TPTP Problem File: DAT183^1.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : DAT183^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Data Structures
% Problem : Lazy lists II 951
% 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__951.p [Bla16]
% Status : Theorem
% Rating : 0.00 v7.2.0, 0.25 v7.1.0
% Syntax : Number of formulae : 340 ( 120 unt; 54 typ; 0 def)
% Number of atoms : 745 ( 236 equ; 0 cnn)
% Maximal formula atoms : 7 ( 2 avg)
% Number of connectives : 3958 ( 103 ~; 29 |; 51 &;3421 @)
% ( 0 <=>; 354 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 8 avg)
% Number of types : 3 ( 2 usr)
% Number of type conns : 153 ( 153 >; 0 *; 0 +; 0 <<)
% Number of symbols : 53 ( 52 usr; 1 con; 0-5 aty)
% Number of variables : 998 ( 12 ^; 885 !; 53 ?; 998 :)
% ( 48 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 14:50:30.967
%------------------------------------------------------------------------------
%----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 (49)
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_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_Ono__bot,type,
no_bot:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Ono__top,type,
no_top:
!>[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_Orderings_Owellorder,type,
wellorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Odense__order,type,
dense_order:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Olinordered__idom,type,
linordered_idom:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Fields_Olinordered__field,type,
linordered_field:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Lattices_Obounded__lattice__top,type,
bounded_lattice_top:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum,type,
condit1656338222tinuum:
!>[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_Olfinite,type,
coinductive_lfinite:
!>[A: $tType] : ( ( coinductive_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List_Ollast,type,
coinductive_llast:
!>[A: $tType] : ( ( coinductive_llist @ A ) > A ) ).
thf(sy_c_Coinductive__List_Ollist_OLCons,type,
coinductive_LCons:
!>[A: $tType] : ( A > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
thf(sy_c_Coinductive__List_Ollist_OLNil,type,
coinductive_LNil:
!>[A: $tType] : ( coinductive_llist @ A ) ).
thf(sy_c_Coinductive__List_Ollist_Olnull,type,
coinductive_lnull:
!>[A: $tType] : ( ( coinductive_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List_Olmember,type,
coinductive_lmember:
!>[A: $tType] : ( A > ( coinductive_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List_Olstrict__prefix,type,
coindu1478340336prefix:
!>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o ) ).
thf(sy_c_Groups_Ominus__class_Ominus,type,
minus_minus:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_HOL_Oundefined,type,
undefined:
!>[A: $tType] : A ).
thf(sy_c_LList2__Mirabelle__hamjzmohle_Oalllsts,type,
lList2435255213lllsts:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( coinductive_llist @ A ) ) ) ).
thf(sy_c_LList2__Mirabelle__hamjzmohle_Oalllstsp,type,
lList21511617539llstsp:
!>[A: $tType] : ( ( A > $o ) > ( coinductive_llist @ A ) > $o ) ).
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_Ofinlstsp,type,
lList2860480441nlstsp:
!>[A: $tType] : ( ( A > $o ) > ( coinductive_llist @ A ) > $o ) ).
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_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_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_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_r,type,
r: coinductive_llist @ a ).
%----Relevant facts (254)
thf(fact_0_lbutlast__LNil,axiom,
! [A: $tType] :
( ( lList2370560421utlast @ A @ ( coinductive_LNil @ A ) )
= ( coinductive_LNil @ A ) ) ).
% lbutlast_LNil
thf(fact_1_LNil__less__LCons,axiom,
! [A: $tType,A2: A,T: coinductive_llist @ A] : ( ord_less @ ( coinductive_llist @ A ) @ ( coinductive_LNil @ A ) @ ( coinductive_LCons @ A @ A2 @ T ) ) ).
% LNil_less_LCons
thf(fact_2_minf_I11_J,axiom,
! [C: $tType,D: $tType] :
( ( ord @ C @ ( type2 @ C ) )
=> ! [F: D] :
? [Z: C] :
! [X: C] :
( ( ord_less @ C @ X @ Z )
=> ( F = F ) ) ) ).
% minf(11)
thf(fact_3_minf_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z: A] :
! [X: A] :
( ( ord_less @ A @ X @ Z )
=> ~ ( ord_less @ A @ T @ X ) ) ) ).
% minf(7)
thf(fact_4_minf_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z: A] :
! [X: A] :
( ( ord_less @ A @ X @ Z )
=> ( ord_less @ A @ X @ T ) ) ) ).
% minf(5)
thf(fact_5_minf_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z: A] :
! [X: A] :
( ( ord_less @ A @ X @ Z )
=> ( X != T ) ) ) ).
% minf(4)
thf(fact_6_minf_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z: A] :
! [X: A] :
( ( ord_less @ A @ X @ Z )
=> ( X != T ) ) ) ).
% minf(3)
thf(fact_7_minf_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z2: A] :
! [X2: A] :
( ( ord_less @ A @ X2 @ Z2 )
=> ( ( P @ X2 )
= ( P2 @ X2 ) ) )
=> ( ? [Z2: A] :
! [X2: A] :
( ( ord_less @ A @ X2 @ Z2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z: A] :
! [X: A] :
( ( ord_less @ A @ X @ Z )
=> ( ( ( P @ X )
| ( Q @ X ) )
= ( ( P2 @ X )
| ( Q2 @ X ) ) ) ) ) ) ) ).
% minf(2)
thf(fact_8_minf_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z2: A] :
! [X2: A] :
( ( ord_less @ A @ X2 @ Z2 )
=> ( ( P @ X2 )
= ( P2 @ X2 ) ) )
=> ( ? [Z2: A] :
! [X2: A] :
( ( ord_less @ A @ X2 @ Z2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z: A] :
! [X: A] :
( ( ord_less @ A @ X @ Z )
=> ( ( ( P @ X )
& ( Q @ X ) )
= ( ( P2 @ X )
& ( Q2 @ X ) ) ) ) ) ) ) ).
% minf(1)
thf(fact_9_pinf_I11_J,axiom,
! [C: $tType,D: $tType] :
( ( ord @ C @ ( type2 @ C ) )
=> ! [F: D] :
? [Z: C] :
! [X: C] :
( ( ord_less @ C @ Z @ X )
=> ( F = F ) ) ) ).
% pinf(11)
thf(fact_10_pinf_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z: A] :
! [X: A] :
( ( ord_less @ A @ Z @ X )
=> ( ord_less @ A @ T @ X ) ) ) ).
% pinf(7)
thf(fact_11_pinf_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z: A] :
! [X: A] :
( ( ord_less @ A @ Z @ X )
=> ~ ( ord_less @ A @ X @ T ) ) ) ).
% pinf(5)
thf(fact_12_llistE,axiom,
! [A: $tType,Y: coinductive_llist @ A] :
( ( Y
!= ( coinductive_LNil @ A ) )
=> ~ ! [X21: A,X22: coinductive_llist @ A] :
( Y
!= ( coinductive_LCons @ A @ X21 @ X22 ) ) ) ).
% llistE
thf(fact_13_pinf_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z2: A] :
! [X2: A] :
( ( ord_less @ A @ Z2 @ X2 )
=> ( ( P @ X2 )
= ( P2 @ X2 ) ) )
=> ( ? [Z2: A] :
! [X2: A] :
( ( ord_less @ A @ Z2 @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z: A] :
! [X: A] :
( ( ord_less @ A @ Z @ X )
=> ( ( ( P @ X )
& ( Q @ X ) )
= ( ( P2 @ X )
& ( Q2 @ X ) ) ) ) ) ) ) ).
% pinf(1)
thf(fact_14_pinf_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z2: A] :
! [X2: A] :
( ( ord_less @ A @ Z2 @ X2 )
=> ( ( P @ X2 )
= ( P2 @ X2 ) ) )
=> ( ? [Z2: A] :
! [X2: A] :
( ( ord_less @ A @ Z2 @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z: A] :
! [X: A] :
( ( ord_less @ A @ Z @ X )
=> ( ( ( P @ X )
| ( Q @ X ) )
= ( ( P2 @ X )
| ( Q2 @ X ) ) ) ) ) ) ) ).
% pinf(2)
thf(fact_15_pinf_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z: A] :
! [X: A] :
( ( ord_less @ A @ Z @ X )
=> ( X != T ) ) ) ).
% pinf(3)
thf(fact_16_pinf_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z: A] :
! [X: A] :
( ( ord_less @ A @ Z @ X )
=> ( X != T ) ) ) ).
% pinf(4)
thf(fact_17_llist_Oinject,axiom,
! [A: $tType,X212: A,X222: coinductive_llist @ A,Y21: A,Y22: coinductive_llist @ A] :
( ( ( coinductive_LCons @ A @ X212 @ X222 )
= ( coinductive_LCons @ A @ Y21 @ Y22 ) )
= ( ( X212 = Y21 )
& ( X222 = Y22 ) ) ) ).
% llist.inject
thf(fact_18_neq__LNil__conv,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( Xs
!= ( coinductive_LNil @ A ) )
= ( ? [X3: A,Xs2: coinductive_llist @ A] :
( Xs
= ( coinductive_LCons @ A @ X3 @ Xs2 ) ) ) ) ).
% neq_LNil_conv
thf(fact_19_llist_Odistinct_I1_J,axiom,
! [A: $tType,X212: A,X222: coinductive_llist @ A] :
( ( coinductive_LNil @ A )
!= ( coinductive_LCons @ A @ X212 @ X222 ) ) ).
% llist.distinct(1)
thf(fact_20_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 ) )
=> ( ! [A4: A] :
( ( member @ A @ A4 @ A3 )
=> ( P @ ( coinductive_LCons @ A @ A4 @ ( coinductive_LNil @ A ) ) ) )
=> ( ! [A4: A,L2: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ L2 @ ( lList22096119349pslsts @ A @ A3 ) )
=> ( ( P @ L2 )
=> ( ( member @ A @ A4 @ A3 )
=> ( P @ ( coinductive_LCons @ A @ A4 @ L2 ) ) ) ) )
=> ( P @ L ) ) ) ) ).
% fps_induct
thf(fact_21_lbutlast__LCons,axiom,
! [A: $tType,R: coinductive_llist @ A,A3: set @ A,A2: A] :
( ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ A3 ) )
=> ( ( ( R
= ( coinductive_LNil @ A ) )
=> ( ( lList2370560421utlast @ A @ ( coinductive_LCons @ A @ A2 @ R ) )
= ( coinductive_LNil @ A ) ) )
& ( ( R
!= ( coinductive_LNil @ A ) )
=> ( ( lList2370560421utlast @ A @ ( coinductive_LCons @ A @ A2 @ R ) )
= ( coinductive_LCons @ A @ A2 @ ( lList2370560421utlast @ A @ R ) ) ) ) ) ) ).
% lbutlast_LCons
thf(fact_22_alllstsp_Ocoinduct,axiom,
! [A: $tType,X4: ( coinductive_llist @ A ) > $o,X5: coinductive_llist @ A,A3: A > $o] :
( ( X4 @ X5 )
=> ( ! [X2: coinductive_llist @ A] :
( ( X4 @ X2 )
=> ( ( X2
= ( coinductive_LNil @ A ) )
| ? [L3: coinductive_llist @ A,A5: A] :
( ( X2
= ( coinductive_LCons @ A @ A5 @ L3 ) )
& ( ( X4 @ L3 )
| ( lList21511617539llstsp @ A @ A3 @ L3 ) )
& ( A3 @ A5 ) ) ) )
=> ( lList21511617539llstsp @ A @ A3 @ X5 ) ) ) ).
% alllstsp.coinduct
thf(fact_23_finlstsp_Oinducts,axiom,
! [A: $tType,A3: A > $o,X5: coinductive_llist @ A,P: ( coinductive_llist @ A ) > $o] :
( ( lList2860480441nlstsp @ A @ A3 @ X5 )
=> ( ( P @ ( coinductive_LNil @ A ) )
=> ( ! [L2: coinductive_llist @ A,A4: A] :
( ( lList2860480441nlstsp @ A @ A3 @ L2 )
=> ( ( P @ L2 )
=> ( ( A3 @ A4 )
=> ( P @ ( coinductive_LCons @ A @ A4 @ L2 ) ) ) ) )
=> ( P @ X5 ) ) ) ) ).
% finlstsp.inducts
thf(fact_24_finlstsp_Osimps,axiom,
! [A: $tType] :
( ( lList2860480441nlstsp @ A )
= ( ^ [A6: A > $o,A7: coinductive_llist @ A] :
( ( A7
= ( coinductive_LNil @ A ) )
| ? [L4: coinductive_llist @ A,B2: A] :
( ( A7
= ( coinductive_LCons @ A @ B2 @ L4 ) )
& ( lList2860480441nlstsp @ A @ A6 @ L4 )
& ( A6 @ B2 ) ) ) ) ) ).
% finlstsp.simps
thf(fact_25_finlstsp_Ocases,axiom,
! [A: $tType,A3: A > $o,A2: coinductive_llist @ A] :
( ( lList2860480441nlstsp @ A @ A3 @ A2 )
=> ( ( A2
!= ( coinductive_LNil @ A ) )
=> ~ ! [L2: coinductive_llist @ A,A4: A] :
( ( A2
= ( coinductive_LCons @ A @ A4 @ L2 ) )
=> ( ( lList2860480441nlstsp @ A @ A3 @ L2 )
=> ~ ( A3 @ A4 ) ) ) ) ) ).
% finlstsp.cases
thf(fact_26_alllstsp_Osimps,axiom,
! [A: $tType] :
( ( lList21511617539llstsp @ A )
= ( ^ [A6: A > $o,A7: coinductive_llist @ A] :
( ( A7
= ( coinductive_LNil @ A ) )
| ? [L4: coinductive_llist @ A,B2: A] :
( ( A7
= ( coinductive_LCons @ A @ B2 @ L4 ) )
& ( lList21511617539llstsp @ A @ A6 @ L4 )
& ( A6 @ B2 ) ) ) ) ) ).
% alllstsp.simps
thf(fact_27_alllstsp_Ocases,axiom,
! [A: $tType,A3: A > $o,A2: coinductive_llist @ A] :
( ( lList21511617539llstsp @ A @ A3 @ A2 )
=> ( ( A2
!= ( coinductive_LNil @ A ) )
=> ~ ! [L2: coinductive_llist @ A,A4: A] :
( ( A2
= ( coinductive_LCons @ A @ A4 @ L2 ) )
=> ( ( lList21511617539llstsp @ A @ A3 @ L2 )
=> ~ ( A3 @ A4 ) ) ) ) ) ).
% alllstsp.cases
thf(fact_28_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_29_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_30_fpslsts__cases,axiom,
! [A: $tType,R: coinductive_llist @ A,A3: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ R @ ( lList22096119349pslsts @ A @ A3 ) )
=> ~ ! [A4: A,Rs: coinductive_llist @ A] :
( ( R
= ( coinductive_LCons @ A @ A4 @ Rs ) )
=> ( ( member @ A @ A4 @ A3 )
=> ~ ( member @ ( coinductive_llist @ A ) @ Rs @ ( lList2236698231inlsts @ A @ A3 ) ) ) ) ) ).
% fpslsts_cases
thf(fact_31_finlsts_OLCons__fin,axiom,
! [A: $tType,L: coinductive_llist @ A,A3: set @ A,A2: A] :
( ( member @ ( coinductive_llist @ A ) @ L @ ( lList2236698231inlsts @ A @ A3 ) )
=> ( ( member @ A @ A2 @ A3 )
=> ( member @ ( coinductive_llist @ A ) @ ( coinductive_LCons @ A @ A2 @ L ) @ ( lList2236698231inlsts @ A @ A3 ) ) ) ) ).
% finlsts.LCons_fin
thf(fact_32_finlsts_OLNil__fin,axiom,
! [A: $tType,A3: set @ A] : ( member @ ( coinductive_llist @ A ) @ ( coinductive_LNil @ A ) @ ( lList2236698231inlsts @ A @ A3 ) ) ).
% finlsts.LNil_fin
thf(fact_33_finlstsp_OLCons__fin,axiom,
! [A: $tType,A3: A > $o,L: coinductive_llist @ A,A2: A] :
( ( lList2860480441nlstsp @ A @ A3 @ L )
=> ( ( A3 @ A2 )
=> ( lList2860480441nlstsp @ A @ A3 @ ( coinductive_LCons @ A @ A2 @ L ) ) ) ) ).
% finlstsp.LCons_fin
thf(fact_34_alllstsp_OLCons__all,axiom,
! [A: $tType,A3: A > $o,L: coinductive_llist @ A,A2: A] :
( ( lList21511617539llstsp @ A @ A3 @ L )
=> ( ( A3 @ A2 )
=> ( lList21511617539llstsp @ A @ A3 @ ( coinductive_LCons @ A @ A2 @ L ) ) ) ) ).
% alllstsp.LCons_all
thf(fact_35_alllstsp_OLNil__all,axiom,
! [A: $tType,A3: A > $o] : ( lList21511617539llstsp @ A @ A3 @ ( coinductive_LNil @ A ) ) ).
% alllstsp.LNil_all
thf(fact_36_finlstsp_OLNil__fin,axiom,
! [A: $tType,A3: A > $o] : ( lList2860480441nlstsp @ A @ A3 @ ( coinductive_LNil @ A ) ) ).
% finlstsp.LNil_fin
thf(fact_37_finlsts_Ocases,axiom,
! [A: $tType,A2: coinductive_llist @ A,A3: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ A2 @ ( lList2236698231inlsts @ A @ A3 ) )
=> ( ( A2
!= ( coinductive_LNil @ A ) )
=> ~ ! [L2: coinductive_llist @ A,A4: A] :
( ( A2
= ( coinductive_LCons @ A @ A4 @ L2 ) )
=> ( ( member @ ( coinductive_llist @ A ) @ L2 @ ( lList2236698231inlsts @ A @ A3 ) )
=> ~ ( member @ A @ A4 @ A3 ) ) ) ) ) ).
% finlsts.cases
thf(fact_38_finlsts_Osimps,axiom,
! [A: $tType,A2: coinductive_llist @ A,A3: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ A2 @ ( lList2236698231inlsts @ A @ A3 ) )
= ( ( A2
= ( coinductive_LNil @ A ) )
| ? [L4: coinductive_llist @ A,A7: A] :
( ( A2
= ( coinductive_LCons @ A @ A7 @ L4 ) )
& ( member @ ( coinductive_llist @ A ) @ L4 @ ( lList2236698231inlsts @ A @ A3 ) )
& ( member @ A @ A7 @ A3 ) ) ) ) ).
% finlsts.simps
thf(fact_39_finlsts__induct,axiom,
! [A: $tType,X5: coinductive_llist @ A,A3: set @ A,P: ( coinductive_llist @ A ) > $o] :
( ( member @ ( coinductive_llist @ A ) @ X5 @ ( lList2236698231inlsts @ A @ A3 ) )
=> ( ! [L2: coinductive_llist @ A] :
( ( L2
= ( coinductive_LNil @ A ) )
=> ( P @ L2 ) )
=> ( ! [A4: A,L2: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ L2 @ ( lList2236698231inlsts @ A @ A3 ) )
=> ( ( P @ L2 )
=> ( ( member @ A @ A4 @ A3 )
=> ( P @ ( coinductive_LCons @ A @ A4 @ L2 ) ) ) ) )
=> ( P @ X5 ) ) ) ) ).
% finlsts_induct
thf(fact_40_finlsts_Oinducts,axiom,
! [A: $tType,X5: coinductive_llist @ A,A3: set @ A,P: ( coinductive_llist @ A ) > $o] :
( ( member @ ( coinductive_llist @ A ) @ X5 @ ( lList2236698231inlsts @ A @ A3 ) )
=> ( ( P @ ( coinductive_LNil @ A ) )
=> ( ! [L2: coinductive_llist @ A,A4: A] :
( ( member @ ( coinductive_llist @ A ) @ L2 @ ( lList2236698231inlsts @ A @ A3 ) )
=> ( ( P @ L2 )
=> ( ( member @ A @ A4 @ A3 )
=> ( P @ ( coinductive_LCons @ A @ A4 @ L2 ) ) ) ) )
=> ( P @ X5 ) ) ) ) ).
% finlsts.inducts
thf(fact_41_LList2__Mirabelle__hamjzmohle_Ollast__LCons,axiom,
! [B: $tType,R: coinductive_llist @ B,A3: set @ B,A2: B] :
( ( member @ ( coinductive_llist @ B ) @ R @ ( lList2236698231inlsts @ B @ A3 ) )
=> ( ( ( R
= ( coinductive_LNil @ B ) )
=> ( ( lList2170638824_llast @ B @ ( coinductive_LCons @ B @ A2 @ R ) )
= A2 ) )
& ( ( R
!= ( coinductive_LNil @ B ) )
=> ( ( lList2170638824_llast @ B @ ( coinductive_LCons @ B @ A2 @ R ) )
= ( lList2170638824_llast @ B @ R ) ) ) ) ) ).
% LList2_Mirabelle_hamjzmohle.llast_LCons
thf(fact_42_lbutlast__snoc,axiom,
! [A: $tType,Xs: coinductive_llist @ A,A3: set @ A,X5: A] :
( ( member @ ( coinductive_llist @ A ) @ Xs @ ( lList2236698231inlsts @ A @ A3 ) )
=> ( ( lList2370560421utlast @ A @ ( coinductive_lappend @ A @ Xs @ ( coinductive_LCons @ A @ X5 @ ( coinductive_LNil @ A ) ) ) )
= Xs ) ) ).
% lbutlast_snoc
thf(fact_43_Coinductive__List_Ofinite__lprefix__nitpick__simps_I3_J,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Y: A,Ys: coinductive_llist @ A] :
( ( coindu328551480prefix @ A @ Xs @ ( coinductive_LCons @ A @ Y @ Ys ) )
= ( ( Xs
= ( coinductive_LNil @ A ) )
| ? [Xs2: coinductive_llist @ A] :
( ( Xs
= ( coinductive_LCons @ A @ Y @ Xs2 ) )
& ( coindu328551480prefix @ A @ Xs2 @ Ys ) ) ) ) ).
% Coinductive_List.finite_lprefix_nitpick_simps(3)
thf(fact_44_lstrict__prefix__code_I3_J,axiom,
! [B: $tType,X5: B,Xs: coinductive_llist @ B] :
~ ( coindu1478340336prefix @ B @ ( coinductive_LCons @ B @ X5 @ Xs ) @ ( coinductive_LNil @ B ) ) ).
% lstrict_prefix_code(3)
thf(fact_45_mem__Collect__eq,axiom,
! [A: $tType,A2: A,P: A > $o] :
( ( member @ A @ A2 @ ( collect @ A @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: $tType,A3: set @ A] :
( ( collect @ A
@ ^ [X3: A] : ( member @ A @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X2: A] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_48_ext,axiom,
! [B: $tType,A: $tType,F2: A > B,G: A > B] :
( ! [X2: A] :
( ( F2 @ X2 )
= ( G @ X2 ) )
=> ( F2 = G ) ) ).
% ext
thf(fact_49_lstrict__prefix__code_I2_J,axiom,
! [B: $tType,Y: B,Ys: coinductive_llist @ B] : ( coindu1478340336prefix @ B @ ( coinductive_LNil @ B ) @ ( coinductive_LCons @ B @ Y @ Ys ) ) ).
% lstrict_prefix_code(2)
thf(fact_50_finlsts__rec__LCons__def,axiom,
! [B: $tType,A: $tType,F2: ( coinductive_llist @ A ) > B,C2: B,D2: A > ( coinductive_llist @ A ) > B > B,R: coinductive_llist @ A,A3: set @ A,A2: A] :
( ( F2
= ( lList21916056377ts_rec @ B @ A @ C2 @ D2 ) )
=> ( ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ A3 ) )
=> ( ( F2 @ ( coinductive_LCons @ A @ A2 @ R ) )
= ( D2 @ A2 @ R @ ( F2 @ R ) ) ) ) ) ).
% finlsts_rec_LCons_def
thf(fact_51_finlsts__rec__LCons,axiom,
! [B: $tType,A: $tType,R: coinductive_llist @ A,A3: set @ A,C2: B,D2: A > ( coinductive_llist @ A ) > B > B,A2: A] :
( ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ A3 ) )
=> ( ( lList21916056377ts_rec @ B @ A @ C2 @ D2 @ ( coinductive_LCons @ A @ A2 @ R ) )
= ( D2 @ A2 @ R @ ( lList21916056377ts_rec @ B @ A @ C2 @ D2 @ R ) ) ) ) ).
% finlsts_rec_LCons
thf(fact_52_llast__singleton,axiom,
! [A: $tType,X5: A] :
( ( coinductive_llast @ A @ ( coinductive_LCons @ A @ X5 @ ( coinductive_LNil @ A ) ) )
= X5 ) ).
% llast_singleton
thf(fact_53_lmember__code_I1_J,axiom,
! [A: $tType,X5: A] :
~ ( coinductive_lmember @ A @ X5 @ ( coinductive_LNil @ A ) ) ).
% lmember_code(1)
thf(fact_54_lappend__code_I2_J,axiom,
! [A: $tType,Xa: A,X5: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_lappend @ A @ ( coinductive_LCons @ A @ Xa @ X5 ) @ Ys )
= ( coinductive_LCons @ A @ Xa @ ( coinductive_lappend @ A @ X5 @ Ys ) ) ) ).
% lappend_code(2)
thf(fact_55_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_56_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_57_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_58_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_59_lappend__LNil2,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( coinductive_lappend @ A @ Xs @ ( coinductive_LNil @ A ) )
= Xs ) ).
% lappend_LNil2
thf(fact_60_lappend__code_I1_J,axiom,
! [A: $tType,Ys: coinductive_llist @ A] :
( ( coinductive_lappend @ A @ ( coinductive_LNil @ A ) @ Ys )
= Ys ) ).
% lappend_code(1)
thf(fact_61_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_62_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_63_llast__LCons2,axiom,
! [A: $tType,X5: A,Y: A,Xs: coinductive_llist @ A] :
( ( coinductive_llast @ A @ ( coinductive_LCons @ A @ X5 @ ( coinductive_LCons @ A @ Y @ Xs ) ) )
= ( coinductive_llast @ A @ ( coinductive_LCons @ A @ Y @ Xs ) ) ) ).
% llast_LCons2
thf(fact_64_lstrict__prefix__code_I4_J,axiom,
! [B: $tType,X5: B,Xs: coinductive_llist @ B,Y: B,Ys: coinductive_llist @ B] :
( ( coindu1478340336prefix @ B @ ( coinductive_LCons @ B @ X5 @ Xs ) @ ( coinductive_LCons @ B @ Y @ Ys ) )
= ( ( X5 = Y )
& ( coindu1478340336prefix @ B @ Xs @ Ys ) ) ) ).
% lstrict_prefix_code(4)
thf(fact_65_lstrict__prefix__code_I1_J,axiom,
! [A: $tType] :
~ ( coindu1478340336prefix @ A @ ( coinductive_LNil @ A ) @ ( coinductive_LNil @ A ) ) ).
% lstrict_prefix_code(1)
thf(fact_66_llast__snoc,axiom,
! [A: $tType,Xs: coinductive_llist @ A,A3: set @ A,X5: A] :
( ( member @ ( coinductive_llist @ A ) @ Xs @ ( lList2236698231inlsts @ A @ A3 ) )
=> ( ( lList2170638824_llast @ A @ ( coinductive_lappend @ A @ Xs @ ( coinductive_LCons @ A @ X5 @ ( coinductive_LNil @ A ) ) ) )
= X5 ) ) ).
% llast_snoc
thf(fact_67_llist__less__induct,axiom,
! [A: $tType,P: ( coinductive_llist @ A ) > $o,Xs: coinductive_llist @ A] :
( ! [Xs3: coinductive_llist @ A] :
( ! [Ys2: coinductive_llist @ A] :
( ( coindu1478340336prefix @ A @ Ys2 @ Xs3 )
=> ( P @ Ys2 ) )
=> ( P @ Xs3 ) )
=> ( P @ Xs ) ) ).
% llist_less_induct
thf(fact_68_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_69_lappend__LNil__LNil,axiom,
! [A: $tType] :
( ( coinductive_lappend @ A @ ( coinductive_LNil @ A ) @ ( coinductive_LNil @ A ) )
= ( coinductive_LNil @ A ) ) ).
% lappend_LNil_LNil
thf(fact_70_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_71_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_72_lappend__snocL1__conv__LCons2,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Y: A,Ys: coinductive_llist @ A] :
( ( coinductive_lappend @ A @ ( coinductive_lappend @ A @ Xs @ ( coinductive_LCons @ A @ Y @ ( coinductive_LNil @ A ) ) ) @ Ys )
= ( coinductive_lappend @ A @ Xs @ ( coinductive_LCons @ A @ Y @ Ys ) ) ) ).
% lappend_snocL1_conv_LCons2
thf(fact_73_finlsts__rec__LNil__def,axiom,
! [A: $tType,B: $tType,F2: ( coinductive_llist @ A ) > B,C2: B,D2: A > ( coinductive_llist @ A ) > B > B] :
( ( F2
= ( lList21916056377ts_rec @ B @ A @ C2 @ D2 ) )
=> ( ( F2 @ ( coinductive_LNil @ A ) )
= C2 ) ) ).
% finlsts_rec_LNil_def
thf(fact_74_finlsts__rec__LNil,axiom,
! [B: $tType,A: $tType,C2: A,D2: B > ( coinductive_llist @ B ) > A > A] :
( ( lList21916056377ts_rec @ A @ B @ C2 @ D2 @ ( coinductive_LNil @ B ) )
= C2 ) ).
% finlsts_rec_LNil
thf(fact_75_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_76_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_77_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 ) )
=> ( ! [X2: A,Xs3: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ Xs3 @ ( lList2236698231inlsts @ A @ A3 ) )
=> ( ( P @ Xs3 )
=> ( ( member @ A @ X2 @ A3 )
=> ( P @ ( coinductive_lappend @ A @ Xs3 @ ( coinductive_LCons @ A @ X2 @ ( coinductive_LNil @ A ) ) ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% lrev_induct
thf(fact_78_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 ) )
=> ~ ! [A4: A,L2: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ L2 @ ( lList2236698231inlsts @ A @ A3 ) )
=> ( ( member @ A @ A4 @ A3 )
=> ( T
!= ( coinductive_lappend @ A @ L2 @ ( coinductive_LCons @ A @ A4 @ ( coinductive_LNil @ A ) ) ) ) ) ) ) ) ).
% finlsts_rev_cases
thf(fact_79_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_80_lmember__code_I2_J,axiom,
! [A: $tType,X5: A,Y: A,Ys: coinductive_llist @ A] :
( ( coinductive_lmember @ A @ X5 @ ( coinductive_LCons @ A @ Y @ Ys ) )
= ( ( X5 = Y )
| ( coinductive_lmember @ A @ X5 @ Ys ) ) ) ).
% lmember_code(2)
thf(fact_81_lrev__LCons,axiom,
! [A: $tType,R: coinductive_llist @ A,A3: set @ A,A2: A] :
( ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ A3 ) )
=> ( ( lList2281150353e_lrev @ A @ ( coinductive_LCons @ A @ A2 @ R ) )
= ( coinductive_lappend @ A @ ( lList2281150353e_lrev @ A @ R ) @ ( coinductive_LCons @ A @ A2 @ ( coinductive_LNil @ A ) ) ) ) ) ).
% lrev_LCons
thf(fact_82_LList2__Mirabelle__hamjzmohle_Ollast__lappend,axiom,
! [A: $tType,X5: coinductive_llist @ A,Y: coinductive_llist @ A,A2: A] :
( ( member @ ( coinductive_llist @ A ) @ X5 @ ( lList2236698231inlsts @ A @ ( top_top @ ( set @ A ) ) ) )
=> ( ( member @ ( coinductive_llist @ A ) @ Y @ ( lList2236698231inlsts @ A @ ( top_top @ ( set @ A ) ) ) )
=> ( ( lList2170638824_llast @ A @ ( coinductive_lappend @ A @ X5 @ ( coinductive_LCons @ A @ A2 @ Y ) ) )
= ( lList2170638824_llast @ A @ ( coinductive_LCons @ A @ A2 @ Y ) ) ) ) ) ).
% LList2_Mirabelle_hamjzmohle.llast_lappend
thf(fact_83_llast__lappend__LCons,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Y: A,Ys: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ Xs )
=> ( ( coinductive_llast @ A @ ( coinductive_lappend @ A @ Xs @ ( coinductive_LCons @ A @ Y @ Ys ) ) )
= ( coinductive_llast @ A @ ( coinductive_LCons @ A @ Y @ Ys ) ) ) ) ).
% llast_lappend_LCons
thf(fact_84_llimit__induct,axiom,
! [A: $tType,P: ( coinductive_llist @ A ) > $o,Xs: coinductive_llist @ A] :
( ( P @ ( coinductive_LNil @ A ) )
=> ( ! [X2: A,Xs3: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ Xs3 )
=> ( ( P @ Xs3 )
=> ( P @ ( coinductive_LCons @ A @ X2 @ Xs3 ) ) ) )
=> ( ( ! [Ys2: coinductive_llist @ A] :
( ( coindu1478340336prefix @ A @ Ys2 @ Xs )
=> ( P @ Ys2 ) )
=> ( P @ Xs ) )
=> ( P @ Xs ) ) ) ) ).
% llimit_induct
thf(fact_85_ex__gt__or__lt,axiom,
! [A: $tType] :
( ( condit1656338222tinuum @ A @ ( type2 @ A ) )
=> ! [A2: A] :
? [B3: A] :
( ( ord_less @ A @ A2 @ B3 )
| ( ord_less @ A @ B3 @ A2 ) ) ) ).
% ex_gt_or_lt
thf(fact_86_linorder__neqE__linordered__idom,axiom,
! [A: $tType] :
( ( linordered_idom @ A @ ( type2 @ A ) )
=> ! [X5: A,Y: A] :
( ( X5 != Y )
=> ( ~ ( ord_less @ A @ X5 @ Y )
=> ( ord_less @ A @ Y @ X5 ) ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_87_dual__order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B4: A,A2: A] :
( ( ord_less @ A @ B4 @ A2 )
=> ( A2 != B4 ) ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_88_linordered__field__no__ub,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type2 @ A ) )
=> ! [X: A] :
? [X1: A] : ( ord_less @ A @ X @ X1 ) ) ).
% linordered_field_no_ub
thf(fact_89_top__apply,axiom,
! [C: $tType,D: $tType] :
( ( top @ C @ ( type2 @ C ) )
=> ( ( top_top @ ( D > C ) )
= ( ^ [X3: D] : ( top_top @ C ) ) ) ) ).
% top_apply
thf(fact_90_lfinite__LCons,axiom,
! [A: $tType,X5: A,Xs: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ ( coinductive_LCons @ A @ X5 @ Xs ) )
= ( coinductive_lfinite @ A @ Xs ) ) ).
% lfinite_LCons
thf(fact_91_lfinite__code_I2_J,axiom,
! [B: $tType,X5: B,Xs: coinductive_llist @ B] :
( ( coinductive_lfinite @ B @ ( coinductive_LCons @ B @ X5 @ Xs ) )
= ( coinductive_lfinite @ B @ Xs ) ) ).
% lfinite_code(2)
thf(fact_92_lfinite__code_I1_J,axiom,
! [A: $tType] : ( coinductive_lfinite @ A @ ( coinductive_LNil @ A ) ) ).
% lfinite_code(1)
thf(fact_93_lfinite__lappend,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
= ( ( coinductive_lfinite @ A @ Xs )
& ( coinductive_lfinite @ A @ Ys ) ) ) ).
% lfinite_lappend
thf(fact_94_lrev__LNil,axiom,
! [A: $tType] :
( ( lList2281150353e_lrev @ A @ ( coinductive_LNil @ A ) )
= ( coinductive_LNil @ A ) ) ).
% lrev_LNil
thf(fact_95_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_96_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_97_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_98_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_99_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_100_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_101_top_Onot__eq__extremum,axiom,
! [A: $tType] :
( ( order_top @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( A2
!= ( top_top @ A ) )
= ( ord_less @ A @ A2 @ ( top_top @ A ) ) ) ) ).
% top.not_eq_extremum
thf(fact_102_top_Oextremum__strict,axiom,
! [A: $tType] :
( ( order_top @ A @ ( type2 @ A ) )
=> ! [A2: A] :
~ ( ord_less @ A @ ( top_top @ A ) @ A2 ) ) ).
% top.extremum_strict
thf(fact_103_lfinite__LConsI,axiom,
! [A: $tType,Xs: coinductive_llist @ A,X5: A] :
( ( coinductive_lfinite @ A @ Xs )
=> ( coinductive_lfinite @ A @ ( coinductive_LCons @ A @ X5 @ Xs ) ) ) ).
% lfinite_LConsI
thf(fact_104_lfinite__LNil,axiom,
! [A: $tType] : ( coinductive_lfinite @ A @ ( coinductive_LNil @ A ) ) ).
% lfinite_LNil
thf(fact_105_lappend__inf,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ~ ( coinductive_lfinite @ A @ Xs )
=> ( ( coinductive_lappend @ A @ Xs @ Ys )
= Xs ) ) ).
% lappend_inf
thf(fact_106_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_107_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_108_lstrict__prefix__lfinite1,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coindu1478340336prefix @ A @ Xs @ Ys )
=> ( coinductive_lfinite @ A @ Xs ) ) ).
% lstrict_prefix_lfinite1
thf(fact_109_lfinite_Oinducts,axiom,
! [A: $tType,X5: coinductive_llist @ A,P: ( coinductive_llist @ A ) > $o] :
( ( coinductive_lfinite @ A @ X5 )
=> ( ( P @ ( coinductive_LNil @ A ) )
=> ( ! [Xs3: coinductive_llist @ A,X2: A] :
( ( coinductive_lfinite @ A @ Xs3 )
=> ( ( P @ Xs3 )
=> ( P @ ( coinductive_LCons @ A @ X2 @ Xs3 ) ) ) )
=> ( P @ X5 ) ) ) ) ).
% lfinite.inducts
thf(fact_110_lfinite_Osimps,axiom,
! [A: $tType] :
( ( coinductive_lfinite @ A )
= ( ^ [A7: coinductive_llist @ A] :
( ( A7
= ( coinductive_LNil @ A ) )
| ? [Xs4: coinductive_llist @ A,X3: A] :
( ( A7
= ( coinductive_LCons @ A @ X3 @ Xs4 ) )
& ( coinductive_lfinite @ A @ Xs4 ) ) ) ) ) ).
% lfinite.simps
thf(fact_111_lfinite_Ocases,axiom,
! [A: $tType,A2: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ A2 )
=> ( ( A2
!= ( coinductive_LNil @ A ) )
=> ~ ! [Xs3: coinductive_llist @ A] :
( ? [X2: A] :
( A2
= ( coinductive_LCons @ A @ X2 @ Xs3 ) )
=> ~ ( coinductive_lfinite @ A @ Xs3 ) ) ) ) ).
% lfinite.cases
thf(fact_112_lfinite__rev__induct,axiom,
! [A: $tType,Xs: coinductive_llist @ A,P: ( coinductive_llist @ A ) > $o] :
( ( coinductive_lfinite @ A @ Xs )
=> ( ( P @ ( coinductive_LNil @ A ) )
=> ( ! [X2: A,Xs3: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ Xs3 )
=> ( ( P @ Xs3 )
=> ( P @ ( coinductive_lappend @ A @ Xs3 @ ( coinductive_LCons @ A @ X2 @ ( coinductive_LNil @ A ) ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% lfinite_rev_induct
thf(fact_113_ord__eq__less__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A2: A,F2: B > A,B4: B,C2: B] :
( ( A2
= ( F2 @ B4 ) )
=> ( ( ord_less @ B @ B4 @ C2 )
=> ( ! [X2: B,Y2: B] :
( ( ord_less @ B @ X2 @ Y2 )
=> ( ord_less @ A @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less @ A @ A2 @ ( F2 @ C2 ) ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_114_ord__less__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A2: A,B4: A,F2: A > B,C2: B] :
( ( ord_less @ A @ A2 @ B4 )
=> ( ( ( F2 @ B4 )
= C2 )
=> ( ! [X2: A,Y2: A] :
( ( ord_less @ A @ X2 @ Y2 )
=> ( ord_less @ B @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less @ B @ ( F2 @ A2 ) @ C2 ) ) ) ) ) ).
% ord_less_eq_subst
thf(fact_115_order__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B @ ( type2 @ B ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,F2: B > A,B4: B,C2: B] :
( ( ord_less @ A @ A2 @ ( F2 @ B4 ) )
=> ( ( ord_less @ B @ B4 @ C2 )
=> ( ! [X2: B,Y2: B] :
( ( ord_less @ B @ X2 @ Y2 )
=> ( ord_less @ A @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less @ A @ A2 @ ( F2 @ C2 ) ) ) ) ) ) ).
% order_less_subst1
thf(fact_116_order__less__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C @ ( type2 @ C ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,B4: A,F2: A > C,C2: C] :
( ( ord_less @ A @ A2 @ B4 )
=> ( ( ord_less @ C @ ( F2 @ B4 ) @ C2 )
=> ( ! [X2: A,Y2: A] :
( ( ord_less @ A @ X2 @ Y2 )
=> ( ord_less @ C @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less @ C @ ( F2 @ A2 ) @ C2 ) ) ) ) ) ).
% order_less_subst2
thf(fact_117_lt__ex,axiom,
! [A: $tType] :
( ( no_bot @ A @ ( type2 @ A ) )
=> ! [X5: A] :
? [Y2: A] : ( ord_less @ A @ Y2 @ X5 ) ) ).
% lt_ex
thf(fact_118_gt__ex,axiom,
! [A: $tType] :
( ( no_top @ A @ ( type2 @ A ) )
=> ! [X5: A] :
? [X1: A] : ( ord_less @ A @ X5 @ X1 ) ) ).
% gt_ex
thf(fact_119_neqE,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X5: A,Y: A] :
( ( X5 != Y )
=> ( ~ ( ord_less @ A @ X5 @ Y )
=> ( ord_less @ A @ Y @ X5 ) ) ) ) ).
% neqE
thf(fact_120_neq__iff,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X5: A,Y: A] :
( ( X5 != Y )
= ( ( ord_less @ A @ X5 @ Y )
| ( ord_less @ A @ Y @ X5 ) ) ) ) ).
% neq_iff
thf(fact_121_order_Oasym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B4: A] :
( ( ord_less @ A @ A2 @ B4 )
=> ~ ( ord_less @ A @ B4 @ A2 ) ) ) ).
% order.asym
thf(fact_122_dense,axiom,
! [A: $tType] :
( ( dense_order @ A @ ( type2 @ A ) )
=> ! [X5: A,Y: A] :
( ( ord_less @ A @ X5 @ Y )
=> ? [Z: A] :
( ( ord_less @ A @ X5 @ Z )
& ( ord_less @ A @ Z @ Y ) ) ) ) ).
% dense
thf(fact_123_less__imp__neq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X5: A,Y: A] :
( ( ord_less @ A @ X5 @ Y )
=> ( X5 != Y ) ) ) ).
% less_imp_neq
thf(fact_124_less__asym,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X5: A,Y: A] :
( ( ord_less @ A @ X5 @ Y )
=> ~ ( ord_less @ A @ Y @ X5 ) ) ) ).
% less_asym
thf(fact_125_less__asym_H,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [A2: A,B4: A] :
( ( ord_less @ A @ A2 @ B4 )
=> ~ ( ord_less @ A @ B4 @ A2 ) ) ) ).
% less_asym'
thf(fact_126_less__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X5: A,Y: A,Z3: A] :
( ( ord_less @ A @ X5 @ Y )
=> ( ( ord_less @ A @ Y @ Z3 )
=> ( ord_less @ A @ X5 @ Z3 ) ) ) ) ).
% less_trans
thf(fact_127_less__linear,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X5: A,Y: A] :
( ( ord_less @ A @ X5 @ Y )
| ( X5 = Y )
| ( ord_less @ A @ Y @ X5 ) ) ) ).
% less_linear
thf(fact_128_less__irrefl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X5: A] :
~ ( ord_less @ A @ X5 @ X5 ) ) ).
% less_irrefl
thf(fact_129_ord__eq__less__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A2: A,B4: A,C2: A] :
( ( A2 = B4 )
=> ( ( ord_less @ A @ B4 @ C2 )
=> ( ord_less @ A @ A2 @ C2 ) ) ) ) ).
% ord_eq_less_trans
thf(fact_130_ord__less__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A2: A,B4: A,C2: A] :
( ( ord_less @ A @ A2 @ B4 )
=> ( ( B4 = C2 )
=> ( ord_less @ A @ A2 @ C2 ) ) ) ) ).
% ord_less_eq_trans
thf(fact_131_dual__order_Oasym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B4: A,A2: A] :
( ( ord_less @ A @ B4 @ A2 )
=> ~ ( ord_less @ A @ A2 @ B4 ) ) ) ).
% dual_order.asym
thf(fact_132_less__imp__not__eq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X5: A,Y: A] :
( ( ord_less @ A @ X5 @ Y )
=> ( X5 != Y ) ) ) ).
% less_imp_not_eq
thf(fact_133_less__not__sym,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X5: A,Y: A] :
( ( ord_less @ A @ X5 @ Y )
=> ~ ( ord_less @ A @ Y @ X5 ) ) ) ).
% less_not_sym
thf(fact_134_less__induct,axiom,
! [A: $tType] :
( ( wellorder @ A @ ( type2 @ A ) )
=> ! [P: A > $o,A2: A] :
( ! [X2: A] :
( ! [Y3: A] :
( ( ord_less @ A @ Y3 @ X2 )
=> ( P @ Y3 ) )
=> ( P @ X2 ) )
=> ( P @ A2 ) ) ) ).
% less_induct
thf(fact_135_antisym__conv3,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [Y: A,X5: A] :
( ~ ( ord_less @ A @ Y @ X5 )
=> ( ( ~ ( ord_less @ A @ X5 @ Y ) )
= ( X5 = Y ) ) ) ) ).
% antisym_conv3
thf(fact_136_less__imp__not__eq2,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X5: A,Y: A] :
( ( ord_less @ A @ X5 @ Y )
=> ( Y != X5 ) ) ) ).
% less_imp_not_eq2
thf(fact_137_less__imp__triv,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X5: A,Y: A,P: $o] :
( ( ord_less @ A @ X5 @ Y )
=> ( ( ord_less @ A @ Y @ X5 )
=> P ) ) ) ).
% less_imp_triv
thf(fact_138_linorder__cases,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X5: A,Y: A] :
( ~ ( ord_less @ A @ X5 @ Y )
=> ( ( X5 != Y )
=> ( ord_less @ A @ Y @ X5 ) ) ) ) ).
% linorder_cases
thf(fact_139_dual__order_Oirrefl,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A] :
~ ( ord_less @ A @ A2 @ A2 ) ) ).
% dual_order.irrefl
thf(fact_140_order_Ostrict__trans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B4: A,C2: A] :
( ( ord_less @ A @ A2 @ B4 )
=> ( ( ord_less @ A @ B4 @ C2 )
=> ( ord_less @ A @ A2 @ C2 ) ) ) ) ).
% order.strict_trans
thf(fact_141_less__imp__not__less,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X5: A,Y: A] :
( ( ord_less @ A @ X5 @ Y )
=> ~ ( ord_less @ A @ Y @ X5 ) ) ) ).
% less_imp_not_less
thf(fact_142_dual__order_Ostrict__trans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B4: A,A2: A,C2: A] :
( ( ord_less @ A @ B4 @ A2 )
=> ( ( ord_less @ A @ C2 @ B4 )
=> ( ord_less @ A @ C2 @ A2 ) ) ) ) ).
% dual_order.strict_trans
thf(fact_143_not__less__iff__gr__or__eq,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X5: A,Y: A] :
( ( ~ ( ord_less @ A @ X5 @ Y ) )
= ( ( ord_less @ A @ Y @ X5 )
| ( X5 = Y ) ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_144_order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B4: A] :
( ( ord_less @ A @ A2 @ B4 )
=> ( A2 != B4 ) ) ) ).
% order.strict_implies_not_eq
thf(fact_145_linordered__field__no__lb,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type2 @ A ) )
=> ! [X: A] :
? [Y2: A] : ( ord_less @ A @ Y2 @ X ) ) ).
% linordered_field_no_lb
thf(fact_146_iso__tuple__UNIV__I,axiom,
! [A: $tType,X5: A] : ( member @ A @ X5 @ ( top_top @ ( set @ A ) ) ) ).
% iso_tuple_UNIV_I
thf(fact_147_UNIV__I,axiom,
! [A: $tType,X5: A] : ( member @ A @ X5 @ ( top_top @ ( set @ A ) ) ) ).
% UNIV_I
thf(fact_148_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_149_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_150_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_151_lstrict__prefix__lappend__conv,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coindu1478340336prefix @ A @ Xs @ ( coinductive_lappend @ A @ Xs @ Ys ) )
= ( ( coinductive_lfinite @ A @ Xs )
& ~ ( coinductive_lnull @ A @ Ys ) ) ) ).
% lstrict_prefix_lappend_conv
thf(fact_152_lnull__lappend,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
= ( ( coinductive_lnull @ A @ Xs )
& ( coinductive_lnull @ A @ Ys ) ) ) ).
% lnull_lappend
thf(fact_153_lappend_Odisc__iff_I2_J,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( ~ ( coinductive_lnull @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) ) )
= ( ~ ( coinductive_lnull @ A @ Xs )
| ~ ( coinductive_lnull @ A @ Ys ) ) ) ).
% lappend.disc_iff(2)
thf(fact_154_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_155_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_156_top__set__def,axiom,
! [A: $tType] :
( ( top_top @ ( set @ A ) )
= ( collect @ A @ ( top_top @ ( A > $o ) ) ) ) ).
% top_set_def
thf(fact_157_ltake__ldrop__id,axiom,
! [A: $tType,X5: coinductive_llist @ A,I: nat] :
( ( coinductive_lappend @ A @ ( lList22119844313_ltake @ A @ X5 @ I ) @ ( lList2508575361_ldrop @ A @ X5 @ I ) )
= X5 ) ).
% ltake_ldrop_id
thf(fact_158_lappend_Oexhaust,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( ( coinductive_lnull @ A @ Xs )
=> ~ ( coinductive_lnull @ A @ Ys ) )
=> ( ~ ( coinductive_lnull @ A @ Xs )
| ~ ( coinductive_lnull @ A @ Ys ) ) ) ).
% lappend.exhaust
thf(fact_159_lzip_Oexhaust,axiom,
! [A: $tType,B: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B] :
( ~ ( ( coinductive_lnull @ A @ Xs )
| ( coinductive_lnull @ B @ Ys ) )
=> ~ ( ~ ( coinductive_lnull @ A @ Xs )
=> ( coinductive_lnull @ B @ Ys ) ) ) ).
% lzip.exhaust
thf(fact_160_llist_Odisc_I2_J,axiom,
! [A: $tType,X212: A,X222: coinductive_llist @ A] :
~ ( coinductive_lnull @ A @ ( coinductive_LCons @ A @ X212 @ X222 ) ) ).
% llist.disc(2)
thf(fact_161_llist_OdiscI_I2_J,axiom,
! [A: $tType,Llist: coinductive_llist @ A,X212: A,X222: coinductive_llist @ A] :
( ( Llist
= ( coinductive_LCons @ A @ X212 @ X222 ) )
=> ~ ( coinductive_lnull @ A @ Llist ) ) ).
% llist.discI(2)
thf(fact_162_not__lnull__conv,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( ~ ( coinductive_lnull @ A @ Xs ) )
= ( ? [X3: A,Xs2: coinductive_llist @ A] :
( Xs
= ( coinductive_LCons @ A @ X3 @ Xs2 ) ) ) ) ).
% not_lnull_conv
thf(fact_163_lnull__def,axiom,
! [A: $tType] :
( ( coinductive_lnull @ A )
= ( ^ [Llist2: coinductive_llist @ A] :
( Llist2
= ( coinductive_LNil @ A ) ) ) ) ).
% lnull_def
thf(fact_164_llist_Ocollapse_I1_J,axiom,
! [A: $tType,Llist: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ Llist )
=> ( Llist
= ( coinductive_LNil @ A ) ) ) ).
% llist.collapse(1)
thf(fact_165_llist_OdiscI_I1_J,axiom,
! [A: $tType,Llist: coinductive_llist @ A] :
( ( Llist
= ( coinductive_LNil @ A ) )
=> ( coinductive_lnull @ A @ Llist ) ) ).
% llist.discI(1)
thf(fact_166_llist_Odisc_I1_J,axiom,
! [A: $tType] : ( coinductive_lnull @ A @ ( coinductive_LNil @ A ) ) ).
% llist.disc(1)
thf(fact_167_lappend__lnull2,axiom,
! [A: $tType,Ys: coinductive_llist @ A,Xs: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ Ys )
=> ( ( coinductive_lappend @ A @ Xs @ Ys )
= Xs ) ) ).
% lappend_lnull2
thf(fact_168_lappend__lnull1,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ Xs )
=> ( ( coinductive_lappend @ A @ Xs @ Ys )
= Ys ) ) ).
% lappend_lnull1
thf(fact_169_lappend_Odisc_I1_J,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ Xs )
=> ( ( coinductive_lnull @ A @ Ys )
=> ( coinductive_lnull @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) ) ) ) ).
% lappend.disc(1)
thf(fact_170_lappend_Odisc_I2_J,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( ~ ( coinductive_lnull @ A @ Xs )
| ~ ( coinductive_lnull @ A @ Ys ) )
=> ~ ( coinductive_lnull @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) ) ) ).
% lappend.disc(2)
thf(fact_171_lnull__imp__lfinite,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ Xs )
=> ( coinductive_lfinite @ A @ Xs ) ) ).
% lnull_imp_lfinite
thf(fact_172_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_173_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_174_lappend_Octr_I1_J,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ Xs )
=> ( ( coinductive_lnull @ A @ Ys )
=> ( ( coinductive_lappend @ A @ Xs @ Ys )
= ( coinductive_LNil @ A ) ) ) ) ).
% lappend.ctr(1)
thf(fact_175_Coinductive__List_Ollast__LCons,axiom,
! [A: $tType,Xs: coinductive_llist @ A,X5: A] :
( ( ( coinductive_lnull @ A @ Xs )
=> ( ( coinductive_llast @ A @ ( coinductive_LCons @ A @ X5 @ Xs ) )
= X5 ) )
& ( ~ ( coinductive_lnull @ A @ Xs )
=> ( ( coinductive_llast @ A @ ( coinductive_LCons @ A @ X5 @ Xs ) )
= ( coinductive_llast @ A @ Xs ) ) ) ) ).
% Coinductive_List.llast_LCons
thf(fact_176_UNIV__eq__I,axiom,
! [A: $tType,A3: set @ A] :
( ! [X2: A] : ( member @ A @ X2 @ A3 )
=> ( ( top_top @ ( set @ A ) )
= A3 ) ) ).
% UNIV_eq_I
thf(fact_177_UNIV__witness,axiom,
! [A: $tType] :
? [X2: A] : ( member @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ).
% UNIV_witness
thf(fact_178_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_179_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_180_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_181_top1I,axiom,
! [A: $tType,X5: A] : ( top_top @ ( A > $o ) @ X5 ) ).
% top1I
thf(fact_182_top__empty__eq,axiom,
! [A: $tType] :
( ( top_top @ ( A > $o ) )
= ( ^ [X3: A] : ( member @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ) ) ).
% top_empty_eq
thf(fact_183_top__conj_I1_J,axiom,
! [A: $tType,X5: A,P: $o] :
( ( ( top_top @ ( A > $o ) @ X5 )
& P )
= P ) ).
% top_conj(1)
thf(fact_184_top__conj_I2_J,axiom,
! [A: $tType,P: $o,X5: A] :
( ( P
& ( top_top @ ( A > $o ) @ X5 ) )
= P ) ).
% top_conj(2)
thf(fact_185_Coinductive__List_Ollast__lappend,axiom,
! [A: $tType,Ys: coinductive_llist @ A,Xs: coinductive_llist @ A] :
( ( ( coinductive_lnull @ A @ Ys )
=> ( ( coinductive_llast @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
= ( coinductive_llast @ A @ Xs ) ) )
& ( ~ ( coinductive_lnull @ A @ Ys )
=> ( ( ( coinductive_lfinite @ A @ Xs )
=> ( ( coinductive_llast @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
= ( coinductive_llast @ A @ Ys ) ) )
& ( ~ ( coinductive_lfinite @ A @ Xs )
=> ( ( coinductive_llast @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
= ( undefined @ A ) ) ) ) ) ) ).
% Coinductive_List.llast_lappend
thf(fact_186_LList2__Mirabelle__hamjzmohle_Ollength__LCons,axiom,
! [A: $tType,R: coinductive_llist @ A,A3: set @ A,A2: A] :
( ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ A3 ) )
=> ( ( lList21232602520length @ A @ ( coinductive_LCons @ A @ A2 @ R ) )
= ( suc @ ( lList21232602520length @ A @ R ) ) ) ) ).
% LList2_Mirabelle_hamjzmohle.llength_LCons
thf(fact_187_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_188_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_189_notfin__inf,axiom,
! [A: $tType,X5: coinductive_llist @ A] :
( ( ~ ( member @ ( coinductive_llist @ A ) @ X5 @ ( lList2236698231inlsts @ A @ ( top_top @ ( set @ A ) ) ) ) )
= ( member @ ( coinductive_llist @ A ) @ X5 @ ( lList21612149805nflsts @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% notfin_inf
thf(fact_190_notinf__fin,axiom,
! [A: $tType,X5: coinductive_llist @ A] :
( ( ~ ( member @ ( coinductive_llist @ A ) @ X5 @ ( lList21612149805nflsts @ A @ ( top_top @ ( set @ A ) ) ) ) )
= ( member @ ( coinductive_llist @ A ) @ X5 @ ( lList2236698231inlsts @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% notinf_fin
thf(fact_191_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_192_LList2__Mirabelle__hamjzmohle_Ollast__LNil,axiom,
! [B: $tType] :
( ( lList2170638824_llast @ B @ ( coinductive_LNil @ B ) )
= ( undefined @ B ) ) ).
% LList2_Mirabelle_hamjzmohle.llast_LNil
thf(fact_193_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_194_inflstsI2,axiom,
! [A: $tType,A2: A,A3: set @ A,T: coinductive_llist @ A] :
( ( member @ A @ A2 @ A3 )
=> ( ( member @ ( coinductive_llist @ A ) @ T @ ( lList21612149805nflsts @ A @ A3 ) )
=> ( member @ ( coinductive_llist @ A ) @ ( coinductive_LCons @ A @ A2 @ T ) @ ( lList21612149805nflsts @ A @ A3 ) ) ) ) ).
% inflstsI2
thf(fact_195_inflsts__cases,axiom,
! [A: $tType,S: coinductive_llist @ A,A3: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ S @ ( lList21612149805nflsts @ A @ A3 ) )
=> ~ ! [A4: A,L2: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ L2 @ ( lList21612149805nflsts @ A @ A3 ) )
=> ( ( member @ A @ A4 @ A3 )
=> ( S
!= ( coinductive_LCons @ A @ A4 @ L2 ) ) ) ) ) ).
% inflsts_cases
thf(fact_196_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_197_Coinductive__List_Ollast__LNil,axiom,
! [A: $tType] :
( ( coinductive_llast @ A @ ( coinductive_LNil @ A ) )
= ( undefined @ A ) ) ).
% Coinductive_List.llast_LNil
thf(fact_198_llast__linfinite,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ~ ( coinductive_lfinite @ A @ Xs )
=> ( ( coinductive_llast @ A @ Xs )
= ( undefined @ A ) ) ) ).
% llast_linfinite
thf(fact_199_ltake__LCons__Suc,axiom,
! [A: $tType,A2: A,L: coinductive_llist @ A,I: nat] :
( ( lList22119844313_ltake @ A @ ( coinductive_LCons @ A @ A2 @ L ) @ ( suc @ I ) )
= ( coinductive_LCons @ A @ A2 @ ( lList22119844313_ltake @ A @ L @ I ) ) ) ).
% ltake_LCons_Suc
thf(fact_200_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_201_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_202_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_203_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_204_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_205_lift__Suc__mono__less__iff,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [F2: nat > A,N: nat,M: nat] :
( ! [N2: nat] : ( ord_less @ A @ ( F2 @ N2 ) @ ( F2 @ ( suc @ N2 ) ) )
=> ( ( ord_less @ A @ ( F2 @ N ) @ ( F2 @ M ) )
= ( ord_less @ nat @ N @ M ) ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_206_lift__Suc__mono__less,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [F2: nat > A,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less @ A @ ( F2 @ N2 ) @ ( F2 @ ( suc @ N2 ) ) )
=> ( ( ord_less @ nat @ N @ N3 )
=> ( ord_less @ A @ ( F2 @ N ) @ ( F2 @ N3 ) ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_207_inflstsI,axiom,
! [A: $tType,X5: coinductive_llist @ A,A3: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ X5 @ ( lList2435255213lllsts @ A @ A3 ) )
=> ( ~ ( member @ ( coinductive_llist @ A ) @ X5 @ ( lList2236698231inlsts @ A @ ( top_top @ ( set @ A ) ) ) )
=> ( member @ ( coinductive_llist @ A ) @ X5 @ ( lList21612149805nflsts @ A @ A3 ) ) ) ) ).
% inflstsI
thf(fact_208_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_209_alllsts__UNIV,axiom,
! [A: $tType,S: coinductive_llist @ A] : ( member @ ( coinductive_llist @ A ) @ S @ ( lList2435255213lllsts @ A @ ( top_top @ ( set @ A ) ) ) ) ).
% alllsts_UNIV
thf(fact_210_LConsE,axiom,
! [A: $tType,X5: A,Xs: coinductive_llist @ A,A3: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ ( coinductive_LCons @ A @ X5 @ Xs ) @ ( lList2435255213lllsts @ A @ A3 ) )
= ( ( member @ A @ X5 @ A3 )
& ( member @ ( coinductive_llist @ A ) @ Xs @ ( lList2435255213lllsts @ A @ A3 ) ) ) ) ).
% LConsE
thf(fact_211_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_212_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_213_alllsts_Ocoinduct,axiom,
! [A: $tType,X4: ( coinductive_llist @ A ) > $o,X5: coinductive_llist @ A,A3: set @ A] :
( ( X4 @ X5 )
=> ( ! [X2: coinductive_llist @ A] :
( ( X4 @ X2 )
=> ( ( X2
= ( coinductive_LNil @ A ) )
| ? [L3: coinductive_llist @ A,A5: A] :
( ( X2
= ( coinductive_LCons @ A @ A5 @ L3 ) )
& ( ( X4 @ L3 )
| ( member @ ( coinductive_llist @ A ) @ L3 @ ( lList2435255213lllsts @ A @ A3 ) ) )
& ( member @ A @ A5 @ A3 ) ) ) )
=> ( member @ ( coinductive_llist @ A ) @ X5 @ ( lList2435255213lllsts @ A @ A3 ) ) ) ) ).
% alllsts.coinduct
thf(fact_214_alllsts_Osimps,axiom,
! [A: $tType,A2: coinductive_llist @ A,A3: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ A2 @ ( lList2435255213lllsts @ A @ A3 ) )
= ( ( A2
= ( coinductive_LNil @ A ) )
| ? [L4: coinductive_llist @ A,A7: A] :
( ( A2
= ( coinductive_LCons @ A @ A7 @ L4 ) )
& ( member @ ( coinductive_llist @ A ) @ L4 @ ( lList2435255213lllsts @ A @ A3 ) )
& ( member @ A @ A7 @ A3 ) ) ) ) ).
% alllsts.simps
thf(fact_215_alllsts_Ocases,axiom,
! [A: $tType,A2: coinductive_llist @ A,A3: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ A2 @ ( lList2435255213lllsts @ A @ A3 ) )
=> ( ( A2
!= ( coinductive_LNil @ A ) )
=> ~ ! [L2: coinductive_llist @ A,A4: A] :
( ( A2
= ( coinductive_LCons @ A @ A4 @ L2 ) )
=> ( ( member @ ( coinductive_llist @ A ) @ L2 @ ( lList2435255213lllsts @ A @ A3 ) )
=> ~ ( member @ A @ A4 @ A3 ) ) ) ) ) ).
% alllsts.cases
thf(fact_216_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_217_alllsts_OLCons__all,axiom,
! [A: $tType,L: coinductive_llist @ A,A3: set @ A,A2: A] :
( ( member @ ( coinductive_llist @ A ) @ L @ ( lList2435255213lllsts @ A @ A3 ) )
=> ( ( member @ A @ A2 @ A3 )
=> ( member @ ( coinductive_llist @ A ) @ ( coinductive_LCons @ A @ A2 @ L ) @ ( lList2435255213lllsts @ A @ A3 ) ) ) ) ).
% alllsts.LCons_all
thf(fact_218_alllsts_OLNil__all,axiom,
! [A: $tType,A3: set @ A] : ( member @ ( coinductive_llist @ A ) @ ( coinductive_LNil @ A ) @ ( lList2435255213lllsts @ A @ A3 ) ) ).
% alllsts.LNil_all
thf(fact_219_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_220_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_221_finite__lemma,axiom,
! [A: $tType,X5: coinductive_llist @ A,A3: set @ A,B5: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ X5 @ ( lList2236698231inlsts @ A @ A3 ) )
=> ( ( member @ ( coinductive_llist @ A ) @ X5 @ ( lList2435255213lllsts @ A @ B5 ) )
=> ( member @ ( coinductive_llist @ A ) @ X5 @ ( lList2236698231inlsts @ A @ B5 ) ) ) ) ).
% finite_lemma
thf(fact_222_finsubsetall,axiom,
! [A: $tType,X5: coinductive_llist @ A,A3: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ X5 @ ( lList2236698231inlsts @ A @ A3 ) )
=> ( member @ ( coinductive_llist @ A ) @ X5 @ ( lList2435255213lllsts @ A @ A3 ) ) ) ).
% finsubsetall
thf(fact_223_infsubsetall,axiom,
! [A: $tType,X5: coinductive_llist @ A,A3: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ X5 @ ( lList21612149805nflsts @ A @ A3 ) )
=> ( member @ ( coinductive_llist @ A ) @ X5 @ ( lList2435255213lllsts @ A @ A3 ) ) ) ).
% infsubsetall
thf(fact_224_alllstsE,axiom,
! [A: $tType,X5: coinductive_llist @ A,A3: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ X5 @ ( lList2435255213lllsts @ A @ A3 ) )
=> ( ~ ( member @ ( coinductive_llist @ A ) @ X5 @ ( lList2236698231inlsts @ A @ A3 ) )
=> ( member @ ( coinductive_llist @ A ) @ X5 @ ( lList21612149805nflsts @ A @ A3 ) ) ) ) ).
% alllstsE
thf(fact_225_inflstsE,axiom,
! [A: $tType,X5: coinductive_llist @ A,A3: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ X5 @ ( lList21612149805nflsts @ A @ A3 ) )
=> ~ ( ( member @ ( coinductive_llist @ A ) @ X5 @ ( lList2435255213lllsts @ A @ A3 ) )
=> ( member @ ( coinductive_llist @ A ) @ X5 @ ( lList2236698231inlsts @ A @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% inflstsE
thf(fact_226_inflsts__def,axiom,
! [A: $tType] :
( ( lList21612149805nflsts @ A )
= ( ^ [A6: set @ A] : ( minus_minus @ ( set @ ( coinductive_llist @ A ) ) @ ( lList2435255213lllsts @ A @ A6 ) @ ( lList2236698231inlsts @ A @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% inflsts_def
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_Un__iff,axiom,
! [A: $tType,C2: A,A3: set @ A,B5: set @ A] :
( ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A3 @ B5 ) )
= ( ( member @ A @ C2 @ A3 )
| ( member @ A @ C2 @ B5 ) ) ) ).
% Un_iff
thf(fact_229_UnCI,axiom,
! [A: $tType,C2: A,B5: set @ A,A3: set @ A] :
( ( ~ ( member @ A @ C2 @ B5 )
=> ( member @ A @ C2 @ A3 ) )
=> ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A3 @ B5 ) ) ) ).
% UnCI
thf(fact_230_Diff__idemp,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B5 ) @ B5 )
= ( minus_minus @ ( set @ A ) @ A3 @ B5 ) ) ).
% Diff_idemp
thf(fact_231_Diff__iff,axiom,
! [A: $tType,C2: A,A3: set @ A,B5: set @ A] :
( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A3 @ B5 ) )
= ( ( member @ A @ C2 @ A3 )
& ~ ( member @ A @ C2 @ B5 ) ) ) ).
% Diff_iff
thf(fact_232_DiffI,axiom,
! [A: $tType,C2: A,A3: set @ A,B5: set @ A] :
( ( member @ A @ C2 @ A3 )
=> ( ~ ( member @ A @ C2 @ B5 )
=> ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A3 @ B5 ) ) ) ) ).
% DiffI
thf(fact_233_Un__Diff__cancel2,axiom,
! [A: $tType,B5: set @ A,A3: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ B5 @ A3 ) @ A3 )
= ( sup_sup @ ( set @ A ) @ B5 @ A3 ) ) ).
% Un_Diff_cancel2
thf(fact_234_Un__Diff__cancel,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ( sup_sup @ ( set @ A ) @ A3 @ ( minus_minus @ ( set @ A ) @ B5 @ A3 ) )
= ( sup_sup @ ( set @ A ) @ A3 @ B5 ) ) ).
% Un_Diff_cancel
thf(fact_235_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_236_Un__UNIV__left,axiom,
! [A: $tType,B5: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( top_top @ ( set @ A ) ) @ B5 )
= ( top_top @ ( set @ A ) ) ) ).
% Un_UNIV_left
thf(fact_237_Un__left__commute,axiom,
! [A: $tType,A3: set @ A,B5: set @ A,C3: set @ A] :
( ( sup_sup @ ( set @ A ) @ A3 @ ( sup_sup @ ( set @ A ) @ B5 @ C3 ) )
= ( sup_sup @ ( set @ A ) @ B5 @ ( sup_sup @ ( set @ A ) @ A3 @ C3 ) ) ) ).
% Un_left_commute
thf(fact_238_Un__left__absorb,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ( sup_sup @ ( set @ A ) @ A3 @ ( sup_sup @ ( set @ A ) @ A3 @ B5 ) )
= ( sup_sup @ ( set @ A ) @ A3 @ B5 ) ) ).
% Un_left_absorb
thf(fact_239_Un__commute,axiom,
! [A: $tType] :
( ( sup_sup @ ( set @ A ) )
= ( ^ [A6: set @ A,B6: set @ A] : ( sup_sup @ ( set @ A ) @ B6 @ A6 ) ) ) ).
% Un_commute
thf(fact_240_Un__absorb,axiom,
! [A: $tType,A3: set @ A] :
( ( sup_sup @ ( set @ A ) @ A3 @ A3 )
= A3 ) ).
% Un_absorb
thf(fact_241_Un__assoc,axiom,
! [A: $tType,A3: set @ A,B5: set @ A,C3: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A3 @ B5 ) @ C3 )
= ( sup_sup @ ( set @ A ) @ A3 @ ( sup_sup @ ( set @ A ) @ B5 @ C3 ) ) ) ).
% Un_assoc
thf(fact_242_ball__Un,axiom,
! [A: $tType,A3: set @ A,B5: set @ A,P: A > $o] :
( ( ! [X3: A] :
( ( member @ A @ X3 @ ( sup_sup @ ( set @ A ) @ A3 @ B5 ) )
=> ( P @ X3 ) ) )
= ( ! [X3: A] :
( ( member @ A @ X3 @ A3 )
=> ( P @ X3 ) )
& ! [X3: A] :
( ( member @ A @ X3 @ B5 )
=> ( P @ X3 ) ) ) ) ).
% ball_Un
thf(fact_243_Un__Diff,axiom,
! [A: $tType,A3: set @ A,B5: set @ A,C3: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A3 @ B5 ) @ C3 )
= ( sup_sup @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ C3 ) @ ( minus_minus @ ( set @ A ) @ B5 @ C3 ) ) ) ).
% Un_Diff
thf(fact_244_bex__Un,axiom,
! [A: $tType,A3: set @ A,B5: set @ A,P: A > $o] :
( ( ? [X3: A] :
( ( member @ A @ X3 @ ( sup_sup @ ( set @ A ) @ A3 @ B5 ) )
& ( P @ X3 ) ) )
= ( ? [X3: A] :
( ( member @ A @ X3 @ A3 )
& ( P @ X3 ) )
| ? [X3: A] :
( ( member @ A @ X3 @ B5 )
& ( P @ X3 ) ) ) ) ).
% bex_Un
thf(fact_245_DiffD2,axiom,
! [A: $tType,C2: A,A3: set @ A,B5: set @ A] :
( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A3 @ B5 ) )
=> ~ ( member @ A @ C2 @ B5 ) ) ).
% DiffD2
thf(fact_246_DiffD1,axiom,
! [A: $tType,C2: A,A3: set @ A,B5: set @ A] :
( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A3 @ B5 ) )
=> ( member @ A @ C2 @ A3 ) ) ).
% DiffD1
thf(fact_247_DiffE,axiom,
! [A: $tType,C2: A,A3: set @ A,B5: set @ A] :
( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A3 @ B5 ) )
=> ~ ( ( member @ A @ C2 @ A3 )
=> ( member @ A @ C2 @ B5 ) ) ) ).
% DiffE
thf(fact_248_UnI2,axiom,
! [A: $tType,C2: A,B5: set @ A,A3: set @ A] :
( ( member @ A @ C2 @ B5 )
=> ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A3 @ B5 ) ) ) ).
% UnI2
thf(fact_249_UnI1,axiom,
! [A: $tType,C2: A,A3: set @ A,B5: set @ A] :
( ( member @ A @ C2 @ A3 )
=> ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A3 @ B5 ) ) ) ).
% UnI1
thf(fact_250_UnE,axiom,
! [A: $tType,C2: A,A3: set @ A,B5: set @ A] :
( ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A3 @ B5 ) )
=> ( ~ ( member @ A @ C2 @ A3 )
=> ( member @ A @ C2 @ B5 ) ) ) ).
% UnE
thf(fact_251_psubset__imp__ex__mem,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ( ord_less @ ( set @ A ) @ A3 @ B5 )
=> ? [B3: A] : ( member @ A @ B3 @ ( minus_minus @ ( set @ A ) @ B5 @ A3 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_252_sup__top__right,axiom,
! [A: $tType] :
( ( bounded_lattice_top @ A @ ( type2 @ A ) )
=> ! [X5: A] :
( ( sup_sup @ A @ X5 @ ( top_top @ A ) )
= ( top_top @ A ) ) ) ).
% sup_top_right
thf(fact_253_sup__top__left,axiom,
! [A: $tType] :
( ( bounded_lattice_top @ A @ ( type2 @ A ) )
=> ! [X5: A] :
( ( sup_sup @ A @ ( top_top @ A ) @ X5 )
= ( top_top @ A ) ) ) ).
% sup_top_left
%----Type constructors (31)
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___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_Owellorder,axiom,
wellorder @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Opreorder_3,axiom,
preorder @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
linorder @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Ono__top,axiom,
no_top @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Oorder_4,axiom,
order @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Oord_5,axiom,
ord @ nat @ ( type2 @ nat ) ).
thf(tcon_Set_Oset___Lattices_Obounded__lattice__top_6,axiom,
! [A8: $tType] : ( bounded_lattice_top @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder__top_7,axiom,
! [A8: $tType] : ( order_top @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_8,axiom,
! [A8: $tType] : ( preorder @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_9,axiom,
! [A8: $tType] : ( order @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Otop_10,axiom,
! [A8: $tType] : ( top @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_11,axiom,
! [A8: $tType] : ( ord @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_HOL_Obool___Lattices_Obounded__lattice__top_12,axiom,
bounded_lattice_top @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oorder__top_13,axiom,
order_top @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Opreorder_14,axiom,
preorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Olinorder_15,axiom,
linorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oorder_16,axiom,
order @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Otop_17,axiom,
top @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oord_18,axiom,
ord @ $o @ ( type2 @ $o ) ).
thf(tcon_Coinductive__List_Ollist___Orderings_Opreorder_19,axiom,
! [A8: $tType] : ( preorder @ ( coinductive_llist @ A8 ) @ ( type2 @ ( coinductive_llist @ A8 ) ) ) ).
thf(tcon_Coinductive__List_Ollist___Orderings_Oorder_20,axiom,
! [A8: $tType] : ( order @ ( coinductive_llist @ A8 ) @ ( type2 @ ( coinductive_llist @ A8 ) ) ) ).
thf(tcon_Coinductive__List_Ollist___Orderings_Oord_21,axiom,
! [A8: $tType] : ( ord @ ( coinductive_llist @ A8 ) @ ( type2 @ ( coinductive_llist @ A8 ) ) ) ).
%----Conjectures (1)
thf(conj_0,conjecture,
~ ( ord_less @ ( coinductive_llist @ a ) @ r @ ( coinductive_LNil @ a ) ) ).
%------------------------------------------------------------------------------