TPTP Problem File: DAT187^1.p
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%------------------------------------------------------------------------------
% File : DAT187^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Data Structures
% Problem : Lazy lists II 1278
% Version : [Bla16] axioms : Especial.
% English :
% Refs : [Fri04] Friedrich (2004), Lazy Lists II
% : [RB15] Reynolds & Blanchette (2015), A Decision Procedure for
% : [Bla16] Blanchette (2016), Email to Geoff Sutcliffe
% Source : [Bla16]
% Names : llist2__1278.p [Bla16]
% Status : Theorem
% Rating : 0.00 v7.2.0, 0.25 v7.1.0
% Syntax : Number of formulae : 338 ( 118 unt; 58 typ; 0 def)
% Number of atoms : 788 ( 206 equ; 0 cnn)
% Maximal formula atoms : 13 ( 2 avg)
% Number of connectives : 4584 ( 68 ~; 10 |; 51 &;4086 @)
% ( 0 <=>; 369 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 8 avg)
% Number of types : 3 ( 2 usr)
% Number of type conns : 140 ( 140 >; 0 *; 0 +; 0 <<)
% Number of symbols : 57 ( 56 usr; 6 con; 0-5 aty)
% Number of variables : 1006 ( 39 ^; 895 !; 25 ?;1006 :)
% ( 47 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 14:52:29.673
%------------------------------------------------------------------------------
%----Could-be-implicit typings (5)
thf(ty_t_Coinductive__List_Ollist,type,
coinductive_llist: $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 (53)
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Otop,type,
top:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Oorder,type,
order:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Opreorder,type,
preorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Oorder__top,type,
order_top:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Oordered__ab__group__add,type,
ordered_ab_group_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_c_Coinductive__List_Ofinite__lprefix,type,
coindu328551480prefix:
!>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List_Olappend,type,
coinductive_lappend:
!>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
thf(sy_c_Coinductive__List_Ollast,type,
coinductive_llast:
!>[A: $tType] : ( ( coinductive_llist @ A ) > A ) ).
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_Olstrict__prefix,type,
coindu1478340336prefix:
!>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o ) ).
thf(sy_c_Groups_Ominus__class_Ominus,type,
minus_minus:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_LList2__Mirabelle__hamjzmohle_Oalllsts,type,
lList2435255213lllsts:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( coinductive_llist @ A ) ) ) ).
thf(sy_c_LList2__Mirabelle__hamjzmohle_Ofinlsts,type,
lList2236698231inlsts:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( coinductive_llist @ A ) ) ) ).
thf(sy_c_LList2__Mirabelle__hamjzmohle_Ofinlsts__rec,type,
lList21916056377ts_rec:
!>[B: $tType,A: $tType] : ( B > ( A > ( coinductive_llist @ A ) > B > B ) > ( coinductive_llist @ A ) > B ) ).
thf(sy_c_LList2__Mirabelle__hamjzmohle_Ofinpref,type,
lList21202317876inpref:
!>[A: $tType] : ( ( set @ A ) > ( coinductive_llist @ A ) > ( set @ ( coinductive_llist @ A ) ) ) ).
thf(sy_c_LList2__Mirabelle__hamjzmohle_Ofpslsts,type,
lList22096119349pslsts:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( coinductive_llist @ A ) ) ) ).
thf(sy_c_LList2__Mirabelle__hamjzmohle_Oinfliveness,type,
lList21015763786veness:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( coinductive_llist @ A ) ) > $o ) ).
thf(sy_c_LList2__Mirabelle__hamjzmohle_Oinflsts,type,
lList21612149805nflsts:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( coinductive_llist @ A ) ) ) ).
thf(sy_c_LList2__Mirabelle__hamjzmohle_Oinfsafety,type,
lList21015939545safety:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( coinductive_llist @ A ) ) > $o ) ).
thf(sy_c_LList2__Mirabelle__hamjzmohle_Oinfsuff,type,
lList2649413865nfsuff:
!>[A: $tType] : ( ( set @ A ) > ( coinductive_llist @ A ) > ( set @ ( coinductive_llist @ A ) ) ) ).
thf(sy_c_LList2__Mirabelle__hamjzmohle_Olbutlast,type,
lList2370560421utlast:
!>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
thf(sy_c_LList2__Mirabelle__hamjzmohle_Oldrop,type,
lList2508575361_ldrop:
!>[A: $tType] : ( ( coinductive_llist @ A ) > nat > ( coinductive_llist @ A ) ) ).
thf(sy_c_LList2__Mirabelle__hamjzmohle_Oliveness,type,
lList21805353693veness:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( coinductive_llist @ A ) ) > $o ) ).
thf(sy_c_LList2__Mirabelle__hamjzmohle_Ollast,type,
lList2170638824_llast:
!>[A: $tType] : ( ( coinductive_llist @ A ) > A ) ).
thf(sy_c_LList2__Mirabelle__hamjzmohle_Ollength,type,
lList21232602520length:
!>[A: $tType] : ( ( coinductive_llist @ A ) > nat ) ).
thf(sy_c_LList2__Mirabelle__hamjzmohle_Olrev,type,
lList2281150353e_lrev:
!>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
thf(sy_c_LList2__Mirabelle__hamjzmohle_Oltake,type,
lList22119844313_ltake:
!>[A: $tType] : ( ( coinductive_llist @ A ) > nat > ( coinductive_llist @ A ) ) ).
thf(sy_c_LList2__Mirabelle__hamjzmohle_Opfinpref,type,
lList2467029176inpref:
!>[A: $tType] : ( ( set @ A ) > ( coinductive_llist @ A ) > ( set @ ( coinductive_llist @ A ) ) ) ).
thf(sy_c_LList2__Mirabelle__hamjzmohle_Oposliveness,type,
lList21952340509veness:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( coinductive_llist @ A ) ) > $o ) ).
thf(sy_c_LList2__Mirabelle__hamjzmohle_Oposlsts,type,
lList21148268032oslsts:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( coinductive_llist @ A ) ) ) ).
thf(sy_c_LList2__Mirabelle__hamjzmohle_Opossafety,type,
lList292406316safety:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( coinductive_llist @ A ) ) > $o ) ).
thf(sy_c_LList2__Mirabelle__hamjzmohle_Opprefix__closed,type,
lList21974196564closed:
!>[A: $tType] : ( ( set @ ( coinductive_llist @ A ) ) > $o ) ).
thf(sy_c_LList2__Mirabelle__hamjzmohle_Oprefix__closed,type,
lList21638733016closed:
!>[A: $tType] : ( ( set @ ( coinductive_llist @ A ) ) > $o ) ).
thf(sy_c_LList2__Mirabelle__hamjzmohle_Osafety,type,
lList21350011628safety:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( coinductive_llist @ A ) ) > $o ) ).
thf(sy_c_LList2__Mirabelle__hamjzmohle_Osuff,type,
lList21475143548e_suff:
!>[A: $tType] : ( ( set @ A ) > ( coinductive_llist @ A ) > ( set @ ( coinductive_llist @ A ) ) ) ).
thf(sy_c_LList2__Mirabelle__hamjzmohle_Osuffix__closed,type,
lList2736192599closed:
!>[A: $tType] : ( ( set @ ( coinductive_llist @ A ) ) > $o ) ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Orderings_Oord__class_Oless,type,
ord_less:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Orderings_Otop__class_Otop,type,
top_top:
!>[A: $tType] : A ).
thf(sy_c_Pure_Otype,type,
type2:
!>[A: $tType] : ( itself @ A ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_P,type,
p: set @ ( coinductive_llist @ a ) ).
thf(sy_v_r____,type,
r: coinductive_llist @ a ).
thf(sy_v_s____,type,
s: coinductive_llist @ a ).
thf(sy_v_t____,type,
t: coinductive_llist @ a ).
thf(sy_v_u____,type,
u: coinductive_llist @ a ).
thf(sy_v_v____,type,
v: coinductive_llist @ a ).
%----Relevant facts (256)
thf(fact_0_tP,axiom,
member @ ( coinductive_llist @ a ) @ t @ p ).
% tP
thf(fact_1__092_060open_062t_A_061_Ar_A_064_064_Au_A_064_064_Av_092_060close_062,axiom,
( t
= ( coinductive_lappend @ a @ r @ ( coinductive_lappend @ a @ u @ v ) ) ) ).
% \<open>t = r @@ u @@ v\<close>
thf(fact_2_psafety,axiom,
lList292406316safety @ a @ ( top_top @ ( set @ a ) ) @ p ).
% psafety
thf(fact_3_scons,axiom,
( s
= ( coinductive_lappend @ a @ r @ u ) ) ).
% scons
thf(fact_4__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062v_O_At_A_061_Ar_A_064_064_Au_A_064_064_Av_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [V: coinductive_llist @ a] :
( t
!= ( coinductive_lappend @ a @ r @ ( coinductive_lappend @ a @ u @ V ) ) ) ).
% \<open>\<And>thesis. (\<And>v. t = r @@ u @@ v \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_5_alllsts__UNIV,axiom,
! [A: $tType,S: coinductive_llist @ A] : ( member @ ( coinductive_llist @ A ) @ S @ ( lList2435255213lllsts @ A @ ( top_top @ ( set @ A ) ) ) ) ).
% alllsts_UNIV
thf(fact_6__092_060open_062r_A_092_060in_062_Apfinpref_AUNIV_As_092_060close_062,axiom,
member @ ( coinductive_llist @ a ) @ r @ ( lList2467029176inpref @ a @ ( top_top @ ( set @ a ) ) @ s ) ).
% \<open>r \<in> pfinpref UNIV s\<close>
thf(fact_7_lappT,axiom,
! [A: $tType,S: coinductive_llist @ A,A2: set @ A,T: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ S @ ( lList2435255213lllsts @ A @ A2 ) )
=> ( ( member @ ( coinductive_llist @ A ) @ T @ ( lList2435255213lllsts @ A @ A2 ) )
=> ( member @ ( coinductive_llist @ A ) @ ( coinductive_lappend @ A @ S @ T ) @ ( lList2435255213lllsts @ A @ A2 ) ) ) ) ).
% lappT
thf(fact_8_lapp__all__invT,axiom,
! [A: $tType,R: coinductive_llist @ A,S: coinductive_llist @ A,A2: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ ( coinductive_lappend @ A @ R @ S ) @ ( lList2435255213lllsts @ A @ A2 ) )
=> ( member @ ( coinductive_llist @ A ) @ R @ ( lList2435255213lllsts @ A @ A2 ) ) ) ).
% lapp_all_invT
thf(fact_9__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062u_O_As_A_061_Ar_A_064_064_Au_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [U: coinductive_llist @ a] :
( s
!= ( coinductive_lappend @ a @ r @ U ) ) ).
% \<open>\<And>thesis. (\<And>u. s = r @@ u \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_10_UNIV__I,axiom,
! [A: $tType,X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ).
% UNIV_I
thf(fact_11_iso__tuple__UNIV__I,axiom,
! [A: $tType,X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ).
% iso_tuple_UNIV_I
thf(fact_12_top__apply,axiom,
! [C: $tType,D: $tType] :
( ( top @ C @ ( type2 @ C ) )
=> ( ( top_top @ ( D > C ) )
= ( ^ [X2: D] : ( top_top @ C ) ) ) ) ).
% top_apply
thf(fact_13_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_14_UNIV__eq__I,axiom,
! [A: $tType,A2: set @ A] :
( ! [X3: A] : ( member @ A @ X3 @ A2 )
=> ( ( top_top @ ( set @ A ) )
= A2 ) ) ).
% UNIV_eq_I
thf(fact_15_UNIV__witness,axiom,
! [A: $tType] :
? [X3: A] : ( member @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ).
% UNIV_witness
thf(fact_16_posliveness__def,axiom,
! [A: $tType] :
( ( lList21952340509veness @ A )
= ( ^ [A3: set @ A,P: set @ ( coinductive_llist @ A )] :
! [X2: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ X2 @ ( lList22096119349pslsts @ A @ A3 ) )
=> ? [Y: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ Y @ ( lList2435255213lllsts @ A @ A3 ) )
& ( member @ ( coinductive_llist @ A ) @ ( coinductive_lappend @ A @ X2 @ Y ) @ P ) ) ) ) ) ).
% posliveness_def
thf(fact_17_suff__all,axiom,
! [A: $tType,X: coinductive_llist @ A,A2: set @ A,S: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ X @ ( lList21475143548e_suff @ A @ A2 @ S ) )
=> ( member @ ( coinductive_llist @ A ) @ X @ ( lList2435255213lllsts @ A @ A2 ) ) ) ).
% suff_all
thf(fact_18_safetyD,axiom,
! [A: $tType,A2: set @ A,P2: set @ ( coinductive_llist @ A ),T: coinductive_llist @ A] :
( ( lList21350011628safety @ A @ A2 @ P2 )
=> ( ( member @ ( coinductive_llist @ A ) @ T @ ( lList2435255213lllsts @ A @ A2 ) )
=> ( ! [R2: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ R2 @ ( lList21202317876inpref @ A @ A2 @ T ) )
=> ? [X4: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ X4 @ ( lList2435255213lllsts @ A @ A2 ) )
& ( member @ ( coinductive_llist @ A ) @ ( coinductive_lappend @ A @ R2 @ X4 ) @ P2 ) ) )
=> ( member @ ( coinductive_llist @ A ) @ T @ P2 ) ) ) ) ).
% safetyD
thf(fact_19_safetyE,axiom,
! [A: $tType,A2: set @ A,P2: set @ ( coinductive_llist @ A )] :
( ( lList21350011628safety @ A @ A2 @ P2 )
=> ! [X4: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ X4 @ ( lList2435255213lllsts @ A @ A2 ) )
=> ( ! [Xa: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ Xa @ ( lList21202317876inpref @ A @ A2 @ X4 ) )
=> ? [Xb: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ Xb @ ( lList2435255213lllsts @ A @ A2 ) )
& ( member @ ( coinductive_llist @ A ) @ ( coinductive_lappend @ A @ Xa @ Xb ) @ P2 ) ) )
=> ( member @ ( coinductive_llist @ A ) @ X4 @ P2 ) ) ) ) ).
% safetyE
thf(fact_20_safetyI,axiom,
! [A: $tType,A2: set @ A,P2: set @ ( coinductive_llist @ A )] :
( ! [T2: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ T2 @ ( lList2435255213lllsts @ A @ A2 ) )
=> ( ! [X4: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ X4 @ ( lList21202317876inpref @ A @ A2 @ T2 ) )
=> ? [Xa: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ Xa @ ( lList2435255213lllsts @ A @ A2 ) )
& ( member @ ( coinductive_llist @ A ) @ ( coinductive_lappend @ A @ X4 @ Xa ) @ P2 ) ) )
=> ( member @ ( coinductive_llist @ A ) @ T2 @ P2 ) ) )
=> ( lList21350011628safety @ A @ A2 @ P2 ) ) ).
% safetyI
thf(fact_21_spos,axiom,
( s
!= ( coinductive_LNil @ a ) ) ).
% spos
thf(fact_22_st,axiom,
ord_less_eq @ ( coinductive_llist @ a ) @ s @ t ).
% st
thf(fact_23_top__set__def,axiom,
! [A: $tType] :
( ( top_top @ ( set @ A ) )
= ( collect @ A @ ( top_top @ ( A > $o ) ) ) ) ).
% top_set_def
thf(fact_24_finpref__suff,axiom,
! [A: $tType,R: coinductive_llist @ A,A2: set @ A,T: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ R @ ( lList21202317876inpref @ A @ A2 @ T ) )
=> ( ( member @ ( coinductive_llist @ A ) @ T @ ( lList2435255213lllsts @ A @ A2 ) )
=> ( member @ ( coinductive_llist @ A ) @ T @ ( lList21475143548e_suff @ A @ A2 @ R ) ) ) ) ).
% finpref_suff
thf(fact_25_safety__def,axiom,
! [A: $tType] :
( ( lList21350011628safety @ A )
= ( ^ [A3: set @ A,P: set @ ( coinductive_llist @ A )] :
! [X2: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ X2 @ ( lList2435255213lllsts @ A @ A3 ) )
=> ( ! [Y: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ Y @ ( lList21202317876inpref @ A @ A3 @ X2 ) )
=> ? [Z: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ Z @ ( lList2435255213lllsts @ A @ A3 ) )
& ( member @ ( coinductive_llist @ A ) @ ( coinductive_lappend @ A @ Y @ Z ) @ P ) ) )
=> ( member @ ( coinductive_llist @ A ) @ X2 @ P ) ) ) ) ) ).
% safety_def
thf(fact_26_safety__prefix__closed,axiom,
! [A: $tType,P2: set @ ( coinductive_llist @ A )] :
( ( lList21350011628safety @ A @ ( top_top @ ( set @ A ) ) @ P2 )
=> ( lList21638733016closed @ A @ P2 ) ) ).
% safety_prefix_closed
thf(fact_27_possafety__def,axiom,
! [A: $tType] :
( ( lList292406316safety @ A )
= ( ^ [A3: set @ A,P: set @ ( coinductive_llist @ A )] :
! [X2: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ X2 @ ( lList21148268032oslsts @ A @ A3 ) )
=> ( ! [Y: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ Y @ ( lList2467029176inpref @ A @ A3 @ X2 ) )
=> ? [Z: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ Z @ ( lList2435255213lllsts @ A @ A3 ) )
& ( member @ ( coinductive_llist @ A ) @ ( coinductive_lappend @ A @ Y @ Z ) @ P ) ) )
=> ( member @ ( coinductive_llist @ A ) @ X2 @ P ) ) ) ) ) ).
% possafety_def
thf(fact_28_possafetyI,axiom,
! [A: $tType,A2: set @ A,P2: set @ ( coinductive_llist @ A )] :
( ! [T2: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ T2 @ ( lList21148268032oslsts @ A @ A2 ) )
=> ( ! [X4: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ X4 @ ( lList2467029176inpref @ A @ A2 @ T2 ) )
=> ? [Xa: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ Xa @ ( lList2435255213lllsts @ A @ A2 ) )
& ( member @ ( coinductive_llist @ A ) @ ( coinductive_lappend @ A @ X4 @ Xa ) @ P2 ) ) )
=> ( member @ ( coinductive_llist @ A ) @ T2 @ P2 ) ) )
=> ( lList292406316safety @ A @ A2 @ P2 ) ) ).
% possafetyI
thf(fact_29_possafetyE,axiom,
! [A: $tType,A2: set @ A,P2: set @ ( coinductive_llist @ A )] :
( ( lList292406316safety @ A @ A2 @ P2 )
=> ! [X4: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ X4 @ ( lList21148268032oslsts @ A @ A2 ) )
=> ( ! [Xa: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ Xa @ ( lList2467029176inpref @ A @ A2 @ X4 ) )
=> ? [Xb: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ Xb @ ( lList2435255213lllsts @ A @ A2 ) )
& ( member @ ( coinductive_llist @ A ) @ ( coinductive_lappend @ A @ Xa @ Xb ) @ P2 ) ) )
=> ( member @ ( coinductive_llist @ A ) @ X4 @ P2 ) ) ) ) ).
% possafetyE
thf(fact_30_possafetyD,axiom,
! [A: $tType,A2: set @ A,P2: set @ ( coinductive_llist @ A ),T: coinductive_llist @ A] :
( ( lList292406316safety @ A @ A2 @ P2 )
=> ( ( member @ ( coinductive_llist @ A ) @ T @ ( lList21148268032oslsts @ A @ A2 ) )
=> ( ! [R2: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ R2 @ ( lList2467029176inpref @ A @ A2 @ T ) )
=> ? [X4: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ X4 @ ( lList2435255213lllsts @ A @ A2 ) )
& ( member @ ( coinductive_llist @ A ) @ ( coinductive_lappend @ A @ R2 @ X4 ) @ P2 ) ) )
=> ( member @ ( coinductive_llist @ A ) @ T @ P2 ) ) ) ) ).
% possafetyD
thf(fact_31_pfinpref__iff,axiom,
! [A: $tType,X: coinductive_llist @ A,A2: set @ A,S: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ X @ ( lList2467029176inpref @ A @ A2 @ S ) )
= ( ( member @ ( coinductive_llist @ A ) @ X @ ( lList21202317876inpref @ A @ A2 @ S ) )
& ( X
!= ( coinductive_LNil @ A ) ) ) ) ).
% pfinpref_iff
thf(fact_32_suff__finpref__iff,axiom,
! [A: $tType,R: coinductive_llist @ A,A2: set @ A,T: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( member @ ( coinductive_llist @ A ) @ T @ ( lList2435255213lllsts @ A @ A2 ) )
=> ( ( member @ ( coinductive_llist @ A ) @ R @ ( lList21202317876inpref @ A @ A2 @ T ) )
= ( member @ ( coinductive_llist @ A ) @ T @ ( lList21475143548e_suff @ A @ A2 @ R ) ) ) ) ) ).
% suff_finpref_iff
thf(fact_33_suff__appE,axiom,
! [A: $tType,R: coinductive_llist @ A,A2: set @ A,T: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( member @ ( coinductive_llist @ A ) @ T @ ( lList21475143548e_suff @ A @ A2 @ R ) )
=> ~ ! [S2: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ S2 @ ( lList2435255213lllsts @ A @ A2 ) )
=> ( T
!= ( coinductive_lappend @ A @ R @ S2 ) ) ) ) ) ).
% suff_appE
thf(fact_34_suff__LNil,axiom,
! [A: $tType,A2: set @ A] :
( ( lList21475143548e_suff @ A @ A2 @ ( coinductive_LNil @ A ) )
= ( lList2435255213lllsts @ A @ A2 ) ) ).
% suff_LNil
thf(fact_35_top__empty__eq,axiom,
! [A: $tType] :
( ( top_top @ ( A > $o ) )
= ( ^ [X2: A] : ( member @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ) ) ).
% top_empty_eq
thf(fact_36_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% order_refl
thf(fact_37_llist__le__refl,axiom,
! [A: $tType,S: coinductive_llist @ A] : ( ord_less_eq @ ( coinductive_llist @ A ) @ S @ S ) ).
% llist_le_refl
thf(fact_38_top1I,axiom,
! [A: $tType,X: A] : ( top_top @ ( A > $o ) @ X ) ).
% top1I
thf(fact_39_LNil__is__lappend__conv,axiom,
! [A: $tType,S: coinductive_llist @ A,T: coinductive_llist @ A] :
( ( ( coinductive_LNil @ A )
= ( coinductive_lappend @ A @ S @ T ) )
= ( ( S
= ( coinductive_LNil @ A ) )
& ( T
= ( coinductive_LNil @ A ) ) ) ) ).
% LNil_is_lappend_conv
thf(fact_40_lappend__is__LNil__conv,axiom,
! [A: $tType,S: coinductive_llist @ A,T: coinductive_llist @ A] :
( ( ( coinductive_lappend @ A @ S @ T )
= ( coinductive_LNil @ A ) )
= ( ( S
= ( coinductive_LNil @ A ) )
& ( T
= ( coinductive_LNil @ A ) ) ) ) ).
% lappend_is_LNil_conv
thf(fact_41_lappend__code_I1_J,axiom,
! [A: $tType,Ys: coinductive_llist @ A] :
( ( coinductive_lappend @ A @ ( coinductive_LNil @ A ) @ Ys )
= Ys ) ).
% lappend_code(1)
thf(fact_42_lappend__LNil2,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( coinductive_lappend @ A @ Xs @ ( coinductive_LNil @ A ) )
= Xs ) ).
% lappend_LNil2
thf(fact_43_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_44_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_45_mem__Collect__eq,axiom,
! [A: $tType,A4: A,P2: A > $o] :
( ( member @ A @ A4 @ ( collect @ A @ P2 ) )
= ( P2 @ A4 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: $tType,A2: set @ A] :
( ( collect @ A
@ ^ [X2: A] : ( member @ A @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [A: $tType,P2: A > $o,Q: A > $o] :
( ! [X3: A] :
( ( P2 @ X3 )
= ( Q @ X3 ) )
=> ( ( collect @ A @ P2 )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_48_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X3: A] :
( ( F @ X3 )
= ( G @ X3 ) )
=> ( F = G ) ) ).
% ext
thf(fact_49_same__lappend__eq,axiom,
! [A: $tType,R: coinductive_llist @ A,A2: set @ A,S: coinductive_llist @ A,T: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( ( coinductive_lappend @ A @ R @ S )
= ( coinductive_lappend @ A @ R @ T ) )
= ( S = T ) ) ) ).
% same_lappend_eq
thf(fact_50_lapp__fin__fin__iff,axiom,
! [A: $tType,R: coinductive_llist @ A,S: coinductive_llist @ A,A2: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ ( coinductive_lappend @ A @ R @ S ) @ ( lList2236698231inlsts @ A @ A2 ) )
= ( ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ A2 ) )
& ( member @ ( coinductive_llist @ A ) @ S @ ( lList2236698231inlsts @ A @ A2 ) ) ) ) ).
% lapp_fin_fin_iff
thf(fact_51_le__LNil,axiom,
! [A: $tType,S: coinductive_llist @ A] :
( ( ord_less_eq @ ( coinductive_llist @ A ) @ S @ ( coinductive_LNil @ A ) )
= ( S
= ( coinductive_LNil @ A ) ) ) ).
% le_LNil
thf(fact_52_LNil__le,axiom,
! [A: $tType,S: coinductive_llist @ A] : ( ord_less_eq @ ( coinductive_llist @ A ) @ ( coinductive_LNil @ A ) @ S ) ).
% LNil_le
thf(fact_53_le__lappend,axiom,
! [A: $tType,R: coinductive_llist @ A,S: coinductive_llist @ A] : ( ord_less_eq @ ( coinductive_llist @ A ) @ R @ ( coinductive_lappend @ A @ R @ S ) ) ).
% le_lappend
thf(fact_54_LNil__suff,axiom,
! [A: $tType,A2: set @ A,S: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ ( coinductive_LNil @ A ) @ ( lList21475143548e_suff @ A @ A2 @ S ) )
= ( S
= ( coinductive_LNil @ A ) ) ) ).
% LNil_suff
thf(fact_55_poslsts__UNIV,axiom,
! [A: $tType,S: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ S @ ( lList21148268032oslsts @ A @ ( top_top @ ( set @ A ) ) ) )
= ( S
!= ( coinductive_LNil @ A ) ) ) ).
% poslsts_UNIV
thf(fact_56_fpslsts__iff,axiom,
! [A: $tType,S: coinductive_llist @ A,A2: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ S @ ( lList22096119349pslsts @ A @ A2 ) )
= ( ( member @ ( coinductive_llist @ A ) @ S @ ( lList2236698231inlsts @ A @ A2 ) )
& ( S
!= ( coinductive_LNil @ A ) ) ) ) ).
% fpslsts_iff
thf(fact_57_poslsts__iff,axiom,
! [A: $tType,S: coinductive_llist @ A,A2: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ S @ ( lList21148268032oslsts @ A @ A2 ) )
= ( ( member @ ( coinductive_llist @ A ) @ S @ ( lList2435255213lllsts @ A @ A2 ) )
& ( S
!= ( coinductive_LNil @ A ) ) ) ) ).
% poslsts_iff
thf(fact_58_le__funD,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [F: A > B,G: A > B,X: A] :
( ( ord_less_eq @ ( A > B ) @ F @ G )
=> ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% le_funD
thf(fact_59_le__funE,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [F: A > B,G: A > B,X: A] :
( ( ord_less_eq @ ( A > B ) @ F @ G )
=> ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% le_funE
thf(fact_60_le__funI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [F: A > B,G: A > B] :
( ! [X3: A] : ( ord_less_eq @ B @ ( F @ X3 ) @ ( G @ X3 ) )
=> ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% le_funI
thf(fact_61_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ( ( ord_less_eq @ ( A > B ) )
= ( ^ [F2: A > B,G2: A > B] :
! [X2: A] : ( ord_less_eq @ B @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ).
% le_fun_def
thf(fact_62_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B @ ( type2 @ B ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A4: A,F: B > A,B2: B,C2: B] :
( ( ord_less_eq @ A @ A4 @ ( F @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C2 )
=> ( ! [X3: B,Y2: B] :
( ( ord_less_eq @ B @ X3 @ Y2 )
=> ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq @ A @ A4 @ ( F @ C2 ) ) ) ) ) ) ).
% order_subst1
thf(fact_63_order__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C @ ( type2 @ C ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A4: A,B2: A,F: A > C,C2: C] :
( ( ord_less_eq @ A @ A4 @ B2 )
=> ( ( ord_less_eq @ C @ ( F @ B2 ) @ C2 )
=> ( ! [X3: A,Y2: A] :
( ( ord_less_eq @ A @ X3 @ Y2 )
=> ( ord_less_eq @ C @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq @ C @ ( F @ A4 ) @ C2 ) ) ) ) ) ).
% order_subst2
thf(fact_64_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A4: A,F: B > A,B2: B,C2: B] :
( ( A4
= ( F @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C2 )
=> ( ! [X3: B,Y2: B] :
( ( ord_less_eq @ B @ X3 @ Y2 )
=> ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq @ A @ A4 @ ( F @ C2 ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_65_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A4: A,B2: A,F: A > B,C2: B] :
( ( ord_less_eq @ A @ A4 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X3: A,Y2: A] :
( ( ord_less_eq @ A @ X3 @ Y2 )
=> ( ord_less_eq @ B @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq @ B @ ( F @ A4 ) @ C2 ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_66_eq__iff,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ^ [Y3: A,Z2: A] : Y3 = Z2 )
= ( ^ [X2: A,Y: A] :
( ( ord_less_eq @ A @ X2 @ Y )
& ( ord_less_eq @ A @ Y @ X2 ) ) ) ) ) ).
% eq_iff
thf(fact_67_antisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X: A,Y4: A] :
( ( ord_less_eq @ A @ X @ Y4 )
=> ( ( ord_less_eq @ A @ Y4 @ X )
=> ( X = Y4 ) ) ) ) ).
% antisym
thf(fact_68_linear,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y4: A] :
( ( ord_less_eq @ A @ X @ Y4 )
| ( ord_less_eq @ A @ Y4 @ X ) ) ) ).
% linear
thf(fact_69_eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y4: A] :
( ( X = Y4 )
=> ( ord_less_eq @ A @ X @ Y4 ) ) ) ).
% eq_refl
thf(fact_70_wlog__linorder__le,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P2: A > A > $o,B2: A,A4: A] :
( ! [A5: A,B3: A] :
( ( ord_less_eq @ A @ A5 @ B3 )
=> ( P2 @ A5 @ B3 ) )
=> ( ( ( P2 @ B2 @ A4 )
=> ( P2 @ A4 @ B2 ) )
=> ( P2 @ A4 @ B2 ) ) ) ) ).
% wlog_linorder_le
thf(fact_71_le__cases,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y4: A] :
( ~ ( ord_less_eq @ A @ X @ Y4 )
=> ( ord_less_eq @ A @ Y4 @ X ) ) ) ).
% le_cases
thf(fact_72_order_Otrans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A4: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A4 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ C2 )
=> ( ord_less_eq @ A @ A4 @ C2 ) ) ) ) ).
% order.trans
thf(fact_73_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y4: A,Z3: A] :
( ( ( ord_less_eq @ A @ X @ Y4 )
=> ~ ( ord_less_eq @ A @ Y4 @ Z3 ) )
=> ( ( ( ord_less_eq @ A @ Y4 @ X )
=> ~ ( ord_less_eq @ A @ X @ Z3 ) )
=> ( ( ( ord_less_eq @ A @ X @ Z3 )
=> ~ ( ord_less_eq @ A @ Z3 @ Y4 ) )
=> ( ( ( ord_less_eq @ A @ Z3 @ Y4 )
=> ~ ( ord_less_eq @ A @ Y4 @ X ) )
=> ( ( ( ord_less_eq @ A @ Y4 @ Z3 )
=> ~ ( ord_less_eq @ A @ Z3 @ X ) )
=> ~ ( ( ord_less_eq @ A @ Z3 @ X )
=> ~ ( ord_less_eq @ A @ X @ Y4 ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_74_antisym__conv,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [Y4: A,X: A] :
( ( ord_less_eq @ A @ Y4 @ X )
=> ( ( ord_less_eq @ A @ X @ Y4 )
= ( X = Y4 ) ) ) ) ).
% antisym_conv
thf(fact_75_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A4: A,B2: A,C2: A] :
( ( A4 = B2 )
=> ( ( ord_less_eq @ A @ B2 @ C2 )
=> ( ord_less_eq @ A @ A4 @ C2 ) ) ) ) ).
% ord_eq_le_trans
thf(fact_76_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A4: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A4 @ B2 )
=> ( ( B2 = C2 )
=> ( ord_less_eq @ A @ A4 @ C2 ) ) ) ) ).
% ord_le_eq_trans
thf(fact_77_order__class_Oorder_Oantisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A4: A,B2: A] :
( ( ord_less_eq @ A @ A4 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ A4 )
=> ( A4 = B2 ) ) ) ) ).
% order_class.order.antisym
thf(fact_78_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y4: A,Z3: A] :
( ( ord_less_eq @ A @ X @ Y4 )
=> ( ( ord_less_eq @ A @ Y4 @ Z3 )
=> ( ord_less_eq @ A @ X @ Z3 ) ) ) ) ).
% order_trans
thf(fact_79_dual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A4: A] : ( ord_less_eq @ A @ A4 @ A4 ) ) ).
% dual_order.refl
thf(fact_80_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P2: A > A > $o,A4: A,B2: A] :
( ! [A5: A,B3: A] :
( ( ord_less_eq @ A @ A5 @ B3 )
=> ( P2 @ A5 @ B3 ) )
=> ( ! [A5: A,B3: A] :
( ( P2 @ B3 @ A5 )
=> ( P2 @ A5 @ B3 ) )
=> ( P2 @ A4 @ B2 ) ) ) ) ).
% linorder_wlog
thf(fact_81_dual__order_Otrans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B2: A,A4: A,C2: A] :
( ( ord_less_eq @ A @ B2 @ A4 )
=> ( ( ord_less_eq @ A @ C2 @ B2 )
=> ( ord_less_eq @ A @ C2 @ A4 ) ) ) ) ).
% dual_order.trans
thf(fact_82_dual__order_Oantisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B2: A,A4: A] :
( ( ord_less_eq @ A @ B2 @ A4 )
=> ( ( ord_less_eq @ A @ A4 @ B2 )
=> ( A4 = B2 ) ) ) ) ).
% dual_order.antisym
thf(fact_83_pref__locally__linear,axiom,
! [A: $tType,S: coinductive_llist @ A,X: coinductive_llist @ A,T: coinductive_llist @ A] :
( ( ord_less_eq @ ( coinductive_llist @ A ) @ S @ X )
=> ( ( ord_less_eq @ ( coinductive_llist @ A ) @ T @ X )
=> ( ( ord_less_eq @ ( coinductive_llist @ A ) @ S @ T )
| ( ord_less_eq @ ( coinductive_llist @ A ) @ T @ S ) ) ) ) ).
% pref_locally_linear
thf(fact_84_prefix__closed__def,axiom,
! [A: $tType] :
( ( lList21638733016closed @ A )
= ( ^ [A3: set @ ( coinductive_llist @ A )] :
! [X2: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ X2 @ A3 )
=> ! [S3: coinductive_llist @ A] :
( ( ord_less_eq @ ( coinductive_llist @ A ) @ S3 @ X2 )
=> ( member @ ( coinductive_llist @ A ) @ S3 @ A3 ) ) ) ) ) ).
% prefix_closed_def
thf(fact_85_llist__le__anti__sym,axiom,
! [A: $tType,S: coinductive_llist @ A,T: coinductive_llist @ A] :
( ( ord_less_eq @ ( coinductive_llist @ A ) @ S @ T )
=> ( ( ord_less_eq @ ( coinductive_llist @ A ) @ T @ S )
=> ( S = T ) ) ) ).
% llist_le_anti_sym
thf(fact_86_finlsts_OLNil__fin,axiom,
! [A: $tType,A2: set @ A] : ( member @ ( coinductive_llist @ A ) @ ( coinductive_LNil @ A ) @ ( lList2236698231inlsts @ A @ A2 ) ) ).
% finlsts.LNil_fin
thf(fact_87_llist__le__trans,axiom,
! [A: $tType,R: coinductive_llist @ A,S: coinductive_llist @ A,T: coinductive_llist @ A] :
( ( ord_less_eq @ ( coinductive_llist @ A ) @ R @ S )
=> ( ( ord_less_eq @ ( coinductive_llist @ A ) @ S @ T )
=> ( ord_less_eq @ ( coinductive_llist @ A ) @ R @ T ) ) ) ).
% llist_le_trans
thf(fact_88_llist__le__finT,axiom,
! [A: $tType,R: coinductive_llist @ A,S: coinductive_llist @ A,A2: set @ A] :
( ( ord_less_eq @ ( coinductive_llist @ A ) @ R @ S )
=> ( ( member @ ( coinductive_llist @ A ) @ S @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ A2 ) ) ) ) ).
% llist_le_finT
thf(fact_89_finT__simp,axiom,
! [A: $tType,R: coinductive_llist @ A,A2: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% finT_simp
thf(fact_90_fin__finite,axiom,
! [A: $tType,R: coinductive_llist @ A,A2: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% fin_finite
thf(fact_91_lappfin__finT,axiom,
! [A: $tType,S: coinductive_llist @ A,A2: set @ A,T: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ S @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( member @ ( coinductive_llist @ A ) @ T @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( member @ ( coinductive_llist @ A ) @ ( coinductive_lappend @ A @ S @ T ) @ ( lList2236698231inlsts @ A @ A2 ) ) ) ) ).
% lappfin_finT
thf(fact_92_lapp__fin__fin__lemma,axiom,
! [A: $tType,R: coinductive_llist @ A,S: coinductive_llist @ A,A2: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ ( coinductive_lappend @ A @ R @ S ) @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ A2 ) ) ) ).
% lapp_fin_fin_lemma
thf(fact_93_finite__lemma,axiom,
! [A: $tType,X: coinductive_llist @ A,A2: set @ A,B4: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ X @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( member @ ( coinductive_llist @ A ) @ X @ ( lList2435255213lllsts @ A @ B4 ) )
=> ( member @ ( coinductive_llist @ A ) @ X @ ( lList2236698231inlsts @ A @ B4 ) ) ) ) ).
% finite_lemma
thf(fact_94_finsubsetall,axiom,
! [A: $tType,X: coinductive_llist @ A,A2: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ X @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( member @ ( coinductive_llist @ A ) @ X @ ( lList2435255213lllsts @ A @ A2 ) ) ) ).
% finsubsetall
thf(fact_95_top__greatest,axiom,
! [A: $tType] :
( ( order_top @ A @ ( type2 @ A ) )
=> ! [A4: A] : ( ord_less_eq @ A @ A4 @ ( top_top @ A ) ) ) ).
% top_greatest
thf(fact_96_top_Oextremum__unique,axiom,
! [A: $tType] :
( ( order_top @ A @ ( type2 @ A ) )
=> ! [A4: A] :
( ( ord_less_eq @ A @ ( top_top @ A ) @ A4 )
= ( A4
= ( top_top @ A ) ) ) ) ).
% top.extremum_unique
thf(fact_97_top_Oextremum__uniqueI,axiom,
! [A: $tType] :
( ( order_top @ A @ ( type2 @ A ) )
=> ! [A4: A] :
( ( ord_less_eq @ A @ ( top_top @ A ) @ A4 )
=> ( A4
= ( top_top @ A ) ) ) ) ).
% top.extremum_uniqueI
thf(fact_98_lappend__LNil__LNil,axiom,
! [A: $tType] :
( ( coinductive_lappend @ A @ ( coinductive_LNil @ A ) @ ( coinductive_LNil @ A ) )
= ( coinductive_LNil @ A ) ) ).
% lappend_LNil_LNil
thf(fact_99_llist__le__def,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( coinductive_llist @ A ) )
= ( ^ [S3: coinductive_llist @ A,T3: coinductive_llist @ A] :
? [D2: coinductive_llist @ A] :
( T3
= ( coinductive_lappend @ A @ S3 @ D2 ) ) ) ) ).
% llist_le_def
thf(fact_100_finpref__fin,axiom,
! [A: $tType,X: coinductive_llist @ A,A2: set @ A,S: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ X @ ( lList21202317876inpref @ A @ A2 @ S ) )
=> ( member @ ( coinductive_llist @ A ) @ X @ ( lList2236698231inlsts @ A @ A2 ) ) ) ).
% finpref_fin
thf(fact_101_alllsts_OLNil__all,axiom,
! [A: $tType,A2: set @ A] : ( member @ ( coinductive_llist @ A ) @ ( coinductive_LNil @ A ) @ ( lList2435255213lllsts @ A @ A2 ) ) ).
% alllsts.LNil_all
thf(fact_102_suff__finpref,axiom,
! [A: $tType,T: coinductive_llist @ A,A2: set @ A,R: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ T @ ( lList21475143548e_suff @ A @ A2 @ R ) )
=> ( ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( member @ ( coinductive_llist @ A ) @ R @ ( lList21202317876inpref @ A @ A2 @ T ) ) ) ) ).
% suff_finpref
thf(fact_103_pprefix__closed__def,axiom,
! [A: $tType] :
( ( lList21974196564closed @ A )
= ( ^ [A3: set @ ( coinductive_llist @ A )] :
! [X2: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ X2 @ A3 )
=> ! [S3: coinductive_llist @ A] :
( ( ( ord_less_eq @ ( coinductive_llist @ A ) @ S3 @ X2 )
& ( S3
!= ( coinductive_LNil @ A ) ) )
=> ( member @ ( coinductive_llist @ A ) @ S3 @ A3 ) ) ) ) ) ).
% pprefix_closed_def
thf(fact_104_livenessE,axiom,
! [A: $tType,A2: set @ A,P2: set @ ( coinductive_llist @ A ),S: coinductive_llist @ A] :
( ( lList21805353693veness @ A @ A2 @ P2 )
=> ( ! [T2: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ T2 @ ( lList2435255213lllsts @ A @ A2 ) )
=> ~ ( member @ ( coinductive_llist @ A ) @ ( coinductive_lappend @ A @ S @ T2 ) @ P2 ) )
=> ~ ( member @ ( coinductive_llist @ A ) @ S @ ( lList2236698231inlsts @ A @ A2 ) ) ) ) ).
% livenessE
thf(fact_105_livenessI,axiom,
! [A: $tType,A2: set @ A,P2: set @ ( coinductive_llist @ A )] :
( ! [S2: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ S2 @ ( lList2236698231inlsts @ A @ A2 ) )
=> ? [X4: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ X4 @ ( lList2435255213lllsts @ A @ A2 ) )
& ( member @ ( coinductive_llist @ A ) @ ( coinductive_lappend @ A @ S2 @ X4 ) @ P2 ) ) )
=> ( lList21805353693veness @ A @ A2 @ P2 ) ) ).
% livenessI
thf(fact_106_liveness__def,axiom,
! [A: $tType] :
( ( lList21805353693veness @ A )
= ( ^ [A3: set @ A,P: set @ ( coinductive_llist @ A )] :
! [X2: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ X2 @ ( lList2236698231inlsts @ A @ A3 ) )
=> ? [Y: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ Y @ ( lList2435255213lllsts @ A @ A3 ) )
& ( member @ ( coinductive_llist @ A ) @ ( coinductive_lappend @ A @ X2 @ Y ) @ P ) ) ) ) ) ).
% liveness_def
thf(fact_107_suffix__closed__def,axiom,
! [A: $tType] :
( ( lList2736192599closed @ A )
= ( ^ [A3: set @ ( coinductive_llist @ A )] :
! [X2: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ X2 @ A3 )
=> ! [S3: coinductive_llist @ A] :
( ( ord_less_eq @ ( coinductive_llist @ A ) @ X2 @ S3 )
=> ( member @ ( coinductive_llist @ A ) @ S3 @ A3 ) ) ) ) ) ).
% suffix_closed_def
thf(fact_108_lrev__lappend,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ Xs @ ( lList2236698231inlsts @ A @ ( top_top @ ( set @ A ) ) ) )
=> ( ( member @ ( coinductive_llist @ A ) @ Ys @ ( lList2236698231inlsts @ A @ ( top_top @ ( set @ A ) ) ) )
=> ( ( lList2281150353e_lrev @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
= ( coinductive_lappend @ A @ ( lList2281150353e_lrev @ A @ Ys ) @ ( lList2281150353e_lrev @ A @ Xs ) ) ) ) ) ).
% lrev_lappend
thf(fact_109_LNil__is__lrev__conv,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ Xs @ ( lList2236698231inlsts @ A @ ( top_top @ ( set @ A ) ) ) )
=> ( ( ( coinductive_LNil @ A )
= ( lList2281150353e_lrev @ A @ Xs ) )
= ( Xs
= ( coinductive_LNil @ A ) ) ) ) ).
% LNil_is_lrev_conv
thf(fact_110_lrev__is__LNil__conv,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ Xs @ ( lList2236698231inlsts @ A @ ( top_top @ ( set @ A ) ) ) )
=> ( ( ( lList2281150353e_lrev @ A @ Xs )
= ( coinductive_LNil @ A ) )
= ( Xs
= ( coinductive_LNil @ A ) ) ) ) ).
% lrev_is_LNil_conv
thf(fact_111_top__conj_I2_J,axiom,
! [A: $tType,P2: $o,X: A] :
( ( P2
& ( top_top @ ( A > $o ) @ X ) )
= P2 ) ).
% top_conj(2)
thf(fact_112_lrev__LNil,axiom,
! [A: $tType] :
( ( lList2281150353e_lrev @ A @ ( coinductive_LNil @ A ) )
= ( coinductive_LNil @ A ) ) ).
% lrev_LNil
thf(fact_113_lrevT,axiom,
! [A: $tType,Xs: coinductive_llist @ A,A2: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ Xs @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( member @ ( coinductive_llist @ A ) @ ( lList2281150353e_lrev @ A @ Xs ) @ ( lList2236698231inlsts @ A @ A2 ) ) ) ).
% lrevT
thf(fact_114_lrev__is__lrev__conv,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ Xs @ ( lList2236698231inlsts @ A @ ( top_top @ ( set @ A ) ) ) )
=> ( ( member @ ( coinductive_llist @ A ) @ Ys @ ( lList2236698231inlsts @ A @ ( top_top @ ( set @ A ) ) ) )
=> ( ( ( lList2281150353e_lrev @ A @ Xs )
= ( lList2281150353e_lrev @ A @ Ys ) )
= ( Xs = Ys ) ) ) ) ).
% lrev_is_lrev_conv
thf(fact_115_lrev__lrev__ident,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ Xs @ ( lList2236698231inlsts @ A @ ( top_top @ ( set @ A ) ) ) )
=> ( ( lList2281150353e_lrev @ A @ ( lList2281150353e_lrev @ A @ Xs ) )
= Xs ) ) ).
% lrev_lrev_ident
thf(fact_116_finlsts__mono,axiom,
! [A: $tType,A2: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B4 )
=> ( ord_less_eq @ ( set @ ( coinductive_llist @ A ) ) @ ( lList2236698231inlsts @ A @ A2 ) @ ( lList2236698231inlsts @ A @ B4 ) ) ) ).
% finlsts_mono
thf(fact_117_alllsts__mono,axiom,
! [A: $tType,A2: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B4 )
=> ( ord_less_eq @ ( set @ ( coinductive_llist @ A ) ) @ ( lList2435255213lllsts @ A @ A2 ) @ ( lList2435255213lllsts @ A @ B4 ) ) ) ).
% alllsts_mono
thf(fact_118_fpslsts__mono,axiom,
! [A: $tType,A2: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B4 )
=> ( ord_less_eq @ ( set @ ( coinductive_llist @ A ) ) @ ( lList22096119349pslsts @ A @ A2 ) @ ( lList22096119349pslsts @ A @ B4 ) ) ) ).
% fpslsts_mono
thf(fact_119_poslsts__mono,axiom,
! [A: $tType,A2: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B4 )
=> ( ord_less_eq @ ( set @ ( coinductive_llist @ A ) ) @ ( lList21148268032oslsts @ A @ A2 ) @ ( lList21148268032oslsts @ A @ B4 ) ) ) ).
% poslsts_mono
thf(fact_120_subset__UNIV,axiom,
! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ ( top_top @ ( set @ A ) ) ) ).
% subset_UNIV
thf(fact_121_finpref__mono2,axiom,
! [A: $tType,S: coinductive_llist @ A,T: coinductive_llist @ A,A2: set @ A] :
( ( ord_less_eq @ ( coinductive_llist @ A ) @ S @ T )
=> ( ord_less_eq @ ( set @ ( coinductive_llist @ A ) ) @ ( lList21202317876inpref @ A @ A2 @ S ) @ ( lList21202317876inpref @ A @ A2 @ T ) ) ) ).
% finpref_mono2
thf(fact_122_suff__mono2,axiom,
! [A: $tType,S: coinductive_llist @ A,T: coinductive_llist @ A,A2: set @ A] :
( ( ord_less_eq @ ( coinductive_llist @ A ) @ S @ T )
=> ( ord_less_eq @ ( set @ ( coinductive_llist @ A ) ) @ ( lList21475143548e_suff @ A @ A2 @ T ) @ ( lList21475143548e_suff @ A @ A2 @ S ) ) ) ).
% suff_mono2
thf(fact_123_top__conj_I1_J,axiom,
! [A: $tType,X: A,P2: $o] :
( ( ( top_top @ ( A > $o ) @ X )
& P2 )
= P2 ) ).
% top_conj(1)
thf(fact_124_lrev__LCons,axiom,
! [A: $tType,R: coinductive_llist @ A,A2: set @ A,A4: A] :
( ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( lList2281150353e_lrev @ A @ ( coinductive_LCons @ A @ A4 @ R ) )
= ( coinductive_lappend @ A @ ( lList2281150353e_lrev @ A @ R ) @ ( coinductive_LCons @ A @ A4 @ ( coinductive_LNil @ A ) ) ) ) ) ).
% lrev_LCons
thf(fact_125_take__fin,axiom,
! [A: $tType,T: coinductive_llist @ A,A2: set @ A,I: nat] :
( ( member @ ( coinductive_llist @ A ) @ T @ ( lList2435255213lllsts @ A @ A2 ) )
=> ( member @ ( coinductive_llist @ A ) @ ( lList22119844313_ltake @ A @ T @ I ) @ ( lList2236698231inlsts @ A @ A2 ) ) ) ).
% take_fin
thf(fact_126_infsuff__finpref,axiom,
! [A: $tType,T: coinductive_llist @ A,A2: set @ A,R: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ T @ ( lList2649413865nfsuff @ A @ A2 @ R ) )
=> ( ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( member @ ( coinductive_llist @ A ) @ R @ ( lList21202317876inpref @ A @ A2 @ T ) ) ) ) ).
% infsuff_finpref
thf(fact_127_ltake__fin,axiom,
! [A: $tType,R: coinductive_llist @ A,I: nat] : ( member @ ( coinductive_llist @ A ) @ ( lList22119844313_ltake @ A @ R @ I ) @ ( lList2236698231inlsts @ A @ ( top_top @ ( set @ A ) ) ) ) ).
% ltake_fin
thf(fact_128_lbutlast__LNil,axiom,
! [A: $tType] :
( ( lList2370560421utlast @ A @ ( coinductive_LNil @ A ) )
= ( coinductive_LNil @ A ) ) ).
% lbutlast_LNil
thf(fact_129_subsetI,axiom,
! [A: $tType,A2: set @ A,B4: set @ A] :
( ! [X3: A] :
( ( member @ A @ X3 @ A2 )
=> ( member @ A @ X3 @ B4 ) )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ B4 ) ) ).
% subsetI
thf(fact_130_subset__antisym,axiom,
! [A: $tType,A2: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ A2 )
=> ( A2 = B4 ) ) ) ).
% subset_antisym
thf(fact_131_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_132_lappend__code_I2_J,axiom,
! [A: $tType,Xa2: A,X: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_lappend @ A @ ( coinductive_LCons @ A @ Xa2 @ X ) @ Ys )
= ( coinductive_LCons @ A @ Xa2 @ ( coinductive_lappend @ A @ X @ Ys ) ) ) ).
% lappend_code(2)
thf(fact_133_LConsE,axiom,
! [A: $tType,X: A,Xs: coinductive_llist @ A,A2: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ ( coinductive_LCons @ A @ X @ Xs ) @ ( lList2435255213lllsts @ A @ A2 ) )
= ( ( member @ A @ X @ A2 )
& ( member @ ( coinductive_llist @ A ) @ Xs @ ( lList2435255213lllsts @ A @ A2 ) ) ) ) ).
% LConsE
thf(fact_134_le__LCons,axiom,
! [A: $tType,X: A,Xs: coinductive_llist @ A,Y4: A,Ys: coinductive_llist @ A] :
( ( ord_less_eq @ ( coinductive_llist @ A ) @ ( coinductive_LCons @ A @ X @ Xs ) @ ( coinductive_LCons @ A @ Y4 @ Ys ) )
= ( ( X = Y4 )
& ( ord_less_eq @ ( coinductive_llist @ A ) @ Xs @ Ys ) ) ) ).
% le_LCons
thf(fact_135_LList2__Mirabelle__hamjzmohle_Oltake__LNil,axiom,
! [A: $tType,I: nat] :
( ( lList22119844313_ltake @ A @ ( coinductive_LNil @ A ) @ I )
= ( coinductive_LNil @ A ) ) ).
% LList2_Mirabelle_hamjzmohle.ltake_LNil
thf(fact_136_lbutlast__LCons,axiom,
! [A: $tType,R: coinductive_llist @ A,A2: set @ A,A4: A] :
( ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( ( R
= ( coinductive_LNil @ A ) )
=> ( ( lList2370560421utlast @ A @ ( coinductive_LCons @ A @ A4 @ R ) )
= ( coinductive_LNil @ A ) ) )
& ( ( R
!= ( coinductive_LNil @ A ) )
=> ( ( lList2370560421utlast @ A @ ( coinductive_LCons @ A @ A4 @ R ) )
= ( coinductive_LCons @ A @ A4 @ ( lList2370560421utlast @ A @ R ) ) ) ) ) ) ).
% lbutlast_LCons
thf(fact_137_lbutlast__snoc,axiom,
! [A: $tType,Xs: coinductive_llist @ A,A2: set @ A,X: A] :
( ( member @ ( coinductive_llist @ A ) @ Xs @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( lList2370560421utlast @ A @ ( coinductive_lappend @ A @ Xs @ ( coinductive_LCons @ A @ X @ ( coinductive_LNil @ A ) ) ) )
= Xs ) ) ).
% lbutlast_snoc
thf(fact_138_set__mp,axiom,
! [A: $tType,A2: set @ A,B4: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B4 )
=> ( ( member @ A @ X @ A2 )
=> ( member @ A @ X @ B4 ) ) ) ).
% set_mp
thf(fact_139_in__mono,axiom,
! [A: $tType,A2: set @ A,B4: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B4 )
=> ( ( member @ A @ X @ A2 )
=> ( member @ A @ X @ B4 ) ) ) ).
% in_mono
thf(fact_140_subsetD,axiom,
! [A: $tType,A2: set @ A,B4: set @ A,C2: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B4 )
=> ( ( member @ A @ C2 @ A2 )
=> ( member @ A @ C2 @ B4 ) ) ) ).
% subsetD
thf(fact_141_subsetCE,axiom,
! [A: $tType,A2: set @ A,B4: set @ A,C2: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B4 )
=> ( ( member @ A @ C2 @ A2 )
=> ( member @ A @ C2 @ B4 ) ) ) ).
% subsetCE
thf(fact_142_equalityE,axiom,
! [A: $tType,A2: set @ A,B4: set @ A] :
( ( A2 = B4 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A2 @ B4 )
=> ~ ( ord_less_eq @ ( set @ A ) @ B4 @ A2 ) ) ) ).
% equalityE
thf(fact_143_subset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A3: set @ A,B5: set @ A] :
! [X2: A] :
( ( member @ A @ X2 @ A3 )
=> ( member @ A @ X2 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_144_equalityD1,axiom,
! [A: $tType,A2: set @ A,B4: set @ A] :
( ( A2 = B4 )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ B4 ) ) ).
% equalityD1
thf(fact_145_equalityD2,axiom,
! [A: $tType,A2: set @ A,B4: set @ A] :
( ( A2 = B4 )
=> ( ord_less_eq @ ( set @ A ) @ B4 @ A2 ) ) ).
% equalityD2
thf(fact_146_set__rev__mp,axiom,
! [A: $tType,X: A,A2: set @ A,B4: set @ A] :
( ( member @ A @ X @ A2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A2 @ B4 )
=> ( member @ A @ X @ B4 ) ) ) ).
% set_rev_mp
thf(fact_147_subset__iff,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A3: set @ A,B5: set @ A] :
! [T3: A] :
( ( member @ A @ T3 @ A3 )
=> ( member @ A @ T3 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_148_rev__subsetD,axiom,
! [A: $tType,C2: A,A2: set @ A,B4: set @ A] :
( ( member @ A @ C2 @ A2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A2 @ B4 )
=> ( member @ A @ C2 @ B4 ) ) ) ).
% rev_subsetD
thf(fact_149_subset__refl,axiom,
! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ A2 ) ).
% subset_refl
thf(fact_150_Collect__mono,axiom,
! [A: $tType,P2: A > $o,Q: A > $o] :
( ! [X3: A] :
( ( P2 @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P2 ) @ ( collect @ A @ Q ) ) ) ).
% Collect_mono
thf(fact_151_subset__trans,axiom,
! [A: $tType,A2: set @ A,B4: set @ A,C3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ C3 )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ C3 ) ) ) ).
% subset_trans
thf(fact_152_set__eq__subset,axiom,
! [A: $tType] :
( ( ^ [Y3: set @ A,Z2: set @ A] : Y3 = Z2 )
= ( ^ [A3: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
& ( ord_less_eq @ ( set @ A ) @ B5 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_153_contra__subsetD,axiom,
! [A: $tType,A2: set @ A,B4: set @ A,C2: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B4 )
=> ( ~ ( member @ A @ C2 @ B4 )
=> ~ ( member @ A @ C2 @ A2 ) ) ) ).
% contra_subsetD
thf(fact_154_Collect__mono__iff,axiom,
! [A: $tType,P2: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P2 ) @ ( collect @ A @ Q ) )
= ( ! [X2: A] :
( ( P2 @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_155_llistE,axiom,
! [A: $tType,Y4: coinductive_llist @ A] :
( ( Y4
!= ( coinductive_LNil @ A ) )
=> ~ ! [X212: A,X222: coinductive_llist @ A] :
( Y4
!= ( coinductive_LCons @ A @ X212 @ X222 ) ) ) ).
% llistE
thf(fact_156_neq__LNil__conv,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( Xs
!= ( coinductive_LNil @ A ) )
= ( ? [X2: A,Xs2: coinductive_llist @ A] :
( Xs
= ( coinductive_LCons @ A @ X2 @ Xs2 ) ) ) ) ).
% neq_LNil_conv
thf(fact_157_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_158_finlsts_OLCons__fin,axiom,
! [A: $tType,L: coinductive_llist @ A,A2: set @ A,A4: A] :
( ( member @ ( coinductive_llist @ A ) @ L @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( member @ A @ A4 @ A2 )
=> ( member @ ( coinductive_llist @ A ) @ ( coinductive_LCons @ A @ A4 @ L ) @ ( lList2236698231inlsts @ A @ A2 ) ) ) ) ).
% finlsts.LCons_fin
thf(fact_159_alllsts_OLCons__all,axiom,
! [A: $tType,L: coinductive_llist @ A,A2: set @ A,A4: A] :
( ( member @ ( coinductive_llist @ A ) @ L @ ( lList2435255213lllsts @ A @ A2 ) )
=> ( ( member @ A @ A4 @ A2 )
=> ( member @ ( coinductive_llist @ A ) @ ( coinductive_LCons @ A @ A4 @ L ) @ ( lList2435255213lllsts @ A @ A2 ) ) ) ) ).
% alllsts.LCons_all
thf(fact_160_lappend__snocL1__conv__LCons2,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Y4: A,Ys: coinductive_llist @ A] :
( ( coinductive_lappend @ A @ ( coinductive_lappend @ A @ Xs @ ( coinductive_LCons @ A @ Y4 @ ( coinductive_LNil @ A ) ) ) @ Ys )
= ( coinductive_lappend @ A @ Xs @ ( coinductive_LCons @ A @ Y4 @ Ys ) ) ) ).
% lappend_snocL1_conv_LCons2
thf(fact_161_finlsts_Oinducts,axiom,
! [A: $tType,X: coinductive_llist @ A,A2: set @ A,P2: ( coinductive_llist @ A ) > $o] :
( ( member @ ( coinductive_llist @ A ) @ X @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( P2 @ ( coinductive_LNil @ A ) )
=> ( ! [L2: coinductive_llist @ A,A5: A] :
( ( member @ ( coinductive_llist @ A ) @ L2 @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( P2 @ L2 )
=> ( ( member @ A @ A5 @ A2 )
=> ( P2 @ ( coinductive_LCons @ A @ A5 @ L2 ) ) ) ) )
=> ( P2 @ X ) ) ) ) ).
% finlsts.inducts
thf(fact_162_finlsts__induct,axiom,
! [A: $tType,X: coinductive_llist @ A,A2: set @ A,P2: ( coinductive_llist @ A ) > $o] :
( ( member @ ( coinductive_llist @ A ) @ X @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ! [L2: coinductive_llist @ A] :
( ( L2
= ( coinductive_LNil @ A ) )
=> ( P2 @ L2 ) )
=> ( ! [A5: A,L2: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ L2 @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( P2 @ L2 )
=> ( ( member @ A @ A5 @ A2 )
=> ( P2 @ ( coinductive_LCons @ A @ A5 @ L2 ) ) ) ) )
=> ( P2 @ X ) ) ) ) ).
% finlsts_induct
thf(fact_163_finlsts_Osimps,axiom,
! [A: $tType,A4: coinductive_llist @ A,A2: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ A4 @ ( lList2236698231inlsts @ A @ A2 ) )
= ( ( A4
= ( coinductive_LNil @ A ) )
| ? [L3: coinductive_llist @ A,A6: A] :
( ( A4
= ( coinductive_LCons @ A @ A6 @ L3 ) )
& ( member @ ( coinductive_llist @ A ) @ L3 @ ( lList2236698231inlsts @ A @ A2 ) )
& ( member @ A @ A6 @ A2 ) ) ) ) ).
% finlsts.simps
thf(fact_164_finlsts_Ocases,axiom,
! [A: $tType,A4: coinductive_llist @ A,A2: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ A4 @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( A4
!= ( coinductive_LNil @ A ) )
=> ~ ! [L2: coinductive_llist @ A,A5: A] :
( ( A4
= ( coinductive_LCons @ A @ A5 @ L2 ) )
=> ( ( member @ ( coinductive_llist @ A ) @ L2 @ ( lList2236698231inlsts @ A @ A2 ) )
=> ~ ( member @ A @ A5 @ A2 ) ) ) ) ) ).
% finlsts.cases
thf(fact_165_alllsts_Ocases,axiom,
! [A: $tType,A4: coinductive_llist @ A,A2: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ A4 @ ( lList2435255213lllsts @ A @ A2 ) )
=> ( ( A4
!= ( coinductive_LNil @ A ) )
=> ~ ! [L2: coinductive_llist @ A,A5: A] :
( ( A4
= ( coinductive_LCons @ A @ A5 @ L2 ) )
=> ( ( member @ ( coinductive_llist @ A ) @ L2 @ ( lList2435255213lllsts @ A @ A2 ) )
=> ~ ( member @ A @ A5 @ A2 ) ) ) ) ) ).
% alllsts.cases
thf(fact_166_alllsts_Osimps,axiom,
! [A: $tType,A4: coinductive_llist @ A,A2: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ A4 @ ( lList2435255213lllsts @ A @ A2 ) )
= ( ( A4
= ( coinductive_LNil @ A ) )
| ? [L3: coinductive_llist @ A,A6: A] :
( ( A4
= ( coinductive_LCons @ A @ A6 @ L3 ) )
& ( member @ ( coinductive_llist @ A ) @ L3 @ ( lList2435255213lllsts @ A @ A2 ) )
& ( member @ A @ A6 @ A2 ) ) ) ) ).
% alllsts.simps
thf(fact_167_alllsts_Ocoinduct,axiom,
! [A: $tType,X5: ( coinductive_llist @ A ) > $o,X: coinductive_llist @ A,A2: set @ A] :
( ( X5 @ X )
=> ( ! [X3: coinductive_llist @ A] :
( ( X5 @ X3 )
=> ( ( X3
= ( coinductive_LNil @ A ) )
| ? [L4: coinductive_llist @ A,A7: A] :
( ( X3
= ( coinductive_LCons @ A @ A7 @ L4 ) )
& ( ( X5 @ L4 )
| ( member @ ( coinductive_llist @ A ) @ L4 @ ( lList2435255213lllsts @ A @ A2 ) ) )
& ( member @ A @ A7 @ A2 ) ) ) )
=> ( member @ ( coinductive_llist @ A ) @ X @ ( lList2435255213lllsts @ A @ A2 ) ) ) ) ).
% alllsts.coinduct
thf(fact_168_not__LCons__le__LNil,axiom,
! [A: $tType,A4: A,L: coinductive_llist @ A] :
~ ( ord_less_eq @ ( coinductive_llist @ A ) @ ( coinductive_LCons @ A @ A4 @ L ) @ ( coinductive_LNil @ A ) ) ).
% not_LCons_le_LNil
thf(fact_169_fps__induct,axiom,
! [A: $tType,L: coinductive_llist @ A,A2: set @ A,P2: ( coinductive_llist @ A ) > $o] :
( ( member @ ( coinductive_llist @ A ) @ L @ ( lList22096119349pslsts @ A @ A2 ) )
=> ( ! [A5: A] :
( ( member @ A @ A5 @ A2 )
=> ( P2 @ ( coinductive_LCons @ A @ A5 @ ( coinductive_LNil @ A ) ) ) )
=> ( ! [A5: A,L2: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ L2 @ ( lList22096119349pslsts @ A @ A2 ) )
=> ( ( P2 @ L2 )
=> ( ( member @ A @ A5 @ A2 )
=> ( P2 @ ( coinductive_LCons @ A @ A5 @ L2 ) ) ) ) )
=> ( P2 @ L ) ) ) ) ).
% fps_induct
thf(fact_170_fpslsts__cases,axiom,
! [A: $tType,R: coinductive_llist @ A,A2: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ R @ ( lList22096119349pslsts @ A @ A2 ) )
=> ~ ! [A5: A,Rs: coinductive_llist @ A] :
( ( R
= ( coinductive_LCons @ A @ A5 @ Rs ) )
=> ( ( member @ A @ A5 @ A2 )
=> ~ ( member @ ( coinductive_llist @ A ) @ Rs @ ( lList2236698231inlsts @ A @ A2 ) ) ) ) ) ).
% fpslsts_cases
thf(fact_171_infsuff__mono2,axiom,
! [A: $tType,S: coinductive_llist @ A,T: coinductive_llist @ A,A2: set @ A] :
( ( ord_less_eq @ ( coinductive_llist @ A ) @ S @ T )
=> ( ord_less_eq @ ( set @ ( coinductive_llist @ A ) ) @ ( lList2649413865nfsuff @ A @ A2 @ T ) @ ( lList2649413865nfsuff @ A @ A2 @ S ) ) ) ).
% infsuff_mono2
thf(fact_172_lrev__induct,axiom,
! [A: $tType,Xs: coinductive_llist @ A,A2: set @ A,P2: ( coinductive_llist @ A ) > $o] :
( ( member @ ( coinductive_llist @ A ) @ Xs @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( P2 @ ( coinductive_LNil @ A ) )
=> ( ! [X3: A,Xs3: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ Xs3 @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( P2 @ Xs3 )
=> ( ( member @ A @ X3 @ A2 )
=> ( P2 @ ( coinductive_lappend @ A @ Xs3 @ ( coinductive_LCons @ A @ X3 @ ( coinductive_LNil @ A ) ) ) ) ) ) )
=> ( P2 @ Xs ) ) ) ) ).
% lrev_induct
thf(fact_173_finlsts__rev__cases,axiom,
! [A: $tType,T: coinductive_llist @ A,A2: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ T @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( T
!= ( coinductive_LNil @ A ) )
=> ~ ! [A5: A,L2: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ L2 @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( member @ A @ A5 @ A2 )
=> ( T
!= ( coinductive_lappend @ A @ L2 @ ( coinductive_LCons @ A @ A5 @ ( coinductive_LNil @ A ) ) ) ) ) ) ) ) ).
% finlsts_rev_cases
thf(fact_174_llast__snoc,axiom,
! [A: $tType,Xs: coinductive_llist @ A,A2: set @ A,X: A] :
( ( member @ ( coinductive_llist @ A ) @ Xs @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( lList2170638824_llast @ A @ ( coinductive_lappend @ A @ Xs @ ( coinductive_LCons @ A @ X @ ( coinductive_LNil @ A ) ) ) )
= X ) ) ).
% llast_snoc
thf(fact_175_LList2__Mirabelle__hamjzmohle_Ollast__lappend,axiom,
! [A: $tType,X: coinductive_llist @ A,Y4: coinductive_llist @ A,A4: A] :
( ( member @ ( coinductive_llist @ A ) @ X @ ( lList2236698231inlsts @ A @ ( top_top @ ( set @ A ) ) ) )
=> ( ( member @ ( coinductive_llist @ A ) @ Y4 @ ( lList2236698231inlsts @ A @ ( top_top @ ( set @ A ) ) ) )
=> ( ( lList2170638824_llast @ A @ ( coinductive_lappend @ A @ X @ ( coinductive_LCons @ A @ A4 @ Y4 ) ) )
= ( lList2170638824_llast @ A @ ( coinductive_LCons @ A @ A4 @ Y4 ) ) ) ) ) ).
% LList2_Mirabelle_hamjzmohle.llast_lappend
thf(fact_176_lbutlast__lapp__llast,axiom,
! [A: $tType,L: coinductive_llist @ A,A2: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ L @ ( lList22096119349pslsts @ A @ A2 ) )
=> ( L
= ( coinductive_lappend @ A @ ( lList2370560421utlast @ A @ L ) @ ( coinductive_LCons @ A @ ( lList2170638824_llast @ A @ L ) @ ( coinductive_LNil @ A ) ) ) ) ) ).
% lbutlast_lapp_llast
thf(fact_177_LList2__Mirabelle__hamjzmohle_Ollast__LCons,axiom,
! [B: $tType,R: coinductive_llist @ B,A2: set @ B,A4: B] :
( ( member @ ( coinductive_llist @ B ) @ R @ ( lList2236698231inlsts @ B @ A2 ) )
=> ( ( ( R
= ( coinductive_LNil @ B ) )
=> ( ( lList2170638824_llast @ B @ ( coinductive_LCons @ B @ A4 @ R ) )
= A4 ) )
& ( ( R
!= ( coinductive_LNil @ B ) )
=> ( ( lList2170638824_llast @ B @ ( coinductive_LCons @ B @ A4 @ R ) )
= ( lList2170638824_llast @ B @ R ) ) ) ) ) ).
% LList2_Mirabelle_hamjzmohle.llast_LCons
thf(fact_178_ltake__lappend__llength,axiom,
! [A: $tType,R: coinductive_llist @ A,A2: set @ A,S: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( lList22119844313_ltake @ A @ ( coinductive_lappend @ A @ R @ S ) @ ( lList21232602520length @ A @ R ) )
= R ) ) ).
% ltake_lappend_llength
thf(fact_179_infsuff__finpref__iff,axiom,
! [A: $tType,R: coinductive_llist @ A,A2: set @ A,T: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( member @ ( coinductive_llist @ A ) @ T @ ( lList21612149805nflsts @ A @ A2 ) )
=> ( ( member @ ( coinductive_llist @ A ) @ T @ ( lList21202317876inpref @ A @ A2 @ R ) )
= ( member @ ( coinductive_llist @ A ) @ R @ ( lList2649413865nfsuff @ A @ A2 @ T ) ) ) ) ) ).
% infsuff_finpref_iff
thf(fact_180_Coinductive__List_Ofinite__lprefix__nitpick__simps_I3_J,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Y4: A,Ys: coinductive_llist @ A] :
( ( coindu328551480prefix @ A @ Xs @ ( coinductive_LCons @ A @ Y4 @ Ys ) )
= ( ( Xs
= ( coinductive_LNil @ A ) )
| ? [Xs2: coinductive_llist @ A] :
( ( Xs
= ( coinductive_LCons @ A @ Y4 @ Xs2 ) )
& ( coindu328551480prefix @ A @ Xs2 @ Ys ) ) ) ) ).
% Coinductive_List.finite_lprefix_nitpick_simps(3)
thf(fact_181_lapp__inf,axiom,
! [A: $tType,S: coinductive_llist @ A,A2: set @ A,T: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ S @ ( lList21612149805nflsts @ A @ A2 ) )
=> ( ( coinductive_lappend @ A @ S @ T )
= S ) ) ).
% lapp_inf
thf(fact_182_notfin__inf,axiom,
! [A: $tType,X: coinductive_llist @ A] :
( ( ~ ( member @ ( coinductive_llist @ A ) @ X @ ( lList2236698231inlsts @ A @ ( top_top @ ( set @ A ) ) ) ) )
= ( member @ ( coinductive_llist @ A ) @ X @ ( lList21612149805nflsts @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% notfin_inf
thf(fact_183_notinf__fin,axiom,
! [A: $tType,X: coinductive_llist @ A] :
( ( ~ ( member @ ( coinductive_llist @ A ) @ X @ ( lList21612149805nflsts @ A @ ( top_top @ ( set @ A ) ) ) ) )
= ( member @ ( coinductive_llist @ A ) @ X @ ( lList2236698231inlsts @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% notinf_fin
thf(fact_184_infsuff__LNil,axiom,
! [A: $tType,A2: set @ A] :
( ( lList2649413865nfsuff @ A @ A2 @ ( coinductive_LNil @ A ) )
= ( lList21612149805nflsts @ A @ A2 ) ) ).
% infsuff_LNil
thf(fact_185_llength__take,axiom,
! [A: $tType,T: coinductive_llist @ A,A2: set @ A,I: nat] :
( ( member @ ( coinductive_llist @ A ) @ T @ ( lList21612149805nflsts @ A @ A2 ) )
=> ( ( lList21232602520length @ A @ ( lList22119844313_ltake @ A @ T @ I ) )
= I ) ) ).
% llength_take
thf(fact_186_inflstsI,axiom,
! [A: $tType,X: coinductive_llist @ A,A2: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ X @ ( lList2435255213lllsts @ A @ A2 ) )
=> ( ~ ( member @ ( coinductive_llist @ A ) @ X @ ( lList2236698231inlsts @ A @ ( top_top @ ( set @ A ) ) ) )
=> ( member @ ( coinductive_llist @ A ) @ X @ ( lList21612149805nflsts @ A @ A2 ) ) ) ) ).
% inflstsI
thf(fact_187_infsuff__inf,axiom,
! [A: $tType,X: coinductive_llist @ A,A2: set @ A,S: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ X @ ( lList2649413865nfsuff @ A @ A2 @ S ) )
=> ( member @ ( coinductive_llist @ A ) @ X @ ( lList21612149805nflsts @ A @ A2 ) ) ) ).
% infsuff_inf
thf(fact_188_infT__simp,axiom,
! [A: $tType,R: coinductive_llist @ A,A2: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ R @ ( lList21612149805nflsts @ A @ A2 ) )
=> ( member @ ( coinductive_llist @ A ) @ R @ ( lList21612149805nflsts @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% infT_simp
thf(fact_189_llist__inf__le,axiom,
! [A: $tType,S: coinductive_llist @ A,A2: set @ A,T: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ S @ ( lList21612149805nflsts @ A @ A2 ) )
=> ( ( ord_less_eq @ ( coinductive_llist @ A ) @ S @ T )
= ( S = T ) ) ) ).
% llist_inf_le
thf(fact_190_infsubsetall,axiom,
! [A: $tType,X: coinductive_llist @ A,A2: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ X @ ( lList21612149805nflsts @ A @ A2 ) )
=> ( member @ ( coinductive_llist @ A ) @ X @ ( lList2435255213lllsts @ A @ A2 ) ) ) ).
% infsubsetall
thf(fact_191_inflstsI2,axiom,
! [A: $tType,A4: A,A2: set @ A,T: coinductive_llist @ A] :
( ( member @ A @ A4 @ A2 )
=> ( ( member @ ( coinductive_llist @ A ) @ T @ ( lList21612149805nflsts @ A @ A2 ) )
=> ( member @ ( coinductive_llist @ A ) @ ( coinductive_LCons @ A @ A4 @ T ) @ ( lList21612149805nflsts @ A @ A2 ) ) ) ) ).
% inflstsI2
thf(fact_192_inflsts__cases,axiom,
! [A: $tType,S: coinductive_llist @ A,A2: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ S @ ( lList21612149805nflsts @ A @ A2 ) )
=> ~ ! [A5: A,L2: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ L2 @ ( lList21612149805nflsts @ A @ A2 ) )
=> ( ( member @ A @ A5 @ A2 )
=> ( S
!= ( coinductive_LCons @ A @ A5 @ L2 ) ) ) ) ) ).
% inflsts_cases
thf(fact_193_fin__inf__cases,axiom,
! [A: $tType,R: coinductive_llist @ A] :
( ~ ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ ( top_top @ ( set @ A ) ) ) )
=> ( member @ ( coinductive_llist @ A ) @ R @ ( lList21612149805nflsts @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% fin_inf_cases
thf(fact_194_lapp__fin__infT,axiom,
! [A: $tType,S: coinductive_llist @ A,A2: set @ A,T: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ S @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( member @ ( coinductive_llist @ A ) @ T @ ( lList21612149805nflsts @ A @ A2 ) )
=> ( member @ ( coinductive_llist @ A ) @ ( coinductive_lappend @ A @ S @ T ) @ ( lList21612149805nflsts @ A @ A2 ) ) ) ) ).
% lapp_fin_infT
thf(fact_195_lapp__inv2T,axiom,
! [A: $tType,R: coinductive_llist @ A,S: coinductive_llist @ A,A2: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ ( coinductive_lappend @ A @ R @ S ) @ ( lList21612149805nflsts @ A @ A2 ) )
=> ( ( ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ A2 ) )
& ( member @ ( coinductive_llist @ A ) @ S @ ( lList21612149805nflsts @ A @ A2 ) ) )
| ( member @ ( coinductive_llist @ A ) @ R @ ( lList21612149805nflsts @ A @ A2 ) ) ) ) ).
% lapp_inv2T
thf(fact_196_lapp__infT,axiom,
! [A: $tType,R: coinductive_llist @ A,S: coinductive_llist @ A,A2: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ ( coinductive_lappend @ A @ R @ S ) @ ( lList21612149805nflsts @ A @ A2 ) )
= ( ( ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ A2 ) )
& ( member @ ( coinductive_llist @ A ) @ S @ ( lList21612149805nflsts @ A @ A2 ) ) )
| ( member @ ( coinductive_llist @ A ) @ R @ ( lList21612149805nflsts @ A @ A2 ) ) ) ) ).
% lapp_infT
thf(fact_197_app__invT,axiom,
! [A: $tType,R: coinductive_llist @ A,A2: set @ A,S: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( member @ ( coinductive_llist @ A ) @ ( coinductive_lappend @ A @ R @ S ) @ ( lList21612149805nflsts @ A @ A2 ) )
=> ( member @ ( coinductive_llist @ A ) @ S @ ( lList21612149805nflsts @ A @ A2 ) ) ) ) ).
% app_invT
thf(fact_198_alllstsE,axiom,
! [A: $tType,X: coinductive_llist @ A,A2: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ X @ ( lList2435255213lllsts @ A @ A2 ) )
=> ( ~ ( member @ ( coinductive_llist @ A ) @ X @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( member @ ( coinductive_llist @ A ) @ X @ ( lList21612149805nflsts @ A @ A2 ) ) ) ) ).
% alllstsE
thf(fact_199_prefix__lemma,axiom,
! [A: $tType,X: coinductive_llist @ A,A2: set @ A,Y4: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ X @ ( lList21612149805nflsts @ A @ A2 ) )
=> ( ( member @ ( coinductive_llist @ A ) @ Y4 @ ( lList21612149805nflsts @ A @ A2 ) )
=> ( ! [S2: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ S2 @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( ord_less_eq @ ( coinductive_llist @ A ) @ S2 @ X )
=> ( ord_less_eq @ ( coinductive_llist @ A ) @ S2 @ Y4 ) ) )
=> ( X = Y4 ) ) ) ) ).
% prefix_lemma
thf(fact_200_inf__neqE,axiom,
! [A: $tType,X: coinductive_llist @ A,A2: set @ A,Y4: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ X @ ( lList21612149805nflsts @ A @ A2 ) )
=> ( ( member @ ( coinductive_llist @ A ) @ Y4 @ ( lList21612149805nflsts @ A @ A2 ) )
=> ( ( X != Y4 )
=> ~ ! [S2: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ S2 @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( ord_less_eq @ ( coinductive_llist @ A ) @ S2 @ X )
=> ( ord_less_eq @ ( coinductive_llist @ A ) @ S2 @ Y4 ) ) ) ) ) ) ).
% inf_neqE
thf(fact_201_inflsts__mono,axiom,
! [A: $tType,A2: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B4 )
=> ( ord_less_eq @ ( set @ ( coinductive_llist @ A ) ) @ ( lList21612149805nflsts @ A @ A2 ) @ ( lList21612149805nflsts @ A @ B4 ) ) ) ).
% inflsts_mono
thf(fact_202_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_203_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_204_finpref__infsuff,axiom,
! [A: $tType,R: coinductive_llist @ A,A2: set @ A,T: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ R @ ( lList21202317876inpref @ A @ A2 @ T ) )
=> ( ( member @ ( coinductive_llist @ A ) @ T @ ( lList21612149805nflsts @ A @ A2 ) )
=> ( member @ ( coinductive_llist @ A ) @ T @ ( lList2649413865nfsuff @ A @ A2 @ R ) ) ) ) ).
% finpref_infsuff
thf(fact_205_inflstsE,axiom,
! [A: $tType,X: coinductive_llist @ A,A2: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ X @ ( lList21612149805nflsts @ A @ A2 ) )
=> ~ ( ( member @ ( coinductive_llist @ A ) @ X @ ( lList2435255213lllsts @ A @ A2 ) )
=> ( member @ ( coinductive_llist @ A ) @ X @ ( lList2236698231inlsts @ A @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% inflstsE
thf(fact_206_lapp__allT__iff,axiom,
! [A: $tType,R: coinductive_llist @ A,S: coinductive_llist @ A,A2: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ ( coinductive_lappend @ A @ R @ S ) @ ( lList2435255213lllsts @ A @ A2 ) )
= ( ( ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ A2 ) )
& ( member @ ( coinductive_llist @ A ) @ S @ ( lList2435255213lllsts @ A @ A2 ) ) )
| ( member @ ( coinductive_llist @ A ) @ R @ ( lList21612149805nflsts @ A @ A2 ) ) ) ) ).
% lapp_allT_iff
thf(fact_207_infsuff__appE,axiom,
! [A: $tType,R: coinductive_llist @ A,A2: set @ A,T: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( member @ ( coinductive_llist @ A ) @ T @ ( lList2649413865nfsuff @ A @ A2 @ R ) )
=> ~ ! [S2: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ S2 @ ( lList21612149805nflsts @ A @ A2 ) )
=> ( T
!= ( coinductive_lappend @ A @ R @ S2 ) ) ) ) ) ).
% infsuff_appE
thf(fact_208_infsafety__def,axiom,
! [A: $tType] :
( ( lList21015939545safety @ A )
= ( ^ [A3: set @ A,P: set @ ( coinductive_llist @ A )] :
! [X2: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ X2 @ ( lList21612149805nflsts @ A @ A3 ) )
=> ( ! [Y: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ Y @ ( lList21202317876inpref @ A @ A3 @ X2 ) )
=> ? [Z: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ Z @ ( lList21612149805nflsts @ A @ A3 ) )
& ( member @ ( coinductive_llist @ A ) @ ( coinductive_lappend @ A @ Y @ Z ) @ P ) ) )
=> ( member @ ( coinductive_llist @ A ) @ X2 @ P ) ) ) ) ) ).
% infsafety_def
thf(fact_209_infliveness__def,axiom,
! [A: $tType] :
( ( lList21015763786veness @ A )
= ( ^ [A3: set @ A,P: set @ ( coinductive_llist @ A )] :
! [X2: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ X2 @ ( lList2236698231inlsts @ A @ A3 ) )
=> ? [Y: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ Y @ ( lList21612149805nflsts @ A @ A3 ) )
& ( member @ ( coinductive_llist @ A ) @ ( coinductive_lappend @ A @ X2 @ Y ) @ P ) ) ) ) ) ).
% infliveness_def
thf(fact_210_llength__drop__take,axiom,
! [A: $tType,T: coinductive_llist @ A,I: nat] :
( ( ( lList2508575361_ldrop @ A @ T @ I )
!= ( coinductive_LNil @ A ) )
=> ( ( lList21232602520length @ A @ ( lList22119844313_ltake @ A @ T @ I ) )
= I ) ) ).
% llength_drop_take
thf(fact_211_LList2__Mirabelle__hamjzmohle_Oldrop__LNil,axiom,
! [A: $tType,I: nat] :
( ( lList2508575361_ldrop @ A @ ( coinductive_LNil @ A ) @ I )
= ( coinductive_LNil @ A ) ) ).
% LList2_Mirabelle_hamjzmohle.ldrop_LNil
thf(fact_212_ldrop__fin__iffT,axiom,
! [A: $tType,T: coinductive_llist @ A,I: nat] :
( ( member @ ( coinductive_llist @ A ) @ ( lList2508575361_ldrop @ A @ T @ I ) @ ( lList2236698231inlsts @ A @ ( top_top @ ( set @ A ) ) ) )
= ( member @ ( coinductive_llist @ A ) @ T @ ( lList2236698231inlsts @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% ldrop_fin_iffT
thf(fact_213_ldrop__inf__iffT,axiom,
! [A: $tType,T: coinductive_llist @ A,I: nat] :
( ( member @ ( coinductive_llist @ A ) @ ( lList2508575361_ldrop @ A @ T @ I ) @ ( lList21612149805nflsts @ A @ ( top_top @ ( set @ A ) ) ) )
= ( member @ ( coinductive_llist @ A ) @ T @ ( lList21612149805nflsts @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% ldrop_inf_iffT
thf(fact_214_ldrop__infT,axiom,
! [A: $tType,T: coinductive_llist @ A,A2: set @ A,I: nat] :
( ( member @ ( coinductive_llist @ A ) @ T @ ( lList21612149805nflsts @ A @ A2 ) )
=> ( member @ ( coinductive_llist @ A ) @ ( lList2508575361_ldrop @ A @ T @ I ) @ ( lList21612149805nflsts @ A @ A2 ) ) ) ).
% ldrop_infT
thf(fact_215_ldropT,axiom,
! [A: $tType,T: coinductive_llist @ A,A2: set @ A,I: nat] :
( ( member @ ( coinductive_llist @ A ) @ T @ ( lList2435255213lllsts @ A @ A2 ) )
=> ( member @ ( coinductive_llist @ A ) @ ( lList2508575361_ldrop @ A @ T @ I ) @ ( lList2435255213lllsts @ A @ A2 ) ) ) ).
% ldropT
thf(fact_216_ldrop__finT,axiom,
! [A: $tType,T: coinductive_llist @ A,A2: set @ A,I: nat] :
( ( member @ ( coinductive_llist @ A ) @ T @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( member @ ( coinductive_llist @ A ) @ ( lList2508575361_ldrop @ A @ T @ I ) @ ( lList2236698231inlsts @ A @ A2 ) ) ) ).
% ldrop_finT
thf(fact_217_drop__nonLNil,axiom,
! [A: $tType,T: coinductive_llist @ A,I: nat] :
( ( ( lList2508575361_ldrop @ A @ T @ I )
!= ( coinductive_LNil @ A ) )
=> ( T
!= ( coinductive_LNil @ A ) ) ) ).
% drop_nonLNil
thf(fact_218_ldrop__LNil__less,axiom,
! [A: $tType,J: nat,I: nat,T: coinductive_llist @ A] :
( ( ord_less_eq @ nat @ J @ I )
=> ( ( ( lList2508575361_ldrop @ A @ T @ J )
= ( coinductive_LNil @ A ) )
=> ( ( lList2508575361_ldrop @ A @ T @ I )
= ( coinductive_LNil @ A ) ) ) ) ).
% ldrop_LNil_less
thf(fact_219_ltake__ldrop__id,axiom,
! [A: $tType,X: coinductive_llist @ A,I: nat] :
( ( coinductive_lappend @ A @ ( lList22119844313_ltake @ A @ X @ I ) @ ( lList2508575361_ldrop @ A @ X @ I ) )
= X ) ).
% ltake_ldrop_id
thf(fact_220_lapp__suff__llength,axiom,
! [A: $tType,R: coinductive_llist @ A,A2: set @ A,S: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( lList2508575361_ldrop @ A @ ( coinductive_lappend @ A @ R @ S ) @ ( lList21232602520length @ A @ R ) )
= S ) ) ).
% lapp_suff_llength
thf(fact_221_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_222_lstrict__prefix__code_I2_J,axiom,
! [B: $tType,Y4: B,Ys: coinductive_llist @ B] : ( coindu1478340336prefix @ B @ ( coinductive_LNil @ B ) @ ( coinductive_LCons @ B @ Y4 @ Ys ) ) ).
% lstrict_prefix_code(2)
thf(fact_223_lstrict__prefix__code_I4_J,axiom,
! [B: $tType,X: B,Xs: coinductive_llist @ B,Y4: B,Ys: coinductive_llist @ B] :
( ( coindu1478340336prefix @ B @ ( coinductive_LCons @ B @ X @ Xs ) @ ( coinductive_LCons @ B @ Y4 @ Ys ) )
= ( ( X = Y4 )
& ( coindu1478340336prefix @ B @ Xs @ Ys ) ) ) ).
% lstrict_prefix_code(4)
thf(fact_224_lstrict__prefix__code_I1_J,axiom,
! [A: $tType] :
~ ( coindu1478340336prefix @ A @ ( coinductive_LNil @ A ) @ ( coinductive_LNil @ A ) ) ).
% lstrict_prefix_code(1)
thf(fact_225_llist__less__induct,axiom,
! [A: $tType,P2: ( coinductive_llist @ A ) > $o,Xs: coinductive_llist @ A] :
( ! [Xs3: coinductive_llist @ A] :
( ! [Ys2: coinductive_llist @ A] :
( ( coindu1478340336prefix @ A @ Ys2 @ Xs3 )
=> ( P2 @ Ys2 ) )
=> ( P2 @ Xs3 ) )
=> ( P2 @ Xs ) ) ).
% llist_less_induct
thf(fact_226_finlsts__rec__LCons,axiom,
! [B: $tType,A: $tType,R: coinductive_llist @ A,A2: set @ A,C2: B,D3: A > ( coinductive_llist @ A ) > B > B,A4: A] :
( ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( lList21916056377ts_rec @ B @ A @ C2 @ D3 @ ( coinductive_LCons @ A @ A4 @ R ) )
= ( D3 @ A4 @ R @ ( lList21916056377ts_rec @ B @ A @ C2 @ D3 @ R ) ) ) ) ).
% finlsts_rec_LCons
thf(fact_227_finlsts__rec__LCons__def,axiom,
! [B: $tType,A: $tType,F: ( coinductive_llist @ A ) > B,C2: B,D3: A > ( coinductive_llist @ A ) > B > B,R: coinductive_llist @ A,A2: set @ A,A4: A] :
( ( F
= ( lList21916056377ts_rec @ B @ A @ C2 @ D3 ) )
=> ( ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( F @ ( coinductive_LCons @ A @ A4 @ R ) )
= ( D3 @ A4 @ R @ ( F @ R ) ) ) ) ) ).
% finlsts_rec_LCons_def
thf(fact_228_finlsts__rec__LNil__def,axiom,
! [A: $tType,B: $tType,F: ( coinductive_llist @ A ) > B,C2: B,D3: A > ( coinductive_llist @ A ) > B > B] :
( ( F
= ( lList21916056377ts_rec @ B @ A @ C2 @ D3 ) )
=> ( ( F @ ( coinductive_LNil @ A ) )
= C2 ) ) ).
% finlsts_rec_LNil_def
thf(fact_229_finlsts__rec__LNil,axiom,
! [B: $tType,A: $tType,C2: A,D3: B > ( coinductive_llist @ B ) > A > A] :
( ( lList21916056377ts_rec @ A @ B @ C2 @ D3 @ ( coinductive_LNil @ B ) )
= C2 ) ).
% finlsts_rec_LNil
thf(fact_230_inflsts__def,axiom,
! [A: $tType] :
( ( lList21612149805nflsts @ A )
= ( ^ [A3: set @ A] : ( minus_minus @ ( set @ ( coinductive_llist @ A ) ) @ ( lList2435255213lllsts @ A @ A3 ) @ ( lList2236698231inlsts @ A @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% inflsts_def
thf(fact_231_LList2__Mirabelle__hamjzmohle_Ollength__LCons,axiom,
! [A: $tType,R: coinductive_llist @ A,A2: set @ A,A4: A] :
( ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( lList21232602520length @ A @ ( coinductive_LCons @ A @ A4 @ R ) )
= ( suc @ ( lList21232602520length @ A @ R ) ) ) ) ).
% LList2_Mirabelle_hamjzmohle.llength_LCons
thf(fact_232_DiffI,axiom,
! [A: $tType,C2: A,A2: set @ A,B4: set @ A] :
( ( member @ A @ C2 @ A2 )
=> ( ~ ( member @ A @ C2 @ B4 )
=> ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A2 @ B4 ) ) ) ) ).
% DiffI
thf(fact_233_Diff__iff,axiom,
! [A: $tType,C2: A,A2: set @ A,B4: set @ A] :
( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A2 @ B4 ) )
= ( ( member @ A @ C2 @ A2 )
& ~ ( member @ A @ C2 @ B4 ) ) ) ).
% Diff_iff
thf(fact_234_Diff__idemp,axiom,
! [A: $tType,A2: set @ A,B4: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ B4 ) @ B4 )
= ( minus_minus @ ( set @ A ) @ A2 @ B4 ) ) ).
% Diff_idemp
thf(fact_235_ltake__LCons__Suc,axiom,
! [A: $tType,A4: A,L: coinductive_llist @ A,I: nat] :
( ( lList22119844313_ltake @ A @ ( coinductive_LCons @ A @ A4 @ L ) @ ( suc @ I ) )
= ( coinductive_LCons @ A @ A4 @ ( lList22119844313_ltake @ A @ L @ I ) ) ) ).
% ltake_LCons_Suc
thf(fact_236_DiffE,axiom,
! [A: $tType,C2: A,A2: set @ A,B4: set @ A] :
( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A2 @ B4 ) )
=> ~ ( ( member @ A @ C2 @ A2 )
=> ( member @ A @ C2 @ B4 ) ) ) ).
% DiffE
thf(fact_237_DiffD1,axiom,
! [A: $tType,C2: A,A2: set @ A,B4: set @ A] :
( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A2 @ B4 ) )
=> ( member @ A @ C2 @ A2 ) ) ).
% DiffD1
thf(fact_238_DiffD2,axiom,
! [A: $tType,C2: A,A2: set @ A,B4: set @ A] :
( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A2 @ B4 ) )
=> ~ ( member @ A @ C2 @ B4 ) ) ).
% DiffD2
thf(fact_239_Diff__mono,axiom,
! [A: $tType,A2: set @ A,C3: set @ A,D4: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ C3 )
=> ( ( ord_less_eq @ ( set @ A ) @ D4 @ B4 )
=> ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ B4 ) @ ( minus_minus @ ( set @ A ) @ C3 @ D4 ) ) ) ) ).
% Diff_mono
thf(fact_240_Diff__subset,axiom,
! [A: $tType,A2: set @ A,B4: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A2 @ B4 ) @ A2 ) ).
% Diff_subset
thf(fact_241_double__diff,axiom,
! [A: $tType,A2: set @ A,B4: set @ A,C3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ C3 )
=> ( ( minus_minus @ ( set @ A ) @ B4 @ ( minus_minus @ ( set @ A ) @ C3 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_242_lift__Suc__antimono__le,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [F: nat > A,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_eq @ A @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq @ nat @ N @ N2 )
=> ( ord_less_eq @ A @ ( F @ N2 ) @ ( F @ N ) ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_243_lift__Suc__mono__le,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [F: nat > A,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_eq @ A @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq @ nat @ N @ N2 )
=> ( ord_less_eq @ A @ ( F @ N ) @ ( F @ N2 ) ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_244_diff__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A4: A,B2: A,D3: A,C2: A] :
( ( ord_less_eq @ A @ A4 @ B2 )
=> ( ( ord_less_eq @ A @ D3 @ C2 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ A4 @ C2 ) @ ( minus_minus @ A @ B2 @ D3 ) ) ) ) ) ).
% diff_mono
thf(fact_245_diff__left__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [B2: A,A4: A,C2: A] :
( ( ord_less_eq @ A @ B2 @ A4 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ C2 @ A4 ) @ ( minus_minus @ A @ C2 @ B2 ) ) ) ) ).
% diff_left_mono
thf(fact_246_diff__eq__diff__less__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A4: A,B2: A,C2: A,D3: A] :
( ( ( minus_minus @ A @ A4 @ B2 )
= ( minus_minus @ A @ C2 @ D3 ) )
=> ( ( ord_less_eq @ A @ A4 @ B2 )
= ( ord_less_eq @ A @ C2 @ D3 ) ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_247_diff__right__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A4: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A4 @ B2 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ A4 @ C2 ) @ ( minus_minus @ A @ B2 @ C2 ) ) ) ) ).
% diff_right_mono
thf(fact_248_llast__singleton,axiom,
! [A: $tType,X: A] :
( ( coinductive_llast @ A @ ( coinductive_LCons @ A @ X @ ( coinductive_LNil @ A ) ) )
= X ) ).
% llast_singleton
thf(fact_249_take__inf__less,axiom,
! [A: $tType,T: coinductive_llist @ A,I: nat] :
( ( member @ ( coinductive_llist @ A ) @ T @ ( lList21612149805nflsts @ A @ ( top_top @ ( set @ A ) ) ) )
=> ( ord_less @ ( coinductive_llist @ A ) @ ( lList22119844313_ltake @ A @ T @ I ) @ T ) ) ).
% take_inf_less
thf(fact_250_llast__LCons2,axiom,
! [A: $tType,X: A,Y4: A,Xs: coinductive_llist @ A] :
( ( coinductive_llast @ A @ ( coinductive_LCons @ A @ X @ ( coinductive_LCons @ A @ Y4 @ Xs ) ) )
= ( coinductive_llast @ A @ ( coinductive_LCons @ A @ Y4 @ Xs ) ) ) ).
% llast_LCons2
thf(fact_251_less__LCons,axiom,
! [A: $tType,A4: A,R: coinductive_llist @ A,B2: A,T: coinductive_llist @ A] :
( ( ord_less @ ( coinductive_llist @ A ) @ ( coinductive_LCons @ A @ A4 @ R ) @ ( coinductive_LCons @ A @ B2 @ T ) )
= ( ( A4 = B2 )
& ( ord_less @ ( coinductive_llist @ A ) @ R @ T ) ) ) ).
% less_LCons
thf(fact_252_llist__less__finT,axiom,
! [A: $tType,R: coinductive_llist @ A,S: coinductive_llist @ A,A2: set @ A] :
( ( ord_less @ ( coinductive_llist @ A ) @ R @ S )
=> ( ( member @ ( coinductive_llist @ A ) @ S @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ A2 ) ) ) ) ).
% llist_less_finT
thf(fact_253_LNil__less__LCons,axiom,
! [A: $tType,A4: A,T: coinductive_llist @ A] : ( ord_less @ ( coinductive_llist @ A ) @ ( coinductive_LNil @ A ) @ ( coinductive_LCons @ A @ A4 @ T ) ) ).
% LNil_less_LCons
thf(fact_254_llist__less__def,axiom,
! [A: $tType] :
( ( ord_less @ ( coinductive_llist @ A ) )
= ( ^ [S3: coinductive_llist @ A,T3: coinductive_llist @ A] :
( ( ord_less_eq @ ( coinductive_llist @ A ) @ S3 @ T3 )
& ( S3 != T3 ) ) ) ) ).
% llist_less_def
thf(fact_255_llist__less__le__not__le,axiom,
! [A: $tType] :
( ( ord_less @ ( coinductive_llist @ A ) )
= ( ^ [S3: coinductive_llist @ A,T3: coinductive_llist @ A] :
( ( ord_less_eq @ ( coinductive_llist @ A ) @ S3 @ T3 )
& ~ ( ord_less_eq @ ( coinductive_llist @ A ) @ T3 @ S3 ) ) ) ) ).
% llist_less_le_not_le
%----Type constructors (23)
thf(tcon_fun___Orderings_Oorder__top,axiom,
! [A8: $tType,A9: $tType] :
( ( order_top @ A9 @ ( type2 @ A9 ) )
=> ( order_top @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A8: $tType,A9: $tType] :
( ( preorder @ A9 @ ( type2 @ A9 ) )
=> ( preorder @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A8: $tType,A9: $tType] :
( ( order @ A9 @ ( type2 @ A9 ) )
=> ( order @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
thf(tcon_fun___Orderings_Otop,axiom,
! [A8: $tType,A9: $tType] :
( ( top @ A9 @ ( type2 @ A9 ) )
=> ( top @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A8: $tType,A9: $tType] :
( ( ord @ A9 @ ( type2 @ A9 ) )
=> ( ord @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
thf(tcon_Nat_Onat___Orderings_Opreorder_1,axiom,
preorder @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
linorder @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Oorder_2,axiom,
order @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Oord_3,axiom,
ord @ nat @ ( type2 @ nat ) ).
thf(tcon_Set_Oset___Orderings_Oorder__top_4,axiom,
! [A8: $tType] : ( order_top @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_5,axiom,
! [A8: $tType] : ( preorder @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_6,axiom,
! [A8: $tType] : ( order @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Otop_7,axiom,
! [A8: $tType] : ( top @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_8,axiom,
! [A8: $tType] : ( ord @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_HOL_Obool___Orderings_Oorder__top_9,axiom,
order_top @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Opreorder_10,axiom,
preorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Olinorder_11,axiom,
linorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oorder_12,axiom,
order @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Otop_13,axiom,
top @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oord_14,axiom,
ord @ $o @ ( type2 @ $o ) ).
thf(tcon_Coinductive__List_Ollist___Orderings_Opreorder_15,axiom,
! [A8: $tType] : ( preorder @ ( coinductive_llist @ A8 ) @ ( type2 @ ( coinductive_llist @ A8 ) ) ) ).
thf(tcon_Coinductive__List_Ollist___Orderings_Oorder_16,axiom,
! [A8: $tType] : ( order @ ( coinductive_llist @ A8 ) @ ( type2 @ ( coinductive_llist @ A8 ) ) ) ).
thf(tcon_Coinductive__List_Ollist___Orderings_Oord_17,axiom,
! [A8: $tType] : ( ord @ ( coinductive_llist @ A8 ) @ ( type2 @ ( coinductive_llist @ A8 ) ) ) ).
%----Conjectures (1)
thf(conj_0,conjecture,
? [X4: coinductive_llist @ a] :
( ( member @ ( coinductive_llist @ a ) @ X4 @ ( lList2435255213lllsts @ a @ ( top_top @ ( set @ a ) ) ) )
& ( member @ ( coinductive_llist @ a ) @ ( coinductive_lappend @ a @ r @ X4 ) @ p ) ) ).
%------------------------------------------------------------------------------