,R: A > A > $o] :
( ( X6 @ X3 @ Xa2 )
=> ( ! [X: coinductive_llist @ A,Xa: coinductive_llist @ A] :
( ( X6 @ X @ Xa )
=> ( ? [Xs6: coinductive_llist @ A,Ys5: coinductive_llist @ A,Xb: A] :
( ( X
= ( coinductive_LCons @ A @ Xb @ Xs6 ) )
& ( Xa
= ( coinductive_LCons @ A @ Xb @ Ys5 ) )
& ( ( X6 @ Xs6 @ Ys5 )
| ( coinductive_llexord @ A @ R @ Xs6 @ Ys5 ) ) )
| ? [Xb: A,Y5: A,Xs6: coinductive_llist @ A,Ys5: coinductive_llist @ A] :
( ( X
= ( coinductive_LCons @ A @ Xb @ Xs6 ) )
& ( Xa
= ( coinductive_LCons @ A @ Y5 @ Ys5 ) )
& ( R @ Xb @ Y5 ) )
| ? [Ys5: coinductive_llist @ A] :
( ( X
= ( coinductive_LNil @ A ) )
& ( Xa = Ys5 ) ) ) )
=> ( coinductive_llexord @ A @ R @ X3 @ Xa2 ) ) ) ).
% llexord.coinduct
thf(fact_139_llexord__LNil,axiom,
! [A: $tType,R: A > A > $o,Ys: coinductive_llist @ A] : ( coinductive_llexord @ A @ R @ ( coinductive_LNil @ A ) @ Ys ) ).
% llexord_LNil
thf(fact_140_llexord__antisym,axiom,
! [A: $tType,R: A > A > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_llexord @ A @ R @ Xs @ Ys )
=> ( ( coinductive_llexord @ A @ R @ Ys @ Xs )
=> ( ! [A4: A,B4: A] :
( ( R @ A4 @ B4 )
=> ~ ( R @ B4 @ A4 ) )
=> ( Xs = Ys ) ) ) ) ).
% llexord_antisym
thf(fact_141_llexord__linear,axiom,
! [A: $tType,R: A > A > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ! [X: A,Y: A] :
( ( R @ X @ Y )
| ( X = Y )
| ( R @ Y @ X ) )
=> ( ( coinductive_llexord @ A @ R @ Xs @ Ys )
| ( coinductive_llexord @ A @ R @ Ys @ Xs ) ) ) ).
% llexord_linear
thf(fact_142_llexord__trans,axiom,
! [A: $tType,R: A > A > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A] :
( ( coinductive_llexord @ A @ R @ Xs @ Ys )
=> ( ( coinductive_llexord @ A @ R @ Ys @ Zs )
=> ( ! [A4: A,B4: A,C2: A] :
( ( R @ A4 @ B4 )
=> ( ( R @ B4 @ C2 )
=> ( R @ A4 @ C2 ) ) )
=> ( coinductive_llexord @ A @ R @ Xs @ Zs ) ) ) ) ).
% llexord_trans
thf(fact_143_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_144_ltakeWhile__eq__LNil__iff,axiom,
! [A: $tType,P2: A > $o,Xs: coinductive_llist @ A] :
( ( ( coindu501562517eWhile @ A @ P2 @ Xs )
= ( coinductive_LNil @ A ) )
= ( ( Xs
!= ( coinductive_LNil @ A ) )
=> ~ ( P2 @ ( coinductive_lhd @ A @ Xs ) ) ) ) ).
% ltakeWhile_eq_LNil_iff
thf(fact_145_ltakeWhile_Octr_I1_J,axiom,
! [A: $tType,Xs: coinductive_llist @ A,P2: A > $o] :
( ( ( coinductive_lnull @ A @ Xs )
| ~ ( P2 @ ( coinductive_lhd @ A @ Xs ) ) )
=> ( ( coindu501562517eWhile @ A @ P2 @ Xs )
= ( coinductive_LNil @ A ) ) ) ).
% ltakeWhile.ctr(1)
thf(fact_146_ltl__ltakeWhile,axiom,
! [A: $tType,P2: A > $o,Xs: coinductive_llist @ A] :
( ( ( P2 @ ( coinductive_lhd @ A @ Xs ) )
=> ( ( coinductive_ltl @ A @ ( coindu501562517eWhile @ A @ P2 @ Xs ) )
= ( coindu501562517eWhile @ A @ P2 @ ( coinductive_ltl @ A @ Xs ) ) ) )
& ( ~ ( P2 @ ( coinductive_lhd @ A @ Xs ) )
=> ( ( coinductive_ltl @ A @ ( coindu501562517eWhile @ A @ P2 @ Xs ) )
= ( coinductive_LNil @ A ) ) ) ) ).
% ltl_ltakeWhile
thf(fact_147_lset__ltakeWhileD,axiom,
! [A: $tType,X3: A,P2: A > $o,Xs: coinductive_llist @ A] :
( ( member @ A @ X3 @ ( coinductive_lset @ A @ ( coindu501562517eWhile @ A @ P2 @ Xs ) ) )
=> ( ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs ) )
& ( P2 @ X3 ) ) ) ).
% lset_ltakeWhileD
thf(fact_148_ltakeWhile__cong,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,P2: A > $o,Q: A > $o] :
( ( Xs = Ys )
=> ( ! [X: A] :
( ( member @ A @ X @ ( coinductive_lset @ A @ Ys ) )
=> ( ( P2 @ X )
= ( Q @ X ) ) )
=> ( ( coindu501562517eWhile @ A @ P2 @ Xs )
= ( coindu501562517eWhile @ A @ Q @ Ys ) ) ) ) ).
% ltakeWhile_cong
thf(fact_149_ltakeWhile__all,axiom,
! [A: $tType,Xs: coinductive_llist @ A,P2: A > $o] :
( ! [X: A] :
( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
=> ( P2 @ X ) )
=> ( ( coindu501562517eWhile @ A @ P2 @ Xs )
= Xs ) ) ).
% ltakeWhile_all
thf(fact_150_neq__LNil__conv,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( Xs
!= ( coinductive_LNil @ A ) )
= ( ? [X4: A,Xs3: coinductive_llist @ A] :
( Xs
= ( coinductive_LCons @ A @ X4 @ Xs3 ) ) ) ) ).
% neq_LNil_conv
thf(fact_151_llist_Oexhaust,axiom,
! [A: $tType,Y3: coinductive_llist @ A] :
( ( Y3
!= ( coinductive_LNil @ A ) )
=> ~ ! [X212: A,X222: coinductive_llist @ A] :
( Y3
!= ( coinductive_LCons @ A @ X212 @ X222 ) ) ) ).
% llist.exhaust
thf(fact_152_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_153_llist_Odisc_I1_J,axiom,
! [A: $tType] : ( coinductive_lnull @ A @ ( coinductive_LNil @ A ) ) ).
% llist.disc(1)
thf(fact_154_llist_OdiscI_I1_J,axiom,
! [A: $tType,Llist: coinductive_llist @ A] :
( ( Llist
= ( coinductive_LNil @ A ) )
=> ( coinductive_lnull @ A @ Llist ) ) ).
% llist.discI(1)
thf(fact_155_llist_Ocollapse_I1_J,axiom,
! [A: $tType,Llist: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ Llist )
=> ( Llist
= ( coinductive_LNil @ A ) ) ) ).
% llist.collapse(1)
thf(fact_156_lnull__def,axiom,
! [A: $tType] :
( ( coinductive_lnull @ A )
= ( ^ [Llist5: coinductive_llist @ A] :
( Llist5
= ( coinductive_LNil @ A ) ) ) ) ).
% lnull_def
thf(fact_157_ltl__simps_I1_J,axiom,
! [A: $tType] :
( ( coinductive_ltl @ A @ ( coinductive_LNil @ A ) )
= ( coinductive_LNil @ A ) ) ).
% ltl_simps(1)
thf(fact_158_llexord__LCons__less,axiom,
! [A: $tType,R: A > A > $o,X3: A,Y3: A,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( R @ X3 @ Y3 )
=> ( coinductive_llexord @ A @ R @ ( coinductive_LCons @ A @ X3 @ Xs ) @ ( coinductive_LCons @ A @ Y3 @ Ys ) ) ) ).
% llexord_LCons_less
thf(fact_159_llexord__LCons__eq,axiom,
! [A: $tType,R: A > A > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,X3: A] :
( ( coinductive_llexord @ A @ R @ Xs @ Ys )
=> ( coinductive_llexord @ A @ R @ ( coinductive_LCons @ A @ X3 @ Xs ) @ ( coinductive_LCons @ A @ X3 @ Ys ) ) ) ).
% llexord_LCons_eq
thf(fact_160_llexord__LCons__left,axiom,
! [A: $tType,R: A > A > $o,X3: A,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_llexord @ A @ R @ ( coinductive_LCons @ A @ X3 @ Xs ) @ Ys )
= ( ? [Y4: A,Ys2: coinductive_llist @ A] :
( ( Ys
= ( coinductive_LCons @ A @ Y4 @ Ys2 ) )
& ( ( ( X3 = Y4 )
& ( coinductive_llexord @ A @ R @ Xs @ Ys2 ) )
| ( R @ X3 @ Y4 ) ) ) ) ) ).
% llexord_LCons_left
thf(fact_161_llexord__code_I3_J,axiom,
! [A: $tType,R: A > A > $o,X3: A,Xs: coinductive_llist @ A,Y3: A,Ys: coinductive_llist @ A] :
( ( coinductive_llexord @ A @ R @ ( coinductive_LCons @ A @ X3 @ Xs ) @ ( coinductive_LCons @ A @ Y3 @ Ys ) )
= ( ( R @ X3 @ Y3 )
| ( ( X3 = Y3 )
& ( coinductive_llexord @ A @ R @ Xs @ Ys ) ) ) ) ).
% llexord_code(3)
thf(fact_162_lnull__llexord,axiom,
! [A: $tType,Xs: coinductive_llist @ A,R: A > A > $o,Ys: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ Xs )
=> ( coinductive_llexord @ A @ R @ Xs @ Ys ) ) ).
% lnull_llexord
thf(fact_163_ord_OLNil,axiom,
! [A: $tType,Less_eq: A > A > $o] : ( coinductive_lsorted @ A @ Less_eq @ ( coinductive_LNil @ A ) ) ).
% ord.LNil
thf(fact_164_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_165_llist_Osimps_I4_J,axiom,
! [A: $tType,B: $tType,F1: B,F23: A > ( coinductive_llist @ A ) > B] :
( ( coindu1381640503_llist @ B @ A @ F1 @ F23 @ ( coinductive_LNil @ A ) )
= F1 ) ).
% llist.simps(4)
thf(fact_166_ldistinct_OLNil,axiom,
! [A: $tType] : ( coindu351974385stinct @ A @ ( coinductive_LNil @ A ) ) ).
% ldistinct.LNil
thf(fact_167_unfold__llist_Octr_I1_J,axiom,
! [A: $tType,B: $tType,P: A > $o,A2: A,G21: A > B,G22: A > A] :
( ( P @ A2 )
=> ( ( coindu1441602521_llist @ A @ B @ P @ G21 @ G22 @ A2 )
= ( coinductive_LNil @ B ) ) ) ).
% unfold_llist.ctr(1)
thf(fact_168_gen__lset__code_I1_J,axiom,
! [A: $tType,A5: set @ A] :
( ( coinductive_gen_lset @ A @ A5 @ ( coinductive_LNil @ A ) )
= A5 ) ).
% gen_lset_code(1)
thf(fact_169_lmember__code_I1_J,axiom,
! [A: $tType,X3: A] :
~ ( coinductive_lmember @ A @ X3 @ ( coinductive_LNil @ A ) ) ).
% lmember_code(1)
thf(fact_170_ltakeWhile_Ocode,axiom,
! [A: $tType] :
( ( coindu501562517eWhile @ A )
= ( ^ [P3: A > $o,Xs5: coinductive_llist @ A] :
( if @ ( coinductive_llist @ A )
@ ( ( coinductive_lnull @ A @ Xs5 )
| ~ ( P3 @ ( coinductive_lhd @ A @ Xs5 ) ) )
@ ( coinductive_LNil @ A )
@ ( coinductive_LCons @ A @ ( coinductive_lhd @ A @ Xs5 ) @ ( coindu501562517eWhile @ A @ P3 @ ( coinductive_ltl @ A @ Xs5 ) ) ) ) ) ) ).
% ltakeWhile.code
thf(fact_171_ltakeWhile_Odisc_I2_J,axiom,
! [A: $tType,Xs: coinductive_llist @ A,P2: A > $o] :
( ~ ( coinductive_lnull @ A @ Xs )
=> ( ( P2 @ ( coinductive_lhd @ A @ Xs ) )
=> ~ ( coinductive_lnull @ A @ ( coindu501562517eWhile @ A @ P2 @ Xs ) ) ) ) ).
% ltakeWhile.disc(2)
thf(fact_172_ltakeWhile_Odisc_I1_J,axiom,
! [A: $tType,Xs: coinductive_llist @ A,P2: A > $o] :
( ( ( coinductive_lnull @ A @ Xs )
| ~ ( P2 @ ( coinductive_lhd @ A @ Xs ) ) )
=> ( coinductive_lnull @ A @ ( coindu501562517eWhile @ A @ P2 @ Xs ) ) ) ).
% ltakeWhile.disc(1)
thf(fact_173_lhd__ltakeWhile,axiom,
! [A: $tType,Xs: coinductive_llist @ A,P2: A > $o] :
( ~ ( coinductive_lnull @ A @ Xs )
=> ( ( P2 @ ( coinductive_lhd @ A @ Xs ) )
=> ( ( coinductive_lhd @ A @ ( coindu501562517eWhile @ A @ P2 @ Xs ) )
= ( coinductive_lhd @ A @ Xs ) ) ) ) ).
% lhd_ltakeWhile
thf(fact_174_lset__LNil,axiom,
! [A: $tType] :
( ( coinductive_lset @ A @ ( coinductive_LNil @ A ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% lset_LNil
thf(fact_175_ord_OSingleton,axiom,
! [A: $tType,Less_eq: A > A > $o,X3: A] : ( coinductive_lsorted @ A @ Less_eq @ ( coinductive_LCons @ A @ X3 @ ( coinductive_LNil @ A ) ) ) ).
% ord.Singleton
thf(fact_176_ord_Olsorted_Ocoinduct,axiom,
! [A: $tType,X6: ( coinductive_llist @ A ) > $o,X3: coinductive_llist @ A,Less_eq: A > A > $o] :
( ( X6 @ X3 )
=> ( ! [X: coinductive_llist @ A] :
( ( X6 @ X )
=> ( ( X
= ( coinductive_LNil @ A ) )
| ? [Xa3: A] :
( X
= ( coinductive_LCons @ A @ Xa3 @ ( coinductive_LNil @ A ) ) )
| ? [Xa3: A,Y5: A,Xs6: coinductive_llist @ A] :
( ( X
= ( coinductive_LCons @ A @ Xa3 @ ( coinductive_LCons @ A @ Y5 @ Xs6 ) ) )
& ( Less_eq @ Xa3 @ Y5 )
& ( ( X6 @ ( coinductive_LCons @ A @ Y5 @ Xs6 ) )
| ( coinductive_lsorted @ A @ Less_eq @ ( coinductive_LCons @ A @ Y5 @ Xs6 ) ) ) ) ) )
=> ( coinductive_lsorted @ A @ Less_eq @ X3 ) ) ) ).
% ord.lsorted.coinduct
thf(fact_177_ord_Olsorted_Osimps,axiom,
! [A: $tType] :
( ( coinductive_lsorted @ A )
= ( ^ [Less_eq2: A > A > $o,A7: coinductive_llist @ A] :
( ( A7
= ( coinductive_LNil @ A ) )
| ? [X4: A] :
( A7
= ( coinductive_LCons @ A @ X4 @ ( coinductive_LNil @ A ) ) )
| ? [X4: A,Y4: A,Xs5: coinductive_llist @ A] :
( ( A7
= ( coinductive_LCons @ A @ X4 @ ( coinductive_LCons @ A @ Y4 @ Xs5 ) ) )
& ( Less_eq2 @ X4 @ Y4 )
& ( coinductive_lsorted @ A @ Less_eq2 @ ( coinductive_LCons @ A @ Y4 @ Xs5 ) ) ) ) ) ) ).
% ord.lsorted.simps
thf(fact_178_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 ) )
=> ( ! [X: A] :
( A2
!= ( coinductive_LCons @ A @ X @ ( coinductive_LNil @ A ) ) )
=> ~ ! [X: A,Y: A,Xs2: coinductive_llist @ A] :
( ( A2
= ( coinductive_LCons @ A @ X @ ( coinductive_LCons @ A @ Y @ Xs2 ) ) )
=> ( ( Less_eq @ X @ Y )
=> ~ ( coinductive_lsorted @ A @ Less_eq @ ( coinductive_LCons @ A @ Y @ Xs2 ) ) ) ) ) ) ) ).
% ord.lsorted.cases
thf(fact_179_ord_Olsorted__code_I2_J,axiom,
! [A: $tType,Less_eq: A > A > $o,X3: A] : ( coinductive_lsorted @ A @ Less_eq @ ( coinductive_LCons @ A @ X3 @ ( coinductive_LNil @ A ) ) ) ).
% ord.lsorted_code(2)
thf(fact_180_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_181_ldropWhile__eq__LNil__iff,axiom,
! [A: $tType,P2: A > $o,Xs: coinductive_llist @ A] :
( ( ( coindu218763757pWhile @ A @ P2 @ Xs )
= ( coinductive_LNil @ A ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ ( coinductive_lset @ A @ Xs ) )
=> ( P2 @ X4 ) ) ) ) ).
% ldropWhile_eq_LNil_iff
thf(fact_182_unfold__llist_Ocode,axiom,
! [B: $tType,A: $tType] :
( ( coindu1441602521_llist @ A @ B )
= ( ^ [P4: A > $o,G212: A > B,G222: A > A,A7: A] : ( if @ ( coinductive_llist @ B ) @ ( P4 @ A7 ) @ ( coinductive_LNil @ B ) @ ( coinductive_LCons @ B @ ( G212 @ A7 ) @ ( coindu1441602521_llist @ A @ B @ P4 @ G212 @ G222 @ ( G222 @ A7 ) ) ) ) ) ) ).
% unfold_llist.code
thf(fact_183_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_184_ltakeWhile_Osimps_I4_J,axiom,
! [A: $tType,Xs: coinductive_llist @ A,P2: A > $o] :
( ~ ( coinductive_lnull @ A @ Xs )
=> ( ( P2 @ ( coinductive_lhd @ A @ Xs ) )
=> ( ( coinductive_ltl @ A @ ( coindu501562517eWhile @ A @ P2 @ Xs ) )
= ( coindu501562517eWhile @ A @ P2 @ ( coinductive_ltl @ A @ Xs ) ) ) ) ) ).
% ltakeWhile.simps(4)
thf(fact_185_eq__LConsD,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Y3: A,Ys: coinductive_llist @ A] :
( ( Xs
= ( coinductive_LCons @ A @ Y3 @ Ys ) )
=> ( ( Xs
!= ( coinductive_LNil @ A ) )
& ( ( coinductive_lhd @ A @ Xs )
= Y3 )
& ( ( coinductive_ltl @ A @ Xs )
= Ys ) ) ) ).
% eq_LConsD
thf(fact_186_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_187_ldistinct_Ocoinduct,axiom,
! [A: $tType,X6: ( coinductive_llist @ A ) > $o,X3: coinductive_llist @ A] :
( ( X6 @ X3 )
=> ( ! [X: coinductive_llist @ A] :
( ( X6 @ X )
=> ( ( X
= ( coinductive_LNil @ A ) )
| ? [Xa3: A,Xs6: coinductive_llist @ A] :
( ( X
= ( coinductive_LCons @ A @ Xa3 @ Xs6 ) )
& ~ ( member @ A @ Xa3 @ ( coinductive_lset @ A @ Xs6 ) )
& ( ( X6 @ Xs6 )
| ( coindu351974385stinct @ A @ Xs6 ) ) ) ) )
=> ( coindu351974385stinct @ A @ X3 ) ) ) ).
% ldistinct.coinduct
thf(fact_188_ldistinct_Osimps,axiom,
! [A: $tType] :
( ( coindu351974385stinct @ A )
= ( ^ [A7: coinductive_llist @ A] :
( ( A7
= ( coinductive_LNil @ A ) )
| ? [X4: A,Xs5: coinductive_llist @ A] :
( ( A7
= ( coinductive_LCons @ A @ X4 @ Xs5 ) )
& ~ ( member @ A @ X4 @ ( coinductive_lset @ A @ Xs5 ) )
& ( coindu351974385stinct @ A @ Xs5 ) ) ) ) ) ).
% ldistinct.simps
thf(fact_189_Coinductive__List_Ofinite__lprefix__nitpick__simps_I3_J,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Y3: A,Ys: coinductive_llist @ A] :
( ( coindu328551480prefix @ A @ Xs @ ( coinductive_LCons @ A @ Y3 @ Ys ) )
= ( ( Xs
= ( coinductive_LNil @ A ) )
| ? [Xs3: coinductive_llist @ A] :
( ( Xs
= ( coinductive_LCons @ A @ Y3 @ Xs3 ) )
& ( coindu328551480prefix @ A @ Xs3 @ Ys ) ) ) ) ).
% Coinductive_List.finite_lprefix_nitpick_simps(3)
thf(fact_190_lstrict__prefix__code_I3_J,axiom,
! [B: $tType,X3: B,Xs: coinductive_llist @ B] :
~ ( coindu1478340336prefix @ B @ ( coinductive_LCons @ B @ X3 @ Xs ) @ ( coinductive_LNil @ B ) ) ).
% lstrict_prefix_code(3)
thf(fact_191_lstrict__prefix__code_I2_J,axiom,
! [B: $tType,Y3: B,Ys: coinductive_llist @ B] : ( coindu1478340336prefix @ B @ ( coinductive_LNil @ B ) @ ( coinductive_LCons @ B @ Y3 @ Ys ) ) ).
% lstrict_prefix_code(2)
thf(fact_192_llast__singleton,axiom,
! [A: $tType,X3: A] :
( ( coinductive_llast @ A @ ( coinductive_LCons @ A @ X3 @ ( coinductive_LNil @ A ) ) )
= X3 ) ).
% llast_singleton
thf(fact_193_llast__LCons2,axiom,
! [A: $tType,X3: A,Y3: A,Xs: coinductive_llist @ A] :
( ( coinductive_llast @ A @ ( coinductive_LCons @ A @ X3 @ ( coinductive_LCons @ A @ Y3 @ Xs ) ) )
= ( coinductive_llast @ A @ ( coinductive_LCons @ A @ Y3 @ Xs ) ) ) ).
% llast_LCons2
thf(fact_194_lstrict__prefix__code_I4_J,axiom,
! [B: $tType,X3: B,Xs: coinductive_llist @ B,Y3: B,Ys: coinductive_llist @ B] :
( ( coindu1478340336prefix @ B @ ( coinductive_LCons @ B @ X3 @ Xs ) @ ( coinductive_LCons @ B @ Y3 @ Ys ) )
= ( ( X3 = Y3 )
& ( coindu1478340336prefix @ B @ Xs @ Ys ) ) ) ).
% lstrict_prefix_code(4)
thf(fact_195_lstrict__prefix__code_I1_J,axiom,
! [A: $tType] :
~ ( coindu1478340336prefix @ A @ ( coinductive_LNil @ A ) @ ( coinductive_LNil @ A ) ) ).
% lstrict_prefix_code(1)
thf(fact_196_llist__less__induct,axiom,
! [A: $tType,P2: ( coinductive_llist @ A ) > $o,Xs: coinductive_llist @ A] :
( ! [Xs2: coinductive_llist @ A] :
( ! [Ys5: coinductive_llist @ A] :
( ( coindu1478340336prefix @ A @ Ys5 @ Xs2 )
=> ( P2 @ Ys5 ) )
=> ( P2 @ Xs2 ) )
=> ( P2 @ Xs ) ) ).
% llist_less_induct
thf(fact_197_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_198_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_199_llast__LCons,axiom,
! [A: $tType,Xs: coinductive_llist @ A,X3: A] :
( ( ( coinductive_lnull @ A @ Xs )
=> ( ( coinductive_llast @ A @ ( coinductive_LCons @ A @ X3 @ Xs ) )
= X3 ) )
& ( ~ ( coinductive_lnull @ A @ Xs )
=> ( ( coinductive_llast @ A @ ( coinductive_LCons @ A @ X3 @ Xs ) )
= ( coinductive_llast @ A @ Xs ) ) ) ) ).
% llast_LCons
thf(fact_200_lprefix__coinduct,axiom,
! [A: $tType,P2: ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( P2 @ Xs @ Ys )
=> ( ! [Xs2: coinductive_llist @ A,Ys4: coinductive_llist @ A] :
( ( P2 @ Xs2 @ Ys4 )
=> ( ( ( coinductive_lnull @ A @ Ys4 )
=> ( coinductive_lnull @ A @ Xs2 ) )
& ( ~ ( coinductive_lnull @ A @ Xs2 )
=> ( ~ ( coinductive_lnull @ A @ Ys4 )
=> ( ( ( coinductive_lhd @ A @ Xs2 )
= ( coinductive_lhd @ A @ Ys4 ) )
& ( ( P2 @ ( coinductive_ltl @ A @ Xs2 ) @ ( coinductive_ltl @ A @ Ys4 ) )
| ( coinductive_lprefix @ A @ ( coinductive_ltl @ A @ Xs2 ) @ ( coinductive_ltl @ A @ Ys4 ) ) ) ) ) ) ) )
=> ( coinductive_lprefix @ A @ Xs @ Ys ) ) ) ).
% lprefix_coinduct
thf(fact_201_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_202_llist_Oleq__refl,axiom,
! [A: $tType,X3: coinductive_llist @ A] : ( coinductive_lprefix @ A @ X3 @ X3 ) ).
% llist.leq_refl
thf(fact_203_lprefix__refl,axiom,
! [A: $tType,Xs: coinductive_llist @ A] : ( coinductive_lprefix @ A @ Xs @ Xs ) ).
% lprefix_refl
thf(fact_204_LCons__lprefix__LCons,axiom,
! [A: $tType,X3: A,Xs: coinductive_llist @ A,Y3: A,Ys: coinductive_llist @ A] :
( ( coinductive_lprefix @ A @ ( coinductive_LCons @ A @ X3 @ Xs ) @ ( coinductive_LCons @ A @ Y3 @ Ys ) )
= ( ( X3 = Y3 )
& ( coinductive_lprefix @ A @ Xs @ Ys ) ) ) ).
% LCons_lprefix_LCons
thf(fact_205_lprefix__code_I1_J,axiom,
! [A: $tType,Ys: coinductive_llist @ A] : ( coinductive_lprefix @ A @ ( coinductive_LNil @ A ) @ Ys ) ).
% lprefix_code(1)
thf(fact_206_lprefix__LNil,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( coinductive_lprefix @ A @ Xs @ ( coinductive_LNil @ A ) )
= ( coinductive_lnull @ A @ Xs ) ) ).
% lprefix_LNil
thf(fact_207_Coinductive__List_Ofinite__lprefix__def,axiom,
! [A: $tType] :
( ( coindu328551480prefix @ A )
= ( coinductive_lprefix @ A ) ) ).
% Coinductive_List.finite_lprefix_def
thf(fact_208_lstrict__prefix__def,axiom,
! [A: $tType] :
( ( coindu1478340336prefix @ A )
= ( ^ [Xs5: coinductive_llist @ A,Ys3: coinductive_llist @ A] :
( ( coinductive_lprefix @ A @ Xs5 @ Ys3 )
& ( Xs5 != Ys3 ) ) ) ) ).
% lstrict_prefix_def
thf(fact_209_lprefix__LCons__conv,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Y3: A,Ys: coinductive_llist @ A] :
( ( coinductive_lprefix @ A @ Xs @ ( coinductive_LCons @ A @ Y3 @ Ys ) )
= ( ( Xs
= ( coinductive_LNil @ A ) )
| ? [Xs3: coinductive_llist @ A] :
( ( Xs
= ( coinductive_LCons @ A @ Y3 @ Xs3 ) )
& ( coinductive_lprefix @ A @ Xs3 @ Ys ) ) ) ) ).
% lprefix_LCons_conv
thf(fact_210_lprefix_Ocoinduct,axiom,
! [A: $tType,X6: ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o,X3: coinductive_llist @ A,Xa2: coinductive_llist @ A] :
( ( X6 @ X3 @ Xa2 )
=> ( ! [X: coinductive_llist @ A,Xa: coinductive_llist @ A] :
( ( X6 @ X @ Xa )
=> ( ? [Xs6: coinductive_llist @ A] :
( ( X
= ( coinductive_LNil @ A ) )
& ( Xa = Xs6 ) )
| ? [Xs6: coinductive_llist @ A,Ys5: coinductive_llist @ A,Xb: A] :
( ( X
= ( coinductive_LCons @ A @ Xb @ Xs6 ) )
& ( Xa
= ( coinductive_LCons @ A @ Xb @ Ys5 ) )
& ( ( X6 @ Xs6 @ Ys5 )
| ( coinductive_lprefix @ A @ Xs6 @ Ys5 ) ) ) ) )
=> ( coinductive_lprefix @ A @ X3 @ Xa2 ) ) ) ).
% lprefix.coinduct
thf(fact_211_lprefix_Osimps,axiom,
! [A: $tType] :
( ( coinductive_lprefix @ A )
= ( ^ [A12: coinductive_llist @ A,A23: coinductive_llist @ A] :
( ? [Xs5: coinductive_llist @ A] :
( ( A12
= ( coinductive_LNil @ A ) )
& ( A23 = Xs5 ) )
| ? [Xs5: coinductive_llist @ A,Ys3: coinductive_llist @ A,X4: A] :
( ( A12
= ( coinductive_LCons @ A @ X4 @ Xs5 ) )
& ( A23
= ( coinductive_LCons @ A @ X4 @ Ys3 ) )
& ( coinductive_lprefix @ A @ Xs5 @ Ys3 ) ) ) ) ) ).
% lprefix.simps
thf(fact_212_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,Ys4: coinductive_llist @ A,X: A] :
( ( A1
= ( coinductive_LCons @ A @ X @ Xs2 ) )
=> ( ( A22
= ( coinductive_LCons @ A @ X @ Ys4 ) )
=> ~ ( coinductive_lprefix @ A @ Xs2 @ Ys4 ) ) ) ) ) ).
% lprefix.cases
thf(fact_213_lprefix__code_I2_J,axiom,
! [A: $tType,X3: A,Xs: coinductive_llist @ A] :
~ ( coinductive_lprefix @ A @ ( coinductive_LCons @ A @ X3 @ Xs ) @ ( coinductive_LNil @ A ) ) ).
% lprefix_code(2)
thf(fact_214_lprefixI,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,X6: 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 ) @ X6 )
=> ( ! [Xs2: coinductive_llist @ A,Ys4: coinductive_llist @ A] :
( ( member @ ( product_prod @ ( coinductive_llist @ A ) @ ( coinductive_llist @ A ) ) @ ( product_Pair @ ( coinductive_llist @ A ) @ ( coinductive_llist @ A ) @ Xs2 @ Ys4 ) @ X6 )
=> ( ( coinductive_lnull @ A @ Xs2 )
| ? [X5: A,Xs7: coinductive_llist @ A,Ys6: coinductive_llist @ A] :
( ( Xs2
= ( coinductive_LCons @ A @ X5 @ Xs7 ) )
& ( Ys4
= ( coinductive_LCons @ A @ X5 @ Ys6 ) )
& ( ( member @ ( product_prod @ ( coinductive_llist @ A ) @ ( coinductive_llist @ A ) ) @ ( product_Pair @ ( coinductive_llist @ A ) @ ( coinductive_llist @ A ) @ Xs7 @ Ys6 ) @ X6 )
| ( coinductive_lprefix @ A @ Xs7 @ Ys6 ) ) ) ) )
=> ( coinductive_lprefix @ A @ Xs @ Ys ) ) ) ).
% lprefixI
thf(fact_215_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_216_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_217_llist_Oleq__antisym,axiom,
! [A: $tType,X3: coinductive_llist @ A,Y3: coinductive_llist @ A] :
( ( coinductive_lprefix @ A @ X3 @ Y3 )
=> ( ( coinductive_lprefix @ A @ Y3 @ X3 )
=> ( X3 = Y3 ) ) ) ).
% llist.leq_antisym
thf(fact_218_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_219_llist_Oleq__trans,axiom,
! [A: $tType,X3: coinductive_llist @ A,Y3: coinductive_llist @ A,Z: coinductive_llist @ A] :
( ( coinductive_lprefix @ A @ X3 @ Y3 )
=> ( ( coinductive_lprefix @ A @ Y3 @ Z )
=> ( coinductive_lprefix @ A @ X3 @ Z ) ) ) ).
% llist.leq_trans
thf(fact_220_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_221_ldistinct__lprefix,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coindu351974385stinct @ A @ Xs )
=> ( ( coinductive_lprefix @ A @ Ys @ Xs )
=> ( coindu351974385stinct @ A @ Ys ) ) ) ).
% ldistinct_lprefix
thf(fact_222_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_223_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_224_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_225_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_226_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_227_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_228_Le__LCons,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,X3: A] :
( ( coinductive_lprefix @ A @ Xs @ Ys )
=> ( coinductive_lprefix @ A @ ( coinductive_LCons @ A @ X3 @ Xs ) @ ( coinductive_LCons @ A @ X3 @ Ys ) ) ) ).
% Le_LCons
thf(fact_229_LCons__lprefix__conv,axiom,
! [A: $tType,X3: A,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_lprefix @ A @ ( coinductive_LCons @ A @ X3 @ Xs ) @ Ys )
= ( ? [Ys2: coinductive_llist @ A] :
( ( Ys
= ( coinductive_LCons @ A @ X3 @ Ys2 ) )
& ( coinductive_lprefix @ A @ Xs @ Ys2 ) ) ) ) ).
% LCons_lprefix_conv
thf(fact_230_lprefix__ltakeWhile,axiom,
! [A: $tType,P2: A > $o,Xs: coinductive_llist @ A] : ( coinductive_lprefix @ A @ ( coindu501562517eWhile @ A @ P2 @ Xs ) @ Xs ) ).
% lprefix_ltakeWhile
thf(fact_231_lprefix__imp__llexord,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,R: A > A > $o] :
( ( coinductive_lprefix @ A @ Xs @ Ys )
=> ( coinductive_llexord @ A @ R @ Xs @ Ys ) ) ).
% lprefix_imp_llexord
thf(fact_232_LNil__lprefix,axiom,
! [A: $tType,Xs: coinductive_llist @ A] : ( coinductive_lprefix @ A @ ( coinductive_LNil @ A ) @ Xs ) ).
% LNil_lprefix
thf(fact_233_llimit__induct,axiom,
! [A: $tType,P2: ( coinductive_llist @ A ) > $o,Xs: coinductive_llist @ A] :
( ( P2 @ ( coinductive_LNil @ A ) )
=> ( ! [X: A,Xs2: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ Xs2 )
=> ( ( P2 @ Xs2 )
=> ( P2 @ ( coinductive_LCons @ A @ X @ Xs2 ) ) ) )
=> ( ( ! [Ys5: coinductive_llist @ A] :
( ( coindu1478340336prefix @ A @ Ys5 @ Xs )
=> ( P2 @ Ys5 ) )
=> ( P2 @ Xs ) )
=> ( P2 @ Xs ) ) ) ) ).
% llimit_induct
thf(fact_234_in__inv__imagep,axiom,
! [B: $tType,A: $tType] :
( ( inv_imagep @ A @ B )
= ( ^ [R3: A > A > $o,F4: B > A,X4: B,Y4: B] : ( R3 @ ( F4 @ X4 ) @ ( F4 @ Y4 ) ) ) ) ).
% in_inv_imagep
thf(fact_235_lfinite__LCons,axiom,
! [A: $tType,X3: A,Xs: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ ( coinductive_LCons @ A @ X3 @ Xs ) )
= ( coinductive_lfinite @ A @ Xs ) ) ).
% lfinite_LCons
thf(fact_236_lfinite__code_I2_J,axiom,
! [B: $tType,X3: B,Xs: coinductive_llist @ B] :
( ( coinductive_lfinite @ B @ ( coinductive_LCons @ B @ X3 @ Xs ) )
= ( coinductive_lfinite @ B @ Xs ) ) ).
% lfinite_code(2)
thf(fact_237_lfinite__code_I1_J,axiom,
! [A: $tType] : ( coinductive_lfinite @ A @ ( coinductive_LNil @ A ) ) ).
% lfinite_code(1)
thf(fact_238_lfinite__ltl,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ ( coinductive_ltl @ A @ Xs ) )
= ( coinductive_lfinite @ A @ Xs ) ) ).
% lfinite_ltl
thf(fact_239_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_240_Coinductive__List_Olprefix__nitpick__simps,axiom,
! [A: $tType] :
( ( coinductive_lprefix @ A )
= ( ^ [Xs5: coinductive_llist @ A,Ys3: coinductive_llist @ A] :
( ( ( coinductive_lfinite @ A @ Xs5 )
=> ( coindu328551480prefix @ A @ Xs5 @ Ys3 ) )
& ( ~ ( coinductive_lfinite @ A @ Xs5 )
=> ( Xs5 = Ys3 ) ) ) ) ) ).
% Coinductive_List.lprefix_nitpick_simps
thf(fact_241_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_242_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_243_lfinite__LNil,axiom,
! [A: $tType] : ( coinductive_lfinite @ A @ ( coinductive_LNil @ A ) ) ).
% lfinite_LNil
thf(fact_244_lfinite__LConsI,axiom,
! [A: $tType,Xs: coinductive_llist @ A,X3: A] :
( ( coinductive_lfinite @ A @ Xs )
=> ( coinductive_lfinite @ A @ ( coinductive_LCons @ A @ X3 @ Xs ) ) ) ).
% lfinite_LConsI
thf(fact_245_lnull__imp__lfinite,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ Xs )
=> ( coinductive_lfinite @ A @ Xs ) ) ).
% lnull_imp_lfinite
thf(fact_246_lfinite__ldropWhile,axiom,
! [A: $tType,P2: A > $o,Xs: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ ( coindu218763757pWhile @ A @ P2 @ Xs ) )
= ( ? [X4: A] :
( ( member @ A @ X4 @ ( coinductive_lset @ A @ Xs ) )
& ~ ( P2 @ X4 ) )
=> ( coinductive_lfinite @ A @ Xs ) ) ) ).
% lfinite_ldropWhile
thf(fact_247_lfinite__ltakeWhile,axiom,
! [A: $tType,P2: A > $o,Xs: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ ( coindu501562517eWhile @ A @ P2 @ Xs ) )
= ( ( coinductive_lfinite @ A @ Xs )
| ? [X4: A] :
( ( member @ A @ X4 @ ( coinductive_lset @ A @ Xs ) )
& ~ ( P2 @ X4 ) ) ) ) ).
% lfinite_ltakeWhile
thf(fact_248_lfinite__induct,axiom,
! [A: $tType,Xs: coinductive_llist @ A,P2: ( coinductive_llist @ A ) > $o] :
( ( coinductive_lfinite @ A @ Xs )
=> ( ! [Xs2: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ Xs2 )
=> ( P2 @ Xs2 ) )
=> ( ! [Xs2: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ Xs2 )
=> ( ~ ( coinductive_lnull @ A @ Xs2 )
=> ( ( P2 @ ( coinductive_ltl @ A @ Xs2 ) )
=> ( P2 @ Xs2 ) ) ) )
=> ( P2 @ Xs ) ) ) ) ).
% lfinite_induct
thf(fact_249_lfinite_Ocases,axiom,
! [A: $tType,A2: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ A2 )
=> ( ( A2
!= ( coinductive_LNil @ A ) )
=> ~ ! [Xs2: coinductive_llist @ A] :
( ? [X: A] :
( A2
= ( coinductive_LCons @ A @ X @ Xs2 ) )
=> ~ ( coinductive_lfinite @ A @ Xs2 ) ) ) ) ).
% lfinite.cases
thf(fact_250_lfinite_Osimps,axiom,
! [A: $tType] :
( ( coinductive_lfinite @ A )
= ( ^ [A7: coinductive_llist @ A] :
( ( A7
= ( coinductive_LNil @ A ) )
| ? [Xs5: coinductive_llist @ A,X4: A] :
( ( A7
= ( coinductive_LCons @ A @ X4 @ Xs5 ) )
& ( coinductive_lfinite @ A @ Xs5 ) ) ) ) ) ).
% lfinite.simps
thf(fact_251_lfinite_Oinducts,axiom,
! [A: $tType,X3: coinductive_llist @ A,P2: ( coinductive_llist @ A ) > $o] :
( ( coinductive_lfinite @ A @ X3 )
=> ( ( P2 @ ( coinductive_LNil @ A ) )
=> ( ! [Xs2: coinductive_llist @ A,X: A] :
( ( coinductive_lfinite @ A @ Xs2 )
=> ( ( P2 @ Xs2 )
=> ( P2 @ ( coinductive_LCons @ A @ X @ Xs2 ) ) ) )
=> ( P2 @ X3 ) ) ) ) ).
% lfinite.inducts
thf(fact_252_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_253_llexord__conv,axiom,
! [A: $tType] :
( ( coinductive_llexord @ A )
= ( ^ [R3: A > A > $o,Xs5: coinductive_llist @ A,Ys3: coinductive_llist @ A] :
( ( Xs5 = Ys3 )
| ? [Zs2: coinductive_llist @ A,Xs3: coinductive_llist @ A,Y4: A,Ys2: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ Zs2 )
& ( Xs5
= ( coinductive_lappend @ A @ Zs2 @ Xs3 ) )
& ( Ys3
= ( coinductive_lappend @ A @ Zs2 @ ( coinductive_LCons @ A @ Y4 @ Ys2 ) ) )
& ( ( Xs3
= ( coinductive_LNil @ A ) )
| ( R3 @ ( coinductive_lhd @ A @ Xs3 ) @ Y4 ) ) ) ) ) ) ).
% llexord_conv
%----Type constructors (3)
thf(tcon_fun___Orderings_Obot,axiom,
! [A8: $tType,A9: $tType] :
( ( bot @ A9 @ ( type2 @ A9 ) )
=> ( bot @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
thf(tcon_Set_Oset___Orderings_Obot_1,axiom,
! [A8: $tType] : ( bot @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_HOL_Obool___Orderings_Obot_2,axiom,
bot @ $o @ ( type2 @ $o ) ).
%----Helper facts (3)
thf(help_If_3_1_T,axiom,
! [P2: $o] :
( ( P2 = $true )
| ( P2 = $false ) ) ).
thf(help_If_2_1_T,axiom,
! [A: $tType,X3: A,Y3: A] :
( ( if @ A @ $false @ X3 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X3: A,Y3: A] :
( ( if @ A @ $true @ X3 @ Y3 )
= X3 ) ).
%----Conjectures (1)
thf(conj_0,conjecture,
member @ node @ n @ ( coinductive_lset @ node @ ( f @ ( product_Pair @ node @ ( set @ node ) @ n @ ns ) ) ) ).
%------------------------------------------------------------------------------