TPTP Problem File: DAT185^1.p
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%------------------------------------------------------------------------------
% File : DAT185^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Data Structures
% Problem : Lazy lists II 1155
% 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__1155.p [Bla16]
% Status : Theorem
% Rating : 1.00 v7.5.0, 0.33 v7.2.0, 0.50 v7.1.0
% Syntax : Number of formulae : 351 ( 117 unt; 56 typ; 0 def)
% Number of atoms : 782 ( 209 equ; 0 cnn)
% Maximal formula atoms : 13 ( 2 avg)
% Number of connectives : 4341 ( 62 ~; 11 |; 39 &;3863 @)
% ( 0 <=>; 366 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 8 avg)
% Number of types : 3 ( 2 usr)
% Number of type conns : 155 ( 155 >; 0 *; 0 +; 0 <<)
% Number of symbols : 55 ( 54 usr; 3 con; 0-5 aty)
% Number of variables : 1008 ( 32 ^; 916 !; 12 ?;1008 :)
% ( 48 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 14:51:44.531
%------------------------------------------------------------------------------
%----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 (51)
thf(sy_cl_Lattices_Obounded__lattice,type,
bounded_lattice:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ominus,type,
minus:
!>[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_Groups_Ogroup__add,type,
group_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Ono__bot,type,
no_bot:
!>[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_Lattices_Osemilattice__sup,type,
semilattice_sup:
!>[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_cl_Lattices_Obounded__lattice__top,type,
bounded_lattice_top:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
cancel146912293up_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_Oinflsts,type,
lList21612149805nflsts:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( coinductive_llist @ A ) ) ) ).
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_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_Oposlsts,type,
lList21148268032oslsts:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( coinductive_llist @ A ) ) ) ).
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_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_Lattices_Osup__class_Osup,type,
sup_sup:
!>[A: $tType] : ( A > A > A ) ).
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_s,type,
s: coinductive_llist @ a ).
thf(sy_v_t,type,
t: coinductive_llist @ a ).
thf(sy_v_x,type,
x: coinductive_llist @ a ).
%----Relevant facts (256)
thf(fact_0_local_Ofinite,axiom,
member @ ( coinductive_llist @ a ) @ s @ ( lList2236698231inlsts @ a @ ( top_top @ ( set @ a ) ) ) ).
% local.finite
thf(fact_1_sx,axiom,
ord_less_eq @ ( coinductive_llist @ a ) @ s @ x ).
% sx
thf(fact_2_tx,axiom,
ord_less_eq @ ( coinductive_llist @ a ) @ t @ x ).
% tx
thf(fact_3_llist__le__refl,axiom,
! [A: $tType,S: coinductive_llist @ A] : ( ord_less_eq @ ( coinductive_llist @ A ) @ S @ S ) ).
% llist_le_refl
thf(fact_4_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_5_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_6_prefix__closed__def,axiom,
! [A: $tType] :
( ( lList21638733016closed @ A )
= ( ^ [A2: set @ ( coinductive_llist @ A )] :
! [X: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ X @ A2 )
=> ! [S2: coinductive_llist @ A] :
( ( ord_less_eq @ ( coinductive_llist @ A ) @ S2 @ X )
=> ( member @ ( coinductive_llist @ A ) @ S2 @ A2 ) ) ) ) ) ).
% prefix_closed_def
thf(fact_7_suffix__closed__def,axiom,
! [A: $tType] :
( ( lList2736192599closed @ A )
= ( ^ [A2: set @ ( coinductive_llist @ A )] :
! [X: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ X @ A2 )
=> ! [S2: coinductive_llist @ A] :
( ( ord_less_eq @ ( coinductive_llist @ A ) @ X @ S2 )
=> ( member @ ( coinductive_llist @ A ) @ S2 @ A2 ) ) ) ) ) ).
% suffix_closed_def
thf(fact_8_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X2: A] : ( ord_less_eq @ A @ X2 @ X2 ) ) ).
% order_refl
thf(fact_9__092_060open_062s_A_092_060in_062_AUNIV_092_060_094sup_062_092_060infinity_062_092_060close_062,axiom,
member @ ( coinductive_llist @ a ) @ s @ ( lList2435255213lllsts @ a @ ( top_top @ ( set @ a ) ) ) ).
% \<open>s \<in> UNIV\<^sup>\<infinity>\<close>
thf(fact_10_llist__le__finT,axiom,
! [A: $tType,R: coinductive_llist @ A,S: coinductive_llist @ A,A3: set @ A] :
( ( ord_less_eq @ ( coinductive_llist @ A ) @ R @ S )
=> ( ( member @ ( coinductive_llist @ A ) @ S @ ( lList2236698231inlsts @ A @ A3 ) )
=> ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ A3 ) ) ) ) ).
% llist_le_finT
thf(fact_11_le__funD,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [F: A > B,G: A > B,X2: A] :
( ( ord_less_eq @ ( A > B ) @ F @ G )
=> ( ord_less_eq @ B @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ).
% le_funD
thf(fact_12_le__funE,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [F: A > B,G: A > B,X2: A] :
( ( ord_less_eq @ ( A > B ) @ F @ G )
=> ( ord_less_eq @ B @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ).
% le_funE
thf(fact_13_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_14_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ( ( ord_less_eq @ ( A > B ) )
= ( ^ [F2: A > B,G2: A > B] :
! [X: A] : ( ord_less_eq @ B @ ( F2 @ X ) @ ( G2 @ X ) ) ) ) ) ).
% le_fun_def
thf(fact_15_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B @ ( type2 @ B ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A4: A,F: B > A,B2: B,C: B] :
( ( ord_less_eq @ A @ A4 @ ( F @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C )
=> ( ! [X3: B,Y: B] :
( ( ord_less_eq @ B @ X3 @ Y )
=> ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq @ A @ A4 @ ( F @ C ) ) ) ) ) ) ).
% order_subst1
thf(fact_16_order__subst2,axiom,
! [A: $tType,C2: $tType] :
( ( ( order @ C2 @ ( type2 @ C2 ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A4: A,B2: A,F: A > C2,C: C2] :
( ( ord_less_eq @ A @ A4 @ B2 )
=> ( ( ord_less_eq @ C2 @ ( F @ B2 ) @ C )
=> ( ! [X3: A,Y: A] :
( ( ord_less_eq @ A @ X3 @ Y )
=> ( ord_less_eq @ C2 @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq @ C2 @ ( F @ A4 ) @ C ) ) ) ) ) ).
% order_subst2
thf(fact_17_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A4: A,F: B > A,B2: B,C: B] :
( ( A4
= ( F @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C )
=> ( ! [X3: B,Y: B] :
( ( ord_less_eq @ B @ X3 @ Y )
=> ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq @ A @ A4 @ ( F @ C ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_18_top__apply,axiom,
! [C2: $tType,D: $tType] :
( ( top @ C2 @ ( type2 @ C2 ) )
=> ( ( top_top @ ( D > C2 ) )
= ( ^ [X: D] : ( top_top @ C2 ) ) ) ) ).
% top_apply
thf(fact_19_alllsts__UNIV,axiom,
! [A: $tType,S: coinductive_llist @ A] : ( member @ ( coinductive_llist @ A ) @ S @ ( lList2435255213lllsts @ A @ ( top_top @ ( set @ A ) ) ) ) ).
% alllsts_UNIV
thf(fact_20_alllsts__mono,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ord_less_eq @ ( set @ ( coinductive_llist @ A ) ) @ ( lList2435255213lllsts @ A @ A3 ) @ ( lList2435255213lllsts @ A @ B3 ) ) ) ).
% alllsts_mono
thf(fact_21_finite__lemma,axiom,
! [A: $tType,X2: coinductive_llist @ A,A3: set @ A,B3: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ X2 @ ( lList2236698231inlsts @ A @ A3 ) )
=> ( ( member @ ( coinductive_llist @ A ) @ X2 @ ( lList2435255213lllsts @ A @ B3 ) )
=> ( member @ ( coinductive_llist @ A ) @ X2 @ ( lList2236698231inlsts @ A @ B3 ) ) ) ) ).
% finite_lemma
thf(fact_22_finlsts__mono,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ord_less_eq @ ( set @ ( coinductive_llist @ A ) ) @ ( lList2236698231inlsts @ A @ A3 ) @ ( lList2236698231inlsts @ A @ B3 ) ) ) ).
% finlsts_mono
thf(fact_23_finsubsetall,axiom,
! [A: $tType,X2: coinductive_llist @ A,A3: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ X2 @ ( lList2236698231inlsts @ A @ A3 ) )
=> ( member @ ( coinductive_llist @ A ) @ X2 @ ( lList2435255213lllsts @ A @ A3 ) ) ) ).
% finsubsetall
thf(fact_24_fin__finite,axiom,
! [A: $tType,R: coinductive_llist @ A,A3: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ A3 ) )
=> ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% fin_finite
thf(fact_25_finT__simp,axiom,
! [A: $tType,R: coinductive_llist @ A,A3: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ A3 ) )
=> ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% finT_simp
thf(fact_26_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_27_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_28_top__greatest,axiom,
! [A: $tType] :
( ( order_top @ A @ ( type2 @ A ) )
=> ! [A4: A] : ( ord_less_eq @ A @ A4 @ ( top_top @ A ) ) ) ).
% top_greatest
thf(fact_29_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_30_dual__order_Otrans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B2: A,A4: A,C: A] :
( ( ord_less_eq @ A @ B2 @ A4 )
=> ( ( ord_less_eq @ A @ C @ B2 )
=> ( ord_less_eq @ A @ C @ A4 ) ) ) ) ).
% dual_order.trans
thf(fact_31_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > A > $o,A4: A,B2: A] :
( ! [A5: A,B4: A] :
( ( ord_less_eq @ A @ A5 @ B4 )
=> ( P @ A5 @ B4 ) )
=> ( ! [A5: A,B4: A] :
( ( P @ B4 @ A5 )
=> ( P @ A5 @ B4 ) )
=> ( P @ A4 @ B2 ) ) ) ) ).
% linorder_wlog
thf(fact_32_dual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A4: A] : ( ord_less_eq @ A @ A4 @ A4 ) ) ).
% dual_order.refl
thf(fact_33_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X2: A,Y2: A,Z: A] :
( ( ord_less_eq @ A @ X2 @ Y2 )
=> ( ( ord_less_eq @ A @ Y2 @ Z )
=> ( ord_less_eq @ A @ X2 @ Z ) ) ) ) ).
% order_trans
thf(fact_34_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_35_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A4: A,B2: A,C: A] :
( ( ord_less_eq @ A @ A4 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq @ A @ A4 @ C ) ) ) ) ).
% ord_le_eq_trans
thf(fact_36_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A4: A,B2: A,C: A] :
( ( A4 = B2 )
=> ( ( ord_less_eq @ A @ B2 @ C )
=> ( ord_less_eq @ A @ A4 @ C ) ) ) ) ).
% ord_eq_le_trans
thf(fact_37_antisym__conv,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [Y2: A,X2: A] :
( ( ord_less_eq @ A @ Y2 @ X2 )
=> ( ( ord_less_eq @ A @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ) ).
% antisym_conv
thf(fact_38_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X2: A,Y2: A,Z: A] :
( ( ( ord_less_eq @ A @ X2 @ Y2 )
=> ~ ( ord_less_eq @ A @ Y2 @ Z ) )
=> ( ( ( ord_less_eq @ A @ Y2 @ X2 )
=> ~ ( ord_less_eq @ A @ X2 @ Z ) )
=> ( ( ( ord_less_eq @ A @ X2 @ Z )
=> ~ ( ord_less_eq @ A @ Z @ Y2 ) )
=> ( ( ( ord_less_eq @ A @ Z @ Y2 )
=> ~ ( ord_less_eq @ A @ Y2 @ X2 ) )
=> ( ( ( ord_less_eq @ A @ Y2 @ Z )
=> ~ ( ord_less_eq @ A @ Z @ X2 ) )
=> ~ ( ( ord_less_eq @ A @ Z @ X2 )
=> ~ ( ord_less_eq @ A @ X2 @ Y2 ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_39_order_Otrans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A4: A,B2: A,C: A] :
( ( ord_less_eq @ A @ A4 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ C )
=> ( ord_less_eq @ A @ A4 @ C ) ) ) ) ).
% order.trans
thf(fact_40_le__cases,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X2: A,Y2: A] :
( ~ ( ord_less_eq @ A @ X2 @ Y2 )
=> ( ord_less_eq @ A @ Y2 @ X2 ) ) ) ).
% le_cases
thf(fact_41_eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X2: A,Y2: A] :
( ( X2 = Y2 )
=> ( ord_less_eq @ A @ X2 @ Y2 ) ) ) ).
% eq_refl
thf(fact_42_linear,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X2: A,Y2: A] :
( ( ord_less_eq @ A @ X2 @ Y2 )
| ( ord_less_eq @ A @ Y2 @ X2 ) ) ) ).
% linear
thf(fact_43_antisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X2: A,Y2: A] :
( ( ord_less_eq @ A @ X2 @ Y2 )
=> ( ( ord_less_eq @ A @ Y2 @ X2 )
=> ( X2 = Y2 ) ) ) ) ).
% antisym
thf(fact_44_eq__iff,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ^ [Y3: A,Z2: A] : Y3 = Z2 )
= ( ^ [X: A,Y4: A] :
( ( ord_less_eq @ A @ X @ Y4 )
& ( ord_less_eq @ A @ Y4 @ X ) ) ) ) ) ).
% eq_iff
thf(fact_45_mem__Collect__eq,axiom,
! [A: $tType,A4: A,P: A > $o] :
( ( member @ A @ A4 @ ( collect @ A @ P ) )
= ( P @ A4 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: $tType,A3: set @ A] :
( ( collect @ A
@ ^ [X: A] : ( member @ A @ X @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X3: A] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_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_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A4: A,B2: A,F: A > B,C: B] :
( ( ord_less_eq @ A @ A4 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: A,Y: A] :
( ( ord_less_eq @ A @ X3 @ Y )
=> ( ord_less_eq @ B @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq @ B @ ( F @ A4 ) @ C ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_50_UNIV__I,axiom,
! [A: $tType,X2: A] : ( member @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ).
% UNIV_I
thf(fact_51_iso__tuple__UNIV__I,axiom,
! [A: $tType,X2: A] : ( member @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ).
% iso_tuple_UNIV_I
thf(fact_52_take__fin,axiom,
! [A: $tType,T: coinductive_llist @ A,A3: set @ A,I: nat] :
( ( member @ ( coinductive_llist @ A ) @ T @ ( lList2435255213lllsts @ A @ A3 ) )
=> ( member @ ( coinductive_llist @ A ) @ ( lList22119844313_ltake @ A @ T @ I ) @ ( lList2236698231inlsts @ A @ A3 ) ) ) ).
% take_fin
thf(fact_53_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_54_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_55_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_56_inflstsI,axiom,
! [A: $tType,X2: coinductive_llist @ A,A3: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ X2 @ ( lList2435255213lllsts @ A @ A3 ) )
=> ( ~ ( member @ ( coinductive_llist @ A ) @ X2 @ ( lList2236698231inlsts @ A @ ( top_top @ ( set @ A ) ) ) )
=> ( member @ ( coinductive_llist @ A ) @ X2 @ ( lList21612149805nflsts @ A @ A3 ) ) ) ) ).
% inflstsI
thf(fact_57_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_58_inflstsE,axiom,
! [A: $tType,X2: coinductive_llist @ A,A3: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ X2 @ ( lList21612149805nflsts @ A @ A3 ) )
=> ~ ( ( member @ ( coinductive_llist @ A ) @ X2 @ ( lList2435255213lllsts @ A @ A3 ) )
=> ( member @ ( coinductive_llist @ A ) @ X2 @ ( lList2236698231inlsts @ A @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% inflstsE
thf(fact_59_UNIV__eq__I,axiom,
! [A: $tType,A3: set @ A] :
( ! [X3: A] : ( member @ A @ X3 @ A3 )
=> ( ( top_top @ ( set @ A ) )
= A3 ) ) ).
% UNIV_eq_I
thf(fact_60_subsetI,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ! [X3: A] :
( ( member @ A @ X3 @ A3 )
=> ( member @ A @ X3 @ B3 ) )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ).
% subsetI
thf(fact_61_subset__antisym,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B3 @ A3 )
=> ( A3 = B3 ) ) ) ).
% subset_antisym
thf(fact_62_lrevT,axiom,
! [A: $tType,Xs: coinductive_llist @ A,A3: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ Xs @ ( lList2236698231inlsts @ A @ A3 ) )
=> ( member @ ( coinductive_llist @ A ) @ ( lList2281150353e_lrev @ A @ Xs ) @ ( lList2236698231inlsts @ A @ A3 ) ) ) ).
% lrevT
thf(fact_63_notinf__fin,axiom,
! [A: $tType,X2: coinductive_llist @ A] :
( ( ~ ( member @ ( coinductive_llist @ A ) @ X2 @ ( lList21612149805nflsts @ A @ ( top_top @ ( set @ A ) ) ) ) )
= ( member @ ( coinductive_llist @ A ) @ X2 @ ( lList2236698231inlsts @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% notinf_fin
thf(fact_64_notfin__inf,axiom,
! [A: $tType,X2: coinductive_llist @ A] :
( ( ~ ( member @ ( coinductive_llist @ A ) @ X2 @ ( lList2236698231inlsts @ A @ ( top_top @ ( set @ A ) ) ) ) )
= ( member @ ( coinductive_llist @ A ) @ X2 @ ( lList21612149805nflsts @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% notfin_inf
thf(fact_65_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_66_set__mp,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,X2: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ( member @ A @ X2 @ A3 )
=> ( member @ A @ X2 @ B3 ) ) ) ).
% set_mp
thf(fact_67_in__mono,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,X2: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ( member @ A @ X2 @ A3 )
=> ( member @ A @ X2 @ B3 ) ) ) ).
% in_mono
thf(fact_68_subsetD,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,C: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ( member @ A @ C @ A3 )
=> ( member @ A @ C @ B3 ) ) ) ).
% subsetD
thf(fact_69_subsetCE,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,C: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ( member @ A @ C @ A3 )
=> ( member @ A @ C @ B3 ) ) ) ).
% subsetCE
thf(fact_70_equalityE,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( A3 = B3 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ~ ( ord_less_eq @ ( set @ A ) @ B3 @ A3 ) ) ) ).
% equalityE
thf(fact_71_subset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A2: set @ A,B5: set @ A] :
! [X: A] :
( ( member @ A @ X @ A2 )
=> ( member @ A @ X @ B5 ) ) ) ) ).
% subset_eq
thf(fact_72_equalityD1,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( A3 = B3 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ).
% equalityD1
thf(fact_73_equalityD2,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( A3 = B3 )
=> ( ord_less_eq @ ( set @ A ) @ B3 @ A3 ) ) ).
% equalityD2
thf(fact_74_set__rev__mp,axiom,
! [A: $tType,X2: A,A3: set @ A,B3: set @ A] :
( ( member @ A @ X2 @ A3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( member @ A @ X2 @ B3 ) ) ) ).
% set_rev_mp
thf(fact_75_subset__iff,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A2: set @ A,B5: set @ A] :
! [T2: A] :
( ( member @ A @ T2 @ A2 )
=> ( member @ A @ T2 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_76_rev__subsetD,axiom,
! [A: $tType,C: A,A3: set @ A,B3: set @ A] :
( ( member @ A @ C @ A3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( member @ A @ C @ B3 ) ) ) ).
% rev_subsetD
thf(fact_77_subset__refl,axiom,
! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ A3 @ A3 ) ).
% subset_refl
thf(fact_78_Collect__mono,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X3: A] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% Collect_mono
thf(fact_79_subset__trans,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,C3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B3 @ C3 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ C3 ) ) ) ).
% subset_trans
thf(fact_80_set__eq__subset,axiom,
! [A: $tType] :
( ( ^ [Y3: set @ A,Z2: set @ A] : Y3 = Z2 )
= ( ^ [A2: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B5 )
& ( ord_less_eq @ ( set @ A ) @ B5 @ A2 ) ) ) ) ).
% set_eq_subset
thf(fact_81_contra__subsetD,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,C: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ~ ( member @ A @ C @ B3 )
=> ~ ( member @ A @ C @ A3 ) ) ) ).
% contra_subsetD
thf(fact_82_Collect__mono__iff,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
= ( ! [X: A] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_83_inflsts__mono,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ord_less_eq @ ( set @ ( coinductive_llist @ A ) ) @ ( lList21612149805nflsts @ A @ A3 ) @ ( lList21612149805nflsts @ A @ B3 ) ) ) ).
% inflsts_mono
thf(fact_84_ldrop__infT,axiom,
! [A: $tType,T: coinductive_llist @ A,A3: set @ A,I: nat] :
( ( member @ ( coinductive_llist @ A ) @ T @ ( lList21612149805nflsts @ A @ A3 ) )
=> ( member @ ( coinductive_llist @ A ) @ ( lList2508575361_ldrop @ A @ T @ I ) @ ( lList21612149805nflsts @ A @ A3 ) ) ) ).
% ldrop_infT
thf(fact_85_top__set__def,axiom,
! [A: $tType] :
( ( top_top @ ( set @ A ) )
= ( collect @ A @ ( top_top @ ( A > $o ) ) ) ) ).
% top_set_def
thf(fact_86_infT__simp,axiom,
! [A: $tType,R: coinductive_llist @ A,A3: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ R @ ( lList21612149805nflsts @ A @ A3 ) )
=> ( member @ ( coinductive_llist @ A ) @ R @ ( lList21612149805nflsts @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% infT_simp
thf(fact_87_infsubsetall,axiom,
! [A: $tType,X2: coinductive_llist @ A,A3: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ X2 @ ( lList21612149805nflsts @ A @ A3 ) )
=> ( member @ ( coinductive_llist @ A ) @ X2 @ ( lList2435255213lllsts @ A @ A3 ) ) ) ).
% infsubsetall
thf(fact_88_llist__inf__le,axiom,
! [A: $tType,S: coinductive_llist @ A,A3: set @ A,T: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ S @ ( lList21612149805nflsts @ A @ A3 ) )
=> ( ( ord_less_eq @ ( coinductive_llist @ A ) @ S @ T )
= ( S = T ) ) ) ).
% llist_inf_le
thf(fact_89_ldrop__finT,axiom,
! [A: $tType,T: coinductive_llist @ A,A3: set @ A,I: nat] :
( ( member @ ( coinductive_llist @ A ) @ T @ ( lList2236698231inlsts @ A @ A3 ) )
=> ( member @ ( coinductive_llist @ A ) @ ( lList2508575361_ldrop @ A @ T @ I ) @ ( lList2236698231inlsts @ A @ A3 ) ) ) ).
% ldrop_finT
thf(fact_90_ldropT,axiom,
! [A: $tType,T: coinductive_llist @ A,A3: set @ A,I: nat] :
( ( member @ ( coinductive_llist @ A ) @ T @ ( lList2435255213lllsts @ A @ A3 ) )
=> ( member @ ( coinductive_llist @ A ) @ ( lList2508575361_ldrop @ A @ T @ I ) @ ( lList2435255213lllsts @ A @ A3 ) ) ) ).
% ldropT
thf(fact_91_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_92_alllstsE,axiom,
! [A: $tType,X2: coinductive_llist @ A,A3: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ X2 @ ( lList2435255213lllsts @ A @ A3 ) )
=> ( ~ ( member @ ( coinductive_llist @ A ) @ X2 @ ( lList2236698231inlsts @ A @ A3 ) )
=> ( member @ ( coinductive_llist @ A ) @ X2 @ ( lList21612149805nflsts @ A @ A3 ) ) ) ) ).
% alllstsE
thf(fact_93_inf__neqE,axiom,
! [A: $tType,X2: coinductive_llist @ A,A3: set @ A,Y2: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ X2 @ ( lList21612149805nflsts @ A @ A3 ) )
=> ( ( member @ ( coinductive_llist @ A ) @ Y2 @ ( lList21612149805nflsts @ A @ A3 ) )
=> ( ( X2 != Y2 )
=> ~ ! [S3: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ S3 @ ( lList2236698231inlsts @ A @ A3 ) )
=> ( ( ord_less_eq @ ( coinductive_llist @ A ) @ S3 @ X2 )
=> ( ord_less_eq @ ( coinductive_llist @ A ) @ S3 @ Y2 ) ) ) ) ) ) ).
% inf_neqE
thf(fact_94_prefix__lemma,axiom,
! [A: $tType,X2: coinductive_llist @ A,A3: set @ A,Y2: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ X2 @ ( lList21612149805nflsts @ A @ A3 ) )
=> ( ( member @ ( coinductive_llist @ A ) @ Y2 @ ( lList21612149805nflsts @ A @ A3 ) )
=> ( ! [S3: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ S3 @ ( lList2236698231inlsts @ A @ A3 ) )
=> ( ( ord_less_eq @ ( coinductive_llist @ A ) @ S3 @ X2 )
=> ( ord_less_eq @ ( coinductive_llist @ A ) @ S3 @ Y2 ) ) )
=> ( X2 = Y2 ) ) ) ) ).
% prefix_lemma
thf(fact_95_UNIV__witness,axiom,
! [A: $tType] :
? [X3: A] : ( member @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ).
% UNIV_witness
thf(fact_96_subset__UNIV,axiom,
! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ A3 @ ( top_top @ ( set @ A ) ) ) ).
% subset_UNIV
thf(fact_97_poslsts__mono,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ord_less_eq @ ( set @ ( coinductive_llist @ A ) ) @ ( lList21148268032oslsts @ A @ A3 ) @ ( lList21148268032oslsts @ A @ B3 ) ) ) ).
% poslsts_mono
thf(fact_98_llength__take,axiom,
! [A: $tType,T: coinductive_llist @ A,A3: set @ A,I: nat] :
( ( member @ ( coinductive_llist @ A ) @ T @ ( lList21612149805nflsts @ A @ A3 ) )
=> ( ( lList21232602520length @ A @ ( lList22119844313_ltake @ A @ T @ I ) )
= I ) ) ).
% llength_take
thf(fact_99_infsuff__mono2,axiom,
! [A: $tType,S: coinductive_llist @ A,T: coinductive_llist @ A,A3: set @ A] :
( ( ord_less_eq @ ( coinductive_llist @ A ) @ S @ T )
=> ( ord_less_eq @ ( set @ ( coinductive_llist @ A ) ) @ ( lList2649413865nfsuff @ A @ A3 @ T ) @ ( lList2649413865nfsuff @ A @ A3 @ S ) ) ) ).
% infsuff_mono2
thf(fact_100_suff__mono2,axiom,
! [A: $tType,S: coinductive_llist @ A,T: coinductive_llist @ A,A3: set @ A] :
( ( ord_less_eq @ ( coinductive_llist @ A ) @ S @ T )
=> ( ord_less_eq @ ( set @ ( coinductive_llist @ A ) ) @ ( lList21475143548e_suff @ A @ A3 @ T ) @ ( lList21475143548e_suff @ A @ A3 @ S ) ) ) ).
% suff_mono2
thf(fact_101_fpslsts__mono,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ord_less_eq @ ( set @ ( coinductive_llist @ A ) ) @ ( lList22096119349pslsts @ A @ A3 ) @ ( lList22096119349pslsts @ A @ B3 ) ) ) ).
% fpslsts_mono
thf(fact_102_finpref__mono2,axiom,
! [A: $tType,S: coinductive_llist @ A,T: coinductive_llist @ A,A3: set @ A] :
( ( ord_less_eq @ ( coinductive_llist @ A ) @ S @ T )
=> ( ord_less_eq @ ( set @ ( coinductive_llist @ A ) ) @ ( lList21202317876inpref @ A @ A3 @ S ) @ ( lList21202317876inpref @ A @ A3 @ T ) ) ) ).
% finpref_mono2
thf(fact_103_inflsts__def,axiom,
! [A: $tType] :
( ( lList21612149805nflsts @ A )
= ( ^ [A2: set @ A] : ( minus_minus @ ( set @ ( coinductive_llist @ A ) ) @ ( lList2435255213lllsts @ A @ A2 ) @ ( lList2236698231inlsts @ A @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% inflsts_def
thf(fact_104_Diff__idemp,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) @ B3 )
= ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) ).
% Diff_idemp
thf(fact_105_Diff__iff,axiom,
! [A: $tType,C: A,A3: set @ A,B3: set @ A] :
( ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) )
= ( ( member @ A @ C @ A3 )
& ~ ( member @ A @ C @ B3 ) ) ) ).
% Diff_iff
thf(fact_106_DiffI,axiom,
! [A: $tType,C: A,A3: set @ A,B3: set @ A] :
( ( member @ A @ C @ A3 )
=> ( ~ ( member @ A @ C @ B3 )
=> ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) ) ) ).
% DiffI
thf(fact_107_top1I,axiom,
! [A: $tType,X2: A] : ( top_top @ ( A > $o ) @ X2 ) ).
% top1I
thf(fact_108_infsuff__finpref__iff,axiom,
! [A: $tType,R: coinductive_llist @ A,A3: set @ A,T: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ A3 ) )
=> ( ( member @ ( coinductive_llist @ A ) @ T @ ( lList21612149805nflsts @ A @ A3 ) )
=> ( ( member @ ( coinductive_llist @ A ) @ T @ ( lList21202317876inpref @ A @ A3 @ R ) )
= ( member @ ( coinductive_llist @ A ) @ R @ ( lList2649413865nfsuff @ A @ A3 @ T ) ) ) ) ) ).
% infsuff_finpref_iff
thf(fact_109_DiffD2,axiom,
! [A: $tType,C: A,A3: set @ A,B3: set @ A] :
( ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) )
=> ~ ( member @ A @ C @ B3 ) ) ).
% DiffD2
thf(fact_110_DiffD1,axiom,
! [A: $tType,C: A,A3: set @ A,B3: set @ A] :
( ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) )
=> ( member @ A @ C @ A3 ) ) ).
% DiffD1
thf(fact_111_DiffE,axiom,
! [A: $tType,C: A,A3: set @ A,B3: set @ A] :
( ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) )
=> ~ ( ( member @ A @ C @ A3 )
=> ( member @ A @ C @ B3 ) ) ) ).
% DiffE
thf(fact_112_suff__finpref,axiom,
! [A: $tType,T: coinductive_llist @ A,A3: set @ A,R: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ T @ ( lList21475143548e_suff @ A @ A3 @ R ) )
=> ( ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ A3 ) )
=> ( member @ ( coinductive_llist @ A ) @ R @ ( lList21202317876inpref @ A @ A3 @ T ) ) ) ) ).
% suff_finpref
thf(fact_113_infsuff__finpref,axiom,
! [A: $tType,T: coinductive_llist @ A,A3: set @ A,R: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ T @ ( lList2649413865nfsuff @ A @ A3 @ R ) )
=> ( ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ A3 ) )
=> ( member @ ( coinductive_llist @ A ) @ R @ ( lList21202317876inpref @ A @ A3 @ T ) ) ) ) ).
% infsuff_finpref
thf(fact_114_finpref__suff,axiom,
! [A: $tType,R: coinductive_llist @ A,A3: set @ A,T: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ R @ ( lList21202317876inpref @ A @ A3 @ T ) )
=> ( ( member @ ( coinductive_llist @ A ) @ T @ ( lList2435255213lllsts @ A @ A3 ) )
=> ( member @ ( coinductive_llist @ A ) @ T @ ( lList21475143548e_suff @ A @ A3 @ R ) ) ) ) ).
% finpref_suff
thf(fact_115_finpref__infsuff,axiom,
! [A: $tType,R: coinductive_llist @ A,A3: set @ A,T: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ R @ ( lList21202317876inpref @ A @ A3 @ T ) )
=> ( ( member @ ( coinductive_llist @ A ) @ T @ ( lList21612149805nflsts @ A @ A3 ) )
=> ( member @ ( coinductive_llist @ A ) @ T @ ( lList2649413865nfsuff @ A @ A3 @ R ) ) ) ) ).
% finpref_infsuff
thf(fact_116_double__diff,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,C3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B3 @ C3 )
=> ( ( minus_minus @ ( set @ A ) @ B3 @ ( minus_minus @ ( set @ A ) @ C3 @ A3 ) )
= A3 ) ) ) ).
% double_diff
thf(fact_117_Diff__subset,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) @ A3 ) ).
% Diff_subset
thf(fact_118_Diff__mono,axiom,
! [A: $tType,A3: set @ A,C3: set @ A,D2: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ C3 )
=> ( ( ord_less_eq @ ( set @ A ) @ D2 @ B3 )
=> ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) @ ( minus_minus @ ( set @ A ) @ C3 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_119_suff__finpref__iff,axiom,
! [A: $tType,R: coinductive_llist @ A,A3: set @ A,T: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ A3 ) )
=> ( ( member @ ( coinductive_llist @ A ) @ T @ ( lList2435255213lllsts @ A @ A3 ) )
=> ( ( member @ ( coinductive_llist @ A ) @ R @ ( lList21202317876inpref @ A @ A3 @ T ) )
= ( member @ ( coinductive_llist @ A ) @ T @ ( lList21475143548e_suff @ A @ A3 @ R ) ) ) ) ) ).
% suff_finpref_iff
thf(fact_120_finpref__fin,axiom,
! [A: $tType,X2: coinductive_llist @ A,A3: set @ A,S: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ X2 @ ( lList21202317876inpref @ A @ A3 @ S ) )
=> ( member @ ( coinductive_llist @ A ) @ X2 @ ( lList2236698231inlsts @ A @ A3 ) ) ) ).
% finpref_fin
thf(fact_121_suff__all,axiom,
! [A: $tType,X2: coinductive_llist @ A,A3: set @ A,S: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ X2 @ ( lList21475143548e_suff @ A @ A3 @ S ) )
=> ( member @ ( coinductive_llist @ A ) @ X2 @ ( lList2435255213lllsts @ A @ A3 ) ) ) ).
% suff_all
thf(fact_122_infsuff__inf,axiom,
! [A: $tType,X2: coinductive_llist @ A,A3: set @ A,S: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ X2 @ ( lList2649413865nfsuff @ A @ A3 @ S ) )
=> ( member @ ( coinductive_llist @ A ) @ X2 @ ( lList21612149805nflsts @ A @ A3 ) ) ) ).
% infsuff_inf
thf(fact_123_minus__apply,axiom,
! [B: $tType,A: $tType] :
( ( minus @ B @ ( type2 @ B ) )
=> ( ( minus_minus @ ( A > B ) )
= ( ^ [A2: A > B,B5: A > B,X: A] : ( minus_minus @ B @ ( A2 @ X ) @ ( B5 @ X ) ) ) ) ) ).
% minus_apply
thf(fact_124_top__empty__eq,axiom,
! [A: $tType] :
( ( top_top @ ( A > $o ) )
= ( ^ [X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ).
% top_empty_eq
thf(fact_125_diff__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A4: A,B2: A,D3: A,C: A] :
( ( ord_less_eq @ A @ A4 @ B2 )
=> ( ( ord_less_eq @ A @ D3 @ C )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ A4 @ C ) @ ( minus_minus @ A @ B2 @ D3 ) ) ) ) ) ).
% diff_mono
thf(fact_126_diff__left__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [B2: A,A4: A,C: A] :
( ( ord_less_eq @ A @ B2 @ A4 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ C @ A4 ) @ ( minus_minus @ A @ C @ B2 ) ) ) ) ).
% diff_left_mono
thf(fact_127_diff__right__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A4: A,B2: A,C: A] :
( ( ord_less_eq @ A @ A4 @ B2 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ A4 @ C ) @ ( minus_minus @ A @ B2 @ C ) ) ) ) ).
% diff_right_mono
thf(fact_128_diff__right__commute,axiom,
! [A: $tType] :
( ( cancel146912293up_add @ A @ ( type2 @ A ) )
=> ! [A4: A,C: A,B2: A] :
( ( minus_minus @ A @ ( minus_minus @ A @ A4 @ C ) @ B2 )
= ( minus_minus @ A @ ( minus_minus @ A @ A4 @ B2 ) @ C ) ) ) ).
% diff_right_commute
thf(fact_129_diff__eq__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [A4: A,B2: A,C: A,D3: A] :
( ( ( minus_minus @ A @ A4 @ B2 )
= ( minus_minus @ A @ C @ D3 ) )
=> ( ( A4 = B2 )
= ( C = D3 ) ) ) ) ).
% diff_eq_diff_eq
thf(fact_130_fun__diff__def,axiom,
! [B: $tType,A: $tType] :
( ( minus @ B @ ( type2 @ B ) )
=> ( ( minus_minus @ ( A > B ) )
= ( ^ [A2: A > B,B5: A > B,X: A] : ( minus_minus @ B @ ( A2 @ X ) @ ( B5 @ X ) ) ) ) ) ).
% fun_diff_def
thf(fact_131_diff__eq__diff__less__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A4: A,B2: A,C: A,D3: A] :
( ( ( minus_minus @ A @ A4 @ B2 )
= ( minus_minus @ A @ C @ D3 ) )
=> ( ( ord_less_eq @ A @ A4 @ B2 )
= ( ord_less_eq @ A @ C @ D3 ) ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_132_top__conj_I1_J,axiom,
! [A: $tType,X2: A,P: $o] :
( ( ( top_top @ ( A > $o ) @ X2 )
& P )
= P ) ).
% top_conj(1)
thf(fact_133_top__conj_I2_J,axiom,
! [A: $tType,P: $o,X2: A] :
( ( P
& ( top_top @ ( A > $o ) @ X2 ) )
= P ) ).
% top_conj(2)
thf(fact_134_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_135_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_136_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_137_LNil__le,axiom,
! [A: $tType,S: coinductive_llist @ A] : ( ord_less_eq @ ( coinductive_llist @ A ) @ ( coinductive_LNil @ A ) @ S ) ).
% LNil_le
thf(fact_138_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_139_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_140_lrev__LNil,axiom,
! [A: $tType] :
( ( lList2281150353e_lrev @ A @ ( coinductive_LNil @ A ) )
= ( coinductive_LNil @ A ) ) ).
% lrev_LNil
thf(fact_141_LNil__suff,axiom,
! [A: $tType,A3: set @ A,S: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ ( coinductive_LNil @ A ) @ ( lList21475143548e_suff @ A @ A3 @ S ) )
= ( S
= ( coinductive_LNil @ A ) ) ) ).
% LNil_suff
thf(fact_142_fpslsts__iff,axiom,
! [A: $tType,S: coinductive_llist @ A,A3: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ S @ ( lList22096119349pslsts @ A @ A3 ) )
= ( ( member @ ( coinductive_llist @ A ) @ S @ ( lList2236698231inlsts @ A @ A3 ) )
& ( S
!= ( coinductive_LNil @ A ) ) ) ) ).
% fpslsts_iff
thf(fact_143_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_144_suff__LNil,axiom,
! [A: $tType,A3: set @ A] :
( ( lList21475143548e_suff @ A @ A3 @ ( coinductive_LNil @ A ) )
= ( lList2435255213lllsts @ A @ A3 ) ) ).
% suff_LNil
thf(fact_145_infsuff__LNil,axiom,
! [A: $tType,A3: set @ A] :
( ( lList2649413865nfsuff @ A @ A3 @ ( coinductive_LNil @ A ) )
= ( lList21612149805nflsts @ A @ A3 ) ) ).
% infsuff_LNil
thf(fact_146_poslsts__iff,axiom,
! [A: $tType,S: coinductive_llist @ A,A3: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ S @ ( lList21148268032oslsts @ A @ A3 ) )
= ( ( member @ ( coinductive_llist @ A ) @ S @ ( lList2435255213lllsts @ A @ A3 ) )
& ( S
!= ( coinductive_LNil @ A ) ) ) ) ).
% poslsts_iff
thf(fact_147_finlsts_OLNil__fin,axiom,
! [A: $tType,A3: set @ A] : ( member @ ( coinductive_llist @ A ) @ ( coinductive_LNil @ A ) @ ( lList2236698231inlsts @ A @ A3 ) ) ).
% finlsts.LNil_fin
thf(fact_148_alllsts_OLNil__all,axiom,
! [A: $tType,A3: set @ A] : ( member @ ( coinductive_llist @ A ) @ ( coinductive_LNil @ A ) @ ( lList2435255213lllsts @ A @ A3 ) ) ).
% alllsts.LNil_all
thf(fact_149_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_150_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_151_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_152_pprefix__closed__def,axiom,
! [A: $tType] :
( ( lList21974196564closed @ A )
= ( ^ [A2: set @ ( coinductive_llist @ A )] :
! [X: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ X @ A2 )
=> ! [S2: coinductive_llist @ A] :
( ( ( ord_less_eq @ ( coinductive_llist @ A ) @ S2 @ X )
& ( S2
!= ( coinductive_LNil @ A ) ) )
=> ( member @ ( coinductive_llist @ A ) @ S2 @ A2 ) ) ) ) ) ).
% pprefix_closed_def
thf(fact_153_lbutlast__LNil,axiom,
! [A: $tType] :
( ( lList2370560421utlast @ A @ ( coinductive_LNil @ A ) )
= ( coinductive_LNil @ A ) ) ).
% lbutlast_LNil
thf(fact_154_ltake__lappend__llength,axiom,
! [A: $tType,R: coinductive_llist @ A,A3: set @ A,S: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ A3 ) )
=> ( ( lList22119844313_ltake @ A @ ( coinductive_lappend @ A @ R @ S ) @ ( lList21232602520length @ A @ R ) )
= R ) ) ).
% ltake_lappend_llength
thf(fact_155_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_156_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_157_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_158_same__lappend__eq,axiom,
! [A: $tType,R: coinductive_llist @ A,A3: set @ A,S: coinductive_llist @ A,T: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ A3 ) )
=> ( ( ( coinductive_lappend @ A @ R @ S )
= ( coinductive_lappend @ A @ R @ T ) )
= ( S = T ) ) ) ).
% same_lappend_eq
thf(fact_159_lapp__fin__fin__iff,axiom,
! [A: $tType,R: coinductive_llist @ A,S: coinductive_llist @ A,A3: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ ( coinductive_lappend @ A @ R @ S ) @ ( lList2236698231inlsts @ A @ A3 ) )
= ( ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ A3 ) )
& ( member @ ( coinductive_llist @ A ) @ S @ ( lList2236698231inlsts @ A @ A3 ) ) ) ) ).
% lapp_fin_fin_iff
thf(fact_160_lapp__inf,axiom,
! [A: $tType,S: coinductive_llist @ A,A3: set @ A,T: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ S @ ( lList21612149805nflsts @ A @ A3 ) )
=> ( ( coinductive_lappend @ A @ S @ T )
= S ) ) ).
% lapp_inf
thf(fact_161_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_162_lappfin__finT,axiom,
! [A: $tType,S: coinductive_llist @ A,A3: set @ A,T: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ S @ ( lList2236698231inlsts @ A @ A3 ) )
=> ( ( member @ ( coinductive_llist @ A ) @ T @ ( lList2236698231inlsts @ A @ A3 ) )
=> ( member @ ( coinductive_llist @ A ) @ ( coinductive_lappend @ A @ S @ T ) @ ( lList2236698231inlsts @ A @ A3 ) ) ) ) ).
% lappfin_finT
thf(fact_163_lapp__fin__fin__lemma,axiom,
! [A: $tType,R: coinductive_llist @ A,S: coinductive_llist @ A,A3: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ ( coinductive_lappend @ A @ R @ S ) @ ( lList2236698231inlsts @ A @ A3 ) )
=> ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ A3 ) ) ) ).
% lapp_fin_fin_lemma
thf(fact_164_llist__le__def,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( coinductive_llist @ A ) )
= ( ^ [S2: coinductive_llist @ A,T2: coinductive_llist @ A] :
? [D4: coinductive_llist @ A] :
( T2
= ( coinductive_lappend @ A @ S2 @ D4 ) ) ) ) ).
% llist_le_def
thf(fact_165_lappT,axiom,
! [A: $tType,S: coinductive_llist @ A,A3: set @ A,T: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ S @ ( lList2435255213lllsts @ A @ A3 ) )
=> ( ( member @ ( coinductive_llist @ A ) @ T @ ( lList2435255213lllsts @ A @ A3 ) )
=> ( member @ ( coinductive_llist @ A ) @ ( coinductive_lappend @ A @ S @ T ) @ ( lList2435255213lllsts @ A @ A3 ) ) ) ) ).
% lappT
thf(fact_166_lapp__all__invT,axiom,
! [A: $tType,R: coinductive_llist @ A,S: coinductive_llist @ A,A3: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ ( coinductive_lappend @ A @ R @ S ) @ ( lList2435255213lllsts @ A @ A3 ) )
=> ( member @ ( coinductive_llist @ A ) @ R @ ( lList2435255213lllsts @ A @ A3 ) ) ) ).
% lapp_all_invT
thf(fact_167_lapp__fin__infT,axiom,
! [A: $tType,S: coinductive_llist @ A,A3: set @ A,T: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ S @ ( lList2236698231inlsts @ A @ A3 ) )
=> ( ( member @ ( coinductive_llist @ A ) @ T @ ( lList21612149805nflsts @ A @ A3 ) )
=> ( member @ ( coinductive_llist @ A ) @ ( coinductive_lappend @ A @ S @ T ) @ ( lList21612149805nflsts @ A @ A3 ) ) ) ) ).
% lapp_fin_infT
thf(fact_168_lapp__inv2T,axiom,
! [A: $tType,R: coinductive_llist @ A,S: coinductive_llist @ A,A3: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ ( coinductive_lappend @ A @ R @ S ) @ ( lList21612149805nflsts @ A @ A3 ) )
=> ( ( ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ A3 ) )
& ( member @ ( coinductive_llist @ A ) @ S @ ( lList21612149805nflsts @ A @ A3 ) ) )
| ( member @ ( coinductive_llist @ A ) @ R @ ( lList21612149805nflsts @ A @ A3 ) ) ) ) ).
% lapp_inv2T
thf(fact_169_lapp__infT,axiom,
! [A: $tType,R: coinductive_llist @ A,S: coinductive_llist @ A,A3: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ ( coinductive_lappend @ A @ R @ S ) @ ( lList21612149805nflsts @ A @ A3 ) )
= ( ( ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ A3 ) )
& ( member @ ( coinductive_llist @ A ) @ S @ ( lList21612149805nflsts @ A @ A3 ) ) )
| ( member @ ( coinductive_llist @ A ) @ R @ ( lList21612149805nflsts @ A @ A3 ) ) ) ) ).
% lapp_infT
thf(fact_170_app__invT,axiom,
! [A: $tType,R: coinductive_llist @ A,A3: set @ A,S: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ A3 ) )
=> ( ( member @ ( coinductive_llist @ A ) @ ( coinductive_lappend @ A @ R @ S ) @ ( lList21612149805nflsts @ A @ A3 ) )
=> ( member @ ( coinductive_llist @ A ) @ S @ ( lList21612149805nflsts @ A @ A3 ) ) ) ) ).
% app_invT
thf(fact_171_ltake__ldrop__id,axiom,
! [A: $tType,X2: coinductive_llist @ A,I: nat] :
( ( coinductive_lappend @ A @ ( lList22119844313_ltake @ A @ X2 @ I ) @ ( lList2508575361_ldrop @ A @ X2 @ I ) )
= X2 ) ).
% ltake_ldrop_id
thf(fact_172_lapp__allT__iff,axiom,
! [A: $tType,R: coinductive_llist @ A,S: coinductive_llist @ A,A3: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ ( coinductive_lappend @ A @ R @ S ) @ ( lList2435255213lllsts @ A @ A3 ) )
= ( ( ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ A3 ) )
& ( member @ ( coinductive_llist @ A ) @ S @ ( lList2435255213lllsts @ A @ A3 ) ) )
| ( member @ ( coinductive_llist @ A ) @ R @ ( lList21612149805nflsts @ A @ A3 ) ) ) ) ).
% lapp_allT_iff
thf(fact_173_suff__appE,axiom,
! [A: $tType,R: coinductive_llist @ A,A3: set @ A,T: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ A3 ) )
=> ( ( member @ ( coinductive_llist @ A ) @ T @ ( lList21475143548e_suff @ A @ A3 @ R ) )
=> ~ ! [S3: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ S3 @ ( lList2435255213lllsts @ A @ A3 ) )
=> ( T
!= ( coinductive_lappend @ A @ R @ S3 ) ) ) ) ) ).
% suff_appE
thf(fact_174_infsuff__appE,axiom,
! [A: $tType,R: coinductive_llist @ A,A3: set @ A,T: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ A3 ) )
=> ( ( member @ ( coinductive_llist @ A ) @ T @ ( lList2649413865nfsuff @ A @ A3 @ R ) )
=> ~ ! [S3: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ S3 @ ( lList21612149805nflsts @ A @ A3 ) )
=> ( T
!= ( coinductive_lappend @ A @ R @ S3 ) ) ) ) ) ).
% infsuff_appE
thf(fact_175_lapp__suff__llength,axiom,
! [A: $tType,R: coinductive_llist @ A,A3: set @ A,S: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ A3 ) )
=> ( ( lList2508575361_ldrop @ A @ ( coinductive_lappend @ A @ R @ S ) @ ( lList21232602520length @ A @ R ) )
= S ) ) ).
% lapp_suff_llength
thf(fact_176_lappend__code_I1_J,axiom,
! [A: $tType,Ys: coinductive_llist @ A] :
( ( coinductive_lappend @ A @ ( coinductive_LNil @ A ) @ Ys )
= Ys ) ).
% lappend_code(1)
thf(fact_177_lappend__LNil2,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( coinductive_lappend @ A @ Xs @ ( coinductive_LNil @ A ) )
= Xs ) ).
% lappend_LNil2
thf(fact_178_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_179_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_180_wlog__linorder__le,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > A > $o,B2: A,A4: A] :
( ! [A5: A,B4: A] :
( ( ord_less_eq @ A @ A5 @ B4 )
=> ( P @ A5 @ B4 ) )
=> ( ( ( P @ B2 @ A4 )
=> ( P @ A4 @ B2 ) )
=> ( P @ A4 @ B2 ) ) ) ) ).
% wlog_linorder_le
thf(fact_181_lappend__LNil__LNil,axiom,
! [A: $tType] :
( ( coinductive_lappend @ A @ ( coinductive_LNil @ A ) @ ( coinductive_LNil @ A ) )
= ( coinductive_LNil @ A ) ) ).
% lappend_LNil_LNil
thf(fact_182_lbutlast__snoc,axiom,
! [A: $tType,Xs: coinductive_llist @ A,A3: set @ A,X2: A] :
( ( member @ ( coinductive_llist @ A ) @ Xs @ ( lList2236698231inlsts @ A @ A3 ) )
=> ( ( lList2370560421utlast @ A @ ( coinductive_lappend @ A @ Xs @ ( coinductive_LCons @ A @ X2 @ ( coinductive_LNil @ A ) ) ) )
= Xs ) ) ).
% lbutlast_snoc
thf(fact_183_lrev__LCons,axiom,
! [A: $tType,R: coinductive_llist @ A,A3: set @ A,A4: A] :
( ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ A3 ) )
=> ( ( 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_184_lbutlast__LCons,axiom,
! [A: $tType,R: coinductive_llist @ A,A3: set @ A,A4: A] :
( ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ A3 ) )
=> ( ( ( 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_185_LConsE,axiom,
! [A: $tType,X2: A,Xs: coinductive_llist @ A,A3: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ ( coinductive_LCons @ A @ X2 @ Xs ) @ ( lList2435255213lllsts @ A @ A3 ) )
= ( ( member @ A @ X2 @ A3 )
& ( member @ ( coinductive_llist @ A ) @ Xs @ ( lList2435255213lllsts @ A @ A3 ) ) ) ) ).
% LConsE
thf(fact_186_le__LCons,axiom,
! [A: $tType,X2: A,Xs: coinductive_llist @ A,Y2: A,Ys: coinductive_llist @ A] :
( ( ord_less_eq @ ( coinductive_llist @ A ) @ ( coinductive_LCons @ A @ X2 @ Xs ) @ ( coinductive_LCons @ A @ Y2 @ Ys ) )
= ( ( X2 = Y2 )
& ( ord_less_eq @ ( coinductive_llist @ A ) @ Xs @ Ys ) ) ) ).
% le_LCons
thf(fact_187_neq__LNil__conv,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( Xs
!= ( coinductive_LNil @ A ) )
= ( ? [X: A,Xs2: coinductive_llist @ A] :
( Xs
= ( coinductive_LCons @ A @ X @ Xs2 ) ) ) ) ).
% neq_LNil_conv
thf(fact_188_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_189_llistE,axiom,
! [A: $tType,Y2: coinductive_llist @ A] :
( ( Y2
!= ( coinductive_LNil @ A ) )
=> ~ ! [X212: A,X222: coinductive_llist @ A] :
( Y2
!= ( coinductive_LCons @ A @ X212 @ X222 ) ) ) ).
% llistE
thf(fact_190_alllsts_OLCons__all,axiom,
! [A: $tType,L: coinductive_llist @ A,A3: set @ A,A4: A] :
( ( member @ ( coinductive_llist @ A ) @ L @ ( lList2435255213lllsts @ A @ A3 ) )
=> ( ( member @ A @ A4 @ A3 )
=> ( member @ ( coinductive_llist @ A ) @ ( coinductive_LCons @ A @ A4 @ L ) @ ( lList2435255213lllsts @ A @ A3 ) ) ) ) ).
% alllsts.LCons_all
thf(fact_191_inflstsI2,axiom,
! [A: $tType,A4: A,A3: set @ A,T: coinductive_llist @ A] :
( ( member @ A @ A4 @ A3 )
=> ( ( member @ ( coinductive_llist @ A ) @ T @ ( lList21612149805nflsts @ A @ A3 ) )
=> ( member @ ( coinductive_llist @ A ) @ ( coinductive_LCons @ A @ A4 @ T ) @ ( lList21612149805nflsts @ A @ A3 ) ) ) ) ).
% inflstsI2
thf(fact_192_inflsts__cases,axiom,
! [A: $tType,S: coinductive_llist @ A,A3: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ S @ ( lList21612149805nflsts @ A @ A3 ) )
=> ~ ! [A5: A,L2: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ L2 @ ( lList21612149805nflsts @ A @ A3 ) )
=> ( ( member @ A @ A5 @ A3 )
=> ( S
!= ( coinductive_LCons @ A @ A5 @ L2 ) ) ) ) ) ).
% inflsts_cases
thf(fact_193_finlsts_OLCons__fin,axiom,
! [A: $tType,L: coinductive_llist @ A,A3: set @ A,A4: A] :
( ( member @ ( coinductive_llist @ A ) @ L @ ( lList2236698231inlsts @ A @ A3 ) )
=> ( ( member @ A @ A4 @ A3 )
=> ( member @ ( coinductive_llist @ A ) @ ( coinductive_LCons @ A @ A4 @ L ) @ ( lList2236698231inlsts @ A @ A3 ) ) ) ) ).
% finlsts.LCons_fin
thf(fact_194_lappend__snocL1__conv__LCons2,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Y2: A,Ys: coinductive_llist @ A] :
( ( coinductive_lappend @ A @ ( coinductive_lappend @ A @ Xs @ ( coinductive_LCons @ A @ Y2 @ ( coinductive_LNil @ A ) ) ) @ Ys )
= ( coinductive_lappend @ A @ Xs @ ( coinductive_LCons @ A @ Y2 @ Ys ) ) ) ).
% lappend_snocL1_conv_LCons2
thf(fact_195_finlsts_Ocases,axiom,
! [A: $tType,A4: coinductive_llist @ A,A3: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ A4 @ ( lList2236698231inlsts @ A @ A3 ) )
=> ( ( A4
!= ( coinductive_LNil @ A ) )
=> ~ ! [L2: coinductive_llist @ A,A5: A] :
( ( A4
= ( coinductive_LCons @ A @ A5 @ L2 ) )
=> ( ( member @ ( coinductive_llist @ A ) @ L2 @ ( lList2236698231inlsts @ A @ A3 ) )
=> ~ ( member @ A @ A5 @ A3 ) ) ) ) ) ).
% finlsts.cases
thf(fact_196_finlsts_Osimps,axiom,
! [A: $tType,A4: coinductive_llist @ A,A3: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ A4 @ ( lList2236698231inlsts @ A @ A3 ) )
= ( ( A4
= ( coinductive_LNil @ A ) )
| ? [L3: coinductive_llist @ A,A6: A] :
( ( A4
= ( coinductive_LCons @ A @ A6 @ L3 ) )
& ( member @ ( coinductive_llist @ A ) @ L3 @ ( lList2236698231inlsts @ A @ A3 ) )
& ( member @ A @ A6 @ A3 ) ) ) ) ).
% finlsts.simps
thf(fact_197_finlsts__induct,axiom,
! [A: $tType,X2: coinductive_llist @ A,A3: set @ A,P: ( coinductive_llist @ A ) > $o] :
( ( member @ ( coinductive_llist @ A ) @ X2 @ ( lList2236698231inlsts @ A @ A3 ) )
=> ( ! [L2: coinductive_llist @ A] :
( ( L2
= ( coinductive_LNil @ A ) )
=> ( P @ L2 ) )
=> ( ! [A5: A,L2: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ L2 @ ( lList2236698231inlsts @ A @ A3 ) )
=> ( ( P @ L2 )
=> ( ( member @ A @ A5 @ A3 )
=> ( P @ ( coinductive_LCons @ A @ A5 @ L2 ) ) ) ) )
=> ( P @ X2 ) ) ) ) ).
% finlsts_induct
thf(fact_198_finlsts_Oinducts,axiom,
! [A: $tType,X2: coinductive_llist @ A,A3: set @ A,P: ( coinductive_llist @ A ) > $o] :
( ( member @ ( coinductive_llist @ A ) @ X2 @ ( lList2236698231inlsts @ A @ A3 ) )
=> ( ( P @ ( coinductive_LNil @ A ) )
=> ( ! [L2: coinductive_llist @ A,A5: A] :
( ( member @ ( coinductive_llist @ A ) @ L2 @ ( lList2236698231inlsts @ A @ A3 ) )
=> ( ( P @ L2 )
=> ( ( member @ A @ A5 @ A3 )
=> ( P @ ( coinductive_LCons @ A @ A5 @ L2 ) ) ) ) )
=> ( P @ X2 ) ) ) ) ).
% finlsts.inducts
thf(fact_199_alllsts_Ocases,axiom,
! [A: $tType,A4: coinductive_llist @ A,A3: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ A4 @ ( lList2435255213lllsts @ A @ A3 ) )
=> ( ( A4
!= ( coinductive_LNil @ A ) )
=> ~ ! [L2: coinductive_llist @ A,A5: A] :
( ( A4
= ( coinductive_LCons @ A @ A5 @ L2 ) )
=> ( ( member @ ( coinductive_llist @ A ) @ L2 @ ( lList2435255213lllsts @ A @ A3 ) )
=> ~ ( member @ A @ A5 @ A3 ) ) ) ) ) ).
% alllsts.cases
thf(fact_200_alllsts_Osimps,axiom,
! [A: $tType,A4: coinductive_llist @ A,A3: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ A4 @ ( lList2435255213lllsts @ A @ A3 ) )
= ( ( A4
= ( coinductive_LNil @ A ) )
| ? [L3: coinductive_llist @ A,A6: A] :
( ( A4
= ( coinductive_LCons @ A @ A6 @ L3 ) )
& ( member @ ( coinductive_llist @ A ) @ L3 @ ( lList2435255213lllsts @ A @ A3 ) )
& ( member @ A @ A6 @ A3 ) ) ) ) ).
% alllsts.simps
thf(fact_201_alllsts_Ocoinduct,axiom,
! [A: $tType,X4: ( coinductive_llist @ A ) > $o,X2: coinductive_llist @ A,A3: set @ A] :
( ( X4 @ X2 )
=> ( ! [X3: coinductive_llist @ A] :
( ( X4 @ X3 )
=> ( ( X3
= ( coinductive_LNil @ A ) )
| ? [L4: coinductive_llist @ A,A7: A] :
( ( X3
= ( coinductive_LCons @ A @ A7 @ L4 ) )
& ( ( X4 @ L4 )
| ( member @ ( coinductive_llist @ A ) @ L4 @ ( lList2435255213lllsts @ A @ A3 ) ) )
& ( member @ A @ A7 @ A3 ) ) ) )
=> ( member @ ( coinductive_llist @ A ) @ X2 @ ( lList2435255213lllsts @ A @ A3 ) ) ) ) ).
% alllsts.coinduct
thf(fact_202_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_203_fps__induct,axiom,
! [A: $tType,L: coinductive_llist @ A,A3: set @ A,P: ( coinductive_llist @ A ) > $o] :
( ( member @ ( coinductive_llist @ A ) @ L @ ( lList22096119349pslsts @ A @ A3 ) )
=> ( ! [A5: A] :
( ( member @ A @ A5 @ A3 )
=> ( P @ ( coinductive_LCons @ A @ A5 @ ( coinductive_LNil @ A ) ) ) )
=> ( ! [A5: A,L2: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ L2 @ ( lList22096119349pslsts @ A @ A3 ) )
=> ( ( P @ L2 )
=> ( ( member @ A @ A5 @ A3 )
=> ( P @ ( coinductive_LCons @ A @ A5 @ L2 ) ) ) ) )
=> ( P @ L ) ) ) ) ).
% fps_induct
thf(fact_204_fpslsts__cases,axiom,
! [A: $tType,R: coinductive_llist @ A,A3: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ R @ ( lList22096119349pslsts @ A @ A3 ) )
=> ~ ! [A5: A,Rs: coinductive_llist @ A] :
( ( R
= ( coinductive_LCons @ A @ A5 @ Rs ) )
=> ( ( member @ A @ A5 @ A3 )
=> ~ ( member @ ( coinductive_llist @ A ) @ Rs @ ( lList2236698231inlsts @ A @ A3 ) ) ) ) ) ).
% fpslsts_cases
thf(fact_205_finlsts__rev__cases,axiom,
! [A: $tType,T: coinductive_llist @ A,A3: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ T @ ( lList2236698231inlsts @ A @ A3 ) )
=> ( ( T
!= ( coinductive_LNil @ A ) )
=> ~ ! [A5: A,L2: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ L2 @ ( lList2236698231inlsts @ A @ A3 ) )
=> ( ( member @ A @ A5 @ A3 )
=> ( T
!= ( coinductive_lappend @ A @ L2 @ ( coinductive_LCons @ A @ A5 @ ( coinductive_LNil @ A ) ) ) ) ) ) ) ) ).
% finlsts_rev_cases
thf(fact_206_lrev__induct,axiom,
! [A: $tType,Xs: coinductive_llist @ A,A3: set @ A,P: ( coinductive_llist @ A ) > $o] :
( ( member @ ( coinductive_llist @ A ) @ Xs @ ( lList2236698231inlsts @ A @ A3 ) )
=> ( ( P @ ( coinductive_LNil @ A ) )
=> ( ! [X3: A,Xs3: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ Xs3 @ ( lList2236698231inlsts @ A @ A3 ) )
=> ( ( P @ Xs3 )
=> ( ( member @ A @ X3 @ A3 )
=> ( P @ ( coinductive_lappend @ A @ Xs3 @ ( coinductive_LCons @ A @ X3 @ ( coinductive_LNil @ A ) ) ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% lrev_induct
thf(fact_207_llast__snoc,axiom,
! [A: $tType,Xs: coinductive_llist @ A,A3: set @ A,X2: A] :
( ( member @ ( coinductive_llist @ A ) @ Xs @ ( lList2236698231inlsts @ A @ A3 ) )
=> ( ( lList2170638824_llast @ A @ ( coinductive_lappend @ A @ Xs @ ( coinductive_LCons @ A @ X2 @ ( coinductive_LNil @ A ) ) ) )
= X2 ) ) ).
% llast_snoc
thf(fact_208_LList2__Mirabelle__hamjzmohle_Ollast__lappend,axiom,
! [A: $tType,X2: coinductive_llist @ A,Y2: coinductive_llist @ A,A4: A] :
( ( member @ ( coinductive_llist @ A ) @ X2 @ ( lList2236698231inlsts @ A @ ( top_top @ ( set @ A ) ) ) )
=> ( ( member @ ( coinductive_llist @ A ) @ Y2 @ ( lList2236698231inlsts @ A @ ( top_top @ ( set @ A ) ) ) )
=> ( ( lList2170638824_llast @ A @ ( coinductive_lappend @ A @ X2 @ ( coinductive_LCons @ A @ A4 @ Y2 ) ) )
= ( lList2170638824_llast @ A @ ( coinductive_LCons @ A @ A4 @ Y2 ) ) ) ) ) ).
% LList2_Mirabelle_hamjzmohle.llast_lappend
thf(fact_209_lbutlast__lapp__llast,axiom,
! [A: $tType,L: coinductive_llist @ A,A3: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ L @ ( lList22096119349pslsts @ A @ A3 ) )
=> ( L
= ( coinductive_lappend @ A @ ( lList2370560421utlast @ A @ L ) @ ( coinductive_LCons @ A @ ( lList2170638824_llast @ A @ L ) @ ( coinductive_LNil @ A ) ) ) ) ) ).
% lbutlast_lapp_llast
thf(fact_210_LList2__Mirabelle__hamjzmohle_Ollast__LCons,axiom,
! [B: $tType,R: coinductive_llist @ B,A3: set @ B,A4: B] :
( ( member @ ( coinductive_llist @ B ) @ R @ ( lList2236698231inlsts @ B @ A3 ) )
=> ( ( ( 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_211_Coinductive__List_Ofinite__lprefix__nitpick__simps_I3_J,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Y2: A,Ys: coinductive_llist @ A] :
( ( coindu328551480prefix @ A @ Xs @ ( coinductive_LCons @ A @ Y2 @ Ys ) )
= ( ( Xs
= ( coinductive_LNil @ A ) )
| ? [Xs2: coinductive_llist @ A] :
( ( Xs
= ( coinductive_LCons @ A @ Y2 @ Xs2 ) )
& ( coindu328551480prefix @ A @ Xs2 @ Ys ) ) ) ) ).
% Coinductive_List.finite_lprefix_nitpick_simps(3)
thf(fact_212_lstrict__prefix__code_I3_J,axiom,
! [B: $tType,X2: B,Xs: coinductive_llist @ B] :
~ ( coindu1478340336prefix @ B @ ( coinductive_LCons @ B @ X2 @ Xs ) @ ( coinductive_LNil @ B ) ) ).
% lstrict_prefix_code(3)
thf(fact_213_lstrict__prefix__code_I1_J,axiom,
! [A: $tType] :
~ ( coindu1478340336prefix @ A @ ( coinductive_LNil @ A ) @ ( coinductive_LNil @ A ) ) ).
% lstrict_prefix_code(1)
thf(fact_214_lstrict__prefix__code_I2_J,axiom,
! [B: $tType,Y2: B,Ys: coinductive_llist @ B] : ( coindu1478340336prefix @ B @ ( coinductive_LNil @ B ) @ ( coinductive_LCons @ B @ Y2 @ Ys ) ) ).
% lstrict_prefix_code(2)
thf(fact_215_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_216_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_217_finlsts__rec__LCons,axiom,
! [B: $tType,A: $tType,R: coinductive_llist @ A,A3: set @ A,C: B,D3: A > ( coinductive_llist @ A ) > B > B,A4: A] :
( ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ A3 ) )
=> ( ( lList21916056377ts_rec @ B @ A @ C @ D3 @ ( coinductive_LCons @ A @ A4 @ R ) )
= ( D3 @ A4 @ R @ ( lList21916056377ts_rec @ B @ A @ C @ D3 @ R ) ) ) ) ).
% finlsts_rec_LCons
thf(fact_218_finlsts__rec__LCons__def,axiom,
! [B: $tType,A: $tType,F: ( coinductive_llist @ A ) > B,C: B,D3: A > ( coinductive_llist @ A ) > B > B,R: coinductive_llist @ A,A3: set @ A,A4: A] :
( ( F
= ( lList21916056377ts_rec @ B @ A @ C @ D3 ) )
=> ( ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ A3 ) )
=> ( ( F @ ( coinductive_LCons @ A @ A4 @ R ) )
= ( D3 @ A4 @ R @ ( F @ R ) ) ) ) ) ).
% finlsts_rec_LCons_def
thf(fact_219_finlsts__rec__LNil__def,axiom,
! [A: $tType,B: $tType,F: ( coinductive_llist @ A ) > B,C: B,D3: A > ( coinductive_llist @ A ) > B > B] :
( ( F
= ( lList21916056377ts_rec @ B @ A @ C @ D3 ) )
=> ( ( F @ ( coinductive_LNil @ A ) )
= C ) ) ).
% finlsts_rec_LNil_def
thf(fact_220_finlsts__rec__LNil,axiom,
! [B: $tType,A: $tType,C: A,D3: B > ( coinductive_llist @ B ) > A > A] :
( ( lList21916056377ts_rec @ A @ B @ C @ D3 @ ( coinductive_LNil @ B ) )
= C ) ).
% finlsts_rec_LNil
thf(fact_221_LList2__Mirabelle__hamjzmohle_Ollength__LCons,axiom,
! [A: $tType,R: coinductive_llist @ A,A3: set @ A,A4: A] :
( ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ A3 ) )
=> ( ( lList21232602520length @ A @ ( coinductive_LCons @ A @ A4 @ R ) )
= ( suc @ ( lList21232602520length @ A @ R ) ) ) ) ).
% LList2_Mirabelle_hamjzmohle.llength_LCons
thf(fact_222_llast__singleton,axiom,
! [A: $tType,X2: A] :
( ( coinductive_llast @ A @ ( coinductive_LCons @ A @ X2 @ ( coinductive_LNil @ A ) ) )
= X2 ) ).
% llast_singleton
thf(fact_223_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_224_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_225_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_226_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_227_fin__Un__inf,axiom,
! [A: $tType,A3: set @ A] :
( ( sup_sup @ ( set @ ( coinductive_llist @ A ) ) @ ( lList2236698231inlsts @ A @ A3 ) @ ( lList21612149805nflsts @ A @ A3 ) )
= ( lList2435255213lllsts @ A @ A3 ) ) ).
% fin_Un_inf
thf(fact_228_sup__apply,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_sup @ B @ ( type2 @ B ) )
=> ( ( sup_sup @ ( A > B ) )
= ( ^ [F2: A > B,G2: A > B,X: A] : ( sup_sup @ B @ ( F2 @ X ) @ ( G2 @ X ) ) ) ) ) ).
% sup_apply
thf(fact_229_sup_Oidem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [A4: A] :
( ( sup_sup @ A @ A4 @ A4 )
= A4 ) ) ).
% sup.idem
thf(fact_230_sup__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [X2: A] :
( ( sup_sup @ A @ X2 @ X2 )
= X2 ) ) ).
% sup_idem
thf(fact_231_sup_Oleft__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [A4: A,B2: A] :
( ( sup_sup @ A @ A4 @ ( sup_sup @ A @ A4 @ B2 ) )
= ( sup_sup @ A @ A4 @ B2 ) ) ) ).
% sup.left_idem
thf(fact_232_sup__left__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [X2: A,Y2: A] :
( ( sup_sup @ A @ X2 @ ( sup_sup @ A @ X2 @ Y2 ) )
= ( sup_sup @ A @ X2 @ Y2 ) ) ) ).
% sup_left_idem
thf(fact_233_sup_Oright__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [A4: A,B2: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ A4 @ B2 ) @ B2 )
= ( sup_sup @ A @ A4 @ B2 ) ) ) ).
% sup.right_idem
thf(fact_234_UnCI,axiom,
! [A: $tType,C: A,B3: set @ A,A3: set @ A] :
( ( ~ ( member @ A @ C @ B3 )
=> ( member @ A @ C @ A3 ) )
=> ( member @ A @ C @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) ) ) ).
% UnCI
thf(fact_235_Un__iff,axiom,
! [A: $tType,C: A,A3: set @ A,B3: set @ A] :
( ( member @ A @ C @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) )
= ( ( member @ A @ C @ A3 )
| ( member @ A @ C @ B3 ) ) ) ).
% Un_iff
thf(fact_236_sup_Obounded__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [B2: A,C: A,A4: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ B2 @ C ) @ A4 )
= ( ( ord_less_eq @ A @ B2 @ A4 )
& ( ord_less_eq @ A @ C @ A4 ) ) ) ) ).
% sup.bounded_iff
thf(fact_237_le__sup__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [X2: A,Y2: A,Z: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ X2 @ Y2 ) @ Z )
= ( ( ord_less_eq @ A @ X2 @ Z )
& ( ord_less_eq @ A @ Y2 @ Z ) ) ) ) ).
% le_sup_iff
thf(fact_238_sup__top__left,axiom,
! [A: $tType] :
( ( bounded_lattice_top @ A @ ( type2 @ A ) )
=> ! [X2: A] :
( ( sup_sup @ A @ ( top_top @ A ) @ X2 )
= ( top_top @ A ) ) ) ).
% sup_top_left
thf(fact_239_sup__top__right,axiom,
! [A: $tType] :
( ( bounded_lattice_top @ A @ ( type2 @ A ) )
=> ! [X2: A] :
( ( sup_sup @ A @ X2 @ ( top_top @ A ) )
= ( top_top @ A ) ) ) ).
% sup_top_right
thf(fact_240_Un__subset__iff,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,C3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A3 @ B3 ) @ C3 )
= ( ( ord_less_eq @ ( set @ A ) @ A3 @ C3 )
& ( ord_less_eq @ ( set @ A ) @ B3 @ C3 ) ) ) ).
% Un_subset_iff
thf(fact_241_Un__Diff__cancel2,axiom,
! [A: $tType,B3: set @ A,A3: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ B3 @ A3 ) @ A3 )
= ( sup_sup @ ( set @ A ) @ B3 @ A3 ) ) ).
% Un_Diff_cancel2
thf(fact_242_Un__Diff__cancel,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( sup_sup @ ( set @ A ) @ A3 @ ( minus_minus @ ( set @ A ) @ B3 @ A3 ) )
= ( sup_sup @ ( set @ A ) @ A3 @ B3 ) ) ).
% Un_Diff_cancel
thf(fact_243_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_244_llist__less__finT,axiom,
! [A: $tType,R: coinductive_llist @ A,S: coinductive_llist @ A,A3: set @ A] :
( ( ord_less @ ( coinductive_llist @ A ) @ R @ S )
=> ( ( member @ ( coinductive_llist @ A ) @ S @ ( lList2236698231inlsts @ A @ A3 ) )
=> ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ A3 ) ) ) ) ).
% llist_less_finT
thf(fact_245_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_246_not__less__LNil,axiom,
! [A: $tType,R: coinductive_llist @ A] :
~ ( ord_less @ ( coinductive_llist @ A ) @ R @ ( coinductive_LNil @ A ) ) ).
% not_less_LNil
thf(fact_247_Un__UNIV__left,axiom,
! [A: $tType,B3: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( top_top @ ( set @ A ) ) @ B3 )
= ( top_top @ ( set @ A ) ) ) ).
% Un_UNIV_left
thf(fact_248_Un__UNIV__right,axiom,
! [A: $tType,A3: set @ A] :
( ( sup_sup @ ( set @ A ) @ A3 @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% Un_UNIV_right
thf(fact_249_top_Oextremum__strict,axiom,
! [A: $tType] :
( ( order_top @ A @ ( type2 @ A ) )
=> ! [A4: A] :
~ ( ord_less @ A @ ( top_top @ A ) @ A4 ) ) ).
% top.extremum_strict
thf(fact_250_top_Onot__eq__extremum,axiom,
! [A: $tType] :
( ( order_top @ A @ ( type2 @ A ) )
=> ! [A4: A] :
( ( A4
!= ( top_top @ A ) )
= ( ord_less @ A @ A4 @ ( top_top @ A ) ) ) ) ).
% top.not_eq_extremum
thf(fact_251_ord__eq__less__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A4: A,F: B > A,B2: B,C: B] :
( ( A4
= ( F @ B2 ) )
=> ( ( ord_less @ B @ B2 @ C )
=> ( ! [X3: B,Y: B] :
( ( ord_less @ B @ X3 @ Y )
=> ( ord_less @ A @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less @ A @ A4 @ ( F @ C ) ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_252_ord__less__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A4: A,B2: A,F: A > B,C: B] :
( ( ord_less @ A @ A4 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: A,Y: A] :
( ( ord_less @ A @ X3 @ Y )
=> ( ord_less @ B @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less @ B @ ( F @ A4 ) @ C ) ) ) ) ) ).
% ord_less_eq_subst
thf(fact_253_order__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B @ ( type2 @ B ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A4: A,F: B > A,B2: B,C: B] :
( ( ord_less @ A @ A4 @ ( F @ B2 ) )
=> ( ( ord_less @ B @ B2 @ C )
=> ( ! [X3: B,Y: B] :
( ( ord_less @ B @ X3 @ Y )
=> ( ord_less @ A @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less @ A @ A4 @ ( F @ C ) ) ) ) ) ) ).
% order_less_subst1
thf(fact_254_order__less__subst2,axiom,
! [A: $tType,C2: $tType] :
( ( ( order @ C2 @ ( type2 @ C2 ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A4: A,B2: A,F: A > C2,C: C2] :
( ( ord_less @ A @ A4 @ B2 )
=> ( ( ord_less @ C2 @ ( F @ B2 ) @ C )
=> ( ! [X3: A,Y: A] :
( ( ord_less @ A @ X3 @ Y )
=> ( ord_less @ C2 @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less @ C2 @ ( F @ A4 ) @ C ) ) ) ) ) ).
% order_less_subst2
thf(fact_255_lt__ex,axiom,
! [A: $tType] :
( ( no_bot @ A @ ( type2 @ A ) )
=> ! [X2: A] :
? [Y: A] : ( ord_less @ A @ Y @ X2 ) ) ).
% lt_ex
%----Type constructors (38)
thf(tcon_HOL_Obool___Lattices_Obounded__lattice,axiom,
bounded_lattice @ $o @ ( type2 @ $o ) ).
thf(tcon_Set_Oset___Lattices_Obounded__lattice_1,axiom,
! [A8: $tType] : ( bounded_lattice @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_fun___Lattices_Obounded__lattice_2,axiom,
! [A8: $tType,A9: $tType] :
( ( bounded_lattice @ A9 @ ( type2 @ A9 ) )
=> ( bounded_lattice @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
thf(tcon_fun___Lattices_Obounded__lattice__top,axiom,
! [A8: $tType,A9: $tType] :
( ( bounded_lattice @ A9 @ ( type2 @ A9 ) )
=> ( bounded_lattice_top @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
thf(tcon_fun___Lattices_Osemilattice__sup,axiom,
! [A8: $tType,A9: $tType] :
( ( semilattice_sup @ A9 @ ( type2 @ A9 ) )
=> ( semilattice_sup @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
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_fun___Groups_Ominus,axiom,
! [A8: $tType,A9: $tType] :
( ( minus @ A9 @ ( type2 @ A9 ) )
=> ( minus @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
thf(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add,axiom,
cancel146912293up_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Lattices_Osemilattice__sup_3,axiom,
semilattice_sup @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Opreorder_4,axiom,
preorder @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
linorder @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Oorder_5,axiom,
order @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Oord_6,axiom,
ord @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ominus_7,axiom,
minus @ nat @ ( type2 @ nat ) ).
thf(tcon_Set_Oset___Lattices_Obounded__lattice__top_8,axiom,
! [A8: $tType] : ( bounded_lattice_top @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Lattices_Osemilattice__sup_9,axiom,
! [A8: $tType] : ( semilattice_sup @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder__top_10,axiom,
! [A8: $tType] : ( order_top @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_11,axiom,
! [A8: $tType] : ( preorder @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_12,axiom,
! [A8: $tType] : ( order @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Otop_13,axiom,
! [A8: $tType] : ( top @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_14,axiom,
! [A8: $tType] : ( ord @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Groups_Ominus_15,axiom,
! [A8: $tType] : ( minus @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_HOL_Obool___Lattices_Obounded__lattice__top_16,axiom,
bounded_lattice_top @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_17,axiom,
semilattice_sup @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oorder__top_18,axiom,
order_top @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Opreorder_19,axiom,
preorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Olinorder_20,axiom,
linorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oorder_21,axiom,
order @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Otop_22,axiom,
top @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oord_23,axiom,
ord @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Groups_Ominus_24,axiom,
minus @ $o @ ( type2 @ $o ) ).
thf(tcon_Coinductive__List_Ollist___Orderings_Opreorder_25,axiom,
! [A8: $tType] : ( preorder @ ( coinductive_llist @ A8 ) @ ( type2 @ ( coinductive_llist @ A8 ) ) ) ).
thf(tcon_Coinductive__List_Ollist___Orderings_Oorder_26,axiom,
! [A8: $tType] : ( order @ ( coinductive_llist @ A8 ) @ ( type2 @ ( coinductive_llist @ A8 ) ) ) ).
thf(tcon_Coinductive__List_Ollist___Orderings_Oord_27,axiom,
! [A8: $tType] : ( ord @ ( coinductive_llist @ A8 ) @ ( type2 @ ( coinductive_llist @ A8 ) ) ) ).
%----Conjectures (1)
thf(conj_0,conjecture,
( ( ord_less_eq @ ( coinductive_llist @ a ) @ s @ t )
| ( ord_less_eq @ ( coinductive_llist @ a ) @ t @ s ) ) ).
%------------------------------------------------------------------------------