TPTP Problem File: DAT182^1.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : DAT182^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Data Structures
% Problem : Lazy lists II 857
% 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__857.p [Bla16]
% Status : Theorem
% Rating : 0.67 v8.1.0, 0.50 v7.5.0, 0.33 v7.2.0, 0.50 v7.1.0
% Syntax : Number of formulae : 325 ( 82 unt; 45 typ; 0 def)
% Number of atoms : 880 ( 222 equ; 0 cnn)
% Maximal formula atoms : 13 ( 3 avg)
% Number of connectives : 3808 ( 93 ~; 22 |; 52 &;3156 @)
% ( 0 <=>; 485 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 9 avg)
% Number of types : 3 ( 2 usr)
% Number of type conns : 205 ( 205 >; 0 *; 0 +; 0 <<)
% Number of symbols : 44 ( 43 usr; 2 con; 0-5 aty)
% Number of variables : 1053 ( 51 ^; 918 !; 45 ?;1053 :)
% ( 39 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 14:49:40.110
%------------------------------------------------------------------------------
%----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 (40)
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_Orderings_Odense__linorder,type,
dense_linorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum,type,
condit1656338222tinuum:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder,type,
condit1037483654norder:
!>[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_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_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_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_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_Oordering__top,type,
ordering_top:
!>[A: $tType] : ( ( A > A > $o ) > ( A > A > $o ) > 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_a,type,
a2: a ).
thf(sy_v_l,type,
l: coinductive_llist @ a ).
%----Relevant facts (256)
thf(fact_0_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_1_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_2_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% order_refl
thf(fact_3_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_4_neq__LNil__conv,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( Xs
!= ( coinductive_LNil @ A ) )
= ( ? [X2: A,Xs2: coinductive_llist @ A] :
( Xs
= ( coinductive_LCons @ A @ X2 @ Xs2 ) ) ) ) ).
% neq_LNil_conv
thf(fact_5_llist__less__def,axiom,
( ( ord_less @ ( coinductive_llist @ a ) )
= ( ^ [S: coinductive_llist @ a,T: coinductive_llist @ a] :
( ( ord_less_eq @ ( coinductive_llist @ a ) @ S @ T )
& ( S != T ) ) ) ) ).
% llist_less_def
thf(fact_6_llist__le__def,axiom,
( ( ord_less_eq @ ( coinductive_llist @ a ) )
= ( ^ [S: coinductive_llist @ a,T: coinductive_llist @ a] :
? [D: coinductive_llist @ a] :
( T
= ( coinductive_lappend @ a @ S @ D ) ) ) ) ).
% llist_le_def
thf(fact_7_lbutlast__LNil,axiom,
! [A: $tType] :
( ( lList2370560421utlast @ A @ ( coinductive_LNil @ A ) )
= ( coinductive_LNil @ A ) ) ).
% lbutlast_LNil
thf(fact_8_fps__induct,axiom,
! [A: $tType,L: coinductive_llist @ A,A2: set @ A,P: ( coinductive_llist @ A ) > $o] :
( ( member @ ( coinductive_llist @ A ) @ L @ ( lList22096119349pslsts @ A @ A2 ) )
=> ( ! [A3: A] :
( ( member @ A @ A3 @ A2 )
=> ( P @ ( coinductive_LCons @ A @ A3 @ ( coinductive_LNil @ A ) ) ) )
=> ( ! [A3: A,L2: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ L2 @ ( lList22096119349pslsts @ A @ A2 ) )
=> ( ( P @ L2 )
=> ( ( member @ A @ A3 @ A2 )
=> ( P @ ( coinductive_LCons @ A @ A3 @ L2 ) ) ) ) )
=> ( P @ L ) ) ) ) ).
% fps_induct
thf(fact_9_alllstsp_Ocases,axiom,
! [A: $tType,A2: A > $o,A4: coinductive_llist @ A] :
( ( lList21511617539llstsp @ A @ A2 @ A4 )
=> ( ( A4
!= ( coinductive_LNil @ A ) )
=> ~ ! [L2: coinductive_llist @ A,A3: A] :
( ( A4
= ( coinductive_LCons @ A @ A3 @ L2 ) )
=> ( ( lList21511617539llstsp @ A @ A2 @ L2 )
=> ~ ( A2 @ A3 ) ) ) ) ) ).
% alllstsp.cases
thf(fact_10_alllstsp_Osimps,axiom,
! [A: $tType] :
( ( lList21511617539llstsp @ A )
= ( ^ [A5: A > $o,A6: coinductive_llist @ A] :
( ( A6
= ( coinductive_LNil @ A ) )
| ? [L3: coinductive_llist @ A,B2: A] :
( ( A6
= ( coinductive_LCons @ A @ B2 @ L3 ) )
& ( lList21511617539llstsp @ A @ A5 @ L3 )
& ( A5 @ B2 ) ) ) ) ) ).
% alllstsp.simps
thf(fact_11_finlstsp_Ocases,axiom,
! [A: $tType,A2: A > $o,A4: coinductive_llist @ A] :
( ( lList2860480441nlstsp @ A @ A2 @ A4 )
=> ( ( A4
!= ( coinductive_LNil @ A ) )
=> ~ ! [L2: coinductive_llist @ A,A3: A] :
( ( A4
= ( coinductive_LCons @ A @ A3 @ L2 ) )
=> ( ( lList2860480441nlstsp @ A @ A2 @ L2 )
=> ~ ( A2 @ A3 ) ) ) ) ) ).
% finlstsp.cases
thf(fact_12_finlstsp_Osimps,axiom,
! [A: $tType] :
( ( lList2860480441nlstsp @ A )
= ( ^ [A5: A > $o,A6: coinductive_llist @ A] :
( ( A6
= ( coinductive_LNil @ A ) )
| ? [L3: coinductive_llist @ A,B2: A] :
( ( A6
= ( coinductive_LCons @ A @ B2 @ L3 ) )
& ( lList2860480441nlstsp @ A @ A5 @ L3 )
& ( A5 @ B2 ) ) ) ) ) ).
% finlstsp.simps
thf(fact_13_lappend__code_I2_J,axiom,
! [A: $tType,Xa: A,X: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_lappend @ A @ ( coinductive_LCons @ A @ Xa @ X ) @ Ys )
= ( coinductive_LCons @ A @ Xa @ ( coinductive_lappend @ A @ X @ Ys ) ) ) ).
% lappend_code(2)
thf(fact_14_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_15_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_16_lappend__LNil2,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( coinductive_lappend @ A @ Xs @ ( coinductive_LNil @ A ) )
= Xs ) ).
% lappend_LNil2
thf(fact_17_lappend__code_I1_J,axiom,
! [A: $tType,Ys: coinductive_llist @ A] :
( ( coinductive_lappend @ A @ ( coinductive_LNil @ A ) @ Ys )
= Ys ) ).
% lappend_code(1)
thf(fact_18_lappend__is__LNil__conv,axiom,
! [A: $tType,S2: coinductive_llist @ A,T2: coinductive_llist @ A] :
( ( ( coinductive_lappend @ A @ S2 @ T2 )
= ( coinductive_LNil @ A ) )
= ( ( S2
= ( coinductive_LNil @ A ) )
& ( T2
= ( coinductive_LNil @ A ) ) ) ) ).
% lappend_is_LNil_conv
thf(fact_19_LNil__is__lappend__conv,axiom,
! [A: $tType,S2: coinductive_llist @ A,T2: coinductive_llist @ A] :
( ( ( coinductive_LNil @ A )
= ( coinductive_lappend @ A @ S2 @ T2 ) )
= ( ( S2
= ( coinductive_LNil @ A ) )
& ( T2
= ( coinductive_LNil @ A ) ) ) ) ).
% LNil_is_lappend_conv
thf(fact_20_ord__eq__less__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A4: A,F: B > A,B3: B,C: B] :
( ( A4
= ( F @ B3 ) )
=> ( ( ord_less @ B @ B3 @ C )
=> ( ! [X3: B,Y2: B] :
( ( ord_less @ B @ X3 @ Y2 )
=> ( ord_less @ A @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less @ A @ A4 @ ( F @ C ) ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_21_ord__less__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A4: A,B3: A,F: A > B,C: B] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X3: A,Y2: A] :
( ( ord_less @ A @ X3 @ Y2 )
=> ( ord_less @ B @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less @ B @ ( F @ A4 ) @ C ) ) ) ) ) ).
% ord_less_eq_subst
thf(fact_22_order__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B @ ( type2 @ B ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A4: A,F: B > A,B3: B,C: B] :
( ( ord_less @ A @ A4 @ ( F @ B3 ) )
=> ( ( ord_less @ B @ B3 @ C )
=> ( ! [X3: B,Y2: B] :
( ( ord_less @ B @ X3 @ Y2 )
=> ( ord_less @ A @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less @ A @ A4 @ ( F @ C ) ) ) ) ) ) ).
% order_less_subst1
thf(fact_23_order__less__subst2,axiom,
! [A: $tType,C2: $tType] :
( ( ( order @ C2 @ ( type2 @ C2 ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A4: A,B3: A,F: A > C2,C: C2] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ( ord_less @ C2 @ ( F @ B3 ) @ C )
=> ( ! [X3: A,Y2: A] :
( ( ord_less @ A @ X3 @ Y2 )
=> ( ord_less @ C2 @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less @ C2 @ ( F @ A4 ) @ C ) ) ) ) ) ).
% order_less_subst2
thf(fact_24_lt__ex,axiom,
! [A: $tType] :
( ( no_bot @ A @ ( type2 @ A ) )
=> ! [X: A] :
? [Y2: A] : ( ord_less @ A @ Y2 @ X ) ) ).
% lt_ex
thf(fact_25_gt__ex,axiom,
! [A: $tType] :
( ( no_top @ A @ ( type2 @ A ) )
=> ! [X: A] :
? [X1: A] : ( ord_less @ A @ X @ X1 ) ) ).
% gt_ex
thf(fact_26_neqE,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( X != Y )
=> ( ~ ( ord_less @ A @ X @ Y )
=> ( ord_less @ A @ Y @ X ) ) ) ) ).
% neqE
thf(fact_27_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_28_neq__iff,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( X != Y )
= ( ( ord_less @ A @ X @ Y )
| ( ord_less @ A @ Y @ X ) ) ) ) ).
% neq_iff
thf(fact_29_order_Oasym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ~ ( ord_less @ A @ B3 @ A4 ) ) ) ).
% order.asym
thf(fact_30_dense,axiom,
! [A: $tType] :
( ( dense_order @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ? [Z: A] :
( ( ord_less @ A @ X @ Z )
& ( ord_less @ A @ Z @ Y ) ) ) ) ).
% dense
thf(fact_31_less__imp__neq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( X != Y ) ) ) ).
% less_imp_neq
thf(fact_32_less__asym,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ~ ( ord_less @ A @ Y @ X ) ) ) ).
% less_asym
thf(fact_33_less__asym_H,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ~ ( ord_less @ A @ B3 @ A4 ) ) ) ).
% less_asym'
thf(fact_34_less__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z2: A] :
( ( ord_less @ A @ X @ Y )
=> ( ( ord_less @ A @ Y @ Z2 )
=> ( ord_less @ A @ X @ Z2 ) ) ) ) ).
% less_trans
thf(fact_35_less__linear,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
| ( X = Y )
| ( ord_less @ A @ Y @ X ) ) ) ).
% less_linear
thf(fact_36_less__irrefl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A] :
~ ( ord_less @ A @ X @ X ) ) ).
% less_irrefl
thf(fact_37_ord__eq__less__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A4: A,B3: A,C: A] :
( ( A4 = B3 )
=> ( ( ord_less @ A @ B3 @ C )
=> ( ord_less @ A @ A4 @ C ) ) ) ) ).
% ord_eq_less_trans
thf(fact_38_ord__less__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A4: A,B3: A,C: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ( B3 = C )
=> ( ord_less @ A @ A4 @ C ) ) ) ) ).
% ord_less_eq_trans
thf(fact_39_dual__order_Oasym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B3: A,A4: A] :
( ( ord_less @ A @ B3 @ A4 )
=> ~ ( ord_less @ A @ A4 @ B3 ) ) ) ).
% dual_order.asym
thf(fact_40_less__imp__not__eq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( X != Y ) ) ) ).
% less_imp_not_eq
thf(fact_41_less__not__sym,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ~ ( ord_less @ A @ Y @ X ) ) ) ).
% less_not_sym
thf(fact_42_less__induct,axiom,
! [A: $tType] :
( ( wellorder @ A @ ( type2 @ A ) )
=> ! [P: A > $o,A4: A] :
( ! [X3: A] :
( ! [Y3: A] :
( ( ord_less @ A @ Y3 @ X3 )
=> ( P @ Y3 ) )
=> ( P @ X3 ) )
=> ( P @ A4 ) ) ) ).
% less_induct
thf(fact_43_antisym__conv3,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [Y: A,X: A] :
( ~ ( ord_less @ A @ Y @ X )
=> ( ( ~ ( ord_less @ A @ X @ Y ) )
= ( X = Y ) ) ) ) ).
% antisym_conv3
thf(fact_44_less__imp__not__eq2,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( Y != X ) ) ) ).
% less_imp_not_eq2
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,A2: set @ A] :
( ( collect @ A
@ ^ [X2: A] : ( member @ A @ X2 @ A2 ) )
= A2 ) ).
% 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_less__imp__triv,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,P: $o] :
( ( ord_less @ A @ X @ Y )
=> ( ( ord_less @ A @ Y @ X )
=> P ) ) ) ).
% less_imp_triv
thf(fact_50_linorder__cases,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ~ ( ord_less @ A @ X @ Y )
=> ( ( X != Y )
=> ( ord_less @ A @ Y @ X ) ) ) ) ).
% linorder_cases
thf(fact_51_dual__order_Oirrefl,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A4: A] :
~ ( ord_less @ A @ A4 @ A4 ) ) ).
% dual_order.irrefl
thf(fact_52_fpslsts__mono,axiom,
! [A: $tType,A2: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B4 )
=> ( ord_less_eq @ ( set @ ( coinductive_llist @ A ) ) @ ( lList22096119349pslsts @ A @ A2 ) @ ( lList22096119349pslsts @ A @ B4 ) ) ) ).
% fpslsts_mono
thf(fact_53_order_Ostrict__trans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A4: A,B3: A,C: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ( ord_less @ A @ B3 @ C )
=> ( ord_less @ A @ A4 @ C ) ) ) ) ).
% order.strict_trans
thf(fact_54_alllstsp__mono,axiom,
! [A: $tType,A2: A > $o,B4: A > $o] :
( ( ord_less_eq @ ( A > $o ) @ A2 @ B4 )
=> ( ord_less_eq @ ( ( coinductive_llist @ A ) > $o ) @ ( lList21511617539llstsp @ A @ A2 ) @ ( lList21511617539llstsp @ A @ B4 ) ) ) ).
% alllstsp_mono
thf(fact_55_finlstsp__mono,axiom,
! [A: $tType,A2: A > $o,B4: A > $o] :
( ( ord_less_eq @ ( A > $o ) @ A2 @ B4 )
=> ( ord_less_eq @ ( ( coinductive_llist @ A ) > $o ) @ ( lList2860480441nlstsp @ A @ A2 ) @ ( lList2860480441nlstsp @ A @ B4 ) ) ) ).
% finlstsp_mono
thf(fact_56_less__imp__not__less,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ~ ( ord_less @ A @ Y @ X ) ) ) ).
% less_imp_not_less
thf(fact_57_dual__order_Ostrict__trans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B3: A,A4: A,C: A] :
( ( ord_less @ A @ B3 @ A4 )
=> ( ( ord_less @ A @ C @ B3 )
=> ( ord_less @ A @ C @ A4 ) ) ) ) ).
% dual_order.strict_trans
thf(fact_58_not__less__iff__gr__or__eq,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ~ ( ord_less @ A @ X @ Y ) )
= ( ( ord_less @ A @ Y @ X )
| ( X = Y ) ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_59_order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ( A4 != B3 ) ) ) ).
% order.strict_implies_not_eq
thf(fact_60_dual__order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B3: A,A4: A] :
( ( ord_less @ A @ B3 @ A4 )
=> ( A4 != B3 ) ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_61_order_Onot__eq__order__implies__strict,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A4: A,B3: A] :
( ( A4 != B3 )
=> ( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ord_less @ A @ A4 @ B3 ) ) ) ) ).
% order.not_eq_order_implies_strict
thf(fact_62_dual__order_Ostrict__implies__order,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B3: A,A4: A] :
( ( ord_less @ A @ B3 @ A4 )
=> ( ord_less_eq @ A @ B3 @ A4 ) ) ) ).
% dual_order.strict_implies_order
thf(fact_63_dual__order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less @ A )
= ( ^ [B2: A,A6: A] :
( ( ord_less_eq @ A @ B2 @ A6 )
& ( A6 != B2 ) ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_64_dual__order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less_eq @ A )
= ( ^ [B2: A,A6: A] :
( ( ord_less @ A @ B2 @ A6 )
| ( A6 = B2 ) ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_65_order_Ostrict__implies__order,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ord_less_eq @ A @ A4 @ B3 ) ) ) ).
% order.strict_implies_order
thf(fact_66_dense__le__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z2: A] :
( ( ord_less @ A @ X @ Y )
=> ( ! [W: A] :
( ( ord_less @ A @ X @ W )
=> ( ( ord_less @ A @ W @ Y )
=> ( ord_less_eq @ A @ W @ Z2 ) ) )
=> ( ord_less_eq @ A @ Y @ Z2 ) ) ) ) ).
% dense_le_bounded
thf(fact_67_dense__ge__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A @ ( type2 @ A ) )
=> ! [Z2: A,X: A,Y: A] :
( ( ord_less @ A @ Z2 @ X )
=> ( ! [W: A] :
( ( ord_less @ A @ Z2 @ W )
=> ( ( ord_less @ A @ W @ X )
=> ( ord_less_eq @ A @ Y @ W ) ) )
=> ( ord_less_eq @ A @ Y @ Z2 ) ) ) ) ).
% dense_ge_bounded
thf(fact_68_dual__order_Ostrict__trans2,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B3: A,A4: A,C: A] :
( ( ord_less @ A @ B3 @ A4 )
=> ( ( ord_less_eq @ A @ C @ B3 )
=> ( ord_less @ A @ C @ A4 ) ) ) ) ).
% dual_order.strict_trans2
thf(fact_69_dual__order_Ostrict__trans1,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B3: A,A4: A,C: A] :
( ( ord_less_eq @ A @ B3 @ A4 )
=> ( ( ord_less @ A @ C @ B3 )
=> ( ord_less @ A @ C @ A4 ) ) ) ) ).
% dual_order.strict_trans1
thf(fact_70_order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less @ A )
= ( ^ [A6: A,B2: A] :
( ( ord_less_eq @ A @ A6 @ B2 )
& ( A6 != B2 ) ) ) ) ) ).
% order.strict_iff_order
thf(fact_71_order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less_eq @ A )
= ( ^ [A6: A,B2: A] :
( ( ord_less @ A @ A6 @ B2 )
| ( A6 = B2 ) ) ) ) ) ).
% order.order_iff_strict
thf(fact_72_order_Ostrict__trans2,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A4: A,B3: A,C: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ C )
=> ( ord_less @ A @ A4 @ C ) ) ) ) ).
% order.strict_trans2
thf(fact_73_order_Ostrict__trans1,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A4: A,B3: A,C: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( ord_less @ A @ B3 @ C )
=> ( ord_less @ A @ A4 @ C ) ) ) ) ).
% order.strict_trans1
thf(fact_74_not__le__imp__less,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [Y: A,X: A] :
( ~ ( ord_less_eq @ A @ Y @ X )
=> ( ord_less @ A @ X @ Y ) ) ) ).
% not_le_imp_less
thf(fact_75_less__le__not__le,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ( ( ord_less @ A )
= ( ^ [X2: A,Y4: A] :
( ( ord_less_eq @ A @ X2 @ Y4 )
& ~ ( ord_less_eq @ A @ Y4 @ X2 ) ) ) ) ) ).
% less_le_not_le
thf(fact_76_le__imp__less__or__eq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less @ A @ X @ Y )
| ( X = Y ) ) ) ) ).
% le_imp_less_or_eq
thf(fact_77_le__less__linear,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
| ( ord_less @ A @ Y @ X ) ) ) ).
% le_less_linear
thf(fact_78_dense__le,axiom,
! [A: $tType] :
( ( dense_linorder @ A @ ( type2 @ A ) )
=> ! [Y: A,Z2: A] :
( ! [X3: A] :
( ( ord_less @ A @ X3 @ Y )
=> ( ord_less_eq @ A @ X3 @ Z2 ) )
=> ( ord_less_eq @ A @ Y @ Z2 ) ) ) ).
% dense_le
thf(fact_79_dense__ge,axiom,
! [A: $tType] :
( ( dense_linorder @ A @ ( type2 @ A ) )
=> ! [Z2: A,Y: A] :
( ! [X3: A] :
( ( ord_less @ A @ Z2 @ X3 )
=> ( ord_less_eq @ A @ Y @ X3 ) )
=> ( ord_less_eq @ A @ Y @ Z2 ) ) ) ).
% dense_ge
thf(fact_80_less__le__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z2: A] :
( ( ord_less @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ Z2 )
=> ( ord_less @ A @ X @ Z2 ) ) ) ) ).
% less_le_trans
thf(fact_81_le__less__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z2: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less @ A @ Y @ Z2 )
=> ( ord_less @ A @ X @ Z2 ) ) ) ) ).
% le_less_trans
thf(fact_82_antisym__conv2,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ~ ( ord_less @ A @ X @ Y ) )
= ( X = Y ) ) ) ) ).
% antisym_conv2
thf(fact_83_antisym__conv1,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ~ ( ord_less @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ X @ Y )
= ( X = Y ) ) ) ) ).
% antisym_conv1
thf(fact_84_less__imp__le,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ).
% less_imp_le
thf(fact_85_le__neq__trans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A4: A,B3: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( A4 != B3 )
=> ( ord_less @ A @ A4 @ B3 ) ) ) ) ).
% le_neq_trans
thf(fact_86_not__less,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ~ ( ord_less @ A @ X @ Y ) )
= ( ord_less_eq @ A @ Y @ X ) ) ) ).
% not_less
thf(fact_87_not__le,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ~ ( ord_less_eq @ A @ X @ Y ) )
= ( ord_less @ A @ Y @ X ) ) ) ).
% not_le
thf(fact_88_order__less__le__subst2,axiom,
! [A: $tType,C2: $tType] :
( ( ( order @ C2 @ ( type2 @ C2 ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A4: A,B3: A,F: A > C2,C: C2] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ( ord_less_eq @ C2 @ ( F @ B3 ) @ C )
=> ( ! [X3: A,Y2: A] :
( ( ord_less @ A @ X3 @ Y2 )
=> ( ord_less @ C2 @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less @ C2 @ ( F @ A4 ) @ C ) ) ) ) ) ).
% order_less_le_subst2
thf(fact_89_order__less__le__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B @ ( type2 @ B ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A4: A,F: B > A,B3: B,C: B] :
( ( ord_less @ A @ A4 @ ( F @ B3 ) )
=> ( ( ord_less_eq @ B @ B3 @ C )
=> ( ! [X3: B,Y2: B] :
( ( ord_less_eq @ B @ X3 @ Y2 )
=> ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less @ A @ A4 @ ( F @ C ) ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_90_order__le__less__subst2,axiom,
! [A: $tType,C2: $tType] :
( ( ( order @ C2 @ ( type2 @ C2 ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A4: A,B3: A,F: A > C2,C: C2] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( ord_less @ C2 @ ( F @ B3 ) @ C )
=> ( ! [X3: A,Y2: A] :
( ( ord_less_eq @ A @ X3 @ Y2 )
=> ( ord_less_eq @ C2 @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less @ C2 @ ( F @ A4 ) @ C ) ) ) ) ) ).
% order_le_less_subst2
thf(fact_91_order__le__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B @ ( type2 @ B ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A4: A,F: B > A,B3: B,C: B] :
( ( ord_less_eq @ A @ A4 @ ( F @ B3 ) )
=> ( ( ord_less @ B @ B3 @ C )
=> ( ! [X3: B,Y2: B] :
( ( ord_less @ B @ X3 @ Y2 )
=> ( ord_less @ A @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less @ A @ A4 @ ( F @ C ) ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_92_less__le,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less @ A )
= ( ^ [X2: A,Y4: A] :
( ( ord_less_eq @ A @ X2 @ Y4 )
& ( X2 != Y4 ) ) ) ) ) ).
% less_le
thf(fact_93_le__less,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less_eq @ A )
= ( ^ [X2: A,Y4: A] :
( ( ord_less @ A @ X2 @ Y4 )
| ( X2 = Y4 ) ) ) ) ) ).
% le_less
thf(fact_94_leI,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ~ ( ord_less @ A @ X @ Y )
=> ( ord_less_eq @ A @ Y @ X ) ) ) ).
% leI
thf(fact_95_leD,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ~ ( ord_less @ A @ X @ Y ) ) ) ).
% leD
thf(fact_96_lappend__LNil__LNil,axiom,
! [A: $tType] :
( ( coinductive_lappend @ A @ ( coinductive_LNil @ A ) @ ( coinductive_LNil @ A ) )
= ( coinductive_LNil @ A ) ) ).
% lappend_LNil_LNil
thf(fact_97_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_98_LList2__Mirabelle__hamjzmohle_Ollist__le__def,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( coinductive_llist @ A ) )
= ( ^ [S: coinductive_llist @ A,T: coinductive_llist @ A] :
? [D: coinductive_llist @ A] :
( T
= ( coinductive_lappend @ A @ S @ D ) ) ) ) ).
% LList2_Mirabelle_hamjzmohle.llist_le_def
thf(fact_99_finlstsp_OLCons__fin,axiom,
! [A: $tType,A2: A > $o,L: coinductive_llist @ A,A4: A] :
( ( lList2860480441nlstsp @ A @ A2 @ L )
=> ( ( A2 @ A4 )
=> ( lList2860480441nlstsp @ A @ A2 @ ( coinductive_LCons @ A @ A4 @ L ) ) ) ) ).
% finlstsp.LCons_fin
thf(fact_100_alllstsp_OLCons__all,axiom,
! [A: $tType,A2: A > $o,L: coinductive_llist @ A,A4: A] :
( ( lList21511617539llstsp @ A @ A2 @ L )
=> ( ( A2 @ A4 )
=> ( lList21511617539llstsp @ A @ A2 @ ( coinductive_LCons @ A @ A4 @ L ) ) ) ) ).
% alllstsp.LCons_all
thf(fact_101_finlstsp_OLNil__fin,axiom,
! [A: $tType,A2: A > $o] : ( lList2860480441nlstsp @ A @ A2 @ ( coinductive_LNil @ A ) ) ).
% finlstsp.LNil_fin
thf(fact_102_alllstsp_OLNil__all,axiom,
! [A: $tType,A2: A > $o] : ( lList21511617539llstsp @ A @ A2 @ ( coinductive_LNil @ A ) ) ).
% alllstsp.LNil_all
thf(fact_103_LList2__Mirabelle__hamjzmohle_Ollist__less__def,axiom,
! [A: $tType] :
( ( ord_less @ ( coinductive_llist @ A ) )
= ( ^ [S: coinductive_llist @ A,T: coinductive_llist @ A] :
( ( ord_less_eq @ ( coinductive_llist @ A ) @ S @ T )
& ( S != T ) ) ) ) ).
% LList2_Mirabelle_hamjzmohle.llist_less_def
thf(fact_104_dual__order_Oantisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B3: A,A4: A] :
( ( ord_less_eq @ A @ B3 @ A4 )
=> ( ( ord_less_eq @ A @ A4 @ B3 )
=> ( A4 = B3 ) ) ) ) ).
% dual_order.antisym
thf(fact_105_dual__order_Otrans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B3: A,A4: A,C: A] :
( ( ord_less_eq @ A @ B3 @ A4 )
=> ( ( ord_less_eq @ A @ C @ B3 )
=> ( ord_less_eq @ A @ C @ A4 ) ) ) ) ).
% dual_order.trans
thf(fact_106_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > A > $o,A4: A,B3: A] :
( ! [A3: A,B5: A] :
( ( ord_less_eq @ A @ A3 @ B5 )
=> ( P @ A3 @ B5 ) )
=> ( ! [A3: A,B5: A] :
( ( P @ B5 @ A3 )
=> ( P @ A3 @ B5 ) )
=> ( P @ A4 @ B3 ) ) ) ) ).
% linorder_wlog
thf(fact_107_dual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A4: A] : ( ord_less_eq @ A @ A4 @ A4 ) ) ).
% dual_order.refl
thf(fact_108_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z2: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ Z2 )
=> ( ord_less_eq @ A @ X @ Z2 ) ) ) ) ).
% order_trans
thf(fact_109_order__class_Oorder_Oantisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A4: A,B3: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ A4 )
=> ( A4 = B3 ) ) ) ) ).
% order_class.order.antisym
thf(fact_110_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A4: A,B3: A,C: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( B3 = C )
=> ( ord_less_eq @ A @ A4 @ C ) ) ) ) ).
% ord_le_eq_trans
thf(fact_111_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A4: A,B3: A,C: A] :
( ( A4 = B3 )
=> ( ( ord_less_eq @ A @ B3 @ C )
=> ( ord_less_eq @ A @ A4 @ C ) ) ) ) ).
% ord_eq_le_trans
thf(fact_112_antisym__conv,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( ord_less_eq @ A @ X @ Y )
= ( X = Y ) ) ) ) ).
% antisym_conv
thf(fact_113_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z2: A] :
( ( ( ord_less_eq @ A @ X @ Y )
=> ~ ( ord_less_eq @ A @ Y @ Z2 ) )
=> ( ( ( ord_less_eq @ A @ Y @ X )
=> ~ ( ord_less_eq @ A @ X @ Z2 ) )
=> ( ( ( ord_less_eq @ A @ X @ Z2 )
=> ~ ( ord_less_eq @ A @ Z2 @ Y ) )
=> ( ( ( ord_less_eq @ A @ Z2 @ Y )
=> ~ ( ord_less_eq @ A @ Y @ X ) )
=> ( ( ( ord_less_eq @ A @ Y @ Z2 )
=> ~ ( ord_less_eq @ A @ Z2 @ X ) )
=> ~ ( ( ord_less_eq @ A @ Z2 @ X )
=> ~ ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_114_order_Otrans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A4: A,B3: A,C: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ C )
=> ( ord_less_eq @ A @ A4 @ C ) ) ) ) ).
% order.trans
thf(fact_115_le__cases,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ~ ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ A @ Y @ X ) ) ) ).
% le_cases
thf(fact_116_wlog__linorder__le,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > A > $o,B3: A,A4: A] :
( ! [A3: A,B5: A] :
( ( ord_less_eq @ A @ A3 @ B5 )
=> ( P @ A3 @ B5 ) )
=> ( ( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A4 @ B3 ) ) ) ) ).
% wlog_linorder_le
thf(fact_117_eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( X = Y )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ).
% eq_refl
thf(fact_118_linear,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
| ( ord_less_eq @ A @ Y @ X ) ) ) ).
% linear
thf(fact_119_antisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ X )
=> ( X = Y ) ) ) ) ).
% antisym
thf(fact_120_eq__iff,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ^ [Y5: A,Z3: A] : Y5 = Z3 )
= ( ^ [X2: A,Y4: A] :
( ( ord_less_eq @ A @ X2 @ Y4 )
& ( ord_less_eq @ A @ Y4 @ X2 ) ) ) ) ) ).
% eq_iff
thf(fact_121_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A4: A,B3: A,F: A > B,C: B] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X3: A,Y2: A] :
( ( ord_less_eq @ A @ X3 @ Y2 )
=> ( ord_less_eq @ B @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq @ B @ ( F @ A4 ) @ C ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_122_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A4: A,F: B > A,B3: B,C: B] :
( ( A4
= ( F @ B3 ) )
=> ( ( ord_less_eq @ B @ B3 @ C )
=> ( ! [X3: B,Y2: B] :
( ( ord_less_eq @ B @ X3 @ Y2 )
=> ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq @ A @ A4 @ ( F @ C ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_123_order__subst2,axiom,
! [A: $tType,C2: $tType] :
( ( ( order @ C2 @ ( type2 @ C2 ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A4: A,B3: A,F: A > C2,C: C2] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( ord_less_eq @ C2 @ ( F @ B3 ) @ C )
=> ( ! [X3: A,Y2: A] :
( ( ord_less_eq @ A @ X3 @ Y2 )
=> ( ord_less_eq @ C2 @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq @ C2 @ ( F @ A4 ) @ C ) ) ) ) ) ).
% order_subst2
thf(fact_124_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B @ ( type2 @ B ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A4: A,F: B > A,B3: B,C: B] :
( ( ord_less_eq @ A @ A4 @ ( F @ B3 ) )
=> ( ( ord_less_eq @ B @ B3 @ C )
=> ( ! [X3: B,Y2: B] :
( ( ord_less_eq @ B @ X3 @ Y2 )
=> ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq @ A @ A4 @ ( F @ C ) ) ) ) ) ) ).
% order_subst1
thf(fact_125_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ( ( ord_less_eq @ ( A > B ) )
= ( ^ [F2: A > B,G2: A > B] :
! [X2: A] : ( ord_less_eq @ B @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ).
% le_fun_def
thf(fact_126_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_127_le__funE,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [F: A > B,G: A > B,X: A] :
( ( ord_less_eq @ ( A > B ) @ F @ G )
=> ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% le_funE
thf(fact_128_le__funD,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [F: A > B,G: A > B,X: A] :
( ( ord_less_eq @ ( A > B ) @ F @ G )
=> ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% le_funD
thf(fact_129_alllstsp_Ocoinduct,axiom,
! [A: $tType,X4: ( coinductive_llist @ A ) > $o,X: coinductive_llist @ A,A2: A > $o] :
( ( X4 @ X )
=> ( ! [X3: coinductive_llist @ A] :
( ( X4 @ X3 )
=> ( ( X3
= ( coinductive_LNil @ A ) )
| ? [L4: coinductive_llist @ A,A7: A] :
( ( X3
= ( coinductive_LCons @ A @ A7 @ L4 ) )
& ( ( X4 @ L4 )
| ( lList21511617539llstsp @ A @ A2 @ L4 ) )
& ( A2 @ A7 ) ) ) )
=> ( lList21511617539llstsp @ A @ A2 @ X ) ) ) ).
% alllstsp.coinduct
thf(fact_130_finlstsp_Oinducts,axiom,
! [A: $tType,A2: A > $o,X: coinductive_llist @ A,P: ( coinductive_llist @ A ) > $o] :
( ( lList2860480441nlstsp @ A @ A2 @ X )
=> ( ( P @ ( coinductive_LNil @ A ) )
=> ( ! [L2: coinductive_llist @ A,A3: A] :
( ( lList2860480441nlstsp @ A @ A2 @ L2 )
=> ( ( P @ L2 )
=> ( ( A2 @ A3 )
=> ( P @ ( coinductive_LCons @ A @ A3 @ L2 ) ) ) ) )
=> ( P @ X ) ) ) ) ).
% finlstsp.inducts
thf(fact_131_lbutlast__lapp__llast,axiom,
! [A: $tType,L: coinductive_llist @ A,A2: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ L @ ( lList22096119349pslsts @ A @ A2 ) )
=> ( L
= ( coinductive_lappend @ A @ ( lList2370560421utlast @ A @ L ) @ ( coinductive_LCons @ A @ ( lList2170638824_llast @ A @ L ) @ ( coinductive_LNil @ A ) ) ) ) ) ).
% lbutlast_lapp_llast
thf(fact_132_complete__interval,axiom,
! [A: $tType] :
( ( condit1037483654norder @ A @ ( type2 @ A ) )
=> ! [A4: A,B3: A,P: A > $o] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ( P @ A4 )
=> ( ~ ( P @ B3 )
=> ? [C3: A] :
( ( ord_less_eq @ A @ A4 @ C3 )
& ( ord_less_eq @ A @ C3 @ B3 )
& ! [X5: A] :
( ( ( ord_less_eq @ A @ A4 @ X5 )
& ( ord_less @ A @ X5 @ C3 ) )
=> ( P @ X5 ) )
& ! [D2: A] :
( ! [X3: A] :
( ( ( ord_less_eq @ A @ A4 @ X3 )
& ( ord_less @ A @ X3 @ D2 ) )
=> ( P @ X3 ) )
=> ( ord_less_eq @ A @ D2 @ C3 ) ) ) ) ) ) ) ).
% complete_interval
thf(fact_133_pinf_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T2: A] :
? [Z: A] :
! [X5: A] :
( ( ord_less @ A @ Z @ X5 )
=> ~ ( ord_less_eq @ A @ X5 @ T2 ) ) ) ).
% pinf(6)
thf(fact_134_pinf_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T2: A] :
? [Z: A] :
! [X5: A] :
( ( ord_less @ A @ Z @ X5 )
=> ( ord_less_eq @ A @ T2 @ X5 ) ) ) ).
% pinf(8)
thf(fact_135_minf_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T2: A] :
? [Z: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z )
=> ( ord_less_eq @ A @ X5 @ T2 ) ) ) ).
% minf(6)
thf(fact_136_minf_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T2: A] :
? [Z: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z )
=> ~ ( ord_less_eq @ A @ T2 @ X5 ) ) ) ).
% minf(8)
thf(fact_137_lbutlast__snoc,axiom,
! [A: $tType,Xs: coinductive_llist @ A,A2: set @ A,X: A] :
( ( member @ ( coinductive_llist @ A ) @ Xs @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( lList2370560421utlast @ A @ ( coinductive_lappend @ A @ Xs @ ( coinductive_LCons @ A @ X @ ( coinductive_LNil @ A ) ) ) )
= Xs ) ) ).
% lbutlast_snoc
thf(fact_138_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_139_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_140_lstrict__prefix__code_I3_J,axiom,
! [B: $tType,X: B,Xs: coinductive_llist @ B] :
~ ( coindu1478340336prefix @ B @ ( coinductive_LCons @ B @ X @ Xs ) @ ( coinductive_LNil @ B ) ) ).
% lstrict_prefix_code(3)
thf(fact_141_lbutlast__LCons,axiom,
! [A: $tType,R: coinductive_llist @ A,A2: set @ A,A4: A] :
( ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( ( R
= ( coinductive_LNil @ A ) )
=> ( ( lList2370560421utlast @ A @ ( coinductive_LCons @ A @ A4 @ R ) )
= ( coinductive_LNil @ A ) ) )
& ( ( R
!= ( coinductive_LNil @ A ) )
=> ( ( lList2370560421utlast @ A @ ( coinductive_LCons @ A @ A4 @ R ) )
= ( coinductive_LCons @ A @ A4 @ ( lList2370560421utlast @ A @ R ) ) ) ) ) ) ).
% lbutlast_LCons
thf(fact_142_predicate1I,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X3: A] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq @ ( A > $o ) @ P @ Q ) ) ).
% predicate1I
thf(fact_143_lapp__fin__fin__iff,axiom,
! [A: $tType,R: coinductive_llist @ A,S2: coinductive_llist @ A,A2: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ ( coinductive_lappend @ A @ R @ S2 ) @ ( lList2236698231inlsts @ A @ A2 ) )
= ( ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ A2 ) )
& ( member @ ( coinductive_llist @ A ) @ S2 @ ( lList2236698231inlsts @ A @ A2 ) ) ) ) ).
% lapp_fin_fin_iff
thf(fact_144_same__lappend__eq,axiom,
! [A: $tType,R: coinductive_llist @ A,A2: set @ A,S2: coinductive_llist @ A,T2: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( ( coinductive_lappend @ A @ R @ S2 )
= ( coinductive_lappend @ A @ R @ T2 ) )
= ( S2 = T2 ) ) ) ).
% same_lappend_eq
thf(fact_145_lstrict__prefix__code_I4_J,axiom,
! [B: $tType,X: B,Xs: coinductive_llist @ B,Y: B,Ys: coinductive_llist @ B] :
( ( coindu1478340336prefix @ B @ ( coinductive_LCons @ B @ X @ Xs ) @ ( coinductive_LCons @ B @ Y @ Ys ) )
= ( ( X = Y )
& ( coindu1478340336prefix @ B @ Xs @ Ys ) ) ) ).
% lstrict_prefix_code(4)
thf(fact_146_lstrict__prefix__code_I1_J,axiom,
! [A: $tType] :
~ ( coindu1478340336prefix @ A @ ( coinductive_LNil @ A ) @ ( coinductive_LNil @ A ) ) ).
% lstrict_prefix_code(1)
thf(fact_147_fpslsts__iff,axiom,
! [A: $tType,S2: coinductive_llist @ A,A2: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ S2 @ ( lList22096119349pslsts @ A @ A2 ) )
= ( ( member @ ( coinductive_llist @ A ) @ S2 @ ( lList2236698231inlsts @ A @ A2 ) )
& ( S2
!= ( coinductive_LNil @ A ) ) ) ) ).
% fpslsts_iff
thf(fact_148_LList2__Mirabelle__hamjzmohle_Ollast__LCons,axiom,
! [B: $tType,R: coinductive_llist @ B,A2: set @ B,A4: B] :
( ( member @ ( coinductive_llist @ B ) @ R @ ( lList2236698231inlsts @ B @ A2 ) )
=> ( ( ( R
= ( coinductive_LNil @ B ) )
=> ( ( lList2170638824_llast @ B @ ( coinductive_LCons @ B @ A4 @ R ) )
= A4 ) )
& ( ( R
!= ( coinductive_LNil @ B ) )
=> ( ( lList2170638824_llast @ B @ ( coinductive_LCons @ B @ A4 @ R ) )
= ( lList2170638824_llast @ B @ R ) ) ) ) ) ).
% LList2_Mirabelle_hamjzmohle.llast_LCons
thf(fact_149_llast__snoc,axiom,
! [A: $tType,Xs: coinductive_llist @ A,A2: set @ A,X: A] :
( ( member @ ( coinductive_llist @ A ) @ Xs @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( lList2170638824_llast @ A @ ( coinductive_lappend @ A @ Xs @ ( coinductive_LCons @ A @ X @ ( coinductive_LNil @ A ) ) ) )
= X ) ) ).
% llast_snoc
thf(fact_150_predicate1D,axiom,
! [A: $tType,P: A > $o,Q: A > $o,X: A] :
( ( ord_less_eq @ ( A > $o ) @ P @ Q )
=> ( ( P @ X )
=> ( Q @ X ) ) ) ).
% predicate1D
thf(fact_151_less__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ( ( ord_less @ ( A > B ) )
= ( ^ [F2: A > B,G2: A > B] :
( ( ord_less_eq @ ( A > B ) @ F2 @ G2 )
& ~ ( ord_less_eq @ ( A > B ) @ G2 @ F2 ) ) ) ) ) ).
% less_fun_def
thf(fact_152_rev__predicate1D,axiom,
! [A: $tType,P: A > $o,X: A,Q: A > $o] :
( ( P @ X )
=> ( ( ord_less_eq @ ( A > $o ) @ P @ Q )
=> ( Q @ X ) ) ) ).
% rev_predicate1D
thf(fact_153_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_154_finlsts__mono,axiom,
! [A: $tType,A2: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B4 )
=> ( ord_less_eq @ ( set @ ( coinductive_llist @ A ) ) @ ( lList2236698231inlsts @ A @ A2 ) @ ( lList2236698231inlsts @ A @ B4 ) ) ) ).
% finlsts_mono
thf(fact_155_finlsts_OLCons__fin,axiom,
! [A: $tType,L: coinductive_llist @ A,A2: set @ A,A4: A] :
( ( member @ ( coinductive_llist @ A ) @ L @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( member @ A @ A4 @ A2 )
=> ( member @ ( coinductive_llist @ A ) @ ( coinductive_LCons @ A @ A4 @ L ) @ ( lList2236698231inlsts @ A @ A2 ) ) ) ) ).
% finlsts.LCons_fin
thf(fact_156_finlsts_OLNil__fin,axiom,
! [A: $tType,A2: set @ A] : ( member @ ( coinductive_llist @ A ) @ ( coinductive_LNil @ A ) @ ( lList2236698231inlsts @ A @ A2 ) ) ).
% finlsts.LNil_fin
thf(fact_157_lapp__fin__fin__lemma,axiom,
! [A: $tType,R: coinductive_llist @ A,S2: coinductive_llist @ A,A2: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ ( coinductive_lappend @ A @ R @ S2 ) @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ A2 ) ) ) ).
% lapp_fin_fin_lemma
thf(fact_158_lappfin__finT,axiom,
! [A: $tType,S2: coinductive_llist @ A,A2: set @ A,T2: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ S2 @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( member @ ( coinductive_llist @ A ) @ T2 @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( member @ ( coinductive_llist @ A ) @ ( coinductive_lappend @ A @ S2 @ T2 ) @ ( lList2236698231inlsts @ A @ A2 ) ) ) ) ).
% lappfin_finT
thf(fact_159_finlsts_Ocases,axiom,
! [A: $tType,A4: coinductive_llist @ A,A2: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ A4 @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( A4
!= ( coinductive_LNil @ A ) )
=> ~ ! [L2: coinductive_llist @ A,A3: A] :
( ( A4
= ( coinductive_LCons @ A @ A3 @ L2 ) )
=> ( ( member @ ( coinductive_llist @ A ) @ L2 @ ( lList2236698231inlsts @ A @ A2 ) )
=> ~ ( member @ A @ A3 @ A2 ) ) ) ) ) ).
% finlsts.cases
thf(fact_160_finlsts_Osimps,axiom,
! [A: $tType,A4: coinductive_llist @ A,A2: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ A4 @ ( lList2236698231inlsts @ A @ A2 ) )
= ( ( A4
= ( coinductive_LNil @ A ) )
| ? [L3: coinductive_llist @ A,A6: A] :
( ( A4
= ( coinductive_LCons @ A @ A6 @ L3 ) )
& ( member @ ( coinductive_llist @ A ) @ L3 @ ( lList2236698231inlsts @ A @ A2 ) )
& ( member @ A @ A6 @ A2 ) ) ) ) ).
% finlsts.simps
thf(fact_161_finlsts__induct,axiom,
! [A: $tType,X: coinductive_llist @ A,A2: set @ A,P: ( coinductive_llist @ A ) > $o] :
( ( member @ ( coinductive_llist @ A ) @ X @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ! [L2: coinductive_llist @ A] :
( ( L2
= ( coinductive_LNil @ A ) )
=> ( P @ L2 ) )
=> ( ! [A3: A,L2: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ L2 @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( P @ L2 )
=> ( ( member @ A @ A3 @ A2 )
=> ( P @ ( coinductive_LCons @ A @ A3 @ L2 ) ) ) ) )
=> ( P @ X ) ) ) ) ).
% finlsts_induct
thf(fact_162_finlsts_Oinducts,axiom,
! [A: $tType,X: coinductive_llist @ A,A2: set @ A,P: ( coinductive_llist @ A ) > $o] :
( ( member @ ( coinductive_llist @ A ) @ X @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( P @ ( coinductive_LNil @ A ) )
=> ( ! [L2: coinductive_llist @ A,A3: A] :
( ( member @ ( coinductive_llist @ A ) @ L2 @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( P @ L2 )
=> ( ( member @ A @ A3 @ A2 )
=> ( P @ ( coinductive_LCons @ A @ A3 @ L2 ) ) ) ) )
=> ( P @ X ) ) ) ) ).
% finlsts.inducts
thf(fact_163_fpslsts__cases,axiom,
! [A: $tType,R: coinductive_llist @ A,A2: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ R @ ( lList22096119349pslsts @ A @ A2 ) )
=> ~ ! [A3: A,Rs: coinductive_llist @ A] :
( ( R
= ( coinductive_LCons @ A @ A3 @ Rs ) )
=> ( ( member @ A @ A3 @ A2 )
=> ~ ( member @ ( coinductive_llist @ A ) @ Rs @ ( lList2236698231inlsts @ A @ A2 ) ) ) ) ) ).
% fpslsts_cases
thf(fact_164_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_165_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_166_lrev__induct,axiom,
! [A: $tType,Xs: coinductive_llist @ A,A2: set @ A,P: ( coinductive_llist @ A ) > $o] :
( ( member @ ( coinductive_llist @ A ) @ Xs @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( P @ ( coinductive_LNil @ A ) )
=> ( ! [X3: A,Xs3: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ Xs3 @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( P @ Xs3 )
=> ( ( member @ A @ X3 @ A2 )
=> ( P @ ( coinductive_lappend @ A @ Xs3 @ ( coinductive_LCons @ A @ X3 @ ( coinductive_LNil @ A ) ) ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% lrev_induct
thf(fact_167_finlsts__rev__cases,axiom,
! [A: $tType,T2: coinductive_llist @ A,A2: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ T2 @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( T2
!= ( coinductive_LNil @ A ) )
=> ~ ! [A3: A,L2: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ L2 @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( member @ A @ A3 @ A2 )
=> ( T2
!= ( coinductive_lappend @ A @ L2 @ ( coinductive_LCons @ A @ A3 @ ( coinductive_LNil @ A ) ) ) ) ) ) ) ) ).
% finlsts_rev_cases
thf(fact_168_ex__gt__or__lt,axiom,
! [A: $tType] :
( ( condit1656338222tinuum @ A @ ( type2 @ A ) )
=> ! [A4: A] :
? [B5: A] :
( ( ord_less @ A @ A4 @ B5 )
| ( ord_less @ A @ B5 @ A4 ) ) ) ).
% ex_gt_or_lt
thf(fact_169_pinf_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z4: A] :
! [X3: A] :
( ( ord_less @ A @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P2 @ X3 ) ) )
=> ( ? [Z4: A] :
! [X3: A] :
( ( ord_less @ A @ Z4 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z: A] :
! [X5: A] :
( ( ord_less @ A @ Z @ X5 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P2 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ) ).
% pinf(1)
thf(fact_170_pinf_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z4: A] :
! [X3: A] :
( ( ord_less @ A @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P2 @ X3 ) ) )
=> ( ? [Z4: A] :
! [X3: A] :
( ( ord_less @ A @ Z4 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z: A] :
! [X5: A] :
( ( ord_less @ A @ Z @ X5 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P2 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ) ).
% pinf(2)
thf(fact_171_pinf_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T2: A] :
? [Z: A] :
! [X5: A] :
( ( ord_less @ A @ Z @ X5 )
=> ( X5 != T2 ) ) ) ).
% pinf(3)
thf(fact_172_pinf_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T2: A] :
? [Z: A] :
! [X5: A] :
( ( ord_less @ A @ Z @ X5 )
=> ( X5 != T2 ) ) ) ).
% pinf(4)
thf(fact_173_pinf_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T2: A] :
? [Z: A] :
! [X5: A] :
( ( ord_less @ A @ Z @ X5 )
=> ~ ( ord_less @ A @ X5 @ T2 ) ) ) ).
% pinf(5)
thf(fact_174_pinf_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T2: A] :
? [Z: A] :
! [X5: A] :
( ( ord_less @ A @ Z @ X5 )
=> ( ord_less @ A @ T2 @ X5 ) ) ) ).
% pinf(7)
thf(fact_175_pinf_I11_J,axiom,
! [C2: $tType,D3: $tType] :
( ( ord @ C2 @ ( type2 @ C2 ) )
=> ! [F3: D3] :
? [Z: C2] :
! [X5: C2] :
( ( ord_less @ C2 @ Z @ X5 )
=> ( F3 = F3 ) ) ) ).
% pinf(11)
thf(fact_176_minf_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z4: A] :
! [X3: A] :
( ( ord_less @ A @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P2 @ X3 ) ) )
=> ( ? [Z4: A] :
! [X3: A] :
( ( ord_less @ A @ X3 @ Z4 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P2 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ) ).
% minf(1)
thf(fact_177_minf_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z4: A] :
! [X3: A] :
( ( ord_less @ A @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P2 @ X3 ) ) )
=> ( ? [Z4: A] :
! [X3: A] :
( ( ord_less @ A @ X3 @ Z4 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P2 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ) ).
% minf(2)
thf(fact_178_minf_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T2: A] :
? [Z: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z )
=> ( X5 != T2 ) ) ) ).
% minf(3)
thf(fact_179_minf_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T2: A] :
? [Z: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z )
=> ( X5 != T2 ) ) ) ).
% minf(4)
thf(fact_180_minf_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T2: A] :
? [Z: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z )
=> ( ord_less @ A @ X5 @ T2 ) ) ) ).
% minf(5)
thf(fact_181_minf_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T2: A] :
? [Z: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z )
=> ~ ( ord_less @ A @ T2 @ X5 ) ) ) ).
% minf(7)
thf(fact_182_minf_I11_J,axiom,
! [C2: $tType,D3: $tType] :
( ( ord @ C2 @ ( type2 @ C2 ) )
=> ! [F3: D3] :
? [Z: C2] :
! [X5: C2] :
( ( ord_less @ C2 @ X5 @ Z )
=> ( F3 = F3 ) ) ) ).
% minf(11)
thf(fact_183_lrev__LCons,axiom,
! [A: $tType,R: coinductive_llist @ A,A2: set @ A,A4: A] :
( ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( lList2281150353e_lrev @ A @ ( coinductive_LCons @ A @ A4 @ R ) )
= ( coinductive_lappend @ A @ ( lList2281150353e_lrev @ A @ R ) @ ( coinductive_LCons @ A @ A4 @ ( coinductive_LNil @ A ) ) ) ) ) ).
% lrev_LCons
thf(fact_184_subsetI,axiom,
! [A: $tType,A2: set @ A,B4: set @ A] :
( ! [X3: A] :
( ( member @ A @ X3 @ A2 )
=> ( member @ A @ X3 @ B4 ) )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ B4 ) ) ).
% subsetI
thf(fact_185_psubsetI,axiom,
! [A: $tType,A2: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B4 )
=> ( ( A2 != B4 )
=> ( ord_less @ ( set @ A ) @ A2 @ B4 ) ) ) ).
% psubsetI
thf(fact_186_subset__antisym,axiom,
! [A: $tType,A2: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ A2 )
=> ( A2 = B4 ) ) ) ).
% subset_antisym
thf(fact_187_lrev__LNil,axiom,
! [A: $tType] :
( ( lList2281150353e_lrev @ A @ ( coinductive_LNil @ A ) )
= ( coinductive_LNil @ A ) ) ).
% lrev_LNil
thf(fact_188_lrevT,axiom,
! [A: $tType,Xs: coinductive_llist @ A,A2: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ Xs @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( member @ ( coinductive_llist @ A ) @ ( lList2281150353e_lrev @ A @ Xs ) @ ( lList2236698231inlsts @ A @ A2 ) ) ) ).
% lrevT
thf(fact_189_psubsetD,axiom,
! [A: $tType,A2: set @ A,B4: set @ A,C: A] :
( ( ord_less @ ( set @ A ) @ A2 @ B4 )
=> ( ( member @ A @ C @ A2 )
=> ( member @ A @ C @ B4 ) ) ) ).
% psubsetD
thf(fact_190_psubset__trans,axiom,
! [A: $tType,A2: set @ A,B4: set @ A,C4: set @ A] :
( ( ord_less @ ( set @ A ) @ A2 @ B4 )
=> ( ( ord_less @ ( set @ A ) @ B4 @ C4 )
=> ( ord_less @ ( set @ A ) @ A2 @ C4 ) ) ) ).
% psubset_trans
thf(fact_191_subset__iff__psubset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A5: set @ A,B6: set @ A] :
( ( ord_less @ ( set @ A ) @ A5 @ B6 )
| ( A5 = B6 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_192_subset__psubset__trans,axiom,
! [A: $tType,A2: set @ A,B4: set @ A,C4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B4 )
=> ( ( ord_less @ ( set @ A ) @ B4 @ C4 )
=> ( ord_less @ ( set @ A ) @ A2 @ C4 ) ) ) ).
% subset_psubset_trans
thf(fact_193_subset__not__subset__eq,axiom,
! [A: $tType] :
( ( ord_less @ ( set @ A ) )
= ( ^ [A5: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B6 )
& ~ ( ord_less_eq @ ( set @ A ) @ B6 @ A5 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_194_psubset__subset__trans,axiom,
! [A: $tType,A2: set @ A,B4: set @ A,C4: set @ A] :
( ( ord_less @ ( set @ A ) @ A2 @ B4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ C4 )
=> ( ord_less @ ( set @ A ) @ A2 @ C4 ) ) ) ).
% psubset_subset_trans
thf(fact_195_psubset__imp__subset,axiom,
! [A: $tType,A2: set @ A,B4: set @ A] :
( ( ord_less @ ( set @ A ) @ A2 @ B4 )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ B4 ) ) ).
% psubset_imp_subset
thf(fact_196_Collect__mono__iff,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
= ( ! [X2: A] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_197_contra__subsetD,axiom,
! [A: $tType,A2: set @ A,B4: set @ A,C: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B4 )
=> ( ~ ( member @ A @ C @ B4 )
=> ~ ( member @ A @ C @ A2 ) ) ) ).
% contra_subsetD
thf(fact_198_set__eq__subset,axiom,
! [A: $tType] :
( ( ^ [Y5: set @ A,Z3: set @ A] : Y5 = Z3 )
= ( ^ [A5: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B6 )
& ( ord_less_eq @ ( set @ A ) @ B6 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_199_subset__trans,axiom,
! [A: $tType,A2: set @ A,B4: set @ A,C4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ C4 )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ C4 ) ) ) ).
% subset_trans
thf(fact_200_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_201_subset__refl,axiom,
! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ A2 ) ).
% subset_refl
thf(fact_202_rev__subsetD,axiom,
! [A: $tType,C: A,A2: set @ A,B4: set @ A] :
( ( member @ A @ C @ A2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A2 @ B4 )
=> ( member @ A @ C @ B4 ) ) ) ).
% rev_subsetD
thf(fact_203_subset__iff,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A5: set @ A,B6: set @ A] :
! [T: A] :
( ( member @ A @ T @ A5 )
=> ( member @ A @ T @ B6 ) ) ) ) ).
% subset_iff
thf(fact_204_set__rev__mp,axiom,
! [A: $tType,X: A,A2: set @ A,B4: set @ A] :
( ( member @ A @ X @ A2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A2 @ B4 )
=> ( member @ A @ X @ B4 ) ) ) ).
% set_rev_mp
thf(fact_205_psubset__eq,axiom,
! [A: $tType] :
( ( ord_less @ ( set @ A ) )
= ( ^ [A5: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B6 )
& ( A5 != B6 ) ) ) ) ).
% psubset_eq
thf(fact_206_equalityD2,axiom,
! [A: $tType,A2: set @ A,B4: set @ A] :
( ( A2 = B4 )
=> ( ord_less_eq @ ( set @ A ) @ B4 @ A2 ) ) ).
% equalityD2
thf(fact_207_equalityD1,axiom,
! [A: $tType,A2: set @ A,B4: set @ A] :
( ( A2 = B4 )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ B4 ) ) ).
% equalityD1
thf(fact_208_subset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A5: set @ A,B6: set @ A] :
! [X2: A] :
( ( member @ A @ X2 @ A5 )
=> ( member @ A @ X2 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_209_equalityE,axiom,
! [A: $tType,A2: set @ A,B4: set @ A] :
( ( A2 = B4 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A2 @ B4 )
=> ~ ( ord_less_eq @ ( set @ A ) @ B4 @ A2 ) ) ) ).
% equalityE
thf(fact_210_subsetCE,axiom,
! [A: $tType,A2: set @ A,B4: set @ A,C: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B4 )
=> ( ( member @ A @ C @ A2 )
=> ( member @ A @ C @ B4 ) ) ) ).
% subsetCE
thf(fact_211_psubsetE,axiom,
! [A: $tType,A2: set @ A,B4: set @ A] :
( ( ord_less @ ( set @ A ) @ A2 @ B4 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A2 @ B4 )
=> ( ord_less_eq @ ( set @ A ) @ B4 @ A2 ) ) ) ).
% psubsetE
thf(fact_212_subsetD,axiom,
! [A: $tType,A2: set @ A,B4: set @ A,C: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B4 )
=> ( ( member @ A @ C @ A2 )
=> ( member @ A @ C @ B4 ) ) ) ).
% subsetD
thf(fact_213_in__mono,axiom,
! [A: $tType,A2: set @ A,B4: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B4 )
=> ( ( member @ A @ X @ A2 )
=> ( member @ A @ X @ B4 ) ) ) ).
% in_mono
thf(fact_214_set__mp,axiom,
! [A: $tType,A2: set @ A,B4: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B4 )
=> ( ( member @ A @ X @ A2 )
=> ( member @ A @ X @ B4 ) ) ) ).
% set_mp
thf(fact_215_LList2__Mirabelle__hamjzmohle_Ollast__lappend,axiom,
! [A: $tType,X: coinductive_llist @ A,Y: coinductive_llist @ A,A4: A] :
( ( member @ ( coinductive_llist @ A ) @ X @ ( lList2236698231inlsts @ A @ ( top_top @ ( set @ A ) ) ) )
=> ( ( member @ ( coinductive_llist @ A ) @ Y @ ( lList2236698231inlsts @ A @ ( top_top @ ( set @ A ) ) ) )
=> ( ( lList2170638824_llast @ A @ ( coinductive_lappend @ A @ X @ ( coinductive_LCons @ A @ A4 @ Y ) ) )
= ( lList2170638824_llast @ A @ ( coinductive_LCons @ A @ A4 @ Y ) ) ) ) ) ).
% LList2_Mirabelle_hamjzmohle.llast_lappend
thf(fact_216_poslsts__mono,axiom,
! [A: $tType,A2: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B4 )
=> ( ord_less_eq @ ( set @ ( coinductive_llist @ A ) ) @ ( lList21148268032oslsts @ A @ A2 ) @ ( lList21148268032oslsts @ A @ B4 ) ) ) ).
% poslsts_mono
thf(fact_217_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_218_top__apply,axiom,
! [C2: $tType,D3: $tType] :
( ( top @ C2 @ ( type2 @ C2 ) )
=> ( ( top_top @ ( D3 > C2 ) )
= ( ^ [X2: D3] : ( top_top @ C2 ) ) ) ) ).
% top_apply
thf(fact_219_UNIV__I,axiom,
! [A: $tType,X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ).
% UNIV_I
thf(fact_220_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_221_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_222_poslsts__UNIV,axiom,
! [A: $tType,S2: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ S2 @ ( lList21148268032oslsts @ A @ ( top_top @ ( set @ A ) ) ) )
= ( S2
!= ( coinductive_LNil @ A ) ) ) ).
% poslsts_UNIV
thf(fact_223_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_224_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_225_subset__UNIV,axiom,
! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ ( top_top @ ( set @ A ) ) ) ).
% subset_UNIV
thf(fact_226_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_227_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_228_UNIV__eq__I,axiom,
! [A: $tType,A2: set @ A] :
( ! [X3: A] : ( member @ A @ X3 @ A2 )
=> ( ( top_top @ ( set @ A ) )
= A2 ) ) ).
% UNIV_eq_I
thf(fact_229_UNIV__witness,axiom,
! [A: $tType] :
? [X3: A] : ( member @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ).
% UNIV_witness
thf(fact_230_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_231_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_232_top__greatest,axiom,
! [A: $tType] :
( ( order_top @ A @ ( type2 @ A ) )
=> ! [A4: A] : ( ord_less_eq @ A @ A4 @ ( top_top @ A ) ) ) ).
% top_greatest
thf(fact_233_finT__simp,axiom,
! [A: $tType,R: coinductive_llist @ A,A2: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% finT_simp
thf(fact_234_fin__finite,axiom,
! [A: $tType,R: coinductive_llist @ A,A2: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% fin_finite
thf(fact_235_iso__tuple__UNIV__I,axiom,
! [A: $tType,X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ).
% iso_tuple_UNIV_I
thf(fact_236_top_Oordering__top__axioms,axiom,
! [A: $tType] :
( ( order_top @ A @ ( type2 @ A ) )
=> ( ordering_top @ A @ ( ord_less_eq @ A ) @ ( ord_less @ A ) @ ( top_top @ A ) ) ) ).
% top.ordering_top_axioms
thf(fact_237_top__set__def,axiom,
! [A: $tType] :
( ( top_top @ ( set @ A ) )
= ( collect @ A @ ( top_top @ ( A > $o ) ) ) ) ).
% top_set_def
thf(fact_238_ordering__top_Oextremum__uniqueI,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,Top: A,A4: A] :
( ( ordering_top @ A @ Less_eq @ Less @ Top )
=> ( ( Less_eq @ Top @ A4 )
=> ( A4 = Top ) ) ) ).
% ordering_top.extremum_uniqueI
thf(fact_239_ordering__top_Onot__eq__extremum,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,Top: A,A4: A] :
( ( ordering_top @ A @ Less_eq @ Less @ Top )
=> ( ( A4 != Top )
= ( Less @ A4 @ Top ) ) ) ).
% ordering_top.not_eq_extremum
thf(fact_240_ordering__top_Oextremum__unique,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,Top: A,A4: A] :
( ( ordering_top @ A @ Less_eq @ Less @ Top )
=> ( ( Less_eq @ Top @ A4 )
= ( A4 = Top ) ) ) ).
% ordering_top.extremum_unique
thf(fact_241_ordering__top_Oextremum__strict,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,Top: A,A4: A] :
( ( ordering_top @ A @ Less_eq @ Less @ Top )
=> ~ ( Less @ Top @ A4 ) ) ).
% ordering_top.extremum_strict
thf(fact_242_ordering__top_Oextremum,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,Top: A,A4: A] :
( ( ordering_top @ A @ Less_eq @ Less @ Top )
=> ( Less_eq @ A4 @ Top ) ) ).
% ordering_top.extremum
thf(fact_243_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_244_finlsts__rec__LCons__def,axiom,
! [B: $tType,A: $tType,F: ( coinductive_llist @ A ) > B,C: B,D4: A > ( coinductive_llist @ A ) > B > B,R: coinductive_llist @ A,A2: set @ A,A4: A] :
( ( F
= ( lList21916056377ts_rec @ B @ A @ C @ D4 ) )
=> ( ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( F @ ( coinductive_LCons @ A @ A4 @ R ) )
= ( D4 @ A4 @ R @ ( F @ R ) ) ) ) ) ).
% finlsts_rec_LCons_def
thf(fact_245_top1I,axiom,
! [A: $tType,X: A] : ( top_top @ ( A > $o ) @ X ) ).
% top1I
thf(fact_246_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_247_finlsts__rec__LNil__def,axiom,
! [A: $tType,B: $tType,F: ( coinductive_llist @ A ) > B,C: B,D4: A > ( coinductive_llist @ A ) > B > B] :
( ( F
= ( lList21916056377ts_rec @ B @ A @ C @ D4 ) )
=> ( ( F @ ( coinductive_LNil @ A ) )
= C ) ) ).
% finlsts_rec_LNil_def
thf(fact_248_finlsts__rec__LNil,axiom,
! [B: $tType,A: $tType,C: A,D4: B > ( coinductive_llist @ B ) > A > A] :
( ( lList21916056377ts_rec @ A @ B @ C @ D4 @ ( coinductive_LNil @ B ) )
= C ) ).
% finlsts_rec_LNil
thf(fact_249_finlsts__rec__LCons,axiom,
! [B: $tType,A: $tType,R: coinductive_llist @ A,A2: set @ A,C: B,D4: A > ( coinductive_llist @ A ) > B > B,A4: A] :
( ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( lList21916056377ts_rec @ B @ A @ C @ D4 @ ( coinductive_LCons @ A @ A4 @ R ) )
= ( D4 @ A4 @ R @ ( lList21916056377ts_rec @ B @ A @ C @ D4 @ R ) ) ) ) ).
% finlsts_rec_LCons
thf(fact_250_ltake__lappend__llength,axiom,
! [A: $tType,R: coinductive_llist @ A,A2: set @ A,S2: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( lList22119844313_ltake @ A @ ( coinductive_lappend @ A @ R @ S2 ) @ ( lList21232602520length @ A @ R ) )
= R ) ) ).
% ltake_lappend_llength
thf(fact_251_top__empty__eq,axiom,
! [A: $tType] :
( ( top_top @ ( A > $o ) )
= ( ^ [X2: A] : ( member @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ) ) ).
% top_empty_eq
thf(fact_252_llength__drop__take,axiom,
! [A: $tType,T2: coinductive_llist @ A,I: nat] :
( ( ( lList2508575361_ldrop @ A @ T2 @ I )
!= ( coinductive_LNil @ A ) )
=> ( ( lList21232602520length @ A @ ( lList22119844313_ltake @ A @ T2 @ I ) )
= I ) ) ).
% llength_drop_take
thf(fact_253_lapp__suff__llength,axiom,
! [A: $tType,R: coinductive_llist @ A,A2: set @ A,S2: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( lList2508575361_ldrop @ A @ ( coinductive_lappend @ A @ R @ S2 ) @ ( lList21232602520length @ A @ R ) )
= S2 ) ) ).
% lapp_suff_llength
thf(fact_254_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_255_ldrop__fin__iffT,axiom,
! [A: $tType,T2: coinductive_llist @ A,I: nat] :
( ( member @ ( coinductive_llist @ A ) @ ( lList2508575361_ldrop @ A @ T2 @ I ) @ ( lList2236698231inlsts @ A @ ( top_top @ ( set @ A ) ) ) )
= ( member @ ( coinductive_llist @ A ) @ T2 @ ( lList2236698231inlsts @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% ldrop_fin_iffT
%----Type constructors (23)
thf(tcon_fun___Orderings_Oorder__top,axiom,
! [A8: $tType,A9: $tType] :
( ( order_top @ A9 @ ( type2 @ A9 ) )
=> ( order_top @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A8: $tType,A9: $tType] :
( ( preorder @ A9 @ ( type2 @ A9 ) )
=> ( preorder @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A8: $tType,A9: $tType] :
( ( order @ A9 @ ( type2 @ A9 ) )
=> ( order @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
thf(tcon_fun___Orderings_Otop,axiom,
! [A8: $tType,A9: $tType] :
( ( top @ A9 @ ( type2 @ A9 ) )
=> ( top @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A8: $tType,A9: $tType] :
( ( ord @ A9 @ ( type2 @ A9 ) )
=> ( ord @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder,axiom,
condit1037483654norder @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
wellorder @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Opreorder_1,axiom,
preorder @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
linorder @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Ono__top,axiom,
no_top @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Oorder_2,axiom,
order @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Oord_3,axiom,
ord @ nat @ ( type2 @ nat ) ).
thf(tcon_Set_Oset___Orderings_Oorder__top_4,axiom,
! [A8: $tType] : ( order_top @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_5,axiom,
! [A8: $tType] : ( preorder @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_6,axiom,
! [A8: $tType] : ( order @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Otop_7,axiom,
! [A8: $tType] : ( top @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_8,axiom,
! [A8: $tType] : ( ord @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_HOL_Obool___Orderings_Oorder__top_9,axiom,
order_top @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Opreorder_10,axiom,
preorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Olinorder_11,axiom,
linorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oorder_12,axiom,
order @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Otop_13,axiom,
top @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oord_14,axiom,
ord @ $o @ ( type2 @ $o ) ).
%----Conjectures (1)
thf(conj_0,conjecture,
~ ( ord_less_eq @ ( coinductive_llist @ a ) @ ( coinductive_LCons @ a @ a2 @ l ) @ ( coinductive_LNil @ a ) ) ).
%------------------------------------------------------------------------------